A SHORT HISTORY OF SCIENCE The whole of modern thought is steeped in science. . .. 
The greatest intellectual revolution mankind has yet 
seen is now slowly taking place by her agency. 
— Huxley. . 
The history of science familiarizes us with the ideas 
of evolution and the continuous transformation of hu- 
man things. ... It shows us that if the accomplish- 
ments of mankind as a whole are grand the contribu- 
tion of each is small. _ gARTON> 
The history of science is the real history of mankind. 
— Du Bois Reymond. 
The history of science . . . presents science as the 
constant pursuit of truth ... a growth to which each 
may contribute. . . . Science is international. 
— Libby. A 
SHORT HISTORY OF SCIENCE 
BY 
W. T. SEDGWICK and H. W. TYLEK 
tu 
Professor of Biology Professor of Mathematics 
at the 
Massachusetts Institute of Technology 
Cambridge 
The history of science should be 
the leading thread in the history of civilization. 
— Sakton. 
Neto Iforfe 
THE MACMILLAN COMPANY 
1917 
All rights reserved Copyright, 1917, 
By THE MACMILLAN COMPANY. 
Set up and electrotyped. Published November, 1917. 
Nor&oooh Pre02 
J. 8. Cushing Co. — Berwick & Smith Co. 
Norwood, Mass., U.S.A. PREFACE 
This book is the outgrowth of a lecture course given by the 
authors for several years* to undergraduate classes of the Massa- 
chusetts Institute of Technology, the chief aims of the course 
being to furnish a broad general perspective of the evolution of 
science, to broaden and deepen the range of the students’ interests 
and to encourage the practice of discriminating scientific reading. 
There are of course excellent treatises on the history of partic- 
ular sciences, but these are as a rule addressed to specialists, and 
concern themselves but little with the important relations of the 
sciences one to another or to the general progress of civilization. 
The present work aims to furnish the student and the general 
reader with a concise account of the origin of that scientific knowl- 
edge and that scientific method which, especially within the last 
century, have come to have so important a share in shaping the 
conditions and directing the activities of human life. The 
specialist in any branch of science is finding it more and more 
difficult to keep himself informed, even to the indispensable mini- 
mum extent, as to current progress in his own field, — and hence 
his frequent neglect of all other branches than his own. 
It may reasonably be expected that some attention to the his- 
tory of science on the part of students will give them a better 
understanding of the broad tendencies which have determined the 
general course of scientific progress, will enlarge their apprecia- 
tion of the work of successive generations, and tend to guard them 
against falling into those ancient pitfalls which have bordered 
the paths of progress. In the words of Mach : — 
There is no grander nor more intellectually elevating spectacle than that 
of the utterances of the fundamental investigators in their gigantic power. 
* By the senior author since 1889. 
V VI 
PREFACE 
Possessed as yet of no methods — for these were first created by their labors 
and are only rendered comprehensible to us by their performances — they 
grapple with and subjugate the object of their inquiry and imprint upon it 
the forms of conceptual thought. Those who know the entire course of the 
development of science will... judge more freely and more correctly the sig- 
nificance of any present scientific movement than those who, limited in their 
views to the age in which their own lives have been spent, contemplate merely 
the trend of intellectual events at the present moment. 
At a time when the forces of science are being diverted from the 
promotion and conservation of civilization to its destruction, and 
when attempts are being made to turn the waters now flowing 
in the stream of science back into ancient and so-called classical 
channels, it will be well for the general reader no less than the 
student of science to review its history, and to judge for himself 
concerning its proper place in contemporary life and education. 
Many volumes would be required to depict the lives of the workers, 
— often marked by self-denial and sometimes by persecution, — 
to trace the full significance of their achievements, or to portray 
the spirit animating their labors; — that spirit of science to which, 
regarding it as a critic rather than a votary, impressive tribute 
has been paid by one of our modern seers: — 
A greater gain to the world . . . than all the growth of scientific knowl- 
edge is the growth of the scientific spirit, with its courage and serenity, its 
disciplined conscience, its intellectual morality, its habitual response to any 
disclosure of the truth. 
— F. G. Peabody. 
It has naturally been foreign to the purpose of the authors to 
admit matter too technical for the general student or, on the other 
hand, too slight in its influence on the general progress of science. 
The division of responsibility between them corresponds roughly 
to that implied by the title “mathematical” and “natural 
sciences”, and emphasis has been laid on interrelations rather 
than on distinctions between the various sciences. The mathe- 
matical group from their relatively greater age and higher de- 
velopment afford the best examples of maturity; the natural 
sciences illustrate more clearly recent progress. No attempt PREFACE 
VII 
has been made by the authors to follow an encyclopaedic plan, under 
which all fields should receive proportional space and treatment, 
each by a competent representative, but some fullness of presenta- 
tion has been aimed at in the particular branches with which they 
are themselves familiar, with briefer indication of developments 
along other lines. 
The authors gladly acknowledge their indebtedness to many 
men of science interested in their undertaking, and to the special 
histories already referred to, on which their own work is largely 
based. Many brief typical quotations from the more important 
authorities are given as a basis for wider or more special study, 
but no systematic attempt has been made to examine original 
sources. No one can possibly be more aware than are the authors 
of the shortcomings of their work, and corrections of errors, from 
which a book of this kind cannot hope to have escaped, will be 
welcomed. 
Massachusetts Institute of Technology, 
Cambridge, 1917. TABLE OF CONTENTS 
PAGE 
Early Civilizations 1 
The Antiquity and Ancestry of Man — Archaeology — Prehis- 
toric Man — The Science of Mankind, Anthropology — Primi- 
tive Interpretations of Nature — Prevalence of Animism in 
Antiquity — Sources of Information Concerning Prehistoric and 
Ancient Times — Some Ancient Lands and Peoples — Babylo- 
nia and Assyria — Egypt — Phoenicia — The Hebrews — The 
Emergence of European Civilization — Aegean Civilization in 
the Bronze Age — The Iron Age; The Greeks or Hellenes. 
CHAPTER I 
CHAPTER II 
Early Mathematical Science in Babylonia and Egypt . . 20 
Primitive Astronomical Notions — The Planets — Astrology 
and Cosmology — Primitive Counting — Primitive Geometry — 
Relation of Greek to Older Civilizations — Babylonian Arith- 
metic — Babylonian Astronomy — Babylonian Geometry — 
Mathematical Science in Egypt — The Ahmes Papyrus — Egyp- 
tian Land Measurement — Egyptian Geometry. 
The Beginnings of Science 35 
Geographical Boundaries — Indebtedness of Greece to Baby- 
lonia and Egypt — The Greek Point of View — Sources — The 
Calendar — Time Measurement — Greek Arithmetic — Greek 
Geometry — The Ionian Philosophers — Thales — Milesian Cos- 
mology — Anaximander — Anaximenes — Pythagoras and his 
School — Pythagorean Arithmetic — Pythagorean Geometry — 
Pythagorean Physical Science — Terrestrial Motion: Philolaus, 
Hicetas. 
CHAPTER III 
CHAPTER IV 
Science in the Golden Age op Greece 58 
Literature and Art — Parmenides — Empedocles — Anaxagoras 
— The Atomists — Democritus of Abdera — The Beginnings of 
IX X 
TABLE OF CONTENTS 
PAGE 
Rational Medicine: Hippocrates of Cos — The Sophists — Hip- 
pias of Elis — The Criticism of Zeno — Circle Measurement: 
Antiphon and Bryson, Hippocrates of Chios — Plato and the 
Academy — The Analytic Method — Platonic Cosmology — 
Archylas — Menaachmus: Conic Sections — A New Cosmology: 
Eudoxus — Aristotle — Aristotle’s Mechanics — Aristotelian 
Astronomy — Theophrastus — Epicurus and Epicureanism — 
Heraclides: Rotation of the Earth. 
Greek Science in Alexandria 87 
The Museum at Alexandria — Euclid — Euclid’s Elements 
— Influence of Euclid — Criticism of Euclid — Other Works of 
Euclid — Archimedes — Archimedes and Euclid — Circle Meas- 
urement — Quadrature of the Parabola — Spirals — Sphere and 
Cylinder — Mechanics of Archimedes — Archimedes as an En- 
gineer — Alexandrian Geography; Earth Measurement — Era- 
tosthenes — Apollonius of Perga — Apollonius and Archimedes — 
Medical Science at Alexandria; Beginnings of Human Anatomy. 
CHAPTER V 
The Decline of Alexandrian Science 115 
Orbital Motion of the Earth: Aristarchus — Excentric Cir- 
cular Orbits — Epicycles — Hipparchus: Star Catalogue — Pre- 
cession of the Equinoxes — Other Astronomical Discoveries; 
Planetary Theory — Invention of Trigonometry — Inventions: 
Ctesibus and Hero — Hero’s Triangle Formula — Inductive 
Arithmetic: Nicomachus — Ptolemy and the Ptolemaic Sys- 
tem— The Almagest — Other Works of Ptolemy — Pappus — 
Beginnings of Algebra: Diophantus — Conclusion and Retrospect. 
CHAPTER VI 
CHAPTER VII 
The Roman World. The Dark Ages 141 
The Roman World-Empire — The Roman Attitude towards 
Science — Roman Engineering and Architecture — Slave Labor 
in Antiquity — Julius Caesar and the Julian Calendar — Vitru- 
vius on Architecture — Frontinus on the Waterworks of Rome 
— Roman Natural Science and Medicine — Lucretius — Strabo 
— Pliny the Elder — Galen — Late Roman Mathematical Sci- 
ence — Capella — Boethius — Science and the Early Christian 
Church — The Eastern Empire; Edict of Justinian — The Dark 
Ages — The Establishment of Schools by Charlemagne. TABLE OF CONTENTS 
XI 
CHAPTER VIII 
PAGE 
Hindu and Arabian Science. The Moors in Spain . . . 156 
Alexandria — Hindu Mathematics — Hindu Astronomy — 
Mohammed and the Hegira — Arabian Mathematical Science 
— Arabian Astronomy — Asiatic Observatories — The Moors in 
Spain. 
CHAPTER IX 
Progress of Science to 1450 a.d 172 
The Crusades — Trivium and Quadrivium; Scholasticism — 
Medieval Universities — Transmission of Science through 
Moorish Spain — Dawn of the Renaissance — Mathematical 
Science in the Thirteenth Century — Roger Bacon — Dante 
Alighieri — Computation in the Middle Ages — Mathematics 
in the Medieval Universities — The Renaissance — Humanism 
— Alchemy — The Mariner’s Compass — Clocks — Wool and 
Silk; Textiles in the Middle Ages — The Invention of Printing. 
CHAPTER X 
A New Astronomy and the Beginnings of Modern Natural 
Science 191 
The Age of Discovery — The Reformation — Pioneers of the 
New Astronomy — Conditions Necessary for Progress — Nico- 
laus Copernicus — De Revolutionibus — Influence of Copernicus 
— Tycho Brahe — Uraniborg — Kepler — Galileo — Medical and 
Chemical Sciences — Anatomy: Vesalius — Revival of Interest 
in Natural History. 
CHAPTER XI 
Progress of Mathematics and Mechanics in the Sixteenth 
Century 230 
Aims and Tendencies of Mathematical Progress — Pacioli — 
Geometry in Art — Robert Recorde — Algebraic Equations of 
Higher Degree—Tartaglia, Cardan— Symbolic Algebra : Vieta 
— Development of Trigonometry — Map-making — The Grego- 
rian Calendar — A New Invention for Computation — “Two 
New Branches of Science ” — A Pioneer in Mechanics; Stevinus 
— Giordano Bruno. 
CHAPTER XII 
Natural and Physical Science in the Seventeenth Century • 255 
The Circulation of the Blood : Harvey — Atmospheric Pres- 
sure; Torricelli’s Barometer — Further Studies of the Atmos- XII 
TABLE OF CONTENTS 
PAGE 
phere; Gases — From Philosophy to Experimentation — From 
Alchemy to Chemistry — A False Theory of Combustion; 
Phlogiston — Beginnings of Organic Chemistry — Organization 
of the First Scientific Academies and Societies — The New 
Philosophy: Bacon and Descartes — Progress of Natural and 
Physical Science in the Seventeenth Century. 
CHAPTER XIII 
Beginnings of Modern Mathematical Science .... 273 
Mathematical Philosophy; Analytic Geometry: Descartes 
— Indivisibles: Cavalieri — Projective Geometry: Desargues — 
Theory of Numbers and Probability: Fermat, Pascal — Me- 
chanics and Optics: Huygens — Wallis and Barrow — Isaac 
Newton — Optics — The Theory of Gravitation; Principia — 
Newton’s Mathematics; Fluxions — Leibnitz — Halley: Pre- 
diction of Comets. 
CHAPTER XIV 
Natural and Physical Science in the Eighteenth Century 304 
Chemistry; Decline of the Phlogiston Theory — A New 
Chemistry : Priestley and Lavoisier — The Synthesis of Water 
— Beginnings of Modern Ideas of Sound — The Beginnings of 
Modern Ideas of Heat; Latent and Specific Heat, Calorimetry 
— Eighteenth Century Researches on Light — Beginnings of 
Modern Ideas of Electricity and Magnetism — Beginnings of 
Modern Ideas, of the Earth — Eighteenth Century Progress 
in Botany, Zoology, etc. — Progress in Comparative Anatomy 
and Physiology — The Industrial Revolution; Inventions; 
Power — Influence of Science upon the Spirit of the Eighteenth 
Century. 
CHAPTER XV 
Modern Tendencies in Mathematical Science .... 323 
Mathematics and Mechanics in the Eighteenth Century — 
Progress in Theoretical Mechanics — Celestial Mechanics — 
The Perturbation Problem — The Nebular Hypothesis — Mod- 
ern Astronomy; Telescopic Discoveries — Mathematical Prog- 
ress and Physical Science — Nineteenth Century Mathematics 
— Non-Euclidean Geometry — Imaginary Numbers — The Dis- 
covery of Neptune—Cosmic Evolution — Distance of the Stars 
— Mathematical Physics. TABLE OF CONTENTS 
XIII 
CHAPTER XVI 
PAGE 
Some Advances in Physical Science in the Nineteenth Cen- 
tury. Energy and the Conservation of Energy . . 348 
Modern Physics — Heat, Thermometry : Carnot, Rumford — 
Light; Wave Theory, Velocity: Young, Fresnel — The Spec- 
troscope and Spectrum Analysis — Electricity and Magnetism: 
Faraday, Green, Ampere, Maxwell — Electromagnetic Theory 
of Light — Kinetic Theory of Gases: Clausius — The Concep- 
tion of Energy — Dissipation of Energy — Modern Chemistry 
— Chemical Laboratories: Liebig — Quantitative Relations; 
Atoms, Molecules, Valence — Synthesis of Organic Substances 
— A Periodic Law among the Elements — Chemical Structure 
— Physical Chemistry; Electrolytic and Thermodynamic De- 
velopments of Chemistry. 
CHAPTER XVII 
Some Advances in Natural Science in the Nineteenth Cen- 
tury. Cosmogony and Evolution 366 
Influence of Eighteenth Century Revolutions — The Scientific 
Revolution — Effects of the Rapid Increase of Knowledge — 
Gradual Appreciation of the Permanence and Scope of Natural 
Law —Natural Theology and an Age of Reason — Natural Phi- 
losophy and Natural History; Differentiation and Hybridizing 
of the Sciences — Progress in Zoology — Progress in Botany — 
Progress in Microscopy; the Achromatic Objective — Embry- 
ology— Progress in Physiology: Johannes Muller; Claude Ber- 
nard — Pathology before Pasteur — The Germ Theory of 
Fermentation, Putrefaction and Disease : Pasteur — Antiseptic 
and Aseptic Surgery: Lister — Rise of Bacteriology and Para- 
sitology— Biogenesis versus Spontaneous Generation — Prog- 
ress of Geological Science—'Glaciers and Glacial Theories — 
Rise of Palaeontology — Ancient and Modern Theories of Cos- 
mogony — Relationship of the Heavens and the Earth — The 
Scale of Life and the Phases of Life — General Resemblance of 
Man to the Lower Animals — Anatomical and Microscopical 
Similarity of Animals and Plants; Organs, Tissues, Cells and 
Protoplasms — Fundamental Unity of Nature; Organic versus 
Inorganic World — Treviranus’ Biology and Lamarck’s Zoologi- 
cal Philosophy — Voyages and Explorations of Naturalists — 
Darwin’s Origin of Species — His Descent of Man — Decline 
of the Theory of Special Creation — Influence of an Age of 
Invention and Industry — Science in the Dawn of the Twen- 
tieth Century. XIV 
TABLE OF CONTENTS 
APPENDICES 
PAGE 
A. The Oath of Hippocrates (about 400 b.c.) . . . 399 
B. The Opus Majus of Roger Bacon (1267 a.d.). An Anal- 
ysis of the Sixth Part by J. H. Bridges 400 
C. Dedication of The Revolutions of the Heavenly Bodies 
by Nicolas Copernicus (1543) . ...... 407 
D. William Harvey’s Dedication of his Work on the Circu- 
lation of the Blood (1628) 412 
E. Galileo before the Inquisition (1633) .... 414 
F. Preface to the Philosophic Naturalis Principia Mathe- 
matical,, by Isaac Newton (1686) 420 
G. An Inquiry into the Causes and Effects of the Variolce 
Vaccine, by Edward Jenner (1798) 422 
H. Principles of Geology, by Charles Lyell (1830). . . 429 
I. Some Inventions of the Eighteenth and Nineteenth 
Centuries 438 
Power; Its Sources and Significance — Gunpowder, Nitro- 
glycerine, Dynamite — The Steam-Engine — The Spinning 
Jenny, the Water Frame, and the Mule — The Cotton Gin 
— Steam Transportation — The Achromatic Compound Micro- 
scope — Illuminating Gas — Friction Matches — The Sewing- 
Machine— Photography — Anesthesia; The Ophthalmoscope 
— India-Rubber — Electrical Apparatus; Telegraph, Telephone, 
Electric Lighting, Electric Machinery — Food Preserving by 
Canning and Refrigeration — The Internal-Combustion Engine 
— Aniline — The Manufacture of Steel: Bessemer — Agricultu- 
ral Apparatus and Inventions — Applied Science; Engineering. 
A Table of Important Dates in the History of Science and 
Civilization 449 
A Short List of Books of Reference ..... 459 
Index ............ 469 ILLUSTRATIONS 
PAGE 
Hecatseus’ Map of the World, 517 b.c. 34 
Herodotus’ Map of the Worl l 57 
Behaim’s Globe, 1492 a.d 190 
The Coperniean System 198 
Tycho Brahe’s Quadrant opposite 204 
Uraniborg opposite 206 
Kepler opposite 210 
Galileo opposite 217 
Galileo’s Dialogue opposite 224 
Stevinus’ Triangle 252 
Huygens opposite 286 
Huygens’ Clock opposite 288 
Newton’s Telescope and Newton’s Theory of the Rainbow opposite 292 
Sketch Map of Places Important in Ancient and Medieval 
Science opposite 448 
XV A SHORT HISTORY OF SCIENCE 
CHAPTER I 
EARLY CIVILIZATIONS 
‘The night of time far surpasseth the day’ said Sir Thomas 
Browne; and it is the task of Archaeology to light up some parts of 
this long night. — Charles Eliot Norton. 
The Antiquity and Ancestry of Man. — It is now gen- 
erally agreed that men of some sort have been living upon this 
earth for many thousand years. It is also, though perhaps less 
generally, agreed that mankind has descended from the lower 
animals, precisely as the men of to-day have descended from 
men that lived and died ages ago. 
The history of science, however, is not so much concerned with 
the ancestry or origin of mankind as with its antiquity; for while 
science is a comparatively recent achievement of the human race, 
its roots may be traced far back in practices and processes of pre- 
historic and primitive times. Mankind is very old, but science 
so far as we know had no existence before the beginning of 
history, i.e. about 6000 years ago, and until 2500 years ago it 
occurred if at all only in rudimentary form. The best opinion 
of to-day holds that man has been on this earth at least 250,000 
years, and in spite of wide variations is of one zoological “ kind ” 
or “ species ” and three principal types or “ races,” viz., white or 
Caucasian, yellow or Mongolian, and black or Ethiopian (Negroid). 
These great races are believed to have had a common ancestry 
in a more primitive race, and this in turn to have descended from 
the lower animals. It is furthermore held that there was prob- 
1 2 
A SHORT HISTORY OF SCIENCE 
ably one principal place of origin, or “ cradle,” of the human 
race from which have spread all known varieties of mankind, alive 
or extinct, and that this was probably in “Indo-Malaysia” in 
that remarkable valley which lies between the rivers Tigris and 
Euphrates and in its upper part is known as Mesopotamia (be- 
tween the rivers). 
Mesopotamia, or the broad valley of the Tigris and Euphrates, 
was the cradle of civilization in the remotest antiquity. There can 
be little doubt that man evolved somewhere in southern Asia, possibly 
during the Pleiocene or Miocene times .... [And] as paleolithic man 
was certainly interglacial in Europe, we may assume that he was 
preglacial in Asia. . . . 
The earliest known civilization in the world arose north of the 
Persian Gulf among the Sumerians .... but the Babylonians of 
history were a mixed people, for Semitic influences according to 
Winckler began to flow up the Euphrates Valley from Arabia during 
the fourth millennium b.c. This influence was more strongly felt, 
however, in Akkad than in Sumer, and it was in the north that the 
first Semitic Empire, that of Sargon the Elder (about 2500 b.c. 
according to E. Meyer) had its seat. . . . The supremacy of Babylon 
was first established by the Dynasty of Hamurabi (about 1950 b.c., 
earlier according to Winckler) which was overthrown by the Hittites 
about 1760 b.c. Then followed the Kassite dominion, which lasted 
from about 1760 to 1100 b.c. ... It was probably due to them that 
the horse, first introduced by the Aryans, became common in south- 
west Asia; it was introduced into Babylon about 1900 B.c. but 
was unknown in Hamurabi’s reign. — Haddon. 
Archeology. — The study of antiquity, and especially of 
prehistoric antiquity, is known as archaeology (the science of 
antiquities or beginnings), and is based upon finds of ruins, 
tools, weapons, caves, skeletons, carvings, ornaments, and 
similar remains or evidences of human life and action in pre- 
historic times. It has been well described as “ unwritten history.” 
Remains of all kinds have long been roughly but conveniently 
classified into three groups corresponding to three periods of 
development, viz.: a Stone Age, a Bronze Age, and an EARLY CIVILIZATIONS 
3 
Iron Age, according to the use of stone, bronze and iron 
implements. 
Prehistoric Man. — If therefore we would begin the history 
of science at the very beginning, we must turn far backward in 
imagination to a time when the human race was barely superior 
to the beasts that perish. Absorbed in a fierce struggle for exist- 
ence, the passing generations had little history and left behind 
them no permanent records. In one respect nevertheless mankind 
stood far above the beasts; namely, in possessing the power of 
language, by which they could not only communicate more readily 
one with another, but also convey to their descendants through 
oral tradition something of whatever they might possess of accu- 
mulated knowledge. Eventually, though slowly, the generations 
began to leave behind them more enduring records, — at first 
crude and fragmentary, in the form of tools, cairns, and other 
monuments, or in drawings, paintings, or carvings, on ivory or 
rocks or trees, or on the walls of caverns, — which should serve 
to inform or instruct other men. Finally, but still slowly, and 
especially out of this so-called “picture-writing,” grew the art of 
writing, which furnished a means of keeping permanent records 
of the past and a new and more perfect way of communication 
between living men and races of men. We who have ourselves 
witnessed some of the consequences of improvements in the arts 
of communication between men and nations, such as have recently 
been effected by steam transportation and telegraphy and teleph- 
ony, can to some extent realize how much the introduction of the 
rudiments of the art of writing may have meant in the progress of 
prehistoric and primitive mankind. 
The Science of Mankind. Anthropology. — The various 
steps in the evolution of mankind and in the earliest development 
of civilization and the arts form the subject matter of one of the 
youngest of the sciences, anthropology, to works upon which 
the reader is referred who would pursue these matters further. 
One of the earliest and still one of the most interesting of these, 
Man’s Place in Nature, by Huxley, is now a classic. Another, 
also somewhat out of date but still very valuable, entitled 4 
A SHORT HISTORY OF SCIENCE 
“ Anthropology,” is of special interest because its author, E. B. 
Tylor, was the founder of the science and is still living (in 1916).1 
The Childhood of the Race. — There is reason to believe 
that the human race, in its long and slow development, has passed 
through periods of essential childhood and youth, very much as 
the individual human being passes slowly through infancy onwards; 
and that, precisely as the individual begins his intellectual life in 
wonder, questioning, and curiosity, so the race has advanced from 
a condition of childish wonder, questionings, and interpretations of 
mankind and the external world, — sun, moon, and stars, thunder 
and lightning, wind, rain, and snow, — which have gradually 
developed into more mature and more scientific explanations. 
This principle of an essential parallelism between individual 
development and racial, named by Haeckel “ the biogenetic 
law,” will be found especially pertinent at many stages in the 
history of science. 
Primitive Interpretations of Nature.—As the child thinks 
he sees in almost everything some living agency, —because most of 
the things that happen about him are obviously connected with 
himself, or his parents, or his nurses, or other children, or with 
his pets, — so man in the childhood of the race and in its earlier 
development sees in the wind some hidden being or personality 
bending the tree, or shaking the leaves, or moaning or sighing in 
the forest, or roaring angrily in thunder. Only a slightly different 
imagination is required to see in the sun, moon, and planets super- 
natural beings or gods travelling across the heavens, and by asso- 
ciation, since they seem to visit his heavens daily or monthly 
or at other regular intervals, to believe that they are somehow con- 
cerned with himself and his welfare or destiny. From this primi- 
tive interpretation to the modern astronomical knowledge of the 
immensity, the movements and the paths, the temperatures, and 
1 The latest edition of Sir John Lubbock’s [Lord Avebury’s] “ Prehistoric Times ” 
should also be consulted. Other easily accessible volumes are 'A. C. Haddon’s 
“The Wanderings of Peoples” (Cambridge Manuals of Science and Literature) 
and J. L. Myres’ “The Dawn of History” (Home University Library Series). 
The chapters on “ Modern Savages” in Lord Avebury’s “ Prehistoric Times” are 
especially instructive. Most important of all is Professor H. F. Osborn’s recent 
work, “ Men of the Old Stone Age.” EARLY CIVILIZATIONS 
5 
even the chemical composition, of those enormous lifeless masses 
which we call sun, moon, and stars, has been a long and laborious 
journey, — how long no one can tell. It is still almost always 
possible to find tribes or peoples somewhere on the earth living 
under one. or more of the various conditions which the more 
highly developed peoples have apparently passed through, and 
there is no great difficulty in finding primitive tribes to-day holding 
such childish interpretations of nature as we have just described. 
This circumstance enables anthropologists, ethnologists, and his- 
torians to draw with considerable confidence the broader outlines 
of the probable history of the more highly developed nations, 
such as those of western Europe and North America, — nations 
in the progress of wrhich, since the beginning of the nineteenth 
century, science has played a notable part. 
The first stepping-stones towards scientific knowledge are 
wonder and curiosity, and peoples are still to be found so low in 
intelligence as to be almost destitute of curiosity. As a rule, 
however, most human beings, no matter how primitive, have 
some curiosity concerning, and some sort of explanation for, the 
commonest events, such as day and night, life, death, sickness, 
health, sun, moon, stars, winds, seasons, and the like. And one 
of the commonest, simplest, and probably most natural, is that 
already referred to as the childish or personal interpretation of 
nature; viz., that which assumes everything to be in a sense 
alive and possessed of some sort of being, animation, or personality, 
kindred to man’s own. This primitive interpretation has been 
called animism. At present, however, the term animalism 
finds more favor among certain anthropologists, apparently for 
the reason that the notion of mere diffuse vitality, or general 
“animation,” is even more primitive, as observed in certain 
peoples of low development, than is the idea of a specific “soul” 
(anima) differentiated from the body and possessing a separate 
existence. For example, a tree blown by the wind may seem to 
a man of very low development to be merely quivering with life, 
and bending before some more powerful but invisible influence, 
diffused, hazy, unembodied, and without personality or name 6 
A SHORT HISTORY OF SCIENCE 
(<animalism). Or it may seem to be an individual tree, bent by 
an invisible but powerful being like a man and perhaps having a 
name such as “Boreas” (the Greeks’ name for the north wind). 
In this latter case we have the assumption of personality and, by 
analogy with man, of the presence and influence of a spirit or 
soul (animism). 
Prevalence of Animism in Antiquity. — Judging by the 
opinions and beliefs of races which still exist in very low 
stages of development, prehistoric man when he pondered at 
all, reasoned largely in the direction of animism. He in- 
terpreted himself and his actions by his own ideas, will, feelings, 
and desires, and reasoned that other things were actuated like- 
wise. If, for example, he killed an ox or a man by a blow, and 
later an ox or a man were killed by lightning, it was reasonable 
to assume that some invisible and manlike being had given the 
ox or man an invisible blow. The oldest records of the human 
race confirm this idea. The ancient Assyrians, Babylonians, and 
Egyptians “animated” much of what we today call inanimate, 
i.e. inorganic, nature; and Greek and Hebrew poetry are full of 
survivals of this view of man and nature, which on the higher 
levels passes into personification and anthropomorphism. The 
establishment of a hierarchy of the gods of Greece, such as was 
supposed to dwell upon Mt. Olympus, is merely a further differ- 
entiation of the same kind. “The Hellenic gods and goddesses 
are glorified men and women.” 
Sources of Information concerning Prehistoric and 
Ancient Times. — These are of three kinds, tradition, monuments 
(including tools, implements, pottery, and other objects which 
have survived to the present time, more or less in their original 
form), and inscriptions. Of these tradition, because readily sub- 
ject to perversion, is the least reliable and need not be further 
considered. It is monuments, such as ruins, tombs, weapons, 
pottery, implements, ornaments, furniture, and the like, upon 
which we must chiefly depend for our knowledge of prehistoric 
times, and the evidence which has been gradually accumulated 
from finds of this sort is extensive and trustworthy and corre- EARLY CIVILIZATIONS 
7 
spondingly valuable. With the introduction of inscriptions of all 
sorts, including drawings, pictures, hieroglyphics, and writings 
of every kind, upon tablets, monuments, walls, caves, clay cylin- 
ders, papyri, parchments, and the like, from about the eighth or 
tenth century b.c., we enter upon the historical period. From that 
time forward we have more or less of the raw material from which 
we may reconstruct the beginnings, not only of civilization and 
art, but also of literature and science. 
Some Ancient Lands and Peoples. — From the standpoint 
of European history, and especially the history of science, the most 
important peoples of antiquity were the Babylonians, Assyrians, 
Egyptians, and Phoenicians. The Babylonians and Assyrians 
occupied the fertile valley of the Tigris and Euphrates; the Egyp- 
tians, that of the Nile; and the Phoenicians the eastern slopes of 
the Mediterranean basin (modern Syria). The first three peoples 
were chiefly agricultural; the last, chiefly seafaring, mercantile, and 
industrial. 
Babylonia and Assyria. — These, lying almost side by side, 
may be considered together, although Babylonia furnishes the 
older and the more important civilization. Babylon and Nineveh 
were the chief cities of the two countries, the former in Mesopo- 
tamia on the Euphrates, the latter above and to the northeast, and 
much nearer the mountains, on the Tigris. 
In that part of Asia which borders upon Africa, to the north of 
Arabia and the Persian Gulf, in an almost tropical region at the foot 
of the Armenian highlands, defended by mountains on the east and 
bounded by desert on the west, opens the broad valley of the Tigris 
and the Euphrates rivers which, flowing from the same mountains 
and in the same direction and maintaining for a long distance a parallel 
but independent course, join at last and fall together into the Persian 
Gulf. In the month of April these two rivers, swollen by the melted 
snows in the mountains of Armenia, overflow, sinking again to the 
level of their beds in June. The country around them therefore was 
very similar to the Nile valley. A large number of canals joined the 
Tigris to the Euphrates, and distributed the water rendered by the 
tropical climate necessary for agriculture. 8 
A SHORT HISTORY OF SCIENCE 
The upper part of the country inclosed between the two rivers 
was properly called Mesopotamia, a term used also roughly to desig- 
nate the whole. The valley of the Upper Tigris, or Upper Mesopo- 
tamia, was Assyria, and the lower part of both valleys Babylonia. . . . 
In these two fertile regions flourished two empires, the Chaldean- 
Babylonian and the Assyrian. 
The Chaldeans, says a trustworthy authority, appear to have been 
a branch of the great Hamite race of Akkad, which inhabited Baby- 
lonia from the earliest times. With this race originated the art of 
writing, the building of cities, the institution of a religious system, and 
the cultivation of all science, and of astronomy in particular. In the 
primitive Akkadian tongue were preserved all the scientific treatises 
known to the Babylonians. It was in fact the language of science in 
the East, as the Latin was in Europe during the Middle Ages. When 
Semitic tribes established an empire in Assyria in the thirteenth cen- 
tury b.c., they adopted the alphabet of the Akkad, and with certain 
modifications applied it to their own language. . . . The mythological, 
astronomical, and other scientific tablets found at Nineveh, are ex- 
clusively in the Akkadian language, and are thus shown to belong to 
a priestly class, exactly answering to the Chaldeans of profane history 
and of the Book of Daniel. . . . 
From about 747 b.c., the accession of Nabonassar, the line of 
kings at Babylon is supplied by the well-known work of Ptolemy, 
the geographer. . . . Babylon, according to ancient historians, 
was surrounded by walls over three hundred feet in height and 
eighty in thickness, and was divided into two parts by the river 
Euphrates, which flowed through it. Narrow streets led to the 
river, on which they opened by gates. Quays enclosed the water, 
and towards the centre a bridge crossed it, but the bridge was 
movable and was only used during the day. At night the two sides 
of the river were completely separated. . . . When, at the present 
time, we visit these formerly prosperous countries, we can scarcely 
believe in the universal fertility that so many witnesses have described. 
The carelessness of the Turkish administration has allowed the irri- 
gation canals to be silted up, and the inundations now form unhealthy 
swamps in the delta of the Tigris and Euphrates. Mesopotamia was 
wonderfully productive in wheat and barley, the enormous returns 
obtained by Bablyonian farmers from their corn-lands being un- 
exampled in modern times; but it possessed neither olives, figs, nor EARLY CIVILIZATIONS 
9 
vines; millet and sesame, however, grew luxuriantly. Date-palms 
abounded, and furnished a large part of the food of the inhabitants. 
The people of Assyria and Chaldea were as skilled in manual 
handicrafts as in the cultivation of the earth. They wove cloths of 
brilliant colors; they also ornamented their garments with a pro- 
fusion of embroideries, and wore magnificent tiaras. Babylonian 
embroidery was celebrated even in the days of the Roman empire. 
The manufacture of carpets, one of the chief luxuries in the East, 
attained wonderful perfection at Babylon, as well as the manufacture 
of personal attire. Their furniture, by its richness and shape, dif- 
fered completely from anything we find in present use amongst Ori- 
entals ; the Assyrians used arm-chairs or sat on stools, and dined as 
we do from tables. The tables and chairs were handsomely decorated 
and in good taste, and it is curious to note that the same designs for 
ornamentation were in use then as we have now — lions’ claws, 
animals’ heads, etc.; and even at the present time the ancient models 
might be studied with profit and copied with advantage. They were 
skilful in working hard as well as soft materials. The cylinders of 
jasper and crystal and the bas-reliefs of Khorsabad sculptured in 
gypsum or in basalt equally denote their proficiency. They were 
acquainted with glass and with various kinds of enamel, and they 
knew how to bake clay for the manufacture of bricks or of porcelain 
vases. Moreover, the art of varnishing earthenware and of cover- 
ing it with paintings by means of coloured enamel was well known at 
Nineveh. 
The cuneiform writing — so called because it is formed by pres- 
sure of the stylus on the soft surface of the clay tablets, producing 
a mark like a wedge or arrow-head — is a development of hieratic, 
itself an improvement on the primitive hieroglyphic. The hieratic 
characters had been scratched with the point of the stylus on the 
clay that served the Mesopotamian peoples for paper. The use of 
the stylus in cuneiform, gave a single element, by the employment of 
which in various combinations, all the letters of the alphabet were 
formed. When the Persians conquered Mesopotamia they published 
their decrees, etc., in the three chief dialects of their subjects — the 
Persian, Median, and Assyrian. Hence the trilingual inscriptions 
which have supplied the key to cuneiform interpretation. The dis- 
covery of the interpretation of the famous inscription at Behistun, 
on the Persian frontier, in three languages, Persian, Median, and 10 
A SHORT HISTORY OF SCIENCE 
Assyrian, enabled Sir Henry Rawlinson to find the key to the Assyr- 
ian characters. . . . 
It is very difficult, in spite of the numerous texts deciphered by 
modern savants, to form any idea of Assyrian literature; yet the 
literature must have been considerable, for Layard found a complete 
library founded by King Asshurbanipal in two of the rooms of his 
palace at Nineveh. This library consisted of square tablets of baked 
earth, with flat or slightly convex surface, on which the cuneiform 
writing had been impressed while the clay was soft, before baking. 
The characters were very clearly and sharply defined, but many of 
them so minute as to be read only with the help of a magnifying glass. 
These tablets, which are preserved at the British Museum, contain a 
kind of grammatical encyclopedia of the Assyrio-Babylonian lan- 
guage, divided into treatises; and also fragments of laws, mythology, 
natural history, geography, etc. Treatises on arithmetic were also 
found in the library, proving that mathematical sciences were known, 
with catalogues of observations of the stars and planets. We have 
already mentioned that astronomy was greatly honored amongst the 
Chaldean priesthood, who had studied the course of the moon with 
so much precision that they were able to predict its eclipse. 
Science and literature developed, in spite of a primitive writing 
engraved upon clay tablets; the art of sculpture was already highly 
refined ; monuments, which without being majestic like the Egyptian 
were imposing in their size and splendid in their colours; rare ele- 
gance in clothing and furniture, denoting great wealth, the result of 
active commerce; a cruel, even ferocious character, revealed by their 
treatment of prisoners, and indeed by all their history; a learned 
caste, devoting themselves to the sciences and also to the unscientific 
methods of astrology; a religion elevated by the primitive idea of a 
supreme god, yet degraded by polytheism and often by gross de- 
bauchery ; kings sufficiently intelligent to construct splendid palaces 
and immense cities, and yet inflated with pride and glorying in the 
most stupid cruelty — such is the picture opened to us by the records 
of Assyrian and Babylonian history. When we observe on the 
Assyrian bas-reliefs all the industries and all the arts, we are inclined 
to acknowledge that they were superior to the nations that surrounded 
them, and we understand how the Greeks drew inspiration from 
Assyrian work as well as from Egyptian. — Verschoyle. History of 
Civilization. EARLY CIVILIZATIONS 
11 
If, in a final summing up, the question be asked, What was the 
legacy which Babylonia and Assyria left to the world after an exist- 
ence of more than three millenniums, the answer would be, that through 
the spread of dominion the culture of the Euphrates Valley made its 
way throughout the greater part of the ancient world, leaving its im- 
press in military organization, in the government of people, in com- 
mercial usages, in the spread of certain popular rites such as the 
various forms of divination, in medical practices and in observation 
of the movements of heavenly bodies—albeit that medicine continued 
to be dependent upon the belief in demons as the source of physical 
ills, and astronomy remained in the service of astrology — and lastly 
in a certain attitude towards life which it is difficult to define in 
words, but of which it may be said that, while it lays an undue em- 
phasis on might, is yet not without an appreciation of the deeper 
yearnings of humanity for the ultimate triumph of what is right. 
— Morris Jastrow, Jr. The Civilization of Babylonia and Assyria. 
Egypt. — Another highly important ancient civilization whose 
beginnings are lost for us in the darkness of prehistoric times is 
that which flourished in the valley of the Nile. 
Near the point where Africa approaches Asia lies a narrow valley, 
walled in by two ranges of mountains, enclosed on the farther side by 
two deserts, and fertilized by the periodical inundations of a mighty 
river. This long and narrow strip of verdure, surrounded by moun- 
tains and menaced by the desert sands, is Egypt. ... A few years 
ago, the beginnings of Egyptian history, and even the source of the 
great river that fertilizes the land of Egypt, were hidden in mystery. 
The sources of the Nile have been at last discovered, and archaeo- 
logists have now retraced the commencement of a history which is 
practically the commencement of all authentic history. To Speke 
and Grant in 1862, and to Baker in 1864, we owe the knowledge of 
the lakes Victoria Nyanza and Albert Nyanza, whence come the 
abundant waters, that swollen by the equatorial rains, at fixed in- 
tervals overflow and fertilize with their mud the soil that borders their 
bed, and refresh a land which lies beneath a sky where a rain-cloud is 
seldom seen. We know, too, how the abundant harvests that regu- 
larly result from the inundations of the Nile, returning ample food to 
moderate labour, promoted the development of the Egyptian nation; 12 
A SHORT HISTORY OF SCIENCE 
how the Nile itself supplied to them a highway for communication, 
rendered doubly useful by the north winds that blow up stream 
more than eight months of the year, carrying the traffic up into 
the interior while the current carries it down; how the Arabian 
desert on the east and the Libyan on the west secured to them 
comparative immunity from invasion and opportunity for internal 
progress. . . . 
One of the prizes of Napoleon’s expedition, a black basalt stone, 
disinterred at Rosetta in 1798, and now in the British Museum, bore 
three parallel and horizontal inscriptions, all quite distinct. One was 
in hieroglyphics, the second in the characters called demotic or popu- 
lar, the third in Greek. Although a great many scientific men ex- 
hausted their skill upon this trilingual inscription, which was a triple 
inscription of the same text, no one could make the Greek characters 
exactly apply to the hieroglyphic signs. Champollion was the first 
to obtain any success, as early as 1812, but further progress was largely 
aided by the labors of Thomas Young [a name associated in the 
history of science with those of Fresnel and Helmholtz]. . . . 
Protected from invasion by the same deserts that isolated them, 
the people who came from Asia and settled in the Nile valley applied 
themselves to the regulation of the periodical inundations, and to the 
distribution of the water. They built towns on the hillocks, in order 
that the water should not reach them; and afterwards, with the 
stones that the two mountain ranges of Libya and Arabia contain in 
abundance, and by the means of transit afforded by the Nile, they 
erected monuments that have defied the course of the centuries. . . . 
The paintings in the tombs also show us men at work upon all 
the arts and all the handicrafts. ‘We see there the workers in stone 
and in wood, the painters of sculpture and of architecture, of furni- 
ture and carpenters’ work; the quarrymen hewing blocks of stone; 
all the operations of the potter’s art; workmen kneading the earth 
with their feet, or with their hands; men at work making stocks, oars 
and sculls; curriers, leather-dyers, and shoe-makers, spinners, cloth- 
weavers with various shaped looms, glass-makers; goldsmiths, jew- 
ellers, and blacksmiths.’ Among the antiquities still in a state of 
good preservation there is much pottery, including vessels of simple 
earthenware and enamelled faience, enamelled and sculptured terra- 
cotta, glass, often resembling Venetian, metal work and jewellery, 
and linen cloth as fine as Indian muslin. — Verschoyle. EARLY CIVILIZATIONS 
13 
Phoenicia. — Two other civilizations of importance, the 
Phoenician and the Hebrew, existed in antiquity between the 
Mediterranean Sea and the great Arabian desert, in what are 
to-day called Syria and Palestine. 
By the side of the Hebrew nation, which owed its grandeur to 
its moral and religious development, dwelt the Phoenicians, a people 
who owed their fame to their maritime and commercial enterprise. 
They occupied a narrow strip of land between Lebanon and the 
Mediterranean, Phoenicia proper being but 28 miles long by one to 
five miles broad, and the territory of the Phoenicians being, at the 
utmost, no more than 120 miles long by 20 wide. . . . The forests 
which clothed the chain of Lebanon supplied the Phoenicians with 
timber for their ships, and they soon made the Mediterranean a high 
road for their navy. Enclosed by mountains in a country that 
prevented their acquiring any inland empire, they became a maritime 
power, the first in the ancient world in order of importance as in 
order of time. Egyptian documents mention the Phoenician towns 
of Gebal, Beryta, Sidon, Sarepta, etc., as early as sixteen or seven- 
teen centuries before the Christian era. The Phoenicians served as 
middlemen to the great civilizations of the Nile and the Euphrates, 
their vessels easily coasting along to the mouth of the Nile, and their 
caravans having but a short journey to reach the point where the mid- 
dle Euphrates almost touches Upper Syria, whence the current would 
carry them down to the quays of Babylon. . . . To the westward 
the Phoenicians sailed beyond the Mediterranean and ventured upon 
the Atlantic Ocean. They coasted the western side of Africa, and 
early accounts record their discoveries of wonderful islands of mar- 
vellous fertility and charming climate, the ‘Fortunate Isles/ — 
probably Madeira and the Canaries. They also sailed along the 
coasts of Spain and Western France and reached Northern Europe. 
Gades (Cadiz) was the starting point for these long and dangerous 
voyages, which extended as far as Great Britain, where a considerable 
trade in tin was carried on. . . . The Phoenicians were the great 
mining people of the ancient world. Gold, silver, iron, tin, lead, cop- 
per, and cinnabar were obtained from Spain, still the chief metallif- 
erous country of Southern Europe. The details given by Diodorus 
concerning the Spanish mines are very circumstantial. ‘The cop- 
per, gold, and silver mines are wonderfully productive,’ and ‘those 14 
A SHORT HISTORY OF SCIENCE 
who work the copper mines draw from the rough ore one quarter of 
the weight in pure metal.’ . . . The Phoenicians not only brought 
the mineral wealth of Spain to the Eastern world, but they had also 
a great trade in wheat, wine, oil, fruits of all kinds, and fine wooL 
They provided Asia with the products of Spain and Gaul, Sicily and 
Africa with the products of Asia. But this maritime commerce could 
only be supplied by an inland trade, which served to connect the 
countries that were a long distance from the sea. Phoenicia found 
itself one of the ports of Asia, the merchandise of distant countries 
was brought to it, and from it was exported all the produce of the 
Asian continent. The caravans supplemented the fleets, and the 
fleets distributed the burdens of the caravans. The land trade was 
chiefly in three directions — to the south it followed the route to 
Arabia and India; to the east, that to Assyria and Babylon; to the 
north, that to Armenia and the Caucasus. 
The Phoenicians were not only the great maritime, the great 
commercial, and the great mining power of antiquity, they were also 
one of the chief manufacturing powers. Like the Egyptians and 
Assyrians, they were skilful potters, and they discovered the art of 
making glass. ‘ It is said,’ writes Pliny the elder, ‘ that some Phoeni- 
cian merchants, having landed on the shores of the river Belus, were 
preparing their meal, and not finding suitable stones for raising their 
saucepans, they used lumps of natron, contained in their cargo, for 
the purpose. When the natron was exposed to the action of the fire, 
it melted into the sand lying on the banks of the river, and they saw 
transparent streams of some unknown liquid trickling over the ground ; 
this was the origin of glass.’ No matter how it may have originated, 
there is no doubt that the Phoenicians manufactured glass on a large 
scale, and their glass-work became celebrated all over the world. 
Dyeing works, however, take the first rank among Phoenician indus- 
tries, and Tyrian purple was one of the chief objects of luxury among 
the ancients. The word ‘ purple ’ was not used only for a single colour, 
but for a particular kind of dye, for which animal colours obtained 
from the juice of certain shellfish were used. The dyeing works could 
not be carried on without cloths, for the Phoenicians dyed woollen 
materials chiefly with their famous purple. The wool came from 
Damascus, and the greater part of their export of woollen stuffs was 
doubtless of their own manufacture. Sidon was the first town that 
became noted for these fabrics. Homer often mentions tunics from EARLY CIVILIZATIONS 
15 
that town, but afterwards they were manufactured all over Phoenicia, 
and particularly at Tyre. Among the products of Phoenician in- 
dustry we must also mention the numerous ornaments and the articles 
whose value depends largely on their workmanship. The trade of 
barter which they had so long maintained with barbaric races, amongst 
whom these objects always find an appreciative market, had incited 
the Phoenicians to apply themselves to these industries. Chains of 
artistically worked gold were worn by Phoenician navigators in Homer’s 
time, and Ezekiel mentions their curious work in ivory, which they 
procured through Assyria from India, and from Ethiopia. Accident 
has preserved the names of only a small number of the articles pro- 
duced by the Phoenicians, but the existence of these among a rich 
and luxurious people implies the existence of others. 
The Phoenician religion was a worship of personified forces of 
nature, especially of the male and female principles of reproduction. 
It was in a popular and simple form a worship of the sun, the moon, 
and the five planets, regarded as intelligent powers actively affecting 
human life. . . . And the Phoenician religion not only consecrated 
licentiousness, it also sanctioned cruelty. Living children were 
offered as burnt sacrifices to Baal as well as to Moloch. One can 
scarcely understand how human sacrifices could have been endured 
by an intelligent people; but this abominable ritual was in force in 
all the colonies, and especially at Carthage, where during the siege of 
the city by Agathocles, about 307 b.c., two hundred boys of the best 
families were offered as burnt sacrifices to the planet Saturn. 
Though we have but few fragments of Phoenician antiquities and 
literature, we at least know their system of writing. It is now proved 
that the Phoenicians did not invent writing; they merely communi- 
cated letters to the Greeks. . . . The Greeks adopted the Phoenician 
characters with only a few modifications; the Latin races used the 
same letters designed more simply; they had received them at a 
very remote date, for the Latin tongue was a sister not a daughter of 
the Greek. The French, Spanish, and Italian languages are all 
derived from the Latin and use the same characters, while even the 
Teutonic languages, like English and German, have adopted this 
alphabet. The Phoenicians must, on this ground alone, take high 
rank in the history of civilization. . . . 
The Phoenicians were not only the pioneers of industry, but by 
their commerce they brought together the peoples of the three con- 16 
A SHORT HISTORY OF SCIENCE 
tinents of the Old World. The first carriers by sea, acting as inter- 
mediate agents between the different nations, they exchanged ideas 
as well as merchandise; their exploration of different countries led 
to the discovery of new riches; they endowed the West with the 
products of the East, and the East with the products of the West. 
They proved to the world that cities can attain a high degree of pros- 
perity by labour, activity, and economy, and they remain examples of 
the highest development of purely commercial qualities. — Verschoyle. 
The Hebrews. — South of Phoenicia and lying between the 
Arabian Desert and the Mediterranean was Palestine: — 
The whole literature of the Hebrews is included in the collection 
of prose and poetry which we call the Bible, or, more accurately, the 
Old Testament. The simplicity of its narratives, the enthusiasm of 
its hymns, the joyful or plaintive melody of the psalms, the fiery 
eloquence of the prophets, place the Bible, independently of its re- 
ligious and historical importance, high among the great literary 
monuments of antiquity. Their literature is a proof that the poetic 
imagination was fully developed amongst the Hebrews, and that the 
people were deeply thoughtful as well as passionately religious. . . . 
The Hebrews were not an artistic or an industrial people; but they 
possessed an indisputable superiority to all other nations of antiquity 
in their purely spiritual religion, and in their appreciation of the 
supreme importance of morals as the proper expression of religion. 
Religion was their rule of life, the maker of their laws, the pervading 
spirit of the whole community, as in no other nation before or since. 
— Verschoyle. 
The Emergence of European Civilization. — Until very 
recently little was known of European events before the writings 
of Herodotus (484-425 b.c.). Within the last half century, how- 
ever, and largely as a result of the labors of the archaeologists 
Schliemann and Evans, the existence of a wonderfully rich and 
complete prehistoric civilization has been revealed on the shores 
and islands of and near the Greek Peninsula. 
The recent discoveries in Crete have added a new horizon to 
European civilization. A new standpoint has been at the same time EARLY CIVILIZATIONS 
17 
obtained for surveying not only the ancient classical world of Greece 
and Rome, but also the modern world in which we live. — Sir 
Arthur Evans. 
Civilization in the Bronze Age. — The newer 
European civilization had also its prehistoric times, and the 
investigations and especially the excavations of the last half 
century have revealed such treasures as the site of Homeric Troy, 
the palace and tomb of Agamemnon, and the cities of Minos and 
others of the sea kings of Crete. It is now known that the Trojan 
War was fought about Hissarlik on the eastern shore of the Dar- 
danelles; that Agamemnon’s palace was at Mycenae in Greek 
Argolis; and that Minos had his home and his naval base of 
Mediterranean sea power on the island of Crete. In Mycenae 
and in Crete the arts were highly developed. Painting, sculpture, 
and pottery, tools, weapons, implements of various kinds, with 
systems of water supply and drainage, testify to the remarkable 
degree of civilization attained in the later Bronze Age, although 
this has left behind it no written records and was formerly known 
to us only through the poems of Homer. 
Even to classical students twenty, nay, ten years ago, Crete was 
scarcely more than a land of legendary heroes and rationalized myths. 
It is true that the first reported aeronautical display was made by a 
youth of Cretan parentage, but in the absence of authenticated records 
of the time and circumstances of his flight, scholars were sceptical of 
his performance. And yet within less than ten short years we are 
faced by a revolution hardly more credible than this story; we are 
asked by archaeologists to carry ourselves back from a.d. 1910 to 
1910 b.c., and witness a highly artistic people with palaces and treas- 
ures and letters, of whose existence we had not dreamed. . . . 
The theme is a fresh one, because nothing was known of the subject 
before 1900; it is important, because the Golden Age of Crete was the 
forerunner of the Golden Age of Greece, and hence of all our western 
culture. The connection between Minoan [Cretan] and Hellenic 
civilization is vital, and not one of locality alone, as is the tie between 
the prehistoric and the historic of America, but one of relationship. 
Egypt may have been foster-mother to classical Greece, but the 
mother, never forgotten by her child, was Crete. . . . 18 
A SHORT HISTORY OF SCIENCE 
Members of three foreign nations have worked in friendly rivalry 
to learn the buried history of Crete . . . and Cretan soil may be 
said to have been found to teem with pre-Hellenic antiquities. The 
hopes of archaeologists have been abundantly justified. We have 
followed them and arrived at the home of the first European civili- 
zation. — Hawes. Crete, the Forerunner of Greece. 
Even at this exceedingly early stage of human progress, the va- 
rious branches of industry had become fairly separated and specialized, 
more so, perhaps, than in the Homeric period, and a considerable 
variety of tools was employed in the various crafts. The carpenter 
was evidently a highly skilled craftsman, and the tools which have 
survived show the variety of work which he undertook. At Knossos 
a carefully hewn tomb held, along with the body of the dead artificer, 
specimens of the tools of his trade — a bronze saw, adze, and chisel. 
‘ A whole carpenter’s kit lay concealed in a cranny of a Gournia house 
left behind in the owner’s hurried flight when the town was attacked 
and burned. He used saws long and short, heavy chisels for stone 
and light for wood, awls, nails, files, and axes much battered by use; 
and what is very important to note, they resemble in shape the tools 
of to-day so closely that they furnish one of the strongest links 
between the first great civilization of Europe and our own.’ Such 
tools were, of course, of bronze. Probably the chief industry of the 
island was the manufacture and export of olive oil. The palace at 
Knossos has its Room of the Olive Press, and its conduit for convey- 
ing the product of the press to the place where it was to be stored for 
use; and probably many of the great jars now in the magazines were 
used for the storage of this indispensable article. — Baikie. Sea 
Kings of Crete. 
The Iron Age. The Greeks or Hellenes. — Soon after the 
arrival of the Iron Age, and probably not far from 1200-1000 b.c., 
a new people became prominent on the shores of the JSgean. 
These were the Greeks or, as they called themselves, Hellenes, — 
inhabitants of Greece or Hellas. Their precise origin is unknown, 
but they were undoubtedly of Indo-European stock and probably 
came, in part at least, from the north. It has been conjectured 
that their conquest of the existing inhabitants was facilitated by, 
if not due to, their possession of weapons of iron. Of the earlier EARLY CIVILIZATIONS 
19 
part of this new period (1200-800 b.c.) we have only the legendary 
accounts of the Homeric and Hesiodic poems, which are now gen- 
erally believed to be based upon, if not actually descriptive of, 
episodes of this age. 
The Hellenes soon supplanted the Phoenicians as traders in the 
southern iEgean; and “ if we now leave the monuments of the 
Egyptian temple or the Assyrian palace and turn to the pages of 
the Iliad and the Odyssey ... at once we are in the open air, 
and in the sunshine of a natural life. The human faculties have 
free play in word and deed. . . From the first the Greek is re- 
solved to confront the facts of life.” — Jebb. 
References for Reading 
Osborn, H. F. Men of the Old Stone Age. 
Lubbock, Sir John (Lord Avebury). Prehistoric Times (7th Edition). 
Myres, J. L. The Dawn of History. 
Tylor, E. B. Anthropology and Primitive Cidture. 
Haddon, A. C. History of Anthropology. 
Jastrow, Morris, Jr. The Civilization of Babylonia and Assyria. 
Hawes, C. H. and H. Crete, the Forerunner of Greece. 
Spearing, H. G. The Childhood of Art. CHAPTER II 
EARLY MATHEMATICAL SCIENCE IN BABYLONIA AND 
EGYPT 
In most sciences one generation tears down what another has built 
and what one has established another destroys. In Mathematics alone 
each generation builds a new story to the old structure. — Hankel. 
A history of science may be based on some more or less definite 
logical system of definitions and classifications. As a matter of 
historical evolution, however, such systems and such points of 
view belong to relatively recent and mature periods. Science 
has grown without very much self-consciousness as to how it is 
itself defined, or any great concern as to the distinction between 
pure and applied science, or as to the boundaries between the 
different sciences. Mathematics, for example, has had its roots in 
the human need of exact statement as to both number and form in 
all sorts of affairs, and on the other hand in the analytical faculties 
of the human mind, which have shaped the development of the pure 
science and given it in course of time its deductive stamp. 
The origin of a science can seldom be precisely determined, and 
the more ancient the science the more difficult is the attainment 
of such precision. The periods at which primitive men of different 
races began to have conscious appreciation of the phenomena of 
nature, of number, magnitude, and geometric form, can never be 
known, nor the time at which their elementary notions began to 
be so classified and associated as to deserve the name of science. 
Very early in any civilization, however, mathematics must ob- 
viously have taken its rise in simple processes of counting and 
adding, of time measurement in primitive astronomy, of the geom- 
etry and arithmetic involved in land measurement and in archi- 
tectural design and construction. We can safely sketch certain 
rough outlines of the prehistoric picture, and we can to some extent 
20 BABYLONIA AND EGYPT 
21 
verify these, on the one hand, by archaeological evidence, on the 
other, by present-day observations of backward races — still in 
their prehistoric stage. 
Primitive Astronomical Notions. — On the astronomical side 
the most obvious fact is the division of time into periods of light and 
darkness by the apparent revolution of the sun about the earth. 
With closer attention it must soon have been observed that the rela- 
tive length of day and night gradually changes, and that this 
change is attended by a wide range of remarkable phenomena. At 
the time of shortest days, vegetable and animal life (in the north 
temperate zone) is checked by severe cold. With the gradually 
lengthening days, however, snow and ice sooner or later disap- 
pear, vegetation is revived, birds return from the warmer south, 
all nature is quickened. In the symbolism of the beautiful old 
myth, the sleeping princess, our earth, is aroused by the kiss of the 
sun-prince. The longest days and those which succeed them 
are a period of excessive heat and of luxuriant vegetation, followed 
by harvests as the days shorten, towards the completion of the 
great annual cycle. In time, closer observers, noting the stars, dis- 
covered that corresponding with this great periodic change are 
gradual variations in the starry hemisphere visible at night, that 
in other words the sun’s place among the stars is progressively 
changing, that it is in fact describing a path completed in a large 
number of days, which after repeated counting is found to be 365. 
It is also found that the midday height of the sun above the 
southern horizon shares in the annual cycle. The determination 
of the number of days in the year is a matter of very gradual ap- 
proximation, possible only to men who have already attained 
some command of numbers and the habit of preserving records 
extending over a long series of years. For there is no well-marked 
beginning of the year as of the day. An erroneous determination 
of the number of days becomes apparent only after a number of 
years, increasing with the accuracy of the original approximation. 
If, for example, the year is assumed to be exactly 365 days, that 
is, about six hours too short, the festivals and other dates will slip 
back about 24 days in a century, and thus lose their original cor- 22 
A SHORT HISTORY OF SCIENCE 
respondence with climatic conditions. A revision of the calendar 
will become necessary. 
Still another natural period is introduced by the motion of the 
moon, which seems like the sun to have a daily motion about 
the earth, and also to describe a closed path among the stars in a 
period of about 29 days. Unlike the sun, however, the moon has 
during this period a remarkable change of apparent shape and 
luminosity from “ new ” to “ full ” and back again. The study of the 
day, the year, the month, thus naturally determined by the great 
heavenly bodies has led to the development of the calendar with 
greater and greater accuracy, the most recent rectification of 
the length of the year dating only (in England) from 1752. The 
difficulty of expressing the precise length of the month and the 
year in days, causing the imperfection of early calendars, has, on 
the other hand, reacted to the advantage of mathematical as- 
tronomy by demanding the greatest possible precision both of 
observation and of the computation based upon it. 
The Planets. — Another celestial phenomenon, though less ob- 
vious than the foregoing, must have found wide recognition in pre- 
historic times. The stars vary widely in grouping and individual 
brilliancy, but in general their relative positions are sensibly 
constant. To this constancy, however, five exceptions are easily 
discovered in the wandering motion of the planets Mercury, Venus, 
Mars, Jupiter, and Saturn, which like sun and moon have their 
several paths among the stars but with seemingly irregular mo- 
tions. Corresponding to these seven bodies there was set up by 
prehistoric people an arbitrary division of time into weeks of seven 
days, “the most ancient monument of astronomical knowledge.” 
The correspondence with the planets is still preserved in the 
names of the days of the week in several modern languages.1 The 
French Italian 
1 Sunday dimanche domenica (Sun) 
Monday lundi lunedi (Moon) 
Tuesday mardi martedi (Mars) 
Wednesday mercredi mercoledi (Mercury) 
Thursday jeudi giovedi (Jupiter) 
Friday vendredi venerdi (Venus) 
Saturday samedi sabato (Saturn) BABYLONIA AND EGYPT 
23 
further division of time into hours, minutes, and seconds has 
followed more arbitrarily, and in connection with the develop- 
ment of progressively improved methods of time measurement. 
Astrology and Cosmology. — Side by side with the develop- 
ment of elementary astronomy on its observational and mathemati- 
cal sides were evolved in intimate connection with it, but sometimes 
in extraordinary imaginative forms, astrology and cosmology, 
dealing respectively with the supposed influence of the heavenly 
bodies on human affairs, and with the structure and organization 
of the world. Both these pseudo-sciences were inextricably 
blended, under priestly and literary influences, with a bewildering 
mass of superstition and mythology, legend and invention. In 
their earlier stages, both doubtless contributed powerfully to 
interest and progress in real science. Ultimately both have had 
to be torn away, as the scaffolding from a cathedral, in the never 
ending process of releasing truth from error. 
Primitive Counting. — On the arithmetical side the present 
counting processes of primitive peoples have particular interest. 
The distinction between one and two similar objects, and that 
between two and three or more, belong to a relatively early stage 
of development, but tribes are known to-day in which the entire 
number scale is one, two, many (i.e. more than two). The pro- 
cess of counting is naturally facilitated by the use of fingers and 
toes as counters, their number 10 being the well-known anatomi- 
cal basis for our denary or decimal number system. This may be 
illustrated by the following passages from E. B. Tylor’s Primitive 
Culture: — 
Father Gilij, describing the arithmetic of the Tamanacs on the 
Orinoco, gives their numerals up to 4; when they come to 5, they 
express it by the word amgnaitone, which being translated means ‘a 
whole hand ’; 6 is expressed by a term which translates the proper 
gesture into words itacono amgnapona tevinitpe, ‘one of the other 
hand,’ and so on up to 9. Coming to 10, they give it in words as 
amgna aceponare, ‘both hands.’ To denote 11 they stretch out 
both the hands, and adding the foot they say puitta-pona tevinitpe, 
‘one to the foot,’ and so on up to 15, which is iptaitone, ‘a whole 24 
A SHORT HISTORY OF SCIENCE 
foot.’ Next follows 16, ‘one to the other foot/ and so on to 20, 
tevin itoto, ‘one Indian;’ 21, itacono itoto jamgnar bona tevinitpe, 
‘ one to the hands of the other Indian ’; 40, acciache itoto, ‘ two In- 
dians/ and so on for 60, 80, 100, ‘three, four, five Indians,’ and be- 
yond if needful. South America is remarkably rich in such evidence 
of an early condition of finger-counting recorded in spoken language. 
The Zulu counting on his fingers begins in general with the little 
finger of his left hand. When he comes to 5, this he may call edesanta 
‘ finish hand; ’ then he goes on to the thumb of the right hand, and 
so the word tatisitupa ‘ taking the thumb ’ becomes a numeral for 6. 
Then the verb komba ‘to point,’ indicating the forefinger, or 
‘ pointer/ makes the next numeral, 7. Thus, answering the question 
‘ How much did your master give you ? ’ a Zulu would say ‘ U kom- 
bile’ ‘He pointed with his forefinger’ i.e. ‘He gave me seven,’ 
and this curious way of using the numeral verb is shown in such an ex- 
ample as ‘amahashi akombile’ ‘the horses have pointed’i.e. ‘there 
were seven of them.’ In like manner, kijangalobili ‘keep back 
two fingers/ i.e. 8, and kijangalolunje ‘keep back one finger’ i.e. 9, 
lead on to kumi, 10; at the completion of each ten the two hands with 
open fingers are clapped together. 
The most instructive evidence I have found bearing on the forma- 
tion of numerals, other than digit-numerals, among the lower races, 
appears in the use on both sides of the globe of what may be called 
numeral-names for children. In Australia a well-marked case occurs. 
With all the poverty of the aboriginal languages in numerals, 3 being 
commonly used as meaning ‘ several or many/ the natives in the Ade- 
laide district have for a particular purpose gone far beyond this narrow 
limit, and possess what is to all intents a special numeral system, extend- 
ing perhaps to 9. They give fixed names to their children in order 
of age, which are set down as follows by Mr. Eyre: 1, Kertameru; 
2, Warritya; 3, Kvdnutya; 4, Monaitya; 5, Milaitya; 6, Marru- 
tya; 7, Wangutya; 8, Ngarlaitya; 9, Pouarna. These are the male 
names, from which the female differ in termination. They are given 
at birth, more distinctive appellations being soon afterwards chosen. 
The mathematical advantage of 12 as a base conveniently 
divisible has often been pointed out, but the choice unfortunately 
had to be made long before its real significance could possibly 
be apprehended, and the difficulty of subsequent change would be BABYLONIA AND EGYPT 
25 
prohibitive. Vestiges of the use of 5 and of 20 are familiar; the 
former, for example, in the Roman numerals IV, VI, etc., the latter 
in such expressions as “three score and ten” and in the French 
quatre-vingt. Increasing maturity of a tribe or race, as of an in- 
dividual, is accompanied by gain in the command of larger and 
larger numbers, the rate of progress being very dependent, however, 
on a fortunate choice of notation. However great the capacity 
for inventing number-words, it soon becomes necessary to employ 
some system which shall lead to a regular development of higher 
from lower names. The selection of a point at which dependent 
names, and later, symbols shall begin, is one of the most important 
steps in the history of mathematics. It is difficult for us to 
realize the extent of our indebtedness to the comparatively recent 
so-called Arabic — or more properly, Hindu — notation, in which 
numbers of whatever magnitude may be expressed by means of 
only ten symbols. In any case, however, the appreciation of 
large numbers soon becomes vague. To most of us the word 
million is nearly equivalent to an innumerable multitude. 
Primitive Geometry. — On the geometrical side data are 
naturally more meagre. The notions of a primitive society in 
regard to areas and perimeters and the ratio of a circumference 
to its diameter may quite escape discovery. On the other hand 
skill in making and reading maps is well known — as among the 
Esquimaux. 
Relation of Greek to Older Civilizations. — Mathematical 
science seems to have first assumed definite form in Greece, and it 
is of particular interest to study the indebtedness of the Greeks to 
the older civilizations referred to in the preceding chapter. Some 
degree of civilization doubtless existed further back than any records 
run, in China, in India, in Babylonia, and in Egypt. But of these 
only the latter two exerted a determining influence on the general 
evolution of European science, India making minor though funda- 
mental contributions at a much later stage. Babylonia and Egypt 
exchanged ideas with each other, and, after unnumbered centuries, 
furnished Greece with a certain nucleus of scientific knowledge of 
which the Greeks made enormous use. In practical engineering 26 
A SHORT HISTORY OF SCIENCE 
the achievements of the older civilizations were marvellous, but for 
the creation of real science as systematized, organized knowl- 
edge, containing within itself the seeds of infinite growth, they 
were quite unequal. 
Babylonian Arithmetic.— In Babylonian arithmetic whole 
numbers were expressed in general by only three of the so-called 
cuneiform or wedge-shaped characters employed on the tablets, 
1 = Y, 10 = 100 = Y but the numbers known to have 
been used run into the hundred thousands, this naturally im- 
plying a highly developed command of the fundamental operations 
by means of which large numbers are made to depend upon smaller 
ones. The use of the words for thousand and ten thousand in 
characterizing an indefinite multitude is illustrated in many 
scriptural passages, for example: “ Saul hath slain his thousands, 
and David his ten thousands” (1 Sam. xviii. 7); “a thousand 
thousands ministered unto him, and ten thousand times ten thou- 
sand stood before him ” (Dan. vii. 10). With such expressions 
may be compared: “ I will make thy seed as the dust of the earth ”; 
“He telleth the number of the stars; he calleth them all by 
their names” (Gen. xiii. 16; Ps. cxlvii. 4). The number 40 
also plays a special role in such expressions as the “ forty years 
in the wilderness,” the “forty days and forty nights” of rain 
which caused the flood. 
Of remarkable interest in the Babylonian inscriptions is the oc- 
currence, side by side with a decimal system, of a number system 
based on 60, employed for mathematical and astronomical pur- 
poses. A table of squares of the natural numbers presents, for 
example, nothing novel for the first seven numbers, after which 
follow, however, the equivalent of 
1 4 is the square of 8 
1 21 is the square of 9 
1 40 is the square of 10 
2 1 is the square of 11 
Just as in our notation, for example, 325 means three times the 
square of ten plus twice ten, plus five, so this table must mean: — BABYLONIA AND EGYPT 
27 
once sixty plus four 
once sixty plus twenty-one 
once sixty plus forty 
twice sixty plus one, etc., 
necessarily implying the representation of 60 by 1 in the second 
place. 
In a table of cubes, the perfect cube 4096 is represented similarly 
by 1 8 16, that is 1 X 602 + 8 X 60 + 16 = 4096. The origin 
of this sexagesimal system has been ingeniously attributed to the 
blending of two civilizations, one possessing a system based on 10, 
the other a system based on 6, — a combination suggested by the 
command of the Persian king that the Ionian troops wait 60 
days at the bridge over the Ister; by the splitting of the river 
by Cyrus into 360 rivulets, etc. Fractions were employed to a 
limited extent with denominators 60, and 3600 (= 60 X 60). 
The great step of completing the number system by a character 
for zero seems not to have been successfully made, though there 
are indications of an approach to it in later Babylonian times. 
There is evidence of a mystical or magical use of numbers. Each 
god, for example, was designated by a number from 1 to 60 ac- 
cording to his rank. 
A rational system of weights and measures was introduced, 
the unit of weight depending on that of length, as in the modern 
metric system. 
Babylonian Astronomy. — In connection with astronomical 
observations the Babylonians invented a method of measuring 
time by means of the water clock or clepsydra. From a vessel 
kept full, water was allowed to escape very slowly into a second 
vessel in which it could be weighed. To equal weights of water 
corresponded equal intervals of time. 
Starting the flow at the moment the upper edge of the sun 
first appeared in the east and stopping as soon as the whole sun 
was visible, the amount of water collected was compared with 
that escaping from sunrise to sunrise, and the sun’s diameter 
thus determined as jiv of its whole path in the sky. The time 28 
A SHORT HISTORY OF SCIENCE 
required for traversing the whole path — i.e. the day — was 
then divided into 12 double hours, in one of which the sun’s disk 
advanced by its own diameter multiplied by 60. Their use of 
the number 60 as a base led also to the further subdivision of the 
hour into 60 minutes of 60 seconds each. The year was reckoned 
as 365 days, and even the unequal rate of the sun’s motion at 
different periods was recognized. 
Particularly noteworthy in connection with Chaldean astronomy 
is the discovery of a period of 6585 days, — a little more than 18 
years, — for the recurrence of eclipses. This would appear to have 
been based on a long series of observations, but to have taken no 
account of the region of visibility of eclipses of the Sun. The 
periods of the planets in their orbits were approximately deter- 
mined, but there is no evidence of a systematic geometrical 
theory of celestial motions. 
As to accuracy of direct observation it is said that in later Baby- 
lonian times angles were measured to within 6 minutes and time to 
less than a minute. Quantities obtained indirectly by observa- 
tions extended over long periods, as the length of the lunar month, 
were naturally determined with correspondingly greater precision. 
A list of eclipses of the moon from 747 b.c. was known to 
Ptolemy, while an astrological work prepared about 3700 b.c. con- 
tains evidence of a long series of pre-existing observations. To 
the Romans the Chaldeans were known as star-gazers, and the art 
of augury or divination was much cultivated, making some of the 
earliest known use of geometrical forms. Herodotus ascribes the 
origin of the sun-dial to Babylonia. 
Babylonian Geometry. — In geometry the elementary use of 
the circle quickly leads to the discovery that a chord equal to the 
radius subtends one-sixth of the four right angles at the centre, and 
is thus one side of a regular inscribed hexagon, a figure found on 
Babylonian monuments. A failure to distinguish between the 
length of the arc and that of its chord led to the first approximation 
to the ratio of a circumference to its diameter, tt = 3, which occurs 
in the Old Testament where King Solomon’s molten sea is said to 
be “ ten cubits from the one brim to the other: it was round all BABYLONIA AND EGYPT 
29 
about, . . . and a line of thirty cubits did compass it round 
about.” 
There is some evidence of a knowledge of the fact that a triangle 
of sides 3, 4, and 5 has a right angle, and the trisection of the right 
angle was accomplished. The circle was divided into 360 degrees. 
The sun-dial and its division into degrees are very clearly men- 
tioned in the books of Kings and Isaiah. Parallels, triangles, and 
quadrilaterals were used. 
We may summarize what we know as to the main features of 
Babylonian mathematical science as follows: — 
In astronomy, records of observations extending over many cen- 
turies, the determination of an 18-year eclipse period, the 
approximate determination of the year as 365 days, a good 
system of measuring time, the identification of Mercury, 
Venus, Mars, Jupiter, and Saturn as planets; 
In arithmetic, a well-developed sexagesimal system, tables of 
squares and cubes, arithmetic and geometric progressions, 
the use of large numbers; 
In geometry, the identification of the right triangle of sides 
3, 4, and 5, the inscribed hexagon, the division of the circum- 
ference into 360 degrees, the crude approximation for the ratio 
of circumference to diameter, tt = 3. 
Mathematical Science in Egypt. — Josephus asserts that 
the Egyptians learned arithmetic from Abraham, who brought 
it with astronomy from Chaldea, and that the Egyptians in their 
turn taught the Greeks. The indebtedness of Greek science to 
Egyptians must at any rate have been very considerable. The 
pyramids are monumental evidence of appreciation of geometric 
form and of a relatively high development of engineering con- 
struction nearly 4000 years before the Christian era. Their 
builders must have had precise geometrical and astronomical 
notions. In nearly all of the pyramids, for example, the slope 
of the lateral faces is 52°, and the direction of their base- 
edges is nearly uniform. The regular inscribed hexagon was 
known. After an earlier year of 12 months of 30 days each, the 30 
A SHORT HISTORY OF SCIENCE 
Egyptians added 5 days at the end of each year. According to 
their legend the god Thot won these days at play from the moon 
goddess. An edict of 238 b.c. introduced the leap-year, but the 
innovation was afterwards forgotten. The Egyptian records 
number more than 350 solar, and more than 800 lunar eclipses 
before the Alexandrian period. 
The Ahmes Papyrus. — Our most important source of in- 
formation in regard to early Egyptian mathematics is the so-called 
Ahmes manuscript, dating from some time between 1700 and 
2000 b.c. “Direction for attaining knowledge of all dark things” 
are the opening words of this oldest known mathematical treatise. 
Rules follow for computing the capacity of barns and the area of 
fields. The text consists, however, rather of actual examples than 
of rules, the inferring of these being left to the reader. Reference 
is made to writings some 500 years older, presumably based in 
their turn on centuries of tradition. 
In the computations fractions are used as well as whole numbers, 
but fractions other than § are expressed in terms of fractions 
with unit numerators. The problem of decomposing other frac- 
tions into a limited number of such reciprocals is interestingly 
treated, examples occurring of considerable complexity. It 
would appear that such decompositions, effected by special de- 
vices or hit upon accidentally, were gradually tabulated as rec- 
ords of mathematical experiment. 
The problems discussed by Ahmes include a class equivalent 
to our algebraic equations of the first degree with one unknown 
quantity, — the first known appearance of this important idea. 
Thus, for example : — 
“Heap (or quantity) its f, its |, its y, its whole makes 33.” In 
our notation - z+ - + -+ z = 33. 
3 2 7 
The solution requires the number to be found which multiplying 
1 + I + \ + -f shall produce 33. The result appears in the suffi- 
ciently intricate form 
14 + i + wr “b tht d- + rrr + tts- BABYLONIA AND EGYPT 
31 
Again: “ Rule for dividing 700 loaves among four persons, f for 
one, \ for the second, for the third, for the fourth, . . . Add 
f, and \ that gives 1 + | Divide 1 by 1 + £ + | that 
gives + it- Make \ + it of 700 that is 400.” Thus, to 
modernize this solution, the four persons A, B, C, and D receive 
on one round 1 + f + i = t loaves; the number of rounds is 
700 1 
i i T or i 1 1 X 700 = 400, from which the respective 
1+2+4 1+2+4 
shares are readily obtained. 
Certain problems show an acquaintance with arithmetic and 
geometric progressions. Thus, for example, a series is given of 
the numbers 7, 49, 343, 2401, 16807, the successive powers of 7, 
accompanied by the words person, cat, mouse, barley, measure. 
Almost 4000 years later this was interpreted to mean: 7 persons 
have each 7 cats, each cat catches 7 mice, each mouse eats 7 stalks 
of. barley, each stalk can yield 7 measures of grain; what are the 
numbers and what is their sum ? 
Special symbols are used for addition, subtraction, and equality. 
The Egyptian seems never to have had a multiplication table. 
Multiplication by 13, for example, was accomplished by repeated 
doubling, and then by adding to the number itself, its products by 
4 and by 8. 
Herodotus reports from the fifth century B.c. that the Egyptians 
reckoned with stones, a practise independently developed in many 
lands, notably in the form of the abacus. This little comput- 
ing machine of beads on wires was invented independently in 
different parts of the ancient world. In China and other parts 
of the Orient it is still widely and very skilfully employed. 
The handbook of Ahmes is also rich on the geometrical side. 
It contains information in regard to weights and measures, and 
treats of the conversion from one denomination into another. 
As in case of the progressions, geometrical problems are given, de- 
pending on the use of formulas not derived in the text itself. They 
include computation of areas of fields bounded either by straight 
lines or circular arcs, including in the former case only isosceles tri- 
angles, rectangles, and trapezoids. An isosceles triangle of base 4 32 
A SHORT HISTORY OF SCIENCE 
and side 10 is said to have as its area i X 10 = 20, the actual area 
being of course | X V 100—4(=19.6 approximately). It is 
interesting that this and similar crude methods continued in use 
by surveyors for many centuries, even after Euclid had given geo- 
metrical science its modern form. Another problem amounts to 
finding two squares having a given total area and their sides in 
a given ratio, being thus equivalent to solving the equations 
x2 + y2 = 100 x: y = 1: f 
By trial x = 1, y = f, give x2 + y2 = (f)2. 
Since 100 = (f)2 X 82, the trial values must be multiplied by 8, 
so that x = 8 and y = 6. 
The classical problem of “ squaring the circle ” is attempted, the 
result being equivalent to the approximation ir = = 3.16, as 
against the actual 3.14 — an excellent result for the time. 
Other computations deal with the capacity of storehouses — of 
unknown shape — for grain. A remarkable group of problems 
deals with a certain geometrical ratio in pyramids equivalent to 
a modern cosine or cotangent, and of interest in connection with 
the uniform slope of the great pyramids. 
Egyptian Land Measurement. — Greek writers emphasize 
the methods of land measurement of the Egyptians consequent on 
the obliteration of boundaries by floods of the Nile. Herodotus 
relates that Sesostris had so divided the land among all Egyptians 
that each received a rectangle of the same size, and was taxed ac- 
cordingly. Whoever lost any of his land by the action of the river 
must report to the king, who would then send an overseer to meas- 
ure the loss, and make a proportionate abatement of the tax. 
Thus arose geometry (geometria = earth measurement). Dio- 
dorus, for example, says: “ The Egyptians claim to have intro- 
duced alphabetical writing and the observation of the stars, like- 
wise the theorems of geometry, and most of the arts and sciences.” 
The priests “occupy themselves busily with geometry and arith- 
metic, for as the river annually changes the land, it causes many 
controversies as to boundaries between neighbors. These cannot 
be easily adjusted unless a geometer ascertains the real facts BABYLONIA AND EGYPT 
33 
by direct measurement. Arithmetic serves them in domestic 
affairs and in connection with the theorems of geometry; it is also 
of no slight advantage to those who occupy themselves with the 
stars. For if the position and motions of the stars have been care- 
fully observed by any people it is by the Egyptians; they preserve 
records of particular observations for an incredibly long series of 
years. . . . The motions and times of revolution and stationary 
points of the planets, also the influence of each on the development 
of living things and all their good and evil influences have been 
very carefully observed by them.” 
Egyptian Geometry. — In a passage written about 420 b.c., 
the Greek mathematician, Democritus, boasts that “ In construct- 
ing lines according to given conditions no one has ever surpassed 
me, not even the so-called rope-stretchers of the Egyptians.” 
The exact orientation of the Egyptian temples required the deter- 
mination of the meridian and of a right angle. Both processes 
were naturally an important part of the mathematical lore of the 
priesthood. The first step was accomplished by observation of 
the stars. It is believed that the second step was the function of 
the “ rope-stretchers,” the name being due to their dependence on a 
rope of length 12, divided by two knots into sections of 3, 4, and 5. 
When the two ends of the rope are joined and the three sections 
drawn taut by the knots, the angle opposite the section 5 is a right 
angle. The geometrical knowledge thus attributed to the Egyp- 
tians of a special case of the Pythagorean proposition does not, of 
course, imply knowledge of the proposition itself, or even the ability 
to prove the particular case, which was probably known only em- 
pirically. Egyptian architecture made use of geometrical figures 
as wall decoration and even employed the principle of propor- 
tionality, by dividing a blank wall-space into squares before apply- 
ing the design. The idea of perspective drawing seems, however, 
not to have been attained. 
The existence of such a problem book as that of Ahmes may be 
considered as fairly implying also the existence of comparable 
treatises of a more theoretical character, but other evidence of this 
is lacking. 34 
A SHORT HISTORY OF SCIENCE 
The main features of Egyptian mathematical science are then 
as follows: about 2000 b.c. a well-developed use of whole numbers 
and fractions; a method of solving equations of the first degree 
with one unknown quantity; an approximate method for find- 
ing the circumference of a circle of given radius; approximate 
methods for finding areas of isosceles triangles and trapezoids; 
the rudiments of a theory of similar figures. 
References for Reading 
Ball. A Short History of Mathematics, Chapter I. Cajori. A History of 
Mathematics, pages 1-15. Berry. A History of Astronomy, Chapter I. 
Dreyer. Planetary Systems, Introduction. Gow. History of Greek Mathe- 
matics, Chapters I, II. 
Map op the World by (517 b.c.) 
(From Breasted’s Ancient Times. Courtesy of Messrs. Ginn & Co.) 
Hecatseus, a geographer of Miletus, travelled widely, including a journey up the Nile, and 
wrote a geography of the world. In this book, as in the Map . . . the Mediterranean Sea was 
the centre and the lands about it were all those known to the author. . . . After the Unknown 
Historian of the Hebrews [about 850 b.c.] he was the first historical writer of the early world. 
— Breasted. CHAPTER III 
THE BEGINNINGS OF SCIENCE IN GREECE 
Except the blind forces of Nature nothing moves in this world 
which is not Greek in its origin. — Sir Henry Sumner Maine. 
A spirit breathed of old on Greece and gave birth to poets and 
thinkers. There remains in our classical education I know not what 
of the old Greek soul — something that makes us look ever upward. 
And this is more precious for the making of a man of science than the 
reading of many volumes of geometry. — Poincare. 
Number, the inducer of philosophies, 
The synthesis of letters. — JEschylus. 
Mathematics, considered as a science, owes its origin to the idealistic 
needs of the Greek philosophers, and not as fable has it, to the practical 
demands of Egyptian economics. . . . Adam was no zoologist when 
he gave names to the beasts of the field, nor were the Egyptian sur- 
veyors mathematicians. — Hankel. 
Geographical Boundaries. — From the twilight of civili- 
zation and the first faint suggestions of science in Chaldea and 
Egypt, we pass to the more brilliant dawn of science and civili- 
zation in Greece. Geographically we shall be concerned not 
merely with Greece itself, but, as time passes, with other Hellenic 
countries, especially the Ionian shores and islands of western 
Asia Minor, and the Greek colonies in southern Italy, Sicily, and, 
after its conquest by Alexander the Great, northern Egypt. Greece 
and its civilization seem immeasurably closer to us both in time 
and in spirit than do ancient Babylonia and Egypt. In these 
more remote civilizations science had been cultivated chiefly as a 
tool, either for immediate practical applications or as a part of the 
professional lore of a conservative priesthood. In Greece, on 
the other hand, for the first time in the history of our race, 
35 36 
A SHORT HISTORY OF SCIENCE 
human thought achieved freedom, and real science became pos- 
sible. 
Mathematics as a science commenced when first some one, prob- 
ably a Greek, proved propositions about any things or about some 
things, without specification of definite particular things. — White- 
head. 
Indebtedness of Greece to Babylonia and Egypt. — It is 
plain, nevertheless, that Greek civilization and Greek science 
owed much to Egypt and Chaldea. Herodotus has been quoted 
already, and Theon of Smyrna (second century a.d.) says: — 
In the study of the planetary movements the Egyptians had 
employed constructive methods and drawing, while the Chaldeans 
preferred to compute, and to these two nations the Greek astrono- 
mers owed the beginnings of their knowledge of the subject. 
Again in the third century a.d. Porphyry observes: — 
From antiquity the Egyptians have occupied themselves with 
geometry, the Phoenicians with numbers and reckoning, the Chal- 
deans with theorems. 
The Greek Point of View. — It is not, however, so much the 
achievements of the Greeks in positive science which compel our 
attention and admiration as it is the remarkable spirit which they 
displayed toward man and the universe. Here for the first time 
we meet with a new point of view, and while Shelley’s well-known 
dictum, “We are all Greeks, our laws, our literature, our religion, 
our art have their roots in Greece,” must be dismissed as in- 
correct as well as extravagant, and even Sir Henry Maine’s 
maxim, which stands at the head of this chapter, is undoubtedly 
an exaggeration, these famous sayings serve well to illustrate 
the fact that with the Greeks came into the world a new 
spirit and a new interpretation of Nature. 
In a striking essay entitled “What we owe to Greece,” Butcher 
has portrayed with extraordinary clearness those characteristics 
of the Greeks which lifted them above all of their predecessors and 
above most if not all of those that have come after them; — BEGINNINGS IN GREECE 
37 
The Greeks before any other people of antiquity possessed the 
love of knowledge for its own sake. To see things as they really are, 
to discern their meaning and adjust their relations, was with them 
an instinct and a passion. Their method in science and philosophy 
might be very faulty and their conclusions often absurd, but they 
had that fearlessness of intellect which is the first condition of seeing 
truly. . . . Greece, first smitten with the passion for truth, had 
the courage to put faith in reason and in following its guidance to 
take no account of consequences. ‘Those,5 says Aristotle, ‘who 
would rightly judge the truth must be arbitrators and not litigants.5 
‘Let us follow the argument wheresoever it leads’ may be taken 
not only as the motto of the Platonic philosophy but as expressing one 
side of the Greek genius. . . . 
At the moment when Greece has come into the main current of 
the world’s history, we find a quickened and stirring sense of per- 
sonality and a free people of intellectual imagination. The oppres- 
sive silence with which Nature and her unexplained forces had brooded 
over man is broken. Not that the Greek temper is irreverent or 
strips the universe of mystery. The mystery is still there and felt . . . 
but the sense of mystery has not yet become mysticism. . . . Greek 
thinkers are not afraid lest they should be guilty of prying into hidden 
things of the gods. They hold frank companionship with thoughts 
that had paralyzed Eastern nations into dumbness or inactivity, and 
in their clear gaze there is no ignoble terror. . . . Know thyself, is 
the answer which the Greek offers to the sphinx’s riddle. . . . But 
to the Greeks, ‘know thyself’ meant not only to know man but the 
less pleasing task to know foreigners. . . . The people of ancient 
India did not care to venture beyond their mountain barriers and to 
know their neighbors. The Egyptians, though in certain branches of 
science they had made progress, — in medicine, in geometry, in as- 
tronomy, — had acquired no scientific distinction for they kept to 
themselves, but the Greeks were travellers. . . . Aristotle thought 
it worth his while to analyze and describe the constitutions of 58 
states, including in his survey not only Greek states but those of the 
barbarian world. . . . 
It was the privilege of the Greeks to discover the sovereign efficacy 
of reason. . . . And it was Ionia that gave birth to the idea which 
was foreign to the East but has become the starting-point of modern 
science, the idea that Nature works by fixed laws. . . . Again, in 38 
A SHORT HISTORY OF SCIENCE 
history the Greeks were the first who combined science and art, reason 
and imagination. . . . The application of a clear and fearless intel- 
lect to every domain of life was one of the services rendered by Greece 
to the world. It was connected with an awakening of the lay spirit. 
In the East the priests had generally held the keys of knowledge. . . . 
To Greece then we owe the love of science, the love of art, the love of 
freedom. . . . And in this union we recognize the distinctive features 
of the West. The Greek genius is the European genius in its first 
and brightest bloom. 
Sources. — The sources of our information as to the details of 
the scientific ideas of the Greeks are exceedingly meagre, some of 
the most important historical and scientific treatises being known 
to us only by title or by detached quotations, or indirectly through 
Arabic translations. Among specific ancient sources of infor- 
mation in regard to Greek mathematical science the following 
may be mentioned : — 
About 330 b.c., Eudemus, a disciple of Aristotle, wrote a his- 
tory of geometry of which a summary by Proclus has been 
preserved. 
About 70 b.c., Geminus of Rhodes wrote an Arrangement of 
Mathematics with historical data. This has also been lost, but 
quotations are preserved in some of the later authors. 
About 140 a.d., Theon of Smyrna wrote Mathematical Rules 
necessary for the Study of Plato. 
About 300 a.d., Pappus’ Collections contain much information 
in regard to the previous development of geometry. 
In the fifth century a.d., Proclus published a commentary on 
Euclid’s Elements with valuable historical data. 
The Calendar. — The Greek calendar was based at an early 
period on the lunar month, the year consisting of 12 months of 
30 days each. About 600 b.c. a correction was made by Solon, 
making every two years contain 13 months of 30 days and 12 of 
29 days each, giving thus 369 days per year. In the following 
century a much closer approximation — 365| days — was at- 
tained by confining the thirteenth month to three years out of 
eight. This arrangement naturally failed, however, to meet the BEGINNINGS IN GREECE 
39 
Greek desire that the months begin regularly at or near new moon, 
and Aristophanes makes the Moon complain: 
Chorus of Clouds 
“ The Moon by us to you her greeting sends, 
But bids us say that she’s an ill-used moon, 
And takes it much amiss that you should still 
Shuffle her days, and turn them topsy-turvy; 
And that the gods (who know their feast-days well,) 
By your false count are sent home supperless, 
And scold and storm at her for your neglect.” 
About 400 b.c., Meton the Athenian observed that 19 years 
consist of almost exactly 235 lunar months, and accordingly pro- 
posed a new calendar with 125 months of 30 days and 110 of 29 
days, corresponding to an average year of 365 days, 6 hours and 
19 minutes — only about 30 minutes too long. Of this Meton’s 
cycle the traditional rule for determining the date of Easter still 
preserves traces. On account of so much confusion in the official 
calendar the almanacs of the time even designated the dates for 
agricultural operations by means of the constellations visible at 
the corresponding time. 
Time Measurement. — While sun and moon suffice for large- 
scale measurement of time, the approximate determination of its 
subdivisions early became important, and this problem has been 
solved with continually increasing precision to our own day. 
Early time measurement depended either on some form of sun- 
dial as a natural means, or on an apparatus analogous to the 
hour-glass as an artificial method. 
In Isaiah xxxviii. 8, in connection with a promise of prolonged 
life to Hezekiah, it is said 
And this shall be a sign unto thee from the Lord, that the Lord 
will do this thing that he hath spoken; behold, I will bring again the 
shadow of the degrees, which is gone down in the sun-dial of Ahaz, 
ten degrees backward. So the sun returned ten degrees, by which 
degrees it was gone down. 40 
A SHORT HISTORY OF SCIENCE 
The first sun-dial of which a description is preserved belongs to 
the time of Alexander the Great, and consisted of a hollow hemi- 
sphere with its rim horizontal and a bead at the centre to cast the 
shadow. Curves drawn on the concave interior divided the period 
from sunrise to sunset into twelve parts, these lengths being thus 
proportionate to the lengths of the daylight period. 
The use of the clepsydra, or water clock, in Greece dates from the 
fifth century b.c. It consisted there of a spherical bottle with a 
minute outlet for the gradual escape of water. Its use in regulat- 
ing public speaking is illustrated by Demosthenes’ demand when 
interrupted, “You there: stop the water.” • 
For the sake of conformity with the sun-dial division of each 
day and each night into twelve equal parts, the rate of flow in the 
clepsydra required continual adjustment. Ingenious improvements 
were made in the mechanism in course of time, but in considering 
the work of the Greek astronomers, the impossibility of what we 
should consider accurate time measurement must not be for- 
gotten. 
Greek Arithmetic. — In Greek arithmetic the earliest known 
numerals are merely the initials of the respective number words. 
Two other systems came into use later. In one of these the 
numbers from 1 to 24 are represented by the 24 letters of the 
Ionian alphabet; in the other the letters represent numbers, but 
no longer in consecutive order. This use of letters for numbers 
was not confined to Greece, but appears to have originated there. 
The Greeks had no zero, and never discovered the immense ad- 
vantage of a position-system, such as that by which we are able 
to express all numbers by only ten symbols. Fractions occur not 
infrequently. The change from the earlier notation to that with 
24 characters was a disastrous one. There were not only more 
characters to memorize, but computation became materially more 
complicated. These disadvantages far more than offset the su- 
perior compactness, the sole merit of the new system. The 
special importance of such compactness for coins has led to the 
suggestion that they were the medium through which this nota- 
tion was introduced. BEGINNINGS IN GREECE 
41 
A simple numerical computation of late date in the Greek 
alphabetic numerals and its modern equivalent are 
tr | e 265 
a £ e 265 
~T~a 40 000, 12 000, 1000 
MM ja 
a 
M ,n’T 12 000, 3 600, 300 
ta t Ice 1 000, 300, 25 
i 70 225 
M a k e 
— Gow. 
Division was an exceedingly laborious process of repeated sub- 
traction. 
Probably nothing in the modern world would have more aston- 
ished a Greek mathematician than to learn that, under the influence 
of compulsory education, the whole population of Western Europe, 
from the highest to the lowest, could perform the operation of division 
for the largest numbers. — Whitehead. 
Approximate square roots were found by the later Greeks. 
Theon in the fourth century a.d. for example gives the following 
rule: — 
When we seek a square-root, we take first the root of the nearest 
square-number. We then double this and divide with it the re- 
mainder reduced to minutes and subtract the square of the quotient, 
then we reduce the remainder to seconds and divide by twice the 
degrees and minutes (of the whole quotient). We thus obtain nearly 
the root of the quadratic. 
The reckoning board, or abacus, — known in so many different 
forms throughout the world, — came into very early use, but 
actual evidence in regard to its form is meagre. A sharp dis- 
tinction was made between the art of calculation (logistica), and 
the science of numbers {arithmetical). The former was deemed 
unworthy the attention of philosophers, and to their attitude may 
be fairly attributed the fact that Greek mathematics was always 42 
A SHORT HISTORY OF SCIENCE 
weak on the analytical side, and seemed in a few centuries to 
reach the limit of its possible development. 
Greek Geometry. — It was in geometry that Greek mathe- 
matics chiefly developed, and for several fundamental reasons. 
The Greek mind had a strong predilection for formal logic, a keen 
aesthetic appreciation of beauty of form, and, on the other hand, 
with no adequate symbolism for arithmetic or algebra, a distinct 
disdain, at any rate among the educated, for the commercialized 
mathematics of computation. The history of Greek mathematics 
is therefore to a great extent the history of geometry. Formal 
geometry as distinguished from the solving of particular geo- 
metrical problems, had, indeed, no previous existence, and we 
have to do with the beginnings of elementary geometry as we now 
know it. 
The Ionian Philosophers. — The sense of curiosity, the feel- 
ing of wonder, the spirit of inquiry, — these are the common ele- 
ments of philosophy and science. It is thus not strange that the 
earliest names in science are likewise the earliest in philosophy. 
In the childhood and youth of the race specialization has not 
begun, all knowledge lies invitingly open to the expanding mind. 
We have seen how much had been accumulated in Egypt and 
Babylonia of knowledge and skill in observing and recording the 
phenomena of the heavens, in irrigation and in measurement of 
land. Much of the same general character was doubtless true of 
the Phoenicians, the Trojans, the Cretans, and other precursors of 
the Greeks. But nothing deserving the name of science has come 
down to us from the iEgean or Greek civilization before the time 
of Thales of Miletus, chief of the Ionian philosophers, and one 
of the “seven wise men of Greece.” 
Thales.—'The ancient and fragmentary register of Greek 
mathematicians, or history of Greek geometry before Euclid, 
attributed to Eudemus, begins: 
As it is now necessary to consider also the beginnings of the arts 
and sciences in the present period, we report that, according to the 
evidence of most, geometry was invented by the Egyptians, taking 
its origin from the measurement of land. This last was necessary BEGINNINGS IN GREECE 
43 
for them on account of the inundation of the Nile, which obliterated 
every man’s boundaries. It is however, nothing wonderful that the 
invention of this as of the other sciences has grown out of necessity, 
as everything in its beginnings proceeds from the incomplete to the 
complete. A regular transition takes place from perception to thought- 
ful consideration, from this to rational knowledge. Just as now with 
the Phoenicians an exact knowledge of numbers took its rise in the 
needs of trade and commerce, so geometry began with the Egyptians 
for the reason mentioned. Thales, who went to Egypt, first brought 
this science into Greece. Much he discovered himself, of much 
however he transmitted the beginnings to his successors. Some 
things he made more general, some more comprehensible. 
The significance packed into this terse quotation may well be 
emphasized. The mathematics of the Chaldeans, the Egyptians, 
the Phoenicians, was merely a tool, crudely shaped to meet vital 
concrete needs; it had little possibility of development. The 
Greek intellect, seizing upon the fragmentary knowledge of these 
practical races, refined from it the germs of a new pure science, 
making the knowledge “more general” and “more comprehen- 
sible,” and at the same time discovering much that was new. On 
the other hand, inclining in its zeal for pure science to the op- 
posite extreme of disregard for the concrete applications, Greek 
science eventually reached its own limit of possible growth. In 
the long run scientific progress must depend on due appreciation 
of the complementary importance of both pure and applied science. 
Thales was of Phoenician descent, and was born about 624 b.c. 
in Miletus, a city of Ionia, at that time a flourishing Greek colony 
in what is now Asia Minor. As an engineer he was employed to 
construct an embankment for the river Halys. As a merchant 
he dealt in salt and oil, and, visiting Egypt, learned there 
something of the wisdom of the Egyptian priesthood. He oc- 
cupied himself with the study of the stars as well as of geometry, 
and in particular, 
announced to the inhabitants of Miletus that night would enter 
upon the day, the sun hide himself, the moon place herself in front, 
so that his light and radiance would be intercepted. 44 
A SHORT HISTORY OF SCIENCE 
Herodotus says that there was a war between the Lydians and 
the Medes, and after various turns of fortune 
in the sixth year a conflict took place, and on the battle being joined, 
it happened that the day suddenly became night. And this change, 
Thales of Miletus had predicted to them, definitely naming this year, 
in which the event really took place. The Lydians and the Medes, 
when they saw the day turned into night, ceased from fighting, and 
both sides were desirous of peace. 
This eclipse is supposed to have taken place in 585 b.c. The 
prediction of the year of an eclipse gained Thales a great repu- 
tation with his contemporaries, though his designation with six 
others as “wise men of Greece” appears to have had a primarily 
political significance. None of the other six at any rate had any 
scientific standing. He taught that the year has 365 days; that 
the equinoxes divide the year unequally; that the moon is illu- 
minated by the sun. The mathematical attainments attributed 
to Thales include the following theorems of elementary geometry; 
the angles at the base of an isosceles triangle are equal; when two 
straight lines cut each other the opposite angles are equal; the 
first proof that the circle is bisected by its diameter; the in- 
scription of the right triangle in the semicircle; the measurement 
of height by shadow, involving the principle of similar triangles. 
Plutarch relates that Niloxenus, conversing with Thales con- 
cerning King Amasis, says: — 
Although he also admires you on account of other things, he 
prizes above everything the measurement of the pyramids, in that 
you have without any trouble and without needing an instrument, 
merely placed your staff at the end of the shadow cast by the pyramid, 
showing from the two triangles formed by the contact of the solar 
rays that one shadow has the same relation to the other as the pyra- 
mid to the staff. 
Some writers even attribute to Thales a knowledge that the 
sum of the angles of a triangle is two right angles, also of the 
idea of a circle as a locus of a point having a certain property, 
but conclusive evidence can hardly be adduced. Even the im- BEGINNINGS IN GREECE 
45 
plied knowledge of similar triangles is doubtful. In connection 
with his shadow measurements it is interesting that his scholar 
Anaximander, born 611 b.c., introduced the sun-dial into Greece. 
While our knowledge of Thales and his work is extremely 
meagre, the mathematical results above mentioned have consid- 
erable significance in connection with the comparison between 
Greece and Egypt. The Egyptian standpoint was fundamentally 
practical, specific, inductive; the Greek shows already its char- 
acteristic tendencies to abstract generalization, to logical proof, 
and to the methods of deductive science. Most of the facts as- 
cribed to Thales may well have been known to the Egyptians. 
For them these facts would have remained unrelated; for the 
Greeks they were the beginnings of an extraordinary develop- 
ment of the science of geometry. 
Milesian Cosmology. — The cosmological ideas of the Milesian 
philosophers were sufficiently ingenious and picturesque. To 
Thales the earth is a circular disk floating in an ocean of water. 
This water is the fundamental element of the whole. Ice, snow, 
and frost turn readily into water, even rocks wear away and 
disappear in it. Man himself seems capable of turning into it, 
while the waters of sea and land shrink into solid residues. By 
evaporation of the water air is formed, its agitation causes earth- 
quakes. The stars between their setting and rising pass behind 
the earth. 
The following passages (Fairbanks’ translation) indicate the 
estimation in which Thales was held by later Greek philosophers. 
As to the quantity and form of this first principle or element, 
there is a difference of opinion; but Thales, the founder of this sort 
of philosophy, says that it is water (accordingly he declares that the 
earth rests on water), getting the idea I suppose because he saw that 
the nourishment of all beings is moist, and that warmth itself is 
generated from moisture and persists in it (for that from which all 
things spring is the first principle of them); and getting the idea also 
from the fact that the germs of all beings are of a moist nature, while 
water is the first principle of the nature of what is moist. . . . 
‘Some say that the earth rests on water. I have ascertained that 46 
A SHORT HISTORY OF SCIENCE 
the oldest statement of this character is the one credited to Thales, 
the Milesian, to the effect that it rests on water, floating like a piece 
of wood or something else of that sort.’ . . . And Thales, according 
to what is related of him, seems to have regarded the soul as some- 
thing endowed with the power of motion, if indeed he said that the 
loadstone has a soul because it moves iron. . . . Some say that soul is 
diffused throughout the whole universe; and it may have been this 
which led Thales to think that all things are full of gods.—Aristotle. 
Of those who say that the first principle is one and movable, to 
whom Aristotle applies the distinctive name of physicists, some say 
that it is limited; as for instance Thales of Miletos . . . who 
seems also to have lost belief in the gods. These say that the first 
principle is water, and they are led to this result by things that appear 
to the senses; for warmth lives in moisture and dead things wither 
up and all germs are moist and all nutriment is moist .... Thales 
is the first to have set on foot the investigation of nature by the 
Greeks; although so many others preceded him, he so fai surpassed 
them as to cause them to be forgotten. It is said that he left nothing 
in writing except a book entitled Nautical Astronomy. — Theo- 
phrastus. 
It is said that Thales of Miletos, one of the seven wise men, was 
the first to undertake the study of Physical Philosophy. He said 
that the beginning (the first principle) and the end of all things is 
water. All things acquire firmness as this solidifies, and again, as it 
melts, their existence is threatened; to this are due earthquakes and 
whirlwinds and movements of the stars .... Thales was the first 
of the Greeks to devote himself to the study and investigation of the 
stars and was the originator of this branch of science; on one occa- 
sion he was looking up at the heavens and was just saying he was in- 
tent on studying what was overhead, when he fell into a well; where- 
upon a maid-servant named Thratta laughed at him and said: ‘ In 
his zeal for things in the sky he does not see what is at his feet.’ And 
he lived in the time of Kroesos. — Hippolytus. 
Thales of Miletos regards the first principle and the element as 
the same thing. ... So we call earth, water, air, fire, elements. . . . 
Thales declared that the first principle of things is water. The 
Physicists, followers of Thales, all recognize that the void is really a 
void. The earth is one and spherical in form. It is in the midst 
of the universe. Thales and Democritus find in water the cause BEGINNINGS IN GREECE 
47 
of earthquakes. . . . Thales thinks that the Etesian winds blowing 
against Egypt raise the mass of the Nile, because its outflow is beaten 
back by the swelling of the sea which lies over its mouth. — JEtius. 
Anaximander. — A second native of Miletus, Anaximander 
(about 611-545 b.c.) had a different interpretation of nature, 
holding that the fundamental stuff, out of which all things are 
made, is something between air and water. He believed the earth 
to be balanced in the centre of the world, because being in the 
centre and having the same relation to all parts of the circum- 
ference, it ought not to tend to fall in one direction rather than in 
any other. This point of view, not easily taken by the layman, 
illustrates the natural tendency of the Greek philosopher to em- 
phasize geometrical symmetry. 
Among those who say that the first principle is one and movable 
and infinite is Anaximander of Miletos, son of Praxiades, pupil and 
successor of Thales. He said that the first principle and element of 
all things is infinite, and he was the first to apply this word to the 
first principle; and he says that it is neither water nor any other one 
of the things called elements, but the infinite is something of a dif- 
ferent nature from which came all the heavens and the worlds in 
them; and from what source things arise, but that they return of 
necessity when they are destroyed. . . . Evidently when he sees the 
four elements changing into one another, he does not deem it right to 
make any one of these the underlying substance, but something else 
besides them. — Theophrastus. 
The earth is a heavenly body, controlled by no other power and 
keeping its position because it is the same distance from all things. 
The form of it is curved, cylindrical, like a stone column. It has two 
faces. One of these is the ground beneath our feet and the other is 
opposite to it. The stars are the circle of fire, separated from the 
fire about the world, and surrounded by air. There are certain 
breathing-holes like the holes of a flute through which we see the 
stars; so that when the holes are stopped up there are eclipses. The 
moon is sometimes full and sometimes in other phases, as these holes 
are stopped up or open. The circle of the sun is 27 times that of the 
moon. . . . Man came into being from another animal, namely the 
fish, for at first he was like a fish. — Hippolytus (on Anaximander). 48 
A SHORT HISTORY OF SCIENCE 
Anaximander, collecting data from the Ionian sailors frequent- 
ing Miletus, constructed a map of the earth, and speculated on the 
relative distances of the heavenly bodies. 
Herodotus relates that 
during the reign of Cleomenes, Aristagoras, prince of Miletus, ar- 
rived at Sparta; the Lacedaemonians affirm, that desiring to have 
a conference with their sovereign, he appeared before him with a 
tablet of brass in his hand, on which was inscribed every known part 
of the habitable world, the seas, and the rivers. 
Anaximenes. — A third Ionian Greek, often associated with 
those just mentioned, is Anaximenes (sixth century b.c.), like them 
a native of Miletus. For him the stars are fixed upon the celes- 
tial vault, and pass behind the northern (highest) part of the earth 
on setting. Air, not water, is the first cause of all things, the 
others being formed by its compression or rarefaction. The heat 
of the sun is due to its rapid motion, but the stars are too remote 
to give out heat. 
Anaximenes arrived at the conclusion that air is the one movable, 
infinite, first principle of all things. For he speaks as follows: ‘ Air 
is the nearest to an immaterial thing; for since we are generated in 
the flow of air, it is necessary that it should be infinite and abundant, 
because it is never exhausted.’ (A fragment accredited to. Anax- 
imenes.) 
Most of the earlier students of the heavenly bodies believed that 
the sun did not go underneath the earth but rather around the earth 
and this region, and that it disappeared from the view and produced 
night because the earth was so high toward the north. . . . Anaxim- 
enes and Anaxagoras and Democritus say that the breadth of the 
earth is the reason why it remains where it is. . . . Anaximenes says 
that the earth was wet, and when it dried it broke apart, and that 
earthquakes are due to the breaking and falling of hills. — Aristotle. 
The school of Thales and his successors in this Ionian outpost 
of Greek civilization was soon succeeded by developments of still 
greater importance in the more remote Italian colonies. BEGINNINGS IN GREECE 
49 
Pythagoras and his School. — The register of mathemati- 
cians proceeds: — “ After these Pythagoras transformed the occu- 
pation with this branch into a true science, by considering the 
foundation of it from a higher standpoint, and investigated its 
theorems in a more abstract and intellectual way. It is he also 
who invented the theory of the irrational and the construction 
of the cosmical bodies.” These few words like those quoted of 
Thales are full of meaning. The Egyptian priests knew 
geometrical facts, the raw material of mathematical science; 
Thales adapted this material to building purposes, Pythagoras 
began the systematic foundations of the structure. Both in 
name and in substance mathematics as a science begins with 
Pythagoras. 
Pythagoras founded in the Greek cities of southern Italy a 
school which had much of the character of a fraternity or secret 
society, this with political tendencies ultimately arousing hos- 
tility which proved destructive to it. Beyond these undisputed 
facts his life and work are obscured by a great mass of tradition 
and myth, even the date of his birth being doubtful. A native 
of the island of Samos not far from Miletus, he appears to have 
been much affected by Egyptian influences during a residence in 
that country. A visit to Babylon even is alleged, but with doubt- 
ful authority. The etiquette of the Pythagorean school required 
that all discoveries should be attributed to the “ Master ” and 
not revealed to outsiders. To Pythagoras himself must probably 
be ascribed the so-called Pythagorean theorem, this forming the 
necessary basis for the theory of the irrational mentioned in the 
register. A similar inference may be drawn in regard to the reg- 
ular polyhedra. On the other hand, Pythagoras appears to have 
interested himself in the theory of numbers, particularly in con- 
nection with music and geometry. He is said to have first in- 
troduced weights and measures among the Greeks. 
The attribution of particular results or beliefs to individuals of 
this period is however very doubtful on account of the fact that 
Pythagoras left no writings whatever, that his school was es- 
sentially a secret society, and that in later centuries it became 50 
A SHORT HISTORY OF SCIENCE 
the custom to credit its founder with all sorts of knowledge which 
he could not possibly have possessed. 
Pythagoras makes the classification, arithmetic (numbers 
absolute), music (numbers applied), geometry (magnitudes at 
rest), astronomy (magnitudes in motion), this fourfold division or 
“ quadrivium ” continuing in vogue for some two thousand years. 
The distinction between abstract and concrete arithmetic had been 
emphasized among the Greeks in comparatively early times. 
Arithmetic and geometry were distinguished on one side from 
mechanics, astronomy, optics, surveying, music, and computation 
on the other. The aim of Greek arithmetic “was entirely differ- 
ent from that of the ordinary calculator, and it was natural that 
the philosopher who sought in numbers to find the plan on which 
the Creator worked, should begin to regard with contempt the 
merchant who wanted only to know how many sardines, at 10 for 
an obol, he could buy for a talent.” 
The limited mathematics of the practical Egyptians had con- 
sisted of numerical cases. It was an easy step for Pythagoras to 
make number in a somewhat mystical sense the central element 
in his philosophy. 
Pythagorean Arithmetic. — In pure arithmetic or number 
theory as We should call it, the Pythagoreans enunciated such 
dicta as, for example, “Unity is the origin and beginning of all 
numbers but not itself a number.” Prime and composite num- 
bers were also distinguished, and theorems of considerable alge- 
braic complexity discovered. There is naturally no algebraic 
symbolism, but “unknown” and “given” quantities are employed 
in the modern sense. Odd and even numbers received special 
names, and besides the series of squares and cubes and the arith- 
metic and geometric progressions previously known, 
other series were derived from these, for example, 
the triangular numbers: 1, 3, 6, 10, 15, etc., by 
successive addition of the natural numbers. The 
reason for the name triangular will be clear if one 
counts the dots in the triangle formed by taking one, two, three or 
more rows beginning at the top of the figure. BEGINNINGS IN GREECE 
51 
The senes of squares is formed by adding the odd numbers 
successively; 1+3 = 4, 1+3+5 = 9, etc. The series 2, 6, 
12, 20, 30, etc. is formed by adding the even 
numbers, or again by multiplying adjacent 
natural numbers. If we construct a series of 
squares or parallelograms with a common angle 
and sides of length 1, 2, 3, 4, 5, etc. the figure 
which must be added to any one to produce 
the next larger was called by the Greeks a 
gnomon, the area of which would be repre- 
sented by one of the series of odd numbers, — an interesting and 
typical example of the Greek habit of combining geometry with 
number-theory. As products of two numbers were associated with 
areas — “square” or “oblong” — so products of three factors 
were interpreted as volumes. A later Pythagorean calls the cube 
the “ geometrical harmony ” — an expression embodying the as- 
sociation of mathematics with music. The cube has indeed 6 
faces, 8 vertices, 12 edges; 6, 8, and 12 are in harmonic progres- 
sion, that is, 8 is the harmonic mean between 6 and 12. 
Pythagorean Geometry. — In geometry the Pythagoreans 
formulated definitions of the fundamental elements, line, sur- 
face, angle, etc. They are credited with a number of theorems 
depending on the application of one surface to another,1 and im- 
plying a knowledge of methods of determining area and of the 
properties of parallel lines. They developed a fairly complete 
theory of the triangle, including the fundamental proof that the 
sum of the angles of a triangle is two right angles, by a method 
not very different from our own. The theory of the “cosmic-al 
bodies” mentioned in the register is of special in- 
terest. Any solid angle must have at least three 
faces. If three equal equilateral triangles have a 
common vertex they will when cut or folded so 
that their edges are brought together, form a solid 
angle, and a fourth equal triangle will complete a regular tet- 
rahedron. Similarly, if we start with four triangles, we may 
1 Some of these are equivalent to the solution of the quadratic equation. 52 
A SHORT HISTORY OF SCIENCE 
build up with four others a regular octahedron, or starting with five, 
an icosahedron with 20 faces. Six triangles, however, will fill the 
angular space about a point, and thus not permit the formation of 
a regular polyhedron. Using squares instead of triangles, we 
obtain only the cube; using pentagons (angle 108°), the regular 
dodecahedron — 12 faces, 3 at each vertex. The Egyptians must 
have been familiar with the cube, the regular tetrahedron, and the 
octahedron. To these, with the icosahedron, the Pythagoreans as- 
sociated the four cosmical elements — earth, air, fire, and water. 
Their discovery of an additional body, the regular dodecahedron, 
formed by 12 pentagons, made a break in the correspondence, and 
the need was met by the addition of the universe, or, according 
to others, the ether, as a fifth term in the cosmical series. This 
correspondence was not merely symbolical, but physical, the 
earth being supposed to consist of cubical particles, etc. We 
cannot infer that the impossibility of a sixth regular polyhedron 
was known. That only these five regular polyhedra are pos- 
sible was in fact first proved by Euclid. There is a tradition 
that the Pythagorean discoverer of the dodecahedron was 
drowned at sea on account of the sacrilege of announcing his 
discovery publicly. A later commentator records a similar 
tradition that the discoverer of the irrational perished by 
shipwreck, since the inexpressible should remain forever con- 
cealed, and that he who touched and opened up this picture 
of life was transported to the place of creation and there washed 
in eternal floods. 
The regular polygons were naturally studied, and in particu- 
lar the decomposition of them into right triangles of 45° and 
30°-60°. With the pentagon the attempt naturally failed, but 
the five-pointed star formed by drawing diagonals was a special 
emblem of the Pythagoreans. With the inscribed pentagon 
connects itself naturally the division of a line in extreme and 
mean ratio, or, as it was later characterized, the “golden sec- 
tion.” This division, by which the square on the greater segment 
of a line is equivalent to the rectangle whose sides are the other 
segment and the whole line, occurs repeatedly in Greek archi- BEGINNINGS IN GREECE 
53 
tecture of the fifth century, with fine effect, and must have been 
systematically employed. 
As to the celebrated theorem which bears the name of Pythag- 
oras, he may well have learned from the Egyptian rope-stretchers 
that a right angle is formed by taking sides of lengths 3 and 4 and 
separating the other ends a distance 5, while his study of numbers 
would easily have led to the discovery that in the series of squares 
the adjacent 9 and 16 make 25. It would naturally be investi- 
gated whether a similar relation could be verified for other right 
triangles. In the most familiar case of the isosceles right tri- 
angle it soon appears that the length of the equal sides being taken 
as 1, the length of the hypotenuse could be only approximately 
expressed. It cannot indeed be exactly expressed by any whole 
number, or fraction; it is irrational. 
If it is true as Whewell says, that the essence of the triumphs of 
science and its progress consists in that it enables us to consider evident 
and necessary, views which our ancestors held to be unintelligible and 
were unable to comprehend, then the extension of the number concept 
to include the irrational, and we will at once add, the imaginary, 
is the greatest forward step which pure mathematics has ever taken. 
— Hankel. 
In this case the proof of the Pythagorean theorem is easily 
effected by a simple graphical construction, involving merely the 
drawing of diagonals of squares. The smaller 
triangles in the figure are evidently all equal. 
The larger square contains four of them, the 
smaller squares, two each. It seems possible 
that this was the Pythagorean method, but as 
to how the proof was accomplished in other cases 
we have no information, the simpler proof of Euclid having com- 
pletely superseded the earlier. On the other hand, for the cor- 
responding arithmetical problem of finding three whole numbers 
which can be the sides of a right triangle, Pythagoras is said to 
have given a correct solution, equivalent in our notation to 
(2 a + l)2 + (2a2 + 2a)2 = (2a2 + 2a + l)2, 54 
A SHORT HISTORY OF SCIENCE 
a denoting any positive integer. How this method was discovered 
remains a matter of conjecture. 
We may recognize here the characteristic elements of the in- 
ductive method, first, observation of the particular fact that in a 
certain right triangle, with sides 3, 4, and 5, the sum of the squares 
on the two sides is equal to that on the hypotenuse; second, the 
formation of the hypothesis that this may be true also for right 
triangles in general; third, the verification of the hypothesis in 
other particular cases. Then follows the deductive confirmation 
of the hypothesis as a law for all right triangles. 
Pythagorean Physical Science. — It has been already noted 
that one of the most fundamental principles of the Pythagorean 
school was the significance attached to number in connection 
with all sorts of phenomena, the regular motions of the heavenly 
bodies, the musical tones, etc. There is a tradition that Pythag- 
oras, walking one day, meditating on the means of measuring 
musical notes, happened to pass near a blacksmith’s shop, and 
had his attention arrested by hearing the hammers as they struck 
the anvil produce sounds which had a musical relation to each 
other. It was found that vibrating cords emitted tones de- 
pendent in a simple way on their length; for example, cords of 
lengths 2, 3, and 4 giving a tone, its fifth and its octave re- 
spectively. The monochord used in studying these numerical 
relations is said to have been the first apparatus of experimental 
physics. It was even supposed that each of the various 
heavenly bodies and the sphere of the fixed stars had a char- 
acteristic tone, these all uniting to produce the so-called “ music 
of the spheres.” 
Terrestrial Motion ; Philolaus, Hicetas. — The universe 
was believed to consist of the four elements, — earth, air, fire, water, 
— to be a sphere with a spherical earth at its centre, and to have 
life. Pythagoras identified the morning and evening stars, and 
attributed the moon’s light to reflection. It is of peculiar interest 
that later Pythagoreans, in particular Philolaus, about 400 b.c., 
attributed the apparent daily motion of the heavenly bodies from 
east to west not to their own actual motion but to a motion of BEGINNINGS IN GREECE 
55 
the earth in the opposite direction. This latter motion, however, 
was thought of, not as a rotation, but as an orbital motion about a 
so-called “central fire.” Just as the moon revolved about the 
earth, always turning the same face towards the latter, so the 
earth might revolve about the central fire which would be forever 
invisible to the inhabitants of the other side of the earth. While 
we say that the moon rotates about its axis in the same time in 
which it revolves about the earth, to the ancients such a motion 
was not considered to include rotation at all. A further essen- 
tially arbitrary assumption introduced between the earth and 
the central fire a counter-earth (antichthon), which was required to 
make up the supposed number of the heavenly bodies, and which 
would hide the central fire from dwellers in the antipodes. 
Aristotle, criticising this theory, says of the Pythagoreans: — 
They do not with regard to the phenomena seek for their reasons 
and causes, but forcibly make the phenomena fit their opinions and 
preconceived notions. . . . When they anywhere find a gap in the 
numerical ratios of things, they fill it up in order to complete the sys- 
tem. As ten is a perfect number and is supposed to comprise the 
whole nature of numbers, they maintain that there must be ten bodies 
moving in the universe, and as only nine are visible, they make the 
antichthon the tenth. 
All the other heavenly bodies describe orbits, each in its own 
hollow sphere about the central fire, the generally adopted order, 
based on the apparent rate of motion among the stars, being 
Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn. Pythagorean 
speculations as to relative distances of the different planets were 
naturally mystical notions merely. The sun was said to move 
around the central fire in an “oblique circle,” i.e. the ecliptic. 
The moon was believed to be inhabited by plants and animals. 
The moon might be eclipsed either by the earth or by the 
counter-earth. This remarkable system, admitting the earth to 
move and not to be the centre of the universe, was not generally 
or long accepted, but had a share in securing the acceptance of 
the theories of Copernicus nearly 2000 years later. One at least 56 
A SHORT HISTORY OF SCIENCE 
of the Pythagoreans made the great further step, somewhat loosely 
described by Cicero in the words : — 
Hicetas of Syracuse, according to Theophrastus, believes that the 
heavens, the sun, moon, stars, and all heavenly bodies are standing 
still, and that nothing in the universe is moving except the earth, 
which, while it turns and twists itself with the greatest velocity round 
its axis, produces all the same phenomena as if the heavens were moved 
and the earth were standing still. 
The activity of the Pythagorean school continued to be im- 
portant until about 400 b.c., that is, until the rise of the Athenian 
school under Plato and his successors. It had not only created 
the science of mathematics; it had developed, however vaguely 
and imperfectly, the idea of a world of physical phenomena 
governed by mathematical laws. 
Dr. Allman says of Pythagoras: — 
In establishing the existence of the regular solids he showed 
his deductive power; in investigating the elementary laws of 
sound he proved his capacity for induction; and in combining 
arithmetic with geometry ... he gave an instance of his philosophic 
power. 
These services, though great, do not form, however, the chief title of 
the Sage to the gratitude of mankind. He resolved that the knowl- 
edge which he had acquired with so great labour, and the doctrine 
which he had taken such pains to elaborate, should not be lost; 
and . . . devoted himself to the formation of a society d’elite, which 
would be fit for the reception and transmission of his science and 
philosophy; and thus became one of the chief benefactors of human- 
ity, and earned the gratitude of countless generations. 
In medicine, we meet before the fifth century only with the 
anatomist Alcmaeon (508 b.c.) of the early medical school at 
Crotona, in Italy, and in natural philosophy (besides Thales and 
others already mentioned) with Xenophanes, who, like Pythagoras, 
held that fossils are in fact what they appear to be, and not mere 
“ freaks of nature,” as was generally believed. BEGINNINGS IN GREECE 
57 
References for Reading 
Allman. Greek Geometry. Chapters I, II. 
Ball. A Short History of Mathematics. Chapter II. 
Berry. A History of Astronomy. Chapter II to page 26. 
Cajori. A History of Mathematics. Pages 16-23. 
Dreyer. Planetary Systems. Chapters I, II. 
Gomperz. Greek Thinkers. Vol. I, pp. 1-164. 
Gow. History of Greek Mathematics. Chapters III, IV, VI. 
Heath. Aristarchus of Samos. 
Butcher. Aspects of the Greek Genius. 
Map of the World according to Herodotus 
(From Breasted’a Ancient Times. Courtesy of Messrs. Ginn & Co.) 
From long journeys in Egypt and other Eastern Countries Herodotus returned with much 
information regarding these lands. His map showed that the Red Sea connected with the 
Indian Ocean, a fact unknown to his predecessor, Hecataeus. [See p. 34,] 
— Breasted. CHAPTER IV 
SCIENCE IN THE GOLDEN AGE OF GREECE 
Our science, in contrast with others, is not founded on a single 
period of human history, but has accompanied the development of 
culture through all its stages. Mathematics is as much interwoven 
with Greek culture as with the most modern problems in engineer- 
ing. She not only lends a hand to the progressive natural sciences, but 
participates at the same time in the abstract investigations of logicians 
and philosophers. — Klein. 
There still remain three studies suitable for freemen. Calcu- 
lation in arithmetic is one of them; the measurement of length, 
surface, and depth is the second; and the third has to do with the 
revolutions of the stars in reference to one another . . . there is in 
them something that is necessary and cannot be set aside ... if 
I am not mistaken, [something of] divine necessity. — Plato. 
Literature and Art. — The fifth century b.c. witnessed that 
astonishing flowering of the Greek genius in literature and mili- 
tary glory which has made it ever since famous. The battles 
of Marathon and Salamis had flung back the Asiatic hosts which 
threatened to overrun and enslave Europe, and had transformed 
the Greeks from a group of jealous and parochial city states into 
a great democratic nation. Trade prospered, wealth increased, 
and for about a century letters, art, and science flourished as never 
before and never since. History began to be written by Herodotus 
and Thucydides. The drama was developed by TSschylus, 
Sophocles, and Euripides to such a pitch that even to-day, after 
the lapse of nearly 2500 years, crowds listen with eager interest 
to the (Edipus of Sophocles and the Iphigenia of Euripides, while 
the poetry of Pindar and the wit of Aristophanes have never lost 
their charm. In architecture and the plastic arts the Parthenon 
and its sculptures still testify to Greek supremacy. 
58 THE GOLDEN AGE OF GREECE 
59 
In science, also, great names testify to memorable deeds. No 
such perfection, to be sure, was attained in science as in literature 
and in sculpture, but vast progress was made in mathematical 
science beyond anything hitherto accomplished, and the founda- 
tions were securely laid for a rational interpretation of man and 
of nature. Literature, architecture, sculpture, and the drama re- 
quire no special apparatus or reagents. Mathematical science also 
is not dependent upon such externals, being in this respect like 
literature and art, and we find geometry and arithmetic at the 
outset moving forward far more rapidly than natural or physi- 
cal science. 
Parmenides. — The recognition of the spherical shape of the 
earth and its division into zones are attributed not only to the 
Pythagoreans, but also to Parmenides of Elis, who lived in the 
early part of the fifth century. He introduced a system of concen- 
tric spheres analogous to that soon to be so highly developed 
by Eudoxus. He identified the evening and the morning stars, 
and attributed the moon’s brightness to reflected light. He 
regarded the sun as consisting of hot and subtle matter detached 
from the Milky Way, the moon chiefly of the dark and cold. 
Empedocles. — Passing over the guesses of Heraclitus and 
Parmenides at the riddle of existence and of man and nature, we 
may pause for a moment to examine the speculations of Empedocles 
(about 455 b.c.). A native of Agrigentum in southern Sicily, 
Empedocles was regarded as poet, philosopher, seer, and im- 
mortal god. He appears to have been a close observer of nature, 
understanding the true cause of solar eclipses and believing the 
moon to be twice as far from the sun as from the earth. The 
latter is held in place by the rapidly rotating heavens “as the 
water remains in a goblet which is swung quickly round in a 
circle.” Aristotle attributes to Empedocles that analysis of the 
universe into the four “ elements,” earth, air, fire, and water, which 
until comparatively recent times was universally accepted as 
fundamental. It is, nevertheless, not only misleading but absurd 
to hold with Gomperz (“Greek Thinkers,” I, 230) that Em- 
pedocles’ theory of the four elements “takes us at a bound into 60 
A SHORT HISTORY OF SCIENCE 
the heart of modern chemistry.” The facts seem rather to be 
that Empedocles put together and hospitably accepted and 
clarified the theories of his various predecessors. He is the 
first sanitarian of whom we have any record, for Empedocles 
is credited with having cut down a hill of his native city and 
thus cured a plague by letting in the north wind, and to have 
done a similar service to the neighboring “parsley” city of Selinus 
(Selinunte) by simply draining a local marsh. 
The following is a fragment from the writings of Empedocles: — 
So all beings breathe in and out; all have bloodless tubes of 
flesh spread over the outside of the body, and at the openings of these 
the outer layers of skin are pierced all over with close-set ducts, so 
that the blood remains within, while a facile opening is cut for the air 
to pass through. Then whenever the soft blood speeds away from 
these, the air speeds bubbling in with impetuous wave, and when- 
ever the blood leaps back the air is breathed out; as when a girl, 
playing with a clepsydra of shining brass, takes in her fair hand the 
narrow opening of the tube and dips it in the soft mass of silvery 
water, the water does not at once flow into the vessel, but the body of 
air within pressing on the close-set holes checks it till she uncovers 
the compressed stream; but then when the air gives way the de- 
termined amount of water enters. And so in the same way when the 
water occupies the depths of the bronze vessel, as long as the narrow 
opening and passage is blocked up by human flesh, the air outside, 
striving eagerly to enter,&olds back the water inside behind the gates 
of the resounding tube, keeping control of its end, until she lets go 
with her hand. Then, on the other hand, the very opposite takes 
place to what happened before; the determined amount of water 
runs off as the air enters. Thus in the same way when the soft blood, 
surging violently through the members, rushes back into the interior, 
a swift stream of air comes in with hurrying wave, and whenever it 
[the blood] leaps back, the air is breathed out again in equal quantity. 
— Fairbanks. 
Anaxagoras. — (500-428 b.c.) For the student of science 
Anaxagoras, a native of Clazomene in Asia Minor, is more im- 
portant than Empedocles. Turning aside from wealth and civic 
distinction in his enthusiasm for science, he seems to have occupied THE GOLDEN AGE OF GREECE 
61 
himself with the problem of squaring the circle, a problem at- 
tacked even by the Egyptians with some degree of success, and 
destined to exercise great influence on the development of Greek 
geometry. The beginnings of perspective are also attributed to 
him, in connection with studies of the stage. He was particu- 
larly interested in a great meteorite — the appearance of which 
he was afterwards said to have predicted — supposing it to have 
fallen from the sun, and inferring that the latter was a “ mass of 
red-hot iron greater than the Peloponnesus,” not very distant 
from the earth. Like the Pythagoreans he assigned as the 
order of distances: — moon, sun, Venus, Mercury, Mars, Jupiter, 
Saturn. The earth’s axis was inclined, in order that there 
might be variations of climate and habitability. He explained 
the moon’s phases correctly, also solar and lunar eclipses, but 
he misinterpreted the Milky Way as due to the shadow cast 
by the earth. His theory of the nature and origin of the cosmos, 
viz. that it was material and had come by the combination and 
differentiation of primitive elementary substances or “seeds” of 
matter, was repugnant to those holding the polytheistic dogmas 
of his time and brought him into popular disfavor. Convicted 
of impiety, he died in exile, 428 b.c. By his insistence upon the 
importance of minute invisible “seeds” or particles of matter he 
paved the way for the “atomism” of Leucippus and Democritus. 
The Atomists. — A very little observation of external nature 
shows that disintegration is forever going on. Ice turns to water, 
water to vapor, rocks to sand and sand to dust — in other words, 
masses to particles. Furthermore, dust vanishes and vapor dis- 
appears, while clouds and fogs, rain and snow, make their appear- 
ance without obvious cause, and dust accumulates from invisible 
sources. What is more reasonable than to suppose that visible 
things — rocks and ice and water — become gradually resolved into 
invisible particles, and that these in their turn condense into new 
visible substances at some later time? For these or similar 
ideas the material “seeds” of Anaxagoras had, as stated above, 
paved the way, when later emphasized by Leucippus and his 
more famous pupil Democritus. Of the life of Leucippus 62 
A SHORT HISTORY OF SCIENCE 
almost nothing is known, but he was probably a contem- 
porary of Empedocles and Anaxagoras, and possibly a pupil 
of Zeno. Leucippus assumed the existence of empty space as 
well as of matter, and held that of atoms all things are consti- 
tuted. Space is infinite in magnitude, atoms infinite in number 
and indivisible, with only quantitative differences. Atoms are 
always in activity, and worlds are produced by atoms variously 
shaped and weighted, falling in empty space and giving rise to an 
eddying motion by mutual impact. 
Democritus of Abdera was a pupil and associate of Leu- 
cippus, whose theories of empty space and material atoms he de- 
veloped and made so famous that his own name alone is often 
associated with them. Of his life, his works, and his death little 
is certainly known, but he may be regarded as marking the culmina- 
tion and conclusion of the Ionian school; and his reputation, both 
in antiquity and in mediaeval times, was immense. Like contem- 
porary and preceding philosophers, his writings were in verse, and 
Cicero is said to have deemed his style worthy of comparison with 
that of Plato. His somewhat boastful comparison of his own 
geometrical power with that of the Egyptian rope-stretchers has 
been quoted. 
Democritus appears to have agreed closely in his interpretation 
of nature with Leucippus, and regarded empty space and atoms 
as cosmic elements. He also held that by the motion of the atoms 
was produced the world with all that it contains. Soul and fire 
are of one nature, their atoms small, smooth, and round. By 
inhaling them life is maintained. Hence the soul perishes with 
and in the same sense as the body, — a doctrine which made 
Democritus odious to later generations. Dante, for example, 
places him far down in hell as “ascribing the world to chance.” 
The atomic theory of perception held that from every object 
“images” of that object are being given off in all directions, some 
of which enter the organs of sense and cause “sensations.” De- 
mocritus further held that sensations are the only sources of our 
knowledge. He was regarded as one of the extreme sceptics of 
antiquity, as e.g. in this saying, “We know nothing: not even THE GOLDEN AGE OF GREECE 
63 
if there is anything to know.” Galileo, himself of a highly scepti- 
cal turn of mind, refers with approval to Democritus, and it is 
probably on this side, i.e. by exemplification of the critical spirit, 
that Democritus rendered his greatest service. His positive con- 
tributions to science, even in atomism, were apparently neither 
novel nor important. Democritus explained the Milky Way 
as composed of a vast number of small stars, but to his dis- 
ciple, Metrodorus of Chios, it was a former path of the Sun. 
The Beginnings of Rational Medicine. Hippocrates of 
Cos. — Before the middle of the fifth century b.c., science in 
the healing art had no existence. Excepting among a few of the 
more enlightened, sacrifices and other appeals to the gods still 
characterized medicine as a priestly rather than a scientific pro- 
fession, while the prevailing ignorance of anatomy and physiology 
made rational treatment of the sick difficult if not impossible. 
Alcmseon, in the previous century, had taken some steps in the 
right direction, proving for example that the sperm does not 
originate, as was currently believed, in the spinal marrow, and that 
the brain is the organ of mind, and advancing a naturalistic theory 
of disease which seems to foreshadow that of his great successor 
Hippocrates. 
Two island centres of medical lore (they can hardly be called 
medical schools), both of the cult of Asclepias, existed in the 
southeastern .Egean, viz. Cos and Cnidus, and on the former was 
born, in 460 b.c., Hippocrates, “the Father of Medicine,” in the 
next century already characterized by Aristotle as “the Great.” 
Of his life, education, practice, and writings comparatively little is 
certainly known. Many of the writings attributed to him are of 
doubtful authenticity and are more safely assigned to the Hip- 
pocratic “school.” Enough remain, however, especially when 
added to the references by later authors to him and to his sayings 
and to his methods of practice, to make it clear that in every re- 
spect Hippocrates was worthy of the lofty reputation with which 
his name has come down to us after five and twenty centuries. 
And yet it is not for the practical arts of medicine or any of its 
basic sciences that Hippocrates did his most famous work. It 64 
A SHORT HISTORY OF SCIENCE 
was rather in his attitude toward health and disease that his 
real greatness lay. For, as far as we know, it was Hippocrates 
who first insisted on regarding disease as a natural rather than a 
supernatural process, and Hippocrates who first urged that care- 
ful observation and study of the patient which entitles him to 
rank as the original “clinician” of medical science. Again, it 
was Hippocrates who first insisted on the existence and importance 
of those processes of self-repair which are to-day recognized as 
fundamental properties of living matter, — processes summed up 
in that famous phrase of his which has come down to us through 
the Latin of the middle ages, — vis medicatrix natures, — “the 
healing power of nature,” one of the finest and truest of the tenets 
of scientific medicine to-day. Finally, by advancing his famous 
theory of the four humors, a theory which with minor modifications 
was for some two thousand years afterwards the prevailing theory 
of pathology, or the nature of disease, among the most enlight- 
ened, Hippocrates still further established his right to be regarded 
as the “father” of medicine, and the first (and only) medical man 
ever authoritatively entitled “the Great.” This theory — crude 
enough to-day — held that health consists in the right mixture, 
and disease in the wrong mixture, of four “humors” (juices) 
of the body, viz. blood, phlegm, yellow bile, and black bile. Here 
again the great merit of Hippocrates’ idea was that it directed 
attention to the body itself, and hence to natural rather than 
supernatural phenomena. 
The tone of the Hippocratic writings is well illustrated by the 
titles of those accepted as probably genuine, e.g. On Airs, Waters, 
and Places; On Epidemics; On Regimen in Acute Diseases; 
On Fractures; On Injuries of the Head; etc. The so-called 
Hippocratic Oath is rightly described by Gomperz as “a 
monument of the highest rank in the history of civilization.” 
That this oath is still administered to graduates about to enter on 
the practice of medicine, is sufficient evidence of the high char- 
acter and far-sighted wisdom of its originator. (See Appendix.) 
The Sophists. — In the fifth century b.c. political events fol- 
lowing war with Persia made Athens supreme in Greece — the THE GOLDEN AGE OF GREECE 
65 
finest and richest city in the world. Its citizens aspired to suc- 
cess in public life, and sought training to that end from the soph- 
ists. While science was not generally cultivated as a leading 
subject in the educational system thus developed,1 mathematics 
could not fail to be esteemed as a means of discipline, and several 
of the sophists made notable contributions to its development. 
Hippias of Elis is the first sophist to be mentioned for impor- 
tant mathematical work. About 420 b.c. Hippias invented a 
curve called the quadratrix, serving for the solution of two of the 
three celebrated problems of Greek geometry; viz. the quadrature 
of the circle and the trisection of an angle. By means of straight 
line and circle constructions, the solution of the quadratic equation 
had been accomplished, though without algebraic symbolism, 
or any recognition of negative or imaginary results. The tri- 
section problem, like that of duplicating the cube, was equivalent 
to the solution of the cubic equation, and could therefore not be 
accomplished by line and circle methods. 
The quadratrix was generated by the inter- 
section P of two moving straight lines, one 
MQ always parallel to its initial position 
OA, the other OR revolving uniformly about 
a centre 0. By means of this curve the 
trisection problem is reduced to that of tri- 
secting a straight line, which is elementary.2 
The curve meets the perpendicular lines OA and OB at C and B 
respectively so that OC: OB = 2:7r, where tt is the ratio of the 
circumference of a circle to its diameter. To this quadrature 
solution the name of the curve is due. 
Dinostratus showed that the assumptions OC: OB > 2 : tt and 
OC: OB <2:7r both lead to contradictions, therefore OC: OB = 2:ir 
— a good example of the Greek reductio ad absurdum. The 
study of a problem not capable of solution by elementary means 
1 See Freeman, “ Schools of Hellas.” 
s To trisect any angle as AOR, draw MQ parallel to OA and divide OM into three 
equal parts by lines parallel to OA, meeting the curve in D and E respectively. 
The radii OS and OT will then trisect the angle AOQ, by the definition of the curve. 66 
A SHORT HISTORY OF SCIENCE 
thus led to the invention of this new curve, the first of which 
we have any definite record. 
The Criticism of Zeno. — The Stoic philosopher Zeno, teach- 
ing in Athens about this time, though not himself a mathematician, 
represents an important phase of philosophical criticism of mathe- 
matics. Every manifold, he says, is a number of units, but a 
true unit is indivisible. Each of the many must thus be itself 
an indivisible unit, or consist of such units. That which is in- 
divisible however can have no magnitude, for everything which 
has magnitude is divisible to infinity. The separate parts have 
therefore no magnitude, etc. Again, as to the possibility of mo- 
tion, he maintains that before the body can reach its destination 
it must reach the middle point, before it can arrive there it must 
traverse the quarter, and so on without end. Motion is thus 
impossible; so the tortoise, if he have any start, cannot be over- 
taken by the swift runner Achilles, for while Achilles is covering 
that distance the tortoise will have attained a second distance, and 
so on. Such specious criticism was naturally, and in a measure 
justly, evoked by misguided efforts of certain mathematicians to 
show that a line consists of a multitude of points, etc. These or 
similar controversies as to the interpretation of the infinite and 
the infinitesimal have persisted till our own day, resembling in that 
respect the classical problems of circle squaring and angle tri- 
section to which reference has been made above. The more or less 
mystical statements about the new discoveries of the Pythagoreans 
also invited sceptical epigrams. 
Zeno was concerned with three problems. . . . These are the 
problem of the infinitesimal, the infinite, and continuity. . . . From 
him to our own day, the finest intellects of each generation in turn 
attacked these problems, but achieved, broadly speaking, nothing. . . . 
— B. Russell. 
Aristotle accordingly solves the problem of Zeno the Eleatic, which 
he propounded to Protagoras the Sophist. Tell me, Protagoras, said 
he, does one grain of millet make a noise when it falls, or does the 
ten-thousandth part of a grain? On receiving the answer that it 
does not, he went on: Does a measure of millet grains make a noise THE GOLDEN AGE OF GREECE 
67 
when it falls, or not? He answered, it does make a noise. Well, 
said Zeno, does not the statement about the measure of millet apply 
to the one grain and the ten-thousandth part of a grain? He as- 
sented, and Zeno continued, Are not the statements as to the noise 
the same in regard to each ? For as are the things that make a noise, 
so are the noises. Since this is the case, if the measure of millet makes 
a noise, the one grain and the ten-thousandth part of a grain make 
a noise. 
Circle Measurement : Antiphon and Bryson ; Hippocrates 
of Chios. — Two of the sophists, Antiphon and Bryson, made 
an interesting contribution to the problem of squaring the circle, 
by means of the inscribed and circumscribed regular polygons. 
Antiphon started with a regular polygon inscribed in a circle, 
and constructed by known elementary methods an equivalent 
square. By doubling the number of sides repeatedly he obtained 
polygons which become more and more nearly equivalent to 
the circle, — the first correct attack on this formidable problem. 
Bryson took the important further step of employing both in- 
scribed and circumscribed polygons, making however the not un- 
natural assumption that the area of the circle may be considered 
the arithmetical mean between them. 
Another great step in the development of the theory of the 
circle was accomplished by Hippocrates of Chios, who had rela- 
tions with the now dispersed Pythagoreans during the latter 
half of the fifth century and came to Athens in later life 
after financial reverses. He is said in the register of mathe- 
maticians to have written the first Elements or textbook of 
mathematics, in which he made effective use of the reductio 
ad absurdum as a method of relating one proposition to 
another. 
To Hippocrates is due the theorem that the areas of circles are 
proportional to the squares on their diameters. He appears to 
have employed geometrical figures with letters at the vertices, in 
the modern fashion. From the theorem in regard to areas of 
circles follows naturally a general theorem for similar segments 
and sectors of circles. His work on lunes is remarkable. Start- 68 
A SHORT HISTORY OF SCIENCE 
ing with an isosceles right triangle, he describes a semicircle on 
each of the three sides. By the theorem just quoted the semi- 
circle on the hypotenuse is equal in 
area to the sum of the other two. If 
the larger semicircle is taken away 
from the entire figure, two equal lunes 
remain; if the two smaller semi- 
circles are taken away, the triangle remains. Therefore the 
two lunes are together equivalent to the triangle, and the 
area of each may be determined. The gulf between rectilin- 
ear and curvilinear figures has at last been successfully crossed. 
A second attempt employs three equal chords instead of two, and 
incidentally the theorem that the square on the side of a triangle 
is greater than the sum of the squares on the other two sides when 
the angle opposite the first side is greater than a right angle. 
Other interesting and still more complicated attempts are pre- 
served. 
A third classical problem was that of the so-called “ duplication 
of the cube.” One of the older Greek tragedians attributed to 
King Minos the words referring to a tomb erected at his order: 
Too small thou hast designed me the royal tomb, 
Double it, yet fail not of the cube. 
At a somewhat later period it is related that the Delians, suf- 
fering from a disease, were bidden by the oracle to double the size 
of one of their altars, and invoked the aid of the Athenian geom- 
eters. Hippocrates transformed the problem of solid geometry 
into one in two dimensions by observing that it is equivalent to 
that of inserting two geometrical means between given extremes. 
In our modern algebraic notation, the continued proportion 
x:y = y:z = z: a leads to the equationsy2 = xz, z2 = ya, whence, 
• • • A 2 
eliminating z, y3 = ax2, y = a*x3; y and z are the desired means 
between x and a, and by putting a = 2x the problem is solved. 
No such algebraic notation existed at this time, however, 
and the geometrical methods invented by later Greek mathema- 
ticians were necessarily very complicated, as will appear below. THE GOLDEN AGE OF GREECE 
69 
Plato and the Academy. — One of the greatest names in the 
history of philosophy is that of Plato, and yet with Plato philosophy 
enters upon a new phase in which it almost parts company with 
science. Before Plato philosophy was almost wholly devoted 
to inquiries or speculations touching the earth, the heavens, and 
the universe, and hence was substantially “nature” or “natural” 
philosophy. But with Plato and ever since his time the larger 
part of philosophy has been devoted to observation and specu- 
lation upon the human mind and its products, and has accordingly 
often been called “mental” or “moral” as contrasted with “nat- 
ural” philosophy. It is therefore Thales and Pythagoras, Democ- 
ritus and Aristotle, rather than Plato and his disciples, who are 
the protagonists of science as the word is used to-day. 
As a disciple of Socrates, Plato found it expedient to leave 
Athens after the death of his master, and during the following 
eleven years he travelled widely in the Mediterranean world, 
doubtless familiarizing himself with the learning of Egypt and of 
the Greek Ptolemies. After having been sold as a slave, re- 
deemed and set free, Plato returned to his native city, and es- 
tablished himself as a philosopher. While primarily a philosopher 
rather than a mathematician, Plato, unlike his master Socrates, — 
who desired only enough mathematics for daily needs, — rated 
highly the importance of mathematics and rendered services of 
the greatest value in its development. This was doubtless due in 
part to the influence of Archytas, a friend of the Pythagoreans, with 
whom he had associated during his prolonged exile. 
The register proceeds: “ Plato . . . caused mathematics in gen- 
eral, and geometry in particular, to make great advances, by reason 
of his well known zeal for the study, for he filled his writings with 
mathematical discourses, and on every occasion exhibited the 
remarkable connection between mathematics and philosophy.” 
“Let no one ignorant of geometry enter under my roof” was 
the injunction which confronted Plato’s would-be disciples. His 
respect for mathematics finds interesting expression in the re- 
marks he puts into the mouth of Socrates in the Dialogues, 
and to him it is largely indebted for its place in higher education. 70 
A SHORT HISTORY OF SCIENCE 
In the Laws he advises the study of music or the lyre to last 
from the age of 13 years to 16, followed by mathematics, weights and 
measures, and the astronomical calendar until 17. For a few picked 
boys on the other hand in the Republic, he recommends before they 
are 18, abstract and theoretical mathematics, theory of numbers, 
plane and solid geometry, kinetics, and harmonics. Of arithmetic 
he says, “Those who are born with a talent for it are quick at 
learning, while even those who are slow at it have their general 
intelligence much increased by studying it.” “No branch of 
education is so valuable a preparation for household management 
and politics and all arts and crafts, sciences and professions, as 
arithmetic; best of all by some divine art, it arouses the dull 
and sleepy brain, and makes it studious, mindful, and sharp.” 
The geometrical Greek view of numbers, exemplified in our 
use of square and cube in algebra, is well illustrated by Thesetetus, 
who says to Socrates that his teacher 
was giving us a lesson in roots, with diagrams, showing us that the 
root of 3 and the root of 5 did not admit of linear measurement by the 
foot (that is, were not rational). He took each root separately up to 
17. There as it happened he stopped, so the other pupil and I de- 
termined, since the roots were apparently infinite in number, to try 
to find a single name which would embrace all these roots. We di- 
vided all numbers into two parts. The number which has a square 
root we likened to the geometrical square, and called ‘square and 
equilateral’ (e.g. 4, 9, 16). The intermediate numbers, such as 3 and 
5 and the rest which have no square root, but are made up of unequal 
factors, we likened to the rectangle with unequal sides, and called 
rectangular numbers. 
Under Plato’s influence mathematics first acquired its unified 
significance, as distinguished from geometry, computation, etc. 
Accurate definitions were formulated, questions of possibility 
considered, methods of proof criticized and systematized, logi- 
cal rigor insisted upon. The philosophy of mathematics was 
begun. The point is the boundary of the line; the line is the 
boundary of the surface; the surface is the boundary of the solid. 71 
THE GOLDEN AGE OF GREECE 
Such axioms as “Equals subtracted from equals leave equals” 
date from this period. The analytical method is developed, con- 
necting that which is to be proved with that which is already 
known. Another principle carefully observed is to isolate the 
problem by removing all non-essential elements, and a third con- 
sists in proving that assumptions inconsistent with that which is 
to be proved are impossible. 
The Analytic Method. — The analytic method, proceed- 
ing from the unknown to the known, depends for its validity on 
the reversibility of the steps; the synthetic method on the contrary 
proceeds from the known to the unknown, with unimpeachable 
validity. It was characteristic of the Greek geometers to aim 
at this form for their demonstrations, even if the results had been 
first obtained analytically. The two methods are well illustrated 
by the following: — 
A circle is given and two external points A and B. It is required 
to draw straight lines AC and BC meeting the circle in C, D, and E 
so that DE shall be parallel to AB. It is 
shown that if the construction can be made, 
the tangent to the circle at D will meet AB 
(produced if necessary) in a point F which 
will lie on a new circle passing through A, C, 
and D. This analysis of consequences is the 
desired clue on which the following synthesis 
of the construction is then based. Starting 
again with A, B and the circle, we locate F so that BA X BF = 
BC X BD = square of the tangent BG from B. Then drawing a 
tangent from F to the circle, D is determined and with it the re- 
quired line DE. 
A solution of the “ duplication of the cube ” problem is also 
attributed to Plato, though the mechanical process employed is so 
much at variance with his usual teachings that the correctness of 
the attribution is seriously questioned. 
SPQR is a frame in which SPQ and PQR are always right angles, 
while PQ may be varied, and SQ and PR can be revolved about Q 72 
A SHORT HISTORY OF SCIENCE 
and P respectively. They are to be so revolved if possible that they 
shall cross at right angles at T, and that ST and TR shall be respect- 
ively equal to the lengths between which mean pro- 
portionals are to be inserted. Then by similar triangles 
ST.PT = PT: QT =QT:RT 
PT and QT are the required mean proportionals. If 
ST is taken equal to twice TR the special case of the 
duplication of the cube is represented. 
To Plato is attributed a systematic method for finding numbers 
which may be sides of right triangles, his method being essentially 
an extension of the Pythagorean already described. Plato’s 
Timseus dialogue is indeed an important source of our information 
in regard to Pythagorean mathematics. Plato speaks with em- 
phatic scorn of the shameful ignorance of mensuration on the 
part of his countrymen. 
He is unworthy of the name of man who is ignorant of the fact 
that the diagonal of a square is incommensurable with its side. 
While predominantly interested in geometry, Plato’s arithmetical 
attainments were considerable for his time. He made, for ex- 
ample, a correct statement about the 59 divisions of 5040. 
Arithmetic has a very great and elevating effect, compelling the 
soul to reason about abstract number, and if visible or tangible ob- 
jects are obtruding upon the argument, refusing to be satisfied. 
— Plato, Republic. 
... It would be proper then, Glaucon, to lay down laws for 
this branch of science and persuade those about to engage in the 
most important state-matters to apply themselves to computation, 
and study it, not in the common vulgar fashion, but with the view 
of arriving at the contemplation of the nature of numbers by the in- 
tellect itself, — not for the sake of buying and selling as anxious 
merchants and retailers, but for war also, and that the soul may 
acquire a facility in turning itself from what is in the course of gen- 
eration to truth and real being. — Plato, Republic. THE GOLDEN AGE OF GREECE 
73 
But the mathematical doctrines concerning the parts and ele- 
ments of the Universe are put forward by Plato, not so much as as- 
sertions concerning physical facts, of which the truth or falsehood is 
to be determined by a reference to nature herself. They are rather 
propounded as examples of a truth of a higher kind than any refer- 
ence to observation can give or can test, and as revelations of prin- 
ciples such as must have prevailed in the mind of the Creator of the 
universe; or else as contemplations by which the mind of man is to 
be raised above the region of sense, and brought nearer to the Divine 
Mind. — Whewell. 
Platonic Cosmology. — The spherical figure of the earth was 
now generally accepted in Greece, and the older fanciful cos- 
mogonies gradually disappeared. To Plato, whose interest in 
physical science was indeed but secondary, the earth was a 
sphere at the centre of the universe, requiring no support. He 
supposes the distances of the heavenly bodies to be proportional 
to the numbers: Moon 1, Sun 2, Venus 3, Mercury 4, Mars 8, 
Jupiter 9, Saturn 27, — these numbers being obtained by com- 
bining the arithmetic and geometric progressions, 1, 2, 4, 8 and 
1, 3, 9, 27. 
Plato accepts as a principle that the heavenly bodies move with 
a uniform and regular circular motion; he then proposes to the mathe- 
maticians this problem: ‘ What are the uniform and regular circular 
motions which may properly be taken as hypotheses in order that 
we may save the appearances presented by the planets ? ’ 
His general conception of the world as expressed in the Timseus 
and in the tenth book of the Republic is decidedly mystical. 
In the latter a soul returning to its body after 12 days in the other 
world relates its experiences in imaginative language: — 
Everyone had to depart on the eighth day and to arrive at a 
place on the fourth day after, whence they from above perceived ex- 
tended through the whole heaven and earth a light as a pillar, mostly 
resembling the rainbow, only more splendid and clearer, at which 
they arrived in one day’s journey; and there they perceived in the 
neighborhood of the middle of the light of heaven, the extremities of 74 
A SHORT HISTORY OF SCIENCE 
the ligatures of heaven extended; for this light was the band of 
heaven, like the hawsers of triremes, keeping the whole circumference 
of the universe together. 
Aristotle sums up Plato’s theories — not too clearly — in the 
words: 
In a similar manner the Timaens shows how the soul moves the 
body because it is interwoven with it. For consisting of the elements 
and divided according to the harmonic numbers, in order that it 
might have an innate perception of harmony and that the universe 
might move in corresponding movements, He bent its straight line 
into a circle, and having by division made two doubly joined circles 
out of the one circle, He again divided one of them into seven circles 
in such a manner that the motions of the heavens are the motions of 
the soul. 
Plato probably had no real knowledge of those deviations of 
the planets from uniform circular motion, which were to engross 
the attention of succeeding philosophers and astronomers. His 
system is consistently geocentric, and assumes a stationary 
earth. According to Plutarch: — 
Theophrastus states that Plato, when he was old, repented of 
having given the earth the central place in the universe which did 
not belong to it, 
this presumably indicating an inclination towards the theories 
of the later Pythagoreans. 
Plato adopts the Pythagorean or Empedoclean hypothesis of 
the four elements, the component particles being assumed to have 
respectively the shapes of the cube (earth), icosahedron (water), 
octahedron (air), and tetrahedron (fire). 
All the heavenly bodies are looked on as divine beings, the first of 
all living creatures, the perfection of whose minds is reflected in their 
orderly motions. 
Summing up an extended discussion of Plato’s astronomical 
theories, Dreyer says: — THE GOLDEN AGE OF GREECE 
75 
There is absolutely nothing in his various statements about 
the construction of the universe tending to show that he had de- 
voted much time to the details of the heavenly motions, as he never 
goes beyond the simplest and most general facts regarding the revo- 
lutions of the planets. Though the conception of the world as Cos- 
mos, the divine work of art, into which the eternal ideas have breathed 
life, and possessing the most godlike of all souls, is a leading feature 
in his philosophy, the details of scientific research had probably no 
great attraction for him, as he considered mathematics inferior to 
pure philosophy in that it assumes certain data as self-evident, for 
which reason he classes it as superior to mere opinion but less clear 
than real science. 
Through his widely read books he helped greatly to spread the 
Pythagorean doctrines of the spherical figure of the earth and the 
orbital motion of the planets from west to east. 
The conjunction of philosophical and mathematical activity 
such as we find, beside Plato, only in Pythagoras, Descartes and 
Leibnitz, has always borne the finest fruits for mathematics. To the 
first we owe scientific mathematics in general. Plato discovered the 
analytical method, through which mathematics was raised above the 
standpoint of the Elements, Descartes created analytic geometry, our 
own celebrated countryman Leibnitz the infinitesimal calculus, — 
and these are the four greatest steps in the development of mathe- 
matics. — Hankel. 
Archytas. — To Archytas, a late Pythagorean, with whom 
Plato had had close relations, was due the earliest solution of the 
duplication problem. This very interesting and somewhat elab- 
orate solution involves a combination of three services, a cone of 
revolution, a cylinder having the vertex of the cone in the cir- 
cumference of its base, and a surface generated by revolving a 
semicircle about an axis passing through one end of its diam- 
eter. It shows remarkable mastery of elementary geometry, 
both plane and solid, and an interesting tendency to employ 
a wider range of methods, including motion, which might, but 
for adverse tendencies, have had important results in connecting 
mathematics with its possible applications to mechanics, etc. 
The influence of Plato in avoiding such connections and asso- 76 
A SHORT HISTORY OF SCIENCE 
ciating geometry with abstract logic and philosophy, undoubtedly 
had compensating advantages in promoting elegance and scien- 
tific rigor, — crystallizing out a more refined product. Archytas 
is said also to have invented the screw and the pulley and to have 
been the first to give a systematic treatment of mechanics, employ- 
ing geometrical theorems for this purpose. 
Meislechmus : Conic Sections. — Even more interesting in 
its foreshadowing of future mathematical developments are the 
solutions of the duplication problem by Menaechmus. The 
problem which we should express in modern algebraic notation 
by the continued proportion a: x: :x:y: :y:b, Menaechmus, with- 
out any such notation or any system of coordinate geometry, shows 
to be equivalent to that of determining the intersection either of a 
parabola and a hyperbola, corresponding to the two proportions 
a: x: :x:y and a: x: :y :b, 
or to the intersection of two parabolas, in case the second pro- 
portion is replaced by x: y : : y: b. The construction of either 
parabola or the hyperbola naturally required some mechanical 
device. 
The Greeks of this period distinguished three types of 
the cone formed by the rotation of the right triangle about one 
of its sides, according as the angle formed by that side with the 
hypotenuse was less than, equal to, or greater than half a right 
angle. A plane perpendicular to an element would cut a cone of 
the first kind in an ellipse, the second in a parabola, the third in 
a hyperbola. These curves wTere named accordingly sections of 
the acute-angled, the right-angled, the obtuse-angled cone. 
The discovery of the conic sections . . . first threw open the 
higher species of form to the contemplation of geometers. But for 
this discovery, which was probably regarded ... as the unprofitable 
amusement of a speculative brain, the whole course of practical phi- 
losophy of the present day, of the science of astronomy, of the theory 
of projectiles, of the art of navigation, might have run in a different 
channel; and the greatest discovery that has ever been made in the 
history of the world, the law of universal gravitation, with its in- THE GOLDEN AGE OF GREECE 
77 
numerable direct and indirect consequences and applications to every 
department of human research and industry, might never to this hour 
have been elicited. — Sylvester. 
Many of Plato’s followers and disciples in the Academy con- 
tinued the development of mathematics. To Xenocrates, for 
example, is attributed the determination of the number of all 
possible syllables as 1,002,000,000,000, a result obtained by 
some unknown method. This whole period is one of great pro- 
ductivity and importance in the history of mathematics. New 
theorems and new methods are discovered, former methods 
are critically scrutinized, loci problems are investigated, these 
and the study of the three classical problems leading to the 
introduction of new curves and a general extension of geo- 
metrical knowledge. Geometry, with emphasis, indeed, on its 
philosophical side, predominates over the theory of numbers, 
and even the latter is given so geometrical a form that mathe- 
matics is unified. 
A New Cosmology. — Eudoxus of Cnidos (408 ?-355 b.c.) 
was a student both of Archytas and, for a time, of Plato. He was 
not only mathematician and astronomer, but also physician. In 
mathematics he is almost a new creator of the science, developing 
the theory of proportion, making a special study of the “golden 
section,” already mentioned in connection with the regular poly- 
gons, and obtaining important results in solid geometry. In the 
words of the register, “Eudoxus of Cnidos . . . first increased 
the number of general theorems, added to the three propor- 
tions three more, and raised to a considerable quantity the learn- 
ing begun by Plato on the subject of the (golden) section, to which 
he applied the analytical method.” 
To him was formerly attributed the proof that the volume of 
a pyramid is one third that of the prism having the same base 
and altitude, as well as the corresponding theorem for cones and 
cylinders. A recently discovered manuscript of Archimedes shows, 
however, that for this Democritus deserves the credit. The 
method of exhaustion, so-called, employed in proving these theo- 78 
A SHORT HISTORY OF SCIENCE 
rems was expressed in the auxiliary theorem: “ When two 
volumes are unequal, it is possible to add their difference to 
itself so many times that the result shall exceed any assigned 
finite volume.” This exceedingly useful and important princi- 
ple, avoiding the difficulties of infinitesimals, was expressed in 
several approximately equivalent forms, and was already im- 
plied in the work of Antiphon and Bryson. A solution of the 
duplication problem which gained Eudoxus the appellation 
“godlike” has been entirely lost. 
There appear to have been no astronomical instruments at this 
time except the simple gnomon and sun-dial, but the more 
obvious irregularities of the planetary motions were beginning to 
attract attention, and under Eudoxus led to the development of 
a new and important theory. Nearest to the central earth is 
the moon, carried on the equator of a sphere revolving from west 
to east in 27 days. The poles of this sphere are themselves car- 
ried on a second sphere, which turns in about 18f years about 
the axis of the zodiac. The angle between the axes of these two 
spheres corresponds with the moon’s variation in latitude. A 
third outer sphere gives the daily east to west motion. Simi- 
larly there are three spheres for the sun. For each of the five 
planets a fourth sphere is necessary to account for the stations 
and retrogressions of its apparent orbital motion — thus making 
with the single sphere of the stars 27 spheres, all having their 
common centre at the centre of the earth. 
How far these spheres were regarded as having concrete exis- 
tence, how far they merely expressed in convenient geometrical 
form the observed relations and motions, we cannot determine 
from extant evidence. The amount of observational data available 
was entirely inadequate to serve as a basis for any quantitatively 
correct theory. The third sphere of the sun was based on an 
erroneous hypothesis as to its motion. For Mercury, Jupiter and 
Saturn the theory was reasonably adequate, for Venus less so, and 
for Mars quite defective. 
Calippus, a follower of Eudoxus, endeavored with some degree 
of success to remedy these defects by adding a fifth sphere for THE GOLDEN AGE OF GREECE 
79 
each of the refractory planets, and at the same time a fourth and 
fifth for the sun, in order to account for the recently discovered 
inequality in the length of the four seasons. 
Reviewing the development of this interesting theory, Dreyer 
says: — 
But with all its imperfections as to detail, the theory of homo- 
centric spheres proposed by Eudoxus demands our admiration as the 
first serious attempt to deal with the apparently lawless motions of the 
planets. . . . Scientific astronomy may really be said to date from 
Eudoxus and Calippus, as we here for the first time meet that mutual 
influence of theory and observation on each other which characterizes 
the development of astronomy from century to century. Eudoxus 
is the first to go beyond mere philosophical reasoning about the con- 
struction of the universe; he is the first to attempt systematically to 
account for the planetary motions. When he has done this the next 
question is how far this theory satisfies the observed phenomena, and 
Calippus at once supplies the observational facts required to test the 
theory, and modifies the latter until the theoretical and observed 
motions agree within the limits of accuracy attainable at that time. 
Philosophical speculation unsupported by steadily pursued obser- 
vations is from henceforth abandoned: the science of astronomy 
has started on its career. 
Eudoxus made the first known proposal for a leap-year, and for 
a star catalogue. A marble celestial globe in the national museum 
at Naples is perhaps a copy of one made by him. 
Aristotle, 384-322 b.c., “the master of those who know,” the 
son of a physician, a student in Plato’s Academy, and tutor of 
Alexander the Great, exercised a mighty and lasting influence on 
the development of Greek science and philosophy. His tenden- 
cies were mainly non-mathematical, but the theorem that the 
sum of the exterior angles of a plane polygon is four right angles 
is ascribed to him. He distinguishes sharply between geodesy as 
an art and geometry as a science; he considers the plane sections 
of the circular cyclinder; he recognizes the physical reason for 
the adoption of ten as the base number of arithmetic; he designates 
unknown quantities by letters. Continuity — an idea so impor- 80 
A SHORT HISTORY OF SCIENCE 
tant in modern mathematical and physical science — he defines 
by saying: — 
A thing is continuous when of any two successive parts, the limits, 
at which they touch, are one and the same, and are, as the word im- 
plies, held together. 
Aristotle’s Mechanics. — In mechanics Aristotle seems 
almost to recognize the principle of virtual velocities. He dis- 
cusses the composition of motions at an angle with each other. 
He enunciates the correct relation between the length of the 
arms of a lever and the loads which will balance each other 
upon it. He even deals with the central and tangential com- 
ponents of circular motion. 
He asks such questions as: “Why are carriages with large 
wheels easier to move than those with small ? ” “ Why do objects 
in a whirlpool move toward the center?” etc. 
He is convinced that the speed of falling bodies is proportional 
to their weight — a belief credulously accepted until Galileo’s 
experiment nineteen centuries later. He illustrates his discussions 
by geometrical figures, and states correctly: — 
If a be a force, /3 the mass to which it is applied, y the distance 
through which it is moved, and 8 the time of the motion, then a will 
move | /3 through 2 y in the time 8, or through y in the time § 8. 
He adds erroneously, however: — 
It does not follow that \ a will move /3 through § y in the time 8, be- 
cause | a may not be able to move /3 at all; for 100 men may drag 
a ship 100 yards, but it does not follow that one man can drag it 
one yard. 
Of the bearing of Aristotle’s physical theories Duhem says: — 
Incapable of any alteration, inaccessible to any violence, the celestial 
essence could manifest no other than its own natural motion, and 
that was uniform rotation about the centre of the universe. 
Aristotle is the author of eight books on Physics, four on the 
Heavens, and four on Meteorology. In physics he explains the 
rainbow, attributes sound to atmospheric motion, and discusses THE GOLDEN AGE OF GREECE 
81 
refraction mathematically. While he undertakes to deal with 
motion, space and time — i.e. with the subject-matter of me- 
chanics — his treatment is too metaphysical to have much real 
value. He declares for example that: — 
The bodies of which the world is composed are solids, and 
therefore have three dimensions. Now, three is the most per- 
fect number, — it is the first of numbers, for of one we do not speak 
as a number, of two we say both, three is the first number of 
which we say all. Moreover, it has a beginning, a middle, and an 
end. 
Francis Bacon in the seventeenth century remarks of 
Aristotle: — 
Nor let any one be moved by this; that in his books Of Animals, 
and in his Problems and in others of his tracts, there is often a quoting 
of experiments. For he had made up his mind beforehand; and 
did not consult experience in order to make right propositions and 
axioms, but when he had settled his system to his will, he twisted ex- 
perience round, and made her bend to his system; so that in this way 
he is even more wrong than his modern followers, the Schoolmen, who 
have deserted experience altogether. 
Aristotelian Astronomy. — Only the second of the four 
books on the Heavens is devoted to astronomy. He considers the 
universe to be spherical, the sphere being the most perfect among 
solid bodies, and the only body which can revolve in its own space. 
Rotation from east to west is more honorable than the reverse. 
He holds that the stars are spherical in form, that they have no 
individual motion, being merely carried all together by their 
one sphere. 
‘ Furthermore, since the stars are spherical, as others maintain and 
we also grant, because we let the stars be produced from that body, 
and since there are two motions of a spherical body, rolling along and 
whirling, then the stars, if they had a motion of their own, ought to 
move in one of these ways. But it appears that they move in 
neither of these ways. For if they whirled (rotated), they would re- 
main at the same spot and not alter their position, and yet they 
manifestly do so, and everybody says they do. It would also be 82 
A SHORT HISTORY OF SCIENCE 
reasonable that all should be moved in the same motion, and yet 
among the stars the sun only seems to do so at its rising or setting, 
and even this one not in itself but only owing to the distance of our 
sight, as this when turned on a very distant object from weakness 
becomes shaky. This is perhaps also the reason why the fixed stars 
seem to twinkle, while the planets do not twinkle. For the planets 
are so near that the eyesight reaches them in its full power, but when 
turned to the fixed stars it shakes on account of the distance, be- 
cause it is aimed at too distant a goal; now its shaking makes the 
motion seem to belong to the star, for it makes no difference whether 
one lets the sight or the seen object be in motion. But that the 
stars have not a rolling motion is evident; for whatever is rolling 
must of necessity be turning, while of the moon only what we call its 
face is visible.’ — Dreyer. 
Aristotle adopts the system of spheres of Eudoxus and 
Calippus, but seems to suppose these spheres to be concrete, and 
not a merely geometrical device for interpreting the phenomena 
or determining the positions. In order however to secure what 
he conceives to be the necessary relation between the motions of 
the spheres, he is obliged to increase their total number from 33 
to not less than 55. The earth is fixed at the centre of the uni- 
verse. That the earth is a sphere is shown logically, and is also 
evident to the senses. During eclipses of the moon, namely, the 
boundary line, which shows the shadow of the earth, is always 
curved. ... If we travel even a short distance south or north, 
the stars over our heads show a great change, some being visible 
in Egypt, but not in more northern lands, and stars are seen to 
set in the south which never do so in the north. It seems therefore 
not incredible that the vicinity of the pillars of Hercules is con- 
nected with that of India, and that there is thus but one ocean. 
The bulk of the earth he considers to be “ not large in compar- 
ison with the size of the other stars.” The estimated circumfer- 
ence of 400,000 stadia — about 39,000 miles — is the earliest known 
estimate of the size of the earth, and is of unknown origin, but 
may quite likely be due to Eudoxus. While the heavens proper 
are characterized by fixed order and circular motion, the space THE GOLDEN AGE OF GREECE 
83 
below the moon’s sphere is subject to continual change, and motions 
within it are in general rectilinear — a theory destined long to 
block progress in mechanics. Of the four elements, earth is near- 
est the centre, water comes next, fire and air form the atmosphere, 
fire predominating in the upper part, air in the lower. In this 
region of fire are generated shooting stars, auroras, and comets, 
the latter consisting of ignited vapors, such as constitute the 
Milky Way. 
Against any orbital motion of the earth Aristotle urges the ab- 
sence of any apparent displacement of the stars. Reviewing his 
astronomical theories, Dreyer says: — 
His careful and critical examination of the opinions of previous phi- 
losophers makes us regret all the more that his search for the causes 
of .'phenomena was often a mere search among words, a series of vague 
and loose attempts to find what was ‘ according to nature ’ and what 
was not; and even though he professed to found his speculations 
on facts, he failed to free his discussion of these from purely metaphys- 
ical and preconceived notions. It is, however, easy to understand 
the great veneration in which his voluminous writings on natural 
science were held for so many centuries, for they were the first, and 
for many centuries the only, attempt to systematize the whole 
amount of knowledge of nature accessible to mankind; while the 
tendency to seek for the principles of natural philosophy by con- 
sidering the meaning of the words ordinarily used to describe the 
phenomena of nature, which to us is his great defect, appealed strongly 
to the mediaeval mind, and, unfortunately, finally helped to retard 
the development of science in the days of Copernicus and Galileo. 
At times Aristotle shows consciousness that his theories are 
based on inadequate knowledge of facts. 
‘ The phenomena are not yet sufficiently investigated. When they 
once shall be, then one must trust more to observation than to spec- 
ulation, and to the latter no farther than it agrees with the phe- 
nomena.’ 
‘An astronomer’ he says ‘must be the wisest of men; his mind 
must be duly disciplined in youth; especially is mathematical study 
necessary; both an acquaintance with the doctrine of number, 84 
A SHORT HISTORY OF SCIENCE 
and also with that other branch of mathematics, which, closely con- 
nected as it is with the science of the heavens, we very absurdly call 
geometry, the measurement of the earth.’ 
Aristotle’s writings include not merely works on scientific sub- 
jects, but treatises of the very first importance On Poetry, On 
Rhetoric, On Metaphysics, On Ethics, and On Politics. Besides 
his scientific works mentioned above, there are others entitled 
On Generation and Destruction, On the Parts of Animals, 
On Generation of Animals, Researches about Animals, On the 
Locomotion of Animals. One of the most important of his 
many services to science is the encyclopedic character of his 
writings, since from time to time he reviews in them the 
opinions of his predecessors whose works are sometimes known to 
us chiefly through his references to them. While standing thus 
upon the shoulders of the past, he shows at the same time both 
vast learning and much originality. He may be truly called the 
founder of zoology. 
Of Aristotle’s contributions to science, the greatest was un- 
questionably that spirit of curiosity, of inquiry, of scepticism, and 
of veracity which he brought to bear on everything about him 
and within him. His observations are often poor, his conclu- 
sions often erroneous, but his interest, his curiosity, his zeal are 
indefatigable. 
Theophrastus. — One of Aristotle’s principal pupils, and his 
successor in his School, was Theophrastus (372-287 b.c.) notable 
in the history of science chiefly as an early student of plants, 
and writer of the most important treatises of antiquity on botany. 
These were two large works, one of ten books and the other of 
eight, On the History of Plants, and On the Causes of Plants, 
respectively. In these, more than 500 species of plants are de- 
scribed, chiefly with reference to their medicinal uses. It is es- 
pecially interesting to note that Theophrastus recognized the 
existence of sex in plants, though he does not appear to have 
known the sex organs. 
Epicurus and Epicureanism. — A few words may be said of 
another philosopher of the fourth century, a follower to some THE GOLDEN AGE OF GREECE 
85 
extent of Democritus and the forerunner and exemplar of the 
Roman Lucretius. This was Epicurus (342-270 b.c.), who, born 
in Samos and educated in Athens and Asia Minor, became a 
famous teacher and the head of a remarkable community “such 
as the ancient world had never seen.” The mode of life in this 
community was not that of the so-called “epicures” of to-day, but 
very plain, — water the general drink, and barley bread the 
general food. The magnetic personality of Epicurus held the 
community together, and his chief work was a treatise on Nature 
in thirty-seven books. Epicureanism is of interest in the history 
of science chiefly because of its effect on its Roman exponent, the 
poet Lucretius. Much of it was even a negation of science and 
the scientific spirit. 
Heraclides. Rotation of the Earth. — To Heraclides of 
Pontus in the fourth century b.c. belongs the distinction of teach- 
ing that the earth turns on its own axis from west to east in 24 
hours. He had been connected with the Pythagoreans, and with 
the schools of Plato and Aristotle. His work is known to us only 
indirectly, none of his own writings having survived. He is said 
also to have advanced the hypothesis that Venus and Mercury 
revolve about the sun, being therefore at a distance from the 
earth sometimes greater than the sun, sometimes less. Geminus 
writing in the first half of the first century b.c. of the different 
fields and points of view of astronomers and physicists, remarks: — 
For why do sun, moon and planets appear to move unequally? 
Because, when we assume their circles to be excentric, or the stars to 
move on an epicycle, the appearing anomaly can be accounted for, 
and it is necessary to investigate in how many ways the phenomena 
can be represented, so that the theory of the wandering stars may be 
made to agree with the etiology in a possible manner. Therefore also 
a certain Heraclides of Pontus stood up and said that also when the 
earth moved in some way and the sun stood still in some way, could 
the irregularity observed relatively to the sun be accounted for. 
In general it is not the astronomer’s business to see what by its nature 
is immovable and of what kind the moved things are, but framing 
hypotheses as to some things being in motion and others being fixed, 86 
A SHORT HISTORY OF SCIENCE 
he considers which hypotheses are in conformity with the phenomena 
in the heavens. He must accept as his principles from the physicist, 
that the motions of the stars are simple, uniform, and regular, of 
which he shows that the revolutions are circular, some along parallels, 
some along oblique circles. 
This contrast between the physical phenomena and the mathe- 
matical theory which corresponds with them, without being true 
or perhaps even possible in all respects, is of continued and in- 
creasing importance in the history of science, as a larger stock 
of facts was accumulated and as theories still imperfect were more 
frequently subjected to critical comparison with observed data, 
instead of being accepted on purely philosophical or metaphysical 
grounds. Heraclides is not credited with any conception of 
orbital or progressive motion of the earth. 
References for Reading 
Allman. Greek Geometry, Chapters III-IX. 
Aristotle. On the Parts of Animals, On Generation, etc. 
Ball. History of Mathematics, Chapter III. 
Butcher, S. H. Aspects of the Greek Genius, Chapter I. 
Berry. History of Astronomy, Chapter II, pp. 26-33. 
Dreyer. Planetary System, Chapters III-V. 
Freeman, K. E. Schools of Hellas. 
Garrison, F. H. A History of Medicine. (For Hippocrates of Cos.) 
Gomperz. Greek Thinkers, Vol. I. 
Gow. History of Greek Mathematics, Chapter VI, Articles 97-116. 
Lewes, G. H. Aristotle, a Chapter in the History of Science. CHAPTER V 
GREEK SCIENCE IN ALEXANDRIA 
There is an astonishing imagination, even in the science of mathe- 
matics. . . . We repeat, there was far more imagination in the head 
of Archimedes than in that of Homer. — Voltaire. 
If the Greeks had not cultivated Conic Sections, Kepler could not 
have superseded Ptolemy; if the Greeks had cultivated Dynamics, 
Kepler might have anticipated Newton. — Whewell. 
If we compare a mathematical problem with an immense rock, 
whose interior we wish to penetrate, then the work of the Greek 
mathematicians appears to us like that of a robust stonecutter, who, 
with indefatigable perseverance, attempts to demolish the rock 
gradually from the outside by means of hammer and chisel; but the 
modern mathematician resembles an expert miner, who first con- 
structs a few passages through the rock and then explodes it with a 
single blast, bringing to light its inner treasures. — Hankel. 
The Museum at Alexandria. — The subjugation of Greece 
by Alexander the Great in 330 b.c. checked the further develop- 
ment of Greek civilization on its native soil. After Alexander’s 
death in 323, his vast empire was divided among his generals, and 
Alexandria, the new Egyptian capital, fell to the lot of Ptolemy. 
The city as such was then barely ten years old, but very soon 
became, under the rule of the Ptolemies, the centre of the learned 
world. By 300 b.c. the Museum (Seat of the Muses) was 
founded, becoming in effect a veritable university of Greek learning. 
To this were attached a great library, a dining hall, and lecture- 
rooms for professors. Here for the next 700 years Greek science 
had its chief abiding place. The fame of Alexandria soon out- 
shone and eventually eclipsed that of Athens, while Romans 
journeyed from Rome — never important in ancient times as a 
87 88 
A SHORT HISTORY OF SCIENCE 
scientific centre — to study at Alexandria the healing art, anatomy, 
mathematical science, geography, and astronomy. Neither 
Athens, Rome, Carthage, nor any other city of the ancient world 
can boast similar distinction as a home of science. 
Euclid. — Three centuries after Thales had introduced the 
rudiments of Egyptian mathematics into Greece, the focus of 
mathematical activity was again transferred to that ancient land, 
but its spirit and aims remained there still for centuries essentially 
Greek. Continuing the ancient register, Proclus writes: — 
Not much younger than these (the Aristotelians) is Euclid, who 
brought the elements together, arranged much of the work of Eudoxus 
in complete form, and brought much which had been begun by 
Thesetetus to completion. Besides he supported what had been only 
partially proved by his predecessors with irrefragable proofs. . . . 
It is related that King Ptolemy asked him once if there were not in 
geometrical matters a shorter way than through the Elements: to 
which he replied that in geometry there is no straight path for 
kings. . . . 
As a recent writer has well said: “ There are royal roads in 
science; but those who first tread them are men of genius and not 
kings.” 
Euclid’s period of activity was about 300 B.c.; his place of birth 
and even his race are unknown; he is said to have been of a 
mild and benevolent disposition, and to have appreciated fully 
the scientific merits of his predecessors. While we know next 
to nothing of his life and personality, his writings have had 
an influence and a prolonged vitality almost, if not quite, unpar- 
alleled. 
Euclid’s “Elements.” — Scientifically, Euclid is attached to 
the Platonic philosophy. Thus he makes the goal of his Elements 
the construction of 'the so-called “Platonic bodies” i.e., the five 
regular polyhedrons. This treatise, which served as the basis of 
practically all elementary instruction for the following 2000 years, 
is naturally his best-known work, and appears to have been ac- 
cepted in the Greek world after many previous attempts as a GREEK SCIENCE IN ALEXANDRIA 
89 
finality. It consisted of thirteen books, of which only the 
first six are ordinarily included in modern editions. The whole 
is essentially a systematic introduction to Greek mathematics, 
consisting mainly of a comparative study of the properties and 
relations of those geometrical figures, both plane and solid, which 
can be constructed with ruler and compass. The comparison of 
unequal figures leads to arithmetical discussion, including the 
consideration of irrational numbers corresponding to incommensu- 
rable lines. The contents may be briefly summarized as follows: 
Book I deals with triangles and the theory of parallels: Book II 
with applications of the Pythagorean theorem, many of the prop- 
ositions being equivalent to algebraic identities, or solutions of 
quadratic equations, which seem to us more simple and obvious 
than to the Greeks. It should be noted however that the geomet- 
rical treatment is relatively advantageous for oral presentation. 
Book III deals with the circle, Book IV with inscribed and cir- 
cumscribed polygons. These first four books thus contain a 
general treatment of the simpler geometrical figures, together 
with an elementary arithmetic and algebra of geometrical magni- 
tudes. In Book V, for lack of an independent Greek arithmetical 
analysis, a theory of proportion (which has thus far been 
avoided) is worked out, with the various possible forms of the 
equation y = - • The results are applied in Book VI to the com- 
b a 
parison of similar figures. This contains the first known problem 
in maxima and minima, — the square is the greatest rectangle 
of given perimeter, — also geometrical equivalents of the solution 
of quadratic equations. The next three Books are devoted to the 
theory of numbers, including for example the study of prime and 
composite numbers, of numbers in proportion, and the determina- 
tion of the greatest common divisor. He shows how to find the 
sum of a geometrical progression, and proves that the number of 
prime numbers is infinite. 
If there were a largest prime number n then the product 1 X 2 X 
3 ... X n increased by 1 would always leave a remainder 1 when di- 
vided by n or by any smaller number. It would thus either be prime 90 
A SHORT HISTORY OF SCIENCE 
itself, or a product of prime factors greater than n, either of 
which suppositions is contrary to the hypothesis that n itself is the 
greatest prime number. 
Book X deals with the incommensurable on the basis of the 
theorem: If two unequal magnitudes are given, and if one takes 
from the greater more than its half, and from the remainder more 
than its half and so on, one arrives sooner or later at a remainder 
which is less than the smaller given magnitude. Books XI, XII, 
and XIII are devoted to solid geometry, leading up to our familiar 
theorems on the volume of prism, pyramid, cylinder, cone, and 
sphere, but in every case without computation, emphasizing the 
habitual distinction between geometry and geodesy or mensura- 
tion ... a distinction expressed by Aristotle in the form: “ One 
cannot prove anything by starting from another species, for ex- 
ample, anything geometrical by means of arithmetic. Where 
the objects are so different as arithmetic and geometry one cannot 
apply the arithmetical method to that which belongs to magni- 
tudes in general, unless the magnitudes are numbers, which can 
happen only in certain cases.” Book XIII passes from the 
regular polygons to the regular polyhedrons, remarking in con- 
clusion that only the known five are possible. 
The extent to which Euclid’s Elements represent original work 
rather than compilation of that of earlier writers cannot be deter- 
mined. It would appear, for example, that much of Books I and 
II is-due to Pythagoras, of III to Hippocrates, of V to Eudoxus, 
and of IV, VI, XI, and XII, to later Greek writers; but the work 
as a whole constitutes an immense advance over previous similar 
attempts. 
Proclus (410-485 a.d.) is the earliest extant source of informa- 
tion about Euclid. Theon of Alexandria edited the Elements 
nearly 700 years after Euclid, and until comparatively recent 
times modern editions have been based upon his. 
Like other Greek learning, Euclid has come down to later times 
through Arab channels. There is a doubtful tradition that an 
English monk, Adelhard of Bath, surreptitiously made a Latin GREEK SCIENCE IN ALEXANDRIA 
91 
translation of the Elements at a Moorish university in Spain 
in 1120. Another dates from 1185, printed copies from 1482 on- 
ward, and an English version from 1570. After Newton’s time 
it found its way from the universities into the lower schools. 
Different versions vary widely as to the axioms and postulates 
on which the work as a whole is based. It is believed that Euclid 
originally wrote five postulates, of which the fourth and fifth are 
now known as Axioms 11 and 12, — “All right angles are equal ” ; 
and the famous parallel axiom: — “If a straight line meets two 
straight lines, so as to make the two interior angles on the same 
side of. it together less than two right angles, these straight lines 
will meet if produced on that side.” The necessarily unsuccess- 
ful attempts which have since been made to prove this as a 
proposition rather than a postulate constitute an important 
chapter in the history of mathematics, leading in the last century 
to the invention of the generalized geometry known as non- 
Euclidean, in which this axiom is no longer valid. 
Influence of Euclid. — The Elements of Euclid have exerted 
an immense influence on the development of mathematics, and 
particularly of mathematical pedagogy. Aside from their sub- 
stance of geometrical facts, they are characterized by a strict 
conformity to a definite logical form, the formulation of what is 
to be proved, the hypothesis, the construction, the progressive 
reasoning leading from the known to the unknown, ending with 
the familiar q.e.d. There is a careful avoidance of whatever 
is not geometrical. No attempt is made to develop initiative 
or invention on the part of the student; the manner in which the 
results have been discovered is rarely evident and is even some- 
times concealed; each proposition has a degree of completeness 
in itself. This treatise translated into the languages of modern 
Europe has been a remarkable means of disciplinary training in 
its special form of logic. No other science has had any such single 
permanently authoritative treatise. 
Criticism of Euclid. — On the other hand, its narrowness of 
aim, its deliberate exclusion of the concrete, its laborious methods 
of dealing with such matters as infinity, the incommensurable or 92 
A SHORT HISTORY OF SCIENCE 
irrational, its imperfect substitutes for algebra, as in the theory 
of proportion, have diminished its usefulness, and have in com- 
paratively recent times (in English-speaking countries) led to the 
substitution of modernized texts. Still, no other mathematical 
treatise has had even approximately the deservedly far-reaching 
influence of Euclid. Its subject-matter is so nearly complete 
that its author’s name is still a current synonym for elementary 
geometry. 
His elements are particularly admired for the order which con- 
trols them, for the choice of theorems and problems selected as funda- 
mental (for he has by no means inserted all which he might give, 
but only those which are really fundamental), and for the varied 
argumentation, producing conviction now by starting from causes, 
now by going back to facts, but always irrefutable, exact and of most 
scientific character. . . . Shall we mention the constantly main- 
tained invention, economy and orderliness, the force with which he 
establishes every point? If one adds to or takes from it, one will 
recognize that he departs thereby from science, tending towards error 
or ignorance. . . . 
Elsewhere Proclus: — 
It is difficult in every science to choose and dispose in suitable 
order the elements from which all the rest may be derived. Of those 
who have attempted this some have increased their collection, others 
have diminished it; some have employed abridged demonstrations, 
others have expanded their presentation indefinitely, etc. 
In such a treatise it is necessary to avoid everything superfluous 
. . . to combine all that is essential, to consider principally and 
equally clearness and brevity, to give theorems their most general 
form, — for the detail of teaching particular cases only makes the 
acquisition of knowledge more difficult. From all these points of 
view, Euclid’s Elements will be found superior to every other. 
In a recent interesting discussion of Euclid’s Elements, F. Klein 
(Elementar-Mathematik vom Hoheren Standpunkt aus. II) says in 
substance: “A false estimation of the Elements finds its source 
in the general misunderstanding of Greek genius which long pre- GREEK SCIENCE IN ALEXANDRIA 
93 
vailed and still finds popular acceptance, namely that Greek 
culture was confined to relatively few fields, but in them reached 
a high degree of perfection and finality. The fact is, however, that 
the Greeks occupied themselves with the greatest versatility in all 
directions, and made in all directions wonderful progress. Never- 
theless, from our modern standpoint, they fell short of the pos- 
sibly attainable in all, and in some directions made only a begin- 
ning. 
“ In mathematics, for example, it has become a tradition that 
Greek geometry reached unique development, while in reality 
many other branches of mathematics were successfully cultivated. 
The development of Greek mathematics was particularly ham- 
pered by the lack of a convenient number-system and notation 
as a basis for an independent arithmetic, and by ignorance of 
negative and imaginary numbers. Euclid’s intention in the Ele- 
ments was by no means to write an encyclopedia of current geom- 
etry, which must have included conic sections and other curves, 
but rather to write for mature readers an introduction to mathe- 
matics in general, the latter being regarded in its turn, in the 
Platonic sense, as necessary preparation for general philosophic 
studies. Hence the emphasis on formal order and logical method, 
as well as the omission of all practical applications. He aims at 
the flawless logical derivation of all geometrical theorems from 
premises completely stated in advance.” 
Allowing for grave uncertainties of text, Klein’s view is summed 
up as follows: 
“(1) The great historical significance of Euclid’s Elements 
consists in the fact that through it the ideal of a flawless logical 
treatment of geometry was first transmitted to future times. 
“ (2) As to the execution, much is very finely done, but much 
remains fundamentally imperfect from our present standpoint. 
“ (3) Numerous details of importance, especially at the begin- 
ning, remain completely doubtful on account of uncertainties of 
the text. 
“(4) The whole development is often needlessly clumsy, as 
Euclid has no arithmetic ready to his hand. 94 
A SHORT HISTORY OF SCIENCE 
“(5) In general the one-sided emphasis on the logical makes 
it difficult to understand the subject-matter as a whole, and its 
internal relations.” 
The Elements of the great Alexandrian remain for all time the 
first, and one may venture to assert, the only perfect model of logical 
exactness of principles, and of rigorous development of theorems. 
If one would see how a science can be constructed and developed to 
its minutest details from a very small number of intuitively per- 
ceived axioms, postulates, and plain definitions, by means of rigorous, 
one would almost say chaste, syllogism, which nowhere makes use 
of surreptitious or foreign aids, if one would see how a science may 
thus be constructed, one must turn to the Elements of Euclid. 
— Hankel. 
Euclid always contemplates a straight line as drawn between two 
definite points, and is very careful to mention when it is to be pro- 
duced beyond this segment. He never thinks of the line as an entity 
given once for all as a whole. This careful definition and limitation, 
so as to exclude an infinity not immediately apparent to the senses, 
was very characteristic of the Greeks in all their many activities. 
It is enshrined in the difference between Greek architecture and Gothic 
architecture, and between the Greek religion and the modern religion. 
The spire on a Gothic cathedral and the importance of the unbounded 
straight line in modern geometry are both emblematic of the trans- 
formation of the modern world. — Whitehead. 
The universally admired perfection of the work of Euclid is re- 
vealed to the historians as the natural product of a long criticism 
which was developed in the constructive period of rational geometry, 
from Pythagoras to Eudoxus. Then commenced to appear the signifi- 
cation of those methods and principles by means of which the Greeks 
themselves attempted to interpret and conquer the paradoxes con- 
cerning infinity. These are the same difficulties which reappeared 
at the time the infinitesimal calculus was founded, and are now again 
asserting themselves in the most refined analysis. — Enriques. 
Other Works of Euclid. — Besides the “Elements” Euclid 
wrote several other mathematical treatises, including one on 
Porisms, a special type of geometrical proposition; and one on 
Data, containing such theorems as the following: GREEK SCIENCE IN ALEXANDRIA 
95 
Given magnitudes have a given ratio to each other. 
When two lines given in position cut each other their point of 
intersection is given. 
When in a circle of given magnitude a line of given magnitude 
is given, it bounds a segment which contains a given angle. 
A work on Fallacies is designed to safeguard the student against 
erroneous reasoning. Still other treatises are devoted to Division 
of Figures, Loci, and Conic Sections; finally there are works on 
Phenomena, on Optics, and on Catoptrics dealing with applica- 
tions of geometry. 
The Phenomena gives a geometrical theory of the universe, the 
Optics is an unsuccessful attempt to deal with problems of vision 
on the hypothesis that light proceeds from the eye to the object 
seen. The fundamental assumptions are, for example: “ Rays 
emitted from the eye are carried in straight lines, distant by an 
interval from one another,” etc. 
The Catoptrics deals in 31 propositions with reflections in plane, 
concave, and convex mirrors. It is remarked that a ring placed 
in a vase so as to be invisible from a certain position, may be made 
visible by filling the vase with water. The authenticity of this 
work is however questionable. 
These two works constitute the earliest known attempt to 
apply geometry systematically to the phenomena of light-rays. 
The law of reflection is correctly applied. Just as geometry is 
based on a definite list of axioms, so Euclid makes his optics 
depend on eight fundamental facts of experience. For example, 
the light rays are straight lines. The figure inclosed by the rays 
is a cone with its vertex at the eye, while the boundary of the 
object corresponds to the base, etc. This work, though in very 
imperfect form, continued in use until Kepler’s time. 
Archimedes. — The second great name in the Alexandrian 
school and one of the greatest in the whole history of science is 
that of Archimedes. He was both geometer and analyst, mathe- 
matician and engineer. He enriched even the highly developed 
Euclidean geometry, made important progress in algebra, laid the 
foundations of mechanics, and even anticipated the infinitesimal 96 
A SHORT HISTORY OF SCIENCE 
calculus, reaching thus a level which was not surpassed for 2000 
years. Born in Syracuse, probably 287 b.c., the greater part of 
his life was spent in his native city, to which he rendered on oc- 
casion invaluable services as a military engineer. According to 
Livy it was due to the efforts of Archimedes that the Romans 
under Marcellus were held in check during the protracted siege 
of Syracuse. On the fall of the city in 212 b.c. the venerable 
mathematician, absorbed in a geometrical problem, was killed by 
a Roman soldier, much to the regret of Marcellus, who appre- 
ciated and would have spared him. The conqueror carried out 
the wish of Archimedes by erecting a monument with a mathe- 
matical figure, and this was with some difficulty rediscovered and 
put in order by Cicero, during his official residence in Sicily, 75 b.c. 
Nothing afflicted Marcellus so much as the death of Archimedes, 
who was then, as fate would have it, intent upon working out some 
problem by a diagram, and having fixed his mind alike and his eyes 
upon the subject of his speculation, he never noticed the incursion 
of the Romans, nor that the city was taken. In this transport of 
study and contemplation, a soldier, unexpectedly coming up to him 
commanded him to follow to Marcellus, which he declined to do be- 
fore he had worked out his problem to a demonstration; the soldier, 
enraged, drew his sword and ran him through. Others write, that a 
Roman soldier, running upon him with a drawn sword, offered to kill 
him; and that Archimedes, looking back, earnestly besought him to 
hold his hand a little while, that he might not leave what he was 
at work upon inconclusive and imperfect; but the soldier, nothing 
moved by his entreaty, instantly killed him. Others again relate, 
that as Archimedes was carrying to Marcellus mathematical instru- 
ments, dials, spheres, and angles, by which the magnitude of the sun 
might be measured to the sight, some soldiers seeing him, and think- 
ing that he carried gold in a vessel, slew him. Certain it is, that his 
death was very afflicting to Marcellus; and that Marcellus ever after 
regarded him that killed him as a murderer; and that he sought for 
his kindred and honored them with signal favours. — Plutarch. 
The known works of Archimedes include the following: two 
books on the Equilibrium of Planes, with an interpolated treatise GREEK SCIENCE IN ALEXANDRIA 
97 
on the Quadrature of the Parabola, two books on the Sphere 
and the Cylinder, the Circle Measurement, the Spirals, the book 
of Conoids and Spheroids, the Sand Number, two books on 
Floating Bodies, Choices. Unlike Euclid’s Elements, these are 
for the most part original papers on new mathematical discoveries, 
which were also often communicated to his contemporaries in the 
form of letters. Pappus quotes Geminus as saying of Archimedes: 
“He is the only man who has known how to apply to all things 
his varied natural gifts and inventive genius.” 
Archimedes and Euclid. — In contrasting the limitations of 
Euclid’s Elements with the broad range of Greek mathematics, 
Klein characterizes the work of Archimedes somewhat as follows: 
(1) Quite in contrast to the spirit controlling Euclid’s Elements, 
Archimedes has a strongly developed sense for numerical com- 
putation. One of his greatest achievements indeed is the calcu- 
lation of the ratio t of the circumference of a circle to its diameter, 
by approximations with regular polygons. There is no trace of 
interest for such numerical results with Euclid, who merely men- 
tions that the areas of two circles are proportional to the squares 
of the radii, two circumferences as the radii, regardless of the 
actual proportionality factor. 
(2) A far-reaching interest in applications of all sorts is char- 
acteristic of Archimedes, including the most varied physical and 
technical problems. Thus he discovered the principles of hydro- 
statics and constructed engines of war. Euclid on the contrary 
does not even mention ruler or compass, merely postulating that 
a straight line can be drawn through two points, or a circle de- 
scribed about a point. Euclid shares the view of certain ancient 
schools of philosophy, — a view unfortunately extant in certain 
quarters,—that the practical application of a science is something 
mechanical and unworthy. The very greatest mathematicians, 
Archimedes, Newton, Gauss, have combined theory and applica- 
tions consistently.1 
1 Plutarch, however, says : “Archimedes possessed so high a spirit, so profound 
a soul, and such treasures of highly scientific knowledge, that though these inven- 
tions (used to defend Syracuse against the Romans) had now obtained him the re- 98 
A SHORT HISTORY OF SCIENCE 
(3) Finally, Archimedes was a great investigator and pioneer, 
who in each of his works carries knowledge a step forward. This 
affects materially the form of presentation. In a most recently 
discovered manuscript, the procedure is essentially modern as 
contrasted with the rigid formalism of the Elements. 
Circle Measurement. — In this Archimedes proves three 
theorems. 
(1) Every circle is equivalent to a right triangle having the 
sides adjacent to the right angle equal respectively to the radius 
and circumference of the circle. 
(2) The circle has to the square on its diameter approximately 
the ratio 11:14. 
(3) The circumference of any circle is three times as great as the 
diameter and somewhat more, namely less than y but more than 
He proves the first theorem by showing that the assumption 
that the circle is either larger or smaller than the triangle leads to 
a contradiction. The second he bases on the third, at which he 
arrives by computing successively the perimeters of both inscribed 
and circumscribed polygons of 3, 6, 12, 24, 48 and 96 sides. All 
this is contrary to the spirit of Euclid and essentially modern in 
its method of successive approximation. The difficulty of the 
achievement in view of the imperfect arithmetical notation avail- 
able can hardly be overrated. 
Quadrature of the Parabola. — Of special interest is his 
quadrature of the parabola. A segment is formed by drawing any 
chord PQ of the parabola : it is known that if a line is drawn from 
the middle point R of the chord parallel to the axis of the parabola, 
the tangent at the point S where this line meets the curve will be 
parallel to the chord, and the perpendicular from S to the chord 
is greater than any other which can be drawn from a point of the 
nown of more than human sagacity; he yet would not deign to leave behind him 
any commentary or writing on such subjects; but, repudiating as sordid and ignoble 
the whole trade of engineering, and every sort of art that lends itself to mere use 
and profit, he placed his whole affection and ambition in those purer speculations 
where there can be no reference to the vulgar needs of life ; studies, the superiority 
of which to all others is unquestioned, and in which the only doubt can be whether 
the beauty and grandeur of the subjects examined, or the precision and cogency of 
the methods and means of proof, most deserve our admiration.” GREEK SCIENCE IN ALEXANDRIA 
99 
arc. The triangle formed by joining the same point S to the ends 
of the original chord being wholly contained within the segment, 
the area of the latter will be greater than that of the triangle and 
less than that of a parallelogram having the same base and alti- 
tude. Now the segment exceeds the triangle 
by two smaller segments, in each of which 
triangles STQ and SPU are again inscribed. 
It is a known property of the parabola that 
each of these triangles has one-eighth the area 
of the triangle PSQ. The area of each of the 
two smaller segments is therefore greater than 
one-eighth and less than one-fourth that of the triangle PSQ. 
The area of the original segment therefore is less than three-halves 
and greater than five-fourths that of triangle PSQ. The construc- 
tion may evidently be repeated any number of times, and the 
ratio of the segment to the triangle will lie between numbers which 
converge towards four-thirds. Archimedes also succeeded in 
determining the area of the ellipse. 
Spirals. — The discussion of spirals is based on the definition, 
“If a straight line moves with uniform velocity in a plane about 
one of its extremities which remains fixed, until it returns to its 
original position, and if at the same time a point moves with uni- 
form velocity starting at the fixed point, the moving point de- 
scribes a spiral.” With the simple resources at his command, 
he also succeeds in obtaining the quadrature of this spiral, and in 
drawing a tangent at any point. In these quadratures he approx- 
imates the summation principle of the modern integral calculus. 
Supplementing Euclid’s treatment of the regular polyhedrons, 
Archimedes investigates the semi-regular solids formed by com- 
bining regular polygons of more than one kind. Of these he finds 
13, ten of which have two kinds of bounding polygons, the others 
three kinds. 
Sphere and Cylinder. — In his important treatise on “ The 
Sphere and the Cylinder” he derives three new theorems: 
(1) That the surface of a sphere is four times the area of its 
great circle. 100 
A SHORT HISTORY OF SCIENCE 
(2) That the convex surface of a segment of a sphere is equal 
to the area of a circle whose radius is equal to the straight line 
from the vertex of the segment to any point in the perimeter of 
its base. 
(3) That the cylinder having a great circle of the sphere for its 
base and the diameter of the sphere for its altitude exceeds the 
sphere by one-half, both in volume and in surface. It was the 
figure for this last proposition which was at his wish carved upon 
his tombstone. 
In attempting to solve the problem of passing a plane 
through a sphere so that the segments thus formed shall have 
either their surfaces or their volumes in an assigned ratio, he is 
led to a cubic equation; he appears to have given both a solution 
and a criterion for the existence of a positive root, but the work 
is lost. 
In his Conoids and Spheroids he deals with the bodies formed 
by the revolution of the ellipse, parabola, and hyperbola, by means 
of plane cross-sections, ascertains the volume of these solids by 
comparing the portion between two neighboring planes with an 
inscribed and a circumscribed cylinder, — much in the modern 
manner. 
It is not possible to find in all geometry more difficult and more 
intricate questions or more simple and lucid explanations (than those 
given by Archimedes). Some ascribe this to his natural genius; 
while others think that incredible effort and toil produced these, to 
all appearance, easy and unlabored results. No amount of inves- 
tigation of yours would succeed in attaining the proof, and yet, once 
seen, you immediately believe you would have discovered it; by so 
smooth and so rapid a path he leads you to the conclusion required. 
— Plutarch. 
In other branches of mathematical science than geometry the 
work of Archimedes was relatively even more important. 
The so-called Cattle Problem, for example, is a notable per- 
formance in the algebra of linear equations. 
“ The sun had a herd of bulls and cows, all of which were either GREEK SCIENCE IN ALEXANDRIA 
101 
white, gray, dun, or piebald; the number of piebald bulls was 
less than the number of white bulls by f) of the number of 
gray bulls, it was less than the number of gray bulls by (| + §) 
of the number of dun bulls, and it was less than the number of 
dun bulls by (•£ + t) of the number of white bulls. The number 
of white cows was (| + f) of the number of gray cattle (bulls and 
cows), the number of gray cows was (j + -§■) of the number of 
dun cattle, the number of dun cows was (i + i) of the number of 
piebald cattle, and the number of piebald cows was + t) of the 
number of white cattle.” The seven equations are insufficient to 
determine the eight unknown quantities. The solution attributed 
to Archimedes consists of numbers of nine figures each. 
Again he succeeds in summing the series of squares: 1, 4, 9, 15, 
25, 36, etc., to n terms, expressing the result in geometrical form. 
Both proof and formulation are of course much more complicated 
by reason of the entire lack of an algebraic symbolism, the same 
remark naturally applying also to the preceding cattle problem 
and to the cubic equation referred to above. This last was in- 
deed to Archimedes not primarily an equation at all, but a pro- 
portion 
a — x : b :: a2: x2. 
In his Circle Measurement already outlined, he showed mastery 
of square root, and the comparison of irrational numbers with 
fractions, showing for example that 
1351 \ -v/o 265 
78 0“ > V O > T37 
How these fractions were obtained cannot be certainly deter- 
mined, but it was presumably by a process analogous at least to 
the modern method of continued fractions, though such fractions 
themselves could not have been known to him. 
In the Sand Counting, Archimedes undertakes to give a number 
which shall exceed the number of grains of sand in a sphere with a 
radius equal to the distance from the earth to the starry firma- 
ment. The treatise begins : “ Many people believe, King Gelon, 
that the number of sand grains is infinite. I mean not the sand 102 
A SHORT HISTORY OF SCIENCE 
about Syracuse, nor even that in Sicily, but also that .on the whole 
mainland, inhabited and uninhabited. There are others again 
who do not indeed assume this number to be infinite, but so great 
that no number is ever named which exceeds this. ... I will 
attempt to show however by geometrical proofs which you will 
accept that among the numbers which I have named . . . some 
not only exceed the number of a sand-heap of the size of the earth, 
but also of that of a pile of the size of the universe.” He assumes 
that 10,000 grains of sand would make the size of a poppy-seed, 
that the diameter of a poppy-seed is not less than one-fortieth of a 
finger-breadth, that the diameter of the earth is less than a million 
stadia, that the diameter of the universe is less than 10,000 di- 
ameters of the earth. To express the vast number which results 
from these assumptions — 1063 in our notation — he employs 
an ingenious system of units of higher order comparable with the 
modern use of exponents, an immense advance on current arith- 
metical symbolism. 
Mechanics of Archimedes. — In mechanics Archimedes is 
a pioneer, giving the first mathematical proofs known. In two 
books on Equiponderance of Planes or Centres of Plane Gravities, 
he deals with the problem of determining the centres of gravity 
of a variety of plane figures, including the parabolic segment. 
A treatise on levers and perhaps on machines in general has been 
lost, as also a work on the construction of a celestial sphere. A 
sphere of the stars and an orrery constructed by him were long 
preserved at Rome. He describes an original apparatus for deter- 
mining the angular diameter of the sun, discussing its degree of 
accuracy. 
The lever and the wedge had been practically known from 
remote antiquity, and Aristotle had discussed the practice of 
dishonest tradesmen shifting the fulcrum of scales towards the 
pan in which the weights lay, but no previous attempt at exact 
mathematical treatment is known. 
Archimedes assumes as evident at the outset: 
(1) Magnitudes of equal weight acting at equal distances from 
their point of support are in equilibrium; GREEK SCIENCE IN ALEXANDRIA 
103 
(2) Magnitudes of equal weight acting at unequal distances 
from their point of support are not in equilibrium, but the one 
acting at the greater distance sinks. 
From these he deduces: 
(3) Commensurable magnitudes are in equilibrium when they 
are inversely proportional to their distances from the point of 
support. 
In a work on Floating Bodies, extant in a Latin version by 
Tartaglia, Archimedes defines a fluid as follows: “Let it be 
assumed that the nature of a fluid is such that, all its parts lying 
evenly and continuous with one another, the part subject to less 
pressure is expelled by the part subject to greater pressure. But 
each part is pressed perpendicularly by the fluid above it, if the 
fluid is falling or under any pressure.” “Every solid body lighter 
than a liquid in which it floats sinks so deep that the mass of liquid 
which has the same volume with the submerged part weighs just 
as much as the floating body.” The specific gravity of heavier 
bodies was of course employed in his solution of the crown problem, 
which with his achievements as a military engineer gave him a 
great reputation among his contemporaries. Vitruvius in his 
De Architectura says: 
Though Archimedes discovered many curious matters that 
evinced great intelligence, that which I am about to mention is the 
most extraordinary. Hiero, when he obtained the regal power in 
Syracuse, having, on the fortunate turn of his affairs, decreed a votive 
crown of gold to be placed in a certain temple to the immortal gods, 
commanded it to be made of great value, and assigned for this purpose 
an appropriate weight of the metal to the manufacturer. The latter, 
in due time, presented the work to the king, beautifully wrought; 
and the weight appeared to correspond with that of the gold which 
had been assigned for it. 
But a report having been circulated, that some of the gold had 
been abstracted, and that the deficiency thus caused had been sup- 
plied by silver, Hiero was indignant at the fraud, and, unacquainted 
with the method by which the theft might be detected, requested 
Archimedes would undertake to give it his attention. Charged with 104 
A SHORT HISTORY OF SCIENCE 
this commission, he by chance went to a bath, and on jumping into 
the tub, perceived that, just in the proportion that his body became 
immersed, in the same proportion the water ran out of the vessel. 
Whence, catching at the method to be adopted for the solution of the 
proposition, he immediately followed it up, leapt out of the vessel in 
joy, and returning home naked, cried out with a loud voice that he 
had found that of which he was in search, for he continued exclaiming, 
‘ I have found it, I have found it V — Vitruvius. 
Archimedes, who combined a genius for mathematics with a physical 
insight, must rank with Newton, who lived nearly two thousand 
years later, as one of the founders of mathematical physics. . . . The 
day (when having discovered his famous principle of hydrostatics 
he ran through the streets shouting Eureka! Eureka!) ought to be 
celebrated as the birthday of mathematical physics; the science came 
of age when Newton sat in his orchard. — Whitehead. 
The recently discovered New Manuscript1 of Archimedes 
throws a very interesting light on his methods of attacking prob- 
lems in mechanics, as well as on his use of mechanical methods 
for geometrical problems. Naturally his mathematical methods 
are highly developed in comparison with the relatively simple 
problems of mechanics with which he deals. 
‘ Certain things first became clear to me by a mechanical method, 
although they had to be demonstrated by geometry afterwards because 
their investigation by the said method did not furnish an actual 
demonstration. But it is of course easier, when we have previously 
acquired, by the method, some knowledge of the questions, to supply 
the proof than it is to find it without any previous knowledge. I 
apprehend that some, either of my contemporaries or of my succes- 
sors, will, by means of the method when once established, be able to 
discover other theorems . . . which have not yet occurred to me.’ 
Our admiration of the genius of the greatest mathematician of 
antiquity must surely be increased, if that were possible, by a perusal 
of the work before us. — Heath. 
Archimedes as an Engineer. — His engineering skill, which 
has gained from an eminent German historian the appellation of 
1 See Reviews by C. S. Slichter and D. E. Smith. Bulletin, American Mathe- 
matical Society, May, 1908, Feb. 1913. GREEK SCIENCE IN ALEXANDRIA 
105 
“the technical Yankee of antiquity,” may be inferred from Plu- 
tarch’s account of the siege of Syracuse : — 
Now the Syracusans, seeing themselves assaulted by the Romans, 
both by sea and by land, were marvellously perplexed, and could not 
tell what to say, they were so afraid; imagining it was impossible for 
them to withstand so great an army. But when Archimedes fell to 
handling his engines, and set them at liberty, there flew in the air 
infinite kinds of shot, and marvellous great stones, with an incredible 
noise and force on the sudden, upon the footmen that came to assault 
the city by land, bearing down, and tearing in pieces all those which 
came against them, or in what place soever they lighted, no earthly 
body being able to resist the violence of so heavy a weight; so that 
all their ranks were marvellously disordered. And as for the galleys 
that gave assault by sea, some were sunk with long pieces of timber 
like unto the yards of ships, whereto they fasten their sails, which 
were suddenly blown over the walls with force of their engines into 
their galleys, and so sunk them by their over great weight. 
These machines (used in the defense of the Syracusans against 
the Romans under Marcellus) he (Archimedes) had designed and 
contrived, not as matters of any importance, but as mere amusements 
in geometry; in compliance with king Hiero’s desire and request, 
some time before, that he should reduce to practice some part of his 
admirable speculation in science, and by accommodating the theo- 
retic truth to sensation and ordinary use, bring it more within the 
appreciation of people in general. Eudoxus and Archytas had been 
the first originators of this far-famed and highly-prized art of me- 
chanics, which they employed as an elegant illustration of geometrical 
truths, and as means of sustaining experimentally, to the satisfaction 
of the senses, conclusions too intricate for proof by words and dia- 
grams. As, for example, to solve the problem, so often required in 
constructing geometrical figures, given the two extremes, to find the 
two mean lines of a proportion, both these mathematicians had re- 
course to the aid of instruments, adapting to their purpose certain 
curves and sections of lines. But what with Plato’s indignation at 
it, and his invectives against it as the mere corruption and annihila- 
tion of the one good of geometry, — which was thus shamefully turn- 
ing its back upon the unembodied objects of pure intelligence to recur 
to sensation, and to ask help (not to be obtained without base super- 106 
A SHORT HISTORY OF SCIENCE 
visions and depravation) from matter; so it was that mechanics came 
to be separated from geometry, and, repudiated and neglected by 
philosophers, took its place as a military art. 
One of his most famous inventions was the water-screw used 
for irrigation, in Egypt, and for pumping. On occasion of diffi- 
culty in the launching of a certain ship he successfully applied 
a cogwheel apparatus with an endless screw. 
Archimedes . . . had stated that given the force, any given 
weight might be moved, and even boasted, we are told, relying on 
the strength of demonstration, that if there were another earth, by 
going into it he could remove this. Hiero being struck with amaze- 
ment at this, and entreating him to make good this problem by actual 
experiment, and show some great weight moved by a small engine, 
he fixed accordingly upon a ship of burden out of the king’s arsenal, 
which could not be drawn out of the dock without great labor and 
many men; and, loading her with many passengers and a full freight, 
sitting himself the while far off with no great endeavor, but only 
holding the head of the pulley in his hand and drawing the cords by 
degrees, he drew the ship in a straight line, as smoothly and evenly, 
as if she had been in the sea. The king, astonished at this, and con- 
vinced of the power of the art, prevailed upon Archimedes to make 
him engines accommodated to all the purposes, offensive and defensive, 
of a siege . . . the apparatus was, in most opportune time, ready at 
hand for the Syracusans, and with it also the engineer himself. 
— Plutarch. 
In astronomy his orrery has been mentioned; he also attempted 
to determine the length of the year more closely. 
To the critical estimates already cited may be added as typical 
of countless others: — 
Whoever gets to the bottom of the works of Archimedes will 
admire the discoveries of the moderns less. — Leibnitz. 
His discoveries are forever memorable for their novelty and the 
difficulty which they presented at that time, and because they are 
the germ of a great part of those which have since been made, chiefly 
in all branches of geometry which have for their object the measure- GREEK SCIENCE IN ALEXANDRIA 
107 
ment of the dimensions of lines and curved surfaces and which require 
the consideration of the infinite. — Mach. 
The genius of Archimedes created the theory of the composition 
of parallel forces, of centres of gravity, and of equilibrium of float- 
ing bodies. But antiquity went no farther; not only were the first 
principles of dynamics unsuspected, but the statistical composition of 
concurrent forces was unknown, and the explanation of machines 
was confined to extension of the principles of the lever, which is the 
starting-point of the works of Archimedes, but may nevertheless 
have been recognized before him. — Tannery. 
Alexandrian Geography : Earth Measurement. — The far 
reaching conquests of Alexander and the resulting migrations and 
colonizations naturally gave a powerful stimulus to geography 
as a branch of descriptive knowledge. Chaldean records became 
accessible to the Alexandrian Greeks and a more accurate system 
of time-measurement was introduced. Until about this period 
it had been customary to make appointments at the time when 
a person’s shadow should have a certain length. 
Eratosthenes, — 275-194 b.c., librarian of the great library at 
Alexandria, making a systematic quantitative study of the data 
thus collected, laid the foundations of mathematical geography— 
a transformation quite analogous to that taking place in as- 
tronomy. After a historical review he gives numerical data about 
the inhabited earth, which he estimates to have a length of 78,000 
stadia and a breadth of 38,000. In connection with this he gives 
also a remarkably successful determination of the circumference 
of the earth. This was based on his observation that a gnomon at 
Syene (Assouan) threw no shadow at noon of the summer solstice, 
while at Alexandria the zenith distance of the sun at noon was 
of the circumference of the heavens. Assuming the two places 
to lie on the same meridian and taking their distance apart as 
5000 stadia, he infers that the whole circumference must be 250,000 
stadia. He or some successor afterwards substituted 252,000, 
perhaps in order to obtain a round number, 700 stadia, for the 
length of one degree. 
This result, subject to some uncertainty as to the length of 108 
A SHORT HISTORY OF SCIENCE 
the stadium, was a close approximation to the real circumference, 
but we may suppose that this degree of accuracy was to some 
extent a matter of accident. Posidonius, a noted Stoic philosopher, 
born in 136 b.c., stated that the bright star Canopus culminated 
just on the horizon at Rhodes, while its meridian altitude at 
Alexandria was “ a quarter of a sign, that is, one forty-eighth part 
of the zodiac.” This would correspond with a circumference of 
240,000 stadia, the method being quite inferior in accuracy to that 
of Eratosthenes, on account of the impossibility of determining 
when a star is just on the horizon. Eratosthenes is also credited 
with measuring the obliquity of the ecliptic with an error of but 
about seven minutes. 
A student of the Athenian Platonists and a man of extraor- 
dinary versatility, philosopher, philologian, mathematician, ath- 
lete, Eratosthenes wrote on many subjects. He may well have 
been responsible for the introduction of leap-year into the Egyp- 
tian calendar by the “ Decree of Canopus ” in 238 b.c., 
in order that the seasons may continually render their service accord- 
ing to the present order and that it may not happen that some of 
the public festivals which are celebrated in the winter come to be 
observed sometimes in the summer. . . . 
He invented a method and a mechanical apparatus for duplicat- 
ing the cube.1 Such a mechanical solution is naturally obnoxious 
to the principles of Plato and Euclid. 
His so-called “sieve” is a method for systematically separating 
out the prime numbers by arranging all the natural numbers 
in order, and then striking out first all multiples of 2, then of 3, 
and so forth, thus sifting out all but the primes 1, 2, 3, 5, 7, 11, 
13, 17, etc. 
Apollonius of Perga, about 260-200 b.c., “the great geom- 
eter,” was the last of this famous Alexandrian group of mathema- 
ticians, and owes his reputation to his important work on the conic 
sections. His predecessors had in general recognized only those sec- 
tions formed from right circular cones by planes normal to an ele- 
1 See Gow, p. 245. GREEK SCIENCE IN ALEXANDRIA 
109 
ment. Archimedes, indeed, and Euclid obtained ellipses by passing 
other planes through right cones, but Apollonius first showed that 
any cone and any section could be taken, and introduced the 
names ellipse, parabola, and hyperbola. In the prefatory letter to 
Book I, Apollonius says to the friend to whom it is addressed: — 
‘Apollonius to Eudemus, greeting. When I was in Pergamum 
with you, I noticed that you were eager to become acquainted with 
my Conics; so I send you now the first book with corrections and will 
forward the rest when I have leisure. I suppose you have not for- 
gotten that I told you that I undertook these investigations at the 
request of Naucrates the geometer, when he came to Alexandria and 
stayed with me; and that, having arranged them in eight books, I 
let him have them at once, not correcting them very carefully (for 
he was on the point of sailing) but setting down everything that 
occurred to me, with the intention of returning to them later. Where- 
fore I now take the opportunity of publishing the needful emendations. 
But since it has happened that other people have obtained the first 
and second books of my collections before correction, do not wonder 
if you meet with copies which are different from this.’ — Gow. 
Of the eight books, the first four are devoted to an elementary 
introduction. In Book I he defines the cone as generated by a 
straight line passing through a point on the circumference of a 
circle and a fixed point not in the same plane; he fixes the manner 
in which sections are to be taken and defines diameters and ver- 
tices of the curves, also the latus rectum and centre, conjugate 
diameters and axes. The other branch of the hyperbola is taken 
due account of for the first time. In Book II asymptotes are 
defined by the statement: “ One draws a tangent at a point of the 
hyperbola, measures on it the length of the diameter parallel to it, 
and connects the point thus determined with the centre of the 
hyperbola.” Book III contains numerous theorems on tangents 
and secants and introduces foci with the definition: “ A focus is 
a point which divides the major axis into two parts whose rectangle 
is one-fourth that of the latus rectum and the major axis,” or the 
square on the minor axis. The focus of the parabola however is 
not recognized, nor has he any knowledge of the directrix of a 110 
A SHORT HISTORY OF SCIENCE 
conic section, these omissions being first filled by Pappus in the 
third century a.d. It is shown that the normal makes equal angles 
with the focal radii to the point of contact, and that the latter 
have a constant sum for the ellipse, a constant difference for the 
hyperbola. This book, he says in the letter quoted above, “ contains 
many curious theorems, most of them are pretty and new, useful 
for the synthesis of solid loci. ... In the invention of these, I ob- 
served that Euclid had not treated synthetically the locus . . . 
but only a certain small portion of it, and that not happily, nor in- 
deed was a complete treatise possible at all without my discoveries.” 
These three books, which are indeed based largely on the earlier 
work of Euclid and others, contain most of the properties of conic 
sections discussed in modern text-books on analytic geometry. 
Book IV discusses the intersections of conics, treating tangency 
correctly as equivalent to two ordinary intersections. In Book V 
Apollonius even undertakes the difficult problem of determining 
the longest and shortest lines which can be drawn from a given 
point to a conic, identifying this with the problem of drawing 
normals from a given point. He succeeds in discovering the points 
for which two such normals coincide, i. e. what we call the centre 
of curvature. Book VI deals with equal and similar conics, reach- 
ing the problem of passing through a given cone a plane which 
shall cut out a given ellipse. Book VII deals with conjugate 
diameters and the complementary chords parallel to them. Book 
VIII is lost. On the whole, in this remarkable work of some 400 
propositions he achieved nearly all the results which are included 
in our modern elementary analytic geometry, even approximating 
the introduction of a system of coordinates by his use of lines 
parallel to the principal axes. 
It is noteworthy that Fermat, one of the inventors of modern 
analytic geometry, was led to it by attempting to restore certain 
lost proofs of Apollonius on loci. 
Of his other mathematical writings little more than the titles 
are known. Among these are one on burning mirrors, one on 
stations and retrogressions of the planets, and one on the use and 
theory of the screw. In astronomy he is believed to have sug- GREEK SCIENCE IN ALEXANDRIA 
Ill 
gested expressing the motions of the planets by combining uniform 
circular motions, an idea afterwards elaborated by Hipparchus 
and Ptolemy. How far his mathematical results were new, how 
far he merely compiled and coordinated the work of others, notably 
Euclid and Archimedes, cannot be precisely determined, but the 
proportion of original work is certainly very large. 
On the arithmetical side he obtained a closer approximation 
than Archimedes for the value of tt, invented an abridged method 
of multiplication, and employed numbers of higher order in the 
manner of Archimedes. This last experiment if followed out to 
its logical conclusions might have had fundamental significance 
for the future development of computation. In the words of 
Gow: — 
he, as well as Archimedes, lost the chance of giving to the world 
once for all its numerical signs. That honor was reserved by the 
irony of fate for a nameless Indian of an unknown time, and we know 
not whom to thank for an invention which has been as important as 
any to the general progress of intelligence. 
Apollonius and Archimedes. — With Apollonius and Archi- 
medes the ancient mathematics had accomplished whatever was 
possible without the resources of analytic geometry and infinitesi- 
mal calculus, which, though already foreshadowed, were not fully 
realized until the seventeenth century. 
It is not only a decided preference for synthesis and a complete 
denial of general methods which characterize the ancient mathematics 
as against our newer science (modern mathematics): besides this 
external formal difference there is another real, more deeply seated, 
contrast, which arises from the different attitudes which the two as- 
sumed relative to the use of the concept of variability. For while the 
ancients, on account of considerations which had been transmitted to 
them from the philosophic school of the Eleatics, never employed the 
concept of motion, the spatial expression for variability, in their 
rigorous system, and made incidental use of it only in the treatment 
of phoronomically generated curves, modern geometry dates from the 
instant that Descartes left the purely algebraic treatment of equations 112 
A SHORT HISTORY OF SCIENCE 
and proceeded to investigate the variations which an algebraic ex- 
pression undergoes when one of its variables assumes a continuous 
succession of values. — Hankel. 
In one of the most brilliant passages of his Aperpu historique 
Chasles remarks that, while Archimedes and Apollonius were the 
most able geometricians of the old world, their works are distinguished 
by a contrast which runs through the whole subsequent history of 
geometry. Archimedes, in attacking the problem of the quadrature 
of curvilinear areas, established the principles of the geometry which 
rests on measurements; this naturally gave rise to the infinitesimal 
calculus, and in fact the method of exhaustions as used by Archimedes 
does not differ in principle from the method of limits as used by 
Newton. Apollonius, on the other hand, in investigating the proper- 
ties of conic sections by means of transversals involving the ratio of 
rectilineal distances and of perspective, laid the foundations of the 
geometry of form and position. — Ball. 
The works of Archimedes and Apollonius marked the most 
brilliant epoch of ancient geometry. They may be regarded, more- 
over, as the origin and foundation of two questions which have occu- 
pied geometers at all periods. The greater part of their works 
are connected with these and are divided by them into two classes, 
so that they seem to share between them the domain of geometry. 
The first of these two great questions is the quadrature of curvi- 
linear figures, which gave birth to the calculus of the infinite, con- 
ceived and brought to perfection successively by Kepler, Cavalieri, 
Fermat, Leibnitz and Newton. 
The second is the theory of conic sections, for which were in- 
vented first the geometrical analysis of the ancients, afterwards 
the methods of perspective and of transversals. This was the pre- 
lude to the theory of geometrical curves of all degrees, and to that 
considerable portion of geometry which considers, in the general 
properties of extension, only the forms and situations of figures, 
and uses only the intersection of lines or surfaces and the ratios of 
rectilineal distances. 
These two great divisions of geometry, which have each its pe- 
culiar character, may be designated by the names of Geometry of 
Measurements and Geometry of Forms and Situations, or Geometry 
of Archimedes and Geometry of Apollonius. — Chasles (Gow). GREEK SCIENCE IN ALEXANDRIA 
113 
Medical Science at Alexandria. Beginnings of Human 
Anatomy. — Alexandria is famous in the history of medicine for 
many reasons. It was here that human,—as contrasted with com- 
parative, — anatomy was first freely studied (probably favored 
by the Egyptian practice of disemboweling and embalming the 
dead) with the result that many of the grotesque errors of the 
earlier Greeks, including even Aristotle, were corrected. In this 
connection two names, and those of rivals, have come down to 
us as of chief importance, Herophilus and Erasistratus. The 
former, himself a student at Cos, was a close follower of the teach- 
ings of Hippocrates and regarded by the ancient world as his 
worthy successor. Erasistratus, on the contrary, opposed the 
Hippocratic doctrines. Both became distinguished anatomists. 
It is believed that the valves of the heart were first recognized 
and named by Erasistratus, who also studied and described 
the divisions, cavities and membranes of the brain, as well 
as the true origin and nature of the nerves. Herophilus like- 
wise studied the brain, the pulmonary artery and the liver, 
besides giving to the duodenum the name (twelve-inch) 
which it still bears. Physiology, meanwhile, made little or no 
progress, and Cicero, two centuries later, still speaks of the 
arteries as “air tubes.” It appears also that vivisection as 
well as anatomy was practised at Alexandria, and probably 
even upon human beings. 
Pergamum, in Asia Minor, was for a time a rival centre of 
medical learning and medical education, but was eventually 
overshadowed by the more famous Alexandrian school. Of this 
last the most celebrated pupil was Galen (born 130 a.d.), the 
most noted medical man of the ancient Roman world. Galen was 
a native of Pergamum who, having first studied at home and at 
Smyrna, spent some years at Alexandria. He then returned to 
Pergamum, but soon went to Rome, where he became physician 
to the Emperor Commodus. Galen was an original and volu- 
minous writer on anatomy. That his name is still constantly 
linked with that of Hippocrates is probably the best evidence of 
his importance in the history of medical science. 114 
A SHORT HISTORY OF SCIENCE 
References for Reading 
Ball. Chapter IV to page 84. 
Berry. Chapter II, Articles 31-36. 
Garrison, F. H. History of Medicine (On Galen, Herophilus, Erasistratus, 
etc.). 
Gow. Chapter VII. 
Heath, T. L. Euclid’s Elements. The Works of Archimedes. Apollonius of 
Perga. 
Mach, E. Science of Mechanics (on Archimedes). 
Mahaffy, J. Alexander’s Empire. 
The three normals to the ellipse. Apollonius. CHAPTER VI 
THE DECLINE OF ALEXANDRIAN SCIENCE 
The century which produced Euclid, Archimedes and Apollonius 
was . . . the time at which Greek mathematical genius attained its 
highest development. For many centuries afterwards geometry re- 
mained a favorite study, but no substantive work fit to be compared 
with the Sphere and Cylinder or the Conics was ever produced. One 
great invention, trigonometry, remains to be completed, but trigo- 
nometry with the Greeks remained always the instrument of astronomy 
and was not used in any other branch of mathematics, pure or applied. 
The geometers who succeed to Apollonius are professors who signalised 
themselves by this or that pretty little discovery or by some com- 
mentary on the classical treatises. 
The force of nature could go no further in the same direction than 
the ingenious applications of exhaustion by Archimedes and the por- 
tentous sentences in which Apollonius enunciates a proposition in 
conics. A briefer symbolism, an analytical geometry, an infinitesimal 
calculus were wanted, but against these there stood the tremendous 
authority of the Platonic and Euclidean tradition, and no discoveries 
were made in physics or astronomy which rendered them imperatively 
necessary. It remained only for mathematicians, as Cantor says, to 
descend from the height which they had reached and “in the descent 
to pause here and there and look around at details which had been 
passed by in the hasty ascent.” The elements of planimetry were 
exhausted, and the theory of conic sections. In stereometry some- 
thing still remained to be done, and new curves, suggested by the 
spiral of Archimedes, could still be investigated. Finally, the arith- 
metical determination of geometrical ratios, in the style of the Meas- 
urement of the Circle, offered a considerable field of research, and to 
these subjects mathematicians now devoted themselves. — Gow. 
In the second century b.c. Hypsicles developed the theory of 
arithmetical progression and added two books of elements to 
Euclid’s thirteen, but the chief mathematical work of this cen- 
115 116 
A SHORT HISTORY OF SCIENCE 
tury was due to Hipparchus, a great astronomer, and Hero, an 
engineer. 
Orbital Motion of the Earth. Aristarchus. — Before 
dealing with Hipparchus and Hero, however, we have to consider 
the highly interesting and significant astronomical theories of 
Aristarchus of Samos (270 b.c.-?), who was the author of a 
treatise On the Dimensions and Distances of the Sun and Moon. 
He endeavored to determine these distances relatively by ascer- 
taining or estimating the angular distance between the two bodies 
when the moon is just half illuminated, that is, when the lines 
joining sun, earth, and moon form a right angle at the moon — a 
method which may have been due to Eudoxus. The difficulties of 
this determination are so serious, however, that no high degree 
of accuracy could be attained, the actual result of Aristarchus 
ff of a right angle — against the true ffjy — corresponding to a 
ratio of about 1 to 19 of the two distances. Aristarchus had no 
trigonometry, and no other method of attacking this problem seems 
to have been known to the Greeks. 
In his Sand Counting already mentioned, Archimedes says of 
Aristarchus, 
He supposes that the fixed stars and the sun are immovable, but 
that the earth is carried round the sun in a circle which is in the middle 
of the course; but the sphere of the fixed stars, lying with the sun round 
the same centre, is of such a size that the circle, in which he supposes 
the earth to move, has the same ratio to the distance of the fixed stars 
as the centre of the sphere has to the surface. But this is evidently 
impossible, for as the centre of the sphere has no magnitude, it follows 
that it has no ratio to the surface. It is therefore to be supposed that 
Aristarchus meant that as we consider the earth as the centre of the 
world, then the earth has the same ratio to that which we call the world, 
as the sphere in which is the circle, described by the earth according 
to him, has to the sphere of the fixed stars. 
Aristarchus thus meets the objection that motion of the earth 
would cause changes in the apparent positions of the stars by as- 
suming that their distances are so great as to render the motion of 117 
DECLINE OF ALEXANDRIAN SCIENCE 
the earth a negligible factor. Another reference to Aristarchus, 
in Plutarch, mentions an opinion that he 
ought to be accused of impiety for moving the hearth of the world, 
as the man in order to save the phenomena supposed that the heavens 
stand still and the earth moves in an oblique circle at the same time as 
it turns round its axis. 
How far this remarkable anticipation of the Copernican theory 
was a conviction rather than a mere fortunate speculation cannot be 
known, but at any rate it failed of that acceptance necessary to its 
permanence. In the next century the rotation of the earth on 
its axis was indeed taught by Seleucus, an Asiatic astronomer, but 
it was 1700 years before these daring theories were again advanced. 
Seleucus also observed the tides, saying “that the revolution of 
the moon is opposed to the earth’s rotation, but the air between the 
two bodies being drawn forward falls upon the Atlantic Ocean, and 
the sea is disturbed in proportion.” 
Planetary Irregularities. — The earlier theory of homocen- 
tric spheres, while accounting more or less successfully for the ap- 
parent motions of the heavenly bodies, had maintained each of 
them at a constant distance from the earth, and thus quite failed 
to explain the differences of brightness which were soon discovered, 
as well as the variations in the apparent size of the moon. The 
conception of motion in neither a straight line nor a circle was re- 
pugnant to the Greek philosophers, and the difficulty was therefore 
met, first by supposing the earth not to be exactly at the centre of 
the circular orbits about it, second by introducing subsidiary 
circles or epicycles. 
Excentric Circular Orbits. — The complete planetary system 
according to the excentric circle theory was therefore as follows. In 
the centre of the universe the earth, round which moved the moon 
in 27 days, and the sun in a year, probably in concentric circles. 
Mercury and Venus moved on circles, the centres of which were al- 
ways on the straight line from the earth to the sun, so that the earth 
was always outside these circles, for which reason the two planets 
are always within a certain limited angular distance of the sun, from 118 
A SHORT HISTORY OF SCIENCE 
which the ratio of the radius of the excentric to the distance of its 
centre from the earth could easily be determined for either planet. 
Similarly, the three outer planets moved on excentric circles, the 
centres of which lay somewhere on the line from the earth to the sun, 
but these circles were so large as always to surround both the sun and 
the earth. — Dreyer. 
It seems probable that Aristarchus was led through this theory 
to conceive of heliocentric orbits, and then to reflect that the earth, 
too, might revolve about the sun as easily as the sun and planets 
round the earth. 
Epicycles. — Progress in observational astronomy increased 
the number and magnitude of planetary irregularities beyond the 
stationary points, retrograde motions, and variations, known to 
Aristarchus, and apparently far beyond possible explanation by 
the simple theory of excentric circles. The system was therefore 
superseded by, or combined with, that of epicycles, not necessarily 
as physically realized, but as at least a geometrical working hy- 
pothesis, which should conform to and explain the observed phe- 
nomena. 
The system of epicycles consists in superimposing one circular 
motion upon another, and repeating the process to any needful 
extent. The motion of the moon about the earth, for example, is 
explained by assuming first a circle (later called the deferent) on 
which moves the centre of a second smaller circle called the epi- 
cycle, on which the moon itself travels. By varying the dimensions 
of both circles and the velocities of the two motions, the observed 
changes, both of position and bright- 
ness of the moon, may be more or less 
satisfactorily accounted for and even 
computed in advance. In particular, 
the apparent retrograde motions of 
the planets in certain parts of their 
orbits may be explained. 
In the figure E denotes the earth, 
the large circle is the deferent of a 
planet, C the centre of the epicycle, P i, P2, Pz, P\ different pos- DECLINE OF ALEXANDRIAN SCIENCE 
119 
sible positions of the planet in its epicycle. The distance of P 
from E obviously varies; the apparent motion of P being com- 
pounded of a forward motion of C and a backward motion at Pi 
is slower, at P3 faster, than the average. By suitable adjust- 
ment of the dimensions and velocities there may be retrogression 
for a certain length of arc near Pi, bounded by stationary points 
where the two motions seem to an observer at E to neutralize each 
other. 
How far this complicated scheme really departed from the 
original postulate of uniform circular motion is sufficiently in- 
dicated by Proclus’ remark, “The astronomers who have pre- 
supposed uniformity of motions of the celestial bodies were ig- 
norant that the essence of these movements is, on the contrary, 
irregularity.” While in point of fact the theory of epicycles and 
that of excentric circles have much in common, the former gradu- 
ally displaced the latter on account of its greater simplicity. Had 
Aristarchus worked out the earlier system in full detail, the history 
of astronomy might have been considerably modified. 
At the Museum of Alexandria a school of observers of whom 
Aristillus and Timocharis were notable members instituted sys- 
tematic astronomical observations with graduated instruments 
and made a small star catalogue. Thus was laid a foundation 
for the brilliant discoveries of Hipparchus and Ptolemy, while 
astronomy, which had in the work of Eudoxus assumed the 
character of true science, though with a too slender observational 
basis, now became an exact science, gradually shedding its encum- 
brances of speculation and vague generalization. 
Hipparchus. Star Catalogue. — The next great astronomer 
and much the greatest of antiquity is Hipparchus, probably a 
native of Bithynia, but long resident at Rhodes, a city which 
rivalled Alexandria itself in its intellectual activity. All his works 
but one are lost, but his great successor and disciple, Ptolemy, has 
based his famous Almagest on the work of Hipparchus and it is 
possible to determine in a general way how much is to be credited 
to each. Having at his disposal the primitive star catalogue of 
Aristillus and Timocharis, Hipparchus was profoundly impressed— 120 
A SHORT HISTORY OF SCIENCE 
as was Tycho Brahe centuries later — by the sudden appearance 
in 134 b.c. in the supposedly changeless starry firmament of a new 
star of the first magnitude. He accordingly set himself the heavy 
task of making a new catalogue, which ultimately included more 
than 1000 stars, for the part of the sky visible to him, and “re- 
mained, with slight alterations, the standard for nearly sixteen 
centuries.” His list of constellations is the basis of our own. 
Precession of the Equinoxes. — While this great piece of 
routine work was deliberately planned by Hipparchus, not so much 
as an end in itself as a necessary basis for future investigators, it 
nevertheless led to his most remarkable discovery, that of the pre- 
cession of the equinoxes. In comparing, namely, the positions of 
certain stars with those observed about 150 years earlier, he de- 
tected a change of distance from the equinoctial point — where the 
celestial equator and the ecliptic meet — amounting in one case 
to about 2°. By an inspiration of genius, he interpreted this 
correctly as due to a slight progressive shifting of the equinoctial 
points, corresponding to a slow rotation of the earth’s axis, by 
means of which the celestial pole in many thousand years describes 
a complete circle. His estimate of 36" per year was considerably 
below the actual value, which is about 50". 
Other Astronomical Discoveries. Planetary Theory. — 
Striving always for greater accuracy and completeness of data, he 
determined the length of the year within about six minutes. In 
attempting to explain the annual motion of the sun, he was aware 
that the change of direction is not uniform, and its distance from 
the earth, as shown by its apparent size, not constant. He de- 
termined the length of spring as 94 days, that of summer as 
and by a somewhat complicated calculation arrived at the value Tri- 
as the eccentricity of the earth’s position in the sun’s orbit. These 
determinations were naturally very difficult and imperfect on ac- 
count of the entire lack of accurate time measurement. Following 
Apollonius, Hipparchus devised a combination of uniform circular 
motions which should account for the observed facts within the 
limits of probable error of observation, and in this undertaking he 
was successful, the degree of accuracy of his theory corresponding 
to that of which his instruments were capable. DECLINE OF ALEXANDRIAN SCIENCE 
121 
With the more complicated lunar theory he was naturally less 
successful. He is believed, however, to have discovered the more 
important irregularities of the moon’s motion, supposing it to 
have a circular orbit in a plane making an angle of 5° with that of 
the sun’s orbit — the ecliptic. The earth is not at the centre, 
but the latter revolves about the earth in a period of nine years. 
Extending his study of eclipses to the ancient records of the Chal- 
deans, he made substantial improvements in the theory of both 
solar and lunar eclipses, and obtained a close approximation for 
the distance of the moon. He estimated the sun’s radius at about 
twelve times that of the earth, its distance from the earth at about 
2550 earth-radii, the moon’s radius y2A that of the earth, its 
distance about 60 earth-radii. The comparison of these figures 
with Ptolemy’s and with the actual are (in earth-radii) — 
Sun’s Radius 
Sun’s Distance 
Moon’s 
Radius 
Moon’s 
Distance 
Hipparchus . . . 
12 
2550 
.29 
60 
Ptolemy .... 
5.5 
1210 
.29 
59 
Actual .... 
109. 
23,000 
.273 
60* 
Hipparchus realized that he had no adequate method for de- 
termining these numbers for the sun. 
The generally accepted order of the planets had now become 
Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, — an order 
adopted very early in Babylonia, and received as a more or less 
probable hypothesis from this time until that of Copernicus. In 
attempting to deal with the motions of the other planets as he had 
done with that of the sun and moon, Hipparchus was soon baffled 
by lack of adequate data, and set himself steadfastly to supply 
the need, resigning to more fortunate future astronomers the task 
of interpretation. 
Eudoxus, more than two centuries earlier, had developed a logical 
mathematical theory of the planetary motions. The more exact 
methods and data of Hipparchus brought out the entire inadequacy 
of existing theory to furnish anything better than a crude approxi- 122 
A SHORT HISTORY OF SCIENCE 
mation to the motions of the planets, and showed the necessity 
both of a better theory and of more complete observational data. 
It is interesting to speculate on the consequences which might have 
resulted for astronomical science had the genius of Hipparchus 
adopted the daring heliocentric theories of Aristarchus instead of 
adhering to the traditional geocentric ideas. 
Invention of Trigonometry.—Not least important among the 
services of Hipparchus to science was his laying the foundations 
of trigonometry, by constructing for astronomical use a table of 
chords, equivalent to our tables of natural sines. He gave also a 
method for solving spherical triangles. It is said that he first 
indicated position on the earth by latitude and longitude — the 
germ of coordinate geometry — Eratosthenes having merely given 
the latitude by means of the height of the pole-star. For mapping 
the sky he used stereographic projection, for mapping the earth 
orthographic. 
To sum up the chief work of Hipparchus: — he made very effec- 
tive use of extant records of earlier astronomers with critical con- 
sideration of their value; he made a prolonged and systematic 
series of observations with the best available instruments; he 
worked out a consistent mathematical theory of the motions of the 
heavenly bodies so far as his data warranted; he made a new 
catalogue of 1080 stars, with the classification by magnitude still in 
use; he discovered the precession of the equinoxes; he laid the 
foundations of trigonometry. 
Delambre, the great French historian of astronomy, says: — 
When we consider all that Hipparchus invented or perfected and 
reflect upon the number of his works and the mass of calculations 
which they imply, we must regard him as one of the most astonishing 
men of antiquity, and as the greatest of all in the sciences which are not 
purely speculative, and which require a combination of geometrical 
knowledge with a knowledge of phenomena, to be observed only by 
diligent attention and refined instruments. 
In spite of these brilliant achievements, the position of Apollonius 
and Hipparchus had become relatively isolated under the prevalent DECLINE OF ALEXANDRIAN SCIENCE 
123 
Stoic philosophy, which was attended with a reversion to primitive 
cosmical notions. Even in Hipparchus a somewhat critical atti- 
tude, excellent in its immediate results, has been regarded by some 
as foreshadowing the period of decadence which actually followed. 
Astronomy is to remain nearly stationary for sixteen centuries. 
Inventions. Ctesibus and Hero. — In the period of civil 
war following the death of Alexander and followed in turn by 
Roman conquest, much attention was naturally devoted to the 
invention and improvements of military engines. Compressed 
air came into use as a motive power and the foundations of 
pneumatics were laid. 
Ctesibus, a barber of Alexandria, distinguished by his mechanical 
inventions, and his follower Hero (or Heron) who flourished in the 
latter part of the second century b.c., made notable inventions 
and some real contributions to mathematical science. The works 
attributed to Hero, on the basis of a great quantity of confused 
and doubtful material, include: — a Mechanics, treating of centres 
of gravity and of the lever, wedge, screw, pulley, and wheel and 
axle; various works on military engines and mechanical toys, a 
Pneumatics — the oldest work extant on the properties of air 
and vapor — describing many machines, among others a fire-en- 
gine, a water-clock, organs, and in particular a steam-engine which 
we may regard as a remote precursor of our modern steam turbine. 
Many of the machines depend for their action on the flow of water 
into a vacuum, which Hero, having no conception of atmospheric 
pressure, attributed to nature’s “abhorrence” of a vacuum. 
He arrived at the important law for the lever and the pulley: 
“The ratio of the times is equal to the inverse ratio of the forces 
applied.” The Dioptra, a treatise on a kind of rudimentary theod- 
olite, discusses such engineering problems as finding differences 
of level, cutting a tunnel through a hill, sinking a vertical shaft to 
meet a horizontal tunnel, measuring a field without entering it, etc. 
The instrument employed is described as a straight plank, 8 or 9 
feet long, mounted on a stand but capable of turning through a 
semicircle. It was adjusted by screws, turning cog-wheels. 
There was an eye-piece at each end and a water level at the side. 124 
A SHORT HISTORY OF SCIENCE 
With it two poles, bearing disks, were used, exactly as by modern 
surveyors. A cyclometer for a carriage is also described, with a 
series of cog-wheels and an index. 
In optics he shows that under the law of equal angles of incidence 
and reflection, the path described by the ray is a minimum. 
Hero’s Triangle Formula. — His Geodesy,—also the Dioptra 
— contains the well-known formula for the area of a triangle 
v _ fa+b+cK, a+b — c„b+c — a„c+a — b 
V 2 X 2 X 2 X 2 
which, since it involves the multiplication of four lengths together, 
is heterodox from the Euclidean standpoint. 
ABC is the given triangle of sides a, b, c, touching its inscribed circle 
at D, E, and F. Taking BJ = AD, we have CJ = %(a + b + c) and 
area ABC = twice area CJM. 
Draw perpendiculars to CM 
at M and to CJ at B, meet- 
ing in H. A semicircle on the 
diameter CH will pass through 
both M and B. The sum 
of the angles CHB and CMB 
is 180°; the triangles BCH 
and MAD are therefore simi- 
lar, 
so that BC: BH = AD : MD, or BC: BJ = BE: ME. 
Also the triangles BGH and EGM are similar, 
so that BH: ME = BG: EG and BC: BJ = BG: EG, 
whence BC + B J: B J = BG +EG: EG, that is, 
CJ: BJ — BE: EG and CJ2: BJ X CJ = CE X BE: CE X EG, 
that is, CJ2: BJ X CJ — BE X CE: EM2, which is equivalent to 
BE xCE-.CJ X EM = CJ X EM: BJ X CJ. 
But CE = $(a + b - c), etc., 
Zi 
whence 4 K2 = (a + b + c) (a + b - c) (b + c - a) (c + a - b). DECLINE OF ALEXANDRIAN SCIENCE 
125 
A triangle with sides 13, 14, 15 is selected as an illustration. Its 
area is 
V21 X 6 X 7 X 8 = 84. 
This work seems to have become a standard authority for 
generations of surveyors, and thus in course of time to have lost 
much of its identity by successive changes. The whole spirit of 
the work is rather Egyptian than Greek, that of the practical 
engineer as distinguished from that of the mathematician, thus in 
a measure a reversion to the aims of the Ahmes manuscript. “ Let 
there be a circle with circumference 22, diameter 7. To find its area. 
Do as follows. 7 X 22 = 154 and = 38|. That is the area.” 
Some of Hero’s methods indicate knowledge of the new trigo- 
nometry of Hipparchus and of the principle of coordinates. Thus 
he finds areas of irregular boundary by counting inscribed rec- 
tangles, a process corresponding to the use of coordinate paper. 
From Hero date such time-honored problems as that of the 
pipes. A vessel is filled by one pipe in time h, by another in 
time t2. How long will it take to fill it when both pipes are used ? 
He defines spherical triangles and proves simple theorems about 
them: — for example, that the angle-sum lies between 180° and 
540°. He determines the volume of irregular solids by measuring 
the water they displace. Having by a blunder introduced V— 63 
he confuses it with V63. 
Inductive Arithmetic. Nicomachus. — As in the case of 
astronomy, progress in geometry now lags and finally ceases alto- 
gether. About 100 a.d. a final era of Greek mathematical science, 
predominantly arithmetical in character, begins with Nicomachus 
of Judea, whose work remained the basis of European arithmetic 
until the introduction of the Arabic arithmetic a thousand years 
later. He enunciates curious theorems about squares and cubes, 
for example : — In the series of odd numbers from 1, the first term 
is the first cube, the sum of the next two is the second, of the 
next three the third, etc., — doubtless simple observation and 
induction. He refers to proportion as very necessary to “ natural 
science, music, spherical trigonometry and planimetry,” and 
discusses various cases in great detail. 126 
A SHORT HISTORY OF SCIENCE 
Mathematics had passed from the study of the philosopher to 
the lecture-room of the undergraduate. We have no more the grave 
and orderly proposition, with its deductive proof. Nicomachus 
writes a continuous narrative, with some attempt at rhetoric, with 
many interspersed allusions to philosophy and history. But more im- 
portant than any other change is this, that the arithmetic of Nico- 
machus is inductive, not deductive. It retains from the old geometrical 
style only its nomenclature. Its sole business is classification, and 
all its classes are derived from, and are exhibited in, actual numbers. 
But since arithmetical inductions are necessarily incomplete, a general 
proposition, though prima facie true, cannot be strictly proved save 
by means of an universal symbolism. Now though geometry was 
competent to provide this to a certain extent, yet it was useless for 
precisely those propositions in which Nicomachus takes most interest. 
The Euclidean symbolism would not show, for instance, that all the 
powers of 5 end in 5 or that the square numbers are the sums of the 
series of odd numbers. What was wanted, was a symbolism similar 
to the ordinary numerical kind, and thus inductive arithmetic led 
the way to algebra. — Gow. 
Ptolemy and the Ptolemaic System. — With Claudius 
Ptolemy, in the second century of our era, Greek astronomy 
reaches its definitive formulation. In the 260 years which had 
elapsed since Hipparchus no progress of consequence had been 
made. 
Of Hipparchus, from whom he inherited so much, Ptolemy 
writes: — 
It was, I believe, for these reasons and especially because he had 
not received from his predecessors as many accurate observations as 
he has left to us, that Hipparchus, who loved truth above everything, 
only investigated the hypotheses of the sun and moon, proving that 
it was possible to account perfectly for their revolutions by combi- 
nations of circular and uniform motions, while for the five planets, 
at least in the writings which he has left, he has not even com- 
menced the theory, and has contented himself with collecting sys- 
tematically the observations, and showing that they did not agree 
with the hypotheses of the mathematicians of his time. He explained 
in fact not only that each planet has two kinds of inequalities but also DECLINE OF ALEXANDRIAN SCIENCE 
127 
that the retrogradations of each are variable in extent, while the other 
mathematicians had only demonstrated geometrically a single in- 
equality and a single arc of retrograde motion; and he believed that 
these phenomena could not be represented by excentric circles nor by 
epicycles carried on concentric circles, but that, it would be necessary 
to combine the two hypotheses. — Dreyer. 
The instruments used by Ptolemy for his astronomical observa- 
tions included: — the “Ptolemaic rule,” consisting of a rod with 
sights pivoted to a vertical rod, the angle at the junction being 
measured by the subtended chord; the armillary circle, a copper 
or bronze ring marked in degrees and mounted in the meridian 
plane on a post. A second movable ring is fitted into this with 
pegs diametrically opposite each other, by means of which the sun’s 
midday height could be measured; the armillary sphere, similar in 
principle but somewhat more complicated; the astrolabe or as- 
tronomical ring for measuring either horizontal or vertical angles. 
Like the Chaldeans Ptolemy also used meridian quadrants of 
masonry. Time was still measured by the flow of water, with 
apparatus considerably improved by Ctesibus and Hero. The 
numerous observations of Ptolemy were made during the period 
125-151 a.d. and he was in Alexandria in 139. 
One of his observations he describes as follows: 
In the 2d year of Antoninus, the 9th day of Pharmonthe, the 
sun being near setting, the last division of Taurus being on the 
meridian (that is, 5% equinoctial hours after noon), the moon was in 
3 degrees of Pisces, by her distance from the sun (which was 92 de- 
grees, 8 minutes); and half an hour after, the sun being set, and 
the quarter of Gemini on the meridian, Regulus appeared, by the 
other circle of the astrolabe, 57% degrees more forwards than the moon 
in longitude. — Whewell. 
The Almagest. — In his celebrated Syntaxis, better known 
from Arabic translations as the Almagest, Ptolemy undertakes to 
present for the first time the whole astronomical science of his age. 
In Book I he reviews the fundamental astronomical data thus: — 128 
A SHORT HISTORY OF SCIENCE 
The earth is a sphere, situated in the centre of the heavens; if 
it were not, one side of the heavens would appear nearer to us than 
the other, and the stars would be larger there; if it were on the celes- 
tial axis but nearer to one pole, the horizon would not bisect the equator 
but one of its parallel circles; if the earth were outside the axis, 
the ecliptic would be divided unequally by the horizon. The earth 
is but as a point in comparison to the heavens, because the stars appear 
of the same magnitude and at the same distances inter se, no matter 
where the observer goes on the earth. It has no motion of translation, 
first, because there must be some fixed point to which the motions of the 
others may be referred, secondly, because heavy bodies descend to the 
centre of the heavens which is the centre of the earth. And if there was 
a motion, it would be proportionate to the great mass of the earth and 
would leave behind animals and objects thrown into the air. This 
also disproves the suggestion made by some, that the earth, while 
immovable in space, turns round its own axis, which Ptolemy ac- 
knowledges would simplify matters very much.1 
Chapter IX explains the calculation of a table of chords. Start- 
ing with the chords of 60° and 72°, already known as sides of regular 
polygons, he devises ingenious geometrical methods for finding 
chords of differences and of half-angles. Thus he computes the 
chords for 12°, 6°, 3°, 1|°, and |°. Hipparchus had already com- 
puted such a table, but Ptolemy completes it by showing that 
§ chord 1|° < chord 1° < £ chord f° 
and thence deriving close approximations for the chords of 1° 
and and constructing a table for each half-degree up to 180°. 
His results are expressed in sexagesimal fractions of the radius (of 
which they are thus numerically independent) and are equivalent 
in accuracy to five decimals in our notation. He also employs 
our present method of interpolation skilfully. This chapter is 
the culmination of Greek trigonometry, which owed its further 
development to Indian and Arabic mathematicians. 
1 “For Ptolemy more geometer and astronomer than philosopher, the astronomer 
who seeks hypotheses adapted to save the apparent movements of the stars knows 
no other guide than the rule of greatest simplicity: It is necessary as far as possible 
to apply the simplest hypotheses to the celestial movements, but if they do not 
suffice, it is necessary to take others which fit better. ” — Duhem. DECLINE OF ALEXANDRIAN SCIENCE 
129 
In Books III, IY, and V, Ptolemy discusses the apparent motions 
and distances of the sun and moon by means of excentrics and 
epicycles, his method for determining the moon’s distance being 
substantially the same as the modern. Book V describes the con- 
struction and use of his chief instrument, the astrolabe. Book VI 
deals with eclipses, using a value of tv equivalent to our 3.1416. 
He determines the distance of the sun, following Hipparchus, by 
observing the breadth of the earth’s shadow when the moon 
crosses it at an eclipse. Books VII and VIII contain a catalogue 
of 1028 stars based on that of Hipparchus, and a discussion of 
precession of the equinoxes, with a close determination of the 
unequal intervals between successive vernal and autumnal equi- 
noxes. The remainder of the treatise is devoted to the planets, 
containing Ptolemy’s chief original contributions. 
While Ptolemy did not take advantage of the better data at 
his command to improve the theory of the sun’s motion, he did 
make substantial progress with that of the moon, the discrepancies 
for which rarely exceed 10', which represented about the maximum 
precision of his instruments. Hipparchus had assumed the moon 
to have a motion representable by one circle with the earth as a 
centre and by an epicycle with its centre upon this. Discrepancies 
between observed and computed positions led Ptolemy, bound as 
he was by the Aristotelian dictum that celestial bodies can move 
only in circular paths, to modify this by making the first circle 
excentric to the earth, the line joining the centres of the circle 
and the earth being itself assumed to revolve. This theory, while 
giving results of sufficient accuracy for the observations at certain 
positions of the moon, exaggerated considerably the variation of 
its distance from the earth, making this at times almost twice 
as great as at others. 
For the five planets, or “wandering stars,” he also assumed ex- 
centric deferents, and as a further means of accounting for dis- 
crepancies, an additional point, in line with the centres of earth and 
deferent, called the “equant,” with respect to which the centre of 
the epicycle would have uniform angular velocity. The planes of 
the epicycles were slightly inclined to that of the ecliptic. 130 
A SHORT HISTORY OF SCIENCE 
Thus in the figure, C is the centre of the circular deferent, E the 
earth and E' the equant. The center A of the epicycle travels 
at such a rate that the line E'A has uniform 
angular velocity. The planet J travels in 
an epicycle about A. These assumptions 
afforded the needful freedom for a fairly close 
approximation to observed planetary motions, 
the mathematical computations involved be- 
coming naturally quite elaborate. Ptolemy 
disclaimed the power of determining the distances or even the 
order of the planets. 
That the system as a whole deserves our admiration as a ready 
means of constructing tables of the movements of sun, moon, and 
planets, cannot be denied. Nearly in every detail (except the varia- 
tion of distance of the moon) it represented geometrically these move- 
ments almost as closely as the simple instruments then in use enabled 
observers to follow them, and it is a lasting monument to the great 
mathematical minds by whom it was gradually developed. 
To the modern mind, accustomed to the heliocentric idea, it is 
difficult to understand why it did not occur to a mathematician like 
Ptolemy to deprive all the outer planets of their epicycles, which were 
nothing but reproductions of the earth’s annual orbit transferred 
to each of these planets, and also to deprive Mercury and Venus of 
their deferents, and place the centres of their epicycles in the sun, as 
Heraclides had done. . . . The system of Ptolemy was a mere geo- 
metrical representation of celestial motions, and did not profess to 
give a correct picture of the actual system of the world. ... For 
more than 1400 years it remained the Alpha and Omega of theoretical 
astronomy, and whatever views were held as to the constitution of 
the world, Ptolemy’s system was almost universally accepted as the 
foundation of astronomical science. — Dreyer. 
After Ptolemy we have no record of any important advance in 
astronomy for nearly 1000 years. 
In reviewing Greek astronomy Berry says, 
The Greeks inherited from their predecessors a number of observa- 
tions, many of them executed with considerable accuracy, which were DECLINE OF ALEXANDRIAN SCIENCE 
131 
nearly sufficient for the requirements of practical life, but in the matter 
of astronomical theory and speculation, in which their best thinkers 
were very much more interested than in the detailed facts, they re- 
ceived virtually a blank sheet on which they had to write (at first with 
indifferent success) their speculative ideas. A considerable interval 
of time was obviously necessary to bridge over the gulf separating 
such data as the eclipse observations of the Chaldeans from such 
ideas as the harmonical spheres of Pythagoras; and the necessary 
theoretical structure could not be erected without the use of mathemati- 
cal methods which had gradually to be invented. That the Greeks, 
particularly in early times, paid little attention to making observations, 
is true enough, but it may fairly be doubted whether the collection 
of fresh material for observations would really have carried astronomy 
much beyond the point reached by the Chaldean observers. When 
once speculative ideas, made definite by the aid of geometry, had 
been sufficiently developed to be capable of comparison with observa- 
tion, rapid progress was made. The Greek astronomers of the scientific 
period, such as Aristarchus, Eratosthenes, and above all Hipparchus, 
appear moreover to have followed in their researches the method which 
has always been fruitful in physical science — namely, to frame pro- 
visional hypotheses, to deduce their mathematical consequences, and 
to compare these with the results of observation. There are few better 
illustrations of genuine scientific caution than the way in which Hip- 
parchus, having tested the planetary theories handed down to him 
and having discovered their insufficiency, deliberately abstained from 
building up a new theory on data which he knew to be insufficient, 
and patiently collected fresh material, never to be used by himself, 
that some future astronomer might thereby be able to arrive at an 
improved theory. 
Of positive additions to our astronomical knowledge made by the 
Greeks the most striking in some ways is the discovery of the ap- 
proximately spherical form of the earth, a result which later work has 
only slightly modified. But their explanation of the chief motions 
of the solar system and their resolution of them into a comparatively 
small number of simpler motions was, in reality, a far more important 
contribution, though the Greek epicyclic scheme has been so re- 
modelled, that at first sight it is difficult to recognize the relation be- 
tween it and our modern views. The subsequent history will, however, 
show how completely each stage in the progress of astronomical science 
has depended on those that preceded. 132 
A SHORT HISTORY OF SCIENCE 
When we study the great conflict in the time of Copernicus be- 
tween the ancient and modern ideas, our sympathies naturally go out 
towards those who supported the latter, which are now known to be 
more accurate, and we are apt to forget that those who then spoke 
in the name of the ancient astronomy and quoted Ptolemy were indeed 
believers in the doctrines which they had derived from the Greeks, 
but that their methods of thought, their frequent refusal to face facts, 
and their appeals to authority, were all entirely foreign to the spirit of 
the great men whose disciples they believed themselves to be. 
Other Works of Ptolemy. — In spite of his scientific attain- 
ments Ptolemy did not disdain to write an elaborate treatise on 
astrology. In a lost work on geometry, Ptolemy made the first 
known of the interminable series of attempts to give a formal 
proof of Euclid’s parallel postulate, an attempt naturally fore- 
doomed to failure. 
In a great treatise on geography, hardly less important than the 
Almagest, Ptolemy gave a description of the known earth, locating 
not less than 5000 places by latitude and longitude. He even gave 
in addition to position the maximum length of day for 39 points 
in India, a land probably better known at this period than in the 
time of Mercator, near the end of the sixteenth century. Ptolemy 
reckoned longitude from the “Fortunate Isles,” — the -western 
boundary of the known world. Various methods of projection 
were discussed in connection with directions for map drawing. 
Ptolemy also wrote on sound and on optics, dealing particularly 
in the latter with refraction, with wThat has been called “ the oldest 
extant example of a collection of experimental measures in any 
other subject than astronomy.” He discovered by careful exper- 
iment and induction the law that light-rays passing from a rarer 
to a denser medium are bent towards the perpendicular, and in- 
vented a simple apparatus for measuring angles of incidence 
and reflection. 
Pappus. — The last two of the great Greek mathematicians were 
Pappus and Diophantus, who lived in Alexandria about 300 a.d. 
The most important work of Pappus is his Collections, in eight 
books, of which all but the first and a part of the second are pre- DECLINE OF ALEXANDRIAN SCIENCE 
133 
served. In this he comments fully on the most important Greek 
mathematical works known to him, making his treatise of the 
highest historical value, particularly in its careful summaries of 
books which have been lost. Book I and most of Book II are 
missing, the third reviews the various solutions of the duplication 
of the cube, adding Pappus’ own, and discusses the regular in- 
scribed polyhedrons; the fourth deals with several less simple 
geometrical matters, including the higher curves, spirals, con- 
choid, quadratrix, etc., the problem of describing a circle tan- 
gent to three given circles which touch each other; the fifth is 
also geometrical. In Book VI Pappus gives the mathematical 
basis for the Ptolemaic astronomy, — i.e. trigonometry and 
optics. Book VII contains his well-known theorems, some- 
times mistakenly attributed to Gulden, that the volume of a 
solid of revolution is equal to the product of the area of the re- 
volving figure and the length of the path of its centre of gravity, 
and that the surface generated is equal to the product of the perim- 
eter and the length of the circular path described by its centre of 
gravity. In this final book he undertakes to deal with certain 
mechanical problems “more clearly and truly” than his prede- 
cessors have done. These include, for example, centre of gravity, 
inclined planes, the moving of a given weight by a given power 
with the help of cog-wheels, the determination of the diameter of 
a broken cylinder. The whole is somewhat weak on the arith- 
metical side. 
With the political decline of Greece and the awakening to in- 
tellectual activity of great Semitic and Egyptian populations, 
mathematical science changed radically from the traditional de- 
ductive geometry, to an arithmetical and algebraic science in 
harmony with the aptitudes which have characterized these races. 
Thus Nicomachus as we have seen was of Jewish antecedents, 
Hero an Egyptian in his point of view and his scientific tendencies. 
Beginnings of Algebra. Diophantus. — Diophantus was 
active in Alexandria in the first half of the fourth century a.d., 
though we know so little about him that even his precise name 
is doubtful. His chief work is his Arithmetic, which is extant 134 
A SHORT HISTORY OF SCIENCE 
however only in somewhat mutilated form. It is the first 
known treatise on algebra, and is devoted to the solution of 
equations, employing algebraic symbols and analytical methods. 
Euclid had given the geometrical equivalent of the solution of a 
quadratic equation, and Hero could solve the same problem alge- 
braically but lacked a satisfactory symbolism. The algebra of 
Diophantus was therefore not a sudden invention, but the result 
of gradual evolution during several centuries of increasing interest 
in arithmetical problems, and declining vogue of the abstract 
Euclidean geometry. 
Writers on the history of algebra distinguish three classes or 
methods of algebraic expression: — 
(a) the rhetorical, where no symbols are used, but every term 
and operation is described in full. This was the only method 
known before Diophantus, and was later in vogue in western 
Europe until the fifteenth century; 
(b) the syncopated, which replaces common words and operations 
by abbreviations, but conforms to the ordinary rules of syntax. 
This was the style of Diophantus; 
(c) the symbolical or modern, using symbols only, without words. 
The syncopated method may be illustrated by the following 
passage from Heath’s Diophantus: — 
Let it be proposed then to divide 16 into two squares. And let 
the first be supposed to be 1*S; therefore the second will be 16f7 — IS. 
Thus 16?7 — IS must be equal to a square. I form the square from 
any number of N’s minus as many U’s as there are in the side of 16 ITs. 
Suppose this to be 2N —4:11. Thus the square itself will be 4S 16 TJ — 
16iV etc. 
In his Arithmetic, which is really a treatise on algebra, Diophan- 
tus represents the (single) unknown by the Greek sigma — all the 
other letters of the Greek alphabet standing for definite numbers 
— with successive powers to the sixth inclusive. If he requires 
two unknowns he admits only one at a time. His originality and 
power in the solution of problems are amply shown, though the 
solutions are rarely complete. For quadratic equations, for ex- DECLINE OF ALEXANDRIAN SCIENCE 
135 
ample, he gives but one root, even when both are positive. Nega- 
tive numbers are for him unreal, and he also avoids the irrational. 
He admits fractional results however, and is indeed the first Greek 
for whom a fraction is a number rather than a mere ratio of two 
numbers. For the solution of pure equations his rule is: “If a 
problem leads to an equation containing the same powers of the 
unknown on both sides but not with the same coefficients, you 
must deduct like from like till only two equal terms remain. 
But when on one side or both some terms are negative, you must 
add the negative terms to both sides till all the terms are positive 
and then deduct as before stated.” 
His method for general quadratics is not given. He solves one 
cubic equation, also particular cases of the indeterminate equation 
Ax2 + Bx + C = Y2. The modern so-called Diophantine equa- 
tions involving the solution in integers of one or more indeter- 
minate equations, do not occur in his own extant work. 
In 130 indeterminate equations, which Diophantus treats, there 
are more than 50 different classes. ... It is therefore difficult for a 
modern, after studying 100 Diophantic equations, to solve the 101st; 
and if we have made the attempt, and after some vain endeavours read 
Diophantus * own solution, we shall be astonished to see how suddenly 
he leaves the broad high-road, dashes into a side-path and with a quick 
turn reaches the goal, often enough a goal with reaching which we 
should not be content; we expected to have to climb a toilsome path, 
but to be rewarded at the end by an extensive view; instead of which, 
our guide leads by narrow, strange, but smooth ways to a small emi- 
nence ; he has finished ! He lacks the calm and concentrated energy 
for a deep plunge into a single important problem; and in this way the 
reader also hurries with inward unrest from problem to problem, as 
in a game of riddles, without being able to enjoy the individual one. 
Diophantus dazzles more than he delights. He is in a wonderful 
measure shrewd, clever, quick-sighted, indefatigable, but does not 
penetrate thoroughly or deeply into the root of the matter. As his 
problems seem framed in obedience to no obvious scientific necessity, 
but often only for the sake of the solution, the solution itself also 
lacks completeness and deeper signification. He is a brilliant performer 
in the art of indeterminate analysis invented by him, but the science has 136 
A SHORT HISTORY OF SCIENCE 
nevertheless been indebted, at least directly, to this brilliant genius 
for few methods, because be was deficient in the speculative thought 
which sees in the True more than the Correct. That is the general 
impression which I have derived from a thorough and repeated study 
of Diophantus ’ arithmetic. — Hankel. 
On the other hand Euler remarks: — 
Diophantus himself, it is true, gives only the most special solu- 
tions of all the questions which he treats, and he is generally content 
with indicating numbers which furnish one single solution. But it 
must not be supposed that his method was restricted to these very 
special solutions. In his time the use of letters to denote undeter- 
mined numbers was not yet established, and consequently the more 
general solutions which we are now enabled to give by means of such 
notation could not be expected from him. Nevertheless, the actual 
methods which he uses for solving any of his problems are as general 
as those which are in use to-day; nay, we are obliged to admit that 
there is hardly any method yet invented in this kind of analysis of 
which there are not sufficiently distinct traces to be discovered in 
Diophantus. 
With the very important process of reducing problems to equa- 
tions he is relatively successful and often highly ingenious. For 
example, “ to find three numbers, so that the product of any two plus 
the sum of the same two shall be given numbers, for example, 8,15, 
and 24.” We should write: xy + x + y = 8; yz + y + z = 15; 
zx + z + x = 24. 
Hence, by subtraction, x(z — y) + z — y =16, 
16 Q 
£ + 1 = , z(x — y) + x — y = 9, z + 1 = —1— , etc. 
z - y x - y 
He, on the other hand, takes a — 1 for one of the numbers and 
readily obtains - — 1 and — — 1 for the others, and a = —• 
a a 5 
He employs tentative assumptions with great effect. For 
example, “To find a cube and its root such that if the same number 
be added to each, the sums shall also be a cube and its root.” If DECLINE OF ALEXANDRIAN SCIENCE 
137 
2x is the original number and x the number added, (an arbitrary 
and presumably erroneous assumption), 8a-3 + x = 27a:3, giving 
19a;2 = 1. The coefficient 19 not being a square, he now seeks to 
find two cubes whose difference is a square. If (x + l)3 — a;3 is 
equated to (2a; — l)2 the special solution x = 7 is easily obtained. 
Returning to the original problem, the new assumption is made: — 
let x = number to be added, 7x = original number. 
(7a;)3 + x = (8a;)3 whence x = ys 
In another type to find a square between 10 and 11, he multi- 
plies both by successive squares of integers until between the prod- 
ucts (by 16) he finds a square, 169. The number required is 
Such processes naturally give particular, not general, 
solutions. 
His lost Porisms are believed to have “ contained propositions 
in the theory of numbers most wonderful for the time.” Sum- 
marizing his methods of dealing with equations we may say 
that:— 
(1) he solves completely equations of the first degree having 
positive roots, showing remarkable skill in reducing simultaneous 
equations to a single equation in one unknown; 
(2) he has a general method for equations of the second degree 
but employs it only to find one positive root; 
(3) more remarkable than his actual solutions of equations are 
his ingenious methods of avoiding equations which he cannot solve. 
How far his work was original, how far like Euclid in his Elements 
it was the result of compilation, cannot be definitely ascertained. 
As a wThole it is somewhat uneven and makes rather the impression 
of great learning than of exceptional originality. He seems in- 
debted in part to predecessors unknown to us. For him the 
earlier Greek distinction between computation and arithmetic has 
lost its force. 
In reviewing the work of Pappus and Diophantus Gow says: — 
the Collections of Pappus can hardly be deemed really important. . . . 
But among his contemporaries, Pappus is like the peak of Teneriffe in 
the Atlantic. He looks back from a distance of 500 years, to find 138 
A SHORT HISTORY OF SCIENCE 
his peer in Apollonius. . . . His work is only the last convulsive effort 
of Greek geometry, which was now nearly dead, and was never effec- 
tually revived. ... It is not so with Ptolemy or Diophantus. The 
trigonometry of the former is the foundation of a new study which was 
handed on to other nations, indeed, but which has thenceforth a con- 
tinuous history of progress. Diophantus also represents the outbreak 
of a movement which probably was not Greek in its origin, and which 
the Greek genius long resisted, but which was especially adapted to 
the tastes of the people who, after the extinction of Greek schools, 
received their heritage and kept their memory green. But no Indian 
or Arab ever studied Pappus or cared in the least for his style or his 
matter. When geometry came once more up to his level, the inven- 
tion of analytical methods gave it a sudden push which sent it far 
beyond him and he was out of date at the very moment when he seemed 
to be taking a new lease of life. 
A melancholy interest attaches to the fate of Hypatia, daughter 
of Theon an Alexandrian mathematician, herself a teacher of 
Greek philosophy and mathematics, who was torn to pieces by 
a Christian mob, doubtless as a representative of pagan (Greek) 
learning, at Alexandria in 415 a.d. 
Conclusion and Retrospect. — Intellectual interests in the 
Greek world (now really Roman) were by this time so completely 
alienated from mathematics, and indeed from science in general, 
that the brilliant work of Pappus and Diophantus aroused but 
slight and temporary interest. Geometry had reached within the 
possible range of the Euclidean method a relatively complete 
development. Algebra under Diophantus attained in spite of 
hampering notation a level not again approached for many cen- 
turies. 
Little need be said of sciences other than those already dealt 
with. These, even more than mathematics and astronomy, shrank 
under Roman autocracy and Christian hostility. Only the 
works of Galen, Strabo, and Pliny need be mentioned, and with 
them we deal in the next chapter. 
The torch of science now passes from the Greeks to the Indians 
of the far East after their conquest by Alexander, to be in turn DECLINE OF ALEXANDRIAN SCIENCE 
139 
surrendered to the Mohammedan conquerors of Alexandria a.d. 
641. By them it is kept from extinction until in later ages it is 
once more fanned to ever increasing radiance in western Europe. 
In attempting a retrospective estimate of Greek science it is 
fundamentally important to judge the whole background fairly. 
In science the Greeks had to build from the foundations. Other 
peoples had extensive knowledge and highly developed arts. Only 
among the Greeks existed the true scientific method with its char- 
acteristics of free inquiry, rational interpretation, verification or 
rectification by systematic and repeated observation, and con- 
trolled deduction from accepted principles. 
The Assyrians, Babylonians and Egyptians had certainly made 
great progress in the use of mechanical devices for moving heavy loads, 
in the construction of scales, and of pumps. Their measuring in- 
struments were well developed, and acute observations were made, 
but of systematic, scientific investigation there is no evidence. The 
Greeks received many results and suggestions from Asia Minor, Meso- 
potamia, and Egypt, but their achievements are essentially their own. 
— Wiedemann. 
In asking ourselves why these extraordinary beginnings seemed 
after a time to lose their power of continued development, we must 
not forget the effect of external conditions. It is conceivable 
indeed that scientific progress should continue from age to age, 
through the genius of individual teachers and students, regardless 
of political and social conditions. Such, however, is not the 
historic fact. For progress in science men of genius are indis- 
pensable, but in no country or age have they alone been able to 
make science flourish under conditions so unfavorable as were 
those of the early centuries of the Christian era. 
Greek science, however, did not “ fail,” learned and elaborate 
as are the explanations that have been given of its alleged failure. 
Under “the chill breath of Roman autocracy” its growth was in- 
deed checked, its animation suspended, for a full thousand years. 
Then in the Renaissance it renewed its vitality and has ever since 
been advancing more and more magnificently. This is not to say 140 
A SHORT HISTORY OF SCIENCE 
that criticisms as to the imperfections of the Greek scientific 
method are invalid, but rather to assert, as most critics must agree, 
that its merits outweighed its defects, and that the latter would 
not have proved disastrous but for the development of political, 
economic and military conditions under which the free Greek spirit 
could not continue its wonderful achievements. 
References for Reading 
Ball. Chapters IV, V. 
Berry. Chapter II, Articles 37-54. 
Dreyer. Chapters VI-IX. 
Gow. Chapters IV, VIII, IX, X. 
Heath. Diophantus of Alexandria. Aristarchus of Samos. CHAPTER VII 
THE ROMAN WORLD. THE DARK AGES 
Among them [the Greeks] Geometry was held in highest honor: 
nothing was more glorious than Mathematics. But we have limited 
the usefulness of this art to measuring and calculating. — Cicero. 
The Romans were as arbitrary and loose in their ideas as the 
Greeks, without possessing their invention, acuteness and spirit of 
system. — Whew ell. 
The Romans, with their limited peasant horizon and their short- 
sighted practical simplicity, cherished always for true science in their 
inmost hearts that peculiar mixture of suspicion and contempt which 
is so familiar today among the half educated. The arch dilettante 
Cicero boasts, even, that his countrymen, thank God ! are not like 
those Greeks, but confine the study of mathematics and that sort of 
thing to the practically useful. — Heiberg. 
The Roman World-Empire. — For several centuries, during 
the decline of Greek learning both in Greece itself and in Alexan- 
dria, two new and powerful States were developing; one having its 
centre at Carthage on the northern shore of Africa, almost opposite 
Sicily, the other—the Roman Empire — on the western shore of 
Italy in the valley of the Tiber. The latter, at first comparatively 
insignificant, rapidly rose to a position of world-wide power, con- 
quering in turn Carthage, Greece, and the East and eventually 
extending over the greater part of the then known world, from 
Britain on the north to the Cataracts of the Nile on the south, 
from India in the east to the Pillars of Hercules in the west. 
The Roman Attitude towards Science. — One of the most 
striking facts in the history of science is the total lack of any 
evidence of real interest in science or in scientific research among 
the Roman people itself or any people under Roman sway. Alex- 
andrian science, even, though previously flourishing, languished 
and went steadily to its fall after the submission of that city to 
the Romans in the first century b.c. The truth seems to be that 
141 142 
A SHORT HISTORY OF SCIENCE 
the Roman people, while highly gifted in oratory, literature, and 
history (as witness, for example, the works of Cicero, Virgil and 
Tacitus), were not interested and therefore not successful in scien- 
tific work. This is the more impressive when we reflect upon their 
marvellous military genius, and their preeminence in world-wide 
power, dominion and influence. In vain do we look for any Roman 
scientist or philosopher of such originality or range as Aristotle or 
Plato; for any Roman astronomer, like Aristarchus or Hipparchus 
or Ptolemy; for any Roman mathematician or inventor, like 
Archimedes; for any Roman natural philosopher, like Democ- 
ritus ; for any Roman pioneer in medicine, like Hippocrates, — 
for Galen was Roman neither by birth nor education, but only 
by adoption late in life. 
Roman Engineering and Architecture. — There is how- 
ever one marked feature of Roman civilization in which extraor- 
dinary ability was displayed and peculiar excellence achieved 
and in which the Romans were unquestionably far superior to all 
their predecessors and, until very recent times, to all their suc- 
cessors. This feature, which is one of the most characteristic, is 
the Roman genius for both military and civil engineering. It is 
only necessary to mention the surviving remains of Roman walls, 
fortresses, roads, aqueducts, theatres, baths, and bridges. Never 
before and never since has any empire built so many, so splendid, 
and so enduring monuments for the service of its peoples in peace 
and in war. The surface of southern Europe, western Asia and 
northern Africa is still covered after the lapse of twenty centuries 
with Roman remains which bid fair to resist decay and destruc- 
tion for another two thousand years. Roman engineering is almost 
as distinguished as is Roman law. The Emperor Constantine 
in the fourth century wrote: “ We need as many engineers as pos- 
sible. As there is lack of them, invite to this study persons of about 
18 years, who have already studied the necessary sciences. Re- 
lieve the parents of taxes and grant the scholars sufficient means.” 
The land surveyors formed a well-organized gild, but they were 
merely practitioners of a traditional art, perpetuating the errors 
of their ancient Egyptian predecessors, not dreaming of new dis- THE ROMAN WORLD 
143 
coveries, nor even of imparting such knowledge as they had, — 
outside the ranks of their own gild. 
Slave Labor in Antiquity. — It must never be forgotten 
that throughout antiquity, and to a great extent even until very 
recent times, the labor question was wholly different from what 
it is to-day. Instead of the labor-saving machinery which is so 
extraordinary a feature of our time, but which was practically non- 
existent before the end of the eighteenth century, the slave was the 
machine for all heavy labor. It is not likely that he was ever a 
particularly cheap machine, but in the mass he was powerful, and 
it was probably largely by his labor that the fields were cultivated 
and irrigated, and that dams and ditches, walls and towers, roads 
and bridges and pyramids and temples, were built and fortified. 
It is notorious that the so-called “ ships ” of war, the galleys, 
were manned by slaves, even down to modern times. It is difficult 
to determine the efficiency of labor of this kind because we are 
generally ignorant as to the time factor, but whether from our 
modern point of view inefficient or not, the results were often re- 
markable and sometimes, as in the case of the Pyramids, stupendous. 
Julius Cassar and the Julian Calendar. — Julius Caesar 
himself undertook two great problems of practical mathematical 
science: — the rectification of the highly confused calendar, and 
a survey of the whole Roman empire. In the year 47 b.c. the 
accumulated calendar error amounted to not less than 85 days. 
Reform was accomplished by a decree making the year con- 
sist of 365 days with an additional day in February once in four 
years. The survey, of which the results were to be shown in a 
great fresco map, was not carried out until the reign of Augustus. 
Vitruvius on Architecture. — The most famous ancient work 
on building and kindred topics, including building materials, is 
that entitled De Architectural, by Vitruvius, a Roman architect 
and engineer living (about 14 b.c.) in the age of Augustus. This 
celebrated work was the only one of importance on architecture 
known to the Middle Ages, and was the guide and text-book of 
the builders of that period as well as of those of the Renaissance. 
The book (now easily accessible in translation) is in part a 144 
A SHORT HISTORY OF SCIENCE 
compilation from earlier, and especially Greek, authors, and in 
part original. Vitruvius uses for 7r the value 3|, — less ac- 
curate than that of Archimedes, but displaced later by the crude 
approximation 3. Of Vitruvius’s life and work almost nothing is 
known, but no other ancient treatise of a similar technical nature 
has had in its own field so much influence on posterity. 
Frontinus on the Waterworks of Rome (c. 40-103 a.d.). 
At about the end of the first century of our era, Sextus Julius 
Frontinus, a Roman soldier and engineer, wrote a highly interest- 
ing and valuable account of the waterworks of Rome. Frontinus 
served as praetor under Vespasian; was afterwards sent to Britain 
as Roman governor of that island; was superseded by Agricola in 
78 a.d. and was appointed in 97 a.d. Curator Aquarum, “an office 
never conferred except upon persons of very high standing.” 
Roman Natural Science and Medicine. — Among the Roman 
workers and authors of importance in the history of natural science 
and medicine only a few require more than passing notice. This 
is the more remarkable when we reflect upon the vast extension of 
the Roman empire and the novel and hitherto unequalled op- 
portunities afforded for observation and collection in natural 
history, and for the study of anthropology, geography, geology, 
meteorology, climatology, zoology, botany and the like, — not to 
mention military surgery, and the hygiene and sanitation of camp- 
life. 
Lucretius (98-55 b.c.) is to-day regarded not only as a great 
Roman poet but also as the most perfect exponent in his time of 
the natural philosophy of the Greeks who preceded him. He was 
a contemporary and a few years the junior of Cicero and Julius 
Caesar. The first two books and the fifth of his Be Rerum 
Natura (On the Nature of Things) are of interest to the modern 
scientific student, because of their dealing with problems of per- 
manent importance to mankind. He was a disciple of Epicurus, 
and apparently also well acquainted with the works of Empedocles, 
Democritus, Anaxagoras, and many other of the great Greek writers 
such as Homer, Hippocrates, Thucydides, and especially Euripides. 
The title of his famous poem shows his interest in natural philos- THE ROMAN WORLD 
145 
ophy, and there is evidence that he was also a teacher and re- 
former. He is antagonistic to superstition and a strong advocate 
of rationalism, but he is neither irreverent nor revolutionary. 
The following passages are typical: — 
Water in summer time flows cool in wells, 
Because the Earth then rarefied by heat, 
Its proper stores most radiate to the air. 
Hence more the Earth is drained of its heat, 
And colder grow the currents under ground. 
But when by cold in winter ’tis compressed, 
Its heat escaping passes into wells. . . . 
And now to tell by which of Nature’s laws, 
The stone called Magnet by the Greeks, — since first 
’Mong the Magnesians found, — can iron draw. 
Men gaze with wonder on the marvellous stone, 
With pendent chain of rings, oft five or more, 
Light hanging in the air suspensive, while 
One from another feels the influence of the stone 
That sends through all its wonder-working power. 
Here many principles we must first lay down 
And slow approach by long preparative, 
Rightly to solve the rare phenomenon. 
The more exact I then attentive ears. . . . 
How different is fire from piercing frost! 
Yet both composed of atoms toothed and sharp, 
As proved by touch. Touch, O ye sacred powers — 
Touch is the organ whence all knowledge flows; 
Touch is the body’s sense of things extern, 
And of sensations that deep spring within; 
Whether delightsome, as in genial act, 
Or rude collision torturing from without; 
How different, then, must forms of atoms be 
Which such sensation varied can produce! 
Strabo,—a Roman traveller, historian and geographer, lived 
somewhere between 63 b.c. and 24 a.d. His Geography is the 
most important work on that subject surviving from antiquity and, 146 
A SHORT HISTORY OF SCIENCE 
while apparently building on the foundation laid by Eratosthenes, 
is plainly an original work devoted largely to his own explorations 
and observations during years of travel and study in different 
countries, including Italy, Greece, Asia Minor, Egypt, and Ethiopia. 
He himself says: — 
Westward I have journeyed to the parts of Etruria opposite Sar- 
dinia ; towards the South from the Euxine to the borders of Ethiopia, 
and perhaps not one of those who have written geographies has 
visited more places than I have between those limits. 
His work is invaluable as a picture of the limited geographical 
knowledge of the time, but he had no such mathematical knowledge 
of geography as had his great predecessors, Eratosthenes, Hip- 
parchus, and Ptolemy. 
Pliny the Elder (23-79 a.d.), sometimes called Pliny the 
Naturalist, is another Roman of scientific attainments, whose great 
work entitled Natural History, although more an encyclopaedia of 
miscellaneous information than a scientific treatise, is, nevertheless, 
like the works of Herodotus, a landmark in the history of civil- 
ization. It consists of thirty-seven books and is easily accessible 
in English. Pliny deals with the universe, God, nature, and 
natural .phenomena; with earth, stars, earthquakes ; with man, 
beasts, shells, fishes, insects, trees, fruits, gums, perfumes, timber, 
the diseases of plants, metals, stones, precious stones, etc. The 
author met his death in that eruption of Vesuvius which over- 
whelmed Pompeii in 79 a.d. and because of his scientific curiosity 
which led him to approach too near to the volcano. 
Galen (Claudius Galenus) who flourished in the second 
century a.d. was born and partly educated at Pergamum in Asia 
Minor, where, after much travelling, and research, chiefly in 
anatomy and philosophy at Smyrna and at Alexandria, he also 
practised the healing art. Sent for by the Roman emperor, Lucius 
Verus, he was afterward physician to Marcus Aurelius and his 
son Commodus. His writings are voluminous, encyclopedic and 
anatomically important, though not especially original, and his 
name is often linked with that of Hippocrates, partly, no doubt, THE ROMAN WORLD 
147 
because after Galen we find no great name in anatomy until we 
come to Vesalius, some 1400 years later. 
Late Roman Mathematical Science. — Two periods may be 
distinguished in ancient mathematical science, the first beginning 
with Pythagoras and ending with Hero. To these four to five 
centuries belong all the original works in geometry, astronomy, 
mechanics, and music. The period closes with the extension of 
the pax Romana over the Orient. The second extends to the 
sixth century, when Hellenism is proscribed by the new religion, 
the genius of invention is extinct, and men merely study the older 
works, commenting and coordinating. Astronomy gradually 
reverts to astrology, the mathematical geography well begun 
under Eratosthenes and Ptolemy becomes superficial and descrip- 
tive, with Strabo and even with Posidonius. 
Whatever the eminence of the Romans in the practical arts of 
war, politics and engineering, their interest in abstract science 
was almost nil. On the other hand, commercial arithmetic, 
which had been studiously neglected by Greek mathematicians, 
now had the place of honor. The Roman numerals, clumsy as 
they seem to us, were superior to the Greek, and a useful system 
of finger-reckoning was developed, supplementing the skilful use 
of the abacus. If no abacus was at hand, the corresponding lines 
were quickly traced on sand or dust, small stones or calculi — 
whence our words calculation and calculus, — serving as counters. 
A complete Roman abacus — of which no example has come down 
to us — seems to have had eight long and eight short grooves. Of 
the former, one held five counters or buttons, each of the others 
four, each of the short grooves one, these last counting as five 
units each. The grooves with six counters served for computa- 
tions with fractions. Geometry — but of Hero rather than of 
Euclid — was valued for its utility in surveying and architecture. 
Preparation for the engineering art included mathematics, optics, 
astronomy, history, and law. There were also teachers of me- 
chanics and architecture. (See Vitruvius, above.) 
Capella. — Early in the fifth century Martianus Capella wrote 
a compendium of grammar, dialectics, rhetoric, geometry, arith- 148 
A SHORT HISTORY OF SCIENCE 
metic, music, and astronomy, of great and lasting educational 
influence. His classification of these “seven liberal arts” main- 
tained itself throughout the Middle Ages and is not yet wholly 
extinct. Gregory of Tours for example says: — “If thou wilt 
be a priest of God, then let our Martianus instruct thee first in 
the seven sciences.” 
Boethius (480-524) born at Rome on the eve of its fall in 476 
is the author not only of the famous Consolations of Philosophy 
but also of works on Music and on Arithmetic which long served 
to represent Greek mathematics to the medieval world. In the 
course of his public-spirited career, Boethius interested himself 
in the reform of the coinage and in the introduction of water- 
clocks and sun-dials. His geometry consists merely of some of 
the simpler propositions of Euclid, with proofs of the first three 
only, and with applications to mensuration. Yet the intellectual 
poverty of the age was such that this remained long the standard 
for mathematical teaching. Boethius’ Arithmetic begins: — 
By all men of old reputation who following Pythagoras’ reputation 
have distinguished themselves by pure intellect it has always been 
considered settled that no one can reach the highest perfection of 
philosophical doctrines, who does not seek the height of learning at 
a certain crossway — the quadrivium. 
For him the things of the world are either discrete (multitudes), 
or continuous (magnitudes). Multitudes are represented by 
numbers, or in their ratios by music; magnitudes at rest are 
treated by geometry, those in motion by astronomy. These four 
of the seven liberal arts form the quadrivium; grammar, dialec- 
tics and rhetoric, the trivium. A Christian in faith, a pagan in 
culture, Boethius has been called the “ bridge from antiquity to 
modern times.” (See page 50.) 
The scholars of the time were almost without exception men 
whose first interests were theological. Mathematics, having no 
direct moral significance, seemed to them in itself unworthy of 
attention. On the other hand, they attached exaggerated im- 
portance to all sorts of mystical attributes of numbers and to the THE DARK AGES 
149 
interpretation of scriptural numbers. Thus Augustine says the 
science of numbers is not created by men, but merely discovered, 
residing in the nature of things. 
Whether numbers are regarded by themselves or their laws applied 
to figures, lines or other motions, they have always fixed rules, which 
have not been made by men at all, but only recognized by the keen- 
ness of shrewd people. 
Science and the Early Christian Church. — In the earlier 
centuries of our era the history of science gradually enters upon 
a new phase. The more highly developed civilization of Greece 
and Rome, weakened by corruption, has finally yielded to the 
attacks on the one hand of barbarous or semicivilized races, — 
Goths, Vandals, Huns, and Arabs, — and on the other hand 
to a moral revolution of humble Jewish origin. These changes 
were adverse to the development, or even the survival, of Greek 
science. The destructive relation of the northern barbarians to 
scientific progress may be easily imagined. The policy of official 
Christianity was based on antecedent antipathy for the unmoral 
intellectual attitude and the degenerate character which the early 
Christians found in close association with Greek learning, and on 
a too literal interpretation of the Jewish scriptures, with their 
primitive Chaldean theories of cosmogony and the world. 
Justin Martyr, in the second century, says that what is true 
in the Greek philosophy can be learned much better from the 
Prophets. Clement of Alexandria (d. 227) calls the Greek philoso- 
phers robbers and thieves who have given out as their own what 
they have taken from the Hebrew prophets. Tertullian (160-220) 
insists that since Jesus Christ and his gospel, scientific research 
has become superfluous. Isidore of Seville in the seventh century 
declares it wrong for a Christian to occupy himself with heathen 
books, since the more one devotes himself to secular learning, the 
more is pride developed in his soul. Lactantius early in the 
fourth century includes in his “Divine Institutions” a section, 
‘ On the false wisdom of the philosophers*/ of which the 24th chap- 
ter is devoted to heaping ridicule on the doctrine of the spherical 150 
A SHORT HISTORY OF SCIENCE 
figure of the earth and the existence of antipodes. It is unnecessary 
to enter into particulars as to his remarks about the absurdity of be- 
lieving that there are people whose feet are above their heads, and 
places where rain and hail and snow fall upwards, while the wonder 
of the hanging gardens dwindles into nothing when compared with 
the fields, seas, towns, and mountains, supposed by philosophers to 
be hanging without support. He brushes aside the argument of 
philosophers that heavy bodies seek the centre of the earth, as un- 
worthy of serious notice; and he adds that he could easily prove by 
many arguments that it is impossible for the heavens to be lower than 
the earth, but he refrains because he has nearly come to the end of his 
book, and it is sufficient to have counted up some errors, from which 
the quality of the rest may be imagined. 
It was natural that Augustine (354-430), . . . should express him- 
self with . . . moderation, as befitted a man who had been a 
student of Plato as well as of St. Paul in his younger days. With 
regard to antipodes, he says that there is no historical evidence of 
their existence, but people merely conclude that the opposite side of 
the earth, which is suspended in the convexity of heaven, cannot be 
devoid of inhabitants. But even if the earth is a sphere, it does not 
follow that that part is above water, or, even if this be the case, that 
it is inhabited; and it is too absurd to imagine that people from our 
parts could have navigated over the immense ocean to the other 
side, or that people over there could have sprung from Adam. With 
regard to the heavens, Augustine was, like his predecessors, bound 
hand and foot by the unfortunate water above the firmament. He 
says that those who defend the existence of this water point to 
Saturn being the coolest planet, though we might expect it to be 
much hotter than the sun, because it travels every day through a 
much greater orbit; but it is kept cool by the water above it. The 
water may be in a state of vapor, but in any case we must not 
doubt that it is there, for the authority of Scripture is greater than 
the capacity of the human mind. He devotes a special chapter to 
the figure of the heaven, but does not commit himself in any way 
though he seems to think that the allusions in Scripture to the heaven 
above us cannot be explained away by those who believe the world 
to be spherical. But anyhow Augustine did not, like Lactantius, 
treat Greek science with ignorant contempt; he appears to have 
had a wish to yield to it whenever Scripture did not pull him the THE DARK AGES 
151 
other way, and in times of bigotry and ignorance this is deserving 
of credit. — Dreyer. 
Arguing elsewhere that the soul perceives what the bodily eye 
cannot, Augustine avails himself of the geometrical analogy of 
the ideal straight line which shall have length without breadth or 
thickness, but he lapses into mysticism when he passes to the 
circle. 
The biographer of St. Eligius (writing in 760 under Pepin) says 
‘What do we want with the so-called philosophies of Pythagoras, 
Socrates, Plato and Aristotle, or with the rubbish and nonsense of 
such shameless poets as Homer, Virgil and Menander? What serv- 
ice can be rendered to the servants of God by the writings of the 
heathen Sallust, Herodotus, Livy, Demosthenes or Cicero?’ Frede- 
gar . . . complains (about 600) that ‘ The world is in its decrepitude, 
intellectual activity is dead, and the ancient writers have no suc- 
cessors.’ . . . — G. H. Putnam, Books of the Middle Ages. 
The following is a broad survey of the whole period: — 
The soft autumnal calm . . . which lingered up to the Antonines 
over that wide expanse of empire from the Persian Gulf to the Pillars 
of Hercules and from the Nile to the Clyde . . . was only a misleading 
transition to that bitter winter which filled the half of the second 
and the whole of the third century, to be soon followed by the abiding 
dark and cold of the Middle Ages. The Empire was moribund when 
Christianity arose. Rome had practically slain the ancient world 
before the Empire replaced the Republic. The barbarous Roman 
soldier who killed Archimedes absorbed in a problem, is but an in- 
stance and a type of what Rome had done always and everywhere by 
Greek art, civilization and science. The Empire lived upon and con- 
sumed the capital of preceding ages, which it did not replace. Popu- 
lation, production, knowledge, all declined and slowly died. . . . 
The sun of ancient science, which had risen in such splendour 
from Thales to Hipparchus, was now sinking rapidly to the horizon; 
and when it at last disappeared, say, in the fifth century, the long 
night of the Middle Ages began. . . . The pursuit of knowledge for 
knowledge’s sake was out of place. ... All the outlets through 
which modern energy is chiefly expended were then closed; a man 
could not serve the state as a citizen, he could not serve knowledge 152 
A SHORT HISTORY OF SCIENCE 
as a man of science. . . . There was only one thing left for him to 
do, — to serve God. — J. C. Morison, The Service of Man. 
The Eastern Empire. Edict of Justinian. — Only half a 
century after the fall of Rome the Greek schools in Athens were 
closed, in 529 a.d., by order of the emperor Justinian, and intel- 
lectual darkness settled down over Eastern Europe. Theology 
became more than ever the chief pursuit of the educated, and Greek 
learning more than ever neglected. Many Greek manuscripts, 
however, were hidden away, and many Greek scholars, though 
scattered, kept alive the feeble spark of Greek learning. 
The Dark Ages. — After the mighty Roman Empire of the 
West had come to its end, the peoples of Christian Europe and of 
the Graeco-Roman world descended into the great hollow which is 
roughly called the Middle Ages, extending from the fifth to the fifteenth 
century, a hollow in which many great and beautiful and heroic things 
were done and created, but in which knowledge, as we understand it 
and as Aristotle understood it, had no place. The revival of learning 
and the Renaissance are memorable as the first sturdy breasting by 
humanity of the hither slope of that great hollow which lies between 
us and the ancient world. The modern man, reformed and regenerated 
by knowledge, looks across it and recognizes on the opposite ridge, in 
the far-shining cities and stately porticoes, in the art, politics and 
science of antiquity, many more ties of kinship and sympathy than 
in the mighty concave between, wherein dwell his Christian ancestry 
in the dim light of scholasticism and theology. — Morison. 
The “great hollow” here so graphically portrayed may be de- 
scribed as the Middle or Medieval Age (c. 450-1450 a.d.) and of 
these ten centuries the first three, or thereabouts, are often called 
the Dark — as they certainly were the darkest — Ages. 
The darkest time in the Dark Ages was from the end of the sixth 
century to the revival of learning under Charles the Great (Charle- 
magne). Bad grammar was openly circulated and sometimes com- 
mended. St. Gregory the Great quoted the Bible in depreciation of 
the Humanities. (Ps. lxx. 15. 16.) The study of heathen authors 
was discouraged more and more. “ Will the Latin grammar save an 
immortal soul?” “What profit is there in the record of pagan sages, THE DARK AGES 
153 
the labors of Hercules or of Socrates ? ” Books came to be scarce. . . . 
But the decline of education was not universal. If studies failed in 
Gaul or Italy, they flourished in Ireland and afterward in Britain, and 
returned later from these outer borders to the old central lands of the 
Empire. Further, in spite of depression and discouragement, there 
was a continuity of learning even in the darkest ages and countries. 
Certain school books hold their ground . . . Capella . . . Boethius . . . 
Cassiodorus . . . And later Isodorus of Seville with a number of other 
authors are found in the ages of distress and anarchy more or less 
calmly giving their lectures and preserving the standards of a liberal 
education. Much of this work was humble enough, but it was of 
great importance for the times that came after. . . . The darkest 
ages, with all their negligence, kept alive the life of the ancient 
world. 
Boethius [in the sixth century a.d., see p. 148] is the interpreter 
of the ancient world and its wisdom, accepted by all the tribes of 
Europe from one age to another, and never disqualified in his office 
of teacher even by the most subtle and elaborate theories of the later 
schools. . . . Cassiodorus (490-585) is wanting in the graces of 
Boethius, and he is much sooner forgotten; but his enormous industry, 
his organization of literary production, his educational zeal have all 
left their effects indelibly in modern civilization. By his definition of 
the seven Liberal Arts, and by his examples of methods in teaching 
them, he is the spiritual author of the universities, the patron of all 
the available learning in the world. — Ker, Dark Ages. 
The Establishment of Schools by Charlemagne. — We 
have seen above how the schools of Athens were closed by- 
Justinian in 529. Such schools as existed after that time were 
chiefly ecclesiastical and their teachings opposed to pagan or 
heathen (i.e. Greek) learning. At length, however, in 787 
Charlemagne, moved it is said by the troublesome variety of 
writing as well as the general illiteracy of his people, ordered the 
establishment of schools in connection with every abbey of his 
realm, and summoned to take charge of them Peter of Pisa and 
Alcuin of York (735-804) (called by Guizot “the intellectual 
prime minister of Charlemagne”), whose names stand among the 
highest in a revival of learning thus begun in western Europe. 154 
A SHORT HISTORY OF SCIENCE 
In the later part of the eighth century begins the great age of 
medieval learning, the educational work of Charles the Great. . . . 
There was some leisure and freedom and much literary ambition. 
The Latin poets of the court of Charlemagne have an enthusiasm and 
delight in classical poetry. ... In prose there was no less activity. 
Besides the scientific treatises and the commentaries, the edifying 
works of Alcuin and others, there were histories. . . . The scholarly 
spirit of the ninth century ... is not limited to the orthodox routine. 
One of the chief scholars, with more Greek than most others, Erigena, 
is famous for more than his learning, as a philosopher, who, whatever 
his respect for the Church, acknowledged no authority higher than 
reason. — Ker. 
Alcuin himself taught rhetoric, logic, mathematics and di- 
vinity, becoming master of the great school at St. Martin’s of 
Tours. Of his arithmetic the following problem is an illustra- 
tion : — 
If 100 bushels of corn are distributed among 100 people in such 
a manner that each man receives 3 bushels, each woman 2, and each 
child half a bushel; how many men, women and children are there ? 
Of six possible solutions Alcuin gives but one. 
The mathematics taught in Charlemagne’s schools would 
naturally include the use of the abacus, the multiplication table, 
and the geometry of Boethius. Beyond this, a little Latin with 
reading and writing sufficed for the needs of the church and her 
servants, and was supplemented by music and theology for her 
higher officers. The recognized intellectual needs of the world 
were indeed but slight. The civilization of Rome had been 
gradually submerged by successive waves of barbaric invasion 
from the north, as a similar fate was soon to be met by the still 
higher culture of Alexandria. The best intellect of the times was 
perforce drawn into other forms of activity, while such scholars 
as remained found no favorable environment for fruitful study. 
The Benedictine monasteries, indeed, sheltered a few studious 
monks whose scientific interest scarcely extended beyond the 
mathematics necessary for their simple accounts, and the com- 
putation connected with the determination of the date of Easter. THE DARK AGES 
155 
Near the close of the tenth century Gerbert of Aquitaine (940- 
1003), afterwards Pope Sylvester II, devoted his versatile genius 
in part to mathematical science. He constructed not only abaci, 
but terrestrial and celestial globes, and collected a valuable library. 
To him were also attributed a clock, and an organ worked by steam. 
He wrote works on the use of the abacus, on the division of numbers 
and on geometry. The last named contains a solution of the rela- 
tively difficult problem to find the sides of a right triangle whose 
hypotenuse and area are given. Unfortunately the latter part of 
his life was absorbed in political intrigue and his death in 1003 
cut short his plans for attempting the recovery of the Holy Land. 
Out of the schools of Charlemagne gradually grew up that 
subtle, minute and over-refined learning of the later Middle Ages 
which has come to be known as Scholasticism. Based as it 
wras upon authority instead of experiment, and magnifying, as it 
did, details more than principles, it sharpened rather than broad- 
ened the intellect, and was indifferent if not unfavorable to science. 
References for Reading 
Lucretius. On the Nature of Things. 
Strabo. Geography. 
Pliny. Natural History. 
Frontinus. The Waterworks of Rome. (Tr. by C. Hersehel.) 
Vitruvius. On Architecture. 
Ker. The Dark Ages. 
Gibbon. Decline and Fall of the Roman Empire. 
Galen. On the Natural Faculties. CHAPTER VIII 
HINDU AND'ARABIAN SCIENCE. THE MOORS IN SPAIN 
The grandest achievement of the Hindus and the one which, 
of all mathematical investigations, has contributed most to the 
general progress of intelligence, is the invention of the principle of 
position in writing numbers. — Cajori. 
Indeed, if one understands by algebra the application of arith- 
metical operations to composite magnitudes of all kinds, whether they 
be rational or irrational number or space magnitudes, then the learned 
Brahmins of Hindustan are the true inventors of algebra. — Hankel. 
In the ninth century the School of Bagdad began to flourish, just 
when the Schools of Christendom were falling into decay in the West 
and into decrepitude in the East. The newly-awakened Moslem in- 
tellect busied itself at first chiefly with Mathematics and Medical 
Science; afterwards Aristotle threw his spell upon it, and an immense 
system of orientalized Aristotelianism was the result. From the 
East, Moslem learning was carried to Spain ; and from Spain Aristotle 
reentered Northern Europe once more, and revolutionized the intel- 
lectual life of Christendom far more completely than he had revolu- 
tionized the intellectual life of Islam. — Rashdall. 
Alexandria fell to the Arabs in 641 a.d. As a matter of his- 
torical perspective it is noteworthy that the interval between its 
foundation by Alexander the Great and its capture by the 
Mohammedans, — during most of which period it was the in- 
tellectual centre of the world, — is almost equal to that between 
Charlemagne’s time and our own. 
The preservation and transmission of portions of Greek science 
through the Dark Ages to the dawn of science in western Europe 
about 1200 a.d. was mainly effected through three distinct, though 
not quite independent, channels. First, there was to a limited 
extent a direct inheritance of ancient learning within the Italian 
peninsula, through all its political and military turmoil. Second, 
a substantial legacy was received indirectly through the Moors 
in Spain; while, third, additions of great importance came later 
156 HINDU, ARABIAN AND MOORISH SCIENCE 
157 
through Italy from Constantinople. Before following the direct 
Latin-Italian line a brief sketch of Hindu and Arabic science is 
desirable. 
Hindu Mathematics. — The far-reaching conquests of Alex- 
ander the Great (330 b.c.) immensely stimulated communica- 
tion of ideas between the Mediterranean world and Asia, and the 
East was able to make certain great contributions to mathematical 
science just where the Greeks were relatively weakest, namely 
in arithmetic and the rudiments of algebra and trigonometry. 
Several centuries before our era the Pythagorean theorem and an 
excellent approximation for were known in India in connection 
with the rules for the construction of altars. The mathematicians 
however from whom we trace the later development of mathematics 
date from the sixth and following centuries. 
About 530 a.d. Arya-bhata wrote a book in four parts dealing 
with astronomy and the elements of spherical trigonometry, and 
enunciating numerous rules of arithmetic, algebra and plane trigo- 
nometry. He gives the sums of the series 
1 -f 2 d- ...-(- 7i 
12 + 22 + ... + n2 
13 + 23 + ... + n3, 
solves quadratic equations, gives a table of sines of successive mul- 
tiples of 3f° — i.e. twenty-fourths of a right angle, — and even 
uses the value 7r = 3.1416, correct to five places. His geometry 
is in general inferior. 
Some years later, Brahmagupta composed a system of as- 
tronomy in verse, with two chapters on mathematics. In this 
he discusses arithmetical progression, quadratic equations, areas 
of triangles, quadrilaterals and circles, volume and surface of 
pyramids and cones. His value of7ris = 3.16+. Typical 
problems and discussions are the following: — 
Two apes lived at the top of a cliff of height 100, whose base was 
distant 200 from a neighboring village. One descended the cliff, and 
walked to the village, the other flew up a height x and then flew in a 158 
A SHORT HISTORY OF SCIENCE 
straight line to the village. The distance traversed by each was the 
same. Find x. 
Beautiful and dear Lilavati, whose eyes are like a fawn’s! tell me 
what are the numbers resulting from one hundred and thirty-five, 
taken into twelve ? if thou be skilled in multiplication by whole or by 
parts, whether by subdivision or form or separation of digits. Tell 
me, auspicious woman, what is the quotient of the product divided by 
the same multiplier? 
The son of Pritha exasperated in combat, shot a quiver of arrows 
to slay Carna. With half his arrows, he parried those of his an- 
tagonist ; with four times the square-root of the quiver-full, he killed 
his horse; with six arrows, he slew Salya; with three he demolished 
the umbrella, standard and bow; and with one, he cut off the head 
of the foe. How many were the arrows, which Arjuna let fly ? 
For the volume contains a thousand lines including precept and 
example. Sometimes exemplified to explain the sense and bearing 
of a rule; sometimes to illustrate its scope and adaptation; one while 
to show variety of inferences; another while to manifest the principle. 
For there is no end of instances; and therefore a few only are exhibited. 
Since the wide ocean of science is difficultly traversed by men of little 
understanding; and, on the other hand, the intelligent have no occa- 
sion for copious instruction. A particle of tuition conveys science to a 
comprehensive mind; and having reached it, expands of its own im- 
pulse. As oil poured upon water, as a secret entrusted to the vile, as 
alms bestowed upon the worthy, however little, so does science infused 
into a wise mind spread by intrinsic force. 
It is apparent to men of clear understanding, that the rule of three 
terms constitutes arithmetic; and sagacity, algebra. Accordingly I 
have said in the chapter of Spherics : 
‘ The rule of three terms is arithmetic; spotless understanding is 
algebra. What is there unknown to the intelligent ? Therefore, for 
the dull alone, it is set forth.’ 
Five centuries later Bhaskara also wrote an astronomy contain- 
ing mathematical chapters, and the contents of this work soon 
became known through the Arabs to western Europe. While the 
preceding writers had no algebraic symbolism, but depended 
laboriously on words and sentences, Bhaskara made considerable 
progress in abbreviated notation. A partial list of subjects, treated HINDU, ARABIAN AND MOORISH SCIENCE 
159 
in his first book, includes weights and measures, decimal numera- 
tion, fundamental operations, addition etc., square and cube root, 
fractions, equations of the first and second degrees, rule of three, 
progressions, approximate value of tt, volumes. Applications are 
made to interest, discount, partnership, and the time of filling 
a cistern by several fountains. While there is reason to believe 
that the decimal system was known as early as the time of Brahma- 
gupta, this work contains the first systematic discussion of it, 
including the so-called Arabic numerals and zero. 
As an intermediate stage between the earlier use of entire words 
and our modern employment of single letters, he employs abbre- 
viations, but multiplication, equality and inequality have still 
to be written out. The divisor is written under the dividend 
without a line, one member of an equation under the other with 
verbal context to insure clearness. Polynomials are arranged in 
powers, though without our exponents, coefficients follow the un- 
known quantities. In his “rules of cipher” he even gives the 
equivalent of a ± 0 = a, 02 = 0, Vo = 0, a -s- 0 = oo. 
In comparison with Greek mathematics, power and freedom 
are gained at the cost of some sacrifice of logical rigor. Among 
the Greeks, only the greatest appreciated the possibility and the 
importance of an unending series of numbers; but the Hindu 
imagination tended naturally in this direction. A notable achieve- 
ment of the Hindus was the introduction of the idea of negative 
numbers and the illustration of positive and negative by assets 
and debts, etc. 
On the whole, the Hindus, having received a part of their 
mathematics originally from the Greeks, made great contributions 
on the arithmetical and algebraic side, their influence on Euro- 
pean science with which they had little or no direct contact being 
exerted mainly through the Arabs. 
The Hindu mathematicians had no interest in what is termed 
mathematical method. They gave no definitions; preserved little 
logical order; they did not care whether the rules they used were 
properly established or not and were generally indifferent to funda- 
mental principles. They never exalted mathematics as a subject 160 
A SHORT HISTORY OF SCIENCE 
of study and indeed their attitude to learning may be described as 
decidedly unmathematical. — G. R. Kaye. 
Hindu Astronomy. — In astronomy a parallel development 
took place. It seems probable that Greek planetary theory 
was introduced into India between the times of Hipparchus and 
Ptolemy, but Hindu astronomy is characterized as “a curious 
mixture of old fantastic ideas and sober geometrical methods of 
calculation.” Aryabhata says indeed “The sphere of the stars 
is stationary, and the earth, making a revolution, produces the 
daily rising and setting of stars and planets,” an opinion rejected 
by the later Brahmagupta. 
Mohammed and the Hegira. — During the sixth and following 
centuries great events were happening in Arabia, an anciently 
settled country, but up to that time a blank in the history of 
civilization and of science. In 569 a.d. or thereabouts was born, 
probably in Mecca, — an insignificant commercial town 45 miles 
from the middle eastern shore of the Red Sea, — that extraordinary 
man Mohammed, whom millions of his fellow men still regard as 
the Prophet of the Almighty (Allah). In 622 Mohammed fled with 
a small company of his disciples to Medina, an agricultural town 
250 miles to the north of Mecca, where he prosecuted his prop- 
aganda, and completed his Koran, — the Mohammedan Bible. 
Here also he died in 632 a.d. 
In Mecca, Mohammed was “the despised preacher of a small 
congregation,” but after his flight (hejira) to Medina, he became 
the leader of a powerful party and ultimately the autocratic 
ruler of Arabia. Even before his death his followers numbered 
thousands, while the religious zeal with wThich they were fired 
has never been surpassed. Taught by Mohammed to convert 
or kill, they threw themselves upon their neighbors with a 
fanatical fury which overcame all obstacles, so that within one 
short century their religion and its adherents swept like a tidal 
wave from the barren valleys of western Arabia northward and 
eastward through Syria over Asia Minor and Mesopotamia, and 
northwestward along the African shores of the Mediterranean to HINDU, ARABIAN AND MOORISH SCIENCE 
161 
the straits of Gibraltar. Egypt, Alexandria, and Carthage fell 
before the Mohammedans, and the Arabian or Moslem empire 
soon rivalled in extent its great predecessor, the Roman. In 711 
Moslems crossed the straits of Gibraltar and entered Spain, 
soon pushing northward into western France as far as Poitiers, 
where their great western and northern movement was finally 
checked by Charles Martel, in 732. This extraordinary onrush, 
occurring almost within a single century, naturally left the Moslems 
little time for the development of learning or for the arts and 
sciences. But after it was over, the Mohammedan invaders 
settled down in their various conquered countries and in some of 
them cultivated the arts of peace. The successive Arabian rulers 
(beginning with Al-Mansur, in 754) patronized learning, and to 
this end collected Greek manuscripts, which, after the closing of the 
Greek schools by Justinian in 529, had become scattered abroad. 
In particular, certain Nestorian Jews were brought to Bagdad and 
by them translations into Arabic were made of some of the works 
of Aristotle, Euclid, Ptolemy, and other Greek authors. The 
learning of India was also drawn upon, especially for the so-called 
Arabic numerals. Thus began a kind of Arabian science, chiefly 
imported at the outset, but destined within the next three centuries 
to take on characteristics of its own. It was, however, under the 
Caliph Al-Mamun (813-833), who has been called, as regards 
schools and learning, the Charlemagne of his people, that Aristotle 
was first translated into Arabic. Al-Mamun caused works on 
mathematics, astronomy, medicine, and philosophy to be translated 
from the Greek, and founded in Bagdad a kind of academy called 
the “House of Science,” with a library and an observatory. 
Arabian Mathematical Science. — While the Arabs them- 
selves were not in general much addicted to scientific pursuits, 
their relations to the Greeks and Hindus, and subsequently to the 
nations of western Europe are of very great importance in the 
history of science. Even if we accept as typical the traditional 
dictum attributed to the Caliph Omar, that whatever in the library 
of Alexandria agreed with the Koran was superfluous, whatever 
disagreed was worse, and all should therefore be destroyed, it was 162 
A SHORT HISTORY OF SCIENCE 
inevitable that individuals in this active-minded race should fall 
under the spell of Greek mathematical science. Their religion 
was in fact more tolerant towards science than was contemporary 
Christianity. 
It would appear that by 900 a.d. the Arabs were familiar on the 
one hand with Brahmagupta’s arithmetic and algebra, including 
the decimal system, and on the other hand with the chief works of 
the great Greek mathematicians, some of which have come down 
to us only through Arabic translations. 
The Algebra of Alkarismi written about 830 was based on the 
work of Brahmagupta, and served in turn as the foundation for 
many later treatises. From its title is derived our word “ algebra,” 
from the author’s name our “algorism.” The book begins: — 
The love of the sciences with which God has distinguished Al- 
Mamun, ruler of the faithful, and his benevolence to scholars, have 
encouraged me to write a short work on computation by completion 
and reduction. Herein I limited myself to the simplest matters, and 
those which are most needed in problems of distribution, inheritance, 
partnership, land measurement, etc. 
The first book contains a discussion of five types of quadratic 
equations: 
ax2 = bx, ax2 = c, ax2 + bx = c, ax2 + c = bx, ax2 = bx + c; 
only real positive roots are accepted, but, unlike the Greeks, he 
recognizes the existence of two roots. He gives a geometri- 
cal solution of the quadratic equation 
analogous to those of Euclid. Suppose 
x2 + 10z = 39 and let AB = BC = x, 
AH = CF = 5; then the areas are 
AC = x2, AK = BF = bx. 
The sum of these is x2 + 10x. Com- 
plete the square HF by adding KE = 25. 
HF = (x + 5)2 = 64, whence x = 3.' 
The series ln + 2n + 3n -|- ... + mn was summed for n = 1,2,3,4, 
and about 1000 a.d. Alkayami is said to have asserted the im- 
Dossibilitv of finding two cubes whose sum should be a cube. Even HINDU, ARABIAN AND MOORISH SCIENCE 
163 
a cubic equation was solved by the aid of intersecting conic sec- 
tions. There is no definite separation of algebra and arithmetic, 
and the former, in spite of relatively rapid development, remains 
entirely rhetorical. The division line of fractions is introduced 
and the check of computation by “casting out nines.” 
In Physics, Al-Hazen (965? — 1038) wrote a work on optics 
enunciating the law of reflection and making a study of spherical 
and parabolic mirrors. He also devised an apparatus for studying 
refraction, being probably the first physicist to note the magnify- 
ing power of spherical segments of glass — i.e. lenses. He gave 
a detailed account of the human eye, and attempted to explain 
the change of apparent shape of the sun and the moon when ap- 
proaching the horizon. The Arabs employed the pendulum for 
time measurement, and tabulated specific gravities of metals, etc. 
In the words of a modern physicist: — 
The Arabs have always reproduced what came down to them from 
the Greeks in thoroughly intelligible form, and applied it to new prob- 
lems, and thus built up the theorems, at first only obtained for par- 
ticular cases, into a greater system, adding many of their own. They 
have thus, rendered an extraordinarily great service, such as would 
correspond in modern times to the investigations which have grown 
out of the pioneer work of such men as Newton, Faraday and Rontgen. 
— Wiedemann. 
Arabian Astronomy. — In connection with astronomy the 
Arabs, following Greek precedents, developed trigonometry, in- 
troducing sines and other functions since current. They used 
masonry quadrants of large size, and even a combination of a 
horizontal circle with two revolving quadrants mounted upon it, 
foreshadowing the modern theodolite. Better and more com- 
plete observational data facilitated, and at the same time de- 
manded, improved mathematical methods, while the necessary 
computations were accomplished much more economically, through 
the use of the decimal number system. Haroun Al-Raschid sent 
to Charlemagne an ingenious water clock, while under his suc- 
cessor, Al-Mamun, two learned mathematicians were commis- 
sioned to measure a degree of the earth’s circumference. 164 
A SHORT HISTORY OF SCIENCE 
* Choose a place in a level desert and determine its latitude. Then 
draw the meridian line and travel along it towards the pole-star. 
Measure the distance in yards. Then measure the latitude of the 
second place. Subtract the latitude of the first and divide the differ- 
ence into the distance of the places in parasangs. The result multi- 
plied by 360 gives the circumference of the earth in parasangs.’ 
— Wiedemann. 
The writer just quoted describes a second method involving 
the measurement of the angle of depression of the horizon as 
seen from the top of a high mountain. 
It is not improbable that western Europe acquired from 
eastern Asia, through Arab channels, the mariner’s compass and 
gunpowder. 
‘ When the night is so dark that the captains can perceive no star to 
orient themselves, they fill a vessel with water and place it in the in- 
terior of the ship, protected from wind; then they take a needle and 
stick it into a straw, forming a cross. They throw this upon the water 
in the vessel mentioned and let it swim on the surface. Hereupon they 
take a magnet, put it near the surface of the water, and turn their 
hands. The needle turns upon the water; then they draw their 
hands suddenly and rapidly back, whereupon the needle points in 
two directions, namely north and south.’ 1232 a. d. 
— Wiedemann. 
The astronomical theory of the Arabs was merely that of 
Ptolemy. But they “were not content to consider the Ptolemaic 
system merely as a geometrical aid to computation; they required 
a real and physically true system of the world, and had therefore 
to assume solid crystal spheres after the manner of Aristotle.” 
The various attempts to devise a better system all miscarried, 
their authors having no new guiding principle, nor superior mathe- 
matical power, and being more or less hampered by Aristotelian 
traditions, though Greek theories of the rotation of the earth 
seem not to have been unknown. 
Asiatic Observatories. — Besides the work of the Arabian 
astronomers themselves, it is an interesting fact that their bar- 
barian Mongol conquerors in the East acquired a temporarily HINDU, ARABIAN AND MOORISH SCIENCE 
165 
active scientific interest, founding a fine observatory at Meraga 
near the northwest frontier of modern Persia. The instruments 
used here are said to have been superior to any used in Europe until 
the time of Tycho Brahe in the sixteenth century. The principal 
achievement of this observatory was the issue of a revised set of 
astronomical tables for computing the motions of the planets, 
together with a new star catalogue. The excellence of their work 
may be inferred from a determination of the precession of the 
equinoxes within 1". This development lasted only a few years 
in the latter half of the thirteenth century. A similar brief out- 
burst of astronomical activity occurred among the Tartars at 
Samarcand (Russian Turkestan) nearly 200 years later, that is, a 
little before the time of Copernicus, and here the first new star 
catalogue since that of Ptolemy was compiled. It is noteworthy 
that there was no hostility between science and the Moham- 
medan church. One of the uses of astronomy indeed was to de- 
termine the direction of Mecca. 
No great original idea can be attributed to any of the Arab and 
other astronomers here discussed. They had, however, a remark- 
able aptitude for absorbing foreign ideas, and carrying them slightly 
further. They were patient and accurate observers, and skilful 
calculators. We owe to them a long series of observations, and 
the invention or introduction of several important improvements 
in mathematical methods. . . . More important than the actual 
contributions of the Arabs to astronomy was the service that they 
performed in keeping alive interest in the science and preserving the 
discoveries of their Greek predecessors. — Berry. 
The Moors in Spain.—We have already touched above upon 
the rapid spread of Mohammedanism westward from its home 
in Arabia, and the remarkable conquests of its followers in Spain 
and western France. These western Mohammedans included 
not only some of pure Arabian stock, but more of mixed descent, 
especially from that part of northern Africa once known as Maure- 
tania, — whence the term Moors, generally applied to the con- 
quering Mohammedans of the west. The Moors entered Spain 166 
A SHORT HISTORY OF SCIENCE 
early in the eighth century, bringing after them the learning of the 
Arabs, so that Hindu and Arabian science, and to some extent 
Greek science, were making their way into southwestern Europe 
even before the schools of Charlemagne were established toward 
the end of the same century in central (Christian) Europe. In the 
ninth and tenth centuries a remarkable civilization arose in Spain, — 
the highest that the Arabian race has ever reached. The develop- 
ments of science in Mohammedan Spain are more or less typical 
of what occurred throughout the whole Arabian empire, and in 
such cities as Cordova, Toledo, and Seville a type of civilization 
and a stage of learning were reached higher in many respects than 
existed at the same time and even for centuries afterward any- 
where in Christian Europe. 
Scarcely had the Arabs become firmly settled in Spain when they 
commenced a brilliant career. Adopting what had now become the 
established policy of the Commanders of the Faithful in Asia, the 
Emirs of Cordova distinguished themselves as patrons of learning, 
and set an example of refinement strongly contrasting with the con- 
dition of the native European princes. Cordova, under their ad- 
ministration, at its highest point of prosperity, boasted of more than 
two hundred thousand houses, and more than a million of inhabitants. 
After sunset, a man might walk in a straight line for ten miles by the 
light of the public lamps. Seven hundred years after this time there 
was not so much as one public lamp in London. Its streets were 
solidly paved. In Paris, centuries subsequently, whoever stepped over 
his threshold on a rainy day stepped up to his ankles in mud. 
— Draper. 
The Mohammedans made some additions to medical science, ; 
and yet their medicine hardly goes beyond that of Galen, whom 
they specially revered. In alchemy they are notable, though 
more by their attempts than their achievements. Too often 
it was simply a search for “potable gold” or other “ elixirs of 
life,” “the philosopher’s stone,” and the like. In the arts and 
industries, however, the Moors deserve special mention. Cordovan 
and Morocco leather are well known. Toledo and Damascus 
blades (swords) were long famous. Arabian horses fur- HINDU, ARABIAN AND MOORISH SCIENCE 
167 
nished Europe with one of the most serviceable strains of that 
useful animal, and many Arabian words have been adopted 
into our language, e.g. alcohol, elixir, algebra, alembic, zenith, 
nadir, etc. 
“. . . Under the caliphs, Moslem Spain became the richest, most 
populous, and most enlightened country in Europe. The palaces, 
the mosques, bridges, aqueducts, and private dwellings reached a 
luxury and beauty of which a shadow still remains in the great mosque 
of Cordova. New industries, particularly silk weaving, flourished 
exceedingly, 13,000 looms existing in Cordova alone. Agriculture, 
aided by perfect systems of irrigation for the first time in Europe, 
was carried to a high degree of perfection, many fruits, trees and 
vegetables hitherto unknown being introduced from the East. Mining 
and metallurgy, glass making, enamelling, and damascening kept whole 
populations busy and prosperous. From Malaga, Seville, and Almeria 
went ships to all parts of the Mediterranean loaded with the rich 
produce of Spanish Moslem taste and industry, and of the natural and 
cultivated wealth of the land. Caravans bore to farthest India and 
darkest Africa the precious tissues, the marvels of metal work, the 
enamels, and precious stones of Spain. All the luxury, culture, and 
beauty that the Orient could provide in return, found its way to the 
Moslem cities of the Peninsula. The schools and libraries of Spain were 
famous throughout the world; science and learning were cultivated 
and taught as they never had been before. Jew and Moslem, in the 
friendly rivalry of letters, made their country illustrious for all time 
by the productions of their study. . . . The schools of Cordova, 
Toledo, Seville, and Saragossa attained a celebrity which subsequently 
attracted to them students from all parts of the world. At first the 
principal subjects of study were literary, such as rhetoric, poetry, 
history, philosophy, and the like, for the fatalism of the faith of Islam 
to some extent retarded the adoption of scientific studies. To these, 
however, the Spanish Jews opened the way, and when the barriers were 
broken down, the Arabs themselves entered with avidity into the 
domain of science. Cordova then became the centre of scientific 
investigation. Medicine and surgery especially were pursued with 
intense diligence and success, and veterinary surgery may be said to 
have there first crystallized into a science. Botany and pharmacy 
also had their famous professors, and astronomy was studied and 168 
A SHORT HISTORY OF SCIENCE 
taught as it had never been before; algebra and arithmetic were 
applied to practical uses, the mariner’s compass was invented, and 
science as applied to the arts and manufactures made the products of 
Moslem Spain — the fine leather, the arms, the fabrics, and the metal 
work — esteemed throughout the world. . . . Canals and water 
wheels for irrigation carried marvellous fertility throughout the south 
of Spain, where the one thing previously wanting to make the land a 
paradise was water. Rice, sugar, cotton, and the silkworm were all 
introduced and cultivated with prodigious success; the silks, brocades, 
velvets, and pottery of Valencia, the beautiful damascened steel of 
Seville, Toledo, Murcia, and Granada, the stamped embossed leather 
of Cordova, and the fine cloths of Seville brought prosperity to 
Moslem and Mozarab alike under the rule of the Omeyyad caliphs, 
while the systematic working of the silver mines of Jaen, the corals on 
the Andalusian coasts, and the pearls of Catalonia supplied the ma- 
terial for the lavish splendor which the rich Arabs affected in their 
attire and adornment. 
The Moors of Andalusia and Valencia acclimatized and cultivated 
a large number of semitropical fruits and plants hitherto little known 
in Europe, and studied arboriculture and horticulture not only practi- 
cally but scientifically. The famous work on the subject by Abu 
Zacaria Al-Awan was the foundation of such books, and of the applica- 
tion of science to gardening. It was mainly derived from Chaldean, 
Greek, and Carthaginian manuscripts now lost. Curiously, Spain had 
produced under the Romans a famous book on agriculture by Colu- 
mella; but for scientific knowledge it cannot be compared to the 
Treatise on Agriculture by Abu Zacaria. . . . From the earliest 
times the wool of Spain had been the finest in the world. . . . Vast 
herds of stunted, ill-looking, but splendidly fleeced sheep belonged to 
the nobles and ecclesiastical lords, and quite early in the period of re- 
conquest, when these classes were all-powerful, a confederacy of sheep 
owners was formed, which by the fourteenth and fifteenth centuries 
had developed into a corporation of immense wealth. This was called 
the Mesta. . . . The fleeces were extremely fine, often weighing 
12 pounds per animal, and the wool was sought after throughout the 
world, especially by Flemish and French cloth workers. Even in the 
ninth century Spanish wool was famous in Persia and in the East; 
and as early as the time of the Phoenicians it was considered the 
finest in the world. — Hume. HINDU, ARABIAN AND MOORISH SCIENCE 
169 
The tenth century was the golden age of Moorish science in 
Spain. Another hundred years and it had gone down forever. 
Its permanent importance, even in conserving the work of the 
ancients, has been questioned, and a recent writer cleverly com- 
pares the whole western movement of the Arabians to the sands 
of their deserts, — now fierce and pitiless when driven by some 
force such as the wind, — now sinking into inert, helpless, infertile 
heaps when left to themselves. 
An Arab renaissance as early as the eighth century had revived 
something of classic knowledge. The poetry and philosophy of Greece 
were studied, and the taste for learning was cultivated with the en- 
thusiasm which the Arabs infused into all their undertakings. Every 
one knows the fascinating account of these things in the pages of 
Gibbon. How the tide of progress flowed from Samarcand and Bo- 
khara to Fez and Cordova. How a vizir consecrated 200,000 pieces of 
gold to the foundation of a college. How the transport of a doctor’s 
books required four hundred camels. How a single library in Spain 
contained 600,000 volumes, while seventy public libraries were opened 
in Andalusia alone. How the Arabian schools of Spain and Italy 
were resorted to by scholars from every country in Europe. . . . 
Here was a state of luxury and learning which contrasted strongly 
enough with the barbarism of the age. But under all this show where 
was the substantial basis? How much of all this was real? Arab 
architecture, in so far as it was Arab, and not built for them by the 
Greeks, was a concoction of whim and fantasy. In those nervous 
hands every strong and simple feature was distorted into endless com- 
plications, and, as always happens, lost in stability what it gained 
in eccentricity. Their learning was of the same character. Though 
they disputed interminably on the rival merits of the Greek philos- 
ophers, they were content to receive all their knowledge of them 
through indifferent translations. When the real revival of learning 
came, and a genuine Renaissance set in, the six or seven centuries of 
Arab civilization were simply ignored and passed over. . . . 
The Arab mind seems to turn by a sort of instinct to the occult, 
the mystical, the fantastic. It is always sighing for new worlds to 
conquer before it has made good the ground it stands on. It has the 
curious gift of turning everything it touches from substance to shadow. 170 
A SHORT HISTORY OF SCIENCE 
Astronomy changes into astrology, and the main business of the science 
becomes the casting of horoscopes. The study of medicine changes 
into the composition of philtres and talismans and the reciting of incan- 
tations. Chemistry changes into a search for the secret of the trans- 
mutation of metals and the elixir of immortal health. In short, the 
tendency always was to shift the appeal from the intellect and reason 
to the fancy and imagination; and their zeal, instead of being devoted 
to laying firm foundations, evaporated in vague aspirations after the 
unintelligible or the unobtainable. . . . 
And the consequence is that not only has the Arab left us little 
or nothing, but his wrhole history seems already more legendary than 
real. Other civilizations abide our question. Not the Greek and 
Roman only, but the remote Assyrian and Egyptian, are definite and 
real in comparison with the Arabian. This seems of another texture. 
It is such stuff as dreams are made of. 
Those so-called conquests of his [the Arab’s] were really the taking 
advantage of a unique opportunity for destroying and pulling down. 
The collapse of the Western Empire, and weakness and paralysis of 
the Eastern, afforded the Arab a fine field for the display of his peculiar 
proweSs. He took to the lumber and debris of these crumbling empires 
as fire takes to rotten wood. But if in the void that separates ancient 
civilisation from modern the Arab appears to advantage, there no 
sooner entered on the scene nations of solid character and creative 
genius than he retired before them, and yielded to their advance. 
— March Phillips. 
The Golden Age of Moorish learning in the tenth century came 
and went, leaving behind it singularly few permanent results. 
Owing to the racial and religious hatreds of the time the Christian 
conquerors of the Moslems, like their Roman prototypes in the 
first few centuries after Christ, had small respect for Greek — and 
less for Mohammedan — learning. Hence, doubtless, it came 
about that to-day in Cordova, for example, almost no traces re- 
main of that Arabian learning of which it was once the celebrated 
seat. Even the site of its illustrious university has faded from 
memory and only its great mosque (of which the heart is occupied 
by a Christian church) remains to bear visible witness to Moham- 
medan Cordova. The same is true of other once famous centres HINDU, ARABIAN AND MOORISH SCIENCE 
171 
of Spanish Mohammedan-Greek learning. Toledo still possesses 
some of its Arabian walls and gateways, and Seville its lovely 
Giralda — “ the first astronomical observatory in Europe ” — and 
its Tower of Gold; but it is only in the Alhambra of Granada 
that any adequate vision can be had of Mohammedan life and 
influence in Spain. Here the quiet, the seclusion, the rich orna- 
mentation, and the music of abundant running waters, still com- 
municate an impression of wealth, taste, and power, and suggest 
possibilities of uninterrupted study and an intellectual life. Else- 
where, evidences of the Mohammedan love of inquiry, of libraries, 
of decoration, and even of fruits and gardens, have been almost 
wholly blotted out. 
References for Reading 
Ball. Chapter IX. 
Berry. Chapter III. 
Draper. History of the Intellectual Development of Europe. Vol. II. 
Dreyer. Chapter XI. 
Hume, M. History of the Spanish People. 
March Phillips. In the Desert. 
Gibbon. Decline and Fall. CHAPTER IX 
PROGRESS OF SCIENCE TO 1450 A.D. 
It cannot be too emphatically stated that there is no historical 
evidence for the theory which connects the new birth of Europe with 
the passing away of the fateful millennial year and with it, the awful 
dread of a coming end of all things. Yet, although there was no breach 
of historical continuity at the year 1000, the date will serve as well as i 
any other that could be assigned to represent the turning-point of 
European history, separating an age of religious terror and theological 
pessimism from an age of hope and vigor and active religious en- 
thusiasm. . . . The change which began to pass over the schools of 
France in the eleventh century and culminated in the great intellectual 
Renaissance of the following age, was but one effect of that general 
revivification of the human spirit which should be recognized as con- 
stituting an epoch in the history of European civilization not less 
momentous than the Reformation or the French Revolution. . . . 
The schools of Christendom became thronged as they were never 
thronged before. A passion for inquiry took the place of the old 
routine. The Crusades brought different parts of Europe into con- 
tact with one another and into contact with the new world of the East, 
— with a new Religion and a new Philosophy, with the Arabic Aris- 
totle, with the Arabic commentators on Aristotle, and eventually even 
with Aristotle in the original Greek. . . . Whatever the causes of 
the change, the beginning of the eleventh century represents, as nearly 
as it is possible to fix it, the turning-point in the intellectual history of 
Europe. — Rashdall. 
The Crusades. — From the time of Mohammed’s hegira 
from Mecca to Medina in 622 a.d. to the siege of Vienna by his 
followers in 1683 — a period of more than 1000 years — Europe 
stood in constant dread of Mohammedan conquest. Fifteen years 
after the hegira, Jerusalem was captured by Omar, and remained 
under Mohammedan control till the end of the first Crusade, since 
which time it has been sometimes in Christian, sometimes in 
Mohammedan, possession. Toleration of Christians in the Holy 
172 PROGRESS OF SCIENCE TO 1450 A.D. 
173 
Land was, however, the rule until the eleventh century, and 
between 700 and 1000 a.d. pilgrimages to Jerusalem were fre- 
quently undertaken by Christians in the West. But after 1010 
such pilgrimages began to be seriously interfered with, and matters 
steadily grew worse, until in 1071 Seljukian Turks displaced 
Arabian Mohammedans as rulers of Jerusalem. These Turks, 
though more rough than intolerant, eventually interfered with both 
trade and pilgrimages, until for this and other reasons the conquest 
of the Holy Land became a passion with the Christian nations. 
In the spring of 1097, after several years of widespread prepara- 
tion, a great host of western Christians, variously estimated at 
150,000 to 600,000, gathered at Constantinople charged with war- 
like and religious zeal and bent on wresting Jerusalem and the 
Holy Land from the possession of the Mohammedan “Infidel.” 
This was the beginning of those expeditions under the banner of 
the Cross, — hence known as the Crusades, — which may be 
regarded as intermittent reactions of the Christian West against 
the pressure of the Mohammedan East. The spirit of Christian 
Europe in the Middle Ages being essentially religious and ec- 
clesiastical, it was natural that its more bold and adventurous 
youth should regard with jealousy and indignation the wide 
extent of the Mohammedan empire and especially its possession 
of Jerusalem and other holy places. In all, eight such Crusades 
are recognized by historians, and of these the influence upon 
Christian Europe must have been immense. In the first place, 
the expansion of the intellectual outlook due to the mere experiences 
of travel, for men born and bred under the parochial limitations 
of feudalism and monasticism, must have been great. Then, too, 
the arts and appliances observed abroad, the different stand- 
ards of all sorts, the wealth and luxury of the distant East, doubt- 
less had a powerful effect upon Europe when reported or intro- 
duced by the Crusaders upon their return. When we reflect upon 
the ages of darkness which had rested upon Christian Europe from 
the fall of Rome into the hands of the barbarians to the fall of 
Jerusalem into the hands of the Turks — a period of almost 
exactly six hundred years — we may agree with those who are 174 
A SHORT HISTORY OF SCIENCE 
disposed to look upon the Crusades as an age of discovery com- 
parable with that of the new world by Columbus and his follow- 
ers,— but a discovery of the East instead of the West. 
The period of the Crusades extends over about two centuries, 
viz: — from 1090 to 1290, and thus immediately precedes the 
Renaissance, of which it was apparently one of the most im- 
portant factors. 
Trivium : Quadrivium. Scholasticism. — Meantime, follow- 
ing the mandate of Charlemagne establishing schools in connection 
with all the abbeys and monasteries of his vast domain of central 
Europe, a characteristic technical and essentially verbal scholarship 
gradually arose which, although chiefly ecclesiastical in substance, 
and so narrow in its range as almost completely to neglect natural 
science, was often thorough and sometimes profound. This 
learning in its later development is known as “Scholasticism,” of 
which the foundation and essence was the famous curriculum of 
“the seven liberal arts,” founded upon the educational doctrines 
of Plato, but adapted to the fashion of the Middle Ages. These 
consisted of a quadrivium — geometry, astronomy, music and arith- 
metic— and a trivium — grammar, logic, and rhetoric, (p. 148.) 
In the introduction to the Logic of Aristotle which was in the 
hands of every student even in the Dark Ages, the Isagoge of Por- 
phyry, the question was explicitly raised in a very distinct and 
emphatic manner. The words in which this writer states, without 
resolving, the problem of the Scholastic Philosophy, have played per- 
haps a more momentous part in the history of thought, than any 
other passage of equal length in all literature outside the canonical 
Scriptures. They are worth quoting at length: 
‘Next, concerning genera and species, the question indeed whether 
they have a substantial existence, or whether they consist in bare 
intellectual concepts only, or whether, if they have a substantial ex- 
istence, they are corporeal or incorporeal, and whether they are sep- 
arable from the sensible properties of the things (or particulars of 
sense), or are only in those properties and subsisting about them, I 
shall forbear to determine. For a question of this kind is a very deep 
one and one that requires a longer investigation.’ — Rashdall. PROGRESS OF SCIENCE TO 1450 A.D. 
175 
To show the low state of natural history it suffices to refer to 
an extraordinary work, the so-called Physiologus or Bestiary, a 
kind of scriptural allegory of animal life, originally Alexandrian, 
but surviving in mutilated forms and widely used in medieval 
times. The childish and grotesque character of this curious 
compendium shows how ill-adapted were the centuries of crusad- 
ing to the calm pursuits of science; they were indeed almost barren 
in this direction. 
Scholasticism, nevertheless, lingered long after the Crusades 
were ended, and abundant survivals of it exist even today. 
Medieval Universities. — The origin of the universities which 
play so great a part in the cultivation and dissemination of learn- 
ing in the later middle ages is involved in obscurity. The medical 
school at Salerno in southern Italy seems to have become known 
in the ninth century, so that the University of Salerno is some- 
times called the oldest in Europe. It was still famous in the 
thirteenth century. The law school at Bologna, in northern 
Italy, became well known about 1000 a.d., though the date of 
the University of Bologna is usually given as near the end of the 
twelfth century. The University of Paris is often dated from 
the early part of the same century. None of these early univer- 
sities was much more than an association or gild of masters and 
pupils. Laboratories for instruction were of course unknown. 
In the eleventh and twelfth centuries there was a gradual de- 
velopment from the previous monastic schools to the beginnings 
of modern universities at Paris, Bologna, Salerno, Oxford, and 
Cambridge, the schools themselves however continuing along 
their previous lines; and from that time onward to our own, the 
universities have played the chief part in the advancement of 
learning in general and of science in particular. In their develop- 
ment theological influences were naturally dominant, and it is 
interesting to observe that the use of Aristotle’s Natural Philoso- 
phy, which became later the stronghold of orthodox conserva- 
tism, was prohibited in the thirteenth century. 
Medieval academic standards were naturally low. The univer- 
sity was a voluntary and privileged society of scholars. Not until 176 
A SHORT HISTORY OF SCIENCE 
1426 is there a record of the refusal of a degree for poor scholarship, 
and the victim then sought redress by legal proceedings, though 
in vain. In most of the early universities logic, philosophy, and 
theology were cultivated rather than even mathematical science. 
Transmission of Science through Moorish Spain. — The 
meagre rivulet of classical science derived directly from Greek 
and Roman sources is now mingled with the current which found 
its way through northern Africa and Spain under the Moors. 
Boethius’ rudimentary work was supplanted, and before 1400, 
the first five books of Euclid were taught at many universities. 
Ptolemy’s Almagest was also translated from the Arabic into 
Latin early in the twelfth century, probably with the use of 
Arabic numerals. Near the close of the Moorish domination of 
Spain, King Alfonso X of Castile (1223-1284) collected at Toledo 
a body of Christian and Jewish scholars who under his direc- 
tion prepared the celebrated Alfonsine Tables, using the new 
Arabic numerals. These enjoyed a high reputation for three 
centuries, though first printed in 1483. 
While we thus owe to the Arabs a considerable debt for pre- 
serving for the use of later ages the precious heritage of Greek 
learning, the revival of learning in the fourteenth century came 
chiefly from other quarters and would probably have come in due 
time even if Arabic influences had not been at work. Yet it is 
noteworthy that early in the twelfth century re-translations of the 
Greek classics began to be made from the Arabic, and these may 
well have supplied the very limited demand for them tolerated by 
the church for the next hundred years. In spite of jealous ex- 
clusiveness the learning of the great schools of Granada, Cordova, 
and Seville gradually found its way to Paris, Oxford, and Cam- 
bridge. 
During the course of the twelfth century a struggle had been going 
on in the bosom of Islam between the Philosophers and the Theo- 
logians. It was just at the moment when, through the favor of the 
Caliph Al-Mansur, the Theologians had succeeded in crushing the 
Philosophers, that the torch of Aristotelian thought was handed on to 
Christendom. . . . PROGRESS OF SCIENCE TO 1450 A.D. 
177 
It was from this time and from this time only (though the change 
had been prepared in the region of pure Theology by Peter the Lom- 
bard) that the Scholastic Philosophy became distinguished by that 
servile deference to authority with which it has been in modern times 
too indiscriminately reproached. And the discovery of the new Aris- 
totle was by itself calculated to check the originality and speculative 
freedom which, in the paucity of books, had characterized the active 
minds of the twelfth century. The tendency of the sceptics was to 
transfer to Aristotle or Averroes the authority which the orthodox 
had attributed to the Bible and the Fathers of the Church. 
— Raskdall. 
Dawn of the Renaissance. — In the thirteenth century 
it becomes plain that a new spirit is arising in Europe. We 
cannot fail to detect at this time the existence, even at places as 
far apart as Oxford and Bologna — infinitely further apart then 
than now, — of a widespread desire for knowledge and a zeal 
for learning such as had not been known for centuries. Arabic 
mathematical science is introduced from northern Africa by 
Leonardo Pisano. A fresh and notable philosopher — Albertus 
Magnus — appears. Thomas Aquinas writes his famous Imi- 
tatio Christi. Great Gothic cathedrals arise, more universities are 
founded, and, most noteworthy of all for the history of science, 
an original student of nature appears, in Roger Bacon. 
By the beginning of the thirteenth century, in consequence of 
the opening up of communications with the East — through inter- 
course with the Moors in Spain, through the conquest of Constanti- 
nople, through the Crusades, through the travels of enterprising scholars 
— the whole of the works of Aristotle were gradually making their 
way into the Western world. Some became known in translations 
direct from the Greek; more in Latin versions of older Syriac or 
Arabic translations. And now the authority which Aristotle had 
long enjoyed as a logician — nay, it may almost be said the authority 
of logic itself — communicated itself in a manner to all that he wrote. 
Aristotle was accepted as a well-nigh final authority upon Meta- 
physics, upon Moral Philosophy, and with far more disastrous results 
upon Natural Science. The awakened intellect of Europe busied 178 
A SHORT HISTORY OF SCIENCE 
itself with expounding, analysing and debating the new treasures 
unfolded before its eyes. . . . 
And of the scientific side of this revival Italy was the centre. 
This branch of the movement began, indeed, before the twelfth 
century. It was in Italy that the Latin world first came into con- 
tact with the half-forgotten treasures of Greek wisdom, with the 
wisdom which the Arabs had borrowed from the Greeks and with 
original products of the remoter East. Of the Medical School of Sa- 
lerno we have already spoken. It was probably in Italy and through 
the Arabic that the Englishman Adelard of Bath translated Euclid 
into Latin during the first half of the eleventh century. At about 
the same time modern musical notation originated with the discoveries 
of the Camaldulensian monk, Guido of Arezzo. In the first years of 
the following century the Algebra and the Arithmetic which the 
Arabs had borrowed from the Hindus were introduced into Italy 
by the Pisan merchant, Leonardo Fibonacci. ... It was to this 
Arabo-Greek influence that Bologna owed its very important School 
of Medicine and Mathematics—two subjects more closely connected 
then than now through their common relationship to Astrology. 
— Rashdall. 
Mathematical Science in the Thirteenth Century. — 
Increasing activity in mathematical science was due largely to 
Leonardo Pisano of Italy, Jordanus Nemorarius of Saxony, and 
Roger Bacon of England. 
Leonardo Pisano or Fibonacci (born 1175) was educated in 
Barbary, where his father was in charge of the custom-house, and 
thus became familiar with Alkarismi’s algebra, and the Arabic 
decimal system. He appreciated their advantages and on his 
return to Italy published in his Liber Abaci an account which 
gave them currency in Europe “in order that the Latin race 
might no longer be deficient in that knowledge.” As the mathe- 
matical masterpiece of the Middle Ages, it remained a standard 
for more than two centuries. His algebra is rhetorical, but gains 
by the employment of geometrical methods. He discusses the 
fundamental operations with whole numbers and fractions, using 
the present line for division. Fractions are decomposed into parts 
with unit numerators as in early Egypt. Through the Arabs PROGRESS OF SCIENCE TO 1450 A.D. 
179 
Leonardo inherits Egyptian as well as Greek traditions, for ex- 
ample, the type of fraction just mentioned, square and cube root, 
progressions, the method of false assumption. It would appear 
that when the Arabs conquered Alexandria some of the old 
Egyptian culture was preserved. The rule of three, partner- 
ship, powers and roots, and the solution of equations are also 
included. 
In 1225 the emperor, impressed by the accounts of Pisano’s 
mathematical power, arranged a mathematical tournament of 
which the challenge questions are preserved: 
‘ To find a number of which the square, when either increased or 
diminished by 5, would remain a square. 
‘To find by the methods used in the tenth book of Euclid a line 
whose length x should satisfy the equation x3 + 2x2 + 10a: = 20. 
‘ Three men, A, B, C, possess a sum of money u, their shares being 
in the ratio 3:2:1. A takes away x, keeps half of it, and deposits 
the remainder with D; B takes away y, keeps § of it, and deposits 
the remainder with D; C takes away all that is left, namely z, keeps 
§ of it, and deposits the remainder with D. This deposit is found 
to belong to A, B, and C in equal proportions. Find u, x, y and z.’ 
Leonardo gave a correct solution of the first and third, also a root 
of the cubic equation correct to nine decimals. — Ball. 
Jordanus Nemorarius wrote important Latin works on arith- 
metic, geometry, and astronomy. His De Triangulis — the most 
important of these — consists of four books dealing not only 
with triangles, but with polygons and circles. He generally uses 
Arabic numerals, and denotes quantities known or unknown by 
letters. He solves the problem of finding two numbers having a 
given sum and product, by a method equivalent to our elemen- 
tary algebra. This is practically the first European syncopated 
algebra, but seems to have become too little known to have far- 
reaching results in a time not yet ripe for this invention. A book 
on Weights contains elements of mechanics. 
Albertus Magnus, born near the end of the twelfth century, 
became an ardent champion of the newly discovered but pro- 
scribed works of Aristotle. In particular he interpreted the Milky 180 
A SHORT HISTORY OF SCIENCE 
Way as an accumulation of small stars, and ridiculed the current 
objections to antipodes, striving, however, always to harmonize 
the ancient science with the theology of his church. 
Two Oxford scholars, John of Holywood (Sacrobosco) and Roger 
Bacon, have next to be mentioned. Sacrobosco lectured at Paris 
on arithmetic and algebra, and wrote standard books on the former 
with rules but no proofs, and an astronomy of which more than 
sixty editions were afterwards printed. 
Roger Bacon (1214-1294?). — In the history of natural science 
one thirteenth century name stands out before all others, viz.: 
that of Roger or “Friar” Bacon, a member of the Franciscan 
order, born at Ilchester, England, in 1214. He was a pupil of 
Robert Grosseteste “ who had especially devoted himself to 
mathematics and experimental science,” and had studied the 
works of the Arabian authors. Bacon also travelled abroad and 
studied at the University of Paris, — at that time the centre 
of European learning. Here he took the degree of Doctor of 
Theology and probably also here became a Franciscan friar. He 
taught at Oxford, where he had a kind of laboratory for alchemical 
experiments. Doubtless it was for this that he became reputed 
as a worker in “magic” and the “black arts,” for in 1257 he wTas 
forbidden by the head of his order to teach, and was sent to Paris, 
where he underwent great privations. In 1266 he was invited 
by Pope Clement IV to prepare and send to him a treatise on the 
sciences, and within 18 months he had written and sent three 
important works — his Opus Majus, Opus Minus, and Opus 
Tertium. In 1268 he returned to Oxford and there composed 
several more works, but under a later Pope his books were con- 
demned and he was thrown into prison where he remained until 
about a year before his death. 
In Paris, Bacon devoted himself particularly to physical science 
and mathematics. His Opus Majus (1267) contains both a 
summary of ancient and current physical science, and a philosophy 
of learning based on Greek, Roman, and Arabic authorities. He 
insisted that natural science must have an experimental basis, 
and that astronomy and the physical sciences must be founded on PROGRESS OF SCIENCE TO 1450 A.D. 
181 
mathematics, “the alphabet of all philosophy.” On the other 
hand he says: — 
We must consider that words exercise the greatest influence. Al- 
most all wonders are accomplished through speech. In words the 
highest enthusiasm expresses itself. Therefore words, deeply thought 
. . . keenly realized, well calculated, and spoken with emphasis, have 
notable power. 
Bacon enunciated the essential principles of calendar reform, 
recognizing that the current plan of 365| days led to an error 
of one day in 130 years. He made an acute criticism of the arbi- 
trary assumptions and the artificial complexity of the Ptolemaic 
astronomy; he discussed reflection and refraction, spherical 
aberration, rainbows, magnifying glasses, and shooting stars; 
he attributed the tides to the action of the lunar rays. In a 
chapter on geography he “ comes to the conclusion that the ocean 
between the east coast of Asia and Europe is not very broad. 
This . . . was quoted by Columbus in 1498. ... It is pleasant 
to think that the persecuted English monk, then two hundred 
years in his grave, was able to lend a powerful hand in widening 
the horizon of mankind.” (See Appendix.) 
Most of this remarkable work — not printed for nearly 500 
years — was so far in advance of the age that it not only failed of 
appreciation, but exposed the author to accusations of magic, and 
even to imprisonment. In spite of his many attainments he 
believed in astrology, in the doctrine of “ signatures ” and in the 
“ philosopher’s stone,” and “ knew ” that the circle had been 
squared. He prophesied ships propelled swiftly by mechanical 
means and carriages without horses. He repudiated belief in witch- 
craft,1 and paid the penalty for his courage by many years in prison. 
Dante Alighieri (1265-1321).—Another notable scholar of 
the thirteenth century is Dante, the greatest poetical genius 
of the Middle Ages, who requires our notice not only because of 
his influence in awakening and stimulating the minds of his own 
and later times, but also as the author of a treatise On Water 
1 Not merely astrology and alchemy but even magic and necromancy were at 
this time the subjects of university lecture courses. 182 
A SHORT HISTORY OF SCIENCE 
and the Earth (De Aqua et Terra) which, according to himself, 
was delivered at Mantua in 1320 as a contribution to the question, 
then much discussed, “ whether on any part of the earth’s surface 
water is higher than the earth.” In his cosmology, Dante seems 
to derive from Aristotle and Pliny, without having attained 
familiarity with the Ptolemaic system. 
Computation in the Middle Ages. — During the fourteenth 
century there was continued activity in the gradual dissemination 
of Arabic learning, largely through the medium of almanacs and 
calendars, so that Arabic computations, Euclidean geometry, and 
Ptolemaic astronomy became widely known. Some of these 
calendars emphasized the religious side and gave dates of church 
festivals for a series of years, others specialized in astrology, 
medicine, or astronomy. For ecclesiastical purposes Roman 
numerals were preferred, but at least an explanation of the new 
Arabic characters and their use was generally given. 
The arithmetic of Boethius, based on Roman numerals, retained 
its vogue in northern Europe as late as about 1600. Arabic 
arithmetic, or algorism, based on the Liber Abaci of Leonardo 
Pisano, employing the decimal scale and including the elements 
of algebra, came into general use among the Italian merchants 
in the thirteenth and fourteenth centuries, though not without 
meeting serious opposition. Outside of Italy, however, accounts 
were kept in Roman numerals till about 1550, and in the more 
conservative religious and educational institutions, for a hundred 
years longer. The Florentines at the same time considerably 
simplified the classification of arithmetical operations, in accord- 
ance with our modern list: — numeration, addition, subtraction, 
multiplication, division, involution and evolution. 
Addition and subtraction were begun at the left. The multi- 
plication table, at first little known, ended with 5x5. For 
further products up to 10 X 10, a system of finger reckoning was 
widely used, the rule running: — 
Let the number five be represented by the open hand; the number 
six by the hand with one finger closed; the number seven by the hand 
with two fingers closed; the number eight by the hand with three PROGRESS OF SCIENCE TO 1450 A.D. 
183 
fingers closed; and the number nine by the hand with four fingers 
closed. To multiply one number by another let the multiplier be 
represented by one hand, and the number multiplied by the other, 
according to the above convention. Then the required answer is 
the product of the number of fingers (counting the thumb as a finger) 
open in the one hand by the number of fingers open in the other to- 
gether with ten times the total number of fingers closed.1 
Long division naturally required the skill of a mathematical 
expert. For example, if it is necessary to divide 1330 by 84 
{Ball, p. 191) the Arabic or Persian method may be represented 
as follows, the right hand figure summing up the whole process: 
13 3 0 
8 4 
0 
1 3 3 0 
8 
5 3 0 
4 
TTT 
8 4 
XX 
| 0 I 1 
1 | 3 | 3 | 0 
_ 8 
_5_ 3 0 
4 
4 9 0~ 
4 0 
9 5“ 
2 0 
7 (T 
8 4 
8 4 
_8_ _4 
0 1 5 
The galley or “ scratch ” method generally employed in Italy 
would take for the same problem, the successive forms {Ball, p. 
192): T 
4 4 09 00 
5 09 09 00 00 
;03O(i ;00O(i ;00O(i5 ;00O(i5 ;00O(i5 
04 00 004 004 000 
8 0 0 
1 In modern notation: If x is the number of fingers closed in one hand, y the 
number closed in the other, then 
(5 + x) (5 + y) = (5 - x) (5 — y) + 10 (x + y). 184 
A SHORT HISTORY OF SCIENCE 
This method was considered simpler than our modern long 
division, and remained in use till the seventeenth century. 
The signs +, —, -5-, and the use of decimal fractions belong to 
a somewhat later period. 
The characteristics of the algoristic arithmetic are: (1) the use 
of the Hindu-Arabic system of notation; (2) the system of local value; 
(3) the use of the zero; (4) the entire discarding of the abacus; (5) 
the combined use of symbols and numbers (in reality a combination 
of algebra and arithmetic, as these terms are understood to-day); 
and (6) the introduction into Western Europe of a vast amount of 
arithmetical material from the East by means of Latin translations 
from Arabian sources. While the general tendency of this period 
was to approach the study of arithmetic from its practical and 
scientific sides, the mystical aspects of the subject — so popular 
in the earlier periods — are by no means neglected. The fantas- 
tic treatment of the properties of numbers is still common in this 
age. . . . 
Thus the beginning of the thirteenth century marks the introduc- 
tion of the Arabian system of notation and its adoption in place of 
both the Roman notation and the abacus. This fundamental revolu- 
tion was brought about only gradually, and that of the algorism can 
be traced in the translated literature of the Hindu-Arabian arith- 
metic. — Abelson. 
Mathematics in the Medieval Universities. — The state of 
mathematics in the universities toward the close of the fourteenth 
century may be inferred from the requirements for the master’s 
degree at Prague (1384) and Vienna (1389). The former included 
Sacrobosco’s Sphere, Euclid Books I-VI, optics, hydrostatics, 
theory of the lever, and astronomy. Lectures were given on 
arithmetic, finger-reckoning, almanacs, and-Ptolemy’s Almagest. 
At Vienna, Euclid I-V, perspective, proportional parts, mensu- 
ration, and a recent version of Ptolemy were required. In Leipsic, 
however, in 1437 and 1438 mathematical (?) lectures were confined 
to astrology, and conditions seem to have been much the same at 
the Italian universities, while Oxford and Paris probably occupied 
an intermediate level. PROGRESS OF SCIENCE TO 1450 A.D. 
185 
There can be no doubt that at all times medieval schools taught 
all that their respective generations knew of arithmetic; that the 
teachers of arithmetic in the schools were often the famous mathe- 
maticians of their day; that this teaching, since it kept pace with the 
increase in the knowledge of the subject, was progressive in character, 
and that at no time, not even in the barren generations at the close of 
the Middle Ages, when the scholastic education had outlived its use- 
fulness, did arithmetic cease to be a subject of study in the arts facul- 
ties of the medieval universities. — Abelson. 
The Renaissance. — With the fourteenth century we enter 
upon one of the most interesting and noteworthy periods of human 
history; viz. the Renaissance. Neither the term nor the period 
is, however, sharply defined, the former signifying an awakening 
or “new birth,” the latter covering loosely the fourteenth to the 
sixteenth centuries. It is only necessary to recapitulate briefly 
some of the phenomena touched upon in the present chapter, to 
realize that the civilization of the later Middle Ages has been under- 
going great changes. The Crusades marked the first and perhaps 
most important of these, while the rediscovery or recovery of the 
classics from Arabian and other sources in the eleventh to the thir- 
teenth centuries, followed by the revival of (classical) learning in 
the fourteenth must have been powerful ferments of the medieval 
scholastic mind, expanded and uplifted as it was by the poetical 
philosophy of Dante and challenged by the naturalism and ration- 
alism of Roger Bacon. 
The great events of the fourteenth century were in part new, 
and in part the natural extension and development of those of 
the thirteenth. A strange and appalling natural phenomenon was 
the famous epidemic known as the “ black death,” a quickly fatal 
disease which carried off from one quarter to one half of all the 
inhabitants of Europe, producing social changes — such as the 
rise of wages — which are still felt. 
Humanism. — The development of better education begun in 
the thirteenth century was marked in the fourteenth by the found- 
ing of many now famous universities and colleges and by that 
revival of ancient learning which is associated especially with the 186 
A SHORT HISTORY OF SCIENCE 
name of Petrarch (1304-1374). This revival, while at first chiefly 
literary and philosophical, brought with it translations into Latin 
— the current language of scholars at that time — of Aristotle and 
other classical writers of scientific importance, and thus aided in 
bringing on a new birth or renaissance in science as well as in 
other branches. 
Precisely as there is one great name in thirteenth century 
literature, viz. that of Dante, which must be regarded with 
attention by all students of history, so in the fourteenth the name 
and work of Petrarch require careful consideration. Francesco 
Petrarca, commonly called Petrarch, a gifted Italian poet and 
scholar, greatly promoted the revival of ancient learning by insisting 
on the importance and merits of the Greek and Roman authors. 
Petrarch was less eminent as an Italian poet than as the founder 
of Humanism, the inaugurator of the Renaissance in Italy. . . . 
Standing within the kingdom of the Middle Ages, he surveyed the 
kingdom of the modern spirit and, by his own inexhaustible indus- 
try in the field of scholarship and study, he determined what we call 
the revival of learning. By bringing the men of his own generation 
into sympathetic contact with antiquity, he gave a decisive impulse 
to that European movement which restored freedom, self-conscious- 
ness and the faculty of progress to the human intellect. . . . He 
was the first man to collect libraries, to accumulate coins, to advocate 
the preservation of antique monuments, and to collate manuscripts. 
Though he knew no Greek, he was the first to appreciate its vast im- 
portance ; and through his influence, Boccaccio laid the earliest founda- 
tions of its study. ... For him the authors of the Greek and Latin 
world were living men, — more real in fact than those with whom he 
corresponded; and the rhetorical epistles he addressed to Cicero, 
Seneca and Varro prove that he dwelt with them on terms of 
sympathetic intimacy. — Symonds. 
Rich as the fourteenth and fifteenth centuries are in mathe- 
matical science and geographical discovery, and in art and inven- 
tion, they are almost destitute of positive achievement in natural 
science. Doubtless the scientific spirit of curiosity and inquiry 
was alive and active, but thus far it had taken other directions. PROGRESS OF SCIENCE TO 1450 A.D. 
187 
Alchemy. — What astrology was to astronomy, alchemy was 
to chemistry; viz. the crude and often magic-working predecessor. 
The search for such will o’ the wisps as the “ philosopher’s stone,” 
the “elixir of life,” “potable gold” and the “transmutation of 
elements,” is probably as old as human history. The ancients 
seem to have dabbled in it, the Arabs to have been devoted to it, 
and the men of the Middle Ages, and even of the fourteenth and 
fifteenth centuries, to have spent much time upon it. Alembics 
and receivers, “Moors’ Heads” and “Moors’ Noses,” calci- 
fication, distillation, and the like typify interesting and by no 
means fruitless gropings after the real composition of things. 
The names of Albertus Magnus, Bernard of Treviso, Eck of Salz- 
burg, and Basil Valentine are some which have come down to us 
as most important at this time, and as we read of the prepara- 
tion of the “ spirits of salt ” (hydrochloric acid), the calcification 
(oxidation) of mercury, etc., we realize that their labors, though 
often misdirected, were the prelude to better things. 
The Mariner’s Compass. — The loadstone was certainly known 
to antiquity as a stone having the power of attracting and 
carrying a load of iron, but its directive property seems to have 
been first recognized and used for guidance on land or sea by the 
Chinese, since according to Humboldt, Chinese ships navigated 
the Indian Ocean with the magnetic needle in the third century of 
our era. The Arabs are also credited with its invention and use, 
as stated in the preceding chapter. The first reference to it in 
Christian Europe is said to be in a poem by Guyot of Provence, 
dated 1190, while references are also made to the compass in 
works of the thirteenth century. One of these runs: — 
No master mariner dares to use it lest he should be suspected of 
being a magician; nor would the sailors venture to go to sea under 
the command of a man using an instrument which so much appeared 
to be under the influence of the powers below. 
It is probable, however, that the compass was first made 
commonly useful to western Europe early in the fourteenth 
century, by Flavio Gioja, a native of Amalfi, a small port 188 
A SHORT HISTORY OF SCIENCE 
near Naples in Italy, who first poised the needle on a pivot 
instead of a card floating on water, as had been the custom 
before his time. (See page 164.) 
Clocks. — Clocks with wheels seem to have come into occa- 
sional use from the twelfth to the fourteenth centuries, and one of 
the first is said to have been sent by the Sultan of Egypt in 
1232 to the Emperor Frederick II. 
It resembled a celestial globe, in which the sun, moon and planets 
moved, being impelled by weights and wheels so that they pointed 
out the hour, day and night, with certainty. 
Another is mentioned as in Canterbury cathedral, while still an- 
other at St. Albans, made by R. Wallingford who was abbot there 
in 1326, is said to have been so notable “that all Europe could not 
produce such another.” It remained for Huygens in the seven- 
teenth century to apply pendulums to clocks. 
Wool and Silk. Textiles in the Middle Ages. — As an 
example of the industrial history of the times the following account 
of conditions in Spain is given: — 
The cloth manufactures in Spain continued to be of the coarsest 
character until after the marriage of Catharine of Lancaster to the 
heir of Castile (1388) when finer cloths were manufactured and 
improved methods adopted. Up to that time the cloths used by 
people of the higher class came from Bruges, from London, and from 
Montpellier. James II of Aragon — the sovereign of Barcelona, 
where there were at the time hundreds of looms at work making a 
coarse woolen — wished to send a present to the Sultan of Egypt 
(1314 and 1322), and chose green cloths from Chalons and red cloths 
from Rheims and Douai, but sent no Spanish stuff; while the stew- 
ard’s accounts of Fernando V show that all his household were dressed 
in garments of imported stuffs. The great centre for the sale of wool 
was at Medina del Campo, and the cloth factories of Segovia and 
Toledo were the most active and celebrated in Castile, while those 
of Barcelona were the principal in the east of Spain. It is asserted 
that the improvement in the qualities of the Spanish cloth after 
the coming of the Plantagenet princess to Spain was partly owing 
to the fact that some herds of English sheep formed part of her PROGRESS OF SCIENCE TO 1450 A.D. 
189 
dowry, and the blending of staples enabled a better cloth to be 
made. The Flemish weavers mixed Spanish with English wool for 
their best textures. 
During the Arab domination of the south, Jaen, Granada, Valencia, 
and Seville had been great centres of silk culture and manufacture. 
Edrisi says that in the kingdom of Jaen in the thirteenth century 
there were 3000 villages where the cultivation of the silkworm was 
carried on, while in Seville there were 6000 silk looms, and Almeria 
had 800 looms for the manufacture of fancy brocades, etc. We are 
also told that a minister of Pedro the Cruel owned 125 chests of silk 
and gold tissue. In the twelfth century, a very flourishing trade in 
silks, velvets, and brocades was carried on with Constantinople and 
the East generally. Even in the fourteenth and early fifteenth cen- 
turies, the silks of Valencia and the bullion embroideries and gold and 
silver tissues of Cordova and Toledo were unsurpassed in Christen- 
dom, though heavily handicapped by the growing burdens placed 
upon craftsmen by labor laws and racial prejudice, and the dis- 
couragement of luxury by sumptuary regulations. — Hume. 
The Invention of Printing. — Before the middle of the 
fifteenth century, printing was done chiefly from fixed blocks of 
wood, metal, or stone, as is the case to-day in the printing of en- 
gravings, wood cuts and the like. The introduction of movable 
types, capable of an almost infinite variety of combination was 
therefore a forward step of- fundamental importance, since the 
same letter or picture could be used over and over in new com- 
binations where previously it could be used but once. Until quite 
recently, it was generally held that the invention of the art of 
printing from movable types was the work of Johann Gutenberg 
(1397-1468) of Mainz on the Rhine, aided by Johann Faust or 
Fust, a rich citizen of Mainz. Of late, however, the claim of 
Gutenberg has been much disputed. 
The controversy about the person and nationality of the inventor 
[of the art of printing] and the place of invention resembles the rival 
claims of seven cities to be the birthplace of Homer. . . . The best 
authorities agree on Gutenberg. Jacob Wimpheling wrote in 1507 . . . 
‘ Of no art can we Germans be more proud than of the art of printing, 190 
A SHORT HISTORY OF SCIENCE 
which made us the intellectual bearers of the doctrines of Christianity, 
of all divine and earthly sciences, and thus benefactors of the whole 
race.’ — Schaff. 
Abelson. The Seven Liberal Arts. 
Ball. History of Mathematics, Chapters VI, VIII, X. 
Cajori. History of Mathematics. 
Cajori. History of Physics. 
Draper. Intellectual Development of Europe. 
Muir. Alchemy and the Beginnings of Chemistry. 
Rashdall. Universities of the Middle Ages. 
Schaff. The Renaissance. 
Symonds. The Renaissance. 
White. Warfare of Science with Theology. 
References 
for 
Reading 
A Map of the Globe in the Time of Columbus 
(After J. H. Robinson. Courtesy of Messrs. Ginn & Co.) 
In 1492 a German mariner, Behaim, made a globe which is still preserved in Nuremberg. He 
did not know of the existence of the American continents or of the vast Pacific Ocean. . . . 
He places Japan (Cipango) where Mexico lies. In the reproduction many names are omitted, 
and the outlines of North and South America are sketched in. 
— J. H. Robinson, Mediaeval and Modern Times. CHAPTER X 
A NEW ASTRONOMY AND THE BEGINNINGS OF 
MODERN NATURAL SCIENCE 
The breeze from the shores of Hellas cleared the heavy scholastic 
atmosphere. Scholasticism was succeeded by Humanism, by the 
acceptance of this world as a fair and goodly place given to man to 
enjoy and to make the best of. In Italy the reaction became so great 
that it seemed destined to put paganism once more in the place of 
Christianity; and though it produced lasting monuments in art and 
poetry, the earnestness was wanting which in Germany brought about 
the revival of science, and later on the rebellion against spiritual 
tyranny. . . . Astronomy profited more than any other science by 
this revival of learning, and about the middle of the fifteenth century 
the first of the long series of German astronomers arose who paved the 
way for Copernicus and Kepler, though not one of them deserves to 
be called a precursor of these heroes. — Dreyer. 
The silent work of the great Regiomontanus in his chamber at 
Nuremberg computed the ephemerides which made possible the 
discovery of America by Columbus. — Rudio. 
The extension of the geographical field of view over the whole 
earth and the release of thought and feeling from the restrictions of 
the Middle Ages mark a division of equal importance with the fall 
of the ancient world a thousand years earlier. — Dannemann. 
Science begins to dawn, but only to dawn, when a Copernicus, 
and after him a Kepler or a Galileo, sets to work on these raw materials, 
and sifts from them their essence. She bursts into full daylight only 
when a Newton extracts the quintessence. There has been as yet 
but one Newton; there have not been very many Keplers. — Tait. 
The Age of Discovery. — With the end of the fifteenth 
century and the beginning of the sixteenth opens one of the most 
marvellous chapters in all history; viz. the Discovery of the New 
World. At about the same time further explorations of the old 
world attained equal extent and interest. We have referred above 
(p. 174) to the Discovery of the East by the Crusaders, and now 
191 192 
A SHORT HISTORY OF SCIENCE 
with Columbus, Magellan, and their successors, we have an even 
more pregnant Discovery of the West. Meanwhile, Diaz and da 
Gama pushed the explorations of Prince Henry of Portugal, “ the 
Navigator,” to the south, and in rounding the Cape of Good 
Hope completed the Discovery of the South. To the north, ex- 
plorers had already advanced to regions of perpetual snow and 
ice, so that in all directions there were new problems of intense 
interest profoundly moving the imagination of mankind. 
The Reformation. — Another potent element was added to 
the already complex fermentation of medieval ideas when in 1517 
a widespread insurrection began in the Christian Church, the most 
conservative and most powerful institution of the Middle Ages. 
This revolution, — for such it proved to be, — with which the 
name of Luther will always be chiefly associated, soon aroused 
a wave of determined opposition, naturally strongly conservative, 
known to-day as the “counter-reformation,” of which the In- 
quisition was one instrument. 
The increased importance of the art of navigation reacted 
powerfully on the underlying sciences of mathematics and astron- 
omy, particularly through the demand for improved astronomi- 
cal tables. The Church, even, had a strong, if restricted, interest 
in astronomy on account of the necessity of more accurate data 
for its calendar. 
Pioneers of the New Astronomy. — Nicholas of Cusa 
(1401-1464), later Bishop of Brixen, wrote on Learned Ignorance, 
arguing that the universe, being infinite in extent, could have no 
centre, and that the earth has diurnal rotation. “ It is now clear 
that, the earth really moves, if we do not at once observe it, 
since we perceive motion only through comparison wTith some- 
thing immovable.” In mathematics he follows Euclid and 
Archimedes, cooperating in a translation of the latter from 
Greek into Latin, and dealing with the squaring of the circle. 
He makes a map of the known world, using central projection. 
He is said to have determined areas of irregular boundary by the 
then novel method of cutting them out and weighing, and is one 
of the first to emphasize the importance of measurement in all A NEW ASTRONOMY 
193 
investigations. He showed independence of thinking, but his 
astronomical theories were too little developed — and too specula- 
tive — to constitute real progress in an age not yet quite ripe for 
their reception. 
Peurbach (1423-1461), who had as a youth met Nicholas of 
Cusa in Rome, became professor of astronomy and mathematics 
in Vienna and has been called “ the founder of observational and 
mathematical astronomy in the West.” Recognizing the imper- 
fections of the Alfonsine tables he published a new edition of 
the Almagest with tables of natural sines — instead of chords — 
computed for every ten minutes. He depended mainly, however, 
on imperfect Arabic translations. 
His more eminent pupil and successor, Johann Muller, of 
Konigsberg, better known as Regiomontanus (1436-1476), was 
the most distinguished scientific man of his time. After the 
fall of Constantinople he was among the first to avail himself of 
the opportunities for more direct acquaintance with the works 
of Archimedes, Apollonius, and Diophantus. For the defective 
version of the Almagest which had come through Arabic 
channels he substituted the Greek original, while his tables, pub- 
lished in 1475, were important both for astronomy and for the 
voyages of discovery of Vasco da Gama, Vespucci, and Colum- 
bus. These tables covered the period 1473 to 1560, giving sines 
for each minute of arc, longitudes for sun and moon, latitude for 
the moon, and a list of predicted eclipses from 1475 to 1530. An- 
other work on astrology includes a table of natural tangents for 
each degree. A wealthy merchant of Nuremberg erected an 
elaborately equipped observatory for Regiomontanus, and the 
printing-press recently established there became the most impor- 
tant in Germany. Accepting, however, a summons to Rome to 
reform the calendar, he was murdered at the age of 40. 
His De Triangulis (1464) is the earliest modern trigonometry. 
Four of its five books are devoted to plane trigonometry, the 
other to spherical. He determines triangles from three given 
conditions, using sines and cosines, and employs quadratic equa- 
tions successfully in some of his solutions. One of his problems 194 
A SHORT HISTORY OF SCIENCE 
is “to determine a triangle when the difference of two sides, the 
perpendicular on the base, and the difference between the seg- 
ments into which the base is divided are given: i.e. a — b, a sin B, 
a cos B — b cos A are known; to find, a, b, c, A, B, C” An- 
other is to construct from four given lines a quadrilateral which 
can be inscribed in a circle. 
Conditions Necessary for Progress. — The genius of 
Hipparchus and Ptolemy had brought Greek astronomy to its 
culmination. Higher it could not rise until three conditions 
should be fulfilled, even though here and there the heliocentric 
hypothesis might be adopted through an unsupported inspiration 
of individuals. First, there must be better astronomical instru- 
ments and more accurate observations, extended over long periods. 
Second, there must be improved methods of mathematical com- 
putation for the reduction and interpretation of these observations. 
Third, there must be substantial progress towards clear thinking 
as to the fundamental facts and laws of motion. These conditions 
were met one after another during the sixteenth and seventeenth 
centuries by an extraordinary series of men of genius, among whom 
the chief were Copernicus, Tycho Brahe, Kepler, Galileo, and 
Newton. Their work constitutes a great part of the history of 
science during these two centuries — and one of the most won- 
derful chapters of all time. 
Of these five, Copernicus and Kepler were predominantly inter- 
ested on the mathematical and theoretical side, Tycho Brahe was 
a great observer, Galileo combined experimental and observa- 
tional skill with a new appreciation of physical laws, while Newton, 
building on the foundation laid by all the others, made a magnifi- 
cent synthesis of their results into a rational and consistent mathe- 
matical theory of the solar system. These five represent Poland, 
South Germany, Denmark, Italy, and England. Scientific progress 
is no longer localized or dependent on princely patronage. It has 
now become international. 
Nicolaus Copernicus (1473-1543) was born in the remote 
little city of Thorn on the Vistula, and having relatives in the 
Church, prepared himself for an ecclesiastical career. This led A NEW ASTRONOMY 
195 
him, after medical study at Cracow, first to the university of 
Vienna, then to the chief Italian universities, Bologna, Padua, 
Ferrara, and Rome, where he found opportunity to cultivate his 
mathematical talents and to master what was then known of 
astronomy. He became canon at Frauenburg in his native land 
in 1497, and from 1512 until his death thirty years later, was 
settled there, rendering varied public services, and practising 
gratuitously, as needful, the medical art he had also learned. At 
the same time he found it possible to devote much attention to 
astronomical studies. 
In his study of the classical writers he came upon a statement 
that certain Pythagorean philosophers explained the phenomena 
of the daily and yearly motions of the heavenly bodies by sup- 
posing the earth itself to rotate on its axis and to have also an 
orbital motion. 
‘ Occasioned by this, I also began to think of a motion of the earth, 
and although the idea seemed absurd, still, as others before me had 
been permitted to assume certain circles in order to explain the motions 
of the stars, I believed it would readily be permitted me to try whether 
on the assumption of some motion of the earth better explanations 
of the revolutions of the heavenly spheres might not be found. And 
thus I have, assuming the motions which I in the following work at- 
tribute to the earth, after long and careful investigation, finally found 
that when the motions of the other planets are referred to the circu- 
lation of the earth and are computed for the revolution of each star, 
not only do the phenomena necessarily follow therefrom, but the 
order and magnitude of the stars and all their orbs and the heaven 
itself are so connected that in no part can anything be transposed 
without confusion to the rest and to the whole universe.’ — Dreyer. 
‘ I made every effort to read anew all the books of philosophers I 
could obtain, in order to ascertain if there were not some one of 
them of the opinion that other motions of the heavenly bodies existed 
than are assumed by those who teach mathematical sciences in the 
schools. So I found first in Cicero that Hicetas of Syracuse believed 
the earth moved. Afterwards I found also in Plutarch that others 
were likewise of this opinion. . . . Starting thence I began to re- 
flect on the mobility of the earth.’— Timer ding. 196 
A SHORT HISTORY OF SCIENCE 
Copernicus was not a great observational astronomer. His 
instruments were poor, his eyesight not keen, his location un- 
favorable for clear skies. His recorded observations are few, 
chiefly of eclipses or oppositions of planets, and of no high degree 
of accuracy. His interest and genius lay rather in the direction 
of profound analysis and careful mathematical revision of the 
current geocentric theory, practically unchanged since its formu- 
lation by Ptolemy thirteen centuries earlier. Unfortunately the 
conditions of the time were adverse to the publication of so radical 
an innovation as a heliocentric theory of the solar system; nor 
was Copernicus ever greatly interested in any publication of his 
results, being both indifferent to reputation and averse to con- 
troversy. 
‘ The scorn, ’ he says, ‘ which I had to fear in consequence of the 
novelty and seeming unreasonableness of my ideas, almost moved me 
to lay the completed work aside.’ 
Moreover, he realized the futility of publishing his revolu- 
tionary theories until he should have buttressed them with a 
planetary system so completely worked out that its superiority 
to the long-intrenched Ptolemaic system should be unquestionable 
— a herculean, if congenial labor. Nevertheless, he gradually 
formulated his astronomical system in manuscript, and about 
1529 issued a Commentariolus giving an outline of his theory, 
which thus became gradually but vaguely known to scholars. 
Ten years later George Joachim — Rheticus — a young professor 
of mathematics from the Lutheran university of Wittenberg, 
visited Copernicus, eager to learn more of the new doctrine. 
The Lutheran church was not more hospitable than the Roman 
Catholic to scientific novelty and Luther himself called Copernicus 
a fool. 
De Revolutionibus.—In 1540 appeared the Prima Narratio 
by Rheticus containing a considerable admixture of astrology, and 
in 1543 the immortal De Revolutionibus Orbiwm Celestium, a copy 
reaching Copernicus, it is said, on his death-bed. He begins with 
certain postulates: first, that the universe is spherical; second, that A NEW ASTRONOMY 
197 
the earth is spherical; third, that the motions of the heavenly bodies 
are uniform circular motions or compounded of such motions. 
•The slender basis for the first and third of these may be inferred 
from his statement in regard to certain hypothetical causes of 
want of uniformity: — 
Both of which things the intellect shrinks from with horror, it 
being unworthy to hold such a view about bodies which are con- 
stituted in the most perfect order. 
He makes the relative character of the motions involved of 
fundamental importance. In his own words: — 
For all change in position which is seen is due to a motion either 
of the observer or of the thing looked at, or to changes in the position 
of both, provided that these are different. For when things are 
moved equally relatively to the same things, no motion is perceived, 
as between the object seen and the observer. 
Thus the daily revolution of sun, moon, and stars about a station- 
ary earth would have the same apparent effect as rotation of the 
earth in the opposite direction about its own axis, and the ap- 
parent yearly motion of the sun about the earth is equivalent to 
an orbital motion of the latter. 
‘It is,’ he says, ‘more probable that the earth turns about its 
axis than that the planets at their various distances, the comets sweep- 
ing through space, and the endless multitude of the fixed stars, describe 
the same regular daily motion about the earth.’ 
The apparent irregularities in the motions of the five known 
planets had been a perpetual stumbling-block to the ancient 
astronomers, requiring more and more complicated hypotheses for 
their explanations as accuracy of observations increased. The 
heliocentric theory of Copernicus, inaccurate as it was in some 
respects, afforded a simple explanation of the fact that Mercury 
and Venus seem merely to oscillate east and west of the sun, 
while Mars, Jupiter, and Saturn recede indefinitely from it, ex- 
hibiting also periodic reversals of the direction of their motion. 198 
A SHORT HISTORY OF SCIENCE 
The new explanation obviously accounted also for the variations 
in the brightness of these planets. 
* It is certain, ’ he says, ‘ that Saturn, Jupiter and Mars are always 
nearest the earth when they rise in the evening, that is when they are 
in opposition to the sun, as the earth is situated between them and 
The Copernican System 
the sun. On the contrary, Mars and Jupiter are farthest from the 
earth when they set in the evening, the sun lying between them and 
us. This proves sufficiently that the sun is the centre of their orbits, 
as of those of Venus and Mercury. Since thus all planets move 
about one centre it is necessary that the space which remains between 
the circles of Venus and Mars, contain the earth and its accompanying 
moon.’ 
He is, therefore, not afraid to maintain that the earth with the 
moon encircling it, traverses a great circle in its annual motion 
among the planets about the sun. The universe, however, is so 
vast, that the distances of the planets from the sun are insignificant A NEW ASTRONOMY 
199 
in comparison with that of the sphere of the stars. He holds 
all this easier of comprehension, than if the mind is confused by 
an almost endless mass of circles, as is necessary for those who put 
the earth in the centre of the universe. 
‘So in fact the sun seated on the royal throne guides the family 
of planets encircling it. We find thus in this arrangement a har- 
monious connection not otherwise realized. For here one can see 
why the forward and backward motions of Jupiter seem greater than 
those of Saturn and smaller than those of Mars.’ 
His adherence to the Greek assumption of uniform circular 
motion leaves him still under the necessity of retaining an elabo- 
rate system of epicycles, but he rejects Ptolemy’s equant. 
. . . He his fabric of the heavens 
Hath left to their disputes, perhaps to move 
His laughter at their quaint opinions wide; 
Hereafter when they come to model heaven 
And calculate the stars, how will they wield 
The mighty frame ! how build, unbuild, contrive 
To save appearances! how gird the sphere 
With centric and eccentric scribbled o’er, 
Cycle in epicycle, orb in orb ! 
— Milton, Paradise Lost, VIII. 
The epicycles of Copernicus numbered however but 34,— 
sufficing “to explain the whole construction of the world and the 
whole dance of the planets ” — against the 79 to which the Ptole- 
maic theory had gradually attained. The completeness of mathe- 
matical detail with which the whole theory is worked out can not 
here be adequately described. He includes so much trigonometry 
as his astronomical work requires, also a revision of Ptolemy’s star 
catalogue. He computes a very accurate value of the equinoctial 
precession, and interprets this correctly as due to a slow conical 
motion of the earth’s axis, like that of a top coming to rest. 
Copernicus estimates the relative sizes of moon, earth and 200 
A SHORT HISTORY OF SCIENCE 
sun as 1:43 :6937, and the distance from earth to sun — according 
to the method of Aristarchus — at about 1200 earth-radii, that 
is about of the actual. 
Revolutionary as were the theories expounded by Copernicus 
they were not clothed in such popular form as to occasion imme- 
diate or general controversy. In dedicating his work to the 
Pope, Copernicus says in substance: — 
It seems to me that the church can derive some advantage from 
my labors. Under Leo X indeed the rectification of the calendar 
was not possible, since the length of the year and the motions of 
the sun and moon were not exactly determined. I have sought to 
determine these more closely. What I have accomplished, I leave 
to the judgment of your Holiness, and of the learned mathemati- 
cians. (See Appendix.) 
Moreover criticism was in considerable measure disarmed by 
a fraudulent preface inserted by Osiander, a Lutheran theologian 
of Nuremberg, to whom the care of publication had been par- 
tially intrusted by Rheticus. In this preface, ostensibly by 
Copernicus himself, it is stated, — 
that though many will take offence at the doctrine of the earth’s 
motion, it will be found on further consideration that the author 
does not deserve blame. For the object of an astronomer is to 
put together the history of the celestial motions from careful ob- 
servations, and then to set forth their causes or hypotheses about 
them, if he cannot find the real causes, so that those motions can be 
computed on geometrical principles. But it is not necessary that 
his hypotheses should be true, they need not even be probable; it 
is sufficient if the calculations founded on them agree with the obser- 
vations. Nobody would consider the epicycle of Venus probable, 
as the diameter of the planet in its perigee ought to be four times as 
great as in the apogee, which is contradicted by the experience of all 
times. Science simply does not know the cause of the apparently 
irregular motions, and an astronomer will prefer the hypothesis which 
is most easily understood. Let us therefore add the following new 
hypotheses to the old ones, as they are admirable and simple, but A NEW ASTRONOMY 
201 
nobody must expect certainty about astronomy, for it cannot give it; 
and whoever takes for truth what has been designed for a different 
purpose, will leave this science as a greater fool than he was when 
he approached it. 
Influence of Copernicus.—The publication of De Revolu- 
tionibus was naturally a powerful stimulus to astronomical and 
mathematical studies. Thus Rheticus, whose relations to Coper- 
nicus had been so fruitful, calculated a new and extensive set 
of mathematical tables, while Reinhold, who had hailed Coper- 
nicus as a new Ptolemy, published astronomical tables — the 
Prutenic or Prussian — on the basis of Copernicus’ work, superior 
to the Alfonsine, previously current. 
Before the new doctrine should be completely justified or the 
reverse, it was necessary that certain mechanical notions should 
be clarified, and that more accurate observational data should be 
systematically collected. Copernicus had based his imposing 
structure on a very slender foundation of actual fact, and had 
professed his complete satisfaction if his theoretical results should 
come within ten minutes of the observed positions of the planets, — 
a degree of accuracy which he did not, in fact, attain. On the other 
hand, he could indeed answer, but not rise entirely above, the 
traditional notions that the four elements of the ancients must 
have rectilinear, the heavenly bodies circular, motion; also, that if 
the earth rotated in twenty-four hours, loose bodies would long 
since have been thrown off, falling bodies would not fall, and clouds 
would always be left behind in the west. 
As suggested by Dreyer: — 
It is interesting, though useless, to speculate on what would have 
been the chances of immediate success of the work of Copernicus if 
it had appeared fifty years earlier. Among the humanists there 
certainly was considerable freedom of thought, and they would not 
have been prejudiced against the new conception of the world because 
it upset the medieval notion of a set of planetary spheres inside the 
empyrean sphere, with places allotted for the hierarchy of angels. 
If one of the leaders of the Church (at least in Italy) at the beginning 202 
A SHORT HISTORY OF SCIENCE 
of the sixteenth century had been asked whether the idea of the earth 
moving through space was not clearly heretical, he would probably 
merely have smiled at the innocence of the enquirer and have answered 
in the words of Pomponazzi that a thing might be true in philosophy 
and yet false in theology. But the times had changed. The sun of 
the Renaissance had set when, in 1527, the hordes of the Constable of 
Bourbon sacked and desecrated Rome; the Reformation had put an 
end to the religious and intellectual solidarity of the nations, and the 
contest between Rome and the Protestants absorbed the mental 
energy of Europe. During the second half of the sixteenth century 
science was therefore very little cultivated, and though astronomy and 
astrology attracted a fair number of students (among whom was one 
of the first rank), still theology was thought of first and last. And 
theology had come to mean the most literal acceptance of every word 
of Scripture; to the Protestants of necessity, since they denied the 
authority of Popes and Councils, to the Roman Catholics from a 
desire to define their doctrines more narrowly and to prove how un- 
justified had been the revolt against the Church of Rome. There 
was an end of all talk of Christian Renaissance and of all hope of rec- 
onciling faith and reason; a new spirit had arisen which claimed 
absolute control for Church authority. Neither side could therefore 
be expected to be very cordial to the new doctrine. 
Robert Recorde, in his Pathway to Knowledge (1551), has his 
“Master” state to a “scholar ”: 
‘Eraelides Ponticus, a great philosopher, and two great clerkes of 
Pythagoras schole, Philolaus and Ecphantus, were of the contrary 
opinion, but also Nicias Syracusius and Aristarchus Samius seem with 
strong arguments to approve it.’ After saying that the matter is too 
difficult and must be deferred till another time, the Master states 
that ‘ Copernicus, a man of great learning, of muche experience and 
of wondrefull diligence in obseruation, hathe renewed the opinion of 
Aristarchus Samius, and affirmeth that the earthe not only moueth 
circularlye about his own centre, but also may be, yea and is con- 
tinually out of the precise centre 38 hundredth thousand miles; but 
bicause the vnderstanding of that controuersy dependeth of profounder 
knowledge than in this introduction may be vttered conueniently, I will 
let it passe tyll some other time.’ A NEW ASTRONOMY 
203 
A little later Francis Bacon writes: — 
* In the system of Copernicus there are many and grave difficulties; 
for the threefold motion with which he encumbers the earth is a serious 
inconvenience, and the separation of the sun from the planets, with 
which he has so many affections in common, is likewise a harsh step; 
and the introduction of so many immovable bodies into nature, as when 
he makes the sun and the stars immovable, the bodies which are pecul- 
iarly lucid and radiant, and his making the moon adhere to the earth 
in a sort of epicycle, and some other things which he assumes, are 
proceedings which mark a man who thinks nothing of introducing fic- 
tions of any kind into nature, provided his calculations turn out well.’ 
Bacon himself was very ignorant of all that had been done by 
mathematics; and, strange to say, he especially objected to astronomy 
being handed over to the mathematicians. Leverrier and Adams, 
calculating an unknown planet into a visible existence by enormous 
heaps of algebra, furnish the last comment of note on this specimen 
of the goodness of Bacon’s view. . . . Mathematics was beginning 
to be the great instrument of exact inquiry; Bacon threw the science 
aside, from ignorance, just at the time when his enormous sagacity, 
applied to knowledge, would have made him see the part it was to play. 
If Newton had taken Bacon for his master, not he, but somebody else, 
would have been Newton. — Be Morgan. 
Copernicus cannot be said to have flooded with light the dark 
places of nature — in the way that one stupendous mind subsequently 
did — but still, as we look back through the long vista of the history 
of science, the dim Titanic figure of the old monk seems to rear itself 
out of the dull flats around it, pierces with its head the mists that over- 
shadow them, and catches the first gleam of the rising sun, . . . 
Like some iron peak, by the Creator 
Fired with the red glow of the rushing morn. 
— E. J. C. Morton. 
Tycho Brahe (1546-1601). —The first great need of the new 
Copernican astronomy — adequate and accurate data — was soon 
to be supplied by Tycho Brahe, born in 1546 of a noble Danish 
family. While a student at the University of Copenhagen his 
interest in astronomy was enlisted by an eclipse, and later, at 
Leipsic, he persisted in devoting to his new avocation the time 204 
A SHORT HISTORY OF SCIENCE 
and attention he was expected to give to subjects more highly 
esteemed for a man of birth and fortune. 
From a lunar eclipse which took place while he was at Leipsic, 
Tycho foretold wet weather, which also turned out to be correct. 
Here, too, he began his life work of procuring and improving the 
best instruments for astronomical observations, at the same time 
testing and correcting their errors. Returning to Denmark from 
travels in Germany, his predilection for astronomy was powerfully 
stimulated by the appearance in the constellation Cassiopeia, in 
November, 1572, of a brilliant new star, which remained visible for 
16 months. The great importance attached to this occurrence by 
Tycho and his contemporaries was due to the evidence it afforded 
against the truth of the Aristotelian conviction that the heavens 
were immutable, since Tycho’s careful observations showed that 
the star must certainly be more distant than the moon, and that it 
had no share in the planetary motions. He reluctantly published 
an account of the new star, expressing still his adherence to 
the current pre-Copernican notions of crystalline spheres for the 
different heavenly bodies and of atmospheric comets, all com- 
bined with astrological reflections and inferences, as illustrated by 
the following passages from Dreyer’s biography: — 
The star was at first like Venus and Jupiter, and its effects will 
therefore first be pleasant; but as it then became like Mars, there will 
next come a period of wars, seditions, captivity, and death of princes 
and destruction of cities, together with dryness and fiery meteors in 
the air, pestilence, and venomous snakes. Lastly, the star became 
like Saturn, and there will therefore, finally, come a time of want, 
death, imprisonment, and all kinds of sad things. 
As the star seen by the wise men foretold the birth of Christ, 
the new one was generally supposed to announce His last coming 
and the end of the world. 
That an unusual celestial phenomenon occurring at that particular 
moment should have been considered as indicating troublous times, 
is extremely natural when we consider the state of Europe in 1573. 
The tremendous rebellion against the Papal supremacy, which for a 
long time had seemed destined to end in the complete overthrow of Tycho Brahe’s Quadrant. A NEW ASTRONOMY 
205 
the latter, appeared now to have reached its limit, and many people 
thought that the tide had already commenced to turn. 
Tycho considered that the new star was formed of ‘celestial 
matter,’ not differing from that of which the other stars are composed, 
except that it was not of such perfection or solid composition as in 
the stars of permanent duration. It was therefore gradually dissolved 
and dwindled away. It became visible to us because it was illuminated 
by the sun, and the matter of which it was formed was taken from the 
Milky Way, close to the edge of which the star was situated, and in 
which Tycho believed he could now see a gap or hole which had not 
been there before. 
But the star had a truer mission than that of announcing the 
arrival of an impossible golden age. It roused to unwearied exertions 
a great astronomer, it caused him to renew astronomy in all its branches 
by showing the world how little it knew about the heavens; his 
work became the foundation on which Kepler and Newton built their 
glorious edifice, and the star of Cassiopeia started astronomical science 
on the brilliant career which it has pursued ever since, and swept 
away the mist that obscured the true system of the world. As Kepler 
truly said, ‘If that star did nothing else, at least it announced and 
produced a great astronomer.’ 
At the same time the book bears witness to the soberness of mind 
which distinguishes him from most of the other writers on the subject 
of the star. His account of it is very short, but it says all there 
could be said about it — that it had no parallax, that it remained 
immovable in the same place, that it looked like an ordinary star — 
and it describes the star’s place in the heavens accurately, and its 
variations in light and color. Even though Tycho made some re- 
marks about the astrological significance of the star, he did so in a 
way which shows that he did not himself consider this the most valu- 
able portion of his work. To appreciate his little book perfectly, it 
is desirable to glance at some of the other numerous books and pam- 
phlets which were written about the star, and of most of which 
Tycho himself has in his later work given a very detailed analysis. 
In 1575 Tycho obtained while travelling a copy of Copernicus’ 
Commentariolus, and in the following year received from King 
Frederick II the island of Hveen, with funds for the maintenance 
of an observatory upon it. As to the former his opinion is that 206 
A SHORT HISTORY OF SCIENCE 
‘ The Ptolemean system was too complicated, and the new one 
which that great man Copernicus had proposed, following in the 
footsteps of Aristarchus of Samos, though there was nothing in it 
contrary to mathematical principles, was in opposition to those of 
physics, as the heavy and sluggish earth is unfit to move, and the 
system is even opposed to the authority of Scripture.’ 
— Dreyer, Tycho Brahe. 
Uraniborg. — The observatory of Uraniborg — the castle of 
the heavens — at Hveen was an extraordinary establishment. 
In a large square inclosure oriented according to the points of 
the compass, were several observatories, a library, laboratory, 
living-rooms and, later, workshops, a paper-mill and printing- 
press, and even underground observatories. The whole estab- 
lishment was administered with lavish extravagance, while Tycho 
was neither careful of his obligations nor free from arbitrary ar- 
rogance in his personal and administrative relations. In spite of 
these difficulties “ a magnificent series of observations, far transcend- 
ing in accuracy and extent anything that had been accomplished 
by his predecessors” was carried on for not less than 21 years. 
At the same time medicine and alchemy were also cultivated. 
Concerned as he was to secure the greatest possible accuracy, 
Tycho constructed instruments of great size; for example, a 
wooden quadrant for outdoor use with a brass scale of some ten 
feet radius, permitting readings to fractions of a minute. 
The best artists in Augsburg, clockmakers, jewellers, smiths, and 
carpenters, were engaged to execute the work, and from the zeal 
which so noble an instrument inspired, the quadrant was completed 
in less than a month. Its size was so great that twenty men could 
with difficulty transport it to its place of fixture. The two principal 
rectangular radii were beams of oak; the arch which lay between 
their extremities was made of solid wood of a particular kind, and the 
whole was bound together by twelve beams. It received additional 
strength from several iron bands, and the arch was covered with 
plates of brass, for the purpose of receiving the 5400 divisions into 
which it was to be subdivided. A large and strong pillar of oak, shod 
with iron, was driven into the ground, and kept in its place by solid Uraniborg. A NEW ASTRONOMY 
207 
mason work. To this pillar the quadrant was fixed in a vertical 
plane, and steps were prepared to elevate the observer, when stars of 
a low altitude required his attention. As the instrument could not 
be conveniently covered with a roof, it was protected from the weather 
by a covering made of skins; but notwithstanding this and other 
precautions, it was broken to pieces by a violent storm, after having 
remained uninjured for the space of five years. — Brewster. 
A smaller but more serviceable azimuth quadrant of brass gave 
angles to the nearest minute. He had a copper globe constructed 
at great expense with the positions of some 1000 stars carefully 
marked upon it. 
The very precision of his observations tended to confirm his 
scepticism of the Copernican hypothesis, as it seemed incredible 
that the earth’s supposed orbital motion should cause no change 
which he could detect in the position and brightness of the stars. 
He was also misled by supposing that the stars had measurable 
angular magnitude. He was not successful in making any funda- 
mental improvement in the relatively crude methods of time 
measurement, depending himself on wheel-mechanism without 
the regulating pendulum, and an apparatus of the sand-glass or 
clepsydra type. 
In 1577 Tycho made observations on a brilliant comet, and 
drew from them important theoretical inferences; namely, that 
instead of being an atmospheric phenomenon, the comet was at 
least three times as remote as the moon, and that it was revolving 
about the sun at a greater distance than Venus — unimpeded by 
the familiar crystalline spheres. He was even led, in discussing 
apparent irregularities of its motion, to suggest that its orbit 
might be oval — foreshadowing one of Kepler’s great discoveries. 
According to the current view of his time, comets 
were formed by the ascending from the earth of human sins and 
wickedness, formed into a kind of gas, and ignited by the anger of 
God. This poisonous stuff falls down again on people’s heads, and 
causes all kinds of mischief, such as pestilence, Frenchmen (!), sudden 
death, bad weather, etc. — Dreyer, Tycho Brahe. 208 
A SHORT HISTORY OF SCIENCE 
Eleven years later Tycho published a volume on the comet as 
a part of a comprehensive astronomical treatise which was, how- 
ever, never completed. About the same time his royal patron 
died, and the new administration proved less sympathetic with 
the great astronomer’s work and less indulgent with his extrava- 
gance and personal eccentricities. 
After a series of disagreements, Tycho withdrew from his ob- 
servatory in 1597, spent the winter in Hamburg, and after nego- 
tiations with different sovereigns, accepted the invitation of the 
Emperor Rudolph to settle in Prague in 1599. Here he again 
organized a staff of assistants, including, to the great advantage 
of himself and of his science, the young Kepler, but his further 
progress was prematurely terminated by death in 1601, at the age 
of 55. 
Tycho’s chief services to the progress of astronomy consisted 
first, in the superior accuracy of his instruments and observations, 
heightened by repetition and systematic correction of errors; 
second, in the extension of these observations over a long series of 
years. In both respects he departed from current practice, and 
anticipated the modern. In point of accuracy his errors of star- 
places seem rarely to have exceeded 1' to 2', and he even de- 
termined the length of the year within one second. While he 
recomputed almost every important astronomical constant, he 
accepted the traditional distance of the sun. 
Kepler gave striking evidence later of his confidence in Tycho’s 
accuracy by writing: — 
‘ Since the divine goodness has given to us in Tycho Brahe a most 
careful observer, from whose observations the error of 8' is shewn in 
this calculation, ... it is right that we should with gratitude recog- 
nize and make use of this gift of God. ... For if I could have 
treated 8' of longitude as negligible I should have already corrected 
sufficiently the hypothesis . . . discovered in chapter xvi. But as 
they could not be neglected, these 8' alone have led the way towards 
the complete reformation of astronomy, and have made the subject- 
matter of a great part of this work.’ — Berry. A NEW ASTRONOMY 
209 
On the other hand, Tycho was not strong on the theoretical side. 
He was never willing to accept the (5opernican hypothesis of 
rotation and orbital motion of the earth — maintaining, for ex- 
ample, that if the earth moved, a stone dropped from the top of a 
tower must fall at a distance from the foot. Again with refer- 
ence to the apparent displacement of the stars which would be 
expected to result from orbital motion of the earth, he says: — 
A yearly motion would relegate the sphere of the fixed stars to 
such a distance that the path described by the earth must be insig- 
nificant in comparison. Dost thou hold it possible that the space 
between the sun, the alleged centre of the universe, and Saturn 
amounts to not even of that distance ? At the same time this 
space must be void of stars. 
Sensible, however, of the weakness of the Ptolemaic theory, he 
devised an ingenious compromise in which the planets revolved 
about the Sun in their respective periods, and the entire heavens 
about the earth daily — all of which is not mathematically dif- 
ferent from the Copernican theory. 
We see in him at the same time a perfect son of the sixteenth 
century, believing the universe to be woven together by mysterious 
connecting threads which the contemplation of the stars or of the 
elements of nature might unravel, and thereby lift the veil of the 
future; we see that he is still, like most of his contemporaries, a 
believer in the solid spheres and the atmospherical origin of comets, 
to which errors of the Aristotelean physics he was destined a few 
years later to give the death-blow by his researches on comets; 
we see him also thoroughly discontented with his surroundings, and 
looking abroad in the hope of finding somewhere else the place and 
the means for carrying out his plans. 
As a practical astronomer Tycho has not been surpassed by any 
observer of ancient or modern times. The splendor and number of 
his instruments, the ingenuity which he exhibited in inventing new ones 
and in improving and adding to those which were formerly known, 
and his skill and assiduity as an observer, have given a character to 
his labors and a value to his observations which will be appreciated 
to the latest posterity. — Brewster. 210 
A SHORT HISTORY OF SCIENCE 
Kepler. — Pierre de la Ramee, or Petrus Ramus, a French 
mathematician and philosopher, impatient with the cumbrous 
astronomical hypotheses of the ancients, and unsatisfied with 
Copernicus’ proposed simplification, published a work in 1569 
expressing the hope 
‘ that some distinguished German philosopher would arise and found 
a new astronomy on careful observations by means of logic and mathe- 
matics, discarding all the notions of the ancients.’ 
Within a few months he discussed the matter at length with 
Tycho Brahe at Augsburg. Without accepting Ramus’ views, 
the young astronomer did make it his life work to lay the neces- 
sary foundation for such a new astronomy. Thirty years later, 
Mastlin, professor at Tubingen, wrote his former student Kepler 
— then aged 28 — 
that Tycho ‘had hardly left a shadow of what had hitherto been 
taken for astronomical science, and that only one thing was certain, 
which was that mankind knew nothing of astronomical matters.’ 
Born late in 1571 in Wiirtemberg, of Protestant parents in 
very straitened circumstances, Johann Kepler’s whole life was a 
struggle against poverty, ill-health, and adverse conditions. In 
1594, abandoning with some hesitation theological studies, for 
which his acceptance of the new Copernican hypothesis dis- 
qualified him, he was appointed lecturer on mathematics at Gratz. 
Students were few, and his duties included the preparation of a 
yearly almanac, containing, besides what its name implies, a 
variety of weather predictions and astrological information. 
“Mother Astronomy,” he says, “would surely have to suffer 
hunger if the daughter Astrology did not earn their bread.” 
Becoming thus more interested in astronomy, “there were,” he 
says, “three things in particular: viz., the number, the size, and 
the motion of the heavenly bodies, as to which I searched zealously 
for reasons why they were as they were and not otherwise.” The 
first result which seemed to him important, though somewhat 
fantastic from our standpoint, was a crude correspondence be- Kepler (Opera omnia). A NEW ASTRONOMY 
211 
tween the planetary orbits and the five regular solids, published 
in 1596 under a title which may be abridged to Cosmographic 
Mystery. 
The Earth is the circle, the measure of all. Round it describe a 
dodecahedron, the circle including this will be Mars. Round Mars de- 
scribe a tetrahedron, the circle including this will be Jupiter. De- 
scribe a cube round Jupiter, the circle including this will be Saturn. 
Then inscribe in the Earth an icosahedron, the circle inscribed in it 
will be Venus. Inscribe an octahedron in Venus, the circle inscribed 
in it will be Mercury. 
Kepler declared that he would not renounce the glory of this 
discovery “for the whole Electorate of Saxony.” The corre- 
spondence of the dimensions of this fantastic geometrical con- 
struction with the distances of members of our solar system is 
in reality far from close, but both Tycho Brahe and Galileo 
seem to have been favorably impressed by the book. 
The difficulties of Kepler’s position as a Protestant in Gratz 
led him, after a preliminary visit, to accept an engagement as 
Tycho’s assistant at Prague. 
The powers of original genius were then for the first time as- 
sociated with inventive skill and patient observation, and though the 
astronomical data provided by Tycho were sure of finding their ap- 
plication in some future age, yet without them, Kepler’s speculations 
would have been vain and the laws which they enabled him to deter- 
mine would have adorned the history of another century. — Brewster. 
In 1602 Kepler succeeded Tycho as imperial mathematician. 
Most fortunately, also, he secured possession of his chief’s great 
collection of observations, though not of the instruments, — a 
matter of less consequence, since Kepler like Copernicus was a 
mathematician rather than an observer. To the study of these 
records he devoted the next 25 years. Among all the planetary 
observations of Tycho Brahe those of Mars presented the irregu- 
larities most difficult of explanation, and it was these which, having 
been originally assigned to Kepler, engrossed his attention for 
many years, and in the end led to some of his finest discoveries. 212 
A SHORT HISTORY OF SCIENCE 
The Copernican theory like the Ptolemaic involved the resolu- 
tion of the motion of each planet into a main circular motion, 
modified by superimposing other circular motions — epicycles — 
successively upon it, each circle being the path of the centre of the 
next. Even after disentangling the essential irregularities of 
Mars’ orbit from those merely due to irregular motion of the earth, 
he could still obtain no satisfactory agreement with Tycho’s 
records, of which, as has been said, he refused to doubt the ac- 
curacy. Taking advantage of his own failure — as happens to 
men of true genius — he abandoned the restriction of circular 
motions, and experimented with other closed curves, of which 
the ellipse is simplest. Taking the sun at a focus, the problem 
was at last solved, theory and observation reconciled within due 
limits of error. At the same time uniform motion was naturally 
abandoned, for with a non-circular orbit, it was evident that the 
planet could not describe both equal distances and equal areas 
in equal times. Here, again, Kepler’s scientific imagination led 
him to the great discovery that the planet traverses its orbit in 
such a manner that a line joining it to the sun would describe 
sectors of equal area in equal times, the planet thus moving 
fastest when nearest the sun. 
Of Kepler’s celebrated three laws, the first two are: 
The planet describes an ellipse, the sun being in one focus. 
The straight line joining the planet to the sun sweeps out 
equal areas in equal intervals of time. 
These results were published in 1609 as part of extended Com- 
mentaries on the Motions of Mars. 
The great problem was solved at last, the problem which had 
baffled the genius of Eudoxus and had been a stumbling-block to the 
Alexandrian astronomers, to such an extent that Pliny had called 
Mars the inobservabile sidus. The numerous observations made 
by Tycho Brahe, with a degree of accuracy never before attained, 
had in the skilful hand of Kepler revealed the unexpected fact that 
Mars describes an ellipse, in one of the foci of which the sun is situated, 
and that the radius vector of the planet sweeps over equal areas in 
equal times. And the genius and astounding patience of Kepler had A NEW ASTRONOMY 
213 
proved that not only did this new theory satisfy the observations, 
but that no other hypothesis could be made to agree with the obser- 
vations, as every proposed alternative left outstanding errors, such 
as it was impossible to ascribe to errors of observation. Kepler had 
therefore, unlike all his predecessors, not merely put forward a new 
hypothesis which might do as well as another to enable a computer 
to construct tables of the planet’s motion; he had found the actual 
orbit in which the planet travels through space. 
In the history of astronomy there are only two other works of 
equal importance, the book De Revolutionibus of Copernicus and 
the Principia of Newton. The ‘astronomy without hypothesis’ 
demanded by Ramus had at last been produced, and well might 
Kepler proclaim: 
‘ It is well, Ramus, that you have run from this pledge, by quitting 
life and your professorship; if you held it still, I should, with justice, 
claim it.’ 
Resuming later the tendency of his Cosmographic Mystery, 
he published in 1619 his Harmony of the World, containing his 
third law: — 
The squares of the times of revolution of any two planets (in- 
cluding the earth) about the sun are proportional to the cubes of 
their mean distances from the sun. 
In his delight he exclaims ‘Nothing holds me, I will indulge in 
my sacred fury; I will triumph over mankind by the honest confession 
that I have stolen the golden vases of the Egyptians to build up a 
tabernacle for my God, far away from the confines of Egypt.’ — 
‘ What sixteen years ago, I urged as a thing to be sought, that for 
which I joined Tycho Brahe, for which I settled in Prague, for which 
I have devoted the best part of my life to astronomical contempla- 
tions — at length I have brought to light, and recognized its truth 
beyond my most sanguine expectations. It is not eighteen months 
since I got the first glimpse of light, three months since the dawn, 
very few days since the unveiled sun, most admirable to gaze on, 
burst out upon me.’ . . . 
Archimedes of old had said “Give me a place to stand on, and 
I shall move the world.” Tycho Brahe had given Kepler the 
place to stand on, and Kepler did move the world. 214 
A SHORT HISTORY OF SCIENCE 
It should be borne in mind that Kepler’s results depend not 
on a priori theory for their confirmation, but upon actual ob- 
servations supporting them and interpreted by them. The great 
further step of showing that the three laws are not independent 
and empirical, but mathematical consequences of a single me- 
chanical law still awaited the genius of Newton. 
Kepler’s notions in regard to force and motion are still crude. 
Thus, for example, having in mind an analogy with magnetism, 
Kepler says in his Epitome of the Copernican Astronomy, 
(1618-1621): — 
' There is therefore a conflict between the carrying power of the 
sun and the impotence or material sluggishness (inertia) of the planet; 
each enjoys some measure of victory, for the former moves the planet 
from its position and the latter frees the planet’s body to some extent 
from the bonds in which it is thus held . . . but only to be captured 
again by another portion of this rotatory virtue.’ 
Elsewhere he says: — 
‘ We must suppose one of two things: either that the moving spirits, 
in proportion as they are more removed from the sun, are more feeble; 
or that there is one moving spirit in the centre of all the orbits, 
namely, in the sun, which urges each body the more vehemently in 
proportion as it is nearer; but in more distant spaces languishes in 
consequence of the remoteness and attenuation of its virtue.’ 
— Whew ell. 
He recognized the necessity of a force exercised by the sun, but 
believed it inversely proportional to the distance instead of to 
the square of the distance. His notions of gravity are expressed 
in his book on Mars: — 
‘ Every bodily substance will rest in any place in which it is placed 
isolated, outside the reach of the power of a body of the same kind. 
Gravity is the mutual tendency of cognate bodies to join each other 
(of which kind the magnetic force is), so that the earth draws a stone 
much more than the stone draws the earth. Supposing that the 
earth were in the centre of the world, heavy bodies would not seek 
the centre of the world as such, but the centre of a round, cognate A NEW ASTRONOMY 
215 
body, the earth; and wherever the earth is transported heavy bodies 
will always seek it; but if the earth were not round they would not 
from all sides seek the middle of it, but would from different sides be 
carried to different points. If two stones were situated anywhere in 
space near each other, but outside the reach of a third cognate body, 
they would after the manner of two magnetic bodies come together 
at an intermediate point, each approaching the other in proportion 
to the attracting mass. And if the earth and the moon were not kept 
in their orbits by their animal force, the earth would ascend towards 
the moon one fifty-fourth part of the distance, while the moon would 
descend the rest of the way and join the earth, provided that the two 
bodies are of the same density. If the earth ceased to attract the 
water all the seas would rise and flow over the moon. — Dreyer. 
Kepler’s last important published work was his Rudolphine 
Tables (1627), embodying the accumulated results of Tycho’s 
work and his own, and remaining a standard for a century. It is 
noteworthy that during Kepler’s work on these tables, mathe- 
matical computation was peacefully revolutionized by the intro- 
duction of logarithms, newly discovered by Napier and Biirgi. 
In 1628, after vain attempts to collect arrears of his salary as 
imperial mathematician, he even joined Wallenstein as astrologer, 
but died soon after at Regensburg in 1630. 
Kepler also wrote an important work on Dioptrics with a mathe- 
matical discussion of refraction and the different forms of the 
newly invented telescope, the whole constituting the foundation 
of modern optics. In it he develops the first correct theory of 
vision, “Seeing amounts to feeling the stimulation of the retina, 
which is painted with the colored rays of the visible world. The 
picture must then be transmitted to the brain by a mental cur- 
rent, and delivered at the seat of the visual faculty.” He sup- 
poses that color depends on density and transparency, and that 
refraction is due to greater resistance of a dense medium. He 
enunciates the law that intensity of light varies inversely as the 
square of the distance. “ In proportion as the spherical surface 
from whose centre the light proceeds is greater or smaller, so is 
the strength or density of the light-rays which fall on the smaller 216 
A SHORT HISTORY OF SCIENCE 
sphere to the strength of those rays which fall on the larger 
sphere.” He explains the estimation of distance by binocular 
vision. He supposes the velocity of light to be infinite. His 
more purely mathematical work will be mentioned in a later 
chapter. 
Kepler added Plato’s boldness of fancy to his own patient and 
candid habit of testing his fancies by a rigorous and laborious com- 
parison with the phenomena; and thus his discoveries led to those of 
Newton. — Whew ell. 
If Kepler had burnt three-quarters of what he printed, we should 
in all probability have formed a higher opinion of his intellectual 
grasp and sobriety of judgment, but we should have lost to a great 
extent the impression of extraordinary enthusiasm and industry, and 
of almost unequalled intellectual honesty, which we now get from a 
study of his works.— Berry. 
Kepler says: ‘ If Christopher Columbus, if Magellan, if the 
Portuguese, when they narrate their wanderings, are not only ex- 
cused, but if we do not wish these passages omitted, and should lose 
much pleasure if they were, let no one blame me for doing the same.’ 
Kepler’s talents were a kindly and fertile soil, which he cultivated 
with abundant toil and vigor, but with great scantiness of agricultural 
skill and implements. Weeds and the grain throve and flourished 
side by side almost undistinguished; and he gave a peculiar appear- 
ance to his harvest, by gathering and preserving the one class of plants 
with as much care and diligence as the other. — Whew ell. 
Endowed with two qualities, which seemed incompatible with 
each other, a volcanic imagination and a pertinacity of intellect which 
the most tedious numerical calculations could not daunt, Kepler 
conjectured that the movements of the celestial bodies must be con- 
nected together by simple laws, or, to use his own expression, by 
harmonic laws. These laws he undertook to discover. A thousand 
fruitless attempts, errors of calculation inseparable from a colossal 
undertaking, did not prevent him a single instant from advancing 
resolutely toward the goal of which he imagined he had obtained a 
glimpse. Twenty-two years were employed by him in this investiga- 
tion, and still he was not weary of it! What, in reality, are twenty- 
two years of labor to him who is about to become the legislator of 
worlds; who shall inscribe his name in ineffaceable characters upon Galileo (Opere, 1744). A NEW ASTRONOMY 
217 
the frontispiece of an immortal code; who shall be able to exclaim 
in dithyrambic language, and without incurring the reproach of any- 
one, ‘ The die is cast; I have written my book; it will be read either 
in the present age or by posterity, it matters not which; it may well 
await a reader, since God has waited six thousand years for an inter- 
preter of his words.’ — Arago. 
The philosophical significance of Kepler’s discoveries was not 
recognized by the ecclesiastical party at first. It is chiefly this, that 
they constitute a most important step to the establishment of the 
doctrine of the government of the world by law. But it was im- 
possible to receive these laws without seeking for their cause. The 
result to which that search eventually conducted not only explained 
their origin, but also showed that, as laws, they must, in the necessity 
of nature, exist. It may be truly said that the mathematical exposi- 
tion of their origin constitutes the most splendid monument of the 
intellectual power of man. — Draper. 
Galileo. — Columbus discovered America when Copernicus 
was but 19, and before the birth of Tycho Brahe, Magellan had 
completed the proof of the earth’s rotundity by actually 
sailing around it, while Luther had stirred up the great religious 
revolt of Protestantism. The later years of Kepler and Galileo 
fell within the period of the Thirty Years’ War, of which neither 
was to witness the close. Permanent English settlements in 
America had just begun. Galileo (1564-1642), born on the day 
of Michael Angelo’s death, “ nature seeming to signify thereby the 
passing of the sceptre from art to science, ” and in the same year 
with Shakespeare, exerted a mighty influence on the development 
of science in many fields, and in particular laid the foundations 
of modern dynamics. 
It is a remarkable circumstance in the history of science that 
astronomy should have been cultivated at the same time by three such 
distinguished men as Tycho, Kepler and Galileo. While Tycho in 
the 54th year of his age was observing the heavens at Prague, Kepler, 
only 30 years old, was applying his wild genius to the determination 
of the orbit of Mars, and Galileo, at the age of 36, was about to direct 
the telescope to the unexplored regions of space. The diversity of 
gifts which Providence assigned to these three philosophers was no 218 
A SHORT HISTORY OF SCIENCE 
less remarkable. Tycho was destined to lay the foundation of modern 
astronomy by a vast series of accurate observations made with the 
largest and the finest instruments; it was the proud lot of Kepler to 
deduce the laws of the planetary orbits from the observations of his 
predecessors; while Galileo enjoyed the more dazzling honor of 
discovering by the telescope new celestial bodies and new systems 
of worlds. — Brewster. 
Coming into a world still dominated by the Aristotelian tradi- 
tion, Galileo is puzzled by the conflict between his own obser- 
vations and the accepted theories, but firm and fearless in his 
convictions, he eagerly and powerfully controverts the older notions, 
incidentally gaining enemies as well as disciples. What those 
accepted theories were may be exemplified by the following pas- 
sages from a work of Daniel Schwenter (1585-1636), professor of 
mathematics at Altdorf:— 
‘When a body falls it moves faster the nearer it approaches the 
earth. The farther it falls the more power it possesses. For every- 
thing which is heavy, hastens according to the opinion of philosophers 
towards its natural place, that is the centre of the earth, just as man 
returning to his fatherland becomes the more eager the nearer he 
comes, and therefore hastens so much the more. Still another natural 
cause contributes to this. The air which is parted by the falling ball, 
hastens together again behind the ball and drives it always harder.’ 
If the Copernican theory were true, the bullet remaining two 
minutes in the air would be left many miles behind by the revolv- 
ing earth, — a distance which the moving atmosphere could not 
possibly carry it. The rainbow is “a mirror in which the human 
understanding can behold its ignorance in broad day.” The 
powder drives the bullet in an oblique line to the highest point 
of its path, then follows motion in an arc, finally, the natural 
motion vertically downward. 
In his whole point of view and habit of mind Galileo embodied 
the attitude and spirit of modern science. He was keenly alert 
in observing, analyzing, and reflecting on natural phenomena, 
eager and convincing in his expositions, sceptical and intolerant A NEW ASTRONOMY 
219 
of mere authority, whether in science, philosophy, or theology. 
It was a true instinct of the conservatives to recognize in him 
the champion of a principle fatally hostile to their own. Between 
these antagonistic principles no permanent peace was possible. 
While still a mere youth, he discovered the regularity of pendulum 
vibrations by observing the slow swinging of the cathedral lamp 
of Pisa (1582). Before he was 25 he published work on the 
hydrostatic balance (1586), and on the centre of gravity of solids. 
Only a little later he conducted at the leaning tower simple ex- 
periments in falling bodies, which upset world-old notions on 
this everyday matter, showing that the velocity of descent is 
not, as was commonly supposed, proportional to weight. And 
“yet the Aristotelians, who with their own eyes saw the unequal 
weights strike the ground at the same instant, ascribed the effect 
to some unknown cause, and preferred the decision of their master 
to that of nature herself.” 
He further showed that the hypothesis of uniform acceleration 
accounted correctly for the observed relations between space, 
time and velocity, and that the path of a projectile is a parabola. 
In the words of a recent authority, when Galileo 
deduced by experiment, and described with mathematical pre- 
cision, the acceleration of a falling body, he probably contributed 
more to the physical sciences than all the philosophers who had 
preceded him. 
Hearing of the telescope newly invented in Holland, he con- 
structed one for himself, by means of which he discovered sun 
spots, the mountains of the moon, the satellites of Jupiter, the 
rings of Saturn, and the phases of Venus. The sensation created 
by these discoveries is described in the following passages from 
Fahie’s Life of Galileo and Brewster’s Martyrs of Science. 
‘As the news had reached Venice that I had made such an in- 
strument, six days ago I was summoned before their Highnesses, the 
Signoria, and exhibited it to them, to the astonishment of the whole 
senate. Many of the nobles and senators, although of a great age, 
mounted more than once to the top of the highest church tower in 220 
A SHORT HISTORY OF SCIENCE 
Venice, in order to see sails and shipping that were so far off that it 
was two hours before they were seen, without my spy-glass, steering 
full sail into the harbour; for the effect of my instrument is such that 
it makes an object 50 miles off appear as large as if it were only 
five/ 
‘ But the greatest marvel of all is the discovery of four new planets. 
I have observed their motions proper to themselves and in relation 
to each other, and wherein they differ from the motions of the other 
planets. These new bodies move round another very great star, in 
the same way as Mercury and Venus, and, peradventure, the other 
known planets, move round the sun. As soon as my tract is printed, 
which I intend sending as an advertisement to all philosophers and 
mathematicians, I shall send a copy to his Highness, the Grand Duke, 
together with an excellent spy-glass, which will enable him to judge 
for himself of the truth of these novelties/ — Fahie. 
Galileo’s discoveries on the surface of the moon were ill received 
by the followers of Aristotle. According to their preconceived opin- 
ions, the moon was perfectly spherical and absolutely smooth; and to 
cover it with mountains and scoop it out into valleys was an act of 
impiety which defaced the regular forms which Nature herself had 
imprinted. It was in vain that Galileo appealed to the evidence 
of observation and to the actual surface of our own globe. The very 
irregularities on the moon were, in his opinion, a proof of divine wisdom; 
and had its surface been absolutely smooth, it would have been ‘ but 
a vast unblessed desert, void of animals, of plants, of cities, and men 
— the abode of silence and inaction — senseless, lifeless, soulless, and 
stripped of all those ornaments which now render it so varied and so 
beautiful/ 
In examining the fixed stars and comparing them with the planets, 
Galileo observed a remarkable difference in the appearance of their 
discs. All the planets appeared with round globular discs like the 
moon; whereas the fixed stars never exhibited any disc at all but 
resembled lucid points sending forth twinkling rays. Stars of all 
magnitudes he found to have the same appearance; those of the 
fifth and sixth magnitude having the same character, when seen 
through a telescope, as Sirius, the largest of the stars, when seen by 
the naked eye. 
Important and interesting as these discoveries were, they were 
thrown into the shade by those to which he was led during a careful A NEW ASTRONOMY 
221 
examination of the planets with a more powerful telescope. On the 
7th of January, 1610, at one o’clock in the morning, when he directed 
his telescope to Jupiter, he observed three stars near the body of the 
planet, two being to the east and one to the west of him. They were 
all in a straight line, and parallel to the ecliptic and appeared brighter 
than other stars of the same magnitude. Believing them to be fixed 
stars, he paid no great attention to their distances from Jupiter and 
from one another. On the 8th of January, however, when, from some 
cause or other, he had been led to observe the stars again, he found a 
very different arrangement of them; all the three were on the west 
side of Jupiter, nearer one another than before and almost at equal 
distances. Though he had not turned his attention to the extraordi- 
nary fact of the mutual approach of the stars, yet he began to con- 
sider how Jupiter could be found to the east of the three stars, when 
but the day before he had been to the west of two of them. The only 
explanation which he could give of this fact was that the motion of 
Jupiter was direct, contrary to the astronomical calculations and 
that he had got before these two stars by his own motion. 
In this dilemma between the testimony of his senses and the results 
of calculation, he waited for the following night with the utmost 
anxiety, but his hopes were disappointed, for the heavens were wholly 
veiled in clouds. On the 10th, two only of the stars appeared, and 
both on the east side of the planet. As it was obviously impossible 
that Jupiter could have advanced from west to east on the 8th of 
January, and from east to west on the 10th, Galileo was forced to 
conclude that the phenomenon which he had observed arose from the 
motion of the stars, and he set himself to observe diligently their change 
of place. On the 11th there were still only two stars, and both to the 
east of Jupiter, but the more eastern star was now twice as large as 
the other one, though on the preceding night they had been per- 
fectly equal. This fact threw a new light upon Galileo’s difficulties, 
and he immediately drew the conclusion, which he considered to be 
indubitable, ‘ that there were in the heavens three stars which revolve 
around Jupiter, in the same manner as Venus and Mercury revolve 
around the sun.’ On the 12th of January he again observed them in 
new positions, and of different magnitudes; and on the 13th he 
discovered a fourth star, which completed the four secondary planets 
with which Jupiter is surrounded. —Brewster. 
His results were published in ‘The Sidereal Messenger,’ announc- 222 
A SHORT HISTORY OF SCIENCE 
ing ‘great and very wonderful spectacles, and offering them to the 
consideration of every one, but especially of philosophers and as- 
tronomers; which have been observed by Galileo Galilei ... by 
the assistance of a perspective glass lately invented by him; namely 
in the face of the moon, in innumerable fixed stars in the Milky Way, 
in nebulous stars, but especially in four planets which revolve around 
Jupiter at different intervals and periods with a wonderful celerity; 
which, hitherto not known to any one, the author has recently been 
the first to detect, and has decreed to call the Medicean stars.’ 
The reception which these discoveries met with from Kepler is 
highly interesting, and characteristic of the genius of that great man. 
He was one day sitting idle and thinking of Galileo, when his friend 
Wachenfels stopped his carriage at his door to communicate to him 
some intelligence. ‘Such a fit of wonder,’ says he, ‘seized me at a 
report which seemed to be so very absurd, and I was thrown into such 
agitation at seeing an old dispute between us decided in this way, 
that between his joy, my coloring, and the laughter of both, confounded 
as we were by such a novelty, we were hardly capable, he of speaking, 
or I of listening. On our parting, I immediately began to think how 
there could be any addition to the number of the planets without 
overturning my Cosmographic Mystery, according to which Euclid’s 
five regular solids do not allow more than six planets round the sun. . . 
I am so far from disbelieving the existence of the four circumjovial 
planets, that I long for a telescope, to anticipate you, if possible, in 
discovering two round Mars, as the proportion seems to require, six 
or eight round Saturn, and perhaps one each round Mercury and 
Venus.’ 
In a very different spirit did the Aristotelians receive the Sidereal 
Messenger of Galileo. The principal professor of philosophy at 
Padua resisted Galileo’s repeated and urgent entreaties to look at 
the moon and planets through his telescope; and he even labored to 
convince the Grand Duke that the satellites of Jupiter could not pos- 
sibly exist.1 
‘ There are seven windows given to animals in the domicile of the 
head, through which the air is admitted to the tabernacle of the body, 
— Whew ell. 
1 * As I wished to show the satellites of Jupiter to the professors in Florence, they 
would neither see them nor the telescope. These people believe there is no truth 
to seek in nature, but only in the comparison of texts.’ A NEW ASTRONOMY 
223 
to enlighten, to warm, and to nourish it. What are these parts of the 
microcosmos ? Two nostrils, two eyes, two ears, and a mouth. So in 
the heavens, as in a macrocosmos, there are two favorable stars, two 
unpropitious, two luminaries, and Mercury undecided and indifferent. 
From this and many other similarities in nature, such as the seven 
metals, etc. which it were tedious to enumerate, we gather that the 
number of planets is necessarily seven. Moreover, these satellites of 
Jupiter are invisible to the naked eye, and therefore can exercise no 
influence on the earth, and therefore would be useless, and therefore 
do not exist. Besides, the Jews and other ancient nations, as well 
as modern Europeans, have adopted the division of the week into 
seven days, and have named them after the seven planets. Now, 
if we increase the number of the planets, this whole and beautiful 
system falls to the ground.’ — Fahie. 
It was inevitable that such a man as Galileo should accept the 
Copernican hypothesis. He writes to Kepler in 1597: — 
‘I esteem myself fortunate to have found so great an ally in the 
search for truth. It is truly lamentable, that there are so few who 
strive for the true and are ready to turn away from wrong ways of 
philosophizing. But here is no place for bewailing the pitifulness of 
our times, instead of wishing you success in your splendid investiga- 
tions. I do this the more gladly, since I have been for many years 
an adherent of the Copernican theory. It explains to me the cause 
of many phenomena which under the generally accepted theory are 
quite unintelligible. I have collected many arguments for refuting the 
latter, but I do not venture to bring them to publication. 
* That the moon is a body like the earth I have long been assured. 
I have also discovered a multitude of previously invisible fixed stars, 
outnumbering more than ten times those which can be seen by the 
naked eye, — forming the Milky Way. Further I have discovered 
that Saturn consists of three spheres which almost touch each 
other.’ 
While none of Galileo’s astronomical discoveries were either 
necessary or sufficient to confirm the Copernican theory, their 
support was exceedingly important. Thus the slow motion of 
the sun spots across the disc and their subsequent reappearance 224 
A SHORT HISTORY OF SCIENCE 
showed rotation of that body, the satellites of Jupiter and par- 
ticularly the phases of Venus, analogous to those shown by the 
moon, obviously harmonized with the Copernican theory. This 
implied at least that the planets shone by reflected sunlight, and 
it had indeed been insisted against that theory that Venus and 
Mercury under it must show phases till then undiscovered. 
In 1632 Galileo published his celebrated Dialogue on the Two 
Chief Systems of the World, the Ptolemaic and the Copernican, 
a work comparable in magnitude and importance with Copernicus’ 
Revolutions. In the curious preface he says: — 
‘ Judicious reader, there was published some years since in Rome a 
salutiferous Edict, that, for the obviating of the dangerous Scandals 
of the present Age, imposed a reasonable Silence upon the Pythag- 
orean Opinion of the Mobility of the Earth. There want not such as 
unadvisedly affirm, that the Decree was not the production of a sober 
Scrutiny, but of an informed passion; and one may hear some mutter 
that Consultors altogether ignorant of Astronomical observations 
ought not to clipp the wings of speculative wits with rash prohibitions. 
My zeale cannot keep silence when I hear these inconsiderate com- 
plaints. I thought fit, as being thoroughly acquainted with that 
prudent Determination, to appear openly upon the Theatre of the 
World as a Witness of the naked Truth. ... I hope that by these 
considerations the world will know that if other Nations have Navi- 
gated more than we, we have not studied less than they; and that 
our returning to assert the Earth’s stability, and to take the contrary 
only for a Mathematical Capriccio, proceeds not from inadvertency 
of what others have thought thereof, but (had one no other induce- 
ments), from these reasons that Piety, Religion, the Knowledge of the 
Divine Omnipotency, and a consciousness of the incapacity of man’s 
understanding dictate unto us.’ 
In the first of the four conversations into which the work is 
divided, the Aristotelian theory of the peculiar character of the 
heavenly bodies is subjected to destructive criticism, with em- 
phasis on such phenomena as the appearance of new stars, of 
comets and of sun spots, the irregularities of the moon’s surface, 
the phases of Venus, the satellites of Jupiter, etc. Galileo’s Dialogue A NEW ASTRONOMY 
225 
‘When we consider merely the vast dimensions of the celestial 
sphere in comparison with the littleness of our earth . . . and then 
think of the speed of the motion by which a whole revolution of the 
heavens must be accomplished in one day, I cannot persuade myself 
that the heavens turn while the earth stands fast.’ 
Adducing not merely the sun spots themselves, but their rapid 
variation, he insists that the universe is not rigid and permanent, 
but constantly changing or, as science has more and more em- 
phasized since his day, passing through consecutive, related 
phases or evolving. 
‘I can listen only with the greatest repugnance when the quality 
of unchangeability is held up as something preeminent and complete 
in contrast to variability. I hold the earth for most distinguished 
exactly on account of the transformations which take place upon it.’ 
He begins to see the fallacy of the objections that if the earth 
rotated, a body dropped from a masthead would be left behind 
by the ship and that movable objects could be thrown off centrif- 
ugally at the equator. As positive arguments in support of the 
Copernican system, he urges particularly the retrogressions and 
other irregularities of the planets, and also the tides. 
Of the famous controversy of Galileo with the Inquisition, it 
may here suffice to quote the judgment of the court (see Appen- 
dix) : — 
‘The proposition that the sun is in the centre of the world and 
immovable from its place is absurd, philosophically false and formally 
heretical; because it is expressly contrary to the Holy Scriptures,’ etc. 
and a passage from the biographer already cited at so much 
length: — 
For over fifty years he was the knight militant of science, and 
almost alone did successful battle with the hosts of-Churchmen and 
Aristotelians who attacked him on all sides — one man against a 
world of bigotry and ignorance. If then, . . . once, and only once, 
when face to face with the terrors of the Inquisition, he, like Peter, 
denied his Master, no honest man, knowing all the circumstances, 
will be in a hurry to blame him. 226 
A SHORT HISTORY OF SCIENCE 
Of Galileo’s still more remarkable services to physics and 
dynamics, something will be added in a later chapter. 
Medical and Chemical Sciences. — These were still at the 
low medieval level. There was as yet no scientific medicine, and 
no chemistry but alchemy, which was now in its final stage, 
iatro (medical) chemistry. Here one great name is that of 
Paracelsus (1493-1541), erratic and radical Swiss physician and 
alchemist, whose chief merit is his courage in opposing mere 
authority in science, and whose influence long after caused 
“salt, sulphur, and mercury” to be highly regarded and carefully 
studied. He also introduced and insisted upon the importance 
of antimony as a remedy, and is said to have been the first to 
use that tincture of opium which is still known by his name for 
it; viz. laudanum. Paracelsus, on the other hand, in spite of 
the fact that he was a popular surgeon, rejected the study of 
anatomy, taught medical knowledge through scanning of the 
heavens, and considered diseases as spiritual in origin. “The 
true use of chemistry,” he said, “is not to.make gold but to 
prepare medicines.” 
Another name worthy of remembrance in the chemistry of the 
sixteenth century is that of Landmann (Latin, Agricola) whose 
great work on Metallurgy (De Re Metallica, 1546) is the most im- 
portant of this period, and who must also be regarded as the 
first mineralogist of modern times. 
Anatomy. Vesalius.—Hardly less important, meantime, than 
the studies of Copernicus, Tycho Brahe, Galileo and Kepler upon 
the heavenly bodies were those of the Belgian anatomist, Andreas 
Vesalius, upon the human body. For more than 1000 years there 
had been almost no progress in anatomy or medicine, Hippocrates 
and Galen being still regarded as the final authorities in these 
matters up to the middle of the sixteenth century. Vesalius 
(1514-1564), born in Brussels and educated in Paris, was the first 
in modern times to dissect the human body, and to publish excel- 
lent drawings of his dissections. It was said that he opened the 
body of a nobleman before the heart had entirely ceased beating, 
and thereby incurring the displeasure of the Inquisition, was sen- BEGINNINGS OF MODERN NATURAL SCIENCE 
227 
tenced to perform a penitential journey to Jerusalem. At all 
events, he went to Jerusalem and was shipwrecked and lost while 
returning. 
After Vesalius the study of human anatomy was vigorously and 
successfully prosecuted in Italy as was natural, since it was in Italy 
that Humanism and the revival of learning first took firm hold 
of Christian Europe. One of Vesalius’ Italian contemporaries, 
Eustachius, whose name is still familiarly associated with the pas- 
sage or “tube” connecting the throat and the middle ear, is hardly 
less famous in the history of anatomy than is Vesalius himself. 
The name of Fallopius, professor at Pisa in 1548 and at Padua 
in 1551, is also similarly associated with the human oviducts, 
— the so-called Fallopian tubes. His disciple Fabricius of 
Acquapendente discovered the valves in the veins, and was the 
teacher of William Harvey. A Spanish anatomist of note, Michael 
Servetus,—born 1509,—perished as a martyr at the stake in 
1553 because of heretical writings abhorrent alike to the In- 
quisition and to Calvin. 
Of physiology we have as yet little or no account. Doubtless 
all the anatomists just mentioned and many other “philosophers” 
had pondered, as did Aristotle and his predecessors, on the workings 
of the animal, and especially the human, mechanism. But from 
Aristotle (b.c. 322) to William Harvey (1578-1657) no real progress 
was made. It is a melancholy commentary on superstition and 
human prejudice that long after the brilliant work of Vesalius and 
the Italian anatomists, no proper “anatomy acts” existed to make 
lawful dissection either possible or easy, so that for several cen- 
turies afterward anatomists, surgeons, and medical students felt 
themselves at times obliged to resort to “ body-snatching.” 
Natural History and Natural Philosophy. — No great 
progress was made in this field after the observations of Aristotle, 
Theophrastus, Xenophanes, and Pythagoras until the sixteenth 
century. Fossils mostly remained unexplained or were regarded as 
“ freaks ” of nature. Animals and plants were comparatively neg- 
lected and, if studied, considered either as the raw material for sup- 
posed remedies or medicines, or else as treated by Aristotle. The 228 
A SHORT HISTORY OF SCIENCE 
twenty-six books De Animalibus, of Albertus Magnus (d. 1282) 
were not printed until 1478, but were apparently well known in 
manuscript copies. No great worker appears in this almost neg- 
lected field until we come to Conrad Gesner (or Gessner) (1516— 
1565), the first famous naturalist of modern times, on account 
of his vast erudition surnamed “the German Pliny.” Professor of 
Greek at Lausanne and later of Natural History at Basel, he was 
almost as prolific an author as was della Porta fifty years later, 
for he wrote extensively upon plants, animals, milk, medicine, 
and theology, as well as various classical subjects. Yet he ranks 
high in the history of biology, both for the extent and the quality 
of his work in zoology and botany. It is significant that Gesner 
was a Swiss, and as such probably safe from persecution at a 
time when William Turner, an English ornithologist, worked and 
published in Cologne. 
At the end of the fifteenth and beginning of the sixteenth 
centuries Leonardo da Vinci (1452-1519) turned his attention in 
part from art to science, engineering, and inventions, making 
interesting studies in architecture, hydraulics, geology, etc. He 
is regarded as the first engineer of modern times, and has 
been called “the world’s most universal genius.” Palissy, “the 
Potter,” later examined minutely various fossils and took the then 
advanced ground (as Xenophanes and Pythagoras had done, 
however, some two thousand years earlier) that these are in 
reality what they appear to be, i.e. petrified remains of plant and 
animal life, and not “freaks of nature.” Palissy’s bold stand on 
this subject marks one of the first steps in modern times toward 
rational geology. 
It was not until the end of the sixteenth century, when Wil- 
liam Gilbert, an eminent practising physician of Colchester, 
England (1540-1603), published his now famous work on the 
magnet (De Magnete) that further progress was made through 
the first rational treatment of electrical and magnetic phenomena. 
To him is due the name electricity (vis electrica). He regarded 
the earth as a great magnet and, accepting the Copernican 
theory, attributed the earth’s rotation to its magnetic character. BEGINNINGS OF MODERN NATURAL SCIENCE 
229 
He even extended this idea to the heavenly bodies, with an ani- 
mistic tendency. Gilbert is also reputed to have done important 
work in chemistry, but none of this has survived. 
His work is one of the finest examples of inductive philosophy that 
has ever been presented to the world. It is the more remarkable 
because it preceded the Novum Organum of Bacon, in which the 
inductive method of philosophizing was first explained. — Thomas 
Thomson. 
The most prolific writer on natural philosophy and physical 
science of the sixteenth century was G. della Porta (1543-1615), 
a native of Naples and a resident of Rome, founder of an early 
scientific academy there, and afterwards of the famous Accademia 
dei Lincei of Rome. His writings are voluminous and in many 
books, of which we need mention here only his Magia Naturalis, 
(1569), De Refractione (1593), Pneumatica (1691), De Distilla- 
tione (1604), De Munitione (1608) and De Aeris Transmutationi- 
bus (1609). 
In his Natural Magic, della Porta is the first to describe a 
camera obscura, besides touching on many interesting properties 
of lenses, and referring to spectacles, some forms of which had 
long been known. His work On Refraction deals largely with 
binocular vision, and is a criticism of the work of Euclid and 
Galen on that subject. The author hints also at a crude tele- 
scope, and may have known some form of stereoscope. Della 
Porta’s compositions range all the way from natural magic to 
Italian comedies, and entitle him to high rank as a tireless and 
original, if not especially fruitful, thinker and worker. 
References for Reading 
Berry. History of Astronomy. Chapters IV-VII. 
Brewster. Martyrs of Science. 
Dreyer. Tycho Brahe; Planetary Systems. Chapters XII-XVI. 
Fahie. Life of Galileo. 
Gilbert. On the Magnet. 
Locy. Biology and its Makers. 
Lodge. Pioneers of Science. CHAPTER XI 
PROGRESS OF MATHEMATICS AND MECHANICS IN THE 
SIXTEENTH CENTURY 
It was not alone the striving for universal culture which attracted 
the great masters of the Renaissance, such as Brunelleschi, Leonardo 
da Vinci, Raphael, Michael Angelo and especially Albrecht Diirer, with 
irresistible power to the mathematical sciences. They were conscious 
that, with all the freedom of the individual phantasy, art is subject 
to necessary laws and, conversely, with all its rigor of logical structure, 
mathematics follows esthetic laws. — Rudio. 
The miraculous powers of modern calculation are due to three 
inventions: the Arabic Notation, Decimal Fractions and Logarithms. 
— Cajori. 
The invention of logarithms and the calculation of the earlier tables 
form a very striking episode in the history of exact science, and, with 
the exception of the Principia of Newton, there is no mathematical 
work published in the country which has produced such important 
consequences, or to which so much interest attaches as to Napier’s 
Descriptio. — Glaisher. 
It is Italy, which is the fatherland of Archimedes, whose creative 
power embraces all domains of the mechanical science, the land of 
the Renaissance, from out of which those mighty waves of new ideas 
and new impulses in science and art have come forth into the world — 
the fatherland of Galileo the creator of experimental physics, of 
Leonardo da Vinci the engineer, of Lagrange who has given its form 
to modern analytical mechanics. — W. v. Dyck. 
Dynamics is really a product of modern times, and affords the 
rare example of a development fulfilled in a single great personage — 
Galileo. Nothing is finer than how he, beginning in the Aristotelian 
spirit, gradually frees himself from its bondage and, instead of empty 
metaphysics, introduces well-directed methodical investigations of 
nature. — Timer ding. 
The period from the invention of printing about 1450 to that of 
analytic geometry in 1637 was one of very great importance for 
mathematics and mechanics as well as for astronomy. At the 
230 PROGRESS OF MATHEMATICS AND MECHANICS 
231 
beginning, Arabic numerals were known, but the mathematics 
even of the universities hardly extended beyond the early books 
of Euclid and the solution of simple cases of quadratic equations 
in rhetorical form. At the end of the period the foundations 
of modern mathematics and mechanics were securely laid. 
Aims and Tendencies of Mathematical Progress. — In 
the centuries just preceding, the chief applications of mathematics 
had connected themselves with the relatively simple needs of trade, 
accounts and the calendar, with the graphical constructions of the 
architect and the military engineer, and with the sines and tangents 
of the astronomer and the navigator. During the period in ques- 
tion some of these applications became increasingly important, 
and at the same time mathematics was more and more cultivated 
for its own sake. Mathematicians became gradually a more and 
more distinctly differentiated class of scholars; mathematical text- 
books took shape. The beginnings of this evolution have been 
dealt with already; its further progress is now to be traced. 
The larger achievements and tendencies of the period in mathe- 
matical science were the following : — 
In Arithmetic, decimal fractions and logarithms were introduced, 
regulating and immensely simplifying computation; a general 
theory of numbers was developed; in Algebra, a compact and ade- 
quate symbolism was worked out, including the use of the signs 
+, -r-, X, —, =, (), V , and of exponents; equations of the 
third and fourth degree were solved, negative and imaginary roots 
accepted, and many theorems of our modern theory of equations 
discovered. 
In Geometry, the computation of 7r was carried to many deci- 
mals, the beginnings of projective geometry were made, and a 
so-called method of indivisibles developed, foreshadowing the 
integral calculus; in plane and spherical Trigonometry, the 
theorems and processes now in use were worked out, and extensive 
tables computed. 
In Mechanics, ideas about force and motion, equilibrium and 
centre of gravity, were gradually clarified. 
Underlying some of these new developments are the dawning 232 
A SHORT HISTORY OF SCIENCE 
fundamental concepts : function, continuity, limit, derivative, in- 
finitesimal, on which our modern mathematics has been built up. 
Descartes, Newton and Leibnitz are soon to make their revolu- 
tionary discoveries in analytic geometry and the calculus. 
We have seen that up to about 1500 the chief stages in the de- 
velopment of mathematics have been the introduction and im- 
provement of Arabic arithmetic for commercial purposes (though 
accounts were kept in Roman numerals until 1550 to 1650), the 
rediscovery of Greek geometry, and the improvement of trigo- 
nometry in connection with its increasing use in astronomy, navi- 
gation and military engineering. The development of science has 
been powerfully promoted by the general intellectual emancipation 
of the Renaissance, while mathematical progress, beginning earlier, 
has been both a cause and a consequence of the general advance. 
The diffusion and the preservation of scientific knowledge have 
derived immense advantage from the new art of printing and from 
expanding commercial intercourse. Algebra, almost helpless in 
Greek times because, for lack of proper symbolism, expressed only 
in geometrical or rhetorical form, has been converted by a 
process of abbreviation, at first into a syncopated form, inter- 
mediate between the rhetorical and our modern purely symbolic 
notation. 
Pacioli. — The earliest printed book on arithmetic and algebra 
was published at Venice in 1494 by Lucas Pacioli, a Franciscan 
monk born in Tuscany about 1450. Rules are here given for the 
fundamental operations of arithmetic, and for extracting square 
roots. Commercial arithmetic is treated at considerable length 
by the newer algoristic or Arabic methods. The method of arbi- 
trary assumption corrected by proportion is used effectively, for 
example: — 
To find the original capital of a merchant who spent a quarter 
of it in Pisa and a fifth of it in Venice, who received on these transac- 
tions 180 ducats, and who has in hand 224 ducats. 
Assume that his original capital was 100 ducats; then the surplus 
would be 100 — 25 — 20 = 55, but this is f of his actual surplus 
224 — 180, therefore his original capital was f of 100 = 80 ducats. PROGRESS OF MATHEMATICS AND MECHANICS 
233 
Some of Pacioli’s commercial problems are exceedingly compli- 
cated. He solves numerical equations of the first and second 
degree, but admits only positive roots and considers the solution 
of cubic equations, as well as the squaring of the circle, impossible. 
Addition is denoted by p or p, equality sometimes by ae, a begin- 
ning of syncopated algebra. The introduction of the radical sign 
with indices V2> V3 and of the signs + and — date from about 
this time. 
In geometry Pacioli, like Regiomontanus, employs algebraic 
methods. Among other problems he determines a triangle from 
the radius of the inscribed circle and the segments into which 
it divides one of the sides. His solution, though highly esteemed 
at the time, is much less simple than he might have obtained by 
the formulas at his command. 
In the spirit of the Renaissance he brings the feeble mathematics 
of the universities into fruitful relations with the practical mathe- 
matics of the artist and the architect. The inscribed hexagon and 
the equilateral triangle play their part as gild secrets in the develop- 
ment of Gothic architecture. The question is not “ How to prove,” 
but “How to do.” 
On the other hand, the current tendency to drift into mysti- 
cal interpretation is exemplified by the following extract from 
Pacioli: — 
There are three principal sins, avarice, luxury, and pride; three 
sorts of satisfaction for sin, fasting, almsgiving, and prayer; three 
persons offended by sin, God, the sinner himself, and his neighbour; 
three witnesses in heaven, Pater, verbum, and spiritus sanctus; three 
degrees of penitence, contrition, confession, and satisfaction, which 
Dante has represented as the three steps of the ladder that leads to 
purgatory, the first marble, the second black and rugged stone, and the 
third red porphyry. There are three sacred orders in the church 
militant, subdiaconati, diaconati, and presbyterati; there are three parts 
not without mystery, of the most sacred body made by the priest 
in the mass; and three times he says Agnus Dei, and three times, 
Sanctus; and if we well consider all the devout acts of Christian wor- 
ship, they are found in a ternary combination; if we wish rightly to 234 
A SHORT HISTORY OF SCIENCE 
partake of the holy communion, we must three times express our con- 
trition, Domine non sum dignus; but who can say more of the ternary 
number in a shorter compass, than what the prophet says, tu signacidum 
sanctae trinitatis. There are three Furies in the infernal regions; 
three Fates, Atropos, Lachesis, and Clotho. There are three theo- 
logical virtues; Fides, spes, and charitas. Tria sunt pericula mundi: 
Equum currere; navigare, et sub tyranno vivere. There are three 
enemies of the soul: the Devil, the world, and the flesh. There 
are three things which are of no esteem: the strength of a porter, the 
advice of a poor man, and the beauty of a beautiful woman. There 
are three vows of the Minorite Friars; poverty, obedience, and 
chastity. There are three terms in a continued proportion. There 
are three ways in which we may commit sin: corde, ore, ope. Three 
principal things in Paradise: glory, riches, and justice. There are 
three things which are especially displeasing to God: an avaricious 
man, a proud poor man, and a luxurious old man. And all things 
in short, are founded in three; that is, in number, in weight, and in 
measure. 
Geometry in Art. — Brunelleschi (1377-1446), the famous 
architect of the early Renaissance, made a perspective view of the 
Signoria in Florence in a sort of box with clouds. The famous doors 
of the Baptistery by his contemporary Ghiberti show the develop- 
ment of perspective in the marked contrast between the earlier 
and the later panels. Raphael in his School of Athens includes 
himself and Bramante in a group of mathematicians. Painters 
were even called for a time perspectivists — prospettivi. 
Leonardo da Vinci (1452-1519), one of the intellectual giants 
of the Renaissance, eminent alike in art, science and engineer- 
ing, gave the first correct explanation of the partial illumination 
of the darker part of the moon’s disc by reflection from the earth. 
He calls mechanics the paradise of the mathematical sciences, 
because through it one first gains the fruit of these sciences. He 
denies the possibility of perpetual motion, saying “Force is the 
cause of motion and motion the cause of force.” He discusses the 
lever, the wheel and axle, bodies falling freely or on inclined planes, 
foreshadowing Galileo. Contrary to the Aristotelian tradition he 
asserts that everything "tends to continue in its given state, and he PROGRESS OF MATHEMATICS AND MECHANICS 
235 
even enunciates the fundamental principle that for simple machines 
forces in equilibrium are inversely as the virtual velocities. 
‘Whoever,’ he says, ‘appeals to authority applies not his intellect 
but his memory.’ ‘ While Nature begins with the cause and ends with 
the experiment, we must nevertheless pursue the opposite plan, 
beginning with the experiment and by means of it investigating the 
cause.’ ‘No human investigation can call itself true science, unless 
it comes through mathematical demonstration.’ ‘He who scorns 
the certainty of mathematics will not be able to silence sophistical 
theories which end only in a war of words.’ 
Unfortunately his work in this field remained unpublished, and 
therefore relatively unfruitful. 
Leonardo and other great artists of his time — notably Albrecht 
Diirer Of Nuremberg (1471-1528) — developed the geometrical 
theory of perspective. For the purpose of accurately representing 
the human head Diirer made both plans and elevation. “Intelli- 
gent painters and accurate artists,” he says, “ at the sight of works 
painted without regard to true perspective must laugh at the blind- 
ness of these people, because to a right understanding nothing im- 
presses more disagreeably than falsehood in a painting, regardless 
of the diligence with which it has been made. That such painters, 
however, are pleased with their own mistakes is due to the fact 
that they have not learned the art of measurement, without which 
no one can become a true workman.” All this had importance 
both for modern art and modern geometry. 
Characteristic of this period is the so-called Margarita Philo- 
sophica published in many editions from 1503 to 1600. It was the 
first modern encyclopaedia printed, and gives in its twelve books “ a 
compendium of the trivium, the quadrivium, and the natural and 
moral sciences.” 
A younger contemporary of Pacioli, Michael Stifel (1487-1567), 
a German monk converted to Lutheranism, developed a fantastic 
arithmetical interpretation of the Bible, identifying Pope Leo X 
with the beast in Revelation and predicting the immediate end 
of the world, — with results disastrous to his person as well as his 
reputation. 236 
A SHORT HISTORY OF SCIENCE 
He relates . . . that whilst a monk at Esslingen in 1520, and when 
infected by the writings of Luther, he was reading in the library of his 
convent the 13th Chapter of Revelations, it struck his mind that the 
Beast must signify the Pope, Leo X; He then proceeded in pious hope 
to make the calculation of the sum of the numeral letters in Leo 
decimus, which he found to be M, D, C, L, V, I; the sum which these 
formed was too great by M, and too little by X; but he bethought him 
again, that he has seen the name written Leo X; and that there were 
ten letters in Leo decimus, from either of which he could obtain the 
deficient number, and by interpreting the M to mean mysterium, he 
found the number required, a discovery which gave him such un- 
speakable comfort, that he believed that his interpretation must 
have been an immediate inspiration of God. — Peacock. 
Stifel’s writings on arithmetic and algebra embody some improve- 
ments of current notation. He introduced for example the symbols 
1A, 1AA, lAAA for what we should denote by x, x2, x3. 
The low state of computation at this time is illustrated with 
startling clearness by a bulletin on the blackboard at Wittenberg, 
in which Melanchthon urgently invited the academic youth to 
attend a course on arithmetic, adding that the beginnings of the 
science are very easy, and even division can with some diligence be 
comprehended. 
Robert Recorde (1510-1558) studied at Oxford and graduated 
in medicine at Cambridge in 1545, later becoming “royal physi- 
cian.” His “Grounde of Artes” or arithmetic, one of the earliest 
mathematical books printed in English (1540), ran through more 
than 27 editions and exerted a great influence on English education. 
In the “Preface to the Loving Reader” he says: — 
Sore ofttimes have I lamented with myself the unfortunate con- 
dition of England, seeing so many great Clerks to arise in sundry 
other parts of the World, and so few to appear in this our Nation; 
whereas for pregnancy of natural wit (I think) few Nations do excell 
English-men. But I cannot impute the cause to any other thing, 
then to the contempt or misregard of Learning. For as English-men 
are inferiour to no men in mother Wit, so they pass all men in vain 
Pleasures, to which they may attain with great pain and labour; and PROGRESS OF MATHEMATICS AND MECHANICS 
237 
are slack to any never so great commodity, if there hang of it any pain- 
full study or travelsome labour. 
The book itself is in the form of a dialogue or catechism be- 
ginning : — 
The Scholar speaketh. 
‘ Sir, such is your authority in mine estimation, that I am content 
to consent to your saying, and to receive it as truth, though I see 
none other reason that doth lead me thereunto; whereas else in mine 
own conceit it appeareth but vain, to bestow any time privately in 
learning of that thing that every Child may and doth learn at all times 
and hours, when he doth any thing himself alone, and much more when 
he talketh or reasoneth with others.’ 
He employs the symbol + “whyche betokeneth too muche, as 
this line — plaine without a crosse line betokeneth too little.” 
In 1557 he published an algebra under the alluring title “ Whet- 
stone of Witte,” using the sign = for equality, which he says he 
selected because “noe 2 thynges can be moare equalle” than 
two parallel straight lines. 
Algebraic Equations of Higher Degree.—Two great Ital- 
ian mathematicians vied with each other in giving a powerful im- 
petus to the development of algebra in the sixteenth century. 
Niccolo Fontana or Tartaglia (1500-1557) a man of the hum- 
blest origin, lectured at Verona and Venice, and first won fame 
by successfully meeting a challenge to solve mathematical prob- 
lems, all of which proved, as he had anticipated, to involve cubic 
equations. 
His Nova Scienza (1537) discusses falling bodies, and many 
problems of military engineering and fortification, the range 
of projectiles, the raising of sunken galleys, etc. 
The title-page is chiefly occupied by a large plate, which represents 
the courts of Philosophy, to which Euclid is doorkeeper, Aristotle and 
Plato being masters of an inmost court, in which Philosophy sits 
throned, Plato declaring by a label that he will let nobody in who does 
not understand Geometry. In the great court there is a cannon being 
fired, all the sciences looking on in a crowd — such as Arithmetic, 238 
A SHORT HISTORY OF SCIENCE 
Geometry, Music, Astronomy, Cheiromancy, Cosmography, Necro- 
mancy, Astrology, Perspective, and Prestidigitation! A wonderfully 
modest-looking gentleman, with his hand upon his heart, stands among 
the number, with a you-do-me-too-much-honour look upon his coun- 
tenance; Arithmetic and Geometry are pointing to him, and under 
his feet his name is written — Nicolo Tartalea. — Morley, Jerome 
Cardan. 
The Inventioni (1546) gives his solution of the cubic equation. 
A treatise on Numbers and Measures (1556, 1560) gives a method 
for finding the coefficients in the expansion of (1 + x) “ for n = 2, 
...6. It contains also a wide range of problems from commercial 
arithmetic and a collection of mathematical puzzles. The follow- 
ing examples may illustrate these: — 
‘Three beautiful ladies have for husbands three men, who are 
young, handsome, and gallant, but also jealous. The party are 
travelling, and find on the bank of a river, over which they have to 
pass, a small boat which can hold no more than two persons. How 
can they pass, it being agreed that, in order to avoid scandal, no 
woman shall be left in the society of a man unless her husband is 
present ? ’ 
‘A ship carrying as passengers 15 Turks and 15 Christians en- 
counters a storm, and the pilot declares that in order to save the ship 
and crew one half of the passengers must be thrown into the sea. 
To choose the victims, the passengers are placed in a circle, and it is 
agreed that every 9th man shall be cast overboard, reckoning from a 
certain point. In what manner must they be arranged so that the 
lot may fall exclusively upon the Turks?’ 
* Three men robbed a gentleman of a vase containing 24 ounces of 
balsam. Whilst running away they met in a wood with a glass-seller 
of whom in a great hurry they purchased three vessels. On reaching a 
place of safety they wish to divide the booty, but they find that their 
vessels contain 5, 11, and 13 ounces respectively. How can they 
divide the balsam into equal portions ? ’ — Ball. 
There is no other treatise that gives as much information con- 
cerning the arithmetic of the sixteenth century, either as to 
theory or application. The life of the people, the customs of the PROGRESS OF MATHEMATICS AND MECHANICS 
239 
merchants, the struggles to improve arithmetic, are all set forth 
here by Tartaglia in an extended but interesting fashion. 
Tartaglia, anticipating Galileo, taught that falling bodies of 
different weight traverse equal distances in equal times, and that 
a body swung in a circle if released flies off tangentially. 
Girolamo Cardan (1501-1576) led a life of wild and more or 
less disgraceful adventure, strangely combined with various forms 
of scientific or semi-scientific activity, — particularly the practice 
of medicine. He studied at Pavia and Padua, travelled in France 
and England, and became professor at Milan and Pavia. 
His Ars Magna (1545) contains the solution of the cubic equa- 
tion fraudulently obtained from his rival Tartaglia. After its publi- 
cation the aggrieved Tartaglia challenged Cardan to meet him in a 
mathematical duel. This took place in Milan, August 10, 1548, 
but Cardan sent his pupil Ferrari in his place. Tartaglia relates 
that he was accompanied only by his brother, Ferrari by many 
friends. Cardan had left for parts unknown. As Tartaglia began 
to explain to the crowd the origin of the strife and to criticise 
Ferrari’s 31 solutions, he was interrupted by a demand that judges 
be chosen. Knowing no one present he declines to choose; all 
shall be judges. Being finally allowed to proceed he convicts his 
opponent of an erroneous solution, but is then overwhelmed by 
tumultuous clamor with demands that Ferrari must have the floor 
to criticise his solution. In vain he insists that he be allowed to 
finish, after which Ferrari may talk to his heart’s content. Fer- 
rari’s friends are vehement; he gains the floor and chatters about a 
problem which he claims Tartaglia has not been able to solve till 
the dinner hour arrives and Tartaglia, apprehending still worse 
treatment, withdraws in disgust. 
Ferrari (1522-1565), this disciple of Cardan, even succeeded in 
giving a general solution of the equation of the fourth degree, 
beyond which, as has been shown only in quite recent times, the 
solution can in general no longer be similarly expressed. 
Some idea of the difficulty of these sixteenth century achievements 
may be conveyed by the corresponding modernized solutions. If 
the given equation is as? -f- bx2 + cx + d = 0 the coefficient of the 240 
A SHORT HISTORY OF SCIENCE 
first term is made 1 by division and that of the second is made 
0 by the substitution x = y —. The new equation having the form 
3 a 
s $ 
y3 -f- ey +/ = 0 we now put y — z — —, whence z3 — —— +/ = 0, 
o Z Ld i 
a quadratic equation in z3. The solution of the original equation of 
degree three is thus made to depend on that of an equation of degree 
one less. 
Similarly if the given equation of the fourth degree is in our 
notation axA + bx3 + cx2 + dx + e = 0 the coefficient of the 
first term is made 1 by division and that of the second is made 0 by 
t 
the substitution x = y 
4a 
The new equation having the form 
y4 + fy2 + gy + h = 0. 
We put y4 + fy2 + gy + h = (y2 - ay + 0) (y2 + ay + 7) 
whence / = /3 + 7 — a2 
9 = (0 ~ 7) a 
h = fiy. 
We obtain a, (3, 7 from these three equations by eliminating two 
and solving the cubic equation obtained for the other; that is, the solu- 
tion of the original equation of degree four is made to depend on that 
of a new equation of degree one less. 
One of Cardan’s scientific inventions was an improved suspen- 
sion of the compass needle. He was also eminent as an astrologer. 
Symbolic Algebra: Vieta. — Of still greater importance in 
the history of algebra is F. Vieta (1540-1603) a lawyer of the 
French court. He won the interest of Henry IV by solving a com- 
plicated problem proposed by an eminent mathematician, as was 
the custom of the time, as a challenge to the learned world. This 
involved an equation of the 45th degree which he succeeded in 
solving by a trigonometric device. Later he was employed to inter- 
pret the cipher despatches of the hostile Spaniards. His In Artern 
Analyticam Isagoge is the earliest work on symbolic algebra. 
In it known quantities are denoted by consonants, unknown by 
vowels, the use of homogeneous equations is recommended, the PROGRESS OF MATHEMATICS AND MECHANICS 
241 
first six powers of a binomial given, and a special exponential 
notation introduced. He shows that the celebrated classical 
problems of trisecting a given angle and duplicating a cube involve 
the solution of the cubic equation, and makes important discoveries 
in the general theory of equations — for example resolving poly- 
nomials into linear factors and deriving from a given equation 
other equations having roots which differ from those of the first by 
a constant or by a given factor. He solves Apollonius’ famous 
problem of determining the circle tangent to three given circles, 
and expresses tt by an infinite series. He devises systematic 
methods for the solution of spherical triangles. 
Development of Trigonometry. — Many circumstances com- 
bined to promote the development of trigonometry at this period. 
It was needed by the military engineer, the builder of roads, the 
astronomer, the navigator, and the mapmaker whose work was 
tributary to all of these. 
Rheticus (George Joachim, 1514-1576), — “ the great computer 
whose work has never been superseded,” — worked out a table of 
natural sines for every 10 seconds to fifteen places of decimals. 
We owe to him our familiar formulas for sin 2x and sin 3z. The 
notation sin, tan, etc. and the determination of the area of a 
spherical triangle date from about this time. To this period belong 
also the very important work of Mercator on map-making and 
the reform of the calendar by Pope Gregory XIII. 
Map-making. — Mercator (Gerhard Kramer, 1512-1594) de- 
voted himself in his home city, Louvain, to mathematical geogra- 
phy, and gained his livelihood by making maps, globes and 
astronomical instruments, combined in later life with teaching. 
His great world map, completed in 1569, marks an epoch in 
cartography. The first “Atlas” was published by his son in 1595. 
He gives a mathematical analysis of the principles underlying the 
projection of a spherical surface on a plane. 
‘If,’ he says, ‘of the four relations subsisting between any two 
places in respect to their mutual position, namely difference of latitude, 
difference of longitude, direction and distance, only two are regarded, 
the others also correspond exactly, and no error can be committed as 242 
A SHORT HISTORY OF SCIENCE 
must so often be the case with the ordinary marine charts and so much 
the more the higher the latitude/ 
Mercator’s geometrical method amounts to projecting the 
spherical surface of the earth on a cylinder tangent to the earth 
along the equator and having the same axis with the earth. 
Under this method of projection, angles are preserved in magni- 
tude, but areas remote from the equator are disproportionately 
expanded. A straight line on the chart corresponds with the 
course of a ship steering a constant course. 
The Gregorian Calendar.—Until 1582 the Julian calendar 
(p. 143) remained in force with 365| days each year and a gradually 
increasing error amounting at this time to ten days. Under the 
auspices of Pope Gregory the days from October 5 to 15, 1572, 
were dropped and the number of leap-years in 400 reduced from 
100 to 97. Religious jealousies prevented the adoption of this 
reform in Protestant Germany for a century, while England 
postponed it until 1752. 
A New Invention for Computation. — The invention of 
logarithms would appear to have been a natural sequel of any 
adequate theory and notation for exponents. Thus Stifel in his 
arithmetic (1544) had tabulated small integral powers of 2 — from 
j to 64 — and shown the correspondence between multiplication 
of these powers and addition of the indices or exponents, but his 
use of exponents was too limited, he lacked the apparatus of deci- 
mal fractions necessary for the practical application of the method 
and probably had no conception of the vast labor-saving possi- 
bilities so near at hand. 
In 1614 John Napier published at Edinburgh his Mirifici Loga- 
rithmorum Canonis Descriptio, for which the time was so fully ripe 
that an enthusiastic reception was at once assured. Napier as a 
devout Protestant, stimulated by fear of an impending Spanish 
invasion, busied himself with inventions “ proffitabill & necessary 
in theis dayes for the defence of this Hand & withstanding of 
strangers enemies of God’s truth & relegion.” Among these were 
a mirror for burning distant ships, and a sort of armored chariot. 
Impressed by the tremendous calculations then in progress by PROGRESS OF MATHEMATICS AND MECHANICS 
243 
Rheticus, Kepler, and others in connection with the development of 
the new astronomy, Napier made a vastly more important inven- 
tion. His definition of a logarithm rests on the following kinetic 
basis: — 
T S is a straight line of definite length; TXS\ extends to the right 
indefinitely. Moving points P and Px start from T and Tx with 
equal initial speeds; the latter continues at the same rate, the 
former is retarded so that its speed is always proportional to its 
distance from S. If equal intervals are taken on Tx Sx the cor- 
responding intervals in TS will grow smaller to the right. When P 
is at any position Q the logarithm of QS is represented by 
the corresponding length TXQX on the other line. It may be 
shown in fact that if in our notation PS = x, TXPX = y, TS = l, 
This conception involving a functional relation be- 
dy l 
tween two variables went much deeper than the comparison 
of discrete numbers by Stifel. 
Napier’s conception of a logarithm involved a perfectly clear 
apprehension of the nature and consequences of a certain functional 
relationship, at a time when no general conception of such a relation- 
ship had been formulated, or existed in the minds of mathematicians, 
and before the intuitional aspect of that relationship had been clarified 
by means of the great invention of coordinate geometry made later 
in the century by Rene Descartes. A modern mathematician re- 
gards the logarithmic function as the inverse of an exponential func- 
tion; and it may seem to us, familiar as we all are with the use of 
operations involving indices, that the conception of a logarithm 
would present itself in that connection as a fairly obvious one. We 
must however remember that, at the time of Napier, the notion of an 
index, in its generality, was no part of the stock of ideas of a mathe- 
matician, and that the exponential notation was not yet in use. 
— Hobson. 244 
A SHORT HISTORY OF SCIENCE 
Independent tables were computed by the astronomer Biirgi 
and published at Prague in 1620. Both Napier and Biirgi, basing 
their work on the relation which we should express by the equiva- 
lent equations x = a? and y = loga x, avoid fractional values of y by 
taking values of a near 1, their actual values being a = .9999999 
and a = 1.0001 respectively. In choosing a base less than 1, Napier 
is also influenced by his desire that sines and cosines as proper 
fractions shall have positive 
logarithms. If we intro- 
duce our modern graphical 
interpretation of y — loga£, 
Biirgi is concerned with the 
determination of abscissas 
of points where the exponen- 
tial curve is met by the 
horizontal straight lines y 
= c where c takes successive 
integral values. Choosing 
a base a near 1 naturally gives values of x near each other. 
Napier’s choice of a base less than 1 would correspond with the 
same curve inverted. 
In 1615 Henry Briggs, afterwards Savilian Professor of Geom- 
etry at Oxford, wrote of Napier “ I hope to see him this summer, 
if it please God, for I never saw book which pleased me better, or 
made me more wonder.” In connection with this and later visits 
it was soon discovered that great simplification in the practical 
use of logarithms would result from taking log 1=0 and log 10 = 1 
and giving up the restriction of logarithms to integral values, thus 
making the decimal parts of all logarithms depend wholly on the 
sequence of digits. Napier had been so predominantly interested in 
trigonometric applications that his table consisted not of logarithms 
of abstract numbers, but of 7-place logarithms of the trigonometric 
functions for each minute. In connection with his change of the 
base, Briggs developed interesting methods of interpolating and 
testing the accuracy of logarithms. He gives the logarithms from 
1 to 20,000 and from 90,000 to 100,000 to 14 places, computing also 
10-place trigonometric tables with an angular interval of 10 seconds. PROGRESS OF MATHEMATICS AND MECHANICS 
245 
Kepler recognized immediately the enormous significance of the 
new logarithmic method and addressed an enthusiastic panegyric 
to Napier in 1620, not knowing that he had died in 1617. What if 
logarithms had been invented in time to save Kepler his vast com- 
putations ? 
A few years ago we have been shown in a rectorial address what the 
telescope has meant for observational astronomy. An equally great 
significance attached to logarithms for the computing astronomer. 
— Gutzmer. 
Vlacq of Leyden soon after filled the gap in Briggs’ table, and 
this is the basis for the tables since published. The first tables to 
base e, commonly called Napierian, were published in 1619. In 
more recent times methods of interpolation have been employed 
which are more powerful and less laborious, while ordinary com- 
putation has been simplified by avoiding the use of too many deci- 
mal places, and by the mechanical device of the slide-rule. The 
modern computing machine naturally tends to supersede the 
logarithmic method. Among the remarkable computations 
characteristic of the sixteenth century may be mentioned Ludolph 
von Ceulen’s achievement in computing 7r to 35 decimal places, 
using regular polygons of 96 and 192 sides. German writers in 
consequence have sometimes attached his name to this important 
constant. 
In England Thomas Harriott (1560-1621) and William Oughtred 
(1575-1660) rendered important services in introducing the most 
recent advances in arithmetic, algebra and trigonometry. The 
former rejected negative and imaginary roots indeed, but used 
the signs > and <, denotes a2 by a a, etc. Oughtred uses the 
symbols X and ::, also the contractions for sine, cosine, etc. 
“Two New Branches of Science.” — Even after Galileo’s 
condemnation by the Inquisition, though old, infirm, and nearly 
blind, his scientific ardor was unquenched, and in 1638 he pub- 
lished (at Leyden) a work on mechanics under the title, Conver- 
sations and Mathematical Demonstrations on two New Branches 
of Science, which constituted the most notable progress in mechan- 
ics since Archimedes. He says: 246 
A SHORT HISTORY OF SCIENCE 
My purpose is to set forth a very new science dealing with a very 
ancient subject. There is, in nature, perhaps nothing older than 
motion, concerning which the books written by philosophers are neither 
few nor small; nevertheless I have discovered by experiment some 
properties of it which are worth knowing and which have not hitherto 
been either observed or demonstrated. Some superficial observations 
have been made, as, for instance, that the free motion (naturalem 
motum) of a heavy falling body is continuously accelerated; but to 
just what extent this acceleration occurs has not yet been announced; 
for so far as I know, no one has yet pointed out that the distances 
traversed, during equal intervals of time, by a body falling from rest, 
stand to one another in the same ratio as the odd numbers beginning 
with unity. 
It has been observed that missiles and projectiles describe a 
curved path of some sort; however no one has pointed out the fact 
that this path is a parabola. But this and other facts, not few in 
number or less worth knowing, I have succeeded in proving; and what 
I consider more important, there have been opened up to this vast and 
most excellent science, of which my work is merely the beginning, ways 
and means by which other minds more acute than mine will explore its 
remote corners. 
This discussion is divided into three parts; the first part deals 
with motion which is steady or uniform; the second treats of motion as 
we find it accelerated in nature; the third deals with the so-called 
violent motions and with projectiles. . . . 
Throughout this work Galileo depends on results of experiment 
rather than on mere speculation. He recognizes that air has 
weight and that water can be raised but a certain height by 
the ordinary pump,1 but he still accepts the ancient notion that 
1 ‘ This pump worked perfectly so long as the water in the cistern stood above 
a certain level; but below this level the pump failed to work. When I first noticed 
this phenomenon I thought the machine was out of order; but the workman whom 
I called in to repair it told me the defect was not in the pump but in the water 
which had fallen too low to be raised through such a height; and he added that it 
was not possible, either by a pump or by any other machine working on the principle 
of attraction, to lift water a hair’s breadth above eighteen cubits; whether the 
pump be large or small this is the extreme limit of the lift. Up to this time I had been 
so thoughtless that, although I knew a rope, or rod of wood, or of iron, if suffi- 
ciently long, would break by its own weight when held by the upper end, it never 
occurred to me that the same thing would happen, only much more easily, to a PROGRESS OF MATHEMATICS AND MECHANICS 
247 
“nature abhors a vacuum” as an explanation. He shows experi- 
mentally that a body descends an inclined plane with uniformly 
accelerated motion. 
In a board 12 ells in length a groove half an inch wide was made. 
It was drawn straight and lined with very smooth parchment. The 
board was then raised at one end, first one ell, then two. Then Galileo 
let a polished brass ball roll through the groove and determined the 
time of descent for the whole length of the groove. If on the other hand 
he let the ball roll through only one quarter of the length, this required 
just half the time. . . . The distances were to each other as the 
squares of the times, 
a law verified by hundredfold repetitions for all sorts of distances 
and slopes. The time was still determined by weighing water 
escaping through a small orifice. He shows by ingenious experi- 
ments the dependence of velocity on height alone, and that a 
freely falling body has the necessary energy to reach its original 
level. The whole theory of the falling body is now easily deduced. 
When, therefore, I observe a stone initially at rest falling from an 
elevated position and continually acquiring new increments of speed, 
why should I not believe that such increases take place in a manner 
which is exceedingly simple and rather obvious to everybody? If 
now we examine the matter carefully we find no addition or increment 
more simple than that which repeats itself always in the same manner. 
This we readily understand when we consider the intimate relation- 
ship between time and motion; for just as uniformity of motion is de- 
fined by and conceived through equal times and equal spaces (thus we 
call a motion uniform when equal distances are traversed during equal 
time-intervals), so also we may, in a similar manner, through equal 
time-intervals, conceive additions of speed as taking place without 
complication; thus we may picture to our mind a motion as uniformly 
and continuously accelerated when, during any equal intervals of time 
whatever, equal increments of speed are given to it. . . . 
Hence the definition of motion which we are about to discuss may 
column of water. And really is not that thing which is attracted in the pump a 
column of water attached at the upper end and stretched more and more until 
finally a point is reached where it breaks, like a rope, on account of its excessive 
weight? . . 248 
A SHORT HISTORY OF SCIENCE 
be stated as follows : A motion is said to be uniformly accelerated, when 
starting from rest, it acquires, during equal time-intervals, equal in- 
crements of speed. . . . 
The time in which any space is traversed by a body starting from 
rest and uniformly accelerated is equal to the time in which that same 
space would be traversed by the same body moving at a uniform speed 
whose value is the mean of the highest speed and the speed just 
before acceleration began. . . . 
The spaces described by a body falling from rest with a uniformly 
accelerated motion are to each other as the squares of the time- 
intervals employed in traversing these distances. . . . 
Galileo passes from falling bodies to pendulums, in which the fric- 
tion of the inclined plane is absent and air resistance negligible. 
He appreciates the possibility of utilizing the pendulum for time 
measurement, and devises a simple apparatus for the purpose, 
foreshadowing the invention of the clock. He discovers that the 
time of vibration of the pendulum varies as the square root of 
the length. 
He analyzes correctly the component motions of a projectile, 
recognizing the law of the parallelogram of motion, as distinguished 
from the parallelogram of forces discovered by Newton. He shows 
that whether the initial direction of aim is horizontal or not, the 
path described is a parabola with axis vertical, explicitly neglecting 
air resistance and change of direction of vertical force. 
I now propose to set forth those properties which belong to a body 
whose motion is compounded of two other motions, namely, one 
uniform and one naturally accelerated; these properties, well worth 
knowing, I propose to demonstrate in a rigid manner. This is the kind 
of motion seen in a moving projectile; its origin I conceive to be as 
follows: . . . 
A projectile which is carried by a uniform horizontal motion com- 
pounded with a naturally accelerated vertical motion describes a 
path which is a semi-parabola. — Galileo, Two New Sciences. 
All this Dynamics was practically pioneer work of enormous im- 
portance for the future of mechanics. PROGRESS OF MATHEMATICS AND MECHANICS 
249 
In Statics Galileo had somewhat more from the ancients to 
build upon. To him we owe the formulation of the law of virtual 
velocities — applying dynamical ideas to problems of statics. If 
two forces are in equilibrium they are proportional to the cor- 
responding paths, or: What one by any machine gains in power 
is lost in distance. The parallelogram or triangle of forces in 
equilibrium however escapes him, and his ideas about impulse 
though remarkably in advance of his time were not fully worked 
out. 
He investigates strength of materials under tension and fracture, 
with reference to practical applications in construction. He draws 
just inferences in regard to the relation between strength and 
size of plants and animals as well as machines, comparing for 
example hollow bones and straws with solid bodies of similar mass. 
He derives an important formula for the stiffness of a horizontal 
beam supported at one end and regarded as a lever. He discusses 
the curve formed by a cord suspended between two points, recog- 
nizing that it is not a parabola. 
In Hydrostatics he reviews the known work of Archimedes and 
corrects the error of the Aristotelians in regard to the dependence 
of floating on specific gravity. He develops the modern theory 
that the fundamental factor in the mechanics of fluids is that they 
consist of freely moving particles yielding to the slightest force. 
He makes effective application of the principle of virtual velocities 
to fluids. At the close of his third conversation he expresses his 
modest confidence in the great future of his new ideas. 
The theorems set forth in this brief discussion will, when they 
come into the hands of other investigators, continually lead to wonder- 
ful new knowledge. It is conceivable that in such a manner a worthy 
treatment may be gradually extended to all the realms of nature 
— a prediction magnificently fulfilled in succeeding genera- 
tions. 
Among other branches of physics in which Galileo accomplished 
work of value may be mentioned the expansion by heat — the 
beginnings of thermometry, experiments on the acoustics of 250 
A SHORT HISTORY OF SCIENCE 
vibrating cords and plates, discovering the dependence of harmony 
on the ratio of the rates of vibration, and the relations of length, 
thickness, and tension of cords. He explains resonance and dis- 
sonance. He assumes light to have a finite velocity, but does 
not succeed in measuring it. 
Let each of two persons take a light contained in a lantern, or 
other receptacle, such that by the interposition of the hand, the one 
can shut off or admit the light to the vision of the other. Next let 
them stand opposite each other at a distance of a few cubits and prac- 
tice until they acquire such skill in uncovering and occulting their 
lights that the instant one sees the light of his companion he will un- 
cover his own. After a few trials the response will be so prompt that 
without sensible error the uncovering of one light is immediately 
followed by the uncovering of the other, so that as soon as one exposes 
his light he will instantly see that of the other. Having acquired skill, 
at this short distance, let the two experimenters, equipped as before, 
take up positions separated by a distance of two or three miles and 
let them perform the same experiment at night, noting carefully 
whether the exposures and occultations occur in the same manner as 
at short distances; if they do, we may safely conclude that the prop- 
agation of light is instantaneous; but if time is required at a distance 
of three miles which, considering the going of one light and the coming 
of the other, really amounts to six, then the delay ought to be easily 
observable. If the experiment is to be made at still greater distances, 
say eight or ten miles, telescopes may be employed, each observer 
adjusting one for himself at the place where he is to make the experi- 
ment at night; then although the lights are not large and are 
therefore invisible to the naked eye at so great a distance, they can 
readily be covered and uncovered since by aid of the telescopes, once 
adjusted and fixed, they will become easily visible. . . . 
He seeks to apply to astronomical phenomena the new discoveries 
in magnetism. 
Everywhere the mathematical and inductive method became 
manifest in this man. Almost all domains of science received there- 
from the most powerful impulse. And above all the whole field of 
science was freed from the outgrowths of metaphysical modes of 
thought with which it had been previously so overrun. Galileo’s PROGRESS OF MATHEMATICS AND MECHANICS 
251 
individual method consisted namely in always conforming to the limits 
of scientific investigation, and confining his attention to seizing the 
phenomena sharply in their progress and in their relation with allied 
processes, without wandering into a fruitless search after the ultimate 
bases of the phenomena. —Dannemann. 
Such a limitation has been of the highest value for the renewTal of 
natural science as it followed at the beginning of the seventeenth 
century. 
Galileo was not chiefly interested in mathematics, but he em- 
phasizes the dependence of other sciences upon it. 
True philosophy expounds nature to us; but she can be under- 
stood only by him who has learned the speech and symbols in 
which she speaks to us. This speech is mathematics, and its sym- 
bols are mathematical figures. Philosophy is written in this greatest 
book, which continually stands open here to the eyes of all, but can- 
not be understood unless one first learns the language and characters 
in which it is written. This language is mathematics and the 
characters are triangles, circles and other mathematical figures. 
He gives an acute discussion of infinite, infinitesimal and con- 
tinuous quantities leading up to the conclusion “that the attri- 
butes ‘larger,’ ‘smaller,’ and ‘equal’ have no place either in 
comparing infinite quantities with each other or in comparing in- 
finite with finite quantities.” Again “the finite parts of a con- 
tinuum are neither finite nor infinite but correspond to every as- 
signed number.” 
In commenting on Galileo’s achievements, Lagrange the great 
mathematician of the eighteenth century says: — 
These discoveries did not bring to him while living as much 
celebrity as those which he had made in the heavens; but to-day his 
work in mechanics forms the most solid and the most real part of the 
glory of this great man. The discovery of Jupiter’s satellites, of the 
phases of Venus, and the Sun-spots, etc., required only a telescope and 
assiduity; but it required an extraordinary genius to unravel the laws 
of nature in phenomena which one has always under the eye, but the 
explanation of which, nevertheless, had always baffled the researches 
of philosophers. 252 
A SHORT HISTORY OF SCIENCE 
Leonardo da Vinci likens a scientific conquest to a military victory- 
in which theory is the field marshal, experimental facts the soldiers. 
The philosophers who preceded Galileo had, in the main, been trying 
to fight battles without soldiers. — Crew. 
A Pioneer in Mechanics. Stevinus. — Even before Galileo, 
Stevinus of Bruges (1548-1620), a man who thought independently 
on mechanical problems, made the first really important advances 
since Archimedes, eighteen centuries earlier. Besides engaging 
in mercantile pursuits he was quartermaster-general of the 
Dutch army, and an authority on military engineering. He was 
influential in improving methods of public statistics and account- 
ing, and advocated decimal weights and measures. Appreciating 
the possibilities of the decimal fraction he asserted (1585) that 
fractions are quite superfluous, and every computation can be 
made with whole numbers, but he did not realize the simplest 
notation. The honor of this great invention he shares with Biirgi 
of Cassel. Another of his inventions was a sailing carriage carry- 
ing 28 people and outstripping horses. 
In a treatise on Statics and Hydrostatics (1586) he introduced 
comparatively new and powerful geometrical methods for dealing 
with mechanical problems. 
Among the most interesting is 
his discussion of the inclined 
plane by means of an endless 
chain hanging freely over a tri- 
angle with unequal sides. Ex- 
cluding the inadmissible hy- 
pothesis of perpetual motion, 
the uniform chain must be in 
equilibrium in any position. 
The hanging portion is by itself 
in equilibrium, therefore the 
two inclined sections must bal- 
ance each other, and either 
would be balanced by a verti- 
cal force corresponding to the 
Stevinus’ Triangle. PROGRESS OF MATHEMATICS AND MECHANICS 
253 
altitude of the triangle. Arriving thus at the parallelogram of 
forces in equilibrium, he expresses his astonishment by exclaim- 
ing “Here is a wonder and yet no wonder.” 
In studying pulleys and their combinations he arrives at the 
far-reaching result that in a system of pulleys in equilibrium “ the 
products of the weights into the displacements they sustain are 
respectively equal” — a remark containing the principle of virtual 
displacement. He reaches correct results in regard to basal and 
lateral pressure by reasoning analogous to that about the chain, 
and by assuming on occasion that a definite portion of the 
liquid is temporarily solidified. By ingenious experiments he 
proves the dependence of fluid pressure on area and depth, 
and takes proper account of upward and lateral pressure. He 
studies the conditions of equilibrium for floating bodies, show- 
ing that the centre of gravity of the body in question must lie 
in a perpendicular with that of the water displaced by it, and 
that the deeper the centre of gravity of the floating body the 
more stable is the equilibrium. 
In analyzing the lateral pressure of a fluid Stevinus anticipates 
the calculus point of view by dividing the surface into elements on 
each of which the pressure lies between ascertainable values. In- 
creasing the number of divisions, he says it is manifest that one 
could carry this process so far that the difference between the con- 
taining values should be made less than any given quantity how- 
ever small — all quite in harmony with our present definitions of 
a limit. 
Stevinus’ work and that of Galileo seem to have been quite 
independent of each other, the former confining his theory to 
statics, the latter laying a solid foundation for the new science 
of dynamics. Torricelli, a disciple of Galileo best known for his 
invention of the mercurial barometer, extended dynamics to 
liquids, studying the character of a jet issuing from the side 
of a vessel. 
Throughout this period the universities lagged. In Italy 
Galileo lectured to medical students who were supposed to need 
astronomy for medical purposes — i.e. astrology. At Wittenberg 254 
A SHORT HISTORY OF SCIENCE 
there was a professor for arithmetic and the sphere, and one for 
Euclid, Peurbach’s planetary theory and the Almagest, but their 
students were few. . . . So, says a German writer, we face the 
extraordinary fact that the most educated of the nation were as 
helpless in the problems of daily life as a marketwoman of to-day. 
The university lectures in mathematics were mainly confined 
to the most elementary computation, — matters taught more 
thoroughly in the commercial schools, particularly after the 
invention of printing. 
Giordano Bruno (1548-1600).—In the Appendix will be 
found the judgment and sentence of the Inquisition upon 
Galileo, together with his recantation, — one of the darkest pages 
in the history of Science. Another victim of the Inquisition was 
Bruno, an Italian philosopher, who, having joined the Dominican 
order at the age of fifteen, was later accused of impiety and sub- 
jected to persecution. Bruno fled from Rome to France, and 
later to England, where at Oxford he disputed on the rival merits 
of the Copernican and the so-called Aristotelian systems of the 
universe. In 1584 he published an exposition of the Copernican 
theory. Bruno, moreover, attacked the established religion, jeered 
at the monks, scoffed at the Jewish records, miracles, etc., and 
after revisiting Paris, and residing for a time in Wittenberg, rashly 
returned to Italy, where he was apprehended by the Inquisition 
and thrown into prison. After seven years of confinement he 
was excommunicated and, on Feb. 17, 1600, burnt at the stake. 
In 1889 a statue in his honor was unveiled in Rome at the place 
of his execution, the Square of the Flower Market. Thus was 
the end of the sixteenth century illuminated by the flames of 
martyrdom. 
Ball. Short History of Mathematics, Chapters XII, XIII. 
Fahie. Life of Galileo. 
Galileo Galilei. Two New Sciences. (Translated by Crew and De Salvio.) 
Hobson. John Napier and the Invention of Logarithms. 
Lodge. Pioneers of Science (Galileo). 
Mach. Science of Mechanics (for Galileo and Stevinus). 
Morley. Life of Cardan. 
References for Reading CHAPTER XII 
NATURAL AND PHYSICAL SCIENCE IN THE SEVEN 
TEENTH CENTURY 
The Circulation of the Blood : Harvey (1578-1657).—The 
blood has always been regarded as one of the principal parts of 
the body. Hippocrates considered it one of his four great 
“humors,” and in the Hebrew Scriptures it is stated that “the 
blood ... is the life.” Yet up to the seventeenth century nothing 
definite was known of its movements throughout the body. That 
it was under pressure must have been known, for it flowed or 
“escaped” freely from wounds, and flow results only from pressure 
of some sort, while escape is relief from detention. The arteries 
had been misinterpreted for centuries and were early considered to 
be air tubes, because they were studied only after death when as 
we now know they are empty. Even the dissections of the anato- 
mists of the sixteenth century had failed to reveal the complete and 
true office of the arteries, and it remained for Harvey, an English 
pupil of the Italian anatomist Fabricius, to make — largely 
through the vivisection of animals and observation of the heart 
and arteries in actual operation — discoveries of basic importance 
in anatomy, physiology, embryology, and medicine (see Appendix). 
While working in Italy, Harvey learned of and doubtless saw 
the valves in the veins which were discovered by Fabricius. These 
valves are thin flaps of tissue so placed as to check the flow of blood 
in one direction while offering no resistance to that flowing the other 
way. On his return to England, Harvey apparently pondered on 
the function of these valves and saw that they could be of use only 
by permitting the flow of the blood in one direction while prevent- 
ing its movement in the opposite direction. At this time it was 
supposed that the blood simply oscillated, or moved back and forth 
255 256 
A SHORT HISTORY OF SCIENCE 
like a pendulum, a view which, if the valves had any meaning, 
was now plainly untenable. Harvey therefore set to work to 
study the beating of the heart and the flowing of the blood, and 
soon came to the conclusion that there must be a steady flow or 
streaming in one direction, and not an oscillation back and forth 
as was generally supposed. But to prove was here, as always, 
harder than to believe, and much time and labor were required to 
settle the question. At length, however, by dissections and vivi- 
sections of the lower animals, and after publishing (in 1628) a bro- 
chure presenting his facts and meeting objections, Harvey suc- 
ceeded, with the result that his name justly stands to-day beside 
those of the Greek and Alexandrian Fathers of Medicine, Hippoc- 
rates and Galen. It is one of the ironies of fate that while Harvey 
rightly reasoned from circumstantial evidence that the blood must 
steadily flow from the arteries to the veins, he himself never actually 
saw that flowing, — a sight which any schoolboy may now see, 
but impossible before the introduction of the microscope, and first 
enjoyed by Malpighi in 1661, only four years after Harvey’s death. 
In embryology, also, Harvey proved himself an original and 
penetrating observer. In his day and earlier it was supposed that 
the embryo, in the hen’s egg, for example, exists even at the very 
outset as a perfect though extremely minute chick, with all its parts 
complete. This “preformation” theory was opposed by Harvey, 
whose doctrine of “epigenesis” was substantially that of modern 
embryology: viz. that the embryo chick is gradually formed by 
processes of growth and differentiation from comparatively simple 
and undifferentiated matter, somehow set apart and prepared in 
the body of the parents. 
Atmospheric Pressure : Torricelli’s Barometer. — The 
problem of the existence and nature of voids and vacua had always 
been an interesting puzzle for philosophers. The Greeks assumed 
the existence of empty spaces or “voids,” and as late as the age of 
Elizabeth it was the orthodox belief that “nature abhors a 
vacuum.” Galileo, even, held to it in 1638. (Cf. p. 246.) 
Evangelista Torricelli (1608-1647), inspired by the Dialogues of 
Galileo (1638), published on Motion and other subjects in 1644. NATURAL SCIENCE IN SEVENTEENTH CENTURY 
257 
He resided with Galileo and acted as his amanuensis from 1641 
until Galileo’s death. In experimenting with mercury he found 
that this did not rise to 33 feet, but instead to hardly as many 
inches. He next proved, by comparing the specific gravity of 
water and mercury, that the same “pressure” was at work in 
both cases, and boldly affirmed that this pressure was that of the 
atmosphere. The tube of mercury used in his experiments was 
what we now call a barometer {baros, weight), but it was for a 
long time called “the Torricellian Tube,” as the empty space 
above the mercury is still called the “Torricellian vacuum.” 
This invention or discovery of Torricelli’s was one of the most 
fertile ever made, for at one blow it demolished the ancient super- 
stition that “nature abhors a vacuum,” explained very simply 
two ancient puzzles (why water rises in a pump, and why it rises 
only 33 feet), determined accurately the weight of the atmosphere, 
proved it possible to make a vacuum, and gave to mankind an 
entirely new and invaluable instrument, the barometer. Torri- 
celli’s results and explanations were received at first with incredu- 
lity, but were soon confirmed, notably by Pascal (1623-1662) 
in a treatise, New Experiments on the Vacuum. In one of these 
Pascal used wine instead of water or mercury in the Torricellian 
tube, with satisfactory results, and in another, reasoning that if 
Torricelli were right, liquids in the tube should stand lower on a 
mountain than in a valley, persuaded his brother-in-law, Perier, 
to ascend the Puy de Dome (near Clermont, France) in September, 
1648, on which mountain the column was found to be much 
shorter. This and other brilliant work by Pascal have given him 
a high rank among natural philosophers. 
Since it was now easy to obtain a vacuum by the Torricellian 
experiment, fresh attempts were made to produce vacua otherwise. 
Von Guericke, burgomaster of Magdeburg in Hannover, after 
many failures, finally succeeded in pumping the air out of a hollow 
metallic globe. It was in this experiment that the air-pump was 
introduced. Guericke found that his globe had to be very strong 
to resist crushing by the atmospheric pressure, and in the popular 
demonstration now known as that of the Magdeburg hemi- 258 
A SHORT HISTORY OF SCIENCE 
spheres he showed that eight horses on either side were unable to 
overcome this pressure on a particular globe which he had con- 
structed and exhausted of air. These various experiments and dis- 
coveries relating to atmospheric pressure led to the investigations 
and laws of Boyle, Mariotte, and others and, less than a. century 
later, to the steam-engine of Watt, in which steam was at first 
used only to produce a vacuum, — atmospheric pressure being 
employed as the moving force. 
Further Studies of the Atmosphere: Gases. — Meantime 
the chemical composition of the atmosphere was being no less 
eagerly studied. Robert Boyle (1627-1691) published at Oxford 
in 1660, New Experiments Physico-Mechanical touching the 
Spring of the Air and its Effects, and in his Sceptical Chymist 
gives an interesting and instructive picture of the chemical ideas 
of his time. He was the first to insist on the difference between 
compounds and mixtures, and probably the first to use the pneu- 
matic trough for the collection and study of gases. 
The word “gas” was introduced by Van Helmont (1577-1644), 
who by virtue of the following remarkable statement deserves to 
be remembered as the principal chemist of the earlier half of the 
seventeenth century: — 
Charcoal and in general those bodies which are not immediately re- 
solved into water, disengage by combustion spiritum sylvestrum. From 
62 lbs. of oak charcoal 1 lb. of ash is obtained, therefore the remaining 
61 lbs. are this spiritum sylvestre. This spirit, hitherto unknown, I 
call by the new name of gas. It cannot be enclosed in vessels or re- 
duced to a visible condition. There are bodies which contain this 
spirit and resolve themselves entirely into it: in these it exists in a 
fixed or solidified form, from which it is expelled by fermentation, as we 
observe in wine, bread, etc. 
It has been well said that 
this passage is remarkable not only for the explicit mention of car- 
bonic acid gas (as we now call it) as a product of fermentation, and for 
the introduction of the word gas for the first time, but also for its ap- 
peal to the balance, NATURAL SCIENCE IN SEVENTEENTH CENTURY 
259 
the formal introduction of which into chemistry was only made a 
century later by Lavoisier. Van Helmont also points out that his 
gas sylvestre is produced by the action of acids on shells, is en- 
gendered in putrefaction and combustion, and is present in caves, 
mines, and mineral waters. In these ideas and passages we find an 
agreeable departure from the mysticism of the alchemists and the 
wild surmises of Paracelsus. At the same time Van Helmont’s 
ideas in other directions were crude enough, since he is credited with 
a recipe for the artificial production of mice from “ corn and sweet 
basil.” 
From Philosophy to Experimentation. — The seventeenth 
century differs from all before it in the increasing attention paid 
to experimental science. From the philosophizing of Paracelsus 
and Gilbert it is agreeable to pass to the experimental work of 
Harvey, Torricelli and in chemical inquiries to Van Helmont, whose 
logical successor is Robert Boyle (1627-1691), already mentioned 
for his work on the resistance, or “spring,” of the atmosphere, etc. 
Among many other ingenious experiments Boyle worked on evapo- 
ration, in air and in vacuo; on boiling and on freezing; and on 
the effects of exposing animals to the diminished atmospheric pres- 
sure produced by the air-pump. In this direction he was the 
first to prove that fishes require air dissolved in the water in 
which they live. He also studied the rusting of metals — a 
problem then widely discussed — and from all his studies con- 
cludes that there is in the atmosphere some vital substance which 
plays a principal part in such phenomena as combustion, respira- 
tion, and fermentation. When this substance has once been con- 
sumed, flame is instantly extinguished, and yet the air from which 
it has gone seems nearly intact. He wrote a treatise entitled, Fire 
and Flame weighed in a Balance, in which he described the increase 
of weight of metals on calcination. But as he got about the same 
results whether the crucible was open or shut, he was misled into 
the belief that the air had little to do with his results, which he 
attributed rather to the fixation of the “fire” by the porous 
crucibles. In these and Boyle’s other experiments it is plain that 
we are rapidly moving from alchemical and iatro-chemical stages 260 
A SHORT HISTORY OF SCIENCE 
toward the modern experimental period of chemistry, of which he 
and Van Helmont are the pioneers. Neither, however, while work- 
ing on air greatly advanced our ideas of atmospheric chemistry. 
The atmosphere in its relation to combustion and respiration was 
further studied by an English physician, Dr. John Mayow (1645- 
1679), who made many experiments upon the shrinkage of air- 
volume during the burning of camphor and other substances and 
during the confinement of mice under a bell-glass. The dying of 
the mice and the cessation of the combustion, which after a time 
ensued, he attributed to the exhaustion of some ingredient in the 
air indispensable to life and combustion. This ingredient, which 
we now call oxygen, Mayow named “fire-air.” 
Very soon, however, a new theory of combustion (and as it 
turned out a false theory) began to absorb the attention of 
natural philosophers. 
From Alchemy to Chemistry. — The saying is attributed to 
Liebig that “Alchemy was never at any time different from 
chemistry.” In one sense this is undoubtedly true. The search 
for “the philosopher’s stone,” “the elixir of life,” “potable gold,” 
and the “transmutation of metals,” consisted of necessity in the 
use of processes such as boiling, baking, wetting, drying, evaporat- 
ing, condensing, burning, calcifying, decalcifying, acidifying, 
freezing, melting, and the like, mostly tending towards chemical 
changes and the formation of new mixtures and compounds. But 
even if Liebig’s saying were true, chemistry has passed through 
three principal stages; viz. first, purely empirical experimenting, 
mostly for practical purposes, whether metallurgical or other; 
second, an iatro-chemical or medico-chemical phase; and finally 
the really scientific period of to-day, the way for which may be 
said to have been cleared by the Sceptical Chymist of Robert 
Boyle,1 first published in English in Oxford in 1661. In this re- 
1 The Hon. Robert Boyle was one of the most active, perhaps the most so, of 
that remarkable group of scientific investigators who, in the reign of Charles II., 
raised England to the foremost place among European nations in the pursuit of 
science, and gave their period a renown which has caused it to be often spoken of, 
and very justly, as the classical age of English science. . . . Boyle had been since 
1646 engaged in chemical researches in London, being then connected with the earlier NATURAL SCIENCE IN SEVENTEENTH CENTURY 
261 
markable little book Boyle by means of a dialogue discusses and 
sharply criticises the chemistry of the “hermetick” {i.e. Aris- 
totelian) natural philosophers, and also “the vulgar Spagyrists” 
{i.e. the medico-chemists of the Paracelsus type) and questions 
the value of terms then hazy in their meaning, such as “element” 
and “principle,” as used in alchemy. He does not himself pro- 
pound any new theories of consequence, but he does insist on 
more knowledge, more experimentation, and less groundless specu- 
lation. We quote from Professor Pattison Muir’s valuable intro- 
ductory essay to the “Everyman” edition: — 
The Sceptical Chymist embodies the reasoned conceptions which 
Boyle had gained from the experimental investigations of many physi- 
cal phenomena. . . . The book is more than an elegant and suggestive 
discourse on chemico-physical matters; it is an elucidation of the true 
method of scientific inquiry. ... At that time the alchemical scheme 
of things dominated most of those who were inquiring into the trans- 
mutations of material substances. That scheme was based on a magi- 
cal conception of the world. . . . When a magical theory of nature 
prevails, the impressions which external events produce on the senses of 
observers are corrected, not by careful reasoning and accurate experi- 
mentation, but by inquiring whether they fit into the scheme of things 
which has already been elaborated and accepted as the truth. Natural 
events become as clay in the hands of the intellectual potter for whom 
‘ there is nothing good or bad but thinking makes it so.’ . . . An al- 
chemical writer of the seventh century said: ‘ Copper is like a man; 
it has a soul and a body.’ . . . It is not possible to attach any definite, 
clear, meanings to alchemical writings about the four elements. Their 
indefiniteness was their strength. ... As the plain man to-day is 
soothed and made comfortable by the assurance that certain phrases 
to which he attaches no definite meanings are really scientific, so, 
when Boyle lived, the plain man rested happily in the belief that the 
four elements were the last word of science regarding the structure of 
the materials of the world. . . . 
group of scientific inquirers in London known as the ‘ Invisible College ’ . . . Boyle, 
too, we must observe, was above all things unprejudiced. He had leanings towards 
alchemy and never quite repudiated a belief in the possibility of transmuting metals. 
In medical matters, which greatly interested him, he showed perfect tolerance 
towards those whom the profession called quacks. — J. F. Payne. 262 
A SHORT HISTORY OF SCIENCE 
Boyle found the same fault with the ‘Principles’ of the ‘Vulgar 
Spagyrists ’ as he found with the ‘ Elements ’ of the ‘ hermetick philos- 
ophers.’ ‘Tell me what you mean by your Principles and your Ele- 
ments,’ he cried; ‘then I can discuss them with you as working in- 
struments for advancing knowledge.’ 
‘Methinks the Chymists in their search after truth are not unlike 
the navigators of Solomon’s Tarshish Fleet, who brought home from 
their long and perilous voyages not only gold and silver and ivory 
but apes and peacocks too: for so the writings of several (I say not all) 
of your hermetick philosophers present us, together with diverse sub- 
stantial and noble experiments, theories which, either like peacocks’ 
feathers, make a great show, but are neither solid nor useful, or else 
like apes, if they have some appearance of being rational, are blemished 
with some absurdity or other that, when they are attentively con- 
sidered, makes them appear ridiculous.’ 
The fact that at the middle of the seventeenth century criticism 
of this sort seemed to Boyle to be needed shows how little real 
progress toward modern scientific chemistry had even then been 
made; and, as often happens, truth had to be reached through 
further error. 
A False Theory of Combustion : Phlogiston. — Two 
German contemporaries of Boyle, Becher (1625-1682), and Stahl 
(1660-1734), as a result of studies on combustion and the calcining 
of metals, departed from the four elements of antiquity and 
assumed the participation in these processes of a something dis- 
pelled by heating. To this something Stahl gave the name 
;phlogiston, “ the combustible substance, a principle of fire, but not 
fire itself.” And because from a metallic calx (oxide) the metal 
could be recovered by burning with charcoal, the metal was held 
to have absorbed “ phlogiston ” in the process from the charcoal, 
which, having mostly disappeared, was regarded as almost pure 
phlogiston. Conversely, when the metal was calcined (or oxi- 
dized) by burning without charcoal, it was held to have lost its 
phlogiston. This theory, which to-day seems bizarre, satisfied 
the chief requirement imposed on any new theory: viz. that of ac- 
counting for the facts (as then known), and was therefore naturally NATURAL SCIENCE IN SEVENTEENTH CENTURY 
263 
accepted and advocated by natural philosophers for the next 
hundred years. It was not until new facts had been accumulated 
which were not explained by the theory of Becher and Stahl, and 
especially the fact revealed by the use of the balance, that sub- 
stances calcined often gained weight (making it necessary to assume 
that phlogiston possessed negative gravity, or “ levity ” since its 
loss increased weight), that the theory became plainly untenable 
and was abandoned. This, however, only happened late in the 
eighteenth century, and before this time much progress had been 
made in chemistry in other directions. Meanwhile, in spite of its 
falsity, the theory of phlogiston had done good service. It had, 
for example, effectually turned the attention of chemists away 
from magic, from potable gold, and from the making of medicines, 
to speculations on composition, decomposition, and chemical 
change, — topics not only more worthy but more fruitful. 
Beginnings of Organic Chemistry. — Meantime, a kind of 
organic chemistry was initiated by Hermann Boerhaave (1668- 
1738), a physician of Leyden. In the seventeenth and eighteenth 
centuries the term “organic” stood more than it does to-day for 
the living world and its products which were then regarded as 
things altogether apart from the lifeless or inorganic world. To- 
day organic chemistry hardly means more than the chemistry of 
the carbon compounds, but at that time it meant the chemistry of 
bodies found in or produced by living things. Medical men had 
long been interested in alchemy, and in more modern times in iatro- 
chemistry, so that it was natural enough that Boerhaave, a physi- 
cian, should undertake to subject organic substances to chemical 
processes. And this he did, though more in the fashion of the 
pharmaceutical, than the analytical, chemist of to-day. Boer- 
haave was a famous teacher of medicine and of botany, and crowds 
of students attended his lectures, thereby testifying to the now 
rapidly growing popularity of scientific learning. His Elements 
of Chemistry, published in 1732, was widely used and marks an 
epoch in the history of chemistry. 
At about the same time, Dr. Stephen Hales (1677-1761), an 
English clergyman of a strongly scientific bent, did similar work 264 
A SHORT HISTORY OF SCIENCE 
in England. In addition, Hales made important studies on the 
atmosphere, and invented the manometer, which he applied to the 
measurement of the arterial blood pressure in the horse, and the 
upward root pressure in plants, besides accomplishing much other 
good work. Hales will perhaps be longest remembered in chem- 
istry for his skilful use of the 'pneumatic trough, a simple but in- 
dispensable laboratory appliance for the easy collection of gases 
in a closed vessel over water, and especially for his studies on air. 
Medical Science and Medical Theory in the Seventeenth 
Century. Thomas Sydenham. — In the middle of the seven- 
teenth century medical theory took a long step forward under 
the influence of Thomas Sydenham (1624-1689), often called “the 
English Hippocrates” because of the naturalism and rationalism 
which he urged in medicine and because of the sanity of his opinions 
and theories. Setting aside magic, mysticism, and the medical 
chemistry of Paracelsus, and insisting on a material basis (materies 
morbi) for the causes of disease, Sydenham laid the foundations 
of modern scientific medical philosophy and practice. He was a 
close friend of Locke, the philosopher, — by whose materialistic 
and rationalistic ideas he was doubtless influenced, — and was 
also a correspondent of Boyle. His famous definition of disease 
as, “An effort of nature, striving with all her might to restore the 
patient by the elimination of morbific matter,” is still interesting 
for its implication of the modern idea of disease as a struggle for 
existence between pathogenic matters (such as microbes) and the 
inner forces of the body. 
It is, however, to Vesalius and Harvey, to Leeuwenhoek and 
Kircher and Malpighi and the other microscopists of the seven- 
teenth century, and their successors, i.e. to the experimenters and 
laboratory workers, rather than to Sydenham or his successors, 
that medical science is chiefly indebted, since no great progress 
could be made in sound medical theory or rational medical prac- 
tice until anatomy, physiology and microscopy had paved the 
way for a more scientific pathology. 
The Beginning of Modern Ideas of Light and Optics. — 
The nature of light, darkness and vision are very old problems. NATURAL SCIENCE IN SEVENTEENTH CENTURY 
265 
It is easy, even for savages, to account for daylight as sunlight, 
and the corresponding nightlight as moonlight and starlight, but 
for more highly developed man to explain just what the light is 
which comes from sun, moon and stars, is not so easy. Obviously, 
since “ luminaries ” — sun, moon, stars, firebrands and torches — 
produce light which is weaker as distance from the source in- 
creases, a kind of “emission” theory of light is natural and reason- 
able. It was even held by the ancients that we see by means of 
light emitted from our own eyes, and that light is a more or less 
palpable substance. A similar error was held concerning heat, 
which until the end of the eighteenth century was generally re- 
garded as a peculiar material body or substance, “caloric,” which 
when absorbed from other bodies produced a state of heat, and 
when emitted caused, by its absence, cold. 
How men could have believed for ages that objects are rendered 
visible by something projected from the eye itself — so that the organ 
of sight was supposed to be analogous to the tentacula of insects, and 
sight itself a mere species of touch — is most puzzling. They seem 
not till about 350 b.c. to have even raised the question: If this is how 
we see, Why cannot we see in the dark? or, more simply; What is 
darkness? The former of these questions seems to have been first 
put by Aristotle. 
The ancients probably understood that light travels in straight 
lines, and they must have known something about reflection and 
refraction of light, for they knew about images in still water, and 
had mirrors of polished metal, and burning glasses of spherical 
glass shells, or balls of rock crystal. To Hero of Alexandria we owe 
the important deduction from the Greek geometers that the course 
of a reflected ray is the shortest possible (p. 123). 
The perfection of gem cutting among the ancients has also been 
held to prove their acquaintance with lenses. But it was not 
until the seventeenth century that modern ideas of light and 
optics began to be formulated, with the work of Snellius, Descartes 
and Newton on reflection and refraction, and of Romer on the 
velocity, of light. 266 
A SHORT HISTORY OF SCIENCE 
Every student should read the earlier parts of Newton’s Optics in 
which are described the fundamental experiments on the decomposi- 
tion of white light. — Lord Rayleigh. 
The work of Christian Huygens, towards the end of the seven- 
teenth century, second only to that of Newton, both in extent and 
importance, touched upon a great variety of subjects, including 
some in the natural sciences. As a young man he wrote upon 
geometry; in early middle life he invented the cycloidal pendu- 
lum. He was the first to apply pendulums to clocks and spiral 
springs to watches, and to devise the achromatic eye-piece which 
still bears his name. He also made a telescope and, finally, at 
the age of fifty, observed the phenomena of polarization and, 
most important of all, proposed the modern wave theory of light. 
The First Scientific Instruments: Telescope, Barom- 
eter, Thermometer, Air-Pump, Microscope, Manometer.— 
The complete history of the origin of the telescope, the ther- 
mometer and the microscope is not known. The account usually 
given of the invention of the telescope makes it accidental and 
due to the children of a Dutch spectacle maker, named Jansen, 
who while at play happened to bring together two lenses in such 
a way that a distant church spire seen through them looked mag- 
nified and near. The father, whose attention was drawn to the 
phenomenon, seeing in the arrangement a source of profit, there- 
upon made and sold the combination as a toy or “wonder,” under 
which form it was on sale in 1609, becoming known to Galileo, who 
instantly realized its importance and made improvements in it. It 
appears that soon after 1609 Galileo had a fairly good instrument, 
magnifying 8 diameters, with which he was quickly and easily 
able to make some of his most splendid astronomical discoveries. 
The early history of the telescope shows that the effect of com- 
bining two lenses was understood by scientists long before any partic- 
ular use was made of this knowledge; and that those who are 
accredited with introducing perspective glasses to the public hit by 
accident upon the invention. Priority was claimed by two firms of 
spectacle-makers in Middelburg, Holland, namely Zacharias, miscalled PHYSICAL SCIENCE IN SEVENTEENTH CENTURY 
267 
Jansen, and Lippershey. Galileo heard of the contrivance in July 
1609 and soon furnished so powerful an instrument of discovery that 
... he was able to make out the mountains in the moon, the satel- 
lites of Jupiter in rotation, the spots on the revolving sun . . . 
About 1639, Gascoigne, a young Englishman, invented the mi- 
crometer which enables an observer to adjust a telescope with very 
great precision. 
The history of the microscope is closely connected with that of 
the telescope. In the first half of the seventeenth century the simple 
microscope came into use. It was developed from the convex lens 
. . . Leeuwenhoek before 1673 had studied the structure of minute 
animal organisms and ten years later had even obtained sight of bac- 
teria. Very early in the same century Zacharias had presented 
Prince Maurice, the commander of the Dutch forces, and the Arch- 
duke Albert, Governor of Holland, with compound microscopes. 
Kircher (1601-1680) made use of an instrument that represented 
microscopic forms at one thousand times larger than their actual 
size. — Libby. Introduction to the History of Science. 
The name of Galileo goes also with the invention of the ther- 
mometer, an air, or more strictly a water, thermometer having 
been introduced by him about 1597. Mercury was not substituted 
for water until 1670, but alcohol thermometers, also introduced by 
Galileo, were used much earlier. The freezing and boiling of 
water were supposed to take place at variable temperatures and 
it was not until the end of the seventeenth century that it was 
realized that the freezing and the boiling points are invariable. 
(For Galileo’s other work in physics, see pp. 246-250.) Pendulum 
clocks, “ aerial ” telescopes and the achromatic eye-pieces which 
bear his name were introduced by Huygens, the first in 1657 
and the others about 1680. 
The invention of the barometer by Torricelli has already been 
described above (p. 257). The air-pump, though merely the appli- 
cation of an ordinary pump to air instead of water, was so rich 
in its results that it deserves a high place among the other and 
more important inventions of this remarkable scientific era. 
About the origin of the (compound) microscope there is much 268 
A SHORT HISTORY OF SCIENCE 
the same obscurity as about that of the telescope. Simple micro- 
scopes such as “ magnifiers,” burning glasses, spectacles, and other 
lenses, had long been known, — some of them from antiquity, — 
but the compound microscope, which consists of two lenses or 
combinations of lenses so placed as to cooperate in the produc- 
tion of one highly magnified image of a near and minute object (the 
telescope doing the same for large and distant objects), first ap- 
pears about 1650. Some of the earliest microscopists are Kircher, 
Leeuwenhoek, Malpighi, and Grew'. The two former apparently 
saw with the microscope and made drawings of bacteria, besides 
many other micro-organisms and cellular structures. The two 
latter are the founders of microscopic anatomy, Malpighi of that 
of animals, Grew of that of plants. Malpighi’s work is especially 
notable, since he for the first time actually observed the passage of 
blood cells from arteries to veins, and that in 1661 only four years 
after Harvey’s death. Malpighi’s name is also familiar to students 
of human anatomy and physiology in connection with those parts 
of the kidneys and the spleen which bear his name. The versatile 
and accomplished Englishman Dr. Robert Hooke (1635-1703), who 
flourished in this century and did ingenious, extensive, and often 
remarkable work at the basis of almost every branch of modern 
science, was the first to discover by the microscope the cellular 
structure of living things. Hooke was one of the original members 
of the Royal Society,with which Leeuwenhoek also corresponded. 
The most remarkable fact connected with the invention of the 
compound microscope is that, because of its physical imperfec- 
tions, and in spite of some use as just described, it was virtually 
abandoned for almost a century and a half, and only re-introduced 
after the invention and perfection of the achromatic objective in the 
first quarter of the nineteenth century. The truth appears to be 
that owing to excessive spherical and chromatic aberration the 
compound microscope of the seventeenth and eighteenth centuries 
was of limited value, and that microscopists often preferred the 
less powerful, but more perfect, simple microscope. 
The manometer was apparently first used by Stephen Hales, 
who measured with it the blood pressure of a horse, the root pres- PHYSICAL SCIENCE IN SEVENTEENTH CENTURY 
269 
sure of plants, etc. It is described in his Statical Essays (1727) 
and Haemostaticks (1733). 
Organization of the First Scientific Academies and So- 
cieties.— The Academy of Plato (fifth century b.c.), and the 
Lyceum of Aristotle, the Museum at Alexandria (third century b.c.), 
and the so-called Academy of Alcuin (in the eighth century a.d.) 
may be regarded as precursors of the academies and societies of 
the Renaissance, but — with the possible exception of an academy 
formed by Leonardo da Vinci in the fifteenth century — the first 
devoted chiefly to science was probably that founded by della 
Porta at Naples in 1560 and named Academia Secretorum Naturae. 
The requirement for membership was to have made some dis- 
covery in natural science. Della Porta fell under ecclesiastical 
suspicion as a practitioner of the black arts, and though acquitted 
was ordered to close his “Academy.” The Accademia dei Lincei 
(of the Lynx), founded at Rome in 1603, included both della Porta 
and Galileo among its early members, and still flourishes. Its de- 
vice is a lynx with upturned eyes. 
The Royal Society of London, like many other societies, was the 
outgrowth of meetings of friends for discussion and was chartered 
in 1662. (For Boyle’s Invisible College see above, p. 261.) 
Among the earlier members of the Royal Society were Boyle 
and Hooke, Mayow, Huygens, Ray, Grew, Malpighi, Leeuwen- 
hoek, and Isaac Newton. A well-known passage quoted by 
Huxley from Dr. Wallis, one of the first members, is of special 
interest since it shows what subjects were most dwelt upon by men 
of science at the time of Cromwell and the Restoration:— 
Some twenty years before the outbreak of the plague (1665), says 
Huxley, a few calm and thoughtful students banded themselves to- 
gether for the purpose, as they phrased it, of ‘ improving natural 
knowledge. ’ The ends they proposed to attain cannot be stated more 
clearly than in the words of one of the founders of the organisation: — 
‘Our business was (precluding matters of theology and state affairs) 
to discourse and consider of philosophical enquiries, and such as re- 
lated thereunto: — as Physick, Anatomy, Geometry, Astronomy, 
Navigation, Staticks, Magneticks, Chymicks, Mechanicks, and 270 
A SHORT HISTORY OF SCIENCE 
Natural Experiments; with the state of these studies and their cultiva- 
tion at home and abroad. We then discoursed of the circulation of the 
blood, the valves in the veins, the vence lactece, the lymphatic vessels, 
the Copernican hypothesis, the nature of comets and new stars, the 
satellites of Jupiter, the oval shape (as it then appeared) of Saturn, 
the spots on the sun and its turning on its own axis, the inequalities 
and selenography of the moon, the several phases of Venus and Mer- 
cury, the improvement of telescopes and grinding of glasses for that 
purpose, the weight of air, the possibility or impossibility of vacuities 
and nature’s abhorrence thereof, the Torricellian experiment in quick- 
silver, the descent of heavy bodies and the degree of acceleration 
therein, with divers other things of like nature, some of which were 
then but new discoveries, and others not so generally known and em- 
braced as now they are; with other things appertaining to what hath 
been called the New Philosophy, which from the times of Galileo at 
Florence, and Sir Francis Bacon (Lord Verulam) in England, hath 
been much cultivated in Italy, France, Germany, and other parts 
abroad, as well as with us in England.’ 
The learned Dr. Wallis, writing in 1696, narrates in these words 
what happened half a century before, or about 1645. 
Among the first publications of the Royal Society of London 
were the works of Malpighi, the Italian microscopical anatomist, 
in 1669, and others by Leeuwenhoek, the Dutch microscopist. 
The French Academy (Academie des sciences) began its meetings 
in 1666, and the corresponding Berlin Academy in 1700. 
The oldest American association for the promotion of science 
is the American Philosophical Society Held at Philadelphia for 
Promoting Useful Knowledge, proposed by Benjamin Franklin 
in 1743 and finally organized in 1769. Franklin himself presided 
over it from 1769 until his death in 1790. 
The New Philosophy: Bacon and Descartes.—It has been 
shown above how the all-inclusive philosophy of their predecessors 
began with Plato and Aristotle to be divisible into general and 
“natural” philosophy, — a differentiation which continued to 
exist and to increase slowly through the Middle Ages and the 
Renaissance. 
We have also shown how the mariner’s compass, the invention BACON AND DESCARTES 
271 
of printing, the discovery of the New World, the heliocentric 
hypothesis, the idea of the earth as a magnet, the exploration of 
the human body, the Reformation, and the progress of mathe- 
matical science, were already widely opening men’s minds, so that 
by the end of the sixteenth century it is not surprising to find 
the new knowledge reacting upon the old philosophy. With this 
movement two great names will always be associated : viz. those 
of Francis Bacon and Rene Descartes. 
Bacon, because of his official position and immense philosophical 
and literary ability, was able to draw universal attention to the 
methods of science and especially to the method of investigation 
by induction, so that his indirect service to science was great. 
Bacon’s true place in science was, however, well understood by 
his contemporaries, for one of the greatest, Harvey, discoverer of 
the circulation of the blood, remarks that, “the Lord Chancellor 
writes of science like — a Lord Chancellor.” 
Descartes, far more important than Bacon in respect to his con- 
tributions to various branches of science, likewise stirred the in- 
tellect of Europe and helped to bring about those changes in the 
old philosophy which in the minds of many made it new. Des- 
cartes was not only a mathematician of the first rank but an in- 
genious and original worker in many branches of scientific inquiry 
such as music, anatomy, physiology, optics, etc. It is to him that 
we owe the first ideas of mechanism in living bodies, his notion 
of a “man machine” being highly original and suggestive. 
Science, says Descartes, may be compared to a tree; metaphysics 
is the root, physics the trunk, and the three chief branches are 
mechanics, medicine, and morals. 
Here are my books, he is reported to have told a visitor, as he 
pointed to the animals which he had dissected. 
The conservation of health, he writes in 1646, has always been the 
principal end of my studies. 
Bacon and Descartes were methodologists, both urging the 
fundamental importance to progress, of method and its right use in 
investigation and inquiry, and Descartes, younger by almost a 272 
A SHORT HISTORY OF SCIENCE 
generation, admired and to some extent imitated his predecessor 
in this direction. 
Progress of Natural and Physical Science in the 
Seventeenth Century. — A mere glance at the Tabular View 
of Chronology in the Appendix will suffice to show the immense 
superiority of the seventeenth to any preceding century in the 
number as well as the productivity of the workers devoted to the 
mathematical, and likewise to the natural and physical, sciences. 
The achievements of this century in natural philosophy are 
especially notable both for their fundamental character and their 
wide range. A century which began with a Galileo and ended with 
a Huygens and a Newton; which witnessed the introduction of 
the telescope, the barometer, the thermometer, the air-pump, the 
manometer, and the microscope, as well as the organization of 
the greatest and most useful scientific societies the world has 
hitherto known, must be forever famous. And when to the names 
and works of Galileo and Huygens and Newton we add those of 
Kepler, Harvey, Torricelli, Halley, Descartes, Boyle, Hales, 
Boerhaave, Leeuwenhoek, and Malpighi, we have a brilliant com- 
pany indeed. 
References for Reading 
Francis Bacon. Essay in Great Englishmen of the Sixteenth Century, by 
Sidney Lee. 
Robert Boyle. Sceptical Chymist. (Everyman’s Library.) 
Brewster’s Life of Newton, and Lives of Eminent Persons. 
R. Descartes. Life, by Haldane. 
R. Descartes, Discourse touching the method of using one’s reason rightly 
and of seeking scientific truth. Cf. Huxley. Methods and Results, 1896. 
G. E. Hale. National Academies and the Progress of Research. 
William Harvey On The Movement of the Heart and the Blood. (Everyman’s 
Library.) 
William Harvey. By D’Arcy Power. (Masters of Medicine Series.) 
Herschel’s Familiar Lectures. 
Thomas Sydenham. By J. F. Payne. (Masters of Medicine Series.) CHAPTER XIII 
BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
.... All the sciences which have for their end investigations 
concerning order and measure, are related to mathematics, it being 
of small importance whether this measure be sought in numbers, forms, 
stars, sounds, or any other object; that, accordingly, there ought to 
exist a general science which should explain all that can be known about 
order and' measure, considered independently of any application to a 
particular subject, and that, indeed, this science has its own proper 
name, consecrated by long usage, to wit, mathematics. And a proof 
that it far surpasses in facility and importance the sciences which de- 
pend upon it is that it embraces at once all the objects to which these 
are devoted and a great many others besides. ... — Descartes. 
As long as algebra and geometry proceeded along separate paths, 
their advance was slow and their applications limited. But when 
these sciences joined company, they drew from each other fresh vitality 
and thenceforward marched on at a rapid pace toward perfection. 
— Lagrange. 
The application of algebra has far more than any of his meta- 
physical speculations, immortalized the name of Descartes, and con- 
stitutes the greatest single step ever made in the progress of the 
exact sciences. — Mill. 
The idea of coordinates which forms the indispensable scheme for 
making all processes visible, with its many-sided and stimulating 
applications in all branches of daily life, — whether medicine, physi- 
cal geography, political economy, statistics, insurance, the technical 
sciences — the first beginnings of the calculus in their historical 
evolution, the development of the ideas of function and limit in 
connection with the elementary theory of curves, these are things 
without which in the present day not the slightest comprehension of 
the phenomena of nature can be attained, of which, however, the 
knowledge enables us as by magic to gain an insight with which in 
depth and range, but above all in certainty, scarcely any other can be 
compared. — Voss. 
How many celebrate the names of Newton and Leibnitz ! How few 
have a real appreciation of that which these men have created of 
permanent value ! Here lie the roots of our present-day knowledge, 
here the true continuation of the strivings of antique wisdom. 
273 
— Lindemann. 274 
A SHORT HISTORY OF SCIENCE 
The invention of the differential calculus marks a crisis in the 
history of mathematics. The progress of science is divided between 
periods characterized by a slow accumulation of ideas and periods, 
when, owing to the new material for thought thus patiently collected, 
some genius by the invention of a new method or a new point of 
view, suddenly transforms the whole subject on to a high level. 
— Whitehead. 
Mathematical Philosophy. Analytic Geometry. Des- 
cartes. —The invention of analytic geometry by Descartes in 1637 
and the almost contemporary introduction of integral calculus as 
the method of “ indivisibles” may be regarded as the real beginning 
of modern mathematical science. Thanks to these fruitful ideas 
the science has during the three centuries that have since elapsed 
made extraordinary progress both in its own internal development 
and in its application throughout the range of the physical sciences. 
Descartes was born in Touraine in 1596, and after the education 
appropriate for a youth of family and some years of fashionable 
life in Paris, entered the army, then in Holland. His military 
career continued till 1621 with incidental opportunity for his 
favorite speculations in mathematics and philosophy. Some of 
his most fruitful ideas dated from dreams and his best thinking 
was habitually done before rising. 
It is impossible not to feel stirred at the thought of the emotions 
of men at certain historic moments of adventure and discovery — 
Columbus when he first saw the Western shore, Franklin when the 
electric spark came from the string of his kite, Galileo when he first 
turned his telescope to the heavens. Such moments are also granted 
to students in the abstract regions of thought, and high among them 
must be placed the morning when Descartes lay in bed and invented 
the method of coordinate geometry. — Whitehead. 
In order to devote himself more completely to his favorite 
studies he settled in Holland in 1629, devoting the next four years 
to writing a treatise, entitled Le Monde, upon the universe. In 1637 
he published his great Discourse on the Method of Good Reasoning 
and of Seeking Truth in Science.1 This begins: — 
1 Discours de la M6thode pour bien conduire sa raison et chercher la v6rite dans 
les sciences. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
275 
If this discourse seems too long to be read all at once, it can be 
divided into six parts. In the first will be found various considera- 
tions concerning the sciences; in the second, the chief rules of the 
method which the author has sought; in the third, some of those 
of ethics which he has deduced by this method; in the fourth, the 
reasons by which he proves the existence of God and of the human 
soul, which are the foundations of his metaphysics; in the fifth, the 
order of questions of physics which he has sought, and particularly 
the explanation of the movement of the heart and of some other 
difficulties which belong to medicine; also the difference which exists 
between our soul and that of the beasts; and in the last, what 
things he believes necessary in order to go farther in the investiga- 
tion of nature than has been done, and what reasons have made 
him write. 
Good sense is the most widely distributed commodity in the 
world, for every one thinks himself so well supplied with it that even 
those who are hardest to satisfy in every other respect are not accus- 
tomed to desire more of it than they have. In this it is not prob- 
able that all men are mistaken, but rather this testifies that the power 
of good judgment and of discriminating between the true and the false, 
which is properly what one calls good-sense or reason, is naturally 
equal in all men; and thus that the diversity of our opinions is not 
due to the fact that some are more reasonable than others, but only 
that we conduct our thought along different channels, and do not 
consider the same things. For it is not enough to have a good mind, 
but the principal thing is to apply it well. The greatest souls are 
capable of the greatest vices as well as of the greatest virtues: and 
those who only progress very slowly can advance much more, if 
they follow always the straight road than do those who run, depart- 
ing from it. 
His four cardinal precepts were : — 
Never to receive anything for true which he did not recognize to be 
evidently so; that is, to avoid carefully precipitancy and prejudg- 
ment. Second, to divide each of the difficulties which he should 
examine into as many pieces as possible. Third, to conduct his 
thoughts in order, beginning with the simplest objects. The last, 
to make everywhere enumerations so complete and reviews so general 
that he should be assured of omitting nothing. 276 
A SHORT HISTORY OF SCIENCE 
Three appendices dealt with optics, meteors, and geometry, 
the last containing the beginnings of analytic geometry. The 
relation of his philosophy to mathematics may be indicated in 
the following passages. 
Considering that, among all those who up to this time made dis- 
coveries in the sciences, it was the mathematicians alone who had 
been able to arrive at demonstrations — that is to say, at proofs cer- 
tain and evident — I did not doubt that I should begin with the same 
truths that they have investigated, although I had looked for no other 
advantage from them than to accustom my mind to nourish itself upon 
truths and not to be satisfied with false reasons. 
When ... I asked myself why was it then that the earliest phi- 
losophers would admit to the study of wisdom only those who had 
studied mathematics, as if this science was the easiest of all and the 
one most necessary for preparing and disciplining the mind to com- 
prehend the more advanced, I suspected that they had knowledge 
of a mathematical science different from that of our time. . . . 
I believe I find some traces of these true mathematics in Pappus 
and Diophantus, who, although they were not of extreme antiquity, 
lived nevertheless in times long preceding ours. But I willingly be- 
lieve that these writers themselves, by a culpable ruse, suppressed the 
knowledge of them; like some artisans who conceal their secret, they 
feared, perhaps, that the ease and simplicity of their method, if 
become popular, would diminish its importance, and they preferred 
to make themselves admired by leaving to us, as the product of 
their art, certain barren truths deduced with subtlety, rather than 
to teach us that art itself, the knowledge of which would end our 
admiration. 
Those long chains of reasoning, quite simple and easy, which geom- 
eters are wont to employ in the accomplishment of their most difficult 
demonstrations, led me to think that everything which might fall 
under the cognizance of the human mind might be connected together 
in a similar manner, and that, provided only that one should take 
care not to receive anything as true which was not so, and if one 
were always careful to preserve the order necessary for deducing 
one truth from another, there would be none so remote at which 
he might not at last arrive, nor so concealed which he might not 
discover. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
277 
Descartes had attempted the solution of a historic geometrical 
problem propounded by Pappus. From a point P perpendiculars 
are dropped on m given straight lines and also on n other given 
lines. The product of the m perpendiculars is in a constant 
ratio to the product of the n; it is required to determine the locus 
of P. Pappus had stated without proof that for m = n = 2 the 
locus is a conic section, Descartes showed this algebraically, — 
Newton afterwards conquering the difficulty by unaided geometry. 
Descartes distinguished geometrical curves for which x and y 
may be regarded as changing at commensurable rates, or as we 
should say, curves for which the slope is an algebraic function 
of the coordinates, from curves which do not satisfy this condition. 
These he called “mechanical,” and did not discuss further. For 
the accepted definition of a tangent as a line between which and 
the curve no other line can be drawn, he introduced the modern 
notion of limiting position of a secant. In connection with this 
he considered a circle meeting the given curve in two consecutive 
points, a perpendicular to the radius of the circle being a common 
tangent to the circle and the given curve. The circle was not 
however that of curvature, but had its centre on an axis of sym- 
metry of the given curve. He recognized the possibility of ex- 
tending his methods to space of three dimensions, but did not work 
out the details. His geometry contained also a discussion of the 
algebra then known, and gave currency to certain important inno- 
vations, in particular the systematic use of a, b, and c, for known, 
x, y, and z, for unknown quantities; the introduction of exponents; 
the collection of all terms of an equation in one member; the free 
use of negative quantities; the use of undetermined coefficients in 
solving equations; and his rule of signs for studying the number 
of positive or negative roots of equations. He even fancied that 
he had found a method for solving an equation of any degree. 
It is important to distinguish just what Descartes contributed 
to mathematics in his analytic geometry. Neither the com- 
bination of algebra with geometry nor the use of coordinates was 
new. From the time of Euclid quadratic equations had been 
solved geometrically, while latitude and longitude involving a 278 
A SHORT HISTORY OF SCIENCE 
system of coordinates are of similar antiquity. The great step 
made by Descartes was his recognition of the equivalence of an 
equation and the geometrical locus of a point whose coordinates 
satisfy that equation. On this foundation facts known or ascer- 
tainable about geometry may be translated into algebra and con- 
versely. The advantage is comparable with that conferred by the 
possession of two arms or eyes, or even two senses, under a common 
will. The intricate but powerful machinery of algebra becomes 
available for solving geometrical problems, while, on the other 
hand, the geometrical illustration makes the algebra visible and 
concrete. 
Later works dealt with philosophy and physical science, in par- 
ticular with a theory of vortices. Descartes enunciates ten 
natural laws, the first two corresponding with the first two of 
Newton’s. He argues that all matter is in motion and that this 
must result in the formation of vortices. The sun is the centre of 
one great vortex, each planet of its own, thus approximating 
vaguely the future nebular hypothesis. Newton thought it worth 
while to refute this theory, which was chiefly notable as a bold 
attempt to interpret the phenomena of the universe by means of 
a single mechanical principle. 
Lord Kelvin has expressed, with all his force, that the sole satis- 
factory explanation of the phenomena of nature is that which leads 
them back in the last analysis to motion in a continuous incompres- 
sible fluid. This however was the guiding thought with Descartes. 
—Timerding. 
Descartes’s achievements in mathematics leave no doubt of his 
exceptional intellectual power. He had neither the data nor the 
scientific method for accomplishing similar results in other branches 
of science, and in mathematics he would doubtless have accom- 
plished much more had he not expended his energies so widely 
in over-confident reliance on his logical method. He died at 
Stockholm in 1650. 
Indivisibles. Cavalieri. — While Descartes was thus as it 
were incidentally laying the foundations of modern geometrical BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
279 
analysis, his Italian contemporary, Cavalieri (1598-1647) was 
rendering a similar service to the integral calculus in developing 
his theory of indivisibles. 
The problem of measuring the length of a curve or the area of 
a figure having a curved boundary, or the volume of a solid bounded 
by a curved surface goes back indeed to comparatively ancient 
Greek times. Most notable in this direction was the work of 
Archimedes. Kepler, attempting to resolve astronomical difficulties 
by the hypothesis of elliptical orbits, is confronted at once with 
the problem of determining the circumference of an ellipse. He 
gives the approximation 7r (a + 6) where a and b are the semi- 
axes. This is close if a and b are nearly equal, as in most of the 
planetary orbits. Interesting himself in current methods of 
measuring the capacity of casks, he published in 1615 his Nova 
Stereometria Doliorum Vinariorum, in which he determines the 
volumes of many solids bounded by surfaces of revolution. The 
Greek method had in case of the circle, etc., depended on an 
“exhaustion” process of inscribing and circumscribing polygons 
differing less and less from the curve both in boundary and in 
area. Kepler however divided his solid into sections, determined 
the area of a section and then sought the sum. He lacked an 
adequate system of coordinates, a clearly defined conception of a 
limit, and an effective method of summation. In view of the 
intrinsic difficulty of this important problem, however, the extent 
of his success is remarkable. 
He also sought to determine the most economical proportions 
for casks, etc., expressing his view of the underlying mathematical 
theory by the theorem “ In points where the transition from a less 
to the greatest and again to a less takes place, the difference is 
always to a certain degree imperceptible.” 
Cavalieri, in 1635, adopted the form of statement that a line 
consists of an infinite number of points, a surface of an infinity of 
lines, a solid of an infinity of surfaces, but later revised this on the 
basis of the assumption “that any magnitude may be divided 
into an infinite number of small quantities which can be made to 
bear any required ratios one to the other.” On this basis, open 280 
A SHORT HISTORY OF SCIENCE 
as it was to criticism, were solved simple area problems involving 
the parabola and the hyperbola. 
The principle of comparing areas by comparing lengths of a 
system of parallel lines crossing them is easily illustrated in the 
case of the ellipse by comparing it with the circle having as its 
diameter the (horizontal) major axis of the ellipse. If a and b 
are the semi-axes of the ellipse the two 
curves are known to be so related that every 
vertical chord of the circle is in a fixed ratio 
a: b to the part of it lying within the ellipse. 
The area of the circle must bear the same 
relation to the area of the ellipse. The 
transition from length to area while not 
rigorously worked out by Cavalieri does not 
necessarily involve the false assumption that 
area consists of the sum of parallel lines. A similar method is 
evidently applicable to volumes. Thus was anticipated one of the 
most interesting and important processes of modern mathematics, 
— integration as a summation. 
Similarly Cavalieri determined volumes by a consideration of 
the thin sections or elements into which they may be resolved by 
parallel planes. The principle that “two bodies have the same 
volume if sections at the same level have the same area” is still 
known by his name. 
Descartes’s work with tangents seems not to have led him to 
develop the fundamental ideas of the differential calculus, and it 
appeared that the integral calculus would be evolved first from the 
work of Cavalieri. 
Projective Geometry : Desargues. — Hardly less interesting 
than the new ideas of Descartes and Cavalieri are those of their 
contemporary Desargues (1593-1662), an engineer and architect 
of Lyons, who made important researches in geometry. But for 
the still more brilliant geometrical achievements of Descartes, 
these might have led to the immediate development of projective 
geometry, the elements of which are contained in Desargues’s 
work. In general this geometry instead of dealing with definite BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
281 
triangles, polygons, circles, etc., in the Euclidean manner, is based 
on a consideration of all points of a straight line, of all lines through 
a common point and of the possible effects of setting up an orderly 
one-to-one correspondence between them. In particular, Des- 
argues makes a comparative study of the different plane sections 
of a given cone, deducing from known properties of the circle anal- 
ogous results for the other conic sections. 
In his chief work Desargues enunciates the propositions: — 
1. A straight line can be considered as produced to infinity 
and then the two opposite extremities are united. 
2. Parallel lines are lines meeting at infinity and conversely. 
3. A straight line and a circle are two varieties of the same 
species. 
On these he bases a general theory of the plane sections of a cone. 
Desargues contented himself with enunciating general princi- 
ples, remarking: — “ He who shall wish to disentangle this prop- 
osition will easily be able to compose a volume.” He met 
Descartes while employed by Cardinal Richelieu at the siege of 
Rochelle, and they with others met regularly in Paris for the 
discussion of the new Copernican theory and other scientific 
problems. 
He says ‘I freely confess that I never had taste for study or re- 
search either in physics or geometry except in so far as they could 
serve as a means of arriving at some sort of knowledge of the proxi- 
mate causes .... for the good and convenience of life, in maintaining 
health, in the practice of some art, . . . having observed that a good 
part of the arts is based on geometry, among others the cutting of 
stones in architecture, that of sun-dials, that of perspective in 
particular.’ 
Perceiving that the practitioners of these arts had to burden them- 
selves with the laborious acquisition of many special facts in 
geometry, he sought to relieve them by developing more general 
methods and printing notes for distribution among his friends. 
An interesting theorem bearing his name and typical of pro- 
jective geometry is as follows: — If two triangles ABC and A’B’C^ 282 
A SHORT HISTORY OF SCIENCE 
are so related that lines joining corresponding vertices meet in a 
point O, then the intersections of corresponding sides will lie 
in a straight line A"B"C". It 
remained for Monge, the inventor of 
descriptive geometry (p. 335) and 
others more than a century later 
to carry this development forward. 
Desargues’s work was indeed prac- 
tically lost until Poncelet in 1822 
proclaimed him the Monge of his 
century. 
Theory of Numbers and Probability : Fermat, Pascal. — 
But little younger than Descartes and Cavalieri was Pierre de 
Fermat (1601-1665) a man of quite exceptional position in mathe- 
matical history. Devoting to mathematics such leisure as his 
public duties afforded, he nevertheless published almost noth- 
ing, many of his results being known to us only in the form of 
brief marginal notes without proof. In editing Diophantus he 
enunciated numerous theorems on integers, for example, 
An odd prime can be expressed as the difference of two square 
integers in one and only one way. 
No integral values of x, y, z can be found to satisfy the equation 
xu + yn — if n be an integer greater than 2. 
This seemingly simple theorem has been verified for so wide a 
range of values of n, that its truth can hardly be doubted, but no 
general proof has yet been given in spite of a prize of 100,000 marks 
awaiting him who either proves or disproves it. Some writers even 
credit Fermat with a substantial share in the invention of the new 
analytic geometry, in which he had certainly done independent 
work for some years before Descartes’s publication. Laplace in- 
deed calls Fermat “the true inventor of the differential calculus.” 
He discusses problems of maxima and minima, and passing to 
concrete phenomena, enunciates the interesting theorem: that 
Nature, the great workman which has no need of our instru- 
ments and machines, lets everything happen with a minimum of BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
283 
outlay, — an idea not indeed strange to some of the Greeks. 
The law of refraction of a ray of light he deals with correctly as 
a particular case of the principle of economy, a principle which 
exerted a potent influence in the scientific philosophy of the 
following century. Thus for example Euler says in 1744: — 
Since the organization of the world is the most excellent, nothing is 
found in it, out of which some sort of a maximum or minimum 
property does not shine forth. Therefore no doubt can exist, that all 
action in the world can be derived by the method of maxima and 
minima as well as from the actual operating causes. 
Fermat’s work in the theory of probability is fundamental. He 
discusses the case of two players, A and B, where A wants two 
points to win and B three points. Then the game will certainly 
be decided in the course of four trials. Take the letters a and b, 
and write down all the combinations that can be formed of four 
letters. These combinations are 16 in number, namely aaaa, 
aaab, aaba, aabb, abaa, abab, abba, abbb, baaa, baab, baba, babb, 
bbaa, bbab, bbba, bbbb. Now every combination in which a oc- 
curs twice or oftener represents a case favorable to A, and every 
combination in which b occurs three times or oftener represents a 
case favorable to B. Thus, on counting them, it will be found 
that there are 11 cases favorable to A, and 5 cases favorable to 
B ; and, since these cases are all equally likely, A’s chance of win- 
ning the game is to B’s chance as 11 is to 5. 
Like Descartes, Pascal (1623-1662) devoted but a fraction of 
his great talent to mathematical science. 
I have spent much time in the study of the abstract sciences, — 
but the paucity of persons with whom you can communicate on such 
subjects gave me a distaste for them. When I began to study man, 
I saw that these abstract studies were not suited to him, and that in 
diving into them, I wandered farther from my real track than those 
who were ignorant of them, and I forgave men for not having at- 
tended to these things. But I thought at least I should find many 
companions in the study of mankind, which is the true and proper study 
of man. Again I was mistaken. There are yet fewer students of Man 
than of Geometry. 284 
A SHORT HISTORY OF SCIENCE 
Learning geometry surreptitiously at 12 years, he had at 18 
written an essay on conic sections and constructed the first com- 
puting machine. While most of his later life was devoted to re- 
ligion, theology, and literature, he undertook a wide range of 
physical experimentation, and made important contributions to 
the then new theories of numbers and probability, besides a 
discussion of the cycloid. The juvenile essay on conic sections 
contains the beautiful theorem since named for him that the 
opposite sides of a hexagon inscribed in a conic section meet in a 
straight line. Of geometry and logic Pascal says: — 
Logic has borrowed the rules of geometry without understanding 
its power. ... I am far from placing logicians by the side of geom- 
eters who teach the true way to guide the reason. . . . The method 
of avoiding error is sought by every one. The logicians profess to 
lead the way, the geometers alone reach it, and aside from their science 
there is no true demonstration. 
His work on probability connected itself with the problem of 
two players of equal skill wishing to close their play, of which 
Fermat’s solution has been given above. 
The following is my method for determining the share of each 
player when, for example, two players play a game of three points 
and each player has staked 32 pistoles. 
Suppose that the first player has gained two points and the second 
player one point; they have now to play for a point on this condition, 
that if the first player gain, he takes all the money which is at stake, 
namely 64 pistoles; while if the second player gain, each player has 
two points, so that they are on terms of equality, and if they leave 
off playing, each ought to take 32 pistoles. Thus if the first player 
gain, then 64 pistoles belong to him, and if he lose, then 32 pistoles 
belong to him. If therefore the players do not wish to play this game, 
but separate without playing it, the first player would say to the 
second, ‘ I am certain of 32 pistoles, even if I lose this point, and as 
for the other 32 pistoles, perhaps I shall have them and perhaps you 
will have them; the chances are equal. Let us then divide these 32 
pistoles equally, and give me also the 32 pistoles of which I am certain.’ 
Thus the first player will have 48 pistoles and the second 16 pistoles. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
285 
By similar reasoning he shows that if the first player has gained 
two points and the second none, the division should be 56 to 8; 
while if the first has gained one point, the second none, it should 
be 44 and 20. 
The calculus of probabilities, when confined within just limits, 
ought to interest, in an equal degree, the mathematician, the experi- 
mentalist, and the statesman. From the time when Pascal and 
Fermat established its first principles, it has rendered, and continues 
daily to render, services of the most eminent kind. It is the calculus 
of probabilities, which, after having suggested the best arrangements 
of the tables of population and mortality, teaches us to deduce from 
those numbers, in general so erroneously interpreted, conclusions of 
a precise and useful character; it is the calculus of probabilities which 
alone can regulate justly the premiums to be paid for assurances; 
the reserve funds for the disbursements of pensions, annuities, dis- 
counts, etc. It is under its influence that lotteries and other shameful 
snares cunningly laid for avarice and ignorance have definitely disap- 
peared. — Arago. 
With this work connected itself his arithmetical triangle in which 
successive diagonals contain the coefficients which occur in ex- 
pansions by the binomial theorem, which Newton was soon to 
generalize. 
111111 
1 2 3 4 5 
1 3 6 10 
1 4 10 
1 5 
1 
He applies the method of indivisibles successfully to the cycloid 
(the curve generated by a point on the rim of a rolling wheel). 
Pascal invented in 1645 an arithmetical machine, writing the 
Chancellor in regard to it: 
Sir: If the public receives any advantage from the invention which 
I have made to perform all sorts of rules of arithmetic in a manner as 
novel as it is convenient, it will be under greater obligation to your 286 
A SHORT HISTORY OF SCIENCE 
Highness than to my small efforts, since I should only have been able 
to boast of having conceived it, while it owes its birth absolutely to 
the honor of your commands. The length and difficulty of the ordi- 
nary means in use have made me think on some help more prompt and 
easy to relieve me in the great calculations with which I have been 
occupied for several years in certain affairs which depend on the 
occupations with which it has pleased you to honor my father for the 
service of his Majesty in Normandy. I employed for this investi- 
gation all the knowledge which my inclination and the labor of my first 
studies in mathematics have gained for me, and after profound re- 
flection, I recognized that this aid was not impossible to find. 
Mechanics and Optics : Huygens. — Most notable among 
the successors of Galileo in mechanics before we reach Newton 
was Huygens of Holland (1629-1695) who combined mathematical 
power with exceptional practical ingenuity. He first (in 1655) 
explained as a ring the excrescences of Saturn which had been 
misunderstood by Galileo and others, publishing his discovery in 
the occult form a7c5d1e5g1h1i7lim2n9o4p2q1r2s1t5u5. (Annulo cingitur 
tenui, piano, nusquam cohcerente ad eclipticam inclinato.) He also 
discovered Saturn’s largest moon. About the same time he made 
his great invention of the pendulum clock. Accepting a call to 
Paris by Colbert at the founding of the French Academy, he 
remained there from 1666 to 1681. 
In optics he developed and maintained even in opposition to 
the authority of Newton the undulatory or wave theory which 
only found general acceptance a century later. The velocity of 
light Galileo had failed to measure by means of signal lanterns, 
and Descartes had likewise been unable to ascertain it by compar- 
ing the observed and computed instants of a lunar eclipse. 
Huygens points out that even this latter test does not prove in- 
stantaneous transmission. Romer’s conclusive report on observa- 
tions of a satellite of Jupiter dates from 1675. On this basis 
Huygens estimated the velocity of light at 600,000 times that of 
sound, — a result about one-third too small. 
The medium in which light waves travel Huygens named the 
ether, attributing to its particles three properties in comparison Huygens from (Euvres Completes, 1899 BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
287 
with air: extreme minuteness, extreme hardness, extreme elas- 
ticity. On this basis he worked out a consistent theory for re- 
flection and refraction. His discussion of the newly discovered 
phenomenon of double refraction in Iceland spar has been char- 
acterized as an “unsurpassed example of the combination of ex- 
perimental investigation and acute analysis.” The attendant 
phenomenon of polarisation did not escape him, but his theory 
of wave motion was not sufficiently developed to enable him to 
explain the matter adequately. 
In 1673 Huygens published his great work on the pendulum 
{Horologium oscillatorium sive de motu pendulorum), displaying 
wonderful skill in his geometrical treatment of the mechanical 
problems involved. The use of wheel mechanisms with weights 
for measuring time had been more or less familiar for several cen- 
turies, but no effective means for regulating this motion had 
been devised. Galileo, for example, observing the regularity of 
pendulum vibrations, had depended on repeated impulses by 
hand to maintain the motion. Huygens first made the fortunate 
combination of the two elements, without however inventing the 
modern escapement. He studied the cycloidal pendulum for 
which the time of an oscillation would be independent of the 
amplitude and made precise determinations of the length of the 
seconds-pendulum at Paris and the corresponding value of the im- 
portant constant g. The most remarkable achievement in his 
treatise on the pendulum is the correct analysis of the compound 
pendulum based on the definition: — 
The centre of oscillation of any figure whatever is that point in the 
line of gravity, whose distance from the point of suspension is the 
same as the length of the simple pendulum having the same time of 
vibration as the figure. 
In the course of the discussion he formulates the important law, 
afterwards somewhat generalized by others: 
Whenever any heavy bodies are set in motion under the action of 
their own weight, their common centre of gravity cannot rise higher 
than it was at the beginning of the motion. 288 
A SHORT HISTORY OF SCIENCE 
In computing the position of the centre of oscillation he arrives 
2TflT^ 
at a fraction of the form ~—> where m denotes the mass of a 
2,mr 
particle, r its distance from the point of suspension. The nu- 
merator is the so-called “moment of inertia,” the denominator the 
“statical moment” of later mechanics. He shows that the point 
of suspension and the centre of oscillation are interchangeable. 
Finally he discusses the theory of centrifugal force, proving 
that it varies as the square of the velocity and inversely as the 
radius. This subject he also treated more fully in a special mono- 
graph, published after his death when Newton had already given 
a more general theory. His theorems are: — 
1. When equal bodies move with the same velocity in unequal 
circles, the centrifugal forces are to each other inversely as the diameters, 
so that in the smaller circle the said force is greater. 
2. When equal movable bodies travel in the same or equal circles 
with unequal velocities, the centrifugal forces are to each other as the 
squares of the velocities. 
By experiments on a revolving sphere of clay which as he antici- 
pated assumed a spheroidal form, he explains the observed polar 
flattening of Jupiter. He infers that the earth must also be 
flattened, and makes a numerical estimate in anticipation of future 
verification. He explains the effect on a clock pendulum of trans- 
porting it from Paris to an equatorial locality, where its weight is 
opposed by an increased centrifugal force. 
Like Wallis (p. 290) and Sir Christopher Wren he accepted the 
invitation of the Royal Society to attack the general problem of 
impact. This led ultimately to the publication eight years after 
his death of his On the Motion of Bodies under Percussion. 
The theorems enunciated deal with various cases of central im- 
pact, one of the most notable being: — 
By mutual impact of two bodies the sum of the products of the 
masses into the squares of their velocities is the same before and after 
impact. FIGI. 
HG.n. 
figiv: 
fig. m. 
Huygens’ Clock 
(Horologium Oscillatorium, 1673) BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
289 
— the first formulation (1669) of the most comprehensive law of 
mechanics, the conservation of vis viva. 
Huygens visited England in 1689, but made no use of Newton’s 
new calculus in his published work. In his History of the Mathe- 
matical Theories of Attraction and the Figure of the Earth, 
Todhunter says of Huygens: — 
To him we owe the important condition of fluid equilibrium, that the 
resultant force at any point of the free surface must be normal to the 
surface at that point; and this has indirectly promoted the knowledge 
of our subject. But Huygens never accepted the great principle of 
the mutual attraction of particles of matter; and thus he contributed 
explicitly only the solution of a theoretical problem, namely the inves- 
tigation of the form of the surface of rotating fluid under the action of 
a force always directed to a fixed point. 
Wallis and Barrow. — Before attempting to discuss the 
extraordinary work of Sir Isaac Newton in the whole field of 
mathematical science a few words should be added concerning two 
slightly older English mathematicians, John Wallis (1616-1703), 
Savilian professor at Oxford, and Isaac Barrow (1630-1677), 
Lucasian professor at Cambridge. 
Wallis in his Arithmetic of The Infinites (1656) developed 
Cavalieri’s summation ideas effectively, employing the new 
Cartesian geometry and a process equivalent to integration for 
simple algebraic cases. In particular, he explains negative and 
fractional exponents in the modern sense, and then proceeds to 
find the area bounded by OX, the curve y = axm, and any ordinate 
x = h, — or as we should say, he integrates the function axm. 
He develops ingenious methods of interpolation. 
In his Treatise on Algebra he says: — 
It is to me a theory unquestionable, That the Ancients had some- 
what of like nature with our Algebra; from whence many of their pro- 
lix and intricate Demonstrations were derived. . . . But this their 
Art of Invention, they seem very studiously to have concealed: content- 
ing themselves to demonstrate by Apagogical Demonstrations, (or re- 
ducing to Absurdity, if denied,) without showing us the method, by 290 
A SHORT HISTORY OF SCIENCE 
which they first found out those Propositions, which they thus demon- 
strate by other ways. . . . Nonius ‘ O how well it had been if those 
Authors, who have written in Mathematics, had delivered to us their 
Inventions, in the same way, and with the same Discourse, as they 
were found out! And not as Aristotle says of Artificers in Mechanics 
who show us the Engines they have made, but conceal the Artifice, 
to make them the more admired ! ’ 
His Analytical Conic Sections (1665) made Descartes’s geometri- 
cal ideas much more intelligible, and his Algebra (1686) marks an 
important step forward in its systematic use of formulas. He 
also wrote A Summary Account ... of the General Laws of Mo- 
tion, enunciating the formulas for velocity after impact of masses 
m\ and ra2 with velocities V\ and v2: — 
WliVi + VI2V2 
v = 
m 1 + m2 
Barrow, after varied adventures, became first Lucasian professor 
at the University of Cambridge, but resigned his chair six years 
later to his pupil Newton. His work on optics and geometry 
contains a notable discussion of the tangent problem and of what 
he calls the differential triangle, so important in modern elementary 
differential calculus. His general point of view is illustrated by 
the following passage: — 
Now as to what pertains to these Surd numbers (which, as it 
were by way of reproach and calumny, having no merit of their own 
are also styled Irrational, Irregular, and Inexplicable) they are by many 
denied to be numbers properly speaking, and are wont to be banished 
from arithmetic to another Science (which yet is no science), viz. 
algebra. 
Isaac Newton, — was born within a year after Galileo’s death, 
a century after that of Copernicus — December 25, 1642 (O.S.). 
Destined at first to become a farmer, he was fortunately sent at 
17 to the university, where he quickly and eagerly mastered the 
mathematical work of Euclid, Descartes and Wallis, and Kepler’s 
Dioptrics. His discovery of the general binomial theorem dates 
from this time, and he even ventured to attack the great problem BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
291 
of gravitation by carefully comparing the motion of the moon 
with that of a falling body near the earth — for which however 
his data were not yet sufficiently accurate. 
Newton took his B. A. degree in the Lent Term, 1665. In 
that spring the plague appeared, and for a couple of years he lived 
mostly at home, though with occasional residence at Cambridge. 
Probably at this time his creative powers were at their highest. His 
use of fluxions may be traced back to May, 1665; his theory of gravi- 
tation originated in 1666; and the foundation of his optical discov- 
eries would seem to be only a little later. In an unpublished mem- 
orandum made some years later (cancelled, but believed to be correct 
in the part here quoted), he thus described his work of this. time: 
‘ In the beginning of the year 1665 I found the method of approximat- 
ing Series and the Rule for reducing any dignity of any Binomial 
into such a series. The same year, in May, I found the method of 
tangents of Gregory and Slusius, and in November had the direct 
method of Fluxions, and the next year in January had the Theory of 
Colours, and in May following I had entrance into the inverse method 
of Fluxions. And the same year I began to think of gravity extend- 
ing to the orb of the Moon, and . . . from Kepler’s Rule of the period- 
ical times of the Planets being in a sesquialterate proportion of their 
distances from the centers of their orbs I deduced that the forces which 
keep the Planets in their orbs must (be) reciprocally as the squares 
of their distances from the centers about which they revolve: and 
thereby compared the force requisite to keep the Moon in her orb 
with the force of gravity at the surface of the earth, and found them 
answer pretty nearly. All this was in the two plague years of 1665 
and 1666, for in those days I was in the prime of my age for invention, 
and minded Mathematicks and Philosophy more than at any time 
since.’ — Ball, Mathematical Gazette, July, 1914. 
Optics. — Interesting himself in the telescope, Newton suc- 
ceeded in eliminating the disturbing chromatic aberration due to 
unequal refraction of the different colors by constructing a reflect- 
ing telescope with a concave mirror in place of a convex lens. On 
the other hand, turning his attention to the colors of the solar 
spectrum, he wrote his Opticks or a Treatise of the Reflections, 
Refractions, Inflections and Colours of Light, published in 1704. 292 
A SHORT HISTORY OF SCIENCE 
Disclaiming any intention of setting up speculative hypotheses, he 
discusses the observed phenomena of refracted light, speaking of 
his discovery of the different refrangibility of the rays of light as 
“in my judgment the oddest if not the most considerable detection 
which hath hitherto been made in the operations of nature.” 
While he does not insist upon it, he seems always to have the 
underlying idea that light itself consists of minute particles — the 
degree of fineness corresponding with the color—a theory which 
held the field — thanks to his potent authority with his too sub- 
servient followers — against the better undulatory theory of Huy- 
gens until the nineteenth century. In the experiments on which 
this work is based Newton not only decomposes light by a refract- 
ing prism or series of prisms, but also succeeds in recombining the 
component colors to reproduce the original white. 
The colors of objects, he says, are nothing more than their power 
to reflect one or another kind of ray. And in the rays again is 
nothing other than the power to transmit this motion into our organ 
of sense, in which last finally arises the sensation of these motions in 
the form of colors. 
He solves at last the problem of the rainbow. All this constitutes 
an immense advance over the current Aristotelian notions. 
The Theory of Gravitation: Principia— In 1682 Newton 
returned to his attempt of 16 years earlier to explain the moon’s 
motion by means of the assumed influence of gravitation. During 
this long interval French geographers, testing the supposedly spher- 
ical shape of the earth, had made a new and more precise triangu- 
lation — with the first use of telescopic instruments. Newton’s 
earlier data had led to a determination of the acceleration due 
to gravity at the distance of the moon as 13| feet per minute. 
The new data changed this result to 15, in agreement with his 
hypothesis that the force varied inversely as the square of the 
distance. Stirred to the inmost depths of his usually calm nature 
by his realization that he was approaching a solution of the great 
problem, he had to beg a friend to complete his calculations. The 
new astronomy founded by Copernicus, built up by Tycho Brahe, Newton’s Telescope (Great Astronomers, R. S. Ball). 
Newton’s Theory of the Rainbow (Opticks, 1704). BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
293 
Kepler, and Galileo, was now to be completely formulated and 
mathematically interpreted by Newton’s crowning discovery of a 
single mechanical principle governing the whole. 
It was now a question of verifying the correctness of this prin- 
ciple by applying it to all measured or measurable astronomical 
phenomena. The investigation was gradually extended to the 
planets, the moons of Jupiter, the tides, and even the comets. 
Everywhere the law was verified that attraction varies as the prod- 
uct of the masses and inversely as the square of the distance. 
The whole theory was elaborated in Newton’s monumental 
Principia Philosophies Naturalis Mathematica published in 1687. 
He begins this treatise with a series of definitions and laws: 
1. The quantity of matter is the measure of the same, arising from 
its density and bulk conjunctly. 
2. The quantity of motion is the measure of the same, arising from 
the velocity and quantity of matter conjunctly. 
3. The innate force of matter is a power of resisting, by which 
every body, as much as in it lies, endeavours to persevere in its present 
state, whether it be of rest, or of moving uniformly forward in a 
right line. 
4. An impressed force is an action exerted upon a body, in order 
to change its state, either of rest, or of moving uniformly forward in 
a right line. 
5. A centripetal force is that by which bodies are drawn or im- 
pelled, or any way tend, towards a point as to a centre. 
These and succeeding definitions are followed by the famous 
Laws of Motion: 
I. Every body perseveres in its state of rest, or of uniform motion 
in a right line, unless it is compelled to change that state by forces 
impressed thereon. 
II. The alteration of motion is ever proportional to the motive 
force impressed, and is made in the direction of the right line in 
which that force is impressed. 
III. To every action there is always opposed an equal reaction: 
or the mutual actions of two bodies upon each other are always equal, 
and directed to contrary parts. Corollary I continues: A body by two 294 
A SHORT HISTORY OF SCIENCE 
forces conjoined will describe the diagonal of a parallelogram, in the 
same time that it would describe the sides, by those forces apart. 
Of these laws, Pearson in his Grammar of Science remarks: 
The Newtonian laws of motion form the starting-point of most 
modern treatises on dynamics, and it seems to me that physical sci- 
ence, thus started, resembles the mighty genius of an Arabian tale 
emerging amid metaphysical exhalations from the bottle in which for 
long centuries it has been corked down. 
Passing to variable forces he discusses in particular the motion 
of a body acted on by a central attractive force — i.e. a force at- 
tracting it towards a fixed point. He derives the law by which 
equal areas are described in equal times, and shows that con- 
versely, if equal areas are so described in a plane, the determining 
force must be a central one. Turning to the consideration of or- 
bits, he deals with the hypothesis of an elliptical orbit with the 
attracting force at one of the foci, and shows that the attractive 
force must vary inversely as the square of the distance from that 
focus. The same result is obtained for the other conic secti6ns. 
These theorems applying to particles, he next shows that the action 
of a homogeneous sphere on an external particle is the same as if 
its mass were concentrated at its centre, so that the action of two 
such spheres on each other is subject to the laws already derived. 
Comparing planetary motions with those of projectiles which 
had been treated by Galileo, Newton says: — 
That the planets can be held in their paths is evident from the 
motions of projectiles. A stone thrown is deflected from the straight 
line by its weight and falls describing a curved line to the earth. If 
thrown with greater velocity it goes farther, and it could happen that 
it described a curve of 10,100,1000 miles, and at last went outside the 
boundaries of the earth, and never fell back. . . . 
The force of gravity for small distances being sensibly constant 
in direction, causes motion in a path approximately parabolic. 
For greater and greater ranges the change of direction of the 
force must be taken account of and the path recognized as ellipti- 
cal or hyperbolic. The laws as stated deal with the relations of BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
295 
only two mutually attracting bodies. Newton of course appreci- 
ates that such a case is purely ideal and that, since every body 
attracts every other, the result of dealing with only two is merely 
a first approximation to the reality. 
All planets he says are mutually heavy, therefore, for example; Jupiter 
and Saturn will attract each other in the vicinity of their conjunction 
and perceptibly disturb each other’s motion. Similarly the Sun will dis- 
turb the motion of the Moon, and Sun and Moon will disturb our ocean. 
Newton prefaced these applications of the theory with four rules 
which should guide scientific men in making hypotheses. These in 
their final shape, are to the following effect: (1) We should not assume 
more causes than are sufficient and necessary for the explanation of 
observed facts. (2) Hence, as far as possible, similar effects must be 
assigned to the same cause; for instance, the fall of stones in Europe 
and America. (3) Properties common to all bodies within reach of 
our experiments are to be assumed as pertaining to all bodies; for 
instance, extension. (4) Propositions in science obtained by wide in- 
duction are to be regarded as exactly or approximately true, until 
phenomena or experiments show that they may be corrected or are 
liable to exceptions. The substance of these rules is now accepted 
as the basis of scientific investigation. Their formal enunciation here 
serves as a landmark in the history of thought. — Mathematical 
Gazette, July, 1914. 
Every new satellite, says Brewster in his Life of Newton, every 
new asteroid, every new comet, every new planet, every new star 
circulating round its fellow, proclaims the universality of Newton’s 
philosophy, and adds fresh lustre to his name. It is otherwise however 
in the general history of science. The reputation achieved by a great 
invention is often transferred to another which supersedes it, and a 
discovery which is the glory of one age is eclipsed by the extension of 
it in another. ... It is the peculiar glory of Newton, however, 
that every discovery in the heavens attests the universality of his 
laws, and adds a greener leaf to the laurel chaplet which he wears. 
Shrinking always from publicity1 and controversy, Newton like 
Copernicus had gradually perfected his great work, but, like Co- 
1 In one instance he authorized publication of one of his works “so it be without 
my name to it: for I see not what there is desirable in public esteem, were I able 296 
A SHORT HISTORY OF SCIENCE 
pernicus, Newton might never have published it but for the for- 
tunate urgency of a faithful disciple, Edmund Halley. 
Newton’s Mathematics : Fluxions. — Newton’s services to 
mathematics itself were not less original and momentous than 
to celestial mechanics. 
His extraordinary abilities . . . enabled him within a few years to per- 
fect the more elementary . . . processes, and to distinctly advance 
every branch of mathematical science then studied, as well as to create 
several new subjects. There is hardly a branch of modern mathe- 
matics which cannot be traced back to him and of which he did not 
revolutionize the treatment. 
In pure geometry Newton did not establish any new methods, 
but no modern writer has ever shown the same power in using those 
of classical geometry, and he solved many problems in it which had 
previously baffled all attempts. In algebra and the theory of equa- 
tions he introduced the system of literal indices, established the bi- 
nomial theorem . . ., and created no inconsiderable part of the theory 
of equations. . . . He always by choice, avoided using trigonometry 
in his analysis, ... In analytical geometry he introduced the 
modern classification of curves into algebraical and transcendental; 
and established many of the fundamental properties of asymptotes, 
multiple points and isolated loops. He illustrated these by an ex- 
haustive discussion of cubic curves. — Ball. 
Newton’s greatest mathematical achievement was of course the 
invention of the fluxional or infinitesimal calculus. In his Treatise 
of the Method of Fluxions and Infinite Series he says : — 
1. Having observed that most of our modern Geometricians 
neglecting the synthetical Method of the Ancients, have applied 
themselves chiefly to the analytical Art, and by the Help of it have 
overcome so many and so great Difficulties, that all the Speculations 
of Geometry seem to be exhausted, except the Quadrature of Curves, 
and some other things of a like Nature which are not yet brought to 
Perfection: To this End I thought it not amiss, for the sake of young 
to acquire and maintain it: it would perhaps increase my acquaintance, the thing 
which I study chiefly to decline.” Again in 1675 he writes ‘‘I was so persecuted 
with discussions arising out of my theory of light, that I blamed my own impru- 
dence for parting with so substantial a blessing as my quiet, to run after a shadow.” BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
297 
Students in this Science, to draw up the following Treatise; wherein 
I have endeavored to enlarge the Boundaries of Analyticks, and to 
make some Improvements in the Doctrine of Curved Lines. 
— surely a sufficiently modest introduction of perhaps the most 
important step in the progress of mathematical science. 
Something further as to the evolution of his theory of Fluxions 
may be indicated, without too much technical detail, by the fol- 
lowing passages from Brewster: — 
Having met with an example of the method of Fermat, in Schoo- 
ten’s Commentary on the Second Book of Descartes, Newton suc- 
ceeded in applying it to affected equations, and determining the pro- 
portion of the increments of indeterminate quantities. These incre- 
ments he called moments, and to the velocities with which the quan- 
tities increase he gave the names of motions, velocities of increase, and 
fluxions. He considered quantities not as composed of indivisibles, 
but as generated by motion; and as the ancients considered rectangles 
as generated by drawing one side into the other, that is, by moving 
one side upon the other to describe the area of the rectangle, so Newton 
regarded the areas of curves as generated by drawing the ordinate into 
the abscissa, and all indeterminate quantities as generated by con- 
tinual increase. Hence, from the flowing of time and the moments 
thereof, he gave the name of flowing quantities to all quantities which 
increase in time, that of fluxions to the velocities of their increase, and 
that of moments to their parts generated in moments of time. 
Newton then proceeds to show the application of the propositions 
to the solution of the twelve following problems, many of which were 
at that time entirely new : 
1. To draw tangents to curve lines. 
2. To find the quantity of the crookedness of lines. 
3. To find the points distinguishing between the concave and 
convex portions of curved lines. 
4. To find the point at which lines are most or least curved. 
5. To find the nature of the curve line whose area is expressed 
by any given equation. 
6. The nature of any curve line being given, to find other lines 
whose areas may be compared to the area of that given line. 
7. The nature of any curve line being given, to find its area when 298 
A SHORT HISTORY OF SCIENCE 
it may be done; or two curved lines being given, to find the relation 
of their areas when it may be. 
8. To find such curved lines whose lengths may be found, and also 
to find their lengths. 
9. Any curve line being given, to find other lines whose lengths 
may be compared to its length, or to its area, and to compare them. 
10. To find curve lines whose areas shall be equal or have any 
given relations to the length of any given curve line drawn into a given 
right line. 
11. To find the length of any curve line when it may be. 
12. To find the nature of a curve line whose length is expressed by 
any given equation when it may be done. 
Such were the improvements in the higher geometry which Newton 
had made before the end of 1666. 
Such is a brief account of the mathematical writings of Sir Isaac 
Newton, not one of which was voluntarily communicated to the world 
by himself. The publication of his Universal Arithmetic is said to 
have been made by Whiston against his will; and, however this may 
be, it was an unfinished work, never designed for the public. The 
publication of his Quadrature of Curves, and of his Enumeration of 
Curve Lines, was in Newton’s opinion rendered necessary, in con- 
sequence of plagiarisms from the manuscripts of them which he 
had lent to his friends, and the rest of his analytical writings did 
not appear till after his death. 
An account of Newton’s very important work in analytic geome- 
try and the theory of algebraic equations lies outside the range of 
the present work. 
Much of Newton’s reluctance to publish his more revolutionary 
theories may be attributed to his distaste for controversy, and he 
was unfortunately involved not only in such issues as to priority 
as his own reticence invited, but also in defending himself against 
attacks on philosophic grounds. The character of some of these 
may be illustrated by the following passages from an eminent 
critic, Bishop Berkeley: — 
He who can digest a second or third fluxion, a second or third 
difference, need not, methinks, be squeamish about any point in Di- 
vinity. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
299 
And what are these fluxions? The velocities of evanescent in- 
crements. And what are these same evanescent increments? They 
are neither finite quantities, nor quantities infinitely small, nor yet 
nothing. May we not call them ghosts of departed quantities ? 
In regard to the controversy between the friends of Newton 
and those of Leibnitz as to priority in the invention of the calculus, 
Newton himself says in a celebrated scholium: — 
The correspondence which took place about ten years ago, be- 
tween that very skilful geometer G. G. Leibnitz and myself, when I 
had announced to him that I possessed a method of determining 
maxima and minima, of drawing tangents, and of performing similar 
operations, which was equally applicable to surds and to rational 
quantities, and concealed the same in transposed letters, involving 
this sentence, (Data Aequatione quotcunque Fluentes quantitates in- 
volvente, Fluxiones invenire, et vice versa), this illustrious man re- 
plied that he also had fallen on a method of the same kind, and 
he communicated his method, which scarcely differed from my own, 
except in the forms of words and notation (and in the idea of the 
generation of quantities). 
Of the controversy as a whole Newton’s biographer Brewster 
remarks: 
The greatest mathematicians of the age took the field, and states- 
men and princes contributed an auxiliary force to the settlement of 
questions upon which, after the lapse of nearly 200 years, a verdict 
has not yet been pronounced. 
Although the honour of having invented the calculus of fluxions, 
or the differential calculus, has been conferred upon Newton and 
Leibnitz, yet, as in every other great invention, they were but the 
individuals who combined the scattered lights of their predecessors, 
and gave a method, a notation, and a name to the doctrine of quanti- 
ties infinitely small. 
In studying this controversy, after the lapse of nearly a century 
and a half, when personal feelings have been extinguished, and national 
jealousies allayed, it is not difficult, we think, to form a correct esti- 
mate of the claims of the two rival analysts, and of the spirit and 
temper with which, they were maintained. The following are the 
results at which we have arrived: 300 
A SHORT HISTORY OF SCIENCE 
1st — That Newton was the first inventor of the Method of Flux- 
ions ; that the method was incomplete in its notation; and that the 
fundamental principle of it was not published to the world till 1687, 
twenty years after he had invented it. 
2d — That Leibnitz communicated to Newton in 1677 his Differential 
Calculus, with a complete system of notation, and that he published 
it in 1684, three years before the publication of Newton’s Method. 
It is said that when the Queen of Prussia asked Leibnitz his opinion 
of Sir Isaac Newton, he replied that taking mathematicians from the 
beginning of the world to the time when Sir Isaac lived, what he had 
done was much the better half; and added that he had consulted all 
the learned in Europe upon some difficult points without having 
any satisfaction and that when he applied to Sir Isaac, he wrote 
him in answer by the first post, to do so and so, and then he would 
find it. 
The exalted estimation in which Newton’s genius has been 
held in later times may be illustrated by the following passages. 
The great Newtonian Induction of Universal Gravitation is in- 
disputably and incomparably the greatest scientific discovery ever 
made, whether we look at the advance which it involved, the extent 
of the truth disclosed, or the fundamental and satisfactory nature of 
this truth. — Whew ell. 
The efforts of the great philosopher . . . were always super- 
human ; the questions which he did not solve were incapable of solu- 
tion in his time. — Arago. 
Newton was the greatest genius that ever existed, and the most 
fortunate, for we cannot find more than once a system of the world 
to establish. — Lagrange. 
His own attitude is sufficiently indicated in his statements: — 
I do not know what I may appear to the world, but, to myself, 
I seem to have been only like a boy playing on the seashore, and divert- 
ing myself in now and then finding a smoother pebble or a prettier 
shell than ordinary, whilst the great ocean of truth lay all undis- 
covered before me. 
If I have seen farther than Descartes, it is by standing on the 
shoulders of giants. (See also Appendix.) 
— Brewster. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
301 
Leibnitz. — Newton’s great contemporary and scientific rival 
Leibnitz has been called the Aristotle of the seventeenth century. 
Born in 1646 at Leipsic, he took his doctor’s degree at 20 and was 
immediately offered a university professorship. Alchemy, diplo- 
macy, philosophy, mathematics, all shared his energetic attention. 
In the last he invented a calculating machine and the differential 
calculus. In regard to the machine Klein says: — 
merely the formal rules of computation are essential, for only these 
can be followed by the machine. It cannot possibly have an intuitive 
conception of the meaning of the numbers. It is thus no accident that 
a man so great as Leibnitz was both father of purely formal mathe- 
matics and inventor of the first calculating machine. 
He became librarian at Hannover, founded the Academy of 
Sciences in Berlin, and was instrumental in the organization of 
similar bodies in St. Petersburg, Dresden, and Vienna. His ad- 
vanced ideas on education may be inferred from his remark — 
We force our youths first to undertake the Herculean labor of 
mastering different languages, whereby the keenness of the intellect 
is often dulled, and condemn to ignorance all who lack knowledge of 
Latin. 
But for the overshadowing genius of Newton, Leibnitz’ service 
to the progress of science would have been even greater than it 
actually was. Comparing their work in mathematics where their 
competition was keenest, it should be appreciated that while New- 
ton’s work in mathematical science was incomparably greater in 
range, it was Leibnitz who gave to the differential calculus the 
better form and notation out of which our own has grown. 
It appears that Fermat, the true inventor of the differential cal- 
culus, considered that calculus as derived from the calculus of finite 
differences by neglecting infinitesimals of higher orders as compared 
with those of a lower order . . . Newton, through his method of 
fluxions, has since rendered the calculus more analytical, he also sim- 
plified and generalized the method by the invention of his binomial 
theorem. Leibnitz has enriched the differential calculus by a very 
happy notation. — Laplace. 302 
A SHORT HISTORY OF SCIENCE 
Leibnitz’ sense of mathematical form was also well exemplified 
by his important algebraic invention of determinants. In regard 
to numbers he says: — 
The imaginary numbers are a fine and wonderful refuge over 
the divine spirit, almost an amphibium between being and not being. 
Leibnitz’s discoveries lay in the direction in which all modern pro- 
gress in science lies, in establishing order, symmetry, and harmony, 
i.e. comprehensiveness and perspicuity, — rather than in dealing 
with single problems, in the solution of which followers soon at- 
tained greater dexterity than himself. — Merz. 
Leibnitz believed he saw the image of creation in his binary arith- 
metic, in which he employed only two characters, unity and zero. 
Since God may be represented by unity, and nothing by zero, he 
imagined that the Supreme Being might have drawn all things from 
nothing, just as in the binary arithmetic all numbers are expressed 
by unity with zero. This idea was so pleasing to Leibnitz, that he 
communicated it to the Jesuit Grimaldi, President of the Mathematical 
Board of China, with the hope that this emblem of the creation might 
convert to Christianity the reigning emperor who was particularly 
attached to the sciences. — Laplace. 
Halley : Prediction of Comets. — In applying Newton’s 
theories to known comets his friend and disciple Halley made the 
astonishing discovery that some of them instead of visiting the 
solar system once for all, actually described elliptical orbits of 
vast extent and great eccentricity about the sun. Among these 
he found one which having appeared in 1531, 1607, and 1682, 
should, if his identifications were correct, return in 1759. This 
bold prediction was fulfilled, and Halley’s comet has not only 
reappeared in 1835 and 1910, but has even been traced back almost 
to the beginning of our era. A second similar prediction of Halley 
awaits verification in the year 2255. 
In physics, Halley enunciated for spherical lenses and mirrors 
the correct formula - = — + — and that for the barometric 
/ «i a2 
determination of altitudes. His mathematical work included 
graphical discussion of the cubic and biquadratic equations, a BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 
303 
method of computing logarithms, and an edition of Apollonius 
from both Greek and Arabic sources. In his paper on An Esti- 
mate of the Degrees of the Mortality of Mankind, drawn from cu- 
rious Tables of the Births and Funerals at the City of Breslau; 
with an Attempt to ascertain the Price of Annuities upon Lives, 
he laid the foundations of a new and important branch of applied 
mathematics. Having in boyhood occupied himself with mag- 
netic experiments, in middle life he travelled in the tropics and 
made the first magnetic map, published in 1701 under the title “A 
general chart, showing at one view the variation of the com- 
pass.” Drawing curves on this chart through points of declina- 
tion, he invented a graphical method of wide future usefulness. 
From naval captain he became professor of geometry at Oxford, 
then astronomer royal till his death in 1742. One of his most 
notable achievements in astronomy was the discovery of actual 
changes in the apparent relative positions of the fixed stars, 
Aldebaran, Arcturus, and Sirius — answering a question centuries 
old. 
REFERENCES FOR READING 
Ball. Short History of Mathematics. Chapters XV, XVI. 
Berry. History of Astronomy. Chapters VIII, IX, X. 
Brewster. Memoir of Sir Isaac Newton. 
Lodge. Pioneers of Science. VII, VIII, IX. 
Mach. Science of Mechanics. 
Newton. Principia. CHAPTER XIV 
NATURAL AND PHYSICAL SCIENCE IN THE EIGHTEENTH 
CENTURY 
The seventeenth and eighteenth centuries mark the period in 
which, owing to the use of the several vernacular languages of Europe 
in the place of the medieval Latin, thought became nationalized. 
Thus it was that . . . people could make journeys of exploration in 
the region of thought from one country to another, bringing home 
with them new and fresh ideas. Such journeys . . . were those of 
Voltaire to England in 1726 ... of Adam Smith in 1765 to France. 
— Merz. 
In the preface to one of his volumes of essays, Lord Morley 
speaks of the eighteenth century as the scientific Renaissance. 
Such it undoubtedly was, for it was in this century and es- 
pecially in its latter half, that chemistry, geology, botany, zoology* 
and physics, began to make deep impression on the learned world, 
while astronomy and mathematics ventured upon bolder and 
more far-reaching generalizations than they had ever before made. 
Science as a special discipline, or as a branch of learning worthy 
of the highest consideration, had as yet scarcely begun to make 
itself felt, but the names of Newton and Descartes were frequently 
heard in the salons of Paris and keen observers like Voltaire per- 
ceived the rising of a new tide in the affairs of men. A growth of 
popular interest might naturally have been expected after the 
great discoveries of the sixteenth and seventeenth centuries. What 
was not looked for was the concurrence of those political and 
social upheavals ever since rightly known as revolutions; viz. the 
French Revolution, the American Revolution and, probably most 
important of all, the Industrial Revolution. 
Chemistry : Decline of the Phlogiston Theory. — We 
have already touched upon the work of Boerhaave and Hales 
in the field of organic chemistry, so-called, and may now pass on 
304 NATURAL AND PHYSICAL SCIENCE, 1700-1800 
305 
to the studies of Black, Bergmann and others on the gas sylvestre 
(carbonic acid) of Van Helmont. Dr. Joseph Black of Edinburgh, 
a physician of note and, as we shall see, one of the first to put 
the science of heat on a sure foundation, seeking to explain the 
phenomena accompanying the making and the slaking of “quick” 
lime, — phenomena now familar to every beginner in chemistry 
but in 1750 puzzling to all, — remembered that Hales had found 
that “air” could be driven off from certain substances by heat- 
ing, and suspected that in the burning of limestone to make 
quicklime, something might be driven off, the loss of which 
would make it lighter. This something he tried to obtain by 
causing acid to act upon limestone (in the ordinary laboratory 
fashion of to-day) and collecting the gas evolved by the aid of 
Hales’ pneumatic trough. He next weighed the gas and the 
remaining limestone and found that the weight of the former 
agreed with the loss of weight of the latter. He then reversed the 
experiment, causing “fixed air” (as he called it) to bubble through 
a solution of lime, whereupon, as he had anticipated, a white, 
chalk-like powder appeared and fell to the bottom. This simple 
experiment proved extremely fruitful, and we can now see that in 
its use of analysis and synthesis, in its partly quantitative char- 
acter, and in the chemical reasoning employed, it was also highly 
instructive. Best of all, it did not require any hypothetical, 
immaterial or mystical “phlogiston” for satisfactory explanation 
of all the hitherto puzzling phenomena involved. Black invented 
for the gas thus driven off by heat or acid the term “fixed air,” be- 
cause it was evidently “fixed” in the limestone or chalky precipi- 
tate, and because any gas or vapor not obviously something else, 
was still supposed to be “ air,” — the true nature and chemical com- 
position of the atmosphere being still (in 1750) quite unknown. 
At this point Bergmann, a Swedish chemist of distinction, by 
the use of litmus (which Boyle had recommended as a test for acids) 
and other means, discovered that the “fixed air” of Black is an 
acid, and accordingly named it “aerial acid.” Bergmann also 
weighed the new gas, finding it heavier than air, and discovered 
that it is very soluble in water. 306 
A SHORT HISTORY OF SCIENCE 
To sum up: It was now known that there exists an invisible, 
odorless gas, resembling air but heavier than air and more soluble 
in water; that it is acid, and capable of attaching itself to lime, 
making a kind of chalk; that it will not support life, yet is present 
in the human breath, as well as in some mineral waters; and that 
it is given off during fermentations. It only remained for Lavoisier 
to discover (in 1779) that this gas is compounded of two very 
common elements — carbon and oxygen — tightly bound together, 
and may therefore be called, as it often is today, “carbonic acid.” 
But before this could happen other investigations had to prepare 
the way, and especially the discovery of the new “element,” 
oxygen. 
A New Chemistry. Priestley and Lavoisier. — We have 
now reached a period of remarkable activity and rapid progress 
in chemical research. While Black was hard at work upon chemi- 
cal problems in Scotland, and Bergmann in Sweden, Cavendish 
was similarly engaged in England, and in 1766 reported to the 
Royal Society his discovery of a new kind of gas to which, for 
the reason that it took fire whenever flame was applied to it, 
and also because he believed it to be the cause of the occasional 
explosions in mines, he gave the name “inflammable air.” 
Cavendish obtained this gas by treating iron, tin, zinc, or other 
metals with sulphuric acid, very much as Black had obtained 
fixed air by treating limestone with acids. Inflammable air 
was, however, obviously quite unlike fixed air, since it was 
lighter than air — not heavier — and was readily burned. It 
resembled it, nevertheless, in that a lighted candle plunged into 
it went out, and animals died in it just as they did in fixed 
air. It had another peculiar property; viz. that of forming with 
air an explosive mixture. This new gas as we now know was 
hydrogen. 
Not long after, in 1772, other new gases were separated 
and studied; viz. nitrogen by Rutherford, and nitric oxide by 
Priestley. It was on August 1, 1774, however, that Priestley 
made his most important discovery, and one that proved to be the 
very corner-stone of the splendid edifice in which modern chem- NATURAL AND PHYSICAL SCIENCE, 1700-1800 
307 
istry now dwells, namely, the discovery of oxygen. Joseph Priestley, 
fearless reformer, Unitarian clergyman, and tireless experimenter in 
natural philosophy, had already made important and interesting 
discoveries when, in 1774, as stated above, he decomposed by 
heat the reddish powder obtained by calcining mercury, and 
collected and examined the gas given off. Candles and glowing 
coals burned in this gas with extraordinary energy, and mice lived 
in it under a bell glass even longer than in ordinary air. And, 
since it was derived from a burnt, i. e. dephlogisticated, metal 
and yet was colorless and odorless like ordinary air, Priestley 
named it “dephlogisticated air.” The following is his own 
account of his work: — 
There are, I believe, very few maxims in philosophy that have 
laid firmer hold upon the mind than that air, meaning atmospheric 
air, is a simple elementary substance, indestructible and unalterable 
at least as much so as water is supposed to be. In the course of my 
inquiries I was, however, soon satisfied that atmospheric dir is not an 
unalterable thing; for that, according to my first hypothesis, the 
phlogiston with which it becomes loaded from bodies burning in it, 
and the animals breathing it, and various other chemical processes, 
so far alters and depraves it as to render it altogether unfit for inflam- 
mation, respiration, and other purposes to which it is subservient; 
and I had discovered that agitation in the water, the process of vege- 
tation, and probably other natural processes, restore it to its original 
purity. . . . 
Having procured a lens of twelve inches diameter and twenty 
inches focal distance, I proceeded with the greatest alacrity, by the 
help of it, to discover what kind of air a great variety of substances 
would yield, putting them into the vessel, which I filled with quick- 
silver, and kept inverted in a basin of the same. . . . With this ap- 
paratus, after a variety of experiments ... on the 1st of August, 
1774, I endeavored to extract air from mercurius calcinatus per se; 
and I presently found that, by means of this lens, air was expelled from 
it very readily. Having got about three or four times as much as 
the bulk of my materials, I admitted water to it, and found that it 
was not imbibed by it. But what surprised me more than I can 
express was that a candle burned in this air with a remarkably vigorous 308 
A SHORT HISTORY OF SCIENCE 
flame, very much like that enlarged flame with which a candle burns 
in nitrous oxide, exposed to iron or liver of sulphur; but as I had got 
nothing like this remarkable appearance from any kind of air besides 
this particular modification of vitreous air, and I knew no vitreous acid 
was used in the preparation of mercurius calcinatus, I was utterly at 
a loss to account for it. . . . 
The flame of the candle, besides being larger, burned with more 
splendor and heat than in that species of nitrous air; and a piece 
of red-hot wood sparkled in it, exactly like paper dipped in a solution 
of nitre, and it consumed very fast; an experiment that I had never 
thought of trying with dephlogisticated nitrous air. 
... I had so little suspicion of the air from the mercurius cal- 
cinatus, etc., being wholesome, that I had not even thought of apply- 
ing it to the test of nitrous air; but thinking (as my reader must 
imagine I frequently must have done) on the candle burning in it 
after long agitation in water, it occurred to me at last to make the 
experiment; and, putting one measure of nitrous air to two measures 
of this air, I found not only that it was diminished, but that it was 
diminished»quite as much as common air, and that the redness of the 
mixture was likewise equal to a similar mixture of nitrous and com- 
mon air. . . . The next day I was more surprised than ever I had 
been before with finding that, after the above-mentioned mixture of 
nitrous air and the air from mercurius calcinatus had stood all night, 
... a candle burned in it, even better than in common air. 
At almost the same time (1775) Scheele, a Swedish chemist, 
independently discovered the same gas. “Scheele remained a 
poor apothecary all his life, yet was really one of the first chemists 
of Europe.” His name for oxygen was “empyreal air.” 
But if Black and Cavendish and Priestley and Scheele and 
others laid the foundations of modern chemistry, it was the yet 
more famous Lavoisier, who, building upon the results of his 
predecessors, began the erection of the present lofty superstruc- 
ture. Lavoisier soon dismissed forever the long-standing, mysti- 
cal theory of phlogiston through his unremitting use of the balance, 
for the introduction and use of which in analysis he has been 
rightly called “ the founder of quantitative chemistry.” By 
means of the balance he proved that when metals are burnt in NATURAL AND PHYSICAL SCIENCE, 1700-1800 
309 
air, the resulting substances weigh more than did the metal; and 
that if burnt in a closed space the loss in weight of the air equals the 
gain in weight of the metal. And when, finally, he reversed the 
experiment, decomposing the red powder of burnt mercury as 
Priestley had done, and weighing as before, he found that the loss 
of weight of the red powder at the end was exactly equal to the 
weight of the “dephlogisticated air” driven off. 
Lavoisier named the gas thus derived oxygen (acid producer), 
because its compounds seemed to him to be chiefly acids. And 
since Black’s “fixed air” was Bergmann’s “aerial acid,” and could 
be made by burning charcoal in air, Lavoisier suspected that 
fixed air contained oxygen as well as carbon. Accordingly, he 
proceeded to burn charcoal in pure oxygen, and obtained fixed 
air or aerial acid, which he thereupon analyzed, proving that 
100 parts contained 72 of oxygen and 28 of carbon. He there- 
fore named it carbonic acid. Afterward, by burning a diamond 
in pure oxygen and obtaining carbonic acid, he proved that one of 
the precious “stones” is really a form of carbon. Priestley died 
clinging to the phlogiston theory, but his dephlogisticated air 
had now become Lavoisier’s oxygen. Lavoisier, one of the most 
brilliant men of science in any age, continued as long as he lived 
to do remarkable work. He repeated with more precision the 
experiment of Cavendish on the synthesis of water from inflam- 
mable air and ordinary air (or oxygen), and named the former 
hydrogen (water producer). But, unfortunately for science, 
Lavoisier, since he had held government office as a farmer-gen- 
eral, found no favor in the eyes of the leaders of the French 
Revolution and on May 18, 1794, at the age of 51, he was guil- 
lotined. Thus was the end of the eighteenth century, otherwise 
in most respects favorable to science and other learning, dis- 
graced by a foul blot on civilization, as had been the end of the 
sixteenth by the burning of Giordano Bruno; the latter a victim to 
misdirected religious conservatism, Lavoisier to equally misdirected 
political radicalism. 
The Synthesis of Water. — In 1784 Cavendish added to his 
other brilliant discoveries that of the composition of water. He 310 
A SHORT HISTORY OF SCIENCE 
had himself discovered inflammable air or hydrogen, and now 
he found that by exploding a mixture of this gas with oxygen, 
water was produced. 
By experiments with the globe it appeared, says Cavendish, 
that when inflammable and common air are exploded in a proper 
proportion, almost all the inflammable air, and near one-fifth the com- 
mon air, lose their elasticity and are condensed into dew. And by 
this experiment it appears that this dew is plain water, and conse- 
quently that almost all the inflammable air is turned into pure water. 
In order to examine the nature of the matter condensed on firing 
a mixture of dephlogisticated and inflammable air, I took a glass globe, 
holding 8800 grain measures, furnished with a brass cock and an ap- 
paratus for firing by electricity. This globe was well exhausted by an 
air-pump, and then filled with a mixture of inflammable and dephlo- 
gisticated air by shutting the cock, fastening the bent glass tube 
into its mouth, and letting up the end of it into a glass jar inverted 
into water and containing a mixture of 19,500 grain measures of 
dephlogisticated air, and 37,000 of inflammable air; so that, upon 
opening the cock, some of this mixed air rushed through the bent tube 
and filled the globe. The cock was then shut and the included air 
fired by electricity, by means of which almost all of it lost its elasticity 
(was condensed into water vapors). The cock was then again opened 
so as to let in more of the same air to supply the place of that de- 
stroyed by the explosion, which was again fired, and the operation 
continued till almost the whole of the mixture was let into the globe 
and exploded. By this means, though the globe held not more than a 
sixth part of the mixture, almost the whole of it was exploded therein 
without any fresh exhaustion of the globe. 
Beginnings of Modern Ideas of Sound. — We have seen 
above that the Greeks were deeply interested in sound, as well 
as in music. The invention of the monochord with the dis- 
covery by Pythagoras of the relation between the length of a 
vibrating string and the sound which it produces was the first 
step in the right direction, although any mystical relation between 
sound and number such as the Pythagoreans inferred has no basis. 
From Pythagoras to Galileo little or no progress was made. 
Galileo recognized that sound is due to vibrations in the air falling NATURAL AND PHYSICAL SCIENCE, 1700-1800 
311 
upon the ear-drum, and in one of his dialogues explains concord 
and dissonance by concurrence or conflict of such vibrations. 
He shows how vibrations causing sound may be made visible, and 
how to measure the relative length of sound waves by scraping 
a brass plate with a chisel, thereby making dust on the plate take 
up positions in parallel lines. That air is really the intermediary 
was proved in 1705 by Hawksbee’s experiment of placing a clock 
in a vacuum. 
After Galileo, the studies, mathematical and experimental, of 
Newton, Euler, and Sauveur (1653-1715) brought acoustics to 
the point where it was taken up and given much of its present 
form by Chladni (1756-1827) — “the Father of Modern Acous- 
tics.” Sauveur, eminent physicist and musician, deserves more 
than passing notice from the fact that he “had neither voice nor 
ear” (being half deaf and dumb) and yet achieved distinction for 
his original researches in both sound and music. Intended by his 
parents for the church, he early manifested delight in mechanical 
contrivances and in arithmetic, but the course of his life was 
determined by a copy of Euclid which accidentally came to his 
notice. He thereupon abandoned an ecclesiastical career and, 
having in consequence lost the support of his relatives, obtained a 
livelihood by teaching mathematics, becoming a professor of 
that subject in 1686. During the remainder of his life the study 
of acoustics, and particularly the scientific theory of music (in 
which he was the first to draw attention to overtones or har- 
monics) occupied much of his attention. Many of his papers 
were published in the Memoirs of the French Academy, 1700-1714. 
The work of Chladni at the beginning of the nineteenth century 
laid broad and deep the foundations of acoustics as we know that 
science to-day, and upon this foundation Helmholtz and Tyndall 
in the middle of that century reared a large part of the modern 
superstructure. Chladni carried much further the experiments of 
Galileo on vibrating plates, substituting a violin bow for the 
chisel, and sand for dust on the plates, obtaining thereby that 
wonderful variety of figures which is nowadays demonstrated 
to beginners by every teacher of natural philosophy. He also 312 
A SHORT HISTORY OF SCIENCE 
devised a simple method of counting the number of vibrations 
corresponding with each note. 
The Beginnings of Modern Ideas of Heat: Latent and 
Specific Heat ; Calorimetry. — The earlier ideas of the nature 
of heat are not unlike those of light. Theories of emission were 
at first preferred for both, and present ideas of vibration or 
undulation have appeared only recently. Heat was even re- 
garded as an imponderable yet material substance, “ caloric,” 
emitted by hot bodies and absorbed by cold ones. High temper- 
ature meant the presence, and low temperature, the absence, of 
caloric. The rise of mercury in the thermometer tube was ex- 
plained as due to the expansion of the mercury not, as to-day, by 
increase of distance between molecules, but by the addition of 
caloric and the consequent increase of total material. Francis 
Bacon remarked on the problem of heat and in an interesting 
passage on the proper method of its study shows that he knew 
many of its phenomena, including its development “in bodies 
heated by rubbing.” But it remained for Black, of Edinburgh, 
whose work on fixed air or carbonic acid we have dwelt upon 
above, to make the fundamental researches which paved the 
way both for the study of the theory of heat by Rumford at the 
end of the eighteenth century, and for its industrial uses by 
Watt in steam engineering in the middle of that century. Black, 
while experimenting on heating and cooling, discovered that 
heat may be applied to boiling water, or to water containing 
melting ice, without raising the temperature. Obviously, such 
applied heat must either be lost or somehow become latent. 
Further experiments showed that once the ice is all melted, or 
once the escape of steam is stopped by covering the water in a 
closed vessel, the temperature begins to rise; the heat is no longer 
concealed, lost, or absorbed, but produces obvious effects. Ap- 
parently the lost heat was somehow used in melting the ice and 
in making the steam. These and other experiments by Black 
proved suggestive to Watt, who at this time was trying to improve 
upon the air-and-steam engine of Newcomen, in which atmos- 
pheric pressure was used to push a piston in a cylinder filled with NATURAL AND PHYSICAL SCIENCE, 1700-1800 
313 
and then emptied of steam, and it was largely by the aid of 
Black’s studies on latent heat that Watt was enabled and en- 
couraged to persevere and push to completion his own epoch- 
making discoveries in steam engineering. 
Black was also the first to recognize and investigate what we 
know today as “specific” heat and, by means of a cavity in a block 
of ice into which various heated bodies were brought, to weigh the 
water each would produce while cooling from the same tempera- 
ture, — in other words to invent and use the calorimeter. 
Eighteenth Century Researches on Light. — These start 
from the great work of Newton, and especially of Huygens, in 
the previous century, and continue with the fruitful inventions 
of the achromatic telescope by Hall in 1733, and the work of 
Dollond, an English optician, upon achromatic lenses, leading 
up to the construction in 1758 of achromatic telescope objectives. 
The achromatic telescope now became a serviceable instrument, 
but the compound microscope had to wait more than half a cen- 
tury longer for correspondingly serviceable achromatic objectives. 
It was not until the opening of the nineteenth century that much 
further progress was made in our knowledge of light. It is 
rather for progress in sound, in heat, and in electricity, that 
eighteenth century physical science is chiefly notable. 
Beginnings of Modern Ideas of Electricity and Mag- 
netism. — The seventeenth century had witnessed no great 
progress in these subjects, and the sixteenth century work of 
William Gilbert stood as almost the only important contribu- 
tion to our knowledge of them until about 1730. Von Guericke, 
following suggestions of Gilbert, had, it is true, made a rude 
electrical machine which he described in a work published in 
1672, and had observed the electric spark, which with his machine 
was so small as to be seen only in the dark and to be heard with 
difficulty. Much more important were the observations of Francis 
Hawksbee (or Hauksbee), “one of the most active experimental 
philosophers of his age,” and one of the first to study capillary 
action, who in 1705 communicated to the Royal Society several 
curious experiments on what he called “the mercurial phosphorus,” 314 
A SHORT HISTORY OF SCIENCE 
showing that light could be produced by passing common air 
through mercury placed in a well-exhausted receiver. These 
phenomena, which had been observed before Hawksbee’s time 
and had been variously explained, were attributed by him to 
electricity, for he remarked their resemblance to lightning. Like 
Newton he used a revolving glass globe, rubbed by the hand, to 
generate electricity. These and other results he published in 
1707-1709. 
Further investigations by Wall, Gray and Wheeler, Desagu- 
liers, Dufay, and many others prepared the way for Watson, 
Franklin, Galvani, and Volta, whose investigations in the latter 
half of the eighteenth century, added to those just referred to, 
would alone make that century forever famous in the history of 
science. On these brilliant discoveries we can only briefly touch. 
Wall (1708) compared the electric spark and its crackling to 
lightning and thunder. Gray and Wheeler (1729) and later Dufay 
created what has been called an epoch in the history of electricity 
by discovering that different bodies differ in electrical conductiv- 
ity, while Desaguliers confirmed and extended their results. 
Dufay (1699-1739) repeated these and other experiments and 
discovered that there are two hinds of electricity, positive and nega- 
tive, or, as he called them from their source of origin, vitreous and 
resinous. Watson undertook with the aid of a party of friends 
from the Royal Society to determine the velocity of the electric 
current and found “that through the whole length of a wire 12,276 
feet long, the velocity of electricity was instantaneous.” Many 
others had also made important, if minor, contributions to the new 
science of the electric “fluid” — for like heat (caloric) electricity 
was regarded as material though imponderable — before that 
science was further developed and widely popularized by Frank- 
lin. 
Benjamin Franklin (1706-1790) was not the first American to do 
good work for science, but he was the first to gain wide renown 
in it together with an international reputation. Even before 
1750, Franklin had argued that all the known phenomena of elec- 
tricity had their counterpart in lightning, but it was not until NATURAL AND PHYSICAL SCIENCE, 1700-1800 
315 
June, 1752, that he made his famous kite experiment, which showed 
that lightning is really an electrical phenomenon, since a Leyden 
jar can be charged from the skies, and at the same time proved that 
atmospheric and machine-made (frictional) electricity are one and 
the same thing. This daring experiment was performed also in 
Europe at about the same time by others, and marks one of the 
greatest triumphs of science in any age, for it simplified and to a 
great extent explained one of the oldest and most awe-inspiring 
phenomena of nature. Furthermore, by correlating the thunder- 
bolts of Zeus and the shocks of a Leyden jar with the sparks of an 
electrical machine, and even with those from a cat’s back, it tended 
mightily to inspire confidence in natural philosophy and to lessen 
correspondingly the universal dread of unseen, mysterious, and 
supposedly supernatural, powers or influences. Other important 
work in electricity was done in this century by Beccaria in Italy, 
Canton and Symmer in England, and many others. 
The Beginnings of Modern Ideas of the Earth. — The 
eighteenth century saw important progress also in biological 
subjects, — although the word biology was not yet born, and 
zoology and botany, even, were still undifferentiated and closely 
associated with geological knowledge under the broad and hospi- 
table Aristotelian term “natural history.” Physiology meanwhile 
retained its close connection with its parent medical scienoe, its 
logical relations to zoology and botany being generally unrecog- 
nized. Inquiries were, however, on foot destined, as we can 
now see, to bring about changes, namely, to differentiate natural 
history into geology, botany, and zoology and, finally, to integrate 
the two latter sciences into one greater than either, viz. biology. 
It must have occurred to the thoughtful reader of the foregoing 
pages that the four elements of the Greeks and their followers 
have by this time lost their primitive character, and become 
highly complex compounds or combinations of phenomena. Air, 
for example, by the end of the eighteenth century was proved to 
be a mixture of various elements and compounds, to possess weight 
and to exert pressure, to have no part in abhorrence of vacua, and 
to be the seat of marvellous aqueous and electrical phenomena. 316 
A SHORT HISTORY OF SCIENCE 
Fire, long a mystery, was by this time regarded, not as an “ele- 
ment,” but as a luminous centre of intense chemical change. 
Water, above all, — abundant, important, susceptible of metamor- 
phism into ice, snow, dew, fog, and steam, — had surrendered the 
secrets of its very being, having been split apart into hydrogen and 
oxygen gases, and created or restored as a liquid by bringing these 
two gases together at high temperature. Here also, as in Frank- 
lin’s kite experiment, mystery, if not dispelled, was at least driven 
back; and the suggestion became natural and reasonable, — If 
only enough were known, might not many other mysteries in na- 
ture be lessened, if not altogether done away ? 
But, while these more comprehensible and more rational ideas 
of air, fire, and water were now the common property of natural 
philosophy, natural history still held unsolved most of its ancient 
problems. The earth, for example, while probably no longer re- 
garded as one of the four elements, was as yet a standing puzzle 
in respect to its origin, both as a whole and as to its parts. 
Astronomy had proved the planetary character of the earth 
but had not yet suggested for it or for its fellows any natural, 
as opposed to supernatural, origin, and was entirely silent as to 
the sources and history of the earth’s crust, so rich in minerals, 
metals, volcanoes, earthquakes, soils, craters, gases and par- 
ticularly fossils, — those mute remains which could no longer 
be disposed of as freaks of nature, but must be looked upon 
as indefeasible witnesses to a prehistoric past. Leonardo in 
the fifteenth and Palissy in the sixteenth century revived the 
ideas of Pythagoras and Xenophanes as to the true nature of fos- 
sils, but no further progress of note was made for upwards of a 
hundred years, when about 1670 Steno, a Dane, and Scilla, an 
Italian, published studies on petrifactions, illustrated with draw- 
ings. Hooke, already referred to (p. 268), Ray, the naturalist, and 
later (1695) Woodward, made collections of chalk, gravel, coal, 
and marble, and gravely discussed their meaning in terms of the 
Flood of Noah. In this unsatisfactory position matters stood at 
the end of the seventeenth century, and it was not until nearly the 
middle of the eighteenth, viz. in 1740, that much further progress NATURAL AND PHYSICAL SCIENCE, 1700-1800 
317 
was made. At that time Lazzaro Moro put forward the view that 
the rocks must have been in process of formation when fossils were 
included in them, and that the earth’s crust evidently consists of 
strata, superimposed one upon another. He also reasoned from 
the character of the included fossils, back to the character of the 
rocks containing them, — arguing that some must have been 
laid down in fresh water, others in salt water, and hence some in 
rivers or lakes, and some in the sea. 
In 1765 the first school of mines of which we have record was 
established at Freiberg, in Saxony, and here appeared in 1775 a 
student of the natural history of minerals and of the earth, viz. 
Abraham Werner, son of an Inspector of Mines at Freiberg, and 
eventually a popular teacher there of mining and geology. Wer- 
ner’s name is associated with a special school — the Neptunists 
— who, following him, held that the crust of the earth had been 
laid down in water. In opposition, another school — the Vul- 
canists—arose, holding that it has come rather by fire, volcanoes, 
and the like. 
Towards the end of the eighteenth century, Dr. Hutton of 
Edinburgh, and William Smith, an English surveyor, made patient, 
accurate, and detailed studies of fossils and their distribution, and 
of erosion and other work of water, over a considerable area, and 
published, the former a Theory of the Earth (1788), the latter a 
geological map of England (1815). These formed a solid basis 
for that epoch-making work by Lyell, in 1830, to which we shall 
refer hereafter. Hutton deserves to be especially remembered 
with honor for his insistence that the best interpreter of the past 
is the present; that if we would know how rocks were formed ages 
ago, we have only to observe how they are being formed today. 
This simple doctrine of “ uniformitarianism ” was not only an 
inspiration to Lyell, but, largely through Lyell, prepared the way 
for Darwinism and other evolutionary ideas requiring time with 
slow change, in the scientific revolution of the nineteenth century. 
Eighteenth Century Progress in Botany, Zoology, etc. — 
The great world of plant and animal life, even at the middle of 
the eighteenth century, was still almost unexplored and unclas- 318 
A SHORT HISTORY OF SCIENCE 
sified. The early work of Aristotle in zoology and of Theophras- 
tus in botany, as well as that of Gesner in the sixteenth century 
and Ray and Grew and Malpighi and Willughby in the seventeenth 
have been referred to above, but until we come to Buff on (1707- 
1785), the French naturalist, and Linnaeus (Carl von Linne) (1707- 
1778), the Swedish, we meet with no other great name, and find 
no important researches on record within the field of natural 
history. Buffon’s special contribution to science was a fine work 
on “Natural History,” and an infectious enthusiasm which so 
popularized him that 20,000 people are said to have mourned at 
his funeral. 
Linnaeus was also an immensely popular writer and teacher of 
natural history, who at the same time advanced the science of 
botany by introducing an order and system into the classifica- 
tion of plants which facilitated their comparative study. It is 
not too much to say that Linnaeus established botany as a 
science. He also did much work upon animals and minerals, but 
his famous dictum, “stones grow, plants grow and live, and 
animals grow, live, and feel,” while emphasizing an important 
and fundamental similarity in natural objects, has long since lost 
whatever standing it may once have had. It is not so much in 
their properties as in their substance that stones, plants, and 
animals agree, and the greatest service done by Linnaeus for 
science was his insistence on the importance of the careful ob- 
servation of likeness and difference, and of clear and accurate 
description. To this end he introduced a binomial system, so 
that closer and more accurate classification in natural history 
was facilitated and ever after employed. His first great work, Sys- 
tema natures, was published in 1735. His collection of plants, 
insects, books, etc. now forms the nucleus of the Linnaean 
Society library in London, founded in 1788. The special pro- 
cedures adopted by him proved to be artificial, and were soon re- 
placed by a more natural basis of classification introduced by de 
Jussieu; but the scientific names applied to many animals (includ- 
ing man himself, Homo sapiens) and many plants, are still in com- 
mon use throughout the scientific world. NATURAL AND PHYSICAL SCIENCE, 1700-1800 
319 
In the time of Aristotle man took his place naturally at the head 
of the other animals. . . . But the influence of religion and phi- 
losophy did not long permit of this association. Man came to be 
regarded as the chef-d’oeuvre of creation, a thing apart ‘ a little lower 
than the angels.’ In the eighteenth century came a startling change, 
man was wrenched from this detached and isolated attitude and 
linked on once more to the beasts of the field. This was the work 
of Linnaeus. . . . 
Buffon did not classify, he described . . . the genius of Linnaeus 
lay in classification. Order and method were with him a passion. 
In his Systema naturae he fixed the place of man in nature, arranging 
Homo sapiens as a distinct species in the order Primates, together 
with the apes, the lemurs and the bats. — Haddon. 
Progress in Comparative Anatomy and Physiology. —The 
seventeenth century was peculiarly rich in physiological and 
anatomical discoveries, largely because it was endowed with a man 
of genius in physiology, Harvey, and with a new and valuable 
instrument, — the compound microscope. It is true that this 
last did not fulfil its promise, because of mechanical defects, but 
it was good enough to enable Malpighi to clinch with positive 
proof Harvey’s theory of the circulation of the blood, besides re- 
vealing certain anatomical features of spleen and kidney, hitherto 
unknown. In 1743 Haller (1707-1777) of Berne proved that the 
muscles do not depend for their contractility, as had been sup- 
posed, upon “vital spirits” sent in through the nerves, but possess 
independent and intrinsic powers of contraction even when sepa- 
arated from the nervous system or from the body itself. This im- 
portant discovery, together with much excellent anatomical work, 
was made by Haller at Gottingen, where he was professor “of 
anatomy, surgery, and botany” and whither he soon drew 
large numbers of enthusiastic pupils. Haller will long be re- 
membered, not only for his great work in physiology and in teaching, 
but also as one of the founders of comparative anatomy. In 
this subject, however, he was soon left far behind by the famous 
John Hunter (1728-1793) and William Hunter, his brother. 
We must not omit to observe that with comparative anatomy, 320 
A SHORT HISTORY OF SCIENCE 
comparative botany, and comparative geology and mineralogy, 
eighteenth century science was now laying solid foundations for 
the great generalizations of the nineteenth century. Compar- 
ative physiology, even, was making a beginning, with the ex- 
periments of Bonnet (1720-1793) upon the reproduction of lost 
parts in the lower animals, and of Spallanzani (1729-1799) upon 
spontaneous generation. To consideration of the labors of these 
last we shall return. 
The Industrial Revolution. Inventions. Power.—Far- 
reaching in their consequences as were the French Revolution of 
1793 and the American Revolution of 1776, it is the Industrial 
Revolution, especially after 1770, with which the student of the 
history of science has chiefly to deal. Before the Industrial Revolu- 
tion, i.e. almost everywhere before 1760 or even 1770, whatever 
machinery existed was run mostly by hand or foot, and was hence 
easily operated in the separate homes of the workers. But within 
the next thirty years the factory system had come, with coopera- 
tive labor in or about some central power-plant, and with ma- 
chinery driven by water power or steam. With this change, which 
increased the output of the individual, and took work and workers 
out of the home, a revolution began which is still affecting every 
country and has modified the very structure of human society. 
The change was probably imminent in any event, for the use 
of water power had begun before the introduction of steam; but 
the perfection of the steam-engine by Watt, who as we have seen, 
was powerfully aided by the scientific studies of his fellow country- 
man, Black, on heat, steam, evaporation, and calorimetry, greatly 
hastened, and soon made almost universal, the mighty change. 
Henceforth machinery was to become the handmaid of toil, and 
to bring with it not only factory industry in place of home indus- 
try but, before long, improved means of transportation, effecting a 
virtual shrinkage of the world and a far closer contact of mankind. 
Almost coinciding with the introduction of water power and 
steam power, came a great burst of invention. The spinning 
“jenny” and the “water frame” came almost hand in hand with 
the “ mule ” and the “ power loom; ” while, as if to meet these on the NATURAL AND PHYSICAL SCIENCE, 1700-1800 
321 
cotton field, the cotton “gin” (engine), was invented (by Eli 
Whitney of Connecticut) to replace the slow and tedious process 
of separating the cotton fibre or staple from its seed — hitherto 
laboriously done by hand. Applied chemistry also began to 
appear, e.g. in the manufacture of illuminating gas by Murdoch 
at Salford, England, in 1792, while the discoveries of Galvani and 
Volta at the very end of the century opened up that new era 
of electricity in the midst of which we dwell to-day. 
The Influence of Science upon the Spirit of the Eigh- 
teenth Century. — Writers on the literature of the eighteenth 
century, after condemning it because of its comparative barrenness 
in great works of art or literature, are apt to find the reason in one 
or another aspect of the growth of science. Professor Dowden, for 
example, in his essay on Goethe, remarks that 
Rousseau’s emancipation of the heart, was felt in the eighteenth 
century to be a blessed deliverance from the eager, yet too arid, 
speculation of the age, 
although he admits that: — 
Humanity, as Voltaire said, had lost its title-deeds, and the task 
of the eighteenth century was to recover them. 
Dowden’s unusually charitable judgment of the century is more 
or less typical of literary opinion generally. 
For the scientist, on the other hand, few centuries in all history 
are more important, for the eighteenth was not only rich in scien- 
tific performance but still more pregnant with promise. And 
even in art — if in that term music be included — and literature, 
a century which produced a Haydn, a Mozart and a Beethoven, 
with a Burns, a Voltaire, a Wordsworth and a Goethe, need not 
fear to hold up its head. 
We have mentioned above the first School of Mines: viz. that 
at Freiberg, in Saxony (1765). The first School of Civil Engineer- 
ing was established in Paris (1747). In this century also were 
established new universities, e.g., Yale (1701), Gottingen (1737), 
Princeton (1746), Bonn (1777) and Brussels (1781). 322 
A SHORT HISTORY OF SCIENCE 
The “physiocrats” and the “encyclopaedists” of the French 
school of practical philosophers also deserve notice, for they were 
professedly inspired by science and seeking to apply it to human 
society. Even in so humble a pursuit as the attempt to overcome 
in time of famine the prejudices of the populace against the potato, 
Turgot and his fellows did good work for applied science. Nor 
should we forget the service to social science of Count Rumford, 
who for the first time grappled boldly with problems as far apart 
as the control of mendicity, of smoky chimneys, and of poverty. 
Much of Rumford’s best work, though done in the nineteenth 
century, had its origin in the scientific spirit and achievements of 
the eighteenth. 
As the century drew to its close, an English physician, Edward 
Jenner, by the use of the basic methods of inductive scientific 
research — accurate observation, skillful experimentation, careful 
generalization and thorough verification—created a new science, 
preventive medicine, and conferred upon mankind the priceless 
blessings of vaccination. (See Appendix G.) 
The nebular hypothesis of Laplace, through its central idea of 
natural development rather than sudden and special (artificial) 
creation of the solar system, was an important preparation of 
men’s minds for theories of transformation or evolution. Hutton’s 
Theory of the Earth enforced the same idea for the familiar earth, 
while the metamorphoses of the parts of the flower, pointed out by 
Goethe, helped to pave the way for acceptance of the idea of 
gradual modification of organs and even of organisms into others. 
To these matters we shall return in our discussion of Evolution 
in the final chapter. 
REFERENCES FOR READING 
(See page 461.) CHAPTER XV 
MODERN TENDENCIES IN MATHEMATICAL SCIENCE 
Mathematics is the queen of the sciences and arithmetic the 
queen of mathematics. She often condescends to render service to 
astronomy and other natural sciences, but in all relations she is en- 
titled to the first rank. — Gauss. 
Thought-economy is most highly developed in mathematics, that 
science which has reached the highest formal development, and on 
which natural science so frequently calls for assistance. Strange as 
it may seem, the strength of mathematics lies in the avoidance of all 
unnecessary thoughts, in the utmost economy of thought-operations. 
The symbols of order, wThich we call numbers, form already a system 
of wonderful simplicity and economy. When in the multiplication 
of a number with several digits we employ the multiplication table 
and thus make use of previously accomplished results rather than 
repeat them each time, when by the use of tables of logarithms we 
avoid new numerical calculations by replacing them by others long 
since performed, when we employ determinants instead of carrying 
through from the beginning the solution of a system of equations, 
when we decompose new integral expressions into others that are 
familiar, — we see in all this but a faint reflection of the intellectual 
activity of a Lagrange or Cauchy, who with the keen discernment of 
a military commander marshals a whole troop of completed operations 
in the execution of a new one. — Mach. 
The iron labor of conscious logical reasoning demands great per- 
severance and great caution; it moves on but slowly, and is rarely 
illuminated by brilliant flashes of genius. It knows little of that 
facility with wdiich the most varied instances come thronging into 
the memory of the philologist or historian. Rather is it an essential 
condition of the methodical progress of mathematical reasoning that 
the mind should remain concentrated on a single point, undisturbed 
alike by collateral ideas on the one hand, and by wishes and hopes on 
323 324 
A SHORT HISTORY OF SCIENCE 
the other, and moving on steadily in the direction it has deliberately 
chosen. — Helmholtz. 
Nature herself exhibits to us measurable and observable quan- 
tities in definite mathematical dependence; the conception of a 
function is suggested by all the processes of nature where we observe 
natural phenomena varying according to distance or to time. Nearly 
all the “known” functions have presented themselves in the attempt 
to solve geometrical, mechanical, or physical problems. — Merz. 
We have now reached a period of maturity in the evolution of 
mathematical science beyond which any attempt to follow its 
details would involve technical discussions outside the range of 
this work. The present chapter will be devoted to a general 
survey of modern tendencies in pure and applied mathematics, 
in mechanics, in mathematical physics and in astronomy. The 
most notable single fact in the centuries under discussion is the 
increasing specialization resulting from the great expansion of 
scientific knowledge. It is no longer possible for the individual 
scholar to command the range at once of philosophy, mathe- 
matics, physics, chemistry, and the natural sciences. It has even 
become more and more difficult to have a general knowledge of 
any one of these broad fields. 
Mathematics and Mechanics in the Eighteenth Century. 
— The invention of the infinitesimal calculus by Newton and 
Leibnitz was comparable in its relations and consequences with 
the discovery of a new world by Columbus two centuries earlier. 
As in that case the discovery was not an absolutely sudden one; 
other explorers had hoped or imagined, but only genius of that 
highest order which we call inspired, gained the complete revela- 
tion. The years next following the great discovery were natu- 
rally a period of eager and wide-ranging exploration, of optimistic 
self-confident pioneering. Such was the power of the new method, 
that one might rashly hope no secret of nature could long resist 
its attack. As circumnavigation of the globe was not long in 
following the discovery of America, so the cycle of mathematical 
knowledge might be completed. The parallel has failed. The 
calculus grew out of the insistent grappling by mathematicians with TENDENCIES IN MATHEMATICAL SCIENCE 
325 
problems which had defied the feebler tools of the earlier mathe- 
matics. One obstacle after another has been gradually sur- 
mounted by the invention of new and more powerful methods of 
ever increasing generality, just as increasingly powerful telescopes 
have revealed unnumbered new suns; and no boundary or limit 
to this evolutionary progress can be foreseen or imagined. On 
the other hand, as the new world has been gradually settled, 
civilized, and cultivated, so the fields of mathematics which were 
opened up in the eighteenth century have been critically ex- 
amined in the nineteenth, with much revision of fundamentals. 
The main features of eighteenth century mathematics were: 
— the working out of the differential and integral calculus into 
substantially the form they have ever since retained; the begin- 
nings of differential equations as a natural outgrowth of integral 
calculus, and the beginnings of the calculus of variations; the 
systematic application of the new ideas to mechanics, and in par- 
ticular to celestial mechanics. The century was also notable for 
important discoveries in astronomy and physics, including for ex- 
ample that of the aberration of light; a vigorous attack on “ the 
problem of three bodies ”; and the earlier telescopic work of the 
Herschels, culminating in the discovery of a new planet, Uranus. 
Among the leading mathematicians of the period were Mac- 
laurin of Scotland, various members of the Swiss Bernoulli family, 
Euler also a native of Switzerland, Lagrange of Italy, and in 
France, Clairaut, d’Alembert, and Laplace. In spite of the unique 
supremacy of Newton, the absence of Britons from this list is 
notable. The bitter personal controversy between Newton’s ad- 
herents and those of Leibnitz produced or aggravated an un- 
fortunate division between the English and the continental 
mathematicians. For the former, persistence in Newton’s in- 
ferior notation became a matter of national pride, and progress 
was correspondingly retarded. Of the mathematicians named 
above, the Bernoullis and Euler on the continent and Maclaurin 
in Scotland bore a leading part in the systematization of the 
calculus, while Lagrange and Laplace were preeminent in the 
development of analytical and celestial mechanics respectively. 326 
A SHORT HISTORY OF SCIENCE 
Maclaurin’s (1698-1746) Treatise of Fluxions (1742) was “ the 
first logical and systematic exposition of the method of fluxions,” 
and the applications to problems contained in it were characterized 
by Lagrange as the “masterpiece of geometry, comparable with 
the finest and most ingenious work of Archimedes.” Maclaurin’s 
point of view may be illustrated by the following passage: — 
Magnitudes were supposed to be generated by motion; and, by 
comparing the increments that were generated in any equal successive 
parts of the time, it was first determined whether the motion was uni- 
form, accelerated, or retarded. . . . When the motion was accelerated, 
this increment was resolved into two parts; that which alone would 
have been generated if the motion had not been accelerated, but had 
continued uniform from the beginning of the time, and that which was 
generated in consequence of the continual acceleration of the motion 
during that time. The latter part was rejected, and the former only 
retained for measuring the motion at the beginning of the time. 
And in like manner, when the motion was retarded, ... so that the 
motion at the time proposed was accurately measured, and the ratio 
of the fluxions always accurately represented. In the method of 
infinitesimals, the element, by which any quantity increases or de- 
creases, is supposed to be infinitely small, and is generally expressed 
by two or more terms, some of which are infinitely less than the rest, 
which being neglected as of no importance, the remaining terms form 
what is called the difference of the proposed quantity. The terms 
that are neglected in this manner, as infinitely less than the other 
terms of the element, are the very same which arise in consequence 
of the acceleration, or retardation, of the generating motion, during 
the infinitely small time in which the element is generated. . . . The 
conclusions are accurately true, without even an infinitely small 
error. . . . 
Daniel Bernoulli (1700-1782) made such good use of the new 
mathematical methods in attacking previously unsolved problems 
of mechanics, that he has been called the founder of mathematical 
physics. He recognized the importance of the principle of the 
conservation of force anticipated in part by Huygens. 
Euler (1707-1783), while Swiss by birth, spent most of his life TENDENCIES IN MATHEMATICAL SCIENCE 
327 
at the courts of St. Petersburg and Berlin. In spite of partial 
and ultimately complete blindness, his scientific productivity was 
enormous, and one of the most ambitious scientific undertakings 
of our own time is the publication of his complete works in 45 
volumes, by international cooperation. Our college mathematics 
— algebra, analytic geometry, and the calculus — owes its pres- 
ent shape largely to his works. 
Euler’s Complete Introduction to Algebra was one of the most 
influential books on algebra in the eighteenth century, and not the 
least because it is written with extraordinary clearness and in easily 
intelligible form. Euler was at that time already totally blind. He 
picked out a young man whom he had brought with him from Berlin 
as an attendant and who could reckon tolerably, but who otherwise 
had no understanding of mathematics. He was a tailor by trade 
and of moderate intellectual capacity. To him Euler dictated this 
book, and the amanuensis not only understood everything well but 
in a short time acquired the power to carry out difficult algebraic 
processes by himself with much facility. It was this book which 
completing the development begun by Vieta made algebra an inter- 
national mathematical shorthand. 
Euler formulated the idea of function which has proved so fun- 
damental in modern mathematics, both pure and applied. His 
work also contains the first systematic treatment of the calculus 
of variations, which is defined as “the method of finding the 
change caused in an expression containing any number of variables 
when one lets all or any of the variables change” or more geo- 
metrically “ a method of finding curves having a particular prop- 
erty in the highest or the lowest degree.” 
In other fields Euler “ was the first to treat the vibrations of light 
analytically and to deduce the equation of the curve of vibration as 
dependent upon elasticity and density. ... He deduced the law of 
refraction analytically and explained that the rays of greater wave- 
length must suffer the least deviation. ... He studied dispersion 
in the search for a corrective for chromatic aberration, which Newton 
had declared unattainable. ... It was this investigation that in- 328 
A SHORT HISTORY OF SCIENCE 
duced Dolland to construct his achromatic lenses. . . . Euler was 
thus the only physicist of the eighteenth century who advanced the 
undulatory theory.” — Bull. Amer. Math. Soc., Dec., 1907. 
Euler gained a share of the prize of £20,000 offered by the 
British parliament for a method of determining longitude at 
sea, half of the same prize falling to Harrison the maker of a ship’s 
chronometer sufficiently accurate for the same purpose. 
He was probably the most versatile as well as the most prolific 
of mathematicians of all time. There is scarcely any branch of 
modern analysis to which he was not a large contributor, and his 
extraordinary powers of devising and applying methods of calculation 
were employed by him with great success in each of the existing 
branches of applied mathematics; problems of abstract dynamics, 
of optics, of the motion of fluids, and of astronomy were all in turn 
subjected to his analysis and solved. — Berry. 
It is the invaluable merit of the great Basle mathematician 
Leonhard Euler, to have freed the analytical calculus from all geo- 
metrical bonds, and thus to have established analysis as an inde- 
pendent science, which from his time on has maintained an unchal- 
lenged leadership in the field of mathematics. — Hankel. 
Progress in Theoretical Mechanics. The rapid develop- 
ment of mechanics in the eighteenth century culminated in the 
great classical treatises of d’Alembert (1717-1783) — Traite de 
dynamique — and Lagrange (1736-1813)—Mecanique analytique, 
systematizing and coordinating the theories and results thus far 
obtained. D’Alembert, working out ideas based on Huygens’ 
theory of the centre of oscillation, formulated a very general dy- 
namical principle since known under his name: — 
On a system of points M, M', M" ... connected with one 
another in any way, the forces P, Pr, P" ... are impressed. These 
forces would impart to the free points of the system certain deter- 
minate motions. To the connected points, however, different motions 
are usually imparted — motions which could be produced by the forces 
W, W', W" ... These last are the motions which we shall study. 
Conceive the force P resolved into W and V, the force Pr into W 
and V', and the force P" into W" and V", and so on. Since, owing TENDENCIES IN MATHEMATICAL SCIENCE 
329 
to the connections, only the components W, Wf, W" ... are effec- 
tive, therefore the forces V, V', V" ... must be equilibrated by the 
connections. We will call the forces P, P', P" ... the impressed 
forces, the forces W, Wr, W" ..., which produce the actual motions, 
the effective forces, and the forces V, V', V" ... the forces gained and 
lost, or the equilibrated forces. We perceive, thus, that if we resolve 
the impressed forces into the effective forces and the equilibrated 
forces, the latter form a system balanced by the connections. — Mach. 
To d’Alembert is attributed the celebrated epigram concerning 
Benjamin Franklin, “He snatched the thunderbolt from heaven, 
the sceptre from tyrants ” (Eripuit coelo fulmen sceptrumque 
tyrannis). 
J. L. Lagrange (1736-1813), a native of Turin, also spent many 
years in Berlin and his later life in Paris, where he became pro- 
fessor at the newly established Ecole poly technique. At the 
age of 25 he was pronounced the greatest mathematician living. 
His chief work, the Mecanique analytique, is a masterly discussion 
of the whole subject, showing by the aid of the new mathematical 
methods its dependence on a few fundamental principles. On the 
death of his royal patron, Frederick the Great, in 1787, he was 
invited from Berlin not only to Paris, but to Spain and to Naples, 
accepting the first-named opportunity. Lagrange’s works include 
also very important contributions to differential equations and 
the calculus of variations, of which any detailed account would 
be too technical for our purpose. The significance and impor- 
tance of Lagrange’s Mecanique analytique are within its field com- 
parable with those of Newton’s Principia. 
Lagrange like Newton has possessed in the highest degree the fortu- 
nate art of discovering the universal principles which constitute the 
essence of science. This art he understands how to unite with a rare 
elegance in the development of the most abstruse theories. — Laplace. 
In contrast with the predominantly geometrical and synthetic 
methods of Newton, Lagrange’s methods are mainly analytical. 
Generality of points of view and of methods, precision and ele- 
gance in presentation, have become, since Lagrange, the common 330 
A SHORT HISTORY OF SCIENCE 
property of all who would lay claim to the rank of scientific mathe- 
maticians. — Ilankel. 
When we have grasped the spirit of the infinitesimal method, 
and have verified the exactness of its results either by the geometrical 
method of prime and ultimate ratios, or by the analytical method of 
derived functions, we may employ infinitely small quantities as a sure 
and valuable means of shortening and simplifying our proofs. 
— Lagrange. 
Lagrange also applied his great powers of analysis to problems 
in astronomy and in cartography. 
Celestial Mechanics. — Pierre Simon Laplace (1749-1827) 
was of humble Norman antecedents which he in later life somewhat 
disdained, and played a great part in the scientific activity under 
Napoleon. In the five volumes of his Mecanique celeste, he pro- 
duced a permanent monument to his own genius. It was his lofty 
ambition 
to offer a complete solution of the great mechanical problem pre- 
sented by the solar system, and bring theory to coincide so closely 
with observation that empirical equations should no longer find a 
place in astronomical tables. 
He regarded analysis merely as a means of attacking physical prob- 
lems, though the ability with which he invented the necessary anal- 
ysis is almost phenomenal. As long as his results were true he took 
very little trouble to explain the steps by which he arrived at them; 
he never studied elegance or symmetry in his processes, and it was 
sufficient for him if he could by any means solve the particular ques- 
tion he was discussing. — Ball. 
Bowditch, the American translator of his great work, remarks 
significantly: — 
I never come across one of Laplace’s ‘ Thus it plainly appears ’ without 
feeling sure that I have hours of hard work before me to fill up the 
chasm and find out and show how it plainly appears. 
In the words of the historian Tod hunter, 
a complete evolution of the history will restore the reputation of La- 
place to its just eminence. The advance of mathematical science is TENDENCIES IN MATHEMATICAL SCIENCE 
331 
on the whole remarkably gradual, for with the single exception of 
Newton there is very little exhibition of great and sudden develop- 
ments ; but the possessions of one generation are received, augmented 
and transmitted by the next. It may be confidently maintained that no 
single person has contributed more to the general stock than Laplace. 
The Perturbation Problem. — Newton had worked out the 
theory of a single planet or satellite revolving about its primary. 
The consequent discrepancies were held by some to indicate 
inexactness in his hypothetical laws. Laplace occupied himself 
with a thorough study of the great problem of three bodies,1 and 
without fully solving it, accounted to a great extent for the dis- 
crepancies in question. In particular he maintained the stability 
of the solar system. His Mecanique celeste has been charac- 
terized as an infinitely extended and enriched edition of Newton’s 
Principia. 
In his confidence in the extending range of mathematical methods 
Laplace says: — 
Given for one instant an intelligence which could comprehend all 
the forces by which nature is animated and the respective positions of 
the beings which compose it, if moreover this intelligence were vast 
enough to submit these data to analysis, it would embrace in the same 
formula both the movements of the largest bodies in the universe 
and those of the lightest atom : to it nothing would be uncertain, and 
the future as the past would be present to its eyes. The human mind 
offers a feeble outline of that intelligence, in the perfection which it 
has given to astronomy. Its discoveries in mechanics and in geom- 
etry, joined to that of universal gravity, have enabled it to com- 
prehend in the same analytical expressions the past and future states 
of the world system. 
The Nebular Hypothesis. — In his Exposition du systeme 
du monde, “one of the most perfect and charmingly written 
at any time the positions and motions of three mutually gravitating 
bodies, to determine their positions and motions at any other time — a particular 
case of the actual more general problem : Given the 18 known bodies of the solar 
system, and their positions and motions at any time, to deduce from their mutual 
gravitation by a process of mathematical calculation their positions and motions 
at any other time; and to show that these agree with those actually observed. * 332 
A SHORT HISTORY OF SCIENCE 
popular treatises on astronomy ever published, in which the great 
mathematician never uses either an algebraical formula or a 
geometrical diagram”, Laplace presents the arguments for his 
nebular hypothesis along the following general lines: — 
In spite of the separation of the planets they bear certain re- 
markable relations to each other; 
All the planets travel about the sun in the same direction and 
almost in the same plane; 
The satellites also travel about their planets in this same 
direction and almost in the same plane; 
Finally, sun, planets and satellites revolve in the same sense 
about their own axes and this rotation is approximately in the 
orbital plane. 
These agreements cannot be accidental. Laplace seeks the 
cause in the existence of an original vast nebulous mass forming 
a sort of atmosphere about the sun and extending beyond the 
outermost planet. Initial or acquired rotation of the nebula at- 
tended by gradual cooling and contraction has caused the cen- 
trifugal separation of masses analogous to Saturn’s rings, out of 
which planets have gradually condensed, throwing off their own 
satellites in the process. This hypothesis had already been 
proposed in substance by Kant in 1755. Its later history will 
be touched on in a following chapter. 
Laplace was also deeply interested in the theory of probability, 
as may be illustrated by the following passages: — 
The most important questions of life are, for the most part, really 
only problems of probability. Strictly speaking one may even say 
that nearly all our knowledge is problematical; and in the small 
number of things which we are able to know with certainty, even in 
the mathematical sciences themselves, induction and analogy, the 
principal means for discovering truth, are based on probabilities, so 
that the entire system of human knowledge is connected with this 
theory. 
It is remarkable that a science (probabilities) which began with 
the consideration of games of chance, should have become the most 
important object of human knowledge. TENDENCIES IN MATHEMATICAL SCIENCE 
333 
The theory of probabilities is at bottom nothing but common 
sense reduced to calculus; it enables us to appreciate with exactness 
that which accurate minds feel with a sort of instinct for which oft- 
times they are unable to account. If we consider the analytical 
methods to which this theory has given birth, the truth of the prin- 
ciples on which it is based, the fine and delicate logic which their em- 
ployment in the solution of problems requires, the public utilities 
whose establishment rests upon it, the extension which it has received 
and which it may still receive through its application to the most 
important problems of natural philosophy and the moral sciences; 
if again we observe that, even in matters which cannot be submitted 
to the calculus, it gives us the surest suggestions for the guidance of 
our judgments, and that it teaches us to avoid the illusions which 
often mislead us, then we shall see that there is no science more worthy 
of our contemplations nor a more useful one for admission to our 
system of public education. 
Modern Astronomy. Telescopic Discoveries. — The im- 
mense impetus given to astronomy by the revolutionary discov- 
eries of Copernicus, Tycho Brahe, Galileo, Kepler, and Newton, 
followed in the eighteenth century by the complete working out of 
the mathematical consequences of the gravitation theory by Laplace 
and others, placed the science in advance of all its rivals and 
seemed to make it a model for their imitation. 
A different and most far-reaching tendency appears with the 
work of the Herschels. Friedrich Wilhelm Herschel (1738-1822), 
a poor German musician emigrating to England and devoting his 
spare time unremittingly to astronomy, with the help of his capable 
sister laid the foundations of modern physical astronomy. In 1781 
he amazed himself as well as the scientific world by discovering be- 
yond Saturn a new planet, Uranus, — taking it at first for a 
comet. Constructing more and more powerful telescopes he dis- 
covered several satellites of Uranus and two of Saturn. He also 
determined a motion of the solar system as a whole, towards a 
point in the constellation Hercules. He catalogued more than 
800 double stars and more than 2000 nebulas, recognizing among 
the latter, as he believed, different stages of the evolution of other 
planetary systems. He observes: 334 
A SHORT HISTORY OF SCIENCE 
This method of viewing the heavens seems to throw them into 
a new kind of light. They are now seen to resemble a luxuriant 
garden, which contains the greatest variety of productions, in different 
flourishing beds; and one advantage we may at least reap from it is, 
that we can, as it were, extend the range of our experience to an im- 
mense duration. For, to continue the simile I have borrowed from 
the vegetable kingdom, is it not almost the same thing, whether we 
live successively to witness the germination, blooming, foliage, fecun- 
dity, fading, withering, and corruption of a plant, or whether a vast 
number of specimens selected from every stage through which the plant 
passes in the course of its existence, be brought at once to our view ? 
With a reminiscence of Descartes, he says: — 
I determined to accept nothing on faith, but to see with my own 
eyes what others had seen before me. . . . When I had carefully 
and thoroughly perfected the great instrument in all its parts I 
made systematic use of it in my observations of the heavens, first 
forming a determination never to pass by any, the smallest, portion 
of them without due investigation. 
To the eighteenth century also belong elaborate and costly 
expeditions — including one organized by the American Philosoph- 
ical Society of Philadelphia — to observe transits of Venus, as a 
means for determining the distance from the sun to the earth. 
Mathematical Progress and Physical Science.—Besides 
the extension of mathematical ideas and methods to mechanics, 
astronomy, optics and other branches of physics, chemistry was 
now also becoming a quantitative science. So Scheele begins a 
work published in 1777: — 
To resolve bodies skilfully into their components, to discover their 
properties and to combine them in different ways, is the chief pur- 
pose of chemistry. 
Richter, in his Stoichiometry (1792-1802), even speaks of 
chemistry as a branch of applied mathematics. Already the 
pioneer Robert Boyle had written: — 
I confess, that after I began ... to discern how useful mathe- 
maticks may be made to physicks, I have often wished that I had TENDENCIES IN MATHEMATICAL SCIENCE 
335 
employed about the speculative part of geometry, and the cultivation 
of the specious Algebra I had been taught very young, a good part 
of that time and industry that I had spent about surveying and forti- 
fication (of which I remember I once wrote an entire treatise) and 
other parts of practick mathematicks. 
Mathematicks may help the naturalists, both to frame hypotheses, 
and to judge of those that are proposed to them, especially such as 
relate to mathematical subjects in conjunction with others. 
Even in natural science Stephen Hales says in 1727: — 
And since we are assured that the all-wise Creator has observed 
the most exact proportions, of number, weight and measure, in the 
make of all things; the most likely way therefore to get any insight 
into the nature of these parts of the creation, must in all reason be 
to number, weigh and measure. And we have much encourage- 
ment to pursue this method, of searching into the nature of things, 
from the great success that has attended any attempts of this 
kind. 
Summing up these tendencies, a recent writer remarks: — 
In the eighteenth century mathematics was regarded by many 
scholars as the ideal, the completeness and exactness of whose methods 
should be arrived at by other less highly developed branches. So 
Laplace’s popularized version of his celestial mechanics met an eager 
need, and even Voltaire undertook the championship of the Newtonian 
philosophy. Logic and even ethics were drawn into the mathematical 
retinue. For Maupertuis the good is a positive quantity, the bad a 
negative. Joys and griefs make up human life according to the laws 
of algebraic addition, and it is the business of statesmen to see that the 
positive balance is as large as possible. The great Buffon adds to his 
natural history a supplement on moral arithmetic. Mathematics 
aims at the leadership both in natural science and in human affairs. 
In spite of this perhaps exaggerated predilection in learned 
and polite society, educational curricula remained weak and 
conservative. Powerful progressive tendencies growing out of 
the French Revolution found expression in the founding of the 
Ecole polytechnique, — under the leadership of Monge, — which 
has ever since been an important centre of mathematical activity. 336 
A SHORT HISTORY OF SCIENCE 
Its curriculum included, in the first year, analytic geometry of 
space and descriptive geometry, in the second, mechanics of 
solids and liquids, in the third, theory of mechanics. 
Reviewing eighteenth century mathematics, a recent German 
writer says: — 
In the science itself there showed itself with the close of the 
eighteenth century a certain exhaustion. ‘The mine is, it seems to 
me, too deep,’ wrote Lagrange in the year 1781 to d’Alembert, ‘and 
unless new veins are discovered it must sooner or later be abandoned.’ 
In the nineteenth century men have dug deeper and struck noble 
ores, but serious obstacles opposed the progress. It appeared that 
the men of genius of the illustrious period had to some extent practised 
bad building and the whole framework threatened to cave in unless 
the passages were newly supported and the oncoming floods of doubt 
conducted away. For two generations a considerable share of the 
efforts of mathematics must be applied to the hard work of security 
and safety, a labor from which even the greatest . . . have not held 
back. 
Nineteenth Century Mathematics. — As in the century 
following Newton France became the great centre of mathe- 
matical activity, so in the nineteenth century the leadership 
passed to Germany, under the inspiration of Gauss and Riemann 
of Gottingen, Jacobi of Konigsberg, Weierstrass of Berlin,— 
to mention but a few of those no longer living. Outside of Ger- 
many conspicuous names are Cauchy, Galois, Hermite, Legendre, 
and Poincare in France, Cayley and Sylvester in England, Abel in 
Sweden, and Lobatchewski in Russia. 
Characteristic of this period are: the development of a 
general theory of functions based on unifying coordinating prin- 
ciples, compensating the powerful specializing tendencies, and a 
profound critical revision of the previously accepted axioms, 
leading for example to the development of a non-Euclidean 
geometry. In the science generally there is systematic devel- 
opment of instruction and research, notably in the German 
universities; of publication, by the establishment of mathemati- 
cal journals, and the preparation of encyclopedias; numerous TENDENCIES IN MATHEMATICAL SCIENCE 
337 
national societies are formed, and international congresses held. 
These tendencies are naturally not confined to the mathemat- 
ical sciences. In some respects mathematics has merely en- 
joyed its share in the prosperity of a more scientific age, in some 
it has perhaps suffered, at any rate relatively, from the powerful 
stimulus given the natural sciences by the working out of evolu- 
tionary theories. From a position of acknowledged primacy 
among a small number of recognized sciences, it has come to be 
regarded as but one of many. 
It is impossible here even to enumerate the different branches 
of mathematical science developed during this period. Certain 
typical features may however be touched upon. 
Non-Euclidean Geometry. — Each century takes over as a 
heritage from its predecessors a number of problems whose solution 
previous generations of mathematicians have arduously but vainly 
sought. It is a signal achievement of the nineteenth century to have 
triumphed over some of the most celebrated of these problems. 
The most ancient of them is the quadrature of the circle, which 
already appears in our oldest mathematical document, the Papyrus 
Rhind, b.c. 2000. Its impossibility was finally shown by Lindemann, 
1882. 
But of all problems which have come down from the past, by far 
the most celebrated and important relates to Euclid’s parallel axiom. 
Its solution has profoundly affected our views of space and given rise to 
questions even deeper and more far-reaching, which embrace the entire 
foundation of geometry and our space conception.— Pierpont (1904). 
I am convinced more and more that the necessary truth of our 
geometry cannot be demonstrated, at least not by the human intellect 
to the human understanding. Perhaps in another world we may gain 
other insights into the nature of space which at present are unattain- 
able to us. Until then we must consider geometry as of equal rank 
not with arithmetic, which is purely a priori, but with mechanics. 
— Gauss (1817). 
There is no doubt that it can be rigorously established that the 
sum of the angles of a rectilinear triangle cannot exceed 180°. But 
it is otherwise with the statement that the sum of the angles cannot 
be less than 180°; this is the real Gordian knot, the rocks which cause 338 
A SHORT HISTORY OF SCIENCE 
the wreck of all. ... I have been occupied with the problem over 
thirty years and I doubt if anyone has given it more serious attention, 
though I have never published anything concerning it.— Gauss (1824). 
I will add that I have recently received from Hungary a little 
paper on Non-Euclidean geometry, in which I rediscover all my own 
ideas and results worked out with great elegance. . . . The writer 
is a very young Austrian officer, the son of one of my early friends, 
with whom I often discussed the subject in 1798, although my ideas 
were at that time far removed from the development and maturity 
which they have received through the original reflections of this 
young man. I consider the young geometer von Bolyai a genius of 
the first rank. — Gauss (1832). 
The gradual adoption of new and revolutionary ideas on this 
subject may be further illustrated by the following passages: — 
The characteristic features of our space are not necessities of 
thought, and the truth of Euclid’s axioms, in so far as they specially 
differentiate our space from other conceivable spaces, must be es- 
tablished by experience and by experience only. — R. S. Ball. 
If the Euclidean assumptions are true, the constitution of those 
parts of space which are at an infinite distance from us, geometry 
upon the plane at infinity, is just as well known as the geometry of 
any portion of this room. In this infinite and thoroughly well-known 
space the Universe is situated during at least some portion of an 
infinite and thoroughly well-known time. So that there we have 
real knowledge of something at least that concerns the Cosmos; 
something that is true throughout the Immensities and the Eternities. 
That something Lobatchewski and his successors have taken away. 
The geometer of today knows nothing about the nature of the actually 
existing space at an infinite distance; he knows nothing about the 
properties of this present space in a past or future eternity. He 
knows, indeed, that the laws assumed by Euclid are true with an 
accuracy that no direct experiment can approach, not only in this 
place where we are, but in places at a distance from us that no as- 
tronomer has conceived; but he knows this as of Here and Now; 
beyond this range is a There and Then of which he knows nothing at 
present, but may ultimately come to know more. — Clifford. 
Everything in physical science, from the law of gravitation to 
the building of bridges, fpom the spectroscope to the art of navigation, TENDENCIES IN MATHEMATICAL SCIENCE 
339 
would be profoundly modified by any considerable inaccuracy in the 
hypothesis that our actual space is Euclidean. The observed truth 
of physical science, therefore, constitutes overwhelming empirical 
evidence that this hypothesis is very approximately correct, even 
if not rigidly true. — Russell. 
The most suggestive and notable achievement of the last century 
is the discovery of Non-Euclidean geometry. — Hilbert. 
What Vesalius was to Galen, what Copernicus was to Ptolemy, 
that was Lobatchewski to Euclid. There is, indeed, a somewhat 
instructive parallel between the last two cases. Copernicus and 
Lobatchewski were both of Slavic origin. Each of them has brought 
about a revolution in scientific ideas so great that it can only be com- 
pared with that wrought by the other. And the reason of the trans- 
cendent importance of these two changes is that they are changes 
in the conception of the Cosmos. . . . And in virtue of these two 
revolutions the idea of the Universe, the Macrocosm, the All, as 
subject of human knowledge, and therefore of human interest, has 
fallen to pieces. — Clifford. 
Geometrical axioms are neither synthetic a priori conclusions 
nor experimental facts. They are conventions: our choice, amongst 
all possible conventions, is guided by experimental facts; but it 
remains free, and is only limited by the necessity of avoiding all 
contradiction. ... In other words, axioms of geometry are only 
definitions in disguise. That being so what ought one to think of 
this question: Is the Euclidean Geometry true? The question is 
nonsense. One might as well ask whether the metric system is 
true and the old measures false; whether Cartesian co-ordinates are 
true and polar co-ordinates false. — Poincare. 
To make non-Euclidean geometry intelligible to laymen the 
following illustration has been given by Helmholtz: — 
Think of the image of the world in a convex mirror. ... A 
well-made convex mirror of moderate aperture represents the objects 
in front of it as apparently solid and in fixed positions behind its 
surface. But the images of the distant horizon and of the sun in the 
sky lie behind the mirror at a limited distance, equal to its focal 
length. Between these and the surface of the mirror are found the 
images of all the other objects before it, but the images are diminished 340 
A SHORT HISTORY OF SCIENCE 
and flattened in proportion to the distance of their objects from the 
mirror. . . . Yet every straight line or plane in the outer world 
is represented by a straight [ ? ] line or plane in the image. The image 
of a man measuring with a rule a straight line from the mirror, would 
contract more and more the farther he went, but with his shrunken 
rule the man in the image would count out exactly the same number 
of centimeters as the real man. And, in general, all geometrical 
measurements of lines and angles made with regularly varying images 
of real instruments would yield exactly the same results as in the 
outer world, all lines of sight in the mirror would be represented by 
straight lines of sight in the mirror. In short, I do not see how men 
in the mirror are to discover that their bodies are not rigid solids and 
their experiences good examples of the correctness of Euclidean 
axioms. But if they could look out upon our world as we look into 
theirs without overstepping the boundary, they must declare it to 
be a picture in a spherical mirror, and would speak of us just as we 
speak of them; and if two inhabitants of the different worlds could 
communicate with one another, neither, as far as I can see, would be 
able to convince the other that he had the true, the other the dis- 
torted, relation. Indeed I cannot see that such a question would 
have any meaning at all, so long as mechanical considerations are 
not mixed up with it. 
Imaginary Numbers. — The solution of algebraic equations 
had always been hampered by the seeming impossibility of per- 
forming the inverse processes involved. The equation x + 5 = 0 
could not be solved before negative numbers were known; and the 
equations 2 x = 5 and x2 = 2 would be equally insoluble without 
fractions and irrational numbers. Such equations as x2 + 1 = 0 
still remained a stumbling-block at the beginning of the nineteenth 
century. Gauss first pierced the mystery and released algebra 
from its traditional restriction, proving that an equation of any 
degree has a corresponding number of roots of the form a + — 1 
— a discovery of far-reaching importance not merely for higher 
mathematics but even for electrical engineering. 
That this subject [of imaginary magnitudes] has hitherto been 
considered from the wrong point of view and surrounded by a mysteri- 
ous obscurity, is to be attributed largely to an ill-adapted notation. TENDENCIES IN MATHEMATICAL SCIENCE 
341 
If for instance, + 1, — 1, V — 1 had been called direct, inverse, and 
lateral units, instead of positive, negative, and imaginary (or even 
impossible) such an obscurity would have been out of the question. 
— Gauss. 
Concluding an address on the history of mathematics in the 
nineteenth century, a recent writer says: — 
What strikes us at once in our survey of mathematics in the last 
century is its colossal proportions and rapid growth in nearly all 
directions, the great variety of its branches, the generality and com- 
plexity of its methods, an inexhaustible creative imagination, the fear- 
less introduction and employment of ideal elements, and an apprecia- 
tion for a refined and logical development of all its parts.— Pierpont. 
Probably no other department of knowledge plays a larger part 
outside its own narrower domain than mathematics. Some of its 
more elementary conceptions and methods have become part of the 
common heritage of our civilization, interwoven in the every-day 
life of the people. Perhaps the greatest labor-saving invention that 
the world has seen belongs to the formal side of mathematics; I 
allude to our system of numerical notation. . . . Without taking 
too literally the celebrated dictum of the great philosopher Kant 
that the amount of real science to be found in any special subject is 
the amount of mathematics contained therein, it must be admitted 
that each branch of science which is concerned with natural phe- 
nomena, when it has reached a certain stage of development, becomes 
accessible to, and has need of, mathematical methods and language; 
this stage has, for example, been reached in our time by parts of the 
science of chemistry. — Hobson. 
I often say that when you can measure what you are speaking 
about and express it in numbers, you know something about it; but 
when you cannot measure it, when you cannot express it in numbers, 
your knowledge is of a meager and unsatisfactory kind; it may be the 
beginning of knowledge, but you have scarcely in your thoughts 
advanced to the stage of science. — Kelvin. 
The Discovery of Neptune. — In a century filled with re- 
markable scientific achievement, no single triumph has been more 
conspicuous, or in some respects more dramatic, than the discovery 
of the planet Neptune by Adams and Leverrier. From the time 342 
A SHORT HISTORY OF SCIENCE 
of Newton the perturbations of the planets had been the subject 
of continual observation and study. Improved telescopes de- 
manded — and at the same time facilitated — more extended 
and refined computations. Discrepancies between computed and 
observed positions indicated disturbing forces of known or in some 
cases unknown origin. In particular, irregularities — never ex- 
ceeding two minutes of arc — in the motion of the most recently 
discovered planet Uranus, led the young Cambridge graduate 
John Couch Adams (1819-1892) and the eminent French astrono- 
mer Leverrier (1811-1877) to independent attacks on the for- 
midable problem of determining the mass and position of a 
hypothetical new planet which could cause the observed effects 
on Uranus. Unfortunately for Adams the necessary cooperation 
on the part of the observatories was not promptly available, so 
that the actual discovery connected itself with the somewhat 
later work of Leverrier. The discovery was naturally accepted 
as an extraordinary illustration of the power of mathematical as- 
tronomy and a convincing proof of the Newtonian theory of 
gravitation. 
The discovery of this planet [Neptune] is justly reckoned as 
the greatest triumph of mathematical astronomy. Uranus failed 
to move precisely in the path which the computers predicted for it, 
and was misguided by some unknown influence to an extent which 
a keen eye might almost see without telescopic aid. . . . These 
minute discrepancies constituted the data which were found sufficient 
for calculating the position of a hitherto unknown planet, and bring- 
ing it to light. Leverrier wrote to Galle, in substance: Direct your 
telescope to a point on the ecliptic in the constellation of Aquarius, in 
longitude 326°, and you will find within a degree of that place a new 
planet, looking like a star of about the ninth magnitude, and having a 
perceptible disc. The planet was found at Berlin on the night of 
Sept. 26, 1846, in exact accordance with this prediction, within half 
an hour after the astronomers began looking for it, and only about 
52' distant from the precise point that Leverrier had indicated. 
— Young. 
While the telescope serves as a means of penetrating space, and 
of bringing its remotest regions nearer us, mathematics, by inductive TENDENCIES IN MATHEMATICAL SCIENCE 
343 
reasoning, has led us onwards to the remotest regions of heaven, 
and brought a portion of them within the range of our possibilities; 
nay, in our own times — so propitious to the extension of knowledge 
— the application of all the elements yielded by the present condi- 
tions of astronomy has even revealed to the intellectual eyes a heavenly 
body, and assigned to it its place, orbit, mass, before a single telescope 
has been directed towards it. — Humboldt. 
Cosmic Evolution. — Reference has been made to the nebular 
hypothesis included by Laplace in his extended discussion of the 
solar system. During the nineteenth century this theory has 
been subjected to searching scrutiny from many points of view 
and much doubt has been cast on its validity. 
The following summary of present opinion is given by Hale in 
his Stellar Evolution; — 
The nebular hypothesis of Laplace still remains as the most seri- 
ous attempt to exhibit the development of the solar system. At- 
tacked on many grounds, and showing signs of weakness that seem to 
demand radical modification of Laplace’s original ideas, it nevertheless 
presents a picture of the solar system which has served to connect in 
a general way a mass of individual phenomena, and to give signifi- 
cance to apparently isolated facts that offer little of interest with- 
out the illumination of this governing principle. 
We are now in a position to regard the study of evolution as that 
of a single great problem, beginning with the origin of the stars in 
the nebulae and culminating in those difficult and complex sciences 
that endeavor to account, not merely for the phenomena of life, but 
for the laws which control a society composed of human beings. 
As a complement to the preceding may be added the following 
from another specialist in planetary evolution: — 
It is to the glory of astronomy that in it were initiated the two 
most fundamental intellectual movements in the history of mankind, 
viz. the establishment of the possibility of science and of the doctrine 
of evolution. Our intellectual ancestors in the valleys of the Euphrates 
and the Nile and on the hills of Greece looked up into the sky at 
night and saw order there and not chaos. By painstaking obser- 344 
A SHORT HISTORY OF SCIENCE 
vations and calculations they discovered the relatively simple laws 
of the motions of the heavenly bodies, whose invariable and exact 
fulfilment led to the belief that the whole universe in all its parts 
is orderly and that science is possible. In the modern world this 
conclusion is so commonplace that its immense value is apt to be 
overlooked, but a study of the superstitions and the hopeless stagna- 
tion of those portions of mankind which have not yet made the dis- 
covery gives us some measure of its worth. The modern supplement 
to the conception that the universe is not a chaos is that not only is 
it an orderly universe at any instant, but that it changes from one 
state to another in a continuous and orderly fashion. This doctrine 
that science is extensive in time, as well as in space, is the funda- 
mental element in the theory of evolution and the completion of the 
conception of science itself. The ideas of evolution in a scientific 
form were first applied to the relatively simple celestial phenomena. 
More than a century before the appearance of Darwin’s ‘Origin of 
Species,’ and the philosophical writings of Spencer, another English- 
man, Thomas Wright, published a book on the origin of worlds. La- 
place’s nebular hypothesis gave the geologists a basis for their work, 
which in turn paved the way for that of Darwin. For half a century 
now, the doctrine of evolution has been a fundamental factor in the 
elaboration of all scientific theories, and its influence has spread to 
every field of intellectual effort. It has been the good fortune of 
mankind that his skies have sometimes been free of clouds and that 
he has been able to observe those relatively simple yet majestic and 
impersonal celestial phenomena which have not only led to so im- 
portant results as the founding of science and the doctrine of evolu- 
tion, but have strongly colored his poetry, philosophy and religion, 
and have stimulated him to the elaboration of some of his most pro- 
found mathematical theories. — Moulton. 
Distance of the Stars. — Among other astronomical dis- 
coveries bearing a notable relation to the history of mathematical 
science is that of measurable stellar parallax by Bessel (1784- 
1846). One of the traditional objections to the Copernican theory 
had been the fact that no change could be detected in the relative 
position of the stars, such as would apparently result from revolu- 
tion of the earth in a vast orbit. Now with more and more 
powerful instruments it turned out that there were stars near TENDENCIES IN MATHEMATICAL SCIENCE 
345 
enough to show precisely the displacement discovered. In 1837 
Bessel attacked this ancient problem successfully by making 
extremely accurate observation of the relative positions of a certain 
double star (71 Cygni) and its celestial neighbors. He obtained 
for the distance of the double star 657,000 times the mean dis- 
tance from the earth to the sun. Such inconceivably vast dis- 
tances have been since conveniently expressed in a unit called the 
light-year, i.e. the distance a ray of light travels in an entire year 
at 186,000 miles per second. 
Mathematical Physics. — The further progress of applied 
mathematics in the nineteenth century has been interestingly 
summarized by Woodward in a presidential address to the Ameri- 
can Mathematical Society, from which the following extracts are 
quoted. 
Next came the splendid contributions of George Green under 
the modest title of ‘An essay on the application of mathematical 
analysis to the theories of electricity and magnetism.’ It is in this 
essay that the term ‘potential function’ first occurs. Herein also 
his remarkable theorem in pure mathematics, since universally known 
as Green’s theorem, and probably the most important instrument 
of investigation in the whole range of mathematical physics, made its 
appearance. We are all now able to understand, in a general way at 
least, the importance of Green’s work, and the progress made since 
the publication of his essay in 1828. But to fully appreciate his 
work and subsequent progress one needs to know the outlook for the 
mathematico-physical sciences as it appeared to Green at this time 
and to realize his refined sensitiveness in promulgating his discoveries. 
‘It must certainly be regarded as a pleasing prospect to analysts,’ 
he says in his preface, ‘ that at a time when astronomy, from the state 
of perfection to which it has attained, leaves little room for further 
applications of their art, the rest of the physical sciences should show 
themselves daily more and more willing to submit to it.’. . . ‘Should 
the present essay tend in any way to facilitate the application of 
analysis to one of the most interesting of the physical sciences, the 
author will deem himself amply repaid for any labor he may have 
bestowed upon it; and it is hoped the difficulty of the subject will 
incline mathematicians to read this work with indulgence, more partic- 346 
A SHORT HISTORY OF SCIENCE 
ularly when they are informed that it was written by a young man who 
has been obliged to obtain the little knowledge he possesses, at such 
intervals and by such means as other indispensable avocations which 
offer but few opportunities of mental improvement, afforded.’ Where 
in the history of science have we a finer instance of that sort of modesty 
which springs from a knowledge of things ? 
Just as the theories of astronomy and geodesy originated in the 
needs of the surveyor and navigator, so has the theory of elasticity 
grown out of the needs of the architect and engineer. From such 
prosaic questions, in fact, as those relating to the stiffness and the 
strength of beams, has been developed one of the most comprehensive 
and most delightfully intricate of the mathematico-physical sciences. 
Although founded by Galileo, Hooke, and Mariotte in the seventeenth 
century, and cultivated by the Bernoullis and Euler in the last 
century, it is, in its generality, a peculiar product of the present 
century. It may be said to be the engineers’ contribution of the cen- 
tury to the domain of mathematical physics, since many of its most 
conspicuous devotees, like Navier, Lame, Rankine, and Saint-Venant, 
were distinguished members of the profession of engineering. . . . 
The theory of elasticity has for its object the discovery of the 
laws which govern the elastic and plastic deformation of bodies or 
media. In the attainment of this object it is essential to pass from 
the finite and grossly sensible parts of media to the infinitesimal and 
faintly sensible parts. Thus the theory is sometimes called molec- 
ular mechanics, since its range extends to infinitely small particles 
of matter if not to the ultimate molecules themselves. It is easy, 
therefore, considering the complexity of matter as we know it in the 
more elementary sciences, to understand why the theory of elasticity 
should present difficulties of a formidable character and require a 
treatment and a nomenclature peculiarly its own. . . . 
It is from such elementary dynamical and kinematical considera- 
tions as these that this theory has grown to be not only an indis- 
pensable aid to the engineer and physicist, but one of the most at- 
tractive fields for the pure mathematician. As Pearson has remarked, 
‘There is scarcely a branch of physical investigation, from the plan- 
ning of a gigantic bridge to the most delicate fringes of color exhibited 
by a crystal, wherein it does not play its part.’ It is, indeed, funda- 
mental in its relations to the theory of structures, to the theory of 
hydromechanics, to the elastic solid theory of light, and to the theory 
of crystalline media. TENDENCIES IN MATHEMATICAL SCIENCE 
347 
REFERENCES FOR READING 
Clerke. History of Astronomy during the Nineteenth Century. 
Clerke. The Herschels and Modem Astronomy. 
Hale. Stellar Evolution. 
Lodge. Pioneers of Science. 
Mach. Science of Mechanics. 
Merz. History of European Thought in the Nineteenth Century, Chapters 
IV, V. 
Whitehead. Introduction to Mathematics. 
Poincare. Science and Hypothesis. CHAPTER XVI 
SOME ADVANCES IN PHYSICAL SCIENCE IN THE 
NINETEENTH CENTURY. ENERGY AND 
THE CONSERVATION OF ENERGY 
About a century after the publication of the Principia, which, by 
propounding the gravitation formula, raised the ancient and indefinite 
notion of Attraction to the rank of a useful and rigorously defined 
expression, another favorite theory [Atomism] of the ancient philoso- 
phers was similarly elevated to the rank of a leading and useful 
scientific idea. 
The law of gravitation embraced cosmical and some molar phe- 
nomena, but led to vagueness when applied to molecular actions. The 
atomic theory led to a complete systematization of chemical com- 
pounds, but afforded no clue to the mysteries of chemical affinity. 
And the kinetic or mechanical theories of light, of electricity and mag- 
netism, led rather to a new dualism, the division of science into sciences 
of matter and of the ether. ... A more general term had to be found 
under which the different terms could be comprised, which would give 
a still higher generalization, a more complete unification of knowl- 
edge. One of the principal performances of the second half of the nine- 
teenth century has been to find this more general term, and to trace 
its all-pervading existence on a cosmical, a molar, and a molecular 
scale . . . this greatest of all exact generalizations — the conception 
of energy. 
Electrified and magnetised bodies attract or repel each other accord- 
ing to laws discovered by men who never doubted that the action 
took place at a distance, without the intervention of any medium, 
and who would have regarded the discovery of such a medium as 
complicating rather than as explaining the undoubted phenomena of 
attraction. — Merz. 
Through metaphysics first; then through alchemy and chemis- 
try, through physical and astronomical spectroscopy, lastly through 
radio-activity, science has slowly groped its way to the atom.— 
Soddy. 
348 PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
349 
There is in nature a certain magnitude of unsubstantial quality, 
which keeps its value under all alterations of the objects observed, 
while its manner of appearance changes most variously. — Mayer. 
I shall lose no time in repeating and extending these experi- 
ments, being satisfied that the grand agents of nature are, by the 
Creator’s fiat, indestructible; and that whatever mechanical force is 
expended, an exact equivalent of heat is always obtained. — Joule. 
Heat and work are equivalent. The entropy of the universe tends 
to a maximum. — Clausius. 
The later eighteenth and the whole of the nineteenth 
centuries are characterized by increasingly rapid development of 
the physical sciences, which become more and more completely 
differentiated, and more and more important in their influence 
upon industry and civilization. While it is evidently impossible 
within our available space to describe all phases of this varied 
development, we shall attempt to enumerate some of those which 
are most general in their character and most far-reaching in their 
consequences. A relatively complete and highly instructive 
review of the whole subject may be found in Merz’s History 
of European Thought in the Nineteenth Century. 
At the beginning of this century mathematics was in a stage of 
triumphant expansion, in which the related sciences of astronomy 
and mechanics participated. General physics and chemistry 
were still in the preliminary stage of collecting and coordinating 
data, with attempts at quantitative interpretation, while in their 
train the natural sciences were following somewhat haltingly. 
The most notable advance in physical science during the century 
is the gradual working out of the great fundamental principle of 
the conservation of energy, affecting profoundly the whole range 
of phenomena. Of equal — or even greater — importance is 
the gradual realization of progressive development — evolution — 
not only in plant and animal life but even in the inorganic world. 
Physics is gradually enriched by experimental researches and 
by the working out of mathematical theories of heat, light, 
magnetism and electricity. Chemistry, largely hitherto a collec- 
tion of unrelated facts, becomes more and more coordinated with 350 
A SHORT HISTORY OF SCIENCE 
physics and mathematics by means of the spectroscope, the 
principle of the conservation of energy, the atomic theory, the 
kinetic theory of gases, and the study of molecular structure. 
On the other hand, its relations with the organic world are 
made more clear through the investigation of the compounds of 
carbon. 
All other sciences, pure and applied, as well as the industries, 
profit unexpectedly and almost inconceivably by these nine- 
teenth century advances in physics and chemistry. The older 
observational and mathematical astronomy achieves a marvellous 
triumph in the discovery of a new planet Neptune, as related in 
Chapter XV, and even this is soon rivalled by the startling 
achievements of the new physical and chemical astronomy. 
Reserving for the following chapter a sketch of the develop- 
ment of the natural sciences under the ultimately dominant in- 
fluence of the theory of evolution, we proceed to outline briefly 
some of the more notable advances in the physical sciences. 
Modern Physics. — Some of the main features in the develop- 
ment of physics in the nineteenth century have been :—the working 
out of consistent theories of light and radiant heat as wave phe- 
nomena of a peculiar hypothetical medium called the “ether”; 
the extensive investigation of electrical and magnetic phenomena 
and the development of an electromagnetic theory even so far as 
to include optics; the working out of a kinetic theory of gases 
with important relations to chemical as well as physical theory; 
the elaboration of general theories of matter, force, and energy, 
all culminating in the crowning discovery of the great unifying 
principle of the Conservation of Energy. 
Heat, Thermometry : Carnot, Rumford. — The invention 
of the thermometer has been traced in Chapter XII. To the 
nineteenth century belongs the determination of an absolute 
scale 1 as distinguished from the arbitrary one previously employed. 
The idea that heat is not a substance but a mode of molecular 
motion arose in the seventeenth and eighteenth centuries, but was 
1 The absolute scale is based on the indirect determination of a temperature 
(— 273° Centigrade = — 459° Fahrenheit) at which the internal activity which 
constitutes heat is supposed to cease. PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
351 
first given a substantial experimental basis by the researches of 
Benjamin Thompson, Count Rumford (1753-1814), who showed 
that by friction of two bodies an unlimited amount of heat could be 
generated. His results were reported to the Royal Society in 1798. 
Rumford made a cylinder of gun-metal rotate in a box contain- 
ing water, and by the friction of a revolving borer driven by horse- 
power the water was heated to boiling in two and a half hours. 
Deeply impressed he exclaims: 
What is heat ? Is there any such thing as an igneous fluid ? . . . 
Anything which any insulated body, or system of bodies, can continue 
to furnish without limitation, cannot possibly be a material substance; 
and it appears to me to be extremely difficult, if not quite impossible, 
to form any distinct idea of anything, capable of being excited, and 
communicated, in the manner the heat was excited and communicated 
in these experiments, except it be motion. 
The “mechanical equivalent of heat”—i.e. the work required 
to heat one pound of water one degree — was roughly calculated. 
Epoch-making in the theory of heat were the researches of 
Sadi Carnot (1796-1832), whose words follow: 
Wherever there is a difference of temperature followed by return to 
equilibrium the generation of power may take place. Water vapor is 
one means, but not the only one. ... A solid body, for example a 
metal bar, gains and loses in length when it is alternately heated and 
cooled, and thus is able to move bodies fastened to its ends. . . . 
The whole process he pictures as a cycle in which a certain 
portion of the heat applied is converted into work, a certain other 
portion being lost. Thus the new science of thermodynamics was 
born. The thorough and complete investigation of the “me- 
chanical equivalent of heat” belongs to J. P. Joule (1818-1889) 
of Manchester, England, a pupil of Dalton the chemist. 
Light; Wave Theory, Velocity: Young, Fresnel.— As 
stated in Chapter XIV Huygens had supported a wave theory of 
light, while Newton accepted an emission theory. That sound 
was propagated by atmospheric waves was well known. There was 
a troublesome contrast however in the phenomenon of shadows. 352 
A SHORT HISTORY OF SCIENCE 
How could wave-propagation be reconciled with sharply defined 
shadows? Why should not light “go round a corner” as well as 
sound ? These difficulties were met by Thomas Young who re- 
vived Huygens’ wave theory, which was definitively established 
by Fresnel’s researches on refraction, beginning in 1815. 
The determination of the velocity of light by observations of 
the moons of Jupiter has been mentioned already (page 286). 
About 1850 this problem was solved by a new method devised by 
Fizeau. A ray of light passes between the teeth of a wheel to a 
mirror and back again. During the time required by the ray to 
pass thus out and back, the gap through which it has passed may 
have been just replaced by a tooth, in which case the light will be 
intercepted. By measuring the speed of the wheel when varied in 
a definite way the speed of the light ray may be determined. The 
result agreed with that obtained by the astronomical method within 
about .5 %. At almost the same time Foucault, by an ingenious 
laboratory device, proved that light travels more slowly in water 
than in air — a result incompatible with the emission theory. 
The sporadic beginnings of a genuine kinetic view of natural 
phenomena, after having been cultivated ... by Huygens and Euler, 
and early in the nineteenth century by Rumford and Young, were 
united into a consistent physical theory by Fresnel, who has been 
termed the Newton of optics, and who consistently, and all but com- 
pletely, worked out one great example of this kind of reasoning. He 
has the glory of having not only established the undulatory theory of 
light on a firm foundation, but still more of having impressed natural 
philosophers with the importance of studying the laws of regular 
vibratory motion and the phenomena of periodicity in the most general 
manner. 
In astronomy and optics the suggestion of common sense, which 
regards the earth as stationary and light as an emission travelling in 
straight lines, had indeed allowed a certain amount of definite know- 
ledge ... to be accumulated. A real physical theory, however, was 
impossible until the notions suggested by common sense were com- 
pletely reversed, and an ideal construction put in the place of a seem- 
ingly obvious theory. This was done in astronomy at one stroke by 
Copernicus; in optics only gradually, tentatively, and hesitatingly. PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
353 
Newton himself had pronounced the pure emission theory to be 
insufficient — and only a preliminary formulation. 
Young boldly generalized the undulatory theory by maintaining that 
“ a luminiferous ether pervades the universe, rare and elastic in a high 
degree,” that the sensation of different colors depends on the different 
frequency of vibration excited by light in the retina. . . . 
In January, 1817, long before Fresnel had made up his mind to adopt 
a similar conclusion . . . Young announced in a letter . . . that in 
the assumption of transverse vibrations, after the manner of the vibra- 
tions of a stretched string, lay the possibility of explaining polariza- 
tion. ... — Merz. 
The Spectroscope and Spectrum Analysis. — For closer 
study of the spectrum of Newton and the “dark lines” observed by 
Fraunhofer in 1815 (and in 1802 by Wollaston) in the spectrum, 
Kirchhoff and Bunsen in 1859-60 perfected the “spectroscope.” 
This is essentially no more than a telescope so attached to the 
prism producing the spectrum from a slit as to facilitate minute 
scrutiny of the latter. It was by these workers and at this time 
that spectrum analysis became firmly established as a means of 
detecting the chemical constituents of celestial bodies. Kirch- 
hoff wrote in 1859: — 
I conclude that colored flames in the spectra of which bright lines 
present themselves, so weaken the rays of the color of these lines, 
when such rays pour through them, that in place of the bright lines, 
dark ones appear as soon as there is brought behind the flame a source 
of light of sufficient intensity in which these lines are otherwise want- 
ing, thus originating two great applications of his principle — the 
search, through the study of the spectra of distant stellar sources of 
light, after the ingredients which are present in those distant lumi- 
naries, and the search, through the study of the flames of terrestrial 
substances, for new spectral lines announcing yet undiscovered ele- 
ments. 
In 1862, only three years after Kirchhoff and Bunsen’s application of 
the spectroscope to the study of the sun, Huggins measured the posi- 
tion of the lines in the spectra of about forty stars, with a small slit 
spectroscope attached to an 8-inch telescope. In 1876 he successfully 
applied photography to a study of the ultra-violet region of stellar 354 
A SHORT HISTORY OF SCIENCE 
spectra, and in 1879 published his paper On the Photographic Spectra 
of Stars. The results were arranged and discussed with reference 
to their bearing on stellar evolution. — Hale. 
The first application of the spectroscope to the corona of the 
sun was made in 1868 by Janssen and Lockyer, independently, re- 
vealing the chemical composition of the solar prominences as 
chiefly hydrogen, calcium, and helium. 
Electricity and Magnetism: Faraday, Green, Ampere, 
Maxwell.—Seebeck (1770-1831) in his work On the Magnetism 
of the Galvanic Circuit published a first account of the magnetic 
field illustrated by magnetized iron filings and later so fruitfully 
investigated by Faraday. The sciences of electricity and mag- 
netism had originated in the latter part of the eighteenth century 
with Coulomb’s use of the torsion-balance, by means of which he 
made accurate comparison between the attractive or repulsive 
forces exercised by electrified and magnetized bodies, and the 
mechanical forces required to twist wires. Thus he found the first 
definite units, a process carried much farther by Gauss and Weber. 
Ampere (1775-1836) stimulated by Oersted’s discovery of the 
effect of the electric current on magnets, published in 1820 a 
fundamental discussion of electrodynamics and soon after enun- 
ciated his celebrated law: — 
Two parallel and like directed currents attract each other, while 
two parallel currents of opposite directions repel each other. 
He also succeeded in expressing the quantitative relations in- 
volved by a mathematical formula. 
Faraday, one of the most distinguished investigators in the 
whole history of physical science, rescued electricity from the 
mysterious notion of currents acting on each other through empty 
space, by the fruitful conception of a magnetic field, of which a 
new and comprehensive mathematical theory was gradually worked 
out by Maxwell. Faraday’s discoveries were so far-reaching that 
they have even been coupled with the law of the conservation of 
energy and Darwin’s theory of descent as the greatest scientific 
ideas of the latter half of the century. He observes: — PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
355 
Atoms and lines of force have become a practical — shall I say a 
popular ? — reality, whereas they were once only the convenient 
method of a single original mind for gathering together and unifying in 
thought a bewildering mass of observed phenomena, or at most capable 
of being utilized for a mathematical description and calculation of 
actual effects. 
Yet Helmholtz says of Faraday: 
It is indeed remarkable in the highest degree to observe how, by a 
kind of intuition, without using a single formula, he found out a 
number of comprehensive theorems, which can only be strictly proved 
by the highest powers of mathematical analysis. ... I know how 
often I found myself despairingly staring at his descriptions of lines of 
force, their number and tension, or looking for the meaning of sen- 
tences in which the galvanic current is defined as an axis of force. . . . 
Faraday apprehended the principle of the conservation of 
energy even before it had come to clear expression as common 
property, saying, for example, in refuting the theory that elec- 
tricity could be generated by metallic contact alone: — 
But in no case, not even in those of [electric fishes], is there a 
pure creation or a production of power without a corresponding ex- 
haustion of something to supply it. 
Like Young, Dalton, and Joule, Faraday did not belong to the 
orthodox Cambridge school then dominant in English mathe- 
matical and physical science, and recognition of the significance 
of his ideas was consequently retarded. 
What the atomic theory has done for chemistry, Faraday’s lines of 
force are now doing for electrical and magnetic phenomena. . . . Yet 
the circumstances under which Faraday’s work was done were those 
of penury. 
Electromagnetic Theory of Light.—In 1845 Faraday writes: 
I . . . . have at last succeeded in magnetising and electrifying a 
ray of light, and in illuminating a magnetic line of force. . . . Em- 
ploying a ray of light, we can tell, by the eye, the direction of the 
magnetic lines through a body: and by the alteration of the ray and 
its optical effect on the eye, can see the course of the lines just as 
we can see the course of a thread of glass. 356 
A SHORT HISTORY OF SCIENCE 
I have deduced the relation between the statical and dynamical 
measures of electricity, and have shown by a comparison of the 
electro-magnetic experiments of Kohlrausch and Weber with the veloc- 
ity of light as found by Fizeau, that the elasticity of the magnetic 
medium in air is the same as that of the luminiferous medium, if these 
two coexistent, coextensive and equally elastic media are not rather 
one medium. . . . We can scarcely avoid the inference that light 
consists in the transverse undulations of the same medium which is 
the cause of electric and magnetic phenomena. — Maxwell. 
We must not listen to any suggestion that we may look upon the 
luminiferous ether as an ideal way of putting the thing. A real matter 
between us and the remoter stars I believe there is, and that light 
consists of real motions of that matter, motions just such as are 
described by Fresnel and Young, motions in the way of transverse 
vibrations. 
— Kelvin, Baltimore lecture. 
Hertz, a pupil of Helmholtz, first proved in 1887 the existence 
of those undulations which now bear his name, showing also 
that these travel with the rapidity of light, and that they are, 
like light and heat waves, susceptible of reflection, refraction, 
and polarization, and until he measured their length and velocity, 
no great progress was made in verifying those relations experi- 
mentally. Such more recent applications of Hertz’s ideas as 
radio-telegraphy and radio-telephony testify to their immense 
practical as well as theoretical importance. 
With the establishment of the electromagnetic theory of light, 
what we may call the undulatory series became complete. Sound 
had long been known to be due to waves or “undulations” and 
the wave theory of heat and of light was accepted, so that it had 
only remained to prove the existence of electrical and magnetic 
undulations, and to show that such waves moved with the 
velocity and other characteristics of light. This it was which 
was done mathematically by Maxwell and experimentally by Hertz. 
The velocity of propagation of an electro-magnetic disturbance 
in air . . . does not differ more from the velocity of light in air . . 
than the several calculated values of these quantities differ among 
each other. — Maxwell. PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
357 
Kinetic Theory of Gases. Clausius. The modern theory of 
gases was born . . . when Joule in 1857 actually calculated the 
velocity with which a particle of hydrogen . . . must be moving, 
assuming that the atmospheric pressure is equilibrated by the 
rectilinear motion and impact of the supposed particles of the gas on 
each other and the walls of the containing vessel. This meant taking 
the atomic view of matter in real earnest, not merely symbolically, 
as chemists had done. — Merz. 
The theory owes its full development, however, to the re- 
searches of Maxwell, Clausius and Boltzmann. 
The great turning-point, indeed, lay in the kinetic theory of gases, 
which . . . had introduced quite novel considerations by showing 
how the dead pressure of gases and vapors could be explained on the 
hypothesis of a very rapid but disorderly translational movement 
of the smallest particles in every possible direction. 
The Conception of Energy. — Newton’s Principia contains 
by implication the modern notion of energy — but the first 
clear and consistent fixing of the modern terminology is found 
in Poncelet’s Mecanique industrielle, 1829. The idea of work was 
thus developed from the standpoint of the engineer — notably 
under the influence of Rankine; while on the other hand, it is a 
not less remarkable fact that Black, Young, Mayer and Helmholtz 
all came to their scientific work through another form of applied 
science — medicine. 
A considerable step toward the general idea of the conservation 
of energy was taken by Rumford in his determination of the 
mechanical equivalent of heat, but the final achievement is due 
mainly to Joule in England and Mayer and Helmholtz in Germany. 
In 1847 Helmholtz read before the Physical Society of Berlin one 
of the most remarkable papers of the century {Die Erhaltung 
der Kraft), in which he says with full justice: — 
I think in the foregoing I have proved that the above mentioned 
law does not go against any hitherto known facts of natural science, 
but is supported by a large number of them in a striking manner. J 
have tried to enumerate as completely as possible what consequences 
result from the combination of other known laws of nature, and how 358 
A SHORT HISTORY OF SCIENCE 
they require to be confirmed by further experiments. The aim of this 
investigation, and what must excuse me likewise for its hypothetical 
sections, was to explain to natural philosophers the theoretical and 
practical importance of the law, the complete verification of which 
may well be looked upon as one of the main problems of physical 
science in the near future. 
Fifteen years later Helmholtz spoke of the principle as follows : — 
The last decades of scientific development have led us to the recogni- 
tion of a new universal law of all natural phenomena, which, from its 
extraordinarily extended range, and from the connection which it 
constitutes between natural phenomena of all kinds, even of the 
remotest times and the most distant places, is especially fitted to give 
us an idea of what I have described as the character of the natural 
sciences, which I have chosen as the subject of this lecture. This law 
is the Law of the Conservation of Force, a term the meaning of which 
I must first explain. It is not absolutely new; for individual domains 
of natural phenomena it was enunciated by Newton and Daniel 
Bernoulli; and Rumford and Humphry Davy have recognised dis- 
tinct features of its presence in the laws of heat. The possibility 
that it was of universal application was first stated by Mayer in 1842, 
while almost simultaneously with, and independently of him, Joule 
made a series of important and difficult experiments on the relation 
of heat to mechanical force, which supplied the chief points in which 
the comparison of the new theory with experience was still wanting. 
The law in question asserts, that the quantity of force which can he 
brought into action in the whole of Nature is unchangeable, and can neither 
be increased nor diminished. 
This doctrine is now so fundamental and so familiar as to require 
no further comment. The indestructibility of matter had already 
become axiomatic. Henceforth, energy also was to be considered 
constant and indestructible. 
Dissipation of Energy. — It remained for William Thomson 
(Lord Kelvin), applying the principle of the conservation of energy 
to the thermodynamic laws of Carnot, to deduce the other great 
principle of the Dissipation of Energy, which recognizes that 
while total energy is constant, useful energy is diminishing by the 
continual degeneration of other forms into non-useful or dis- PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
359 
sipated heat. All workers in this field, from Carnot to Thom- 
son, had appreciated the impossibility of “perpetual motion.” 
Helmholtz expresses his appreciation of Thomson’s contri- 
bution to the theory in a striking passage: 
We must admire the acumen of Thomson, who could read between 
the letters of a mathematical equation, for some time known, which 
spoke only of heat, volume and pressure of bodies, conclusions which 
threaten the universe, though indeed only in infinite time, with eternal 
death. 
Thomson, more than any other thinker, put the problem into 
common-sense language. . . . He saw at once, when adopting Joule’s 
doctrine of the convertibility of heat and mechanical work, that, if 
all processes in the world be reduced to those of a perfect mechanism, 
they will have this property of a perfect machine, namely, that it 
can work backward as well as forward. It is against all reason and 
common sense to carry out this idea in its integrity and completeness. 
If then, the motion of every particle of matter in the universe were 
precisely reversed at any instant, the course of nature would be simply 
reversed forever after. The bursting bubble of foam at the foot of a 
waterfall would reunite and descend into the water; the thermal 
motions would reconcentrate their energy and throw the mass up the 
fall in drops, re-forming into a close column of ascending water. Heat 
which had been generated by the friction of solids and dissipated by 
conduction and radiation with absorption, would come again to the 
place of contact and throw the moving body back against the force 
to which it had previously yielded. Boulders would recover from the 
mud the materials required to rebuild them into their previous jagged 
forms, and would become re-united to the mountain-peak from which 
they had formerly broken away. And also, if the materialistic hypoth- 
esis of life were true, living creatures would grow backwards with 
conscious knowledge of the future, but with no memory of the past, 
and would become again unborn. But the real phenomena of life 
infinitely transcend human science; and speculation regarding con- 
sequences of their imagined reversal is utterly unprofitable. Far 
otherwise, however, is it in respect to the reversal of the motions 
of matter uninfluenced by life, a very elementary consideration of 
which leads to the full explanation of the theory of dissipation of 
energy. 360 
A SHORT HISTORY OF SCIENCE 
Modern Chemistry. — Main features in nineteenth century 
chemistry are: — the discovery of the fundamental quantitative 
relations of chemical reactions; the development of a consistent 
and definite theory of atoms, molecules and valence; the 
synthesis of organic substances; the discovery of periodic 
relations and characteristics; the development of ideas of chemical 
structure; the development of electro-chemistry; the foundation 
of physical chemistry. 
With Lavoisier, “the father of modern chemistry,” the science, 
heretofore descriptive and empirical, had become quantitative 
and productive, seeking like the older sciences of astronomy and 
physics to make itself mathematical — an exact science. Postu- 
lating the existence of indestructible elementary substances, 
Lavoisier controlled and interpreted chemical reactions by careful 
weighing. Until he entered the field there was no generalization 
wide enough to entitle chemistry to be called a science. 
Chemical Laboratories : Liebig. — In the nineteenth 
century chemical studies received a powerful impetus through the 
establishment of teaching laboratories at the universities — in 
which Liebig at Giessen in 1826 was a pioneer. He writes: 
At Giessen all were concentrated in the work, and this was a passion- 
ate enjoyment. . . . The necessity of an institute where the pupil 
could instruct himself in the chemical art, . . . was then in the air, 
and so it came about that on the opening of my laboratory . . . pupils 
came to me from all sides. ... I saw very soon that all progress in 
organic chemistry depended on its simplification. . . . The first 
years of my residence at Giessen were almost exclusively devoted to 
the improvement of organic analysis, and with the first successes 
there began at the small university an activity such as the world had 
not yet seen. ... A kindly fate had brought together in Giessen the 
most talented youths from all countries of Europe. . . . Every one 
was obliged to find his own way for himself. ... We worked from 
dawn to the fall of night. 
To investigate the essence of a natural phenomenon, three 
conditions are necessary. We must first study and know the 
phenomenon itself, from all sides; we must then determine in what 
relation it stands to other natural phenomena; and lastly, when we PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
361 
have ascertained all these relations, we have to solve the problem of 
measuring these relations and the laws of mutual dependence — that 
is, of expressing them in numbers. In the first period of chemistry, 
all the powers of men’s minds were devoted to acquiring a knowledge 
of the properties of bodies. . . . This is the alchemistical period. 
The second period embraces the determination of the mutual relations 
or connections of these properties; this is the period of phlogistic 
chemistry. In the third ... we ascertain by weight and measure 
and express in numbers the degree in which the properties of bodies 
are mutually dependent. The inductive sciences begin with the 
substance itself, then come just ideas, and lastly, mathematics are 
called in, and, with the aid of numbers, complete the work. 
Quantitative Relations : Atoms ; Molecules ; Valence. — 
It took . . . nearly a century . . . before the rule of definite pro- 
portions was generally established, becoming a guide for chemical 
analysis. . . . 
The vaguer terms of chemical affinity and elective attraction, of 
chemical action, of adhesion and elasticity . . . gradually dis- 
appeared, when by the aid of the chemical balance each simple sub- 
stance and each definite compound began to be characterized and 
labelled with a fixed number. — Merz. 
Proust, analyzing various metallic oxides and sulphides, obtained 
constant percentage results, from which however no obvious infer- 
ences could be drawn by him. Dalton (1766-1844) had the happy 
inspiration to interpret these figures in relation to weights of the 
combined oxygen, making the lightest element, hydrogen, the unit or 
measure of his system. His hypothesis that elements combine in 
weights proportional to small whole numbers — the “law of mul- 
tiple proportions,” has since been verified by innumerable analyses. 
It has recently been shown that Dalton was in the habit of 
regarding all physical phenomena as the result of the interaction 
of small particles. He was thus naturally led to the conception 
of definite atomic weights to be determined by experiment. In 
the words of Dalton : 
We can as well undertake to incorporate a new planet in the solar 
system or to annihilate one there as to create or destroy an atom of 362 
A SHORT HISTORY OF SCIENCE 
hydrogen. All the changes we can effect consist in the separation of 
atoms bound together before and in the union of those previously 
separated. 
The atomic theory while highly serviceable has always been 
subjected to severe criticism. In 1840, for example, Dumas 
declared that it did not deserve the confidence placed in it, and that 
if he could he would banish the word “ atom,” convinced that 
science should confine itself to what could be known by experience. 
As late as 1852 Frankland says: 
I had not proceeded far, in the investigation of the organo-metallic 
compounds before the facts brought to light began to impress upon me 
the existence of a fixity in the maximum combining value or capacity 
of saturation in the metallic elements which had not before been sus- 
pected. ... It was evident that the atoms of zinc, tin, arsenic . . . 
had only room, ... for the attachment of a fixed and definite number 
of the atoms of other elements. 
Independent researches have, in combination with the older chemi- 
cal theories, introduced so much definiteness into this line of thought 
that ‘ the Newtonian theory of gravitation is not surer to us now than 
is the atomic or molecular theory in chemistry and physics — so far, 
at all events, as its assertion of heterogeneousness in the minute 
structure of matter, apparently homogeneous to our senses, and to our 
most delicate direct instrumental tests.’ — Kelvin, 1886. 
The three main criticisms of the atomic theory are: — 
(1) that it is based on inference, not on direct observation; 
and is therefore only a provisional hypothesis; (2) that it takes 
no account of chemical forces — ‘‘affinity”; (3) that it over- 
emphasizes analysis. 
The idea of “atomicity” and “valency” . . . was not possible 
without the clear notion of the “molecule” as distinct from the 
“atom.” This idea had lain dormant in the now celebrated but 
long forgotten law of Avogadro, which was established in 1811 
almost immediately after the appearance of Dalton’s atomic theory. 
It had been known since . . . Boyle and Mariotte that equal 
volumes of different gases under equal pressure change their volumes 
equally if the pressure is varied equally, and it was also known . . . PHYSICAL SCIENCE IN THE NINETEENTH. CENTURY 
363 
that equal volumes of different gases under equal pressure change their 
volumes equally with equal rise of temperature. These facts sug- 
gested to Avogadro, and almost simultaneously to Ampere, the very 
simple assumption that this is owing to the fact that equal volumes of 
different gases contain an equal number of the smallest independent 
particles of matter. This is Avogadro’s celebrated hypothesis. It 
was the first step in the direct physical verification of the atomic view 
of matter. — Merz. 
Synthesis of Organic Substances. — Until the middle of the 
nineteenth century there was an apparently fundamental separa- 
tion between organic and inorganic nature. Since then they have 
been brought together by the general laws of energy and to some 
extent by the principles of evolution, as will appear in the following 
chapter. In 1828 Wohler (of Gottingen) had indeed succeeded 
in preparing urea out of inorganic materials, a discovery which 
disproved such difference as was hitherto considered to exist 
between organic and inorganic bodies. 
A Periodic Law among the Elements. — With gradually in- 
creasing knowledge of the fundamental constants of chemistry 
— the atomic weights — attempts were naturally made to connect 
these with the chemical and physical properties of the correspond- 
ing elements: valence, affinity, specific gravity, specific heat, etc. 
In 1869-71 Mendelejeff, a Russian chemist, succeeded in establish- 
ing remarkable relations between these data, and on tabulating 
them enunciated his Periodic Law, which has resulted in the dis- 
covery of several new and hitherto unsuspected elements. As the 
existence of the planet Neptune (page 341) had been predicted to 
fill an apparent gap in a system, so Mendelejeff under the periodic 
law was able to predict the existence of other and missing ele- 
ments in the series of chemical elements. And just as the pre- 
diction of Adams and Leverrier was fulfilled by the actual discovery 
of Neptune, so the prophecy of Mendelejeff was justified by the 
discovery of gallium in 1871, scandium in 1879, and germanium 
in 1886. Furthermore, the periodic law enabled Mendelejeff 
to question the correctness of certain accepted atomic weights, and 
here, also, he was justified by subsequent redeterminations. 364 
A SHORT HISTORY OF SCIENCE 
It may be questioned whether the celebrated periodic law of New- 
lands, Lothar Meyer and Mendelejeff, which has brought some order 
into the atomic and other numbers referring to the different elements, 
and has even made it possible to predict the existence of unknown ele- 
ments with definite properties, stands really in a firmer position than 
the once well-known but now forgotten law of Bode, according to 
which the gap in the series which gives the distances of the planets 
from the sun indicated the existence of a planet between Mars and 
Jupiter. 
Chemical Structure. — Crystallography — a science of the 
nineteenth century — established an important connection be- 
tween chemistry and geometry. Haiiy made mineralogy “as 
precise and methodical as astronomy. . . . He was to Werner and 
Rome de l’lsle, his predecessors, what Newton had been to Kepler 
and Copernicus.” 
In the early years of the atomic theory Wollaston had predicted 
that philosophers would seek a geometrical conception of the 
distribution of the elementary particles in space — a prophecy 
first practically fulfilled by Van’t Hoff’s Chemistry in Space (1875). 
The chemical character is dependent primarily upon the arrange- 
ment and number of the atoms, and in a lesser degree upon their 
chemical nature (V. Meyer). The atomic view first became a scientific 
instrument, when arithmetical relations of a definite and unalterable 
kind were proved to exist; it became a yet more useful instrument, 
when to the arithmetical there were added geometrical conceptions. 
— Merz. 
Physical Chemistry: Electrolytic and Thermodynamic 
Developments of Chemistry. — In the latter part of the nine- 
teenth century much light was thrown on a wide range of physical 
and chemical phenomena by the study of solutions and their 
electrolytic behavior. Much had already been accomplished by 
Davy in the decomposition of substances by the electric current, 
leading for example to the first isolation of the elements, sodium 
and potassium. Faraday showed that for a given substance the 
amount decomposed is dependent solely on the quantity of 
electricity passed through and that for different substances the PHYSICAL SCIENCE IN THE NINETEENTH CENTURY 
365 
amounts set free at the electrodes are proportional to their 
chemical equivalents. To him the name electrolysis is due. A 
closer study of the phenomena of electrolysis led Clausius to the 
hypothesis that the molecules of salts, acids, and bases, pre- 
viously regarded as disintegrated only by the passage of the 
electric current, are already dissociated in ordinary solutions. 
To these electrically charged part-molecules Faraday gave the 
name ions. Arrhenius proved that salts in dilute solution are 
dissociated into their ions almost completely, instead of only 
very slightly as Clausius supposed. This theory of Arrhenius, 
known as the Theory of Electrolytic Dissociation, of which an 
account would be too technical for the present purpose, co- 
ordinates and correlates heterogeneous masses of chemical facts, 
which apparently bore little or no relation to one another, and 
refers them to a common cause. 
During the latter part of the nineteenth century a study of the 
rate and equilibrium conditions of chemical reactions led by de- 
grees to the formulation of the so-called law of mass action and to 
many important thermodynamic relations. Chemistry thus came 
to share with physics the possibility of utilizing the calculus, 
becoming thereby more fully a quantitative science. 
REFERENCES FOR READING 
Mekz, J. T. History of European Thought in the Nineteenth Century. 
Poincare, H. Science and Hypothesis. 
Ramsay, William. The Gases of the Atmosphere and the History of Their 
Discovery. 
Roscoe, H. E. John Dalton. 
Soddy, F. Matter and Energy. 
Thompson, S. P. Michael Faraday: His Life and Work. 
Tilden, W. A. Progress of Scientific Chemistry in Our Own Time. 
Tyndall, John. Faraday as a Discoverer. CHAPTER XVII 
SOME ADVANCES IN NATURAL SCIENCE IN THE NINE- 
TEENTH CENTURY. COSMOGONY AND EVOLUTION 
What the classical renaissance was to men of the fifteenth and 
sixteenth centuries, the scientific movement is to us. It has given 
a new trend to education. It has changed the outlook of the mind. 
It has given a new intellectual background to life. — Sadler. 
The rapid increase of natural knowledge, which is the chief char- 
acteristic of our age, is effected in various ways. The main army of 
science moves to the conquest of new worlds slowly and surely, nor 
ever cedes an inch of the territory gained. But the advance is covered 
and facilitated by the ceaseless activity of clouds of light troops 
provided with a weapon — always efficient, if not always an arm of 
precision — the scientific imagination. It is the business of these 
enfants perdus of science to make raids into the realm of ignorance 
wherever they see, or think they see, a chance; and cheerfully to 
accept defeat, or it may be annihilation, as the reward of error. 
Unfortunately the public, which watches the progress of the cam- 
paign, too often mistakes a dashing incursion . . . for a forward 
movement of the main body; fondly imagining that the strategic 
movement to the rear, which occasionally follows, indicates a battle 
lost by science. — Huxley. 
Influence of Eighteenth Century Revolutions. — If the 
French Revolution had done no more than to upset as it did the 
social equilibrium of the centuries, its effect in stimulating inquiry 
and generating doubt in almost every direction could not have 
failed to further scientific studies and promote wholesome investi- 
gation into the fundamental relations of man and nature. But 
even before that revolution, some of the ablest minds in France, 
keenly alive to the teachings of Descartes and Newton and the 
lessons of seventeenth century science, had rejected the cur- 
rent cosmogony of Moses, although they had nothing with which 
to replace it. In particular, the eighteenth century questioned 
all custom and authority, and the theory of special creation 
possessed no other basis. 
366 NATURAL SCIENCE IN THE NINETEENTH CENTURY 
367 
The American Revolution was likewise an uprising against 
long established custom and authority, and accordingly contrib- 
uted to the doubts and questionings of the time, while the 
Industrial Revolution, by fundamental and world-wide changes, 
such as the introduction of machinery and the factory system, 
and by its tendency to concentrate and urbanize populations 
previously rural and segregated, facilitated intellectual contact, 
promoted discussion, and aroused and excited inquiry and in- 
vestigation. 
The Scientific Revolution. — The most brilliant single 
achievement of nineteenth century science was the detection by 
Adams and Leverrier of the presence in our solar system of Nep- 
tune, a new and hitherto unknown planet. But the most revolu- 
tionary achievement, and probably the most far-reaching, was the 
assembling and formulation of convincing evidence in favor of 
organic evolution, i.e. of the gradual development, rather than 
the sudden creation, of living things. It is difficult to-day to 
realize the commotion into which the intellectual world was 
thrown at the middle of the nineteenth century when a new and 
promising solution of the long-standing problem of the origin of 
the different kinds (species) of plants and animals by means of 
natural rather than supernatural law, was propounded by Darwin 
and Wallace. And while the last half of the eighteenth cen- 
tury was the period of great political and social revolutions, — 
the French, the American, and the Industrial, — the last half 
of the nineteenth century experienced, in its acceptance of a new 
cosmogony, a fourth, even more profound and momentous, the 
Scientific Revolution. The discovery of Neptune was a triumph 
of mathematics and astronomy, the establishment of the theory 
of organic evolution, a triumph of biology. The discovery of 
Adams and Leverrier was immediately accepted and everywhere 
applauded, but the ideas broached by Darwin and his collabo- 
rators encountered widespread and powerful opposition, and were 
accepted only tardily and reluctantly. 
Influence of the Rapid Increase of Knowledge. — The 
invention of printing, the discovery of the new world, and the works 368 
A SHORT HISTORY OF SCIENCE 
of such intellectual giants as Galileo, Kepler, and Newton, followed 
as these were by the rapid increase of knowledge, both of nature 
and of man, in the seventeenth and eighteenth centuries, had 
placed within the reach of all a vast amount of new facts touching 
the familiar heavens and the familiar earth. Moreover, these 
facts were mostly capable of some sort of rational interpretation, 
i.e. of assignment to a place in some category of facts or phenomena 
already understood and regarded as natural rather than super- 
natural. In short, in the eighteenth and nineteenth centuries 
the stock of human knowledge had been not only rapidly and im- 
mensely enlarged and enriched, but at the same time more or 
less successfully correlated with knowledge previously possessed 
and valued. Some of this new knowledge, moreover, was so 
different from the old as to seem like a fresh revelation. 
Gradual Appreciation of the Permanence and Scope of 
Natural Law. — While it had been easy hitherto to assume the 
occasional suspension of, or interference with, the ordinary course 
of events by supernatural or other unseen or unknown influences, 
it gradually became clear that no such suspension or interference 
could happen without upsetting what seemed to be the natural 
and orderly sequence of events, — what we now call “ the order 
of nature.” Hence doubt arose in many minds whether such 
suspensions or interferences do in fact occur, and whether fixed 
and changeless law is not a fundamental phenomenon of nature. 
The vastness and variety also of the heavens, no less than the 
order conspicuous in a mighty system so nicely balanced and so 
perfectly correlated as must be the cosmos explored and described 
by Copernicus, Galileo, Kepler, Newton and their successors, 
gradually dawned upon the human intellect, and profoundly 
impressed mankind. 
Moreover, if the thoughtful turned from the contemplation 
of the macrocosm — the heavens — and the revelations of the 
telescope, to the microcosm — man, — the labors of Vesalius and 
the Italian anatomists, and of Harvey and the microscopists, 
served to show that here also law and order and a kind of mechan- 
ical regularity and perfection held sway, while plants and the lower NATURAL SCIENCE IN THE NINETEENTH CENTURY 
369 
animals had long been observed to be strictly obedient to and 
dependent upon the laws of nature in respect to climate, season, 
food, reproduction, etc. 
Natural Theology and an Age of Reason. — At the end of 
the seventeenth century John Ray, an English zoologist, drew 
attention to the remarkable adaptations everywhere discover- 
able in nature and especially in plants and animals, and suggested 
that these adaptations were sufficient to prove the existence of 
“design” in the universe, — a powerful argument in favor of the 
Mosaic cosmogony. The same idea was urged more at length by 
others in the eighteenth century as an offset to the growing scep- 
ticism of the age, and especially by Butler in his great work on 
the Analogy of Religion, Natural and Revealed, to the Course and 
Constitution of Nature (1736), and by Paley in his famous Nat- 
ural Theology (1802). More popular and more radical influences 
were simultaneously at work in the opposite direction, as for 
example, Paine’s Rights of Man and Age of Reason, while 
Gibbon with prodigious learning, and Hume with searching philo- 
sophical criticism, added to the increasing dissatisfaction of the 
thoughtful with the current cosmogony, — a dissatisfaction 
which had been rapidly growing under the doctrine of the neces- 
sity of doubt emphasized by Descartes. 
Between 1842 and 1846 appeared a revolutionary work entitled 
Vestiges of Creation, by an anonymous author, which aroused in- 
tense interest in scientific circles and a storm of criticism from those 
who held to the old cosmogony. It is now known to have been 
written by Robert Chambers, an Edinburgh publisher who pre- 
ferred to remain unknown from fear of injuring his partners by 
bringing down upon them the wrath of critics for his heterodoxy. 
Chambers was an amateur geologist and in his “ Vestiges ” under- 
takes to treat the genesis of the earth on more rational and more 
natural principles than was possible by following the orthodox 
theory of special creation. 
The publication of Darwin’s Origin of Species in 1859 re- 
sulted, after a period of earnest and sometimes acrimonious 
discussion, in the establishment of what is now known as the 370 
A SHORT HISTORY OF SCIENCE 
theory or doctrine of organic evolution, and in the displacement of 
the prevailing theory of special creation, — a forward step which 
removed the principal stumbling-block in the way of acceptance 
of the theory of general evolution, inorganic as well as organic, 
telluric as well as stellar. In other words, Darwin’s work at one 
blow cleared the way for a new cosmogony. 
Natural Philosophy and Natural History. Differen- 
tiation and Hybridizing of the Sciences. — Mathematics, 
always recognized as a principal branch of the tree of learning, 
at the beginning of the nineteenth century still held its place as the 
mother of the sciences. Astronomy, also, often called the queen 
of the sciences, still occupied its ancient and honorable position, 
having by this time lost all traces of discreditable affiliation with 
astrology. With the physical and natural sciences, as we now 
know them, the case was different. These still existed in a com- 
paratively amorphous and largely undifferentiated condition as 
“natural philosophy” and “natural history,” — the former the 
lineal descendant of the Ionian nature philosophy, now promoted to 
a high place in public esteem by the work of Newton, whose great 
Princi'pia were philosopkiae naturalis. Gradually, as time went 
on, chemistry was more and more differentiated from natural 
philosophy, until about 1875 it became customary to drop the 
term natural philosophy, using instead two terms, chemistry and 
physics. By the end of the century the present practice was fully 
established. 
The primitive condition of the natural history sciences at the 
beginning of the century may be inferred from the remark of an 
eminent geologist (Geikie) that at that time geology and biology 
were not yet inductive sciences. By 1880, however, natural 
history had budded off geology, botany, zoology, and physiology 
as independent sciences, and the parent term, though still em- 
ployed, was rapidly falling into disuse, having become much too 
broad for any single science. 
Meantime, hybridizing as well as differentiation has become 
common, e.g. of physics with chemistry (physical chemistry), 
and of mathematics with physics (mathematical physics) and with NATURAL SCIENCE IN THE NINETEENTH CENTURY 
371 
many other sciences. We now have also fertile hybrids between 
chemistry and biology (physiological chemistry, bio-chemistry, 
chemical biology, etc.), and between physics and geology (geo- 
physics), between electricity and chemistry (electro-chemistry), 
between physics and mineralogy, etc. Hybridizing of this kind 
is one of the most characteristic as well as one of the most fruit- 
ful phenomena of the nineteenth century. 
Botany, zoology and geology, daughters of natural history, 
were the children of its old age, for the term “ natural history ” is 
as ancient as Aristotle, while geology was not fully born until the 
publication of Lyell’s Principles in 1830, and biology not until 
the era of the great generalists of the Victorian Age — Darwin, 
Spencer, Huxley, etc. — soon after the middle of what one of 
them, Wallace, well qualified by his own great work to speak 
with authority, has called “the wonderful century.” Botany 
and zoology as such arose about the beginning of the century. 
Geology, dealing with the natural history of the earth and its 
lifeless contents, and biology, dealing with the world of life, have 
both now very numerous progeny, e.g. from geology, stratigraphy, 
mineralogy, petrography, petrology, palaeontology, etc., and from 
biology, zoology and botany, with their numerous subdivisions, 
— morphology, physiology, cytology, anthropology, bacteriology, 
parasitology, etc. Here also highly prolific crossing has oc- 
curred both among members of each minor group and also be- 
tween members of the two major groups, natural philosophy and 
natural history; as, for example, in palaeontology, which may 
be said to be half geology and half biology, in physiological 
optics, in bio-metrics, etc. To the very beginning of the century 
belongs the first appearance of the term “ biology,” introduced 
by Treviranus (1776-1837) a German naturalist and professor 
of mathematics in Bremen who in 1802-1805 published a work 
entitled Biologie, oder Philosophic der lebenden Natur. 
The greatest achievement of natural history, not only of the 
nineteenth century but of all time, was the bringing about of the 
general acceptance of a new cosmogony — the theory of evolution 
— on the presentation by the naturalist Darwin of convincing 372 
A SHORT HISTORY OF SCIENCE 
evidence in favor of organic evolution together with a plausible 
explanation of the mechanism (natural selection) of its operation. 
To this we shall return. In a century so rich and so varied in 
its achievements in natural science as was the nineteenth, we 
can obviously only touch — and that very briefly — upon a few 
of the more important and fundamental. 
Of the remarkable progress of all the sciences during the 
Victorian era, Huxley has given the best brief and general ac- 
count in his essay entitled Advance of Science in the Last Half 
Century (1887), prepared in celebration of the first fifty years 
of the reign of Queen Victoria. 
Progress in Zoology. — The work of Buffon and Linnaeus 
in the field of biology and of Werner and Hutton and Smith in 
that of geology has been referred to in Chapter XIV. Of these 
only Werner (d. 1817) and Smith (d. 1839) lived on into the nine- 
teenth century. 
The return of the astronomers and the geologists to ancient 
ideas of gradual development or, as this is now called, evolution, 
for the lifeless earth, was foreshadowed for the living world with 
Bonnet (1720-1793) who in 1764, in his Contemplation de la nature, 
advanced the theory that living things form a gradual and natural 
“scale” (ladder), rising from lowest to highest without any break 
in continuity. Buffon, in his great work on natural history, which 
was published between 1749-1804 in 44 quarto volumes, had 
dealt with the animal world very much as Linnaeus had dealt 
with plants, Buffon excelling in description, Linnaeus both in 
description and in classification and holding firmly to the idea of 
the fixity as well as the definite demarcation of species. 
Meantime, epoch-making work in zoology was being done by 
three investigators — Lamarck, Cuvier, and St. Hilaire — at the 
Museum of Natural History in Paris. In 1778 Lamarck (1744- 
1829) published a small book on botany. In 1801 appeared 
his great work On the Organization of Living Bodies, which is 
now a landmark in the history of biology and of the doctrine of 
organic evolution. In this work and in his Philosophic zoologique, 
Lamarck boldly proposes to substitute for special creation — NATURAL SCIENCE IN THE NINETEENTH CENTURY 
373 
the current theory of cosmogony — the idea of gradual develop- 
ment or evolution, an ancient idea thenceforward made the 
keynote of his speculations. Systematic zoology and compara- 
tive anatomy, the latter already well begun by Hunter in the * 
eighteenth century, were immensely advanced by Cuvier (1769- 
1832), who however clung tenaciously to the theory of special 
creation; while Geoffroy St. Hilaire (1772-1844) — also a com- 
parative anatomist, but one whose interests lay rather in the 
functional than in the anatomical resemblances of the parts of 
animals, and who is therefore regarded as “the father of homol- 
ogy”— on the whole opposed Cuvier’s and favored Lamarck’s 
ideas. His Philosophic anatomique appeared in 1818-1822. 
Nature, said St. Hilaire, has formed all living beings on one plan, 
essentially the same in principle, but varied in a thousand ways in all 
the minor parts; all the differences are only a complication and modi- 
fication of the same organs. 
This similarity of structure, or homology as it is called, which runs 
through all animals, was thus first clearly stated by St. Hilaire, and 
has now been carefully worked out and confirmed. . . . Yet Cuvier 
opposed it to the last, for his mind was full of another idea which is 
equally true; namely, how perfectly each part of an animal is made 
to fit all the other parts; and it seemed to him impossible that this 
could be, unless each part was created expressly for the work it had 
to do. 
The discussion between the two friends became so animated that 
all Europe was excited by it. It is said that Goethe, then an old 
man of eighty-one, meeting a friend, exclaimed, ‘Well, what do you 
think of this great event ? the volcano has burst forth, all is in flames. ’ 
His friend thought he spoke of the French Revolution of July, 1830, 
which had just occurred, and he answered accordingly. ‘You do not 
understand me,’ said Goethe, ‘ I speak of the discussion between Cuvier 
and St. Hilaire: the matter is of the highest importance. The 
method of looking at nature which St. Hilaire has introduced can 
now never be lost sight of.’ — Arabella Buckley Fisher. 
The history of zoology in the first half of the nineteenth century 
is chiefly that of the work of Cuvier, St. Hilaire, Lamarck, Agassiz 
and their disciples. 374 
A SHORT HISTORY OF SCIENCE 
Progress in Botany. — The Linnaean system of classification 
for the higher plants was a purely empirical system, based largely 
upon the number of stamens and not involving any ideas of rela- 
tionship through descent. B. de Jussieu (1699-1767), a friend and 
pupil of Linnaeus, had proposed a different system of classifica- 
tion in which weight is given to the totality of resemblances of 
whatever kind, and this, which almost inevitably led to the close 
association of related forms, was an important step toward a 
natural classification, i.e. one based avowedly upon relationship, 
common ancestry, or descent. De Jussieu’s nephew Antoine de 
Jussieu continued and extended his uncle’s work. A. de Candolle 
(1778-1841) later adopted and extended de Jussieu’s system 
which, with his own, now forms the basis of our present natural 
system. It is noteworthy that this change of opinion in regard 
to the relationship of the species of plants was ultimately effected 
without theological protest.1 
The discovery by Goethe of the homologies of the parts, and 
by Linnaeus of the organs of sex, of the flower, were important 
steps toward the modern theory of the evolution of plant life. 
Progress in Microscopy. The Achromatic Objective. — 
Compound microscopes, i.e. microscopes consisting of two lenses, 
an objective and an eyepiece, were probably invented at about 
the same time as telescopes, — which likewise consist of two lenses 
or systems of lenses. But because of their imperfections, in 
respect especially to spherical and chromatic aberration, such 
microscopes were often inferior for use to the best simple micro- 
scopes. It was not until about 1835 that the cdmpound micro- 
scope, though invented in the seventeenth century, became the 
superior instrument that it is to-day, through the accumulated 
improvements of a number of workers, — especially Amici, Lilly, 
Lister, and Chevalier — resulting in the achromatic objective, 
free from both spherical and chromatic aberration. 
Almost immediately thereafter, with the new microscopy, 
began a rich harvest of discoveries, in what Pasteur has called the 
1 For an account of Linnaeus’ attitude to the doctrine of the fixity of species see 
A. D. White, Warfare of Science, Vol. I, pp. 47, 59, 60. NATURAL SCIENCE IN THE NINETEENTH CENTURY 
375 
world of the infinitely little, similar to that reaped after the intro- 
duction of the telescope by Galileo, and his exploration of the 
stars, in the world of the infinitely great. The cell theory of 
Schleiden and Schwann appeared in 1839 and yeast was redis- 
covered (see p. 378) in 1837. The first contagious disease (Mus- 
cardine) traced to a fungus parasite was worked out by Bassi 
in 1837, the first contagious disease (Favus) of man due to a 
fungus, by Schoenlein in 1839. Protoplasm was first described in 
1846. Ehrenberg, in 1838, made numerous and important studies 
on microscopic plants and animals. 
Embryology. — If in 1828 one sharp boundary which had 
always been supposed to stand between the organic and the in- 
organic world was broken down by Wohler’s discovery that urea, 
a substance hitherto exclusively of animal origin, could be ob- 
tained in the laboratory by heating an inorganic substance, 
ammonium cyanate, another well-defined boundary believed to 
exist between the higher and the lower animals had been broken 
down a year earlier, when a Russian zoologist, Karl Ernst von 
Baer (1792-1876), announced that mammals, including man 
himself, reproduce by eggs, precisely as do the lower animals. 
In 1828 von Baer published our first important work on compara- 
tive embryology, — of which science he thus became the founder. 
The discovery by von Baer of the human ovum overthrew 
completely the “animalculists” who for centuries had contended 
that within the earliest embryo of man the future offspring existed 
completely formed, but only in miniature. This theory, because 
it assumed for development a mere unfolding, was known as em- 
bryologic evolution. Harvey, on the other hand, had propounded 
a theory of epigenesis, i.e. development comprising growth and 
differentiation out of an originally minute, simple, and undiffer- 
entiated body. This “body” — the human ovum — was now 
described by von Baer as -gwo' inch in diameter and nowise different 
in appearance from other animal eggs in their earliest stages. 
Comparative anatomy had already shown that Linnaeus was 
right in placing man among the animals, and now embryology con- 
firmed and strengthened this view of man’s place in nature. 376 
A SHORT HISTORY OF SCIENCE 
Progress in Physiology. Johannes Muller. Claude 
Bernard. — To the work upon physiology of Harvey in the 
seventeenth century and of Haller and Bichat and others in the 
eighteenth was now added that of Johannes Muller (1801-1858) 
whose “Elements of Physiology,” appearing between 1837 and 
1840, put the whole subject on a fresh and thoroughly scientific 
basis. Muller has been called the founder of modern physiology. 
Fortunate in his pupils — DuBois Reymond, Helmholtz, Ludwig, 
Volkmann, and Vierordt — these were no less fortunate in their 
master, for Muller was a great teacher, and for the rest of the 
century the teachings of Johannes Muller and his disciples fur- 
nished a powerful stimulus and a safe guide to physiological 
research, especially in Germany. 
In France, also, physiology won renown and recognition through 
the researches of Claude Bernard (1813-1878), a pupil of 
Magendie, whose assistant he became in 1841 and whom he suc- 
ceeded as assistant professor in 1847 and as professor in 1855. 
Bernard was the first occupant of the newly established chair 
of physiology at the Sorbonne. The laboratory was attached to 
his professorship until 1864. On his death in 1878 he was accorded 
by the State the honor of a public funeral, — the first ever be- 
stowed by France upon a man of science and only 84 years after 
the public guillotining of Lavoisier. By his discovery of the 
significance of the pancreatic secretion and especially of the glyco- 
genic function of the liver Bernard opened up the vast field of 
“internal secretions,” the study of which has yielded, and is still 
yielding, some of the most fruitful results of physiological research 
hitherto obtained. Before Bernard, each organ seemed to have 
one function and only one, but since his time this simple, mechan- 
ical concept has given way to a realization of correlations and 
complexities within the animal mechanism such as had not then 
been dreamt of. 
A great step forward in this dark field was taken in 1843 by the 
French physiologist, Claude Bernard, a man whose name should 
be remembered for his striking discoveries, ingenious and skillful 
experiments, his clear thoughts, lofty imagination and the beautiful, NATURAL SCIENCE IN THE NINETEENTH CENTURY 
377 
simple and luminous style in which his books and papers were written. 
— Mathews. 
In England and America the newer physiology made but scant 
progress until Foster published (in 1876) in England an epoch- 
making treatise embodying in a fascinating form the methods 
and results of continental physiology. 
Pathology before Pasteur. — Before the nineteenth cen- 
tury disease was regarded as an inscrutable mystery. Epidemics, 
plagues, and pestilences came and went, without apparent rea- 
son. The most fatal and therefore most famous of these was the 
Black Death of the fourteenth century. Others had been the 
Plague of Athens, the Sweating Sickness, the Dancing Mania, and 
Leprosy. One of the worst and commonest was Scurvy, which 
attacked chiefly sailors, soldiers, prisoners and the poor. 
Attempts to explain disease were manifold. Primitive man 
naturally attributed it to the power of evil spirits (daemons or 
devils) and sought prevention and cure in exorcism and the casting 
out of devils. Hippocrates looked for the sources of disease in 
abnormal mixtures of four great juices or “humors” of the body, 
— blood, yellow bile, phlegm, and black bile; and his theory had 
the merit of being based upon natural rather than supernatural 
ideas, for which reason probably it survived until the time of 
Sydenham in the seventeenth century. But the theory of Hip- 
pocrates failed to account for epidemics, for which the cause had 
to be sought in meteorological disturbances, such as storms, or in 
astronomical phenomena, such as comets or eclipses, or in unusual 
terrestrial happenings, such as earthquakes, volcanic eruptions, 
the flight of birds, the appearance of insects, vermin and what not. 
With the increase of knowledge in the sixteenth and seventeenth 
centuries ideas of this primitive kind no longer sufficed, and 
Sydenham urged that disease must have an independent material 
basis, a materies morbi. Not much progress was made, how- 
ever, even by Sydenham, and the eighteenth century left behind 
it no important contributions to the theory of disease, — the work 
of Hahnemann, for example, bearing upon therapeutics rather 
than pathology. 378 
A SHORT HISTORY OF SCIENCE 
The nineteenth century began as a period of agnosticism in 
pathology. The older theories were discredited, but beyond a 
general belief in the material basis of the agencies of disease 
almost nothing was known. Scurvy, indeed, had been shown 
to be due to lack of certain kinds of food, and smallpox had been 
proved to be preventable both by inoculation and vaccination. 
Boyle, in the seventeenth century, had ventured the guess that 
diseases might be “fermentations,” but as fermentations were 
not yet understood the suggestion had little value. Light finally 
came from two sources, viz., from parasitology and from zymology, 
— the science of fermentations. It had long been recognized that 
the mistletoe was a parasite causing serious disease in trees, 
and that tapeworms might cause disease in animals which they 
infested. It was not, however, until the microscope came into 
use that the itch, long known as a contagious disease, was found 
to be due to a parasitic insect, while flies, fleas, bedbugs and lice 
were still thought to be annoying rather than dangerous. The 
discovery in 1837 that “honeycomb” of the scalp (Favus), an 
infectious disease in which yellow crusts appear on the head, is 
due to a vegetable parasite related to the moulds was a surprise, 
as was the demonstration in 1839 that an infectious disease 
(Muscardine) of silkworms is likewise due to a mould. 
The microscope also served to reveal what have been called 
“ the footprints of disease ” within the cells and tissues, making 
possible the work in cellular pathology by Virchow. 
The Germ Theory of Fermentation, Putrefaction and 
Disease. Pasteur. — At about this time the achromatic com- 
pound microscope was coming into use, and by its aid the alco- 
holic fermentation, hitherto regarded as a purely chemical phe- 
nomenon, was found to be intimately associated with, if not 
actually caused by, a living, growing microorganism, yeast, ob- 
served and figured by Leeuwenhoek in 1680, but in the early 
nineteenth century regarded rather as potent organic matter in 
some peculiar catalytic state or condition (Liebig) than as a 
living thing. Between the rediscovery of yeast in 1837 and 
Pasteur’s epoch-making studies upon it in 1859, fermentation was NATURAL SCIENCE IN THE NINETEENTH CENTURY 
379 
studied by several workers of eminence — among whom was 
Helmholtz — but it was chiefly Pasteur who in a memorable series 
of researches finally proved that yeast is the one and only cause 
of the alcoholic fermentation, — a biological or “germ” theory 
of fermentation, thus displacing Liebig’s chemical or catalytic 
theory, — the germ in this case being yeast. By the use of the 
microscope, combined with new and ingenious methods of culti- 
vation of yeast and other microbes, Pasteur, between 1859 
and 1865, proved beyond doubt that yeast is the agent of the 
alcoholic fermentation and that other microbes are the agents of 
other familiar fermentations, such as the butyric and acetic. His 
work was marked by remarkable precision and refinement. 
From a germ theory of normal fermentations, putrefactions, 
and decay it was a short step to a germ theory of undesirable or 
abnormal fermentations, such as often occur in brewing and wine 
making. In these last, the microscope revealed to Pasteur the 
presence of strange forms foreign to the ordinary fermentation, 
and evidently wild yeast, moulds, or other extraneous microbes 
which by interfering with or supplanting the normal forms, pro- 
duced disagreeable, or abnormal, i.e. “diseased”, beer or wine. 
Similarly, it was only a second step from the diseases of wine 
and beer to those of animals and man. A disastrous epidemic 
disease affecting silkworms in the south of France at this time 
brought to Pasteur an urgent request that he should make an 
investigation. “But,” said Pasteur, “I have never handled a 
silkworm.” “So much the better,” said Dumas, the chemist, who 
insisted that he should thus patriotically enter the field of animal 
pathology. Pasteur yielded and spent three years in studies upon 
the silkworm disease, with results invaluable to science and espe- 
cially to pathology. 
Antiseptic and Aseptic Surgery. Lister. — Meantime 
Lister, an English surgeon resident in Edinburgh, led on by 
Pasteur’s researches, introduced a new and scientific treatment of 
open wounds, based on the germ theory. Open wounds whether 
made by accident or in surgery ordinarily suppurate, i.e. become 
red, swollen and inflamed and eventually produce pus. Lister 380 
A SHORT HISTORY OF SCIENCE 
surmised that this suppuration is due to germs from the air, from 
the surgeon’s hands, from instruments, etc., and acting on this 
theory proposed to prevent such wound diseases by destroying the 
germs in the air and upon the wound by some “ antiseptic,” i.e. some 
substance that should prevent sepsis (putrefaction) or suppuration. 
Carbolic acid (phenol) had recently been introduced into com- 
merce and was highly recommended as a deodorant. This Lister 
used, and with results so satisfactory that his antiseptic surgery 
soon became famous. It has since given way to aseptic surgery, 
which differs from it simply in preventing wound-infection rather 
than intreating it after it has occurred. In battle-fields antiseptic 
surgery must still be used, since the work of the surgeon is done 
only after the wound has been made. Antiseptic and aseptic 
surgery are among the most priceless blessings of the race and 
among the greatest triumphs of nineteenth century science. 
One serious objection stood in the way of the establishment and 
acceptance of the germ theory; viz., the possibility that germs 
were the consequence and not the cause of fermentation, putre- 
faction, or disease; and this objection was frequently urged. In 
1876, however, it was met and overcome by Robert Koch, a 
district physician of Wollstein, in Prussia. By the use of the 
methods of Pasteur, Koch succeeded in making a series of suc- 
cessive cultivations of the microbes of anthrax (splenic fever, 
charbon, or malignant pustule) in such a way that at the end of 
his experiment he had a pure culture of the microbes in question. 
Obviously, if with these he could 'produce the disease by infecting 
a susceptible animal or a wound, they must be its cause and not 
its consequence. In this he was completely successful, thereby 
establishing beyond all possible perad venture the truth of the 
germ theory. 
Rise of Bacteriology and Parasitology. — The labors of 
Pasteur, Lister, Koch and others soon led to the birth of a new 
science, — Bacteriology — of which the first fruit was the dis- 
covery in rapid succession by Koch and his pupils of the hitherto 
unknown germs of some of the worst and most mysterious dis- 
eases afflicting the human race; the bacillus of typhoid fever in NATURAL SCIENCE IN THE NINETEENTH CENTURY 
381 
1879, — more fully worked out in 1884; the bacillus of tuberculosis 
in 1882; the vibrio of Asiatic cholera in 1883; the bacilli of lock- 
jaw and of diphtheria in 1884; the bacillus of bubonic plague in 
1894; and about the same time by others of the microorganisms 
of malaria, sleeping sickness, and several other diseases. 
In some cases such as smallpox and yellow fever no germs have 
yet been observed; but this seems at present to be because they 
are too small to be seen with the microscope or to be held back, 
as most germs are, by pipe-clay filters. If, nevertheless, we review 
the list just given of those plagues in which the causative micro- 
organism was detected and cultivated between 1879 and 1889, we 
cannot avoid the conclusion that the ninth decade of the nine- 
teenth century was the most important hitherto in the history of 
pathology. When we go further, and compare these rich dis- 
coveries and the fruit they have since borne, in preventive medi- 
cine, preventive sanitation and preventive hygiene, with our 
previous ignorance of the nature of disease and of its control, we 
realize that since that decade the world has possessed not only a 
new pathology but also a new science. 
Besides bacteriology another science, parasitology, has also 
become prominent since the decade of the great pathological 
discoveries. The parasitic character of the mistletoe, the tape- 
worm, the flea, the louse, the mosquito and other visible pests 
was long ago evident, but it was only after the discoveries of Pas- 
teur and Koch and their disciples were fully comprehended that 
the germ theory of disease was seen to be at bottom a theory of 
parasitism. Thereupon 'parasitology assumed a new place and a 
new significance as a branch of pathology. 
Biogenesis versus Spontaneous Generation. — The ques- 
tion of the origin and beginnings of life on the earth has always 
been obscure and perplexing to mankind, and up to the middle of 
the nineteenth century the account attributed to Moses, — the 
so-called theory of special creation, — was still predominant 
though about to give way to the theory of evolution. A similar 
obscurity veiled the beginnings of individual life. Omne vivum 
ex ovo (all life from the egg) was the motto of those who thought 382 
A SHORT HISTORY OF SCIENCE 
only of the higher animals. Omne vivum ex vivo was that of those 
who held that living things come only from antecedent life, even 
if not from eggs. Both of these groups were biogenesists, since 
they maintained that living things come only from other living 
things. Opposed to them were the abiogenesists who disputed 
these ideas and believed in “spontaneous” generation (abiogenesis), 
i.e. in the origin of living things from lifeless or non-living matter. 
The dispute was very old, Aristotle, for example, having 
favored the idea of spontaneous generation. In the eighteenth 
century Spallanzani for biogenesis and Needham for abiogenesis 
had fought over again the ancient battle. Lamarck, at the be- 
ginning of the nineteenth century, looked with favor upon abio- 
genesis. The improved microscope of that century seemed at 
first to strengthen the evidence for spontaneous generation by 
revealing almost everywhere the presence of microbic life, and 
the idea of an apparently easy generation of new life was wel- 
comed by some interested in evolution, as accounting naturally 
rather than supernaturally for the origin of life in general. Mean- 
time, the discovery of the mammalian ovum by von Baer in 1827 
had somewhat improved the position of the biogenesists, but the 
whole question remained open and unsettled at the middle of 
the century and until it was attacked by Pasteur, who was thor- 
oughly equipped with the most exact scientific methods of the 
day. For the details of the struggle in which Pasteur battled for 
biogenesis we must refer the reader to the Life of Pasteur, by 
Radot, and to Tyndall’s Floating Matter of the Air. The up- 
shot was that all the evidence advanced by the advocates of 
spontaneous generation was shown by Pasteur, — ably seconded 
by Tyndall, — to be due to defective technique; for when such 
defects were corrected no evidence remained of the generation 
of life by lifeless matter. Thus was triumphantly closed, at least 
for the time, one of the most ancient of controversies. 
Obviously, the question of a possible spontaneous generation 
of living matter from lifeless under special conditions such as 
may have existed during the early history of our globe remains 
open. All that modern science has done is to controvert such NATURAL SCIENCE IN THE NINETEENTH CENTURY 
383 
evidence as has been advanced of its ordinary and frequent 
occurrence under such natural conditions as prevail to-day. 
Progress of Geological Science. — In 1785 Hutton, to 
whom we have already briefly referred, presented to the Royal 
Society of Edinburgh a paper entitled Theory of the Earth, or 
an Investigation of the Laws Observable in the Composition, 
Dissolution and Restoration of Land upon the Globe (p. 317). 
In this remarkable work the doctrine is expounded that geology 
is not cosmogony, but must confine itself to the study of the materials 
of the earth; that everywhere evidence may be seen that the present 
rocks of the earth’s surface have been formed out of the waste of 
older rocks . . . that every portion of the upraised land is subject 
to decay; and that this decay must tend to advance until the whole 
of the land has been worn away. ... In some of these broad general- 
izations Hutton was anticipated by the Italian geologists; but to 
him belongs the credit of having first perceived their mutual relations 
and combined them in a luminous coherent theory everywhere based 
upon observation. ... It is by his Theory of the Earth that Hutton 
will be remembered with reverence while geology continues to be 
cultivated. — Geikie. 
In the early part of the nineteenth century it was, nevertheless, 
firmly held that the earth had undergone various “revolutions,” 
“catastrophes” and the like which, taken together with the Flood 
of Noah, were sufficient to explain its present surface features, such 
as mountains, valleys, plains, boulders, caves, deserts, sea-coasts, etc. 
These views were summed up in the term Catastrophism, i.e. that 
at a number of successive epochs — of which the age of Noah was 
the latest — great revolutions had taken place on the earth’s surface; 
that during each of these cataclysms all living things were destroyed; 
and that, after an interval, the world was restocked with fresh assem- 
blages of plants and animals, to be destroyed in turn and entombed 
in the strata at the next revolution. — Judd. 
Moreover, at the beginning of the century most considerations 
of the earth and of the living world were dominated by two pre- 
conceived ideas: first, that the universe, including the earth and 
its belongings, had originated as described in the first chapter of 384 
A SHORT HISTORY OF SCIENCE 
Genesis, and second, that the present features of the earth are to 
be explained chiefly by the more recent Flood of Noah described 
in the seventh chapter. 
Before geology had attained to the position of an inductive science, 
it was customary to begin all investigations into the history of the 
earth by propounding or adopting some more or less fanciful hypothe- 
sis in explanation of the origin of our planet, or even of the uni- 
verse. . . . To the illustrious James Hutton (1785) geologists are 
indebted for strenuously upholding the doctrine that it is no part of 
the province of geology to discuss the origin of things. He taught 
them that in the materials from which geological evidence is to be 
compiled there can be found £ no traces of a beginning, no prospect of 
an end.’ — Geikie. 
The vast deposits of sand, gravel and clay, with the embedded 
remains of contemporaneous animal and vegetable life with which 
they (glacial torrents) everywhere covered the plains, were viewed 
till recently solely in relation to the Mosaic narrative of a universal 
deluge, and were referred implicitly to that source. — Wilson. 
As late as 1823 Buckland, a distinguished English geologist, 
published a work on extinct animals from a Yorkshire cavern 
entitled Reliquce Diluviance. As for plants and animals, the almost 
universal opinion was that these, like the earth, had been specially 
created and had remained ever since substantially unchanged. 
As for the crust of the earth, composed, as this is often seen to 
be in section, of unlike layers, it was a long time before the current 
idea of sudden creation could be replaced by one so different as 
that of slow and steady deposition. This change, however, was 
finally though only gradually effected, largely through actual 
observation and measurement of the slow deposits of rivers, and 
other geological phenomena of to-day, combined with Lyell’s 
thesis that those of the past were essentially similar. The time 
element, in brief, began to be recognized as a new and an important 
factor in the making of the earth’s crust. 
Reference has been made above to Lyell’s revolutionary trea- 
tise, The Principles of Geology, published in 1830. This great 
work which adopted, emphasized, and extended the works of NATURAL SCIENCE IN THE NINETEENTH CENTURY 
385 
Hutton and Smith, eventually overthrew catastrophism and 
established uniformitarianism in its place. (See Appendix H.) 
Glaciers and Glacial Theories. — The occurrence on the 
earth’s surface, and even on mountain tops, of boulders evidently 
of distant rather than local origin, had long been a puzzle even to 
geologists. The primitive hypothesis of their deposit during the 
Flood — the so-called diluvial theory — no longer sufficed to 
satisfy inquiring minds, and equally inadequate was the idea of 
von Buch that boulders had been thrown up like cannon shot by 
volcanoes and had fallen where found. In 1837 Louis Agassiz 
(1807-1873) advanced the present doctrine: viz. that boulders 
liave been deposited after the melting of masses of ice by which 
they were slowly brought from a distance. Agassiz supported 
this ice or “glacial” theory by personal observations and studies 
of the glaciers of the Alps, and eventually propounded that gen- 
eral theory of glaciation or ice caps at the earth’s poles which is 
now universally accepted. Few theories have ever proved more 
satisfactory, scientifically speaking, more popular, in the best 
sense, or more productive of simple explanations of widespread 
and diverse phenomena. We have only to set over against the 
glacial the diluvial theory, with its fatal weakness in requiring the 
transportation by water of huge masses of rock over long distances, 
or the projectile theory, with its requirement of showers of flying 
boulders falling almost anywhere, to realize the simplicity and 
adequacy of the glacial theory. The word “ boulder,” nevertheless, 
remains as a reminder of the diluvial theory, since it is derived 
from words signifying “the noise of a stone in a stream.” 
Rise of Palaeontology. — Before the nineteenth century 
precise knowledge of extinct animals and plants was almost wholly 
wanting. Fossils had indeed been observed from the earliest 
times but although occasionally correctly interpreted, as for ex- 
ample by Pythagoras and Xenophanes, Leonardo da Vinci and 
Palissy. Hooke (p. 268) at the end of the seventeenth century first 
made the important suggestion that fossils might serve as indicators 
of phases in the earth’s history and as proof of the existence at one 
time of a tropical climate in England. In the eighteenth century 386 
A SHORT HISTORY OF SCIENCE 
these ideas were developed by Woodward and others, while J. 
Gesner introduced into geology the suggestion of great age for 
the earth by estimating the time required for the elevation of 
certain fossiliferous strata in the Apennines at 80,000 years. 
Buffon in France speculated upon the successive emergence and 
depression of the continents, and Werner in Saxony noted in 
successive formations the gradual approach of extinct forms of 
life towards existing forms. 
In the early part of the nineteenth century palaeontology began 
to take on its modern form. Pallas had, indeed, discovered vast 
deposits of extinct mammoths and rhinoceroses in Siberia in 
1768-1774, — Blumenbach had distinguished between the fossil 
mammoth and the living elephant in 1780, — and in 1793 the 
American mastodon was recognized as different from both fossil 
mammoth and living elephant. In 1793 Lamarck recapitulated 
and emphasized the methods and results of his predecessors and 
sought to account for the phenomena, partly by changes in the 
habits and partly by changes in habitat of extinct forms, modi- 
fications from whatever source being held to be conserved and 
accumulated by inheritance, and in 1800 Cuvier published an 
important paper on fossil and living elephants. Not long after, 
remains of huge extinct reptiles were discovered: of the ichthyo- 
saurus and plesiosaurus in 1821; of the mososaurus in 1822; of 
fossil crocodiles in France in 1831; of the iguanodon in 1848. 
These “finds” opened up a new world of buried ancient life almost 
beneath our feet scarcely inferior in interest to the starry world 
far overhead, which had so long excited the curiosity and won- 
der of mankind. 
In 1854 in the caves of Belgium were found remains of lions and 
other animals (including man) which were obviously unlike the 
same species to-day, and have ever since been spoken of as the 
“cave” lion, the “cave” tiger, etc. With the human remains 
were discovered prehistoric implements testifying both to the 
antiquity of man and to the superiority of the cave men to other 
cave animals. Fossil ferns and other plants were also found, and 
even fossil insects, — the latter often in a remarkably good state NATURAL SCIENCE IN THE NINETEENTH CENTURY 
387 
of preservation. Here, also, recognition of the time element be- 
came inevitable and subversive of the idea of sudden and special 
creation. Hitherto belief in the antiquity of man was excep- 
tional, and proofs of such antiquity were almost wholly wanting. 
This discovery, therefore, was profoundly important and of far- 
reaching significance not only in palaeontology but also in the 
foundation of what is to-day known as anthropology. 
Ancient and Modern Theories of Cosmogony. — When 
the wonder and curiosity of primitive man developed into specu- 
lation touching his origin, or the origin of things about him, or 
the origin of the visible universe, cosmogony began. Until the 
middle of the nineteenth century the Jewish or Mosaic cosmogony 
embodied in the first chapter of the Hebrew Scriptures, and ac- 
counting for the origin of the cosmos, both organic and inorganic, 
by a sudden and special creation, was almost universally accepted 
throughout Christendom. Recent investigations indicate that 
this theory was really pre-Jewish in origin and probably Baby- 
lonian. If so, a cosmogony long antedating Greek theories pre- 
vailed throughout Christian Europe and America to the middle of 
the nineteenth century. In the seventeenth century even Galileo, 
Kepler, and Newton raised no question of its essential validity. 
Of the two great minds of the seventeenth century, Newton and 
Leibnitz, both profoundly religious as well as philosophical, one pro- 
duced the theory of gravitation [and] the other objected to that 
theory as subversive of natural religion. — Asa Gray. 
The eighteenth century was an age of doubt. Descartes, who 
“doubted whatever could be doubted,” had been succeeded by 
Voltaire and the encyclopaedists in France, and by Hume and 
Gibbon in England, and incredulity, not to say scepticism, was in 
the air. Yet no new theory of cosmogony appeared, and merely 
to doubt an old hypothesis is neither to destroy nor to supplant 
it. Observations, ideas, and discoveries, however, had long been 
accumulating and were now multiplying, which were destined to 
undermine the Mosaic theory and establish something very dif- 
ferent, and more resembling Greek cosmogonies, in its place. 388 
A SHORT HISTORY OF SCIENCE 
A complete cosmogony should, in theory at least, attempt to 
account both for the origin of the cosmos and for its present as- 
pects. This the theory of special creation failed to do. It de- 
scribed the origin of the cosmos at a period evidently remote, — 
inasmuch as it was stated elsewhere in the Scriptures that many 
generations had come and gone since the Creation, — but was 
silent as to any essential progress or modifications in the mean- 
time. Hence, for those accepting special creation the inference 
naturally was that no great changes either in the heavens or in 
the earth had, in fact, taken place since the initial act of creation, 
and that the present aspect of the cosmos is substantially its 
primitive aspect. On this theory mankind and other living things 
had not developed, but rather stood still or even — as in the case 
of “the fall of man” — actually retrograded from a more perfect 
type. In complete contrast with this ancient, Oriental theory 
the modern theory of Evolution, making no pretence to solve 
the problem of the origin of the cosmos, attempts only to explain 
some of its present aspects. 
Relationship of the Heavens and the Earth. — It had for 
centuries been taken for granted, as a part of the geocentric 
theory, that the heavens and the earth had little if anything in 
common — the earth being the centre of things and of first impor- 
tance. Copernicus, however, had shown that the earth is inferior, 
and tributary to the sun, while his great successors Galileo, 
Kepler, and Newton had proved both earth and sun to be no more 
than members of a huge system of heavenly bodies strictly corre- 
lated by gravitation. Hence, when Franklin drew down light- 
ning from the terrestrial heavens and the spectroscopists not long 
after proved a substantial chemical identity between the earth 
and celestial bodies, the older cosmogony began to seem both 
primitive and parochial. 
Above all, the ideas of Kant and Laplace, which seemed to 
indicate not merely a structural and material kinship between the 
heavens and the earth, — such as that later revealed by the 
spectroscope, — or a functional similarity, — such as that dis- 
covered by Franklin, — but, more remarkable than either, a NATURAL SCIENCE IN THE NINETEENTH CENTURY 
389 
family relationship through a common ancestry, weighed heavily 
against the theory of special and separate creation by an ex- 
traneous will in remote time, with no provision for change to meet 
changing conditions. 
The Scale of Life and the Phases of Life. — It had often 
been commented upon that in the world of life there is always 
present and requiring explanation what Bonnet called the “ scale 
of life,” i.e. the fact that plants and animals not only differ greatly 
in structure and complexity, but that both may be arranged in a 
kind of ascending or descending natural scale (ladder) with highly 
complex forms at the top and relatively simple ones at the bot- 
tom. On the doctrine of special creation it was difficult to find 
any reason for or advantage in the existence of such a scale, 
while the suggestion was obvious that the higher forms had 
somehow come from or passed through lower forms. A rapid 
modification of living forms, such as this suggestion required, is 
obvious in everyday life being exemplified by the so-called phases of 
life, — in which infancy makes way for youth, youth for maturity, 
and maturity for age. To the attentive observer living matter 
appears to be thus forever changing, and on the whole progressing 
or advancing from simplicity to complexity. 
In this respect the stellar universe at first sight appears very 
different, for here permanence rather than change seems to pre- 
vail. As for the earth, changes, indeed, do often occur and some- 
times progressively, as in erosion, glacial action, and the work of the 
tides, so that the earth seems to stand in this respect somewhere 
between changing and advancing organic life and the unchanging 
heavens. When, therefore, the idea broached by Hutton and 
Smith at the end of the eighteenth century that the surface of the 
earth has been made and is still being made what it is to-day, not 
suddenly and once for all by special creation centuries ago, but 
gradually, and by forces and processes similar to those now acting, 
a new and revolutionary notion of the genesis of the earth’s crust 
arose, and one contrary to the idea of special creation. The same 
idea carried further and developed by Lyell in 1830 became thence- 
forward all important: — 390 
A SHORT HISTORY OF SCIENCE 
Amid all the revolutions of the globe the economy of Nature has 
been uniform, and her laws are the only things that have resisted the 
general movement. The rivers and the rocks, the seas and the con- 
tinents have been changed in all their parts; but the laws which 
direct those changes, and the rules to which they are subject, have 
remained invariably the same. — LyelVs Vol. I, Title Page Motto. 
General Resemblance of Man to the Lower Animals. — 
At the end of the eighteenth century the increase of knowledge 
nowhere led to more startling revelations than in comparative 
anatomy, for a very moderate amount of dissection of the various 
types of vertebrates suffices to show that all of these, — including 
man himself, — are built upon the same general plan. Similarity 
extends even into minute details, as in the lungs, aortic arches, 
teeth, eyes and ears, and the complex musculature of the limbs. 
Similarity in the organs and processes of reproduction among the 
higher animals had been recognized ever since the time of Aris- 
totle, and the discovery of the mammalian ovum by von Baer 
in 1827 simply added another link to the long chain of resemblances 
between man and other higher vertebrates and the lower, — such 
as reptiles, frogs, and fishes. Embryology now strengthened this 
chain by showing that the embryos of these various animals are 
more alike than are the adults into which they develop — thus 
suggesting that all were originally similar or even identical, but 
had afterwards become differentiated. Malthus in his startling 
work on the Principle of Population had proved a tendency in 
mankind to multiply, like other animals, without reason and 
beyond the means of subsistence. The antiquity of man, long 
suspected, was established by the finding, in 1854, by Boucher de 
Perthes, of human remains along with those of extinct animals 
in the caves of northern France. Archaeology and linguistic 
studies had already contributed to disprove the conventional 
chronology (which held that the creation of the world occurred 
4004 b.c.) and thus, indirectly, the current cosmogony, by dis- 
covery of the remains of prehistoric culture, and by showing 
that the modern European languages and arts are evidently direct 
and related descendants of earlier and sometimes extinct forms. NATURAL SCIENCE IN THE NINETEENTH CENTURY 
391 
Anatomical and Microscopical Similarity of Animals 
and Plants. Organs, Tissues, Cells, and Protoplasms. — 
The old cosmology emphasized differences rather than resem- 
blances between animals and plants for, barring the one fact of 
life, there is at first sight little in common between them. It 
was always plain that both are provided with organs, and that in 
respect to functions, such as growth, differentiation, the phases of 
life and reproduction, there is great similarity. The improved 
microscope of 1830-1850 now revealed a further and striking sim- 
ilarity, by showing that the organs in both plants and animals 
are made up of tissues, and these in turn of cells; and when about 
1845 it was found that both plant and animal cells contain a slimy, 
colorless substance, apparently similar, if not actually identical, 
in all living things, it was not long before the same name “proto- 
plasm” (first life) was given to this fundamental substance, whether 
it occurred in plant or in animal cells, — in leaves or in muscles. 
It was of this same fundamental protoplasm, common to both 
plants and animals, that Huxley wrote his famous essay entitled 
The Physical Basis of Life. 
Fundamental Unity of Nature. Organic versus Inorganic 
World. — Between things organic and inorganic until the nine- 
teenth century there was supposed to be a great gap. When, 
therefore, in 1828 Wohler produced in the laboratory urea, a 
typical organic substance hitherto unknown except as an animal 
excretion, by merely heating ammonium cyanate, it became evi- 
dent that in chemistry the term “ organic ” had lost its former 
meaning. Since that time many compounds once believed to be 
capable of production only by living things, have been made 
in the laboratory, with the result that the organic and the inorganic 
worlds have been drawn nearer together. The further fact that 
living matter is composed largely of four common chemical 
elements, carbon, hydrogen, oxygen, and nitrogen, and yields on 
analysis no peculiar or mystical substance or element, tended to 
show that life itself might be merely a property of various chemical 
elements in peculiar combination. 
No less surprising than the revelation of the chemical similarity 392 
A SHORT HISTORY OF SCIENCE 
of living and lifeless matter was that by spectrum analysis of the 
chemical composition of the stars, in which various elements 
common on the earth were readily detected. This astounding 
result, taken together with the clearer and more convincing ideas 
of the conservation of matter and energy, and of the similarity 
in nature of heat, light, and sound as undulations, served to 
demonstrate and to emphasize the scope and immanence of 
natural law, as well as the fine adjustment, balance, and economy 
of nature, and to cause interference or governance by anything 
supernatural to seem gratuitous and unwarranted. 
Treviranus’ Biology and Lamarck’s Zoological Phi- 
losophy. — Very early in the nineteenth century the two works 
here mentioned and already referred to appeared, the former 
introducing into science for the first time the word biology, 
and thereby formally recognizing a world of life in contradistinc- 
tion to a world of lifelessness. Both works virtually ignored the 
old cosmogony as applied to plants and animals, and both sought 
after some new and less supernatural theory. Lamarck in par- 
ticular was bold enough to suggest that the elongated neck of 
the giraffe was not specially created, but had been gradually 
developed by constant effort to obtain food beyond its ordi- 
nary reach. Both authors deserve special mention because with 
them biology began consciously and frankly to part company 
with the older cosmogony. Lamarck, for example, after frankly 
accepting the possibility of spontaneous generation for the origin 
of living matter sought to explain its present variety and differ- 
entiation by four laws, which may be stated as follows: — 
1. Life naturally tends to increase and enlarge up to a certain 
self-determined limit. 
2. New organs arise in response to new and reiterated wants 
and to the changes produced by these wants or by efforts to meet 
them. 
3. The development of organs and their functions is determined by 
the use of such organs. 
4. All changes in organization are conserved by generation and 
transmitted to offspring. NATURAL SCIENCE IN THE NINETEENTH CENTURY 
393 
To these ideas Lamarck clung in spite of criticism, for in the 
Introduction to his Natural History of Invertebrates, a much later 
work than his Zoological Philosophy, he again affirms: — 
I conceive that a gasteropod mollusk [e.g. a snail], which, as it 
crawls along, finds the need of touching the bodies in front of it, 
makes efforts to touch those bodies with some of the foremost parts 
of its head, and sends to these every time quantities of nervous fluids 
as well as other liquids. I conceive, I say, that it must result from this 
reiterated afflux towards the points in question that the nerves which 
abut at these points will by slow degrees be extended. Now, as in 
the same circumstances, other fluids of the animal flow also to the 
same places, and especially nourishing fluids, it must follow that 
two or more tentacles will appear and develop insensibly in those 
circumstances at the points referred to. 
The principal significance of these views to-day is in the attempt 
which they embody to explain on natural principles phenomena 
then explained only as supernatural. 
Voyages and Explorations of Naturalists. — In the nine- 
teenth century voyages, expeditions, and explorations for the first 
time were undertaken for the sole and specific purpose of the 
improvement of natural knowledge (to use a phrase associated with 
the origin of the Royal Society). Of these the first were those of 
Alexander von Humboldt who, beginning in 1799, made numerous 
and extensive journeys and observations by land and sea which 
enabled him many years later to publish his monumental Kosmos, 
a work replete with observations and reflections on natural phi- 
losophy and natural history, which eventually gave him a place, in 
this century, among the learned men of Germany second only to 
that occupied by Goethe. 
In 1801, Robert Brown, a British botanist, worthy of remem- 
brance also in connection with the so-called Brownian movement 
of particles under the microscope, accompanied an expedition 
to Australia and brought back representatives of some 4000 new 
species of plants. Most fruitful of all was Darwin’s famous voy- 
age of the Beagle to the Pacific (1831-1836), while in 1838 
Karl Ernst von Baer, the eminent founder of comparative embry- 394 
A SHORT HISTORY OF SCIENCE 
ology, accompanied an exploring expedition to Nova Zembla; 
Dana sailed to the Pacific in 1838-1842; Huxley in the Rattle- 
snake in 1846-1850; and Alfred Russel Wallace visited the 
Malay Archipelago a little later. 
The result of these scientific explorations was to throw a flood 
of new light upon the infinite wealth, variety, creative resources 
and capacity of this earth and its inhabitants, plants as well as 
animals, and to reveal adaptations of organisms to climate, soil, 
and other environmental conditions countless in number and 
marvellous in character, all of which raised again the ancient and 
vexing questions: How did these adaptations arise ? And what 
was the origin of species ? 
Darwin’s Origin of Species (1859). — For this, the most 
influential scientific work of the nineteenth century, and one of 
the most important ever written, the way had now been prepared 
by the publications of Malthus, Treviranus, Lamarck, Lyell, and 
Chambers. In particular, as Huxley pointed out, Lyell’s work, 
by showing that the earth is very old and has been long ages in the 
making had, since 1830, shaken the confidence of men of science in 
the old cosmogony and so paved the way for Darwin; and when in 
1854 remains of prehistoric man were found with those of extinct 
animals, such as the cave bear and the cave lion, even popular 
confidence in the Mosaic theory began to be undermined. 
Darwin’s great work appeared in 1859 and aroused world-wide 
criticism and controversy. In his autobiography Darwin acknowl- 
edges his constant obligation to Lyell and also to Malthus, 
whose emphasis on the struggle for food revealed to Darwin 
the fuller meaning of what in the Origin he termed the struggle 
for existence, a struggle in which by natural selection there 
must be progressively a survival of the fittest. In the long 
and fierce battle which now broke out between the defenders 
of the old cosmogony and the new, and which at first went 
no further than proposing to account for the origin of species of 
living things by natural selection instead of by special creation, 
Darwin was ably supported by his friends the naturalists, — Hux- 
ley, Hooker, and Lyell in England and Asa Gray in America. NATURAL SCIENCE IN THE NINETEENTH CENTURY 
395 
Darwin’s principle of natural selection is stated in his own 
words as follows: — 
As many more individuals of each species are born than can pos- 
sibly survive; and as, consequently, there is a frequently recurring 
struggle for existence, it follows that any being, if it vary however 
slightly in any manner profitable to itself, under the complex and 
sometimes varying conditions of life, will have a better chance of 
surviving, and thus be naturally selected. From the strong principle 
of inheritance, any selected variety will tend to propagate its new 
and modified form. 
The Descent of Man. — The last refuge of the defenders of 
the old cosmogony was man himself, and not a few, among whom 
was Alfred Russel Wallace, co-discoverer with Darwin of the 
principle of natural selection, refused to align man and especially 
man’s mentality, with lower animals and their inferior mental 
powers. Here at least, they affirmed, we see no evidence of evo- 
lution. But as time has gone on, and studies have been extended 
in craniology and in psychology it has become increasingly evi- 
dent that there is little if any more difference in brain weight and 
mental power between the highest apes and the lowest men, 
than between these last and the highest men. Meantime, the 
doctrine of parsimony, in default of any other than a super- 
natural hypothesis, naturally awarded the field to the Darwinian 
theory. 
Decline of the Theory of Special Creation. — Darwin’s 
theory of the method of origin of the various kinds (species) of 
plants and animals by natural processes under natural law soon 
became known as the theory of organic evolution, and before a 
score of years had passed was almost universally accepted among 
naturalists. 
It is the supreme merit of Darwin to have thus pointed out a 
method by which a process of gradual development, or evolution, 
— already accepted for language, art, music, and (after 1830) for 
the earth’s crust, and many years before urged by Laplace for the 
solar system, — could be made no less applicable to, and accept- 
able for, plants and animals, and especially for man. Until this 396 
A SHORT HISTORY OF SCIENCE 
was done, there could be no place for the new cosmogony in the 
general mind. 
The theory of organic evolution by natural selection as main- 
tained by Darwin was naturally subjected forthwith to the severest 
scrutiny, and some of its details have been successfully and 
destructively criticised. In particular, his explanation of the 
mechanism of heredity appears to be untenable, as does also his 
theory that small but incessant variations are gradually accumu- 
lated into departures from the original, ultimately sufficient to 
amount to new species. The studies of Mendel and of Weis- 
mann upon inheritance and of De Vries and others upon varia- 
tion, have supplemented and to some extent supplanted much of 
Darwin’s work upon those subjects. But apart from these and 
other readjustments of details, the Darwinian theory stands 
secure and at present affords the most reasonable explanation 
hitherto proposed of the origin of man and other animals and of 
plants; an explanation, moreover, in harmony with the general 
law of evolution now accepted for the origin of existing forms of 
language, literature, and art; of chemical compounds; of the 
earth; of the solar system ; and of the stars. The law of evolu- 
tion, long ago foreshadowed by the Greeks, is clearly the law of 
the lifeless world, and there seems to be no reason why it should 
not be equally applicable to the world of life,— including man. 
Influence of an Age of Invention and Industry. — Ac- 
ceptance of the theory of evolution was quickened by an increase 
in popular intelligence and a general openmindedness due in part 
to the broadening of education, in part to the ease of travel, and 
in part to the appearance, one after another, of revolutionary inven- 
tions, such as the steam engine and cotton machinery with the 
factory system at the end of the eighteenth century, and the loco- 
motive, the steamboat, the telegraph, the sewing machine, the 
friction match, and many more, in the first half of the nineteenth 
century. All these marvels had been wrought within the brief 
space of a single century, while improved printing-presses, cheaper 
paper, cheaper newspapers, cheaper books, and cheaper illustra- 
tions, as well as more and better schools and schooling had con- NATURAL SCIENCE IN THE NINETEENTH CENTURY 
397 
tributed immensely to popular education, mental receptivity, 
closer contact, industrial cooperation and general intelligence. 
Science in the Dawn of the Twentieth Century. — At the 
beginning of the nineteenth century, science as such had no exist- 
ence either as a branch of learning or as a special discipline, — 
still less as a preparation for practical life. Mathematics, highly 
esteemed largely because of its ancient origin and associations, 
and natural philosophy, had a limited recognition; but the term 
science meant as yet hardly more than knowledge or learn- 
ing. The eighteenth century had, however, sowed broadcast 
the seeds of science and the nineteenth soon began to reap 
the harvest. Before 1850 scientific schools as distinct from 
others had been founded both within and without the older 
colleges and universities. New associations and academies for 
the advancement or promotion of science soon sprang up; 
science courses appeared in some of the public schools; funds 
for scientific research began to be provided; and thousands 
of eager and enthusiastic students began to prefer science, and 
especially applied science, to the older “classical” learning. 
Meantime, the marvellous achievements of invention and of in- 
dustry had caught and fixed public interest and attention, so 
that by the opening of the twentieth century, no branch of learn- 
ing stood in higher favor than science, either for its own sake or 
as a preparation for useful service in contemporary life. 
The master keys of science, now everywhere employed for 
unlocking the problems of the cosmos, are: first, the 'principles 
of mathematics, which admit mankind into the mysteries of the 
relations of number and space — the abstract skeleton of science, 
— and second, the principles of evolution and of energy, which 
reveal some, at least, of the secrets of form and of function, 
not only of the earth and of plants and animals, but of the 
heavens ; something of the prodigious forces of the universe 
and their orderly behavior ; something of that apparently in- 
finite and eternal energy which, while forever changing, is never 
lost; something, though as yet but little, of the nature and the 
processes of life. 398 
A SHORT HISTORY OF SCIENCE 
References for Reading 
Agassiz, L. Life of, by E. C. Agassiz. 
Chambers. Vestiges of Creation. 
Clerke. Modern Cosmogonies. 
Darwin. Origin of Species. Descent of Man. Voyage of the Beagle. 
Darwin, F. Life of Charles Darwin. 
De Vries. Studies in the Theory of Descent. 
Foster. Lectures on the History of Physiology. 
Hale. Stellar Evolution. 
Humboldt, von. Kosmos. 
Huxley, L. Life and Letters of T. H. Huxley. 
H,uxley, T. H. Man’s Place in Nature. Essays. Advance of Science in 
the Last Half Century. 
Judd. The Coming of Evolution. 
Lamarck. Zoological Philosophy. 
Locy. Biology and Its Makers. 
Lyell. Principles of Geology. Antiquity of Man. 
Osborn. From the Greeks to Darwin. 
Radot. Life of Pasteur. 
Scott. The Theory of Evolution. 
Thomson. Heredity and Evolution. 
Tylor. Anthropology. Primitive Culture. 
Wallace. Darwinism. 
Weismann. Essays. Evolution Theory. APPENDICES 
A. THE OATH OF HIPPOCRATES (About 400 b.c.) 
[This, as Gomperz observes, is a monument in the history of civilization. It 
is no less a monument in the history of science, since it proceeds from the Father 
of Medicine more than two thousand years ago and, if we except mathematics, 
medicine is the oldest of the sciences. There are many translations of the 
“Oath,” some more literal and some, like the following, more free.] 
I swear by Apollo the physician [Healer], and /Esculapius, and Health 
[Hygeia], and All-heal [Panaceia], and all the gods and goddesses, that 
according to my ability and judgment I will keep this oath and stip- 
ulation : to reckon him who taught me this art equally dear to me as 
my parents, to share my substance with him and relieve his necessities 
if required; to regard his offspring as on the same footing with my 
own brothers, and to teach them this art if they should wish to learn 
it, without fee or stipulation, and that by precept, lecture and every 
other mode of instruction, I will impart a knowledge of the art to my 
own sons, and to those of my teachers, and to disciples bound by a 
stipulation and oath, according to the law of medicine, but to none 
others. 
I will follow that method of treatment which according to my 
ability and judgment I consider for the benefit of my patients, and 
abstain from whatever is deleterious and mischievous. I will give 
no deadly medicine to anyone if asked, nor suggest any such counsel; 
furthermore, I will not give to a woman an instrument to produce 
abortion. 
With purity and with holiness I will pass my life and practice my 
art. I will not cut a person who is suffering with a stone, but will 
leave this to be done by practitioners of this work. Into whatever 
houses I enter I will go into them for the benefit of the sick, and will 
abstain from every voluntary act of mischief and corruption; and 
further from the seduction of females or males, bond or free. 
399 400 
A SHORT HISTORY OF SCIENCE 
Whatever, in connection with my professional practice or not in 
connection with it, I may see or hear in the lives of men, which ought 
not to be spoken abroad, I will not divulge, as reckoning that all such 
should be kept secret. 
While I continue to keep this oath unviolated, may it be granted 
to me to enjoy life and the practice of the art, respected by all men at 
all times; but should I trespass and violate this oath, may the reverse 
be my lot. 
B. THE OPUS MAJUS OF ROGER BACON (1267 A.D.) 
[An Analysis of the Sixth Part. By J. H. Bridges.] 
Of all the parts of the Opus Majus, the sixth is the most important. It treats 
of experimental science, domina omnium scientiarum et finis totius specula- 
tionis. Without experience, as Bacon constantly repeats, nothing can be known 
with certainty. Even the conclusions of mathematical physics, reached by argu- 
ment from certain principles, must be verified, before the mind can rest satisfied. 
To this great science all the others are subsidiary; they are to it ancillse or hand- 
maids, an expression that curiously reminds one of Francis Bacon. The reason- 
ing in favour of experience is well worth quoting at length: — 
“ There are two modes in which we acquire knowledge, argument and experi- 
ment. Argument shuts up the question, and makes us shut it up too; but it gives 
no proof, nor does it remove doubt and cause the mind to rest in the conscious 
possession of truth, unless the truth is discovered by way of experience, e.g. if any 
man who had never seen fire were to prove by satisfactory argument that fire burns 
and destroys things, the hearer’s mind would not rest satisfied, nor would he avoid 
fire; until by putting his hand or some combustible thing into it, he proved by 
actual experiment what the argument laid down; but after the experiment had been 
made, his mind receives certainty and rests in the possession of truth, which could 
not be given by argument but only by experience. And this is the case even in 
mathematics, where there is the strongest demonstration. For let any one have the 
clearest demonstration about an equilateral triangle without experience of it, his 
mind will never lay hold of the problem until he has actually before him the inter- 
secting circles and the lines drawn from the point of section to the extremities of a 
straight line. He will then accept the conclusion with all satisfaction.” (Op. 
Maj., p. J+J+5 [ed. Bridges, ii. 167].) 
This important passage, it seems to me, marks a distinct advance in the philos- 
ophy of science. The science of that time proceeded wholly per argumentum; 
verification was unknown. Not only, however, does Bacon recognize the necessity 
for experiment, for observation at first-hand, but he has a clear appreciation of the 
true nature of scientific verification. He has already expounded his ideal of 
physical science, the application of mathematics to determine the laws of force 
and to deduce conclusions from these laws; but he is perfectly aware that these APPENDIX B: ROGER BACON 
401 
general conclusions must be tested by comparison with things, must be verified. 
The function of experimental science is, ift a word, Verification. 
“ This Science,” says Bacon, “has three great prerogatives in respect to all 
other sciences. The first is — that it investigates their conclusions by experience. 
For the principles of the other sciences may be known by experience, but the con- 
clusions are drawn from these principles by way of argument. If they require 
particular and complete knowledge of those conclusions, the aid of this science 
must be called in. It is true that mathematics possesses useful experience with 
regard to its own problems of figure and number, which apply to all the sciences 
and to experience itself, for no science can be known without mathematics. But 
if we wish to have complete and thoroughly verified knowledge, we must proceed 
by the methods of experimental science.” (Op. Maj., p. 448 [ed. Bridges, ii. 
172-3].) 
As an example of his method Bacon analyses the phenomena of the rainbow in 
a thoroughly scientific manner. 
The second and third prerogatives (though not of such importance) may also 
be mentioned. The second is — that Experimental Science attains to a knowledge 
of truth which could not be reached by the special sciences; the third — that Ex- 
perimental Science, using and combining the results of the other sciences, is able 
to investigate the secret operations of Nature, to predict what the course of events 
will be, and to invent instruments or machines of wonderful power. 
— Adamson (quoted by A. G. Little). 
PART VI. EXPERIMENTAL SCIENCE 
Chapter I 
Having laid down the general principles of wisdom so far as they 
are found in language, in mathematics, and in optics, I pass to the 
subject of experimental science. . . . 
When Aristotle speaks of knowledge of the cause as a higher kind 
of knowledge than that gained by experience, he is speaking of mere 
empiric knowledge of a fact; I am speaking of experimental knowledge 
of its cause. There are numerous beliefs commonly held in the ab- 
sence of experiment, and wholly false, such as that adamant can be 
broken by goats’ blood, that the beaver when chased throws away 
his testicles, that a vessel of hot water freezes more rapidly than one of 
cold, and so on. Experience is of two kinds: (1) that in which we 
use our bodily senses aided by instruments, and by evidence of trust- 
worthy witnesses; and (2) internal experience of things spiritual, 
which comes of grace, and which often leads to knowledge of earthly 
things. The mind stained with vice is like a rusty or uneven mirror, 402 
A SHORT HISTORY OF SCIENCE 
in which things seem other than they are. Without virtue a man 
may repeat words like a parrot, and imitate other men’s wisdom like 
an ape, and all to no purpose. The intellectual effect of a stainless 
life is well illustrated in the young man who is the bearer of this treatise. 
The degrees of spiritual experience are seven. (1) Spiritual illumina- 
tion ; (2) virtue; (3) the gift of the Holy Spirit described by Isaiah; 
(4) the Beatitudes; (5) spiritual sensibility; (6) Fruits, such as the 
peace of God which passes understanding; (7) states of Rapture. — 
Chapter II 
It is solely by the aid of this science that we shall be able to disabuse 
men of the fraudulent tricks by which magicians have imposed on 
them. As compared with other sciences, this science has three char- 
acteristics (“prerogatives”). Of these the first is, that it constitutes 
a test to which all the conclusions of other sciences are to be subjected. 
In other sciences the principles are discovered by experiment, but the 
conclusion by reasoning. An instance of this is afforded by the rain- 
bow, and by other phenomena of a similar kind, as haloes, etc. The 
natural philosopher forms a judgment on these things: the experi- 
menter proceeds to test the judgment. He seeks for visible objects 
in which the colours of the rainbow appear in the same order. He 
finds this the case with Irish hexagonal crystals when held in the sun’s 
rays. This property, he discovers, is not peculiar to these crystals, 
but is common to all transparent substances of similar shape, simi- 
larly placed. He finds these colours again on the surface of crystals 
when slightly roughened. He finds them in the drops that fall from 
the rower’s oar, when the sun’s rays strike them, or from a water- 
wheel, or in the morning-dew on the grass. They may be seen again 
in sunshine when the eye is half opened, and in many other cases. 
Chapter III 
The shape in which the colours are disposed will vary. Sometimes 
it is rectangular, sometimes circular. 
Armed with these terrestrial facts, the experimenter proceeds to 
examine the celestial phenomenon. He finds, on examining the 
sun’s altitude and that of the summit of the bow, that the two vary 
inversely. The bow is always opposite the sun. A line may be drawn 
Chapter IV APPENDIX B: ROGER BACON 
403 
from the centre of the sun through the eye of the observer and the 
centre of the circle of which the bow is an arc to the sun’s nadir. As 
one extremity of this line is depressed, the other is elevated. It be- 
comes thus possible to compute the altitude of the sun beyond which 
no rainbow is possible, and also the maximum altitude of the bow. 
It will be found both by calculation and experience that this altitude 
in the latitude of Paris is forty-two degrees. 
Chapter V 
Still further investigating the shape of the iris, and the portion of it 
that can be seen, the experimenter conceives a cone of which the 
apex is the eye, the base is the circle of the iris, the axis being the line 
already described drawn from the sun’s centre through the eye to the 
sun’s nadir. In cases where this cone is very short, the whole of the 
base may be above the horizon, as may often be seen in the spray of 
a waterfall. In the sky, however, the cone is too elongated to admit 
of this: the base is bisected in various proportions by the plane of 
the horizon. The arcs visible are not portions of the same circle. 
When the sun is high, and a small arc is visible, it belongs to a larger 
circle than the arc seen when the sun is rising or setting. A bow can 
be seen when the sun is just below the horizon; but owing to terres- 
trial vapours, only the crown of the arch is usually seen. 
Chapter VI 
In some latitudes there can be no rainbow at noon even in the win- 
ter solstice. When the latitude (i.e. the distance from the zenith 
to the equator) is 24° 25', the sun’s altitude at noon in the winter 
solstice will be 42°, therefore there can be no bow. Passing north 
from this latitude, there can always be a noon rainbow till we come 
to latitude 66° 25', when at the winter solstice there is no sun. Similar 
calculations can be made for other latitudes. 
Chapter VII 
We have now to inquire whether the iris comes from incident, 
reflected, or refracted rays. Is the bow an image of the sun? Are 
the colours on the clouds real? Why is the iris of circular form? 
Here we call experiment to our aid. We find on trial that if we move 
in a direction parallel to the rainbow it follows us with a velocity 
exactly equal to our own. The same phenomenon occurs with respect 404 
A SHORT HISTORY OF SCIENCE 
to the sun. We have seen that the sun is always opposite the rain- 
bow ; the line between the centre of the bow and the centre of the 
sun passing through the eye of the observer. If the sun were appar- 
ently stationary, this would involve the bow moving much faster than 
the observer, the latter moving through the same angle, but at less 
distance from the apex. But this is not so. Therefore there is an 
apparent motion of the sun concurrently with that of the bow. The 
case is analogous to what happens when a hundred men are ranged in 
line facing the sun. Each sees the sun in front of him. Their shadows 
seem parallel, though we know in reality they must diverge, yet 
owing to the vast distance of the sun this divergence is imperceptible. 
We are thus brought to the conclusion that, supposing a rainbow to 
occur, each of the hundred men, facing backwards, would see a dif- 
ferent rainbow, to the centre of which his own shadow would point. 
The rays causing the iris are therefore not incident rays, otherwise 
the colour would appear fixed in the cloud. And for the same reason 
they are not refracted rays, for in refraction the image does not follow 
the change of place of the observer, as is the case here. One condi- 
tion of the phenomenon is that the atmosphere shall be more illumi- 
nated at the standpoint of the observer, and less at the position of the 
cloud. The movement of the sun from east to west during the 
appearance of the rainbow may be left out of account. 
Chapter VIII 
The colours in the bow arise from an ocular deception. They are 
analogous to those which appear when the eyes are weak or half-shut. 
They are not due to the same cause as the colours produced when 
light shines through a crystal, since these do not, like the colours 
of the rainbow, shift with the position of the observer. 
Chapter IX 
Each drop of rain in the cloud is to be regarded as a spherical mirror; 
these being small and close together, the effect is that of a continuous 
image rather than of a multitude of images. The colour is due to 
the distortion of the image caused by the sphericity of the mirror. 
Chapter X 
The diversity of colours has been attributed to varieties in the 
texture of the cloud, the denser parts producing violet and blue, the APPENDIX B: ROGER BACON 
405 
lighter parts red and orange. But we see the same colours in the 
dew drops, where there can be no such differences of density; simi- 
larly in the crystal. Aristotle has been wrongly translated and 
interpreted in this matter. Another erroneous belief is that lunar 
rainbows occur only once in fifty years. They may occur at any full 
moon under suitable atmospheric conditions. 
Chapter XI 
The shape of the bow is a difficulty. It cannot be explained by 
refraction. It is to be observed that the same colour is continued all 
round the circle in each ring. All parts of the ring therefore preserve 
the same relation of the solar ray to the eye. This implies circularity 
of form. It is asked why the whole space contained by the circle is 
not occupied with colour. Because from the points in this central 
area rays equal to the angle of incidence are not reflected to the eye. 
Chapter XII 
The cloud therefore is not coloured; the appearance of colour, for 
it is only an appearance, is given by rays reflected from the raindrops. 
Of colours there are five, white, blue, red, green, black; though Aris- 
totle, dividing blue and green into other shades, speaks of seven. 
These colours appear to have some relation to the various structures 
of the eye. In addition to the problem of the rainbow, there is the 
problem of haloes and coronal. On this I give the best explanation 
that as yet occurs to me. I do not however, pretend that it is satis- 
factory. Far more careful experiments, made with properly con- 
structed instruments, are needed before an adequate explanation 
can be given. 
THE SECOND PREROGATIVE OF EXPERIMENTAL SCIENCE 
In all sciences Experiment is able to reveal truths quite unconnected 
with the discussion of principles, and with regard to which it is useless 
in the first instance to assign a reason. The initial state of mind 
should be readiness to believe; this should be followed by experiment: 
reasoning should come last. I subjoin examples of my meaning. 
1. The astronomer constructs his spherical astrolabe, by which he 
can observe the precise longitude and latitude of heavenly bodies 
at different times. But it is not inconceivable that experiment may 
devise means of bringing this instrument into such relation with the 406 
A SHORT HISTORY OF SCIENCE 
revolution of the heavens that it should follow their course. The 
motion of the tides, the periodic changes in certain diseases, the 
diurnal opening and closing of flowers, are facts tending to belief 
that such a discovery is possible. If effected it would supersede all 
other astronomical instruments. 2. My next example relates to the 
act of prolonging human life. As yet we have nothing to rely on but 
ordinary rules of health. These are observed but by few, and usually 
not till the close of life, when it is too late. If a suitable regimen 
were observed by all, no doubt life would be much prolonged. But 
there are special remedies unknown as yet to medicine, but to be found 
by experiment, which may extend the period of life much further. 
Observation of the habits of certain animals may guide us to truths 
in this matter which are as yet hidden. Other indications are given 
in the works of Aristotle, Pliny, Artephius, and others. A combina- 
tion of gold, pearl, flower of sea-dew, spermaceti, aloes, bone of stag’s 
heart, flesh of Tyrian snake and of dragon, properly pre- 
pared in due proportions, might promote longevity to an extent 
hitherto unimagined. 
3. A third example may be found in Alchemy. The problem here 
is not merely to transmute the baser into the more precious metals, 
but to promote gold to its highest degree of perfection. In this per- 
fected gold we should probably have a further aid to the prolongation 
of life. 
THIRD PREROGATIVE OF EXPERIMENTAL SCIENCE 
In this we leave altogether the domain of the sciences now recog- 
nized, and open out entirely new departments of research. At present 
the influences exerted on us by the stars can only be known through 
difficult astronomical calculations. Experimental science may enable 
us to estimate them directly. It may be possible for us to act on the 
character of the inhabitants of any region by altering their environ- 
ment. Inventions of the greatest utility may be discovered, as 
perpetual fire, or explosive substances, or modes of counteracting 
dangerous poisons, and innumerable other properties of matter as 
yet unknown for want of experiment. The Magnet, of which use is 
already made, is but a type of other mutual attractions exerted by 
bodies at a distance. For instance, if a young sapling be longitudinally 
divided and the two divisions be brought near together, held each by 
the middle, the extremities will bend towards each other. In conclu- APPENDIX C: COPERNICUS 
407 
sion, I may point out the influence which the possessors of this science 
may exercise in the promotion of Christianity among the heathen, 
whether in subduing their pride, in disabusing them of false beliefs 
in magic, or in overcoming their material force. 
C. DEDICATION OF 
THE REVOLUTIONS OF THE HEAVENLY BODIES 
BY NICOLAUS COPERNICUS (1543) 
To Pope Paul III 
I can easily conceive, most Holy Father, that as soon as some 
people learn that in this book which I have written concerning the 
revolutions of the heavenly bodies, I ascribe certain motions to the 
Earth, they will cry out at once that I and my theory should be re- 
jected. For I am not so much in love with my conclusions as not to 
weigh what others will think about them, and although I know that 
the meditations of a philosopher are far removed from the judgment 
of the laity, because his endeavor is to seek out the truth in all things, 
so far as this is permitted by God to the human reason, I still believe 
that one must avoid theories altogether foreign to orthodoxy. Ac- 
cordingly, when I considered in my own mind how absurd a perform- 
ance it must seem to those who know that the judgment of many 
centuries has approved the view that the Earth remains fixed as centre 
in the midst of the heavens, if I should on the contrary, assert that the 
Earth moves; I was for a long time at a loss to know whether I should 
publish the commentaries which I have written in proof of its motion, 
or whether it were not better to follow the example of the Pythag- 
oreans and of some others, who were accustomed to transmit the 
secrets of philosophy not in writing but orally, and only to their rela- 
tives and friends, as the letter from Lysis to Hipparchus bears wit- 
ness. They did this, it seems to me, not as some think, because of a 
certain selfish reluctance to give their views to the world, but in 
order that the noblest truths, worked out by the careful study of great 
men, should not be despised by those who are vexed at the idea of 
taking great pains with any form of literature except such as would 
be profitable, or by those who, if they are driven to the study of phi- 
losophy for its own sake by the admonitions and the example of others, 
nevertheless, on account of their stupidity, hold a place among philoso- 408 
A SHORT HISTORY OF SCIENCE 
phers similar to that of drones among bees. Therefore, when I con- 
sidered this carefully, the contempt which I had to fear because of 
the novelty and apparent absurdity of my view, nearly induced me to 
abandon utterly the work I had begun. 
My friends, however, in spite of long delay and even resistance on 
my part, withheld me from this decision. First among these was 
Nicolaus Schonberg, Cardinal of Capua, distinguished in all branches 
of learning. Next to him comes my very dear friend, Tidemann 
Giese, Bishop of Culm, a most earnest student, as he is, of sacred and, 
indeed, of all good learning. The latter has often urged me, at times 
even spurring me on with reproaches, to publish and at last bring to 
light the book which had lain in my study not nine years merely, but 
already going on four times nine. Not a few other very eminent and 
scholarly men made the same request, urging that I should no longer 
through fear refuse to give out my work for the common benefit of 
students of Mathematics. They said I should find that the more 
absurd most men now thought this theory of mine concerning the 
motion of the Earth, the more admiration and gratitude it would com- 
mand after they saw in the publication of my commentaries, the mist 
of absurdity cleared away by most transparent proofs. So, influenced 
by these advisers and this hope, I have at length allowed my friends 
to publish the work, as they had long besought me to do. 
But perhaps your Holiness will not so much wonder that I have 
ventured to publish these studies of mine, after having taken such 
pains in elaborating them that I have not hesitated to commit to 
writing my views of the motion of the Earth, as you will be curious to 
hear how it occurred to me to venture, contrary to the accepted view 
of mathematicians, and well-nigh contrary to common sense, to form 
a conception of any terrestial motion whatsoever. Therefore I would 
not have it unknown to Your Holiness, that the only thing which 
induced me to look for another way of reckoning the movements of 
the heavenly bodies was that I knew that mathematicians by no means 
agree in their investigations thereof. For, in the first place, they are 
so much in doubt concerning the motion of the sun and the moon, 
that they cannot even demonstrate and prove by observation the 
constant length of a complete year; and in the second place, in deter- 
mining the motions both of these and of the other five planets, they 
fail to employ consistently one set of first principles and hypotheses, 
but use methods of proof based only upon the apparent revolutions APPENDIX C: COPERNICUS 
409 
and motions. For some employ concentric circles only; others, eccen- 
tric circles and epicycles; and even by these means they do not com- 
pletely attain the desired end. For, although those who have de- 
pended upon concentric circles have shown that certain diverse mo- 
tions can be deduced from these, yet they have not succeeded thereby 
in laying down any sure principle, corresponding indisputably to the 
phenomena. These, on the other hand, who have devised systems of 
eccentric circles, although they seem in great part to have solved the 
apparent movements by calculations which by these eccentrics are 
made to fit, have nevertheless introduced many things which seem 
to contradict the first principles of the uniformity of motion. Nor 
have they been able to discover or calculate from these the main 
point, which is the shape of the world and the fixed symmetry of its 
parts; but their procedure has been as if someone were to collect 
hands, feet, a head, and other members from various places, all very 
fine in themselves, but not proportionate to one body, and no single 
one corresponding in its turn to the others, so that a monster rather 
than a man would be formed from them. Thus in their process of 
demonstration which they term a “method,” they are found to have 
omitted something essential, or to have included something foreign 
and not pertaining to the matter in hand. This certainly would never 
have happened to them if they had followed fixed principles; for if 
the hypotheses they assumed were not false, all that resulted there- 
from would be verified indubitably. Those things which I am say- 
ing now may be obscure, yet they will be made clearer in their proper 
place. 
Therefore, having turned over in my mind for a long time this un- 
certainty of the traditional mathematical methods of calculating the 
motions of the celestial bodies, I began to grow disgusted that no 
more consistent scheme of the movements of the mechanism of the 
universe, set up for our benefit by that best and most law-abiding 
Architect of all things, was agreed upon by philosophers who other- 
wise investigate so carefully the most minute details of this world. 
Wherefore I undertook the task of rereading the books of all the phi- 
losophers I could get access to, to see whether anyone ever was of the 
opinion that the motions of the celestial bodies were other than those 
postulated by the men who taught mathematics in the schools. And 
I found first, indeed, in Cicero, that Hicetas perceived that the Earth 
moved; and afterward in Plutarch I found that some others were of 410 
A SHORT HISTORY OF SCIENCE’ 
this opinion, whose words I have seen fit to quote here, that they may 
be accessible to all: — 
Some maintain that the Earth is stationary, but Philolaus the Pythag- 
orean says that it revolves in a circle about the fire of the ecliptic, like the 
sun and moon. Heraklides of Pontus and Ekphantus the Pythagorean make 
the Earth move, not changing its position, however, confined in its falling and 
rising around its own centre in the manner of a wheel. 
Taking this as a starting-point, I began to consider the mobility of 
the Earth; and although the idea seemed absurd, yet because I knew 
that the liberty had been granted to others before me to postulate all 
sorts of little circles for explaining the phenomena of the stars, I 
thought I also might easily be permitted to try whether by postulating 
some motion of the Earth, more reliable conclusions could be reached 
regarding the revolution of the heavenly bodies, than those of my 
predecessors. 
And so, after postulating movements, which, farther on in the book, 
I ascribe to the Earth, I have found by many and long observations 
that if the movements of the other planets are assumed for the circular 
motion of the Earth and are substituted for the revolution of each 
star, not only do their phenomena follow logically therefrom, but the 
relative positions and magnitudes both of the stars and all their orbits, 
and of the heavens themselves, become so closely related that in none 
of its parts can anything be changed without causing confusion in 
the other parts and in the whole universe. Therefore, in the course 
of the work I have followed this plan: I describe in the first book all 
the positions of the orbits together with the movements which I 
ascribe to the Earth, in order that this book might contain, as it 
were, the general scheme of the universe. Thereafter in the remaining 
books, I set forth the motions of the other stars and of all their orbits 
together with the movement of the Earth, in order that one may see 
from this to what extent the movements and appearances of the 
other stars and their orbits can be saved, if they are transferred to the 
movement of the Earth. Nor do I doubt that ingenious and learned 
mathematicians will sustain me, if they are willing to recognize and 
weigh, not superficially, but with that thoroughness which Philosophy 
demands above all things, those matters which have been adduced 
by me in this work to demonstrate these theories. In order, how- 
ever, that both the learned and the unlearned equally may see that APPENDIX C: COPERNICUS 
411 
I do not avoid anyone’s judgment, I have preferred to dedicate these 
lucubrations of mine to Your Holiness rather than to any other, be- 
cause, even in this remote corner of the world where I live, you are 
considered to be the most eminent man in dignity of rank and in love 
of all learning and even of mathematics, so that by your authority 
and judmgent you can easily suppress the bites of slanderers, albeit 
the proverb hath it that there is no remedy for the bite of a sycophant. 
If perchance there shall be idle talkers, who, though they are ignorant 
of all mathematical sciences, nevertheless assume the right to pass 
judgment on these things, and if they should dare to criticise and 
attack this theory of mine because of some passage of scripture which 
they have falsely distorted for their own purpose, I care not at all; 
I will even despise their judgment as foolish. For it is not unknown 
that Lactantius, otherwise a famous writer but a poor mathematician, 
speaks most childishly of the shape of the Earth when he makes fun 
of those who said that the Earth has the form of a sphere. It should 
not seem strange then to zealous students, if some such people shall 
ridicule us also. Mathematics are written for mathematicians, to 
whom, if my opinion does not deceive me, our labors will seem to 
contribute something to the ecclesiastical state whose chief office 
Your Holiness now occupies; for when not so very long ago, under 
Leo X, in the Lateran Council the question of revising the ecclesiastical 
calendar was discussed, it then remained unsettled, simply because 
the length of the years and the months, and the motions of the sun 
and moon were held to have been not yet sufficiently determined. 
Since that time, I have given my attention to observing these more 
accurately, urged on by a very distinguished man, Paul, Bishop of 
Fossombrone, who at that time had charge of the matter. But what 
I may have accomplished herein I leave to the judgment of Your 
Holiness in particular, and to that of all other learned mathematicians; 
and lest I seem to Your Holiness to promise more regarding the use- 
fulness of the work than I can perform, I now pass to the work itself. 
(— From the Harvard Classics, Vol. 39, 55-60.) 412 
A SHORT HISTORY OF SCIENCE 
D. HARVEY’S DEDICATION OF HIS WORK ON THE MOTION OF 
THE HEART AND THE CIRCULATION OF THE BLOOD (1628) 
TO HIS VERY DEAR FRIEND, DOCTOR ARGENT, THE EXCELLENT AND AC- 
COMPLISHED PRESIDENT OF THE ROYAL COLLEGE OF PHYSICIANS, AND 
TO OTHER LEARNED PHYSICIANS, HIS MOST ESTEEMED COLLEAGUES 
I have already and repeatedly presented you, my learned friends, 
with my new views of the motion and function of the heart, in my 
anatomical lectures; but having now for more than nine years con- 
firmed these views by multiplied demonstrations in your presence, 
illustrated them by arguments, and freed them from the objections of 
the most learned and skilful anatomists, I at length yield to the re- 
quests, I might say entreaties, of many, and here present them for a 
general consideration in this treatise. 
Were not the work indeed presented through you, my learned 
friends, I should scarce hope that it could come out scatheless and 
complete; for you have in general been the faithful witnesses of 
almost all the instances from which I have either collected the truth 
or confuted error. You have seen my dissections, and at my demon- 
strations of all that I maintained to be objects of sense, you have been 
accustomed to stand by and bear me out with your testimony. And 
as this book alone declares the blood to course and revolve by a new 
route, very different from the ancient and beaten pathway trodden 
for so many ages, and illustrated by such a host of learned and distin- 
guished men, I was greatly afraid lest I might be charged with pre- 
sumption did I lay my work before the public at home, or send it 
beyond seas for impression, unless I had first proposed the subject to 
you, had confirmed its conclusions by ocular demonstrations in your 
presence, had replied to your doubts and objections, and secured the 
assent and support of our distinguished President. For I was most 
intimately persuaded, that if I could make good my proposition before 
you and our College, illustrious by its numerous body of learned in- 
dividuals, I had less to fear from others. I even ventured to hope 
that I should have the comfort of finding all that you had granted me 
in your sheer love of truth, conceded by others who were philosophers 
like yourselves. True philosophers, who are only eager for truth and 
knowledge, never regard themselves as already so thoroughly informed, 
but that they welcome further information from whomsoever and from 
wheresoever it may come; nor are they so narrow minded as to imagine APPENDIX D: HARVEY 
413 
any of the arts or sciences transmitted to us by the ancients, in such 
a state of forwardness or completeness, that nothing is left for the 
ingenuity and industry of others. On the contrary, very many main- 
tain that all we know is still infinitely less than all that still remains 
unknown; nor do philosophers pin their faith to others’ precepts in 
such wise that they lose their liberty, and cease to give credence to 
the conclusions of their proper senses. Neither do they swear such 
fealty to their mistress Antiquity that they openly, and in sight of 
all, deny and desert their friend Truth. But even as they see that 
the credulous and vain are disposed at the first blush to accept and 
believe everything that is proposed to them, so do they observe that 
the dull and unintellectual are indisposed to see what lies before their 
eyes, and even deny the light of the noon-day sun. They teach us 
in our course of philosophy to sedulously avoid the fables of poets 
and the fancies of the vulgar, as the false conclusions of the sceptics. 
And then the studious and good and true, never suffer their minds to 
be warped by the passions of hatred and envy, which unfit men duly 
to weigh the arguments that are advanced in behalf of truth, or to 
appreciate the proposition that is even fairly demonstrated. Neither 
do they think it unworthy of them to change their opinion if truth 
and undoubted demonstration require them to do so. They do not 
esteem it discreditable to desert error, though sanctioned by the 
highest antiquity, for they know full well that to err, to be deceived, 
is human; that many things are discovered by accident and that 
many may be learned indifferently from any quarter, by an old man 
from a youth, by a person of understanding from one of inferior 
capacity. 
My dear colleagues, I had no purpose to swell this treatise into a 
large volume by quoting the names and writings of anatomists, or to 
make a parade of the strength of my memory, the extent of my read- 
ing, and the amount of my pains; because I profess both to learn and 
to teach anatomy, not from books but from dissections; not from 
the positions of philosophers but from the fabric of nature; and then 
because I do not think it right or proper to strive to take from the 
ancients any honor that is their due, nor yet to dispute with the 
moderns, and enter into controversy with those who have excelled in 
anatomy and been my teachers. I would not charge with wilful 
falsehood anyone who was sincerely anxious for truth, nor lay it to 
anyone’s door as a crime that he had fallen into error. I avow myself 414 
A SHORT HISTORY OF SCIENCE 
the partisan of truth alone; and I can indeed say that I have used 
all my endeavors, bestowed all my pains on an attempt to produce 
something that should be agreeable to the good, profitable to the 
learned, and useful to letters. 
Farewell, most worthy Doctors, 
And think kindly of your Anatomist 
William Harvey. 
E. GALILEO BEFORE THE INQUISITION (1633) 
I. His Condemnation 
We, Gasparo del titolo di S. Croce, in Gierusalemme Borgia; 
Fra Felice Centino del titolo di S. Anastasia, detto d’Ascoli; 
Guido del titolo di S. Maria del Popolo Bentivoglio; 
Fra Desiderio Scaglia del titolo di S. Carlo, detto di Cremona; 
Fra Antonio Barberino, detto di S. Onofrio; 
Laudivio Zacchia del titolo di S. Pietro in Vincolo, detto di S. Sisto; 
Berlingero del titolo di S. Agostino, Gessi; 
Fabricio del titolo di S. Lorenzo in pane e perna, Verospi, chiamato 
Prete; 
Francesco di S. Lorenzo, in Damaso Barberino; e 
Martino di S. Maria Nuova, Ginetti Diaconi; 
by the Grace of God, Cardinals of the Holy Roman Church, Inquisi- 
tors-General throughout the whole Christian Republic, Special Depu- 
ties of the Holy Apostolical Chair against heretical depravity, 
Whereas you, Galileo, son of the late Vincenzo Galilei of Florence, 
aged seventy years, were denounced in 1615 to this Holy Office, for 
holding as true the false doctrine taught by many, namely, that the 
sun is immovable in the centre of the world, and that the earth 
moves, and also with a diurnal motion; also for having pupils whom 
you instructed in the same opinions; also for maintaining a corre- 
spondence on the same with some German mathematicians; also for 
publishing certain letters on the solar spots, in which you developed 
the same doctrine as truth; also for answering the objections which 
were continually produced from the Holy Scriptures, by glozing the 
said Scriptures according to your own meaning; and whereas there- 
upon was produced the copy of a writing, in form of a letter, confessedly 
written by you to a person formerly your pupil, in which, following 
the hypotheses of Copernicus, you include several propositions con- 
trary to the true sense and authority of the Holy Scripture: APPENDIX E: GALILEO 
415 
Therefore this holy tribunal being desirous of providing against 
the disorder and mischief thence proceeding and increasing to the 
detriment of the holy faith, by the desire of His Holiness, and of the 
Most Eminent Lords Cardinals of this supreme and universal Inqui- 
sition, the two propositions of the stability of the sun, and the motion 
of the earth, were qualified by the Theological Qualifiers as follows : 
The proposition that the sun is the centre of the world and im- 
movable from its place is absurd, philosophically false, and formally 
heretical, because it is expressly contrary to the Holy Scripture. 
The proposition that the earth is not the centre of the wTorld, nor 
immovable, but that it moves, and also with a diurnal motion, is also 
absurd, philosophically false, and, theologically considered, at least 
erroneous in faith. 
But whereas being pleased at that time to deal mildly with you, 
it was decreed in the Holy Congregation, held before His Holiness on 
the 25th day of February, 1616, that His Eminence, the Lord Cardinal 
Bellarmine, should enjoin you to give up altogether the said false 
doctrine; if you should refuse, that you should be ordered by the 
Commissary of the Holy Office to relinquish it, not to teach it to others, 
not to defend it, nor ever mention it, and in default of acquiescence 
that you should be imprisoned; and in execution of this decree, on 
the following day at the palace, in presence of His Eminence the said 
Lord Cardinal Bellarmine, after you had been mildly admonished by 
the said Lord Cardinal, you were commanded by the acting Com- 
missary of the Holy Office, before a notary and witnesses to relinquish 
altogether the said false opinion, and in future neither to defend nor 
teach it in any manner, neither verbally nor in writing, and upon 
your promising obedience, you were dismissed. 
And in order that so pernicious a doctrine might be altogether 
rooted out, nor insinuate itself further to the heavy detriment of the 
Catholic faith, a decree emanated from the Holy Congregation of the 
Index prohibiting the books which treat of this doctrine; and it was 
declared false and altogether contrary to the Holy and Divine Scripture. 
And whereas a book has since appeared, published at Florence 
last year, the title of which showed that you were the author, which 
is: The Dialogue of Galileo Galilei, on the Two Principal Systems of 
the World, the Ptolemaic and Copernican; and whereas the Holy 
Congregation has heard that, in consequence of the printing of the 
said book, the false opinion of the earth’s motion and stability of the 416 
A SHORT HISTORY OF SCIENCE 
sun is daily gaining ground; the said book has been taken into careful 
consideration, and in it has been detected a glaring violation of the 
said order, which had been intimated to you; inasmuch as in this 
book you have defended the said opinion, already in your presence 
condemned; although in the said book you labor with many circum- 
locutions to produce the belief that it is left by you undecided, and 
in express terms probable; which is equally a very grave error, since 
an opinion can in no way be probable which has been already declared 
and finally determined contrary to the Divine Scripture: 
Therefore by our order you were cited to this Holy Office, where, 
on your examination upon oath, you acknowledged the said book as 
written and printed by you. You also confessed that you began to 
write the said book ten or twelve years ago, after the order aforesaid 
had been given; also, that you demanded licence to publish it, but 
without signifying to those who granted you this permission that you 
had been commanded not to hold, defend or teach the said doctrine 
in any manner. 
You also confessed that the style of the said book was, in many 
places, so composed that the reader might think the arguments 
adduced on the false side to be so worded as more effectually to en- 
tangle the understanding than to be easily solved, alleging in excuse, 
that you have thus run into an error, foreign (as you say) to your in- 
tention, from writing in the form of a dialogue, and in consequence 
of the natural complacency which everyone feels in regard to his own 
subtilities, and in showing himself more skilful than the generality of 
mankind in contriving, even in favor of false propositions, ingenious 
and apparently probable arguments. 
And, upon a convenient time having been given to you for making 
your defense, you produced a certificate in the handwriting of His 
Eminence the Lord Cardinal Bellarmine, procured as you said, by 
yourself, that you might defend yourself against the calumnies of 
your enemies, who reported that you had abjured your opinions, and 
had been punished by the Holy Office; in which certificate it is 
declared that you had not abjured, nor had been punished, but merely 
that the declaration made by His Holiness, and promulgated by the 
Holy Congregation of the Index, had been announced to you, which 
declares that the opinion of the motion of the earth, and the stability 
of the sun, is contrary to the Holy Scriptures, and, therefore, cannot 
be held or defended. Wherefore, since no mention is there made of APPENDIX E: GALILEO 
417 
two articles of the order, to wit, the order “not to teach,” and “in 
any manner,” you argued that we ought to believe that, in the lapse 
of fourteen or sixteen years, they had escaped your memory, and 
that this was also the reason why you were so silent as to the order, 
when you sought permission to publish your book, and that this is 
said by you not to excuse your error, but that it may be attributed 
to vainglorious ambition, rather than to malice. But this very cer- 
tificate, produced on your behalf, has greatly aggravated your offense, 
since it is therein declared that the said opinion is contrary to the 
Holy Scripture; and yet you have dared to treat of it, to defend it, 
and to argue that it is probable; nor is there any extenuation in the 
licence artfully and cunningly extorted by you, since you did not 
intimate the command imposed upon you. 
But whereas it appeared to us that you had not disclosed the whole 
truth in regard to your intentions, We thought it necessary to pro- 
ceed to the rigorous examination of you, in which (without any preju- 
dice to what you had confessed, and which is above detailed against 
you, with regard to your said intention) you answered like a good 
Catholic. Therefore, having seen and maturely considered the 
merits of your cause, with your said confessions and excuses, and 
everything else which ought to be seen and considered, We have 
come to the underwritten final sentence against you. 
Invoking, therefore, the Most Holy name of Our Lord Jesus Christ, 
and of His Most Glorious Virgin Mother Mary, by this our final 
sentence, which, sitting in council and judgment for the tribunal of 
the Reverend Masters of Sacred Theology, and Doctors of Both 
Laws, Our Assessors, We put forth in this writing touching the matters 
and controversies before us, between the Magnificent Charles Sin- 
cerus, Doctor of Both Laws, Fiscal Proctor of this Holy Office, of the 
one part, and you, Galileo Galilei, an examined and confessed criminal 
from this present writing as above, of the other part. We pronounce, 
judge and declare, that you, the said Galileo, by reason of these things 
which have been detailed in the course of this writing, and which, as 
above, you have confessed, have rendered yourself vehemently sus- 
pected by this Holy Office of heresy; that is to say, that you believe 
and hold the false doctrine, and contrary to the Holy and Divine 
Scriptures, namely, that the sun is the centre of the world, and that 
it does not move from the east to west, that the earth does move 
and is not the centre of the world; also that an opinion can be held 418 
A SHORT HISTORY OF SCIENCE 
and supported as probable after it has been declared and finally de- 
creed contrary to the Holy Scripture, and consequently that you have 
incurred all the censures and penalties enjoined and promulgated in 
the sacred canons, and other general and particular constitutions, 
against delinquents of this description. From which it is Our pleas- 
ure that you be absolved, provided that, first, with a sincere heart 
and unfeigned faith, in Our presence, you abjure, curse, and detest 
the said errors and heresies and every other error and heresy contrary 
to the Catholic and Apostolic Church of Rome, in the form now shown 
to you. 
But, that your grievous and pernicious error and transgression may 
not go altogether unpunished, and that you may be made more cautious 
in the future, and may be a warning to others to abstain from delin- 
quencies of this sort, We decree that the book of the Dialogues of 
Galileo Galilei be prohibited by a public edict. 
We condemn you to the formal prison of this Holy Office for a 
period determinable at Our pleasure; and by way of salutary penance, 
We order you, during the next three years, to recite once a week the 
seven Penitential Psalms. 
Reserving to ourselves the power of moderating, commuting, or 
taking off the whole or part of the said punishment or penance. 
And so We say, pronounce, and by Our sentence declare, decree, 
and reserve, in this and every other better form and manner, which 
lawfully we may and can use. 
So we, the undersigned Cardinals, pronounce. 
Felice, Cardinal di Ascoli, 
Guido, Cardinal Bentivoglio, 
Desiderio, Cardinal di Cremona, 
Antonio, Cardinal S. Onofrio, 
Berlingero, Cardinal Gessi, 
Fabricio, Cardinal Verospi, 
Martino, Cardinal Ginetti. 
I Galileo Galilei, son of the late Vincenzo Galilei of Florence, aged 
seventy years, being brought personally to judgment, and kneeling 
before you, Most Eminent and Most Reverend Lords Cardinals, 
General Inquisitors of the Universal Christian Republic against heret- 
II. His Recantation APPENDIX E: GALILEO 
419 
ical depravity, having before my eyes the Holy Gospels, which I 
touch with my own hands, swear, that I have always believed, and 
now believe, and with the help of God will in future believe, every 
article which the Holy Catholic and Apostolic Church of Rome holds, 
teaches and preaches. But because I had been enjoined by this Holy 
Office altogether to abandon the false opinion which maintains that 
the sun is the centre and immovable, and forbidden to hold, defend, 
or teach the said false doctrine in any manner, and after it had been 
signified to me that the said doctrine is repugnant with the Holy 
Scripture, I have written and printed a book, in which I treat of 
the same doctrine now condemned, and adduce reasons with great 
force in support of the same, without giving any solution, and 
therefore, have been judged grievously suspected of heresy; that 
is to say, that I have held and believed that the sun is the centre 
of the world and immovable, and that the earth is not the centre and 
movable. 
Wishing, therefore, to remove from the mind of Your Emi- 
nences, and of every Catholic Christian, this vehement suspicion 
rightfully entertained towards me, with a sincere heart and un- 
feigned faith, I abjure, curse and detest the said errors and here- 
sies, and generally every other error and sect contrary to the said 
Holy Church; and I swear that I will nevermore in future say 
or assert anything verbally, or in writing, which may give rise 
to a similar suspicion of me; but if I shall know any heretic, 
or anyone suspected of heresy, that I will denounce him to this 
Holy Office, or to the Inquisitor and Ordinary of the place in which 
I may be. 
I swear, moreover, and promise, that I will fulfill, and observe 
fully, all the penances which have been or shall be laid to me by this 
Holy Office. But if it shall happen that I violate any of my said 
promises, oaths and protestations (which God avert!), I subject 
myself to all the pains and punishments which have been de- 
creed and promulgated by the sacred canons, and other general 
and particular constitutions, against delinquents of this description. 
So may God help me, and His Holy Gospels, which I touch with my 
own hands. 
I, the above-named Galileo Galilei, have abjured, sworn, promised 
and bound myself, as above, and in witness thereof with my own hand 
have subscribed this present writing of my abjuration, which I have 420 
A SHORT HISTORY OF SCIENCE 
recited word for word. At Rome in the Convent of Minerva, 22nd 
June, 1633. 
I, Galileo Galilei, have abjured as above with my own hand. 
— From Routledge. History of Science. See also K. von Gebler. 
Galileo Galilei and the Roman Curia. London, 1879. 
PHILOSOPHISE NATURALIS PRINCIPIA MATHEMATICA (1686) 
F. PREFACE TO THE 
Isaac Newton 
Since the ancients (as we are told by Pappus) made great account 
of the science of mechanics in the investigation of natural things; 
and the moderns, laying aside substantial forms and occult qualities, 
have endeavored to subject the phenomena of nature to the laws of 
mathematics, I have in this treatise cultivated mathematics so far as 
it regards philosophy. The ancients considered mechanics in a two- 
fold respect; as rational, which proceeds accurately by demonstra- 
tion, and practical. To practical mechanics all the manual arts be- 
long, from which mechanics took its name. But as artificers do not 
work with perfect accuracy, it comes to pass that mechanics is so 
distinguished from geometry, that what is perfectly accurate is called 
geometrical; what is less so is called mechanical. But the errors 
are not in the art but in the artificers. He that works with less ac- 
curacy is an imperfect mechanic: and if any could work with perfect 
accuracy, he would be the most perfect mechanic of all; for the 
description of right lines and circles, upon which geometry is founded, 
belongs to mechanics. Geometry does not teach us to draw these 
lines, but requires them to be drawn; for it requires that the learner 
should first be taught to describe these accurately, before he enters 
upon geometry; then it shows how by these operations problems may 
be solved. To describe right lines and circles are problems, but not 
geometrical problems. The solution of these problems is required 
from mechanics; and by geometry the use of them, when so solved, 
is shown; and it is the glory of geometry that from those few principles, 
fetched from without, it is able to produce so many things. There- 
fore geometry is founded in mechanical practice and is nothing but 
that part of universal mechanics which accurately proposes and 
demonstrates the art of measuring. But since the manual arts are APPENDIX F: NEWTON 
421 
chiefly conversant in the moving of bodies, it comes to pass that 
geometry is commonly referred to their magnitudes, and mechanics 
to their motion. In this sense rational mechanics will be the science 
of motions resulting from any forces whatsoever, and of the forces 
required to produce any motion, accurately proposed and demon- 
strated. This part of mechanics was cultivated by the ancients in 
the five powers which relate to manual arts, who considered gravity 
(it not being a manual power) no otherwise than as it moved weights 
by those powers. Our design, not respecting arts, but philosophy, 
and our subject, not manual, but natural, powers, we consider chiefly 
those things which relate to gravity, levity, elastic force, the resistance 
of fluids, and the like forces, whether attractive or impulsive; and 
therefore we offer this work as mathematical principles of philosophy; 
for all the difficulty of philosophy seems to consist in this — from the 
phenomena of motions to investigate the forces of nature, and then 
from these forces to demonstrate the other phenomena; and to this 
end the general propositions in the first and second book are directed. 
In the third book we give an example of this in the explication of the 
system of the World; for by the propositions mathematically demon- 
strated in the first book, we there derive from the celestial phenomena 
the forces of gravity with which bodies tend to the sun and the several 
planets. Then, from these forces, by other propositions which are 
also mathematical, we deduce the motions of the planets, the comets, 
the moon, and the sea. I wish we could derive the rest of the phe- 
nomena of nature by the same kind of reasoning from mechanical 
principles; for I am induced by many reasons to suspect that they 
may all depend upon certain forces by which the particles of bodies, 
by some causes hitherto unknown, are either mutually impelled 
towards each other, and cohere in regular figures, or are repelled and 
recede from each other; which forces being unknown, philosophers 
have hitherto attempted the search of nature in vain; but I hope 
the principles here laid down will afford some light either to that or 
some truer method of philosophy. 
In the publication of this work, the most acute and universally 
learned Mr. Edmund Halley not only assisted me with his pains in 
correcting the press and taking care of the schemes, but it was to his 
solicitations that its becoming public is owing; for when he had ob- 
tained of me my demonstrations of the figures of the celestial orbits, 
he continually pressed me to communicate the same to the Royal 422 
A SHORT HISTORY OF SCIENCE 
Society, who afterwards, by their kind encouragement and entreaties, 
engaged me to think of publishing them. But after I had begun to 
consider the inequalities of the lunar motions, and had entered upon 
some other things relating to the laws and measures of gravity, and 
other forces; and the figures that would be described by bodies at- 
tracted according to given laws; and the motion of several bodies 
moving among themselves; the motion of bodies in resisting mediums; 
the forces, densities, and motions of mediums- the orbits of the 
comets, and such like; I put off that publication until I had made a 
search into those matters, and could put out the whole together. 
What relates to the lunar motions (being imperfect) I have put all 
together in the corollaries of proposition 66, to avoid being obliged 
to propose and distinctly demonstrate the several things there con- 
tained in a method more prolix than the subject deserved, and in- 
terrupt the series of the several propositions. Some things, found out 
after the rest, I chose to insert in places less suitable, rather than 
change the number of the propositions and the citations. I heartily 
beg that what I have here done may be read with candor; and that 
the defects I have been guilty of upon this difficult subject may be 
not so much reprehended as kindly supplied, and investigated by new 
endeavors of my readers. 
Isaac Newton. 
Cambridge, Trinity College, 
May 8, 1686. 
(— Translation by Andrew Motte. The Harvard Classics, Vol. 39, pp. 157-159.) 
G. AN INQUIRY INTO THE CAUSES AND EFFECTS OF THE 
VARIOLA VACCINdH, A DISEASE DISCOVERED IN SOME 
OF THE WESTERN COUNTIES OF ENGLAND, PARTICU- 
LARLY GLOUCESTERSHIRE AND KNOWN BY THE NAME 
OF THE COW POX 
Edward Jenner, M.D., F.R.S., etc. 
[The first successful attempt — and this wholly empirical — to control smallpox 
in the human subject was the art of Inoculation with the virus of smallpox itself, 
a procedure derived from the East, and introduced about 1720 into Europe and 
America. In 1796 Jenner laid the foundation of experimental medicine, immu- 
nology and serology, by his work on Vaccination, i.e., inoculation with cow-pox, 
in a paper bearing the above title. The first edition was published in 1798 and 
the second, from which the following extracts are taken, in 1800.] APPENDIX G : JENNER 
423 
The deviation of Man from the state in which he was originally 
placed by Nature seems to have proved to him a prolific source of 
Disease. From the love of splendour, from the indulgences of luxury, 
and from his fondness for amusement, he has familiarised himself with 
a great number of animals, which may not originally have been in- 
tended for his associates. The Wolf, disarmed of ferocity, is now 
pillowed in the lady’s lap. The Cat, the little Tyger of our island, 
whose natural home is the forest, is equally domesticated and caressed. 
The Cow, the Hog, the Sheep, and the Horse, are all, for a variety of 
purposes, brought under his care and dominion. 
There is a disease to which the Horse, from his state of domestica- 
tion, is frequently subject. The Farriers have termed it the Grease. 
It is an inflammation and swelling of the heel, from which issues 
matter possessing properties of a very peculiar kind, which seems 
capable of generating a disease in the Human Body (after it has under- 
gone the modification I shall presently speak of), which bears so strong 
a resemblance to the Small Pox, that I think it highly probable it may 
be the source of that disease. 
In this Dairy Country a great number of cows are kept, and the 
office of milking is performed indiscriminately by men and maid 
servants. One of the former having been appointed to apply dress- 
ings to the heels of a horse affected with the Grease, and not paying 
due attention to cleanliness, incautiously bears his part in milking the 
cows, with some particles of the infectious matter adhering to his 
fingers. When this is the case, it commonly happens that a disease 
is communicated to the cows, and from the cows to the dairymaids, 
which spreads through the farm until most of the cattle and domestics 
feel its unpleasant consequences. This disease has obtained the name 
of Cow Pox. It appears on the nipples of the cows in the form of 
irregular pustules. At their first appearance they are commonly of a 
palish blue, or rather of a colour somewhat approaching to livid, and 
are surrounded by an inflammation. These pustules, unless a timely 
remedy be applied, frequently degenerate into phagedenic ulcers, 
which prove extremely troublesome. The animals become indisposed, 
and the secretion of milk is much lessened. Inflamed spots now begin 
to appear on different parts of the hands of the domestics employed 
in milking, and sometimes on the wrists, which quickly run on to 
suppuration, first assuming the appearance of the small vesications 
produced by a burn. Most commonly they appear about the joints 424 
A SHORT HISTORY OF SCIENCE 
of the fingers and at their extremities; but whatever parts are af- 
fected, if the situation will admit, these superficial suppurations put 
on a circular form, with their edges more elevated than their center, 
and of a colour distantly approaching to blue. Absorption takes 
place, and tumours appear in each axilla. The system becomes 
affected, the pulse is quickened; shivering, succeeded by heat, general 
lassitude and pains about the loins and limbs, with vomiting, come on. 
The head is painful, and the patient is now and then affected with 
delirium. These symptoms, varying in their degrees of violence, 
generally continue from one day to three or four, leaving ulcerated 
sores about the hands, which, from the sensibility of the parts, are 
very troublesome, and commonly heal slowly, frequently becoming 
phagedenic, like those from whence they sprung. The lips, nostrils, 
eyelids, and other parts of the body, are somtimes affected with sores; 
but these evidently arise from their being heedlessly rubbed or 
scratched with the patient’s infected fingers. No eruptions on the 
skin have followed the decline of the feverish symptoms in any in- 
stance that has come under my inspection, one only excepted, and in 
his case a very few appeared on the arms: they were very minute, of 
a vivid red colour, and soon died away without advancing to matura- 
tion ; so that I cannot determine whether they had any connection 
with the preceding symptoms. 
Thus the disease makes its progress from the Horse (as I conceive) 
to the nipple of the Cow, and from the Cow to the Human Subject. 
Morbid matter of various kinds, when absorbed into the system, 
may produce effects in some degree similar; but what renders the 
Cow Pox virus so extremely singular, is, that the person who has 
been thus affected is forever after secure from the infection of the 
Small Pox ; neither exposure to the variolous effluvia, nor the insertion 
of the matter into the skin, producing this distemper. 
In support of so extraordinary a fact, I shall lay before my reader 
a great number of instances : but first it is necessary to observe, that 
pustulous sores frequently appear spontaneously on the nipples of 
the cows, and instances have occurred, though very rarely, of the 
hands of the servants employed in milking being affected with sores 
in consequence, and even of their feeling an indisposition from ab- 
sorption. These pustules are of a much milder nature than those 
which arise from that contagion which constitutes the true Cow Pox. 
They are always free from the bluish or livid tint so conspicuous in APPENDIX G: JENNER 
425 
the pustules in that disease. No erysipelas attends them, nor do they 
shew any phagedenic disposition as in the other case, but quickly 
terminate in a scab without creating any apparent disorder in the 
Cow. This complaint appears at various seasons of the year, but 
most commonly in the spring, when the Cows are first taken from 
their winter food and fed with grass. It is very apt to appear also 
when they are suckling their young. But this disease is not considered 
as similar in any respect to that of which I am treating, as it is in- 
capable of producing any specific effects on the human constitution. 
However, it is of the greatest consequence to point it out here, lest 
the want of discrimination should occasion an idea of security from 
the infection of the Small Pox, which might prove delusive. 
[.Hereupon follow detailed descriptions of numerous cases illustrating Jenner’s 
ideas. Of these we quote only two: Case I, illustrating Jenner’s observations of 
immunity to smallpox acquired naturally by accidental inoculation or vaccination 
with cowpox virus, and Case XVII, his first and therefore most famous example 
of experimental inoculation (or vaccination) upon the person of a boy named James 
Phipps. This, the earliest experiment of the kind ever made, occurred on May 14, 
1796.) 
Case I 
Joseph Merret, now an Under Gardener of the Earl of Berkeley, 
lived as a servant with a Farmer near this place in the year 1770, 
and occasionally assisted in milking his master’s cows. Several 
horses belonging to the farm began to have sore heels, which Merret 
frequently attended. The cows soon became affected with the Cow 
Pox, and soon after several sores appeared on his hands. Swellings 
and stiffness in each axilla followed, and he was so much indisposed 
for several days as to be incapable of pursuing his ordinary employ- 
ment. Previously to the appearance of the distemper among the 
cows there was no fresh cow brought into the farm, nor any servant 
employed who was affected with the Cow Pox. 
In April, 1795, a general inoculation taking place here, Merret was 
inoculated with his family; so that a period of twenty-five years had 
elapsed from his having the Cow Pox to this time. However, though 
the variolous matter was repeatedly inserted into his arm, I found it 
impracticable to infect him with it; an efflorescence only, taking on 
an erysipelatous look about the centre, appearing on the skin near 
the punctured parts. During the whole time that his family had the 426 
A SHORT HISTORY OF SCIENCE 
Small Pox, one of whom had it very full, he remained in the house 
with them, but received no injury from exposure to the contagion. 
It is necessary to observe, that the utmost care was taken to as- 
certain, with the most scrupulous precision, that no one whose case 
is here adduced had gone through the Small Pox previous to these 
attempts to produce that disease. . . . 
Case XVII 
The more accurately to observe the progress of the infection, I 
selected a healthy boy, about eight years old, for the purpose of inocu- 
lation for the Cow Pox. The matter was taken from a sore on the 
hand of a dairymaid (Sarah Nelmes), who was infected by her master’s 
cows, and it was inserted, on the 14th of May, 1796, into the arm of 
the boy by means of two superficial incisions, barely penetrating the 
cutis, each about half an inch long. 
On the seventh day he complained of uneasiness in the axilla, and 
on the ninth he became a little chilly, lost his appetite, and had a 
slight head-ache. During the whole of this day he was perceptibly 
indisposed, and spent the night with some degree of restlessness, but 
on the day following he was perfectly well. 
The appearance of the incisions in their progress to a state of 
maturation were much the same as when produced in a similar man- 
ner by variolous matter. The only difference which I perceived was, 
in the state of the limpid fluid arising from the action of the virus, 
which assumed rather a darker hue, and in that of the efflorescence 
spreading round the incisions, which had more of an erysipelatous 
look than we commonly perceive when variolous matter has been 
made use of in the same manner; but the whole died away (leaving 
on the inoculated parts scabs and subsequent eschars) without giving 
me or my patient the least trouble. 
In order to ascertain whether the boy, after feeling so slight an 
affection of the system from the Cow Pox virus, was secure from the 
contagion of the Small Pox, he was inoculated the 1st of July follow- 
ing with variolous matter, immediately taken from a pustule. Several 
slight punctures and incisions were made on both his arms, and the 
matter was carefully inserted, but no disease followed. The same 
appearances were observable on the arms as we commonly see when 
a patient has had variolous matter applied after having either the Cow APPENDIX G: JENNER 
427 
Pox or the Small Pox. Several months afterwards he was again 
inoculated with variolous matter, but no sensible effect was pro- 
duced on the constitution. . . . 
I shall now conclude from this Inquiry with some general observa- 
tions on the subject, and on some others which are interwoven 
with it. 
Although I presume it may be unnecessary to produce further 
testimony in support of my assertion “ that the Cow Pox protects the 
human constitution from the infection of the Small Pox,” yet it affords 
me considerable satisfaction to say, that Lord Somerville, the Presi- 
dent of the Board of Agriculture, to whom this paper was shown by 
Sir Joseph Banks, has found upon inquiry that the statements were 
confirmed by the concurring testimony of Mr. Dollan, a surgeon, 
who resides in a dairy country remote from this, in which these ob- 
servations were made. With respect to the opinion adduced “that 
the source of the infection is a peculiar morbid matter arising in the 
horse,” although I have not been able to prove it from actual experi- 
ments conducted immediately under my own eye, yet the evidence 
I have adduced appears sufficient to establish it. 
They who are not in the habit of conducting experiments may not 
be aware of the coincidence of circumstance necessary for their being 
managed so as to prove perfectly decisive; nor how often men engaged 
in professional pursuits are liable to interruptions which disappoint 
them almost at the instant of their being accomplished. However, 
I feel no room for hesitation respecting the common origin of the 
disease, being well convinced that it never appears among the cows 
(except it can be traced to a cow introduced among the general herd 
which has been previously infected, or to an infected servant) unless 
they have been milked by some one who, at the same time, has the 
care of a horse affected with diseased heels. 
The spring of the year 1797, which I intended particularly to have 
devoted to the completion of this investigation, proved from its dry- 
ness, remarkably adverse to my wishes; for it frequently happens, 
while the farmers’ horses are exposed to the cold rains which fall at 
that season that their heels become diseased, and no Cow Pox then 
appeared in the neighborhood. 
The active quality of the virus from the horses’ heels is greatly 428 
A SHORT HISTORY OF SCIENCE 
increased after it has acted on the nipples of the cow, as it rarely 
happens that the horse affects his dresser with sores, and as rarely that 
a milkmaid escapes the infection when she milks infected cows. It 
is most active at the commencement of the disease, even before it 
has acquired a pus-like appearance; indeed I am not confident 
whether this property in the matter does not entirely cease as soon 
as it is secreted in the form of pus. I am induced to think it does 
cease, and that it is the thin darkish-looking fluid only, oozing from 
the newly-formed cracks in the heels, similar to what sometimes 
appears from erysipelatous blisters, which give the disease. Nor 
am I certain that the nipples of the cows are at all times in a state to 
receive the infection. The appearance of the disease in the spring 
and the early part of the summer, when they are disposed to be af- 
fected with spontaneous eruptions so much more frequently than at 
other seasons, induces me to think, that the virus from the horse must 
be received upon them when they are in this state, in order to produce 
effects: experiments, however, must determine these points. But 
it is clear that when the Cow Pox virus is once generated, that the 
cows cannot resist the contagion, in whatever state their nipples may 
chance to be, if they are milked with an infected hand. * 
Whether the matter, either from the cow or the horse will affect 
the sound skin of the human body, I cannot positively determine; 
probably it will not, unless on those parts where the cuticle is ex- 
tremely thin, as on the lips for example. I have known an instance 
of a poor girl who produced an ulceration on her lip by frequently 
holding her finger to her mouth to cool the raging of a Cow-Pox sore 
by blowing upon it. The hands of the farmers’ servants here, from 
the nature of their employments, are constantly exposed to those 
injuries which occasion abrasions of the cuticle, to punctures from 
thornes and such like accidents; so that they are always in a state to 
feel the consequences of exposure to infectious matter. 
It is singular to observe that the Cow Pox virus, although it renders 
the constitution unsusceptible of the variolous, should, nevertheless, 
leave it unchanged with respect to its own action. I have already 
produced an instance to point out this, and shall now corroborate it 
with another. 
Elizabeth Wynne, who had the Cow Pox in the year 1759, was 
inoculated with variolous matter, without effect, in the year 1797, 
and again caught the Cow Pox in the year 1798. When I saw her, APPENDIX G: JENNER 
429 
which was on the eighth day after she received the infection, I found 
her affected with general lassitude, shiverings, alternating with heat, 
coldness of the extremities, and a quick and irregular pulse. These 
symptoms were preceded by a pain in the axilla. On her hand was 
one large pustulous sore. 
It is curious also to observe, that the virus, which with respect to its 
effects is undetermined and uncertain previously to it passing from the 
horse through the medium of the cow, should then not only become 
more active, but should invariably and completely possess those spe- 
cific properties which induce in the human constitution symptoms sim- 
ilar to those of the variolous fever, and effect in it that peculiar change 
which forever renders it unsusceptible of the variolous contagion. 
May it not then be reasonably conjectured, that the source of the 
Small Pox is morbid matter of a peculiar kind, generated by a disease 
in the horse, and that accidental circumstances may have again and 
again arisen, still working new changes upon it, until it has ac- 
quired the contagious and malignant form under which we now com- 
monly see it making its devastations amongst us? And, from a 
consideration of the change which the infectious matter undergoes 
from producing a disease on the cow, may we not conceive that many 
contagious diseases, now prevalent amongst us, may owe their present 
appearance not to a simple, but to a compound origin ? For example, 
is it difficult to imagine that the measles, the scarlet fever, and the 
ulcerous sore throat with a spotted skin, have all sprung from the 
same source, assuming some variety in their forms according to the 
nature of their new combinations? The same question will apply 
respecting the origin of many other contagious diseases, which bear 
a strong analogy to each other. 
H. PRINCIPLES OF GEOLOGY: BEING AN ATTEMPT TO EXPLAIN 
THE FORMER CHANGES OF THE EARTH’S SURFACE BY 
REFERENCE TO CAUSES NOW IN OPERATION 
[The first edition of this epoch-making work appeared in 1830 and the second 
edition, from which the following excerpts are taken, in 1832. Few if any hooks 
of the nineteenth century have had greater influence upon human thought. The 
first four chapters constitute an invaluable review of previous geological opinion, 
from the earliest times. The following quotations are from the end of the 
fourth and the latter part of the fifth chapters.] 
Charles Lyell, Esq., F.R.S. 430 
A SHORT HISTORY OF SCIENCE 
We have now arrived at the era of living authors, and shall bring 
to a conclusion our sketch of the progress of opinion in Geology. . . . 
A new school at last arose who professed the strictest neutrality, and 
the utmost indifference to the systems of Werner and Hutton, and 
who were resolved diligently to devote their labours to observation. 
The reaction, provoked by the intemperance of the conflicting par- 
ties, now produced a tendency to extreme caution. Speculative 
views were discountenanced, and through fear of exposing themselves 
to the suspicion of a bias towards the dogmas of a party, some geol- 
ogists became anxious to entertain no opinion whatever on the causes 
of phenomena, and were inclined to scepticism even where the con- 
clusions deducible from observed facts scarcely admitted of reasonable 
doubt. 
Geological Society of London. — But although the reluctance to 
theorize was carried somewhat to excess, no measure could be more 
salutary at such a moment than a suspension of all attempts to form 
what were termed “ theories of the earth.” A great body of new data 
were required, and the Geological Society of London, founded in 
1807, conduced greatly to the attainment of this desirable end. To 
multiply and record observations, and patiently to await the result 
at some future period, was the object proposed by them, and it was 
their favourite maxim that the time was not yet come for a general 
system of geology, but that all must be content for many years to be 
exclusively engaged in furnishing materials for future generalizations. 
By acting up to these principles with consistency, they in a few years 
disarmed all prejudice, and rescued the science from the imputation 
of being a dangerous, or at best but a visionary pursuit. 
Modern Progress of Geology 
Study of Organic Remains. — Inquiries were at the same time 
prosecuted with great success by the French naturalists, who devoted 
their attention especially to the study of organic remains. They 
shewed that the specific characters of fossil shells and vertebrated 
animals might be determined with the utmost precision, and by their 
exertions a degree of accuracy was introduced into this department 
of science, of which it had never before been deemed susceptible. It 
was found that, by the careful discrimination of the fossil contents of 
strata, the contemporary origin of different groups could often be APPENDIX H: LYELL 
431 
established, even where all identity of mineralogical character was 
wanting, and where no light could be derived from the order of super- 
position. 
The minute investigation, moreover, of the relics of the animate 
creation of former ages, had a powerful effect in dispelling the illusion 
which had long prevailed concerning the absence of analogy between 
the ancient and modern state of our planet. A close comparison of 
the recent and fossil species, and the inferences drawn in regard to 
their habits, accustomed the geologist to contemplate the earth as 
having been at successive periods the dwelling-place of animals and 
plants of different races, some of which were discovered to have been 
terrestrial, and others aquatic — some fitted to live in seas, others in 
the waters of lakes and rivers. By the consideration of these topics, 
the mind was slowly and insensibly withdrawn from imaginary pic- 
tures of catastrophes and chaotic confusion, such as haunted the 
imagination of the early cosmogonists. Numerous proofs were dis- 
covered of the tranquil deposition of sedimentary matter and the 
slow development of organic life. If many still continued to main- 
tain, that “ the thread of induction was broken, ” yet in reasoning by 
the strict rules of induction from recent to fossil species, they vir- 
tually disclaimed the dogma which in theory they professed. The 
adoption of the same generic, and, in some cases, even the same 
specific, names for the exuviae of fossil animals, and their living ana- 
logues, was an important step towards familiarising the mind with the 
idea of the identity and unity of the system in distant eras. It was 
an acknowledgment, as it were, that a considerable part of the ancient 
memorials of nature were written in a living language. The growing 
importance then of the natural history of organic remains, and its 
general application to geology, may be pointed out as the character- 
istic feature of the progress of the science during the present century. 
This branch of knowledge has already become an instrument of great 
power in the discovery of truths in geology, and is continuing daily 
to unfold new data for grand and enlarged views respecting the former 
changes of the earth. 
When we compare the result of observations in the last thirty years 
with those of the three preceding centuries, we cannot but look 
forward with the most sanguine expectations to the degree of excel- 
lence to which geology may be carried, even by the labours of the 
present generation. Never, perhaps, did any science, with the excep- 432 
A SHORT HISTORY OF SCIENCE 
tion of astronomy, unfold, in an equally brief period, so many novel 
and unexpected truths, and overturn so many preconceived opinions. 
The senses had for ages declared the earth to be at rest, until the 
astronomer taught that it was carried through space with incon- 
ceivable rapidity. In like manner was the surface of this planet 
regarded as having remained unaltered since its creation, until the 
geologist proved that it had been the theatre of reiterated change, 
and was still the object of slow but never-ending fluctuations. The 
discovery of other systems in the boundless regions of space was the 
triumph of astronomy — to trace the same system through various 
transformations — to behold it at successive eras adorned with dif- 
ferent hills and valleys, lakes and seas, and peopled with new inhabi- 
tants, was the delightful meed of geological research. By the geom- 
eter were measured the regions of space, and the relative distances 
of the heavenly bodies — by the geologist myriads of ages were reck- 
oned, not by arithmetical computation, but by a train of physical 
events — a succession of phenomena in the animate and inanimate 
worlds — signs which convey to our minds more definite ideas than 
figures can do, of the immensity of time. 
Whether our investigation of the earth’s history and structure 
will eventually be productive of as great practical benefits to man- 
kind, as a knowledge of the distant heavens, must remain for the 
decision of posterity. It was not till astronomy had been enriched 
by the observations of many centuries, and had made its way against 
popular prejudices to the establishment of a sound theory, that its 
application to the useful arts was most conspicuous. The cultiva- 
tion of geology began at a later period; and in every step which it 
has hitherto made towards sound ethical principles, it has had to 
contend against more violent prepossessions. The practical advan- 
tages already derived from it have not been inconsiderable: but our 
generalizations are yet imperfect, and they who follow may be expected 
to reap the most valuable fruits of our labour. Meanwhile the charm 
of first discovery is our own, and as we explore this magnificent field 
of inquiry, the sentiment of a great historian of our times may con- 
tinually be present to our minds, that “he who calls what has van- 
ished back again into being, enjoys a bliss like that of creating.” . . . APPENDIX H; LYELL 
433 
Assumption of the Discordance of the Ancient and Existing 
Causes of Change Unphilosophical 
... For more than two centuries the shelly strata of the Sub- 
Apennine hills afforded matter of speculation to the early geologists 
of Italy, and few of them had any suspicion that similar deposits 
were then forming in the neighboring sea. They were as unconscious 
of the continued action of causes still producing similar effects, as the 
astronomers, in the case supposed by us, of the existence of certain 
heavenly bodies still giving and reflecting light, and performing their 
movements as in the olden time. Some imagined that the strata, 
so rich in organic remains, instead of being due to secondary agents, 
had been so created in the beginning of things by the fiat of the Al- 
mighty ; and others ascribed the imbedded fossil bodies to some plastic 
power which resided in the earth in the early ages of the world. At 
length Donati explored the bed of the Adriatic, and found the closest 
resemblance between the new deposits there forming, and those 
which constituted hills above a thousand feet high in various parts 
of the peninsula. He ascertained that certain genera of living testacea 
were grouped together at the bottom of the sea in precisely the same 
manner as were their fossil analogues in the strata of the hills, and that 
some species were common to the recent and fossil world. Beds of 
shells, moreover, in the Adriatic, were becoming incrusted with cal- 
careous rock; and others were recently enclosed in deposits of sand 
and clay, precisely as fossil shells were found in the hills. This splen- 
did discovery of the identity of modern and ancient submarine opera- 
tions was not made without the aid of artificial instruments, which, 
like the telescope, brought phenomena into view not otherwise within 
the sphere of human observation. 
In like manner, in the Vicentin, a great series of volcanic and marine 
sedimentary rocks were examined in the early part of the last century; 
but no geologist suspected, before the time of Arduino, that these 
were partly composed of ancient submarine lavas. If, when these 
enquiries were first made, geologists had been told that the mode of 
formation of such rocks might be fully elucidated by the study of 
processes then going on in certain parts of the Mediterranean, 
they would have been as incredulous as geometers would have been 
before the time of Newton, if any one had informed them that, 
by making experiments on the motion of bodies on the earth, they 434 
A SHORT HISTORY OF SCIENCE 
might discover the laws which regulated the movements of distant 
planets. 
The establishment, from time to time, of numerous points of identi- 
fication, drew at length from geologists a reluctant admission, that 
there was more correspondence between the physical constitution 
of the globe, and more uniformity in the laws regulating the changes 
of its surface, from the most remote eras to the present, than they at 
first imagined. If, in this state of the science, they still despaired or 
reconciling every class of geological phenomena to the operations of 
ordinary causes, even by straining analogy to the utmost limits of 
credibility, we might have expected, that the balance of probability 
at least would now have been presumed to incline towards the identity 
of the causes. But, after repeated experience of the failure of attempts 
to speculate on different classes of geological phenomena, as belong- 
ing to a distinct order of things, each new sect persevered systematic- 
ally in the principles adopted by their predecessors. They invariably 
began, as each new problem presented itself, whether relating to the 
animate or inanimate world, to assume in their theories, that the 
economy of nature was formerly governed by rules quite independent 
of those now established. Whether they endeavoured to account 
for the origin of certain igneous rocks, or to explain the forces which 
elevated hills or excavated valleys, or the causes which led to the 
extinction of certain races of animals, they first presupposed an orig- 
inal and dissimilar order of nature; and when at length they approxi- 
mated, or entirely came round to an opposite opinion, it was always 
with the feeling, that they conceded what they were justified a priori 
in deeming improbable. In a word, the same men who, as natural 
philosophers, would have been greatly surprised to find any deviation 
from the usual course of Nature in their own time, were equally sur- 
prised, as geologists, not to find such deviations at every period of 
the past. 
The Huttonians were conscious that no check could be given to 
the utmost license of conjecture in speculating on the causes of geo- 
logical phenomena, unless we can assume invariable constancy in 
the order of Nature. But when they asserted this uniformity with- 
out any limitation as to time, they were considered, by the majority 
of their contemporaries, to have been carried too far, especially as 
they applied the same principle to the laws of the organic, as well as 
of the inanimate world. APPENDIX H: LYELL 
435 
We shall first advert briefly to many difficulties which formerly 
appeared insurmountable, but which, in the last forty years, have 
been partially or entirely removed by the progress of science; and 
shall afterwards consider the objections that still remain to the doc- 
trine of absolute uniformity. 
In the first place, it was necessary for the supporters of this doc- 
trine to take for granted incalculable periods of time, in order to ex- 
plain the formation of sedimentary strata by causes now in diurnal 
action. The time which they required theoretically, is now granted, 
as it were, or has become absolutely requisite, to account for another 
class of phenomena brought to light by more recent investigations. 
It must always have been evident to unbiassed minds, that succes- 
sive strata, containing, in regular order of superposition, distinct 
beds of shells and corals, arranged in families as they grow at the 
bottom of the sea, could only have been formed by slow and insen- 
sible degrees in a great lapse of ages, yet, until organic remains were 
minutely examined and specifically determined, it was rarely possible 
to prove that the series of deposits met with in one country was not 
formed simultaneously with that found in another. But we are now 
able to determine, in numerous instances, the relative dates of sedi- 
mentary rocks in distant regions, and to show, by their organic re- 
mains, that they were not of contemporary origin, but formed in 
succession. We often find, that where an interruption in the consecu- 
tive formations in one district is indicated by a sudden transition 
from one assemblage of fossil species to another, the chasm is filled 
up, in some other district, by other important groups of strata. 
The more attentively we study the European continent, the greater 
we find the extension of the whole series of geological formations. 
No sooner does the calendar appear to be completed, and the signs 
of a succession of physical events arranged in chronological order, 
than we are called upon to intercalate, as it were, some new periods 
of vast duration. A geologist, whose observations have been confined 
to England, is accustomed to consider the superior and newer groups 
of marine strata in our island as modern, and such they are, compara- 
tively speaking; but when he has travelled through the Italian 
peninsula and in Sicily, and has seen strata of more recent origin 
forming mountains several thousand feet high, and has marked a 
long series both of volcanic and submarine operations, all newer than 
any of the regular strata which enter largely into the physical struc- 436 
A SHORT HISTORY OF SCIENCE 
ture of Great Britain, he returns with more exalted conceptions of 
the antiquity of some of our modern deposits, than he before enter- 
tained of the oldest of the British series. 
We cannot reflect on the concessions thus extorted from us, in 
regard to the duration of past time, without foreseeing that the period 
may arrive when part of the Huttonian theory will be combated on 
the ground of its departing too far from the assumption of uniformity 
in the order of nature. On a closer investigation of extinct volcanos, 
we find proofs that they broke out at successive eras, and that the 
eruptions of one group were often concluded long before others had 
commenced their activity. Some were burning when one class of 
organic beings were in existence, others came into action when dif- 
ferent races of animals and plants existed, — it follows, therefore, 
that the convulsions caused by subterranean movements, which are 
merely another portion of the volcanic phenomena, occurred also 
in succession, and their efforts must be divided into separate sums, 
and assigned to separate periods of time; and this is not all: when 
we examine the volcanic products, whether they be lavas which flowed 
out under water or upon dry land, we find that intervals of time, 
often of great length, intervened between their formation, and that 
the effects of one eruption were not greater in amount than that 
which now results during ordinary volcanic convulsions. The ac- 
companying or preceding earthquakes, therefore, may be considered 
to have been also successive, and to have been in like manner inter- 
rupted by intervals of time, and not to have exceeded in violence 
those now experienced in the ordinary course of nature. 
Already, therefore, may we regard the doctrine of the sudden eleva- 
tion of whole continents by paroxysmal eruptions as invalidated ; and 
there was the greatest inconsistency in the adoption of such a tenet 
by the Huttonians, who were anxious to reconcile former changes to 
the present economy of the world. It was contrary to analogy to 
suppose that Nature had been at any former epoch parsimonious of 
time and prodigal of violence — to imagine that one district was not 
at rest while another was convulsed — that the disturbing forces were 
not kept under subjection, so as never to carry simultaneous havoc 
and desolation over the whole earth, or even over one great region. 
If it could have been shown, that a certain combination of circum- 
stances would at some future period produce a crisis in the subter- 
ranean action, we should certainly have had no right to oppose our APPENDIX H: LYELL 
437 
experience for the last three thousand years as an argument against 
the probability of such occurrences in past ages; but it is not pre- 
tended that such a combination can be foreseen. 
In speculating on catastrophes by water, we may certainly antici- 
pate great floods in future, and we may therefore presume that they 
have happened again and again in past times. The existence of 
enormous seas of fresh water such as the North American lakes, the 
largest of which is elevated more than six hundred feet above the 
level of the ocean, and is in parts twelve hundred feet deep, is alone 
sufficient to assure us, that the time will come, however distant, when 
a deluge will lay waste a considerable part of the American continent. 
No hypothetical agency is required to cause the sudden escape of the 
confined waters. Such changes of level, and opening of fissures, as 
have accompanied earthquakes since the commencement of the 
present century, or such excavation of ravines as the receding cataract 
of Niagara is now effecting, might breach the barriers. Notwith- 
standing, therefore, that we have not witnessed within the last three 
thousand years the devastation by deluge of a large continent, yet, 
as we may predict the future occurrence of such catastrophes, we are 
authorized to regard them as part of the present order of Nature, and 
they may be introduced into geological speculations respecting the 
past, provided we do not imagine them to have been more frequent 
or general than we expect them to be in time to come. 
The great contrast in the aspect of the older and newer rocks, in 
their texture, structure, and in the derangement of the strata, ap- 
peared formerly one of the strongest grounds for presuming that the 
causes to which they owed their origin were perfectly dissimilar from 
those now in operation. But this incongruity may now be regarded 
as the natural result of subsequent modifications, since the difference 
of the relative age is demonstrated to have been so immense, that, 
however slow and insensible the change, it must have become im- 
portant in the course of so many ages. In addition to the volcanic 
heat, to which the Vulcanists formerly attributed too much influence, 
we must allow for the effect of mechanical pressure, of chemical 
affinity, of percolation by mineral waters, of permeation by elastic 
fluids, and the action, perhaps, of many other forces less understood, 
such as electricity and magnetism. In regard to the signs of up- 
raising and sinking, of fracture and contortion in rocks, it is evident 
that newer strata cannot be shaken by earthquakes, unless the sub- 438 
A SHORT HISTORY OF SCIENCE 
jacent rocks are also affected; so that the contrast in the relative 
degree of disturbance in the more ancient and the newer strata, is 
one of many proofs that the convulsions have happened in different 
eras, and the fact confirms the uniformity of the action of subter- 
ranean forces, instead of their greater violence in the primeval ages. 
The science of Geology is enormously indebted to Lyell — more so, as I believe, 
than to any other man who ever lived. — Darwin. Autobiography. 
Pour juger de ce qui est arrive, et meme de ce qui arrivera, nous n’avons qu’d, 
examiner ce qui arrive. — Buffon. Theorie de la Terre. 
I. SOME INVENTIONS OF THE EIGHTEENTH AND NINE- 
TEENTH CENTURIES. APPLIED SCIENCE AND ENGINEERING 
He who seeks for immediate practical use in the pursuit of science, may be 
reasonably sure that he will seek in vain. Complete knowledge and complete 
understanding of the action of the forces of nature and of the mind, is the 
only thing that science can aim at. The individual investigator must find 
his reward in the joy of new discoveries ... in the consciousness of having 
contributed to the growing capital of knowledge. . . . Who could have 
imagined, when Galvani observed the twitching of the frog muscles as he 
brought various metals in contact with them, that eighty years later Europe 
would be overspun with wires which transmit messages from Madrid to 
St. Petersburg with the rapidity of lightning, by means of the same principle 
whose first manifestations this anatomist then observed. — Helmholtz. 
The place of inventions in the history of science is hard to define. 
Conditioned as they doubtless are by a favorable environment — 
at least for survival — they do not always obviously arise as a direct 
or logical consequence of preceding discoveries, or even of known 
principles, but seem sometimes to spring almost de novo from the brain 
of the inventor. And yet such an origin is probably more apparent 
than real. The steam-engine could hardly have come from Watt 
without Newcomen and Black as his predecessors, the telegraph from 
Morse or the telephone from Bell except after Franklin, Oersted and 
Faraday. Probably the truth is that if we only knew all the facts, 
instead of only some of them, we should find every invention the 
natural descendant, near or remote, of science already existing. And 
as inheritance often seems to skip a generation or two and children APPENDIX I: INVENTIONS 
439 
sometimes show no discoverable resemblance to their immediate 
forbears, so inventions may come without disclosing any resemblance 
to parent inventions or ideas, while yet really intimately related to 
knowledge that has gone before. 
Nor is it easy to estimate the reciprocal debt of science to inventions 
and the arts. That this debt is large there can be no doubt. To 
illustrate this fact it is hardly necessary to do more than mention 
examples, such as the service of the compass to the sciences of geog- 
raphy, navigation and surveying; of the telescope and the chronom- 
eter to astronomy; of the microscope to biology; of the air pump 
to natural philosophy; or of the abacus or the Arabic numerals to 
arithmetic. 
Among the more notable of the inventions of the nineteenth cen- 
tury were the locomotive, the steamboat, the friction match, the 
sewing-machine, the steel pen, the telegraph, the telephone and the 
phonograph; labor-saving machinery; explosives; and the internal 
combustion engine, with its numerous offspring (motor vehicles, air- 
planes, motor boats, etc.). 
Power : Its Sources and Significance. — The recent progress 
of science and of civilization has been accompanied by a remarkable 
extension of man’s control over his environment, which has come 
largely with his ability to develop, transmit, and utilize chemical, 
gravitational and electrical energy or power. The ancients and the 
men of the Middle Ages used chiefly the power of man and other ani- 
mals and of winds (windmills) and to some extent water (i.e. gravi- 
tation), as in water-wheels, but knew little of heat power or chemical 
power and nothing of electrical power, or of power transmission of 
any kind,—except in moving herds, treadmills, or marching armies. 
In past times the chief store of national power was manual labor: to-day 
it is the machine that does the work. — K. Pearson. 
The first step in the modern direction was apparently toward chemi- 
cal power, in the invention of gunpowder. 
Gunpowder, Nitroglycerine, Dynamite. — Gunpowder is be- 
lieved to have been known to the Chinese long belong it appeared 
in Europe An explosive mixture of charcoal, sulphur, and nitre 
was apparently also known to the Arabians, but the first important 
appearance of gunpowder in Europe was about the fourteenth cen- 
tury, and since the sixteenth it has played an all-important part in 440 
A SHORT HISTORY OF SCIENCE 
war and in peace. Its effects upon society and civilization have been 
profound, and with society and civilization the progress of science 
is always closely bound up. 
The manufacture of gunpowder marks the beginning of the manu- 
facture of power, if we may describe the controlled accumulation, 
storage and liberation of energy by that convenient term. In 1845 
gun-cotton was invented by Schonbein, and in 1847 nitroglycerine by 
Sobrero, and both explosives were found to be far more copious and 
powerful sources of energy than gunpowder. It was Alfred Nobel, 
however, a Swedish engineer, who after mixing nitroglycerine with 
gunpowder first made practical use of this for blasting. It was also 
Nobel who in 1867 made nitroglycerine less dangerous by diluting it 
with inert substances such as silicious earth, — mixtures to which he 
gave the name dynamite.1 
The manufacture of power from gravitational sources, such as 
water-power and wind power, goes back to the earliest times — 
sails, wind-mills and water-wheels being of very ancient origin. 
Power from fuel begins with Newcomen, Watt and the steam- 
engine. Electrical power is at present chiefly derived indirectly 
from gravitational (hydraulic) or from chemical (fuel) sources. 
The Steam-engine.—The last half of the eighteenth century was 
not merely an era of great revolutions: it was also an age of great 
inventions and among these, first in importance as well as first to 
arise, was the steam-engine. 
Various and more or less successful attempts to utilize heat or 
steam as a source of power had been made before Watt’s time, such, 
for example, as those of Hero in Alexandria (120 b.c.) the Marquis 
of Worcester (1663) and Newcomen (1705). Of these only New- 
comen’s need be dwelt upon here. In Newcomen’s engine a vertical 
cylinder with piston was used, the piston-rod, also vertical, being fixed 
above to one end of a walking-beam of which the other end carried a 
parallel rod. Thus the rise and fall of the piston caused a corre- 
sponding fall and rise of a parallel rod, which could be attached to 
anything, e.g. to a pump. The cylinder was connected with a steam 
1 Nobel died in 1896, bequeathing his fortune, estimated at $9,000,000, to the 
founding of a fund which supports the international “prizes” — usually $40,000 
each — which bear his name and are annually awarded to those who have most 
contributed to “the good of humanity.” Five prizes have been usually given: 
viz. one in physics, one in chemistry, one in medicine or physiology, one in 
literature and one for the promotion of peace. APPENDIX I: INVENTIONS 
441 
boiler by a pipe fitted with a stopcock, and was filled with steam below 
the piston by opening the stopcock. The steam pressing upon the 
boiler raised the piston and depressed the parallel (pump) rod. The 
stopcock was then closed, a “vent” in the cylinder was opened, cold 
water was introduced from another pipe to condense the steam, where- 
upon a vacuum formed, and the atmospheric pressure depressed the 
piston and lifted the pump rod. By having the various stopcocks 
carefully worked by hand a certain regularity of operation could 
be obtained, but before long improvements were made and the stop- 
cocks were caused to work automatically. But since the cold (con- 
densing) water chilled the cylinder, much heat was necessarily wasted. 
Watt began by inventing (in 1765) a separate condenser, for cooling 
the steam without cooling the cylinder, — thus saving a vast amount 
of heat. He next abandoned altogether the use of atmospheric pres- 
sure for depressing the piston, employing steam above as well as 
below the piston, to lower as well as to lift it: and with these improve- 
ments, to which he added many others, he soon had in his possession 
a serviceable and automatic steam-engine, rudimentary in many 
respects, but not essentially unlike that of to-day. 
The Spinning Jenny, the Water-Frame and the Mule.—In 
1770 James Hargreaves patented the spinning jenny, a frame with a 
number of spindles side by side, by which many threads could be 
spun at once instead of only one, as in the old, one-thread, distaff or 
the spinning wheel. In 1771 Arkwright operated successfully in a 
mill a patent spinning machine which, because actuated by water 
power, was known as the “water-frame.” In 1779 Crompton com- 
bined the principles involved in Hargreaves’ and Arkwright’s machines 
into one, which, because of this hybrid origin, became known as the 
spinning “mule.” This proved so successful that by 1811 more than 
four and a half million spindles worked as “mules” were in operation 
in England. 
A similar machine* for weaving was soon urgently needed, and in 
1785 the “power loom” of Cartwright appeared, although it required 
much improvement and was not widely used before 1813. 
The Cotton Gin (engine). — With the inventions just described 
facilities arose for the manufacture of cotton as well as woollen, but 
the supply of raw cotton was limited, chiefly because of the difficulty 
of separating the staple (fibres) from the seeds upon which they are 
borne. Cotton had for centuries been grown and manufactured in 442 
A SHORT HISTORY OF SCIENCE 
India, the fibres being separated from the seeds by a rude hand ma- 
chine known as a churka, used by the Chinese and Hindus. By 
this it was impossible to clean cotton rapidly. The invention there- 
fore in 1793 by Eli Whitney of Connecticut of the saw cotton- 
gin which enormously facilitated this separation was one of the 
most important inventions ever made. This consisted in a series 
of saws revolving between the interstices of an iron bed upon which 
the cotton was so placed as to be drawn through while the seeds were 
left behind. The value of the saw gin was instantly recognized and 
the output of cotton in America was rapidly and immensely increased 
by its use. 
Steam Transportation. — Boats and ships propelled by man 
power or by the wind have been used from time immemorial, and 
parallel rails for wheeled conveyors moved by animal power or by 
gravity preceded the steam locomotive. The steamboat and the 
steam vehicle appeared at (or in the case of the latter even before) 
the opening of the nineteenth century. 
The first practically successful steamboat was a tug, the Charlotte 
Dundas, built and operated in Soctland for the towing of canal boats 
by Symmington in 1802. The first commercially successful steam- 
boat was Fulton’s Clermont, on the Hudson, in 1807. The first 
steam-engine to run on roads appears to have been Cugnot’s in France 
in 1769. The first to run on rails was Trevithick’s, in 1804, built to 
fit the rails of a horse railway. This engine also discharged its exhaust 
steam into the funnel to aid the draught of the furnace, — a device 
of fundamental importance to the further development of the loco- 
motive. The first practically successful locomotive was Stephenson’s 
Rocket (1829). 
The compound (double or triple expansion) engine, which dates 
from 1781 (Hornblower), 1804 (Woolf), and 1845 (McNaughton), 
embodies what is perhaps the greatest single improvement in the 
steam-engine in the nineteenth century. The turbine has begun to 
replace the reciprocating engine only very recently (1900). 
The Achromatic Compound Microscope. — The compound 
microscope, after its introduction about the middle of the seventeenth 
century, and its use by Malphigi, Kircher, Leeuwenhoek, Grew, and 
others, was of only limited value because of the spherical, and espe- 
cially the chromatic, aberration of its lenses. This remained true 
until long after Huygens had perfected the eye-piece of the telescope, APPENDIX I: INVENTIONS 
443 
and Hall and Dolland had succeeded in correcting chromatic aberra- 
tion in telescope objectives by the combination of crown and flint 
glass, in the eighteenth century. 
Amici, of Modena, in 1812, Fraunhofer of Munich in 1816, Tully 
of London in 1824, J. J. Lister in 1830 and others gradually per- 
fected the achromatic microscope objective, so that about 1835 really 
excellent instruments became accessible to microscopical investiga- 
tors. The numerous discoveries in cellular biology and in pathology 
which soon followed testify to the extent and importance of these 
improvements. 
Illuminating Gas, — made by the destructive distillation of coal, 
was invented and introduced in 1792 by William Murdock, who in 
1802 had so far perfected the process that even the exterior of his 
factory in Birmingham was illuminated with gas in celebration of the 
peace of Amiens. 
Friction Matches,—were preceded early in the nineteenth century 
by splinters of wood coated with sulphur and tipped with a mixture 
of chlorate of potash and sugar. These when touched with sulphuric 
acid ignited. It was not, however, until 1827 that practical friction 
matches were made and sold. These were known, after their inventor, 
as “Congreves” and consisted of wooden splints coated with sulphur 
and tipped with a mixture of sulphide of antimony, chlorate of potash, 
and gum. When subjected to severe friction, specially arranged for, 
these took fire. The phosphorus friction match was introduced 
commercially in 1833. 
The Sewing-Machine. — Very few labor-saving inventions sur- 
pass in efficiency sewing-machines. These also were invented in 
the nineteenth century and had a gradual development, in which 
various inventors participated. The first which need be mentioned 
was that of a French tailor, named Thimonier, patented in 1830. 
It is said that although made of wood and clumsy, eighty of these 
machines were in use in Paris in 1841, when an ignorant mob wrecked 
the establishment in which they were located and nearly murdered 
the inventor. The most important ideas embodied in modern ma- 
chines are, however, of strictly American origin, the work of Walter 
Hunt of New York, and of Elias Howe of Spencer, Massachusetts 
being of principal importance (1846). Other Americans, especially 
Singer, Grover, Wilson and Gibbs, afterwards contributed to the 
present excellence and variety of the sewing-machine. 444 
A SHORT HISTORY OF SCIENCE 
Photography. — Scheele, the Swedish chemist, appears to have 
been the first to study the effect of sunlight on silver chloride. Others, 
including Rumford and Davy, observed the chemical properties of 
light, but it was Wedgwood who, in 1802, made the first photograph 
by throwing shadows upon white paper moistened with nitrate of 
silver. Wedgwood was unable, however, to fix his prints. 
Daguerreotypes, taken on silver plated copper, date from 1839, 
and were made by covering the copper with a thin film of silver iodide, 
— a compound sensitive to light. The image was developed by mer- 
cury vapor and fixed by sodium hyposulphite. The discovery of the 
fixing power of hyposulphite was in itself alone of immense impor- 
tance. With the name of Daguerre, who began experimenting in 1826, 
that of a fellow countryman and partner, Niepce, is intimately asso- 
ciated. 
The subsequent development of photography is due to a host of 
workers. The collodion film which underlies all modern work was 
first introduced in 1850. It is said to be a practically perfect medium 
because totally unaffected by silver nitrate. 
Anaesthesia. The Ophthalmoscope. — Anaesthesia, or insen- 
sibility to pain, during dental surgical operations was introduced, if 
not discovered, by Wells, a dentist of Hartford, Connecticut, who 
himself took nitrous oxide gas for anaesthesia in 1844. The first 
public demonstration of surgical anaesthesia under ether was made 
by a dentist, Morton, and a surgeon, Jackson, at the Massachusetts 
General Hospital in Boston in 1846. Anaesthesia by chloroform was 
introduced by Simpson of Edinburgh, in 1847. 
The ophthalmoscope, an instrument for examination of the inte- 
rior of the eye, of inestimable value to medicine, was invented by 
Helmholtz in 1851. It is said that when von Graefe, an eminent 
ophthalmologist, first saw with it the interior of the eye he cried out, 
‘‘Helmholtz has unfolded to us a new world.” 
India-rubber, — the coagulated and dried juice of the rubber 
tree, first reported by Herrera, “ who in the second voyage of Columbus 
observed that the inhabitants of Hayti played a game with balls made 
‘ of the gum of a tree ’ and that the balls although large were lighter, 
and bounced better, than the windballs of Castile,” was at the end of 
the eighteenth century still a curiosity, employed by Priestley, among 
others, as an eraser or “rubber.” 
Rubber is a hydrocarbon soft when pure but readily hardened by APPENDIX I: INVENTIONS 
445 
“ vulcanization,” i.e. treatment with sulphur or certain sulphur com- 
pounds (chloride, carbon bisulphide), a process introduced by Good- 
year in 1839. 
Electrical Apparatus ; Telegraph, Telephone, Electric 
Lighting, Electric Machinery. — The first important applica- 
tion of electricity to the service of man was the telegraph. This is 
too well known to require more than the briefest description. An 
electric circuit in a wire “made” or “broken” at one point is likewise 
made or broken at all other points. Hence, it is only necessary to 
employ a preconcerted system of make-and-break signals to dispatch 
messages. This plan was first employed in 1836 by S. F. B. Morse, 
a native of Charlestown, Massachusetts, and the first telegraph line 
between two cities was installed between Baltimore and Washington 
in 1844. The first transatlantic cable was laid in 1858. 
The telephone, invented by Alexander Graham Bell, is even more 
familiar. This, also, depends on the making and breaking of an elec- 
tric circuit, not (as is usual in the telegraph) by a key manipulated by 
the finger, but by sound waves of the human voice impinging upon a 
delicate membrane (the transmitter) and reproduced at a distance by 
corresponding vibrations of another delicate membrane (the receiver). 
Wireless telegraphy and wireless telephony differ from ordinary 
telegraphy and telephony merely in the use of signal waves set up in 
the ether instead of signal waves (i.e. making and breaking) set up in 
the current carried by a wire. Both arts are inventions of very recent 
date. 
The electric light, which had long been known as a laboratory 
experiment, became of practical utility about 1880, with the inven- 
tion of the incandescent lamp, first the carbon arc and then the car- 
bon filament, the former by Brush, the latter by Edison. 
The phonograph was invented by Edison in 1876, and was the 
culmination of attempts extending over many years to record and 
reproduce sound waves. In these attempts Young, Konig, Fleeming 
Jenkin and many others participated. 
Food Preserving by Canning and Refrigeration. — In 1810 
Appert of France succeeded in preserving foods in closed vessels by 
heating and sealing while hot. In 1816 a small amount of food pre- 
served in this way found its way into the British Navy, where its 
value was recognized to some extent as a preventive of scurvy. It was 
not, however, until after the American Civil War that the industry 446 
A SHORT HISTORY OF SCIENCE 
began to assume anything like the vast extent and importance it has 
since reached. 
Refrigeration in various forms has been used for food preserving 
probably from the earliest times, but the present enormous industry 
of cold storage has all grown up since the middle of the nineteenth 
century with the invention and development of refrigerators (domestic 
and commercial) and especially of machines for producing and dis- 
tributing compressed air or other vapors or brine ammonia and other 
liquids at very low temperatures. These have been perfected rather 
rapidly since 1860, but did not become common before 1880. The 
first cargo of fresh meat successfully exported from America to Europe 
was shipped in March, 1879, and from New Zealand to Europe in 
February, 1880, arriving after a passage of 98 days in excellent 
condition. 
The Internal-Combustion Engine. — For a century or there- 
abouts the steam-engine stood without a rival as a thermodynamic 
machine and prime mover. Innumerable attempts had been made 
meantime to construct other kinds of engines to convert heat more 
directly into power for mechanical work; but it was not until 1876 
that the internal-combustion engine as improved by Otto became a 
practical success. 
In the steam-engine, the furnace in which the heat is generated is 
external to the cylinder in which that heat does its work, the steam 
being merely an intermediary. It is therefore an external-combus- 
tion engine. Obviously, if the fuel burned is made to liberate its 
heat in the cylinder instead of the furnace, the steam can be dis- 
pensed with. This is what actually happens in the internal-com- 
bustion engine. The present enormous extent of the use of such 
engines for motors of all kinds, testifies to the importance of this 
invention. 
Aniline, —was first obtained from indigo in 1826 by Unverdorben 
and named by him crystalline. In 1834 Runge prepared a similar 
substance from coal tar, and in 1841 Fritsche obtained from indigo 
an oil which he called aniline, — a word derived from the Sanskrit 
Nila, the indigo plant. The commercial importance of aniline in the 
dye-stuffs industry dates from the discovery of mauve by Perkin in 
1858. This was the first of the notable series of aniline dyes now so 
well known, and the forerunner of the immense color industry of to-day. 
The Manufacture of Steel; Bessemer. — The making of steel APPENDIX I: INVENTIONS 
447 
by the decarbonization of cast-iron, a process which initiated what 
has been called the “age of steel,” was introduced by Bessemer (1813— 
1898) in 1856. Bessemer’s attention was drawn to the subject by 
his recognition of the necessity of improving gun-metal. Bessemer’s 
process was at first only partially successful, but since others have 
shown how to improve it (by the addition of spiegeleisen, etc.) it 
has reached enormous proportions. 
Agricultural Apparatus and Inventions. — Beginning about 
1850 an era of improved agricultural apparatus began, of which one 
result has been the opening of vast tracts of farm lands which might 
otherwise have remained unproductive. Steel plows, better harrows, 
mowing-machines, horse-power rakes, haymaking machinery, and 
especially harvesters of ingenious design for cereal crops (first intro- 
duced by McCormick in 1834), threshing-machines and spraying- 
machines are to-day common, where these were almost unknown 
before 1875. Machinery has also been applied to dairying, first to the 
making of butter and cheese, and more recently even to the milking 
of cows. Progress has also been made in the preservation of milk and 
of eggs by condensing, drying, freezing, etc. by new and economical 
processes invented and applied since that time. 
Applied Science. Engineering. — Very much as discoveries 
and inventions blend together and as both spring from a common 
source, manifested as curiosity, inquiry, experimentation and cor- 
relation (i.e. from science), so applied science, including engineering, 
comes from a common ancestry, i.e. from correlated knowledge,— 
which is science. Both terms are loosely used and both cover to-day 
a multitude of diversified human activities. 
With the progress of science, arts and invention, engineering and 
other forms of applied science have developed so that these frequently 
have their own schools, either with or apart from universities and 
colleges; the school for miners at Freiberg, in Saxony, begun in 1765, 
being now only one of hundreds of technological and scientific schools 
for the training of engineers and others. Up to 1850 most engineers 
in America were trained in military schools and were primarily military 
engineers. But from that time forward the civil, as opposed to the 
military, engineer began to appear, and from the parent stem of civil 
engineering we now have mechanical, mining, electrical, sanitary, 
chemical, marine and other branches of engineering, often highly 
specialized. The term “engineer” is now very widely employed, with 448 
A SHORT HISTORY OF SCIENCE 
more or less appropriateness, to occupations remote from those of the 
military or civil engineer, as for example, the “ illuminating engineer,” 
the “ efficiency engineer,” the “ public health engineer,” etc. We may 
soon expect to have added to these many others, such as the agricul- 
tural engineer, the forest engineer and even the fishery engineer. 
An historical sketch of applied science and engineering would ob- 
viously include the work of Archimedes, Vitruvius, Frontinus, and 
Leonardo, and proceed with the applications made of the discov- 
eries and inventions of the Renaissance and modern times. Some of 
this ground is covered in the present volume, and more of it in the 
series of books by Smiles entitled Lives of the Engineers. 
— There is scarcely a department of science or art which is the same, or at all the 
same, as it was fifty years ago. A new world of inventions — of railways and of 
telegraphs— has grown up around us which we cannot help seeing; a new world 
of ideas is in the air and affects us, though we do not see it. 
— Bagehot. Physics and Politics {1868). 
— Only since continental ideas and influences have gained ground in this country 
{Great Britain) has the word science gradually taken the place of that which used 
to be termed natural philosophy or simply philosophy. One reason why science 
forms such a prominent feature in the culture of this age is the fact that only 
within the last hundred years has scientific research approached the more intricate 
phenomena and the more hidden forces and conditions which make up and govern 
our everyday life. The great inventions of the sixteenth, seventeenth and eighteenth 
centuries were made without special scientific knowledge, and frequently by persons 
who possessed skill rather than learning. They greatly influenced science and 
promoted knowledge, but they were brought about more by accident or by the prac- 
tical requirements of the age than by the power of an unusual insight acquired by 
study. But in the course of the last hundred years the scientific investigation of 
chemical and electric phenomena has taught us to disentangle the intricate web of 
the elementary forces of nature, to lay bare the many interwoven threads, to break up 
the equilibrium of actual existence, and to bring within our power and under our 
control forces of undreamed-of magnitude. The great inventions of former ages 
were made in countries where practical life, industry and commerce were most 
advanced; but the great inventions of the last fifty years in chemistry and electricity 
and the science of heat have been made in the scientific laboratory: the former 
were stimulated by practical wants; the latter themselves produced new practical 
requirements, and created new spheres of labor, industry, and commerce. Science 
and knowledge have in the course of this century overtaken the march of practical 
life in many directions. — Merz. SKETCH MAP SHOWING PLACES 
IMPORTANT in ancient and mediaeval science 
flu.? H Duff HU. , Some Important Names, Dates and Events in the 
History of Science and Civilization 
(For certain earlier events, see Chapters I and II.) 
c. = circa, about. 
ScTRNOF. 
General History, Literature, 
Art, etc. 
c. 2000-1700. Ahmes Papyrus. 
c. 640-546. Thales, 
c. 611-545. Anaximander. 
c. 588-524. Anaximenes, 
c. 582-500. Pythagoras, 
c. 576-480. Xenophanes, 
c. 540-475. Heraclitus, 
c. 539. Parmenides. 
c. 500. Alcmseon. 
c. 500-428. Anaxagoras, 
c. 470. Hippocrates of Chios 
(Mathematician). 
469-399. Socrates, 
c. 465. Empedocles, 
c. 460. Leucippus, 
c. 460-370. Democritus, 
c. 460. Hippocrates of Cos 
(Physician), 
c. 428-347. Archytas. 
427-347. Plato, 
c. 420. Hippias. 
c. 408-? Eudoxus. 
c. 1100. Gades {Cadiz') founded 
by the Phoenicians. 
c. 1000. Homer. David. Solomon, 
c. 850. Carthage founded. 
c. 800. Hesiod, 
c. 753. Rome founded {legend- 
ary). 
c. 700. Nineveh flourishes under 
Sennacherib. 
c. 660. Byzantium founded. 
610. Sappho and other Greek 
poets. 
Necho II undertakes to 
connect River Nile and 
Red Sea by Canal. His 
sailors circumnavigate 
Africa. 
c. 606. Nineveh destroyed. 
c. 600. Marseilles founded. 
c. 560. Croesus and Solon, 
c. 550-478. Confucius, 
c. 538. Babylon taken by Cyms. 
625-456. iEschylus. 
c. 500. Carthaginians explore 
west coast of Africa. 
490-429. Pericles. 
490. Marathon, Battle of. 
c. 484-425. Herodotus. 
480. Thermopylae, Battle of. 
480. Salamis, Battle of. 
480-406. Euripides. Phidias. 
450-385. Aristophanes. 
450-400. Thucydides, 
c. 434-359. Xenophon, 
c. 430. The plague at Athens. 
Before the Sixth Century B.C. 
Sixth Century B.C. 
Fifth Century B.C. 
449 450 
A SHORT HISTORY OF SCIENCE 
General History, Literature, 
Art, etc. 
Science 
c. 400. Meton (Calendar), 
c. 400. Motion of Earth 
(Philolaus). 
384-322. Aristotle. 
375-325. Mensechmus. 
c. 375. Heraclides of Pontus. 
372-287. Theophrastus, 
c. 350-260. Zeno (Stoic). 
342-270. Epicurus, 
c. 330-275. Euclid, 
c. 326. Eudemus. 
384-322. Demosthenes, 
c. 370. Diogenes. Scopas. Prax- 
iteles. 
356-323. Alexander the Great. 
338. Choeronea, Battle of. 
332. Alexandria founded. 
323-30. The Ptolemies, I-VI. 
c. 300. Museum and Library of 
Alexandria. 
300. Epicurus. 
283. The Pharos built at 
Alexandria. 
280. The Colossus of Rhodes. 
c. 285. Theocritus. 
269. Silver money first coined 
in Rome. 
c. 210. The Great Chinese Walt 
begun. 
Paper made in China. 
c. 170. Polybius, 
c. 166. Terence. 
161. Philosophers and Rheto- 
ricians banished from 
Rome. 
146. Carthage destroyed (re- 
built in 123). 
Fourth Century B.C. 
Third Century B.C. 
c. 300. Herophilus. Erasistra- 
tus. 
287-212. Archimedes, 
c. 276-194. Erastosthenes. 
270- Aristarchus, 
c. 260-200. Apollonius. 
238. Decree of Canopus 
{Leap Year). 
Second Century B.C. 
c. 135. Ctesibius. 
c. 146-126. Hipparchus. CHRONOLOGY 
451 
Science 
General History, Literature, 
Art, etc. 
106-43. Cicero. 
102-44. Caesar. 
59 B.C.-17 a.i). 
Livy. 
54 b.c.-39 a.d. 
Seneca. 
47. Ccesar takes Alexandria. 
39. Pollio founds First 
Public Library. 
27. End of Homan Republic. 
Golden Age of Roman 
Literature. (Horace, 
Virgil, Livy, etc.) 
272-337. Constantine (First 
Christian Emperor). 
354-430. St. Augustine. 
98-55. Lucretius, 
c. 70- Geminus. 
c. 63 b.c.-24 a.d. 
Strabo. 
Varro. 
47. Julian Calendar. 
14. Vitruvius. De Archi- 
tectura. 
Nicomachus. 
23-79. Pliny. * 
c. 40-103. Frontinus. 
c. 75. Hero. 
c. 130. Galen. 
c. 140. Ptolemy. {Almagest.) 
c. 140. Theon of Smyrna. 
c. 250. Diophantus. 
c. 300. Pappus. 
c. 370. Theon of Alexandria. 
First Century B.C. 
First Century A.D. 
Second Century 
Third Century A.D. 
Fourth Century 452 
A SHORT HISTORY OF SCIENCE 
Science 
General History, Literature, 
Art, etc. 
Sixth Century A.D. Fifth Century 
410-486. Proclus. 
476. Brahmagupta. 
Martianus Capella 
(Liberal Arts). 
c. 480-524. Boethius, 
c. 630. Arya-bhata. 
781-790. Schools of Alcuin. 
c. 830. Algebra of Alkarismi. 
940-1003. Gerbert (Pope Syl- 
vester II). 
980-1037. Avicenna. 
c. 1000. Bhaskara. 
c. 1038. Alhazen. 
476. Fall of Rome. 
629. Edict of Justinian. 
Schools of Athens 
closed. 
669-632. Mohammed. 
622. The Hegira. 
641. Fall of Alexandria. 
• 
711. Moorish Conquest of 
Spain. 
732. Moorish Invasion of 
Western Europe 
checked by Charles 
Martel. 
c. 742-814. Charlemagne. 
962. Holy Roman Empire. 
Abelard. 
1066. Battle of Hastings. 
1096-1270. Crusades. 
Seventh Century 
Eighth 
Ninth Century 
Tenth 
Eleventh Century CHRONOLOGY 
453 
Science 
General History, Literature, 
Art, etc. 
Arabic numerals. 
1113-1150. Translations of 
Greek Classics 
from Arabic. 
1126-1198. Averroes. 
Jordanus Nemora- 
rius. 
1175. Pisano (Fibonacci)* 
1206-1280. Albertus Magnus. 
1210. Aristotle's Physics 
proscribed in 
Paris. 
1214-1294. Roger Bacon, 
c. 1219. University of Bo- 
logna. 
1249. University College, 
Oxford. 
1284. Peterhouse College, 
Cambridge. 
Mongolian Observa- 
tory at Meraga. 
1364. University of 
Vienna. 
1401-1464. Nicolas of Cusa. 
1423-1461. Peurbach. 
1436-1476. Regiomontanus. 
Tartar Observatory 
(Samar cand). 
1452-1519. Leonardo da Vinci. 
1473-1543. Copernicus. 
1486-1567. Stifel. 
1490-1565. Agricola. 
1493-1541. Paracelsus. 
1501-1576. Cardan. 
1503. Margarita philoso- 
phica. 
Twelfth Century 
1215. Magna Gharta. 
c. 1254-1324. Marco Polo. 
1265-1321. Dante Alighieri. 
c. 1300. Spectacles invented. 
1304-1374. Petrarch. 
1337-1453. The Hundred Tears' 
War. 
c. 1340-1400. Chaucer. 
1340-1450. The Black Death. 
1379-1446. Brunelleschi. 
1444-1511. Bramante. 
c. 1450. Invention of Printing. 
1453. Fall of Constantinople 
to the Turks. 
1471-1528. Diirer. 
1492. Discovery of America. 
1497. Vasco da Gama rounds 
Cape of Good Hope. 
Thirteenth 
Fourteenth 
Fifteenth Century 
Sixteenth 454 
A SHORT HISTORY OF SCIENCE 
Science 
General History, Literature, 
Art, etc. 
1510-1558. Recorde. 
c. 1506-1559. Tartaglia. 
1510-1589. Palissy. 
1512-1594. Mercator. 
1514-1564. Vesalius. 
1514-1576. Rheticus. 
1616-1565. Gesner. 
1522-1565. Ferrari. 
1540-1603. Vieta. 
1543. De Revolutionibus 
of Copernicus. 
1543- Baptista della Porta. 
1544- Gilbert. 
1546-1601. Tycho Brahe. 
1548-1600. Bruno. 
1548-1620. Stevinus. 
1550-1617. Napier. 
1560- Harriott. 
1561- Francis Bacon. 
1664-1642. Galileo. 
1571-1630. Kepler. 
1575-1660. Oughtred. 
1577- Van Helmont. 
1578- W. Harvey. 
1582. Gregorian Calen- 
dar. 
1591-1626. Snellius. 
1593-1662. Desargues. 
1596-1650. Descartes. 
1598-1647. Cavalieri. 
1601- Fermat. 
1602- Von Guericke. 
1608-1647. Torricelli. 
1616-1703. Wallis. 
1623- Pascal. 
1624- Sydenham. 
1627-1691. Boyle. 
1629- Huygens. 
1630- Barrow. 
1635-1703. Hooke. 
1635-1672. Willughby. 
1642-1727. Newton. 
1646-1716. Leibnitz. 
1513. Balboa reaches Pacific 
Ocean. 
1517. Protestant Reforma- 
tion. 
1519-1522. First Circumnaviga- 
tion of the Olobe by 
Magellan. 
1524-1580. Camoens. 
1630. Spinning wheel. 
1547-1616. Cervantes, 
c. 1552-1599. Spenser. 
1564-1616. Shakespeare. 
1673-1637. Ben Jonson. 
1688. Defeat of the Spanish 
Armada. 
1698. Edict of Nantes. 
1600-1681. Calderon. 
1605. Don Quixote. 
1607. First Permanent Eng- 
lish Colony in 
America. 
1608-1774. Milton. 
1618-1648. Thirty Years' War. 
1622-1673. Moltere. 
1631-1700. Dryden. 
1636. Harvard College 
founded. 
1638- Louis XIV. 
1639- Racine. 
Sixteenth Century (Continued) 
Seventeenth Century CHRONOLOGY 
455 
Science 
General History, Literature, 
Art, etc. 
1637. Discours sur la Me- 
thode. (Analytic 
Geometry.) 
1644-1710. Roemer. 
1656-1742. Halley. 
1660-1734. Stahl. 
1668-1738. Boerhaave. 
1677-1761. Hales. 
1687. Principia of New- 
ton. 
1698- Maclaurin. 
1699- Dufay. 
1699- Jussieu. 
1700- Bernouilli, D. 
1705. Newcomen's En- 
gine. 
1706- Franklin. 
1707- Linnaeus. 
1707-1783. Euler. 
1707-1788. Buffon. 
1707-1777. Haller. 
1717-1783. d’Alembert. 
1726-1797. Hutton. 
1728-1793. Hunter. 
1731-1810. Cavendish. 
1733-1804. Priestley. 
1736-1813. Lagrange. 
1736-1819. Watt. 
1738-1822. Herschel, F. W. 
1742- Scheele. 
1743- Lavoisier. 
1744- Lamarck. 
1746-1818. Monge. 
1749- Laplace. 
1750- Werner. 
1752- Legendre. 
1753- Rumford. 
1764. Watt's Steam En- 
gine. 
1766-1844. Dalton. 
1767. Spinning Jenny. 
1769-1832. Cuvier. 
1769-1859. Humboldt. 
1649-1660. English Common- 
wealth. 
1660-1731. De Foe. 
1672-1725. Peter the Great. 
1683. Siege of Vienna by 
Turks. 
1688-1744. Pope. 
1694-1778. Voltaire. 
1709-1784. Samuel Johnson. 
1711-1776. Hume. 
1728-1774. Goldsmith. 
1732-1790. Washington. 
1737-1794. Gibbon. 
1749-1832. Goethe. 
1759-1796. Burns. 
1759-1805. Schiller. 
1769-1821. Napoleon I. 
Seventeenth Century (Continued) 
Eighteenth Century 456 
A SHORT HISTORY OF SCIENCE 
Science 
General History, Literature, 
Art, etc. 
1769. Spinning Frame. 
1773-1829. Young. 
1774. Discovery of Oxy- 
gen. 
1775- Ampere. 
1776- Treviranus. 
1777- Gauss. 
1778- Davy. 
1778- De Candolle. 
1779- Berzelius. 
1781-1848. Stephenson. 
1781. Discovery of Ura- 
nus. 
1783. Air Balloon. 
1784-1846. Bessel. 
Parallax of stars. 
1791-1867. Faraday. 
1791- Morse. 
1792. Cotton Oin. 
1792- von Baer. 
1793- Lobatchewski. 
1794. Ecole Polytech- 
nique. 
1796. Vaccination. 
1796-1832. Carnot. 
1799-1853. St. Hilaire. 
1801-1858. Muller. 
1803-1873. Liebig. 
1807-1873. Agassiz, L. 
1809- Darwin. 
1810- Gray, Asa. 
1811- Leverrier. 
1811-1899. Bunsen. 
1813-1878. Bernard. 
1813-1898. Bessemer. 
1817- Hooker. 
1818- Joule. 
1819- Adams. 
1820- Spencer. 
1820- Florence Nightin- 
gale. 
1821- Helmholtz. 
1822- Mendel. 
1770- Wordsworth. 
1771- Scott. 
1772- Coleridge. 
1773- Metternich. 
1775-1781. American Revolution, 
1789-1794. French Revolution. 
1792-1822. Shelley. 
1795-1821. Keats. 
1795-1881. Carlyle. 
1802-1885. Victor Hugo. 
1802- Kossuth. 
1803- Emerson. 
1804- Cobden. 
1805. Battle of Trafalgar. 
1805- Mazzini. 
1807-1882. Longfellow. 
1807-1882. Garibaldi. 
1807-1892. Whittier. 
1809-1865. Lincoln. 
1809- Tennyson. 
1810- Cavour. 
1811- Thackeray. 
1812- Dickens. 
1815. Battle of Waterloo. 
1815-1898. Bismarck. 
Eighteenth Century (Continued) 
Nineteenth Century CHRONOLOGY 
457 
Science 
General History, Literature, 
Art, etc. 
1822-1888. Clausius. 
1822-1895. Pasteur. 
1822- Gibbs. 
1823- Wallace. 
1824- Kirchhoff. 
1824- Kelvin. 
1825- Huxley. 
1827-1914. Lister. 
1828. Synthesis of Urea. 
1828. Stephenson's 
“ Bocket." 
1829-1896. KekulA 
1830. Lyell's Principles 
of Geology. 
1831-1879. Maxwell. 
1834-1906. Langley. 
1834-1907. Mendelfieff. 
1834- Weismann. 
1835- Newcomb. 
1836. The Telegraph. 
1838-1907. Perkin. 
1839. The Daguerreotype. 
1843-1910. Koch. 
1846. Anaesthesia. 
Discovery of Nep- 
tune. 
1847- Graham Bell. 
1847. Die Erhaltung der 
Kraft. 
1847- Edison. 
1848- De Vries. 
1852- Van’t Hoff. 
1857-1894. Hertz. 
1858. Atlantic Cable. 
1859-1860. Spectroscope. 
1859- Arrhenius. 
1859. Origin of Species. 
1868. Antiseptic Surgery. 
1876. Germ Theory. 
Telephone. 
1877. Phonograph. 
1896. First Successful 
Flight (Langley). 
1834. Poor Law Beform in 
England. 
1837. Accession of Queen 
Victoria. 
1846-1848. War between United 
States and Mexico. 
1846. Bepeal of the Corn 
Laws. 
1848. Abdication of Louis 
Philippe of France. 
1852. Accession of Napoleon 
III. 
1853-1856. Crimean War. 
1854. First (Perry) Treaty 
between United States 
and Japan. 
1859- William II of Germany. 
1859. Peace of Villafranca. 
1861. First Italian Parlia- 
ment. 
1861-1865. Civil War in the United 
States. 
1866. War between Prussia 
and Austria. 
1867. End of the Shogxmate 
of Japan. 
1870-1871. War between Prussia 
and France. 
1890. Promulgation of Con- 
stitution of Japan. 
1894. War between China 
and Japan. 
1898. War between Spain and 
the United States. 
1899. Boer War in South 
Africa. 
1900. Boxer Uprising in 
China. 
Nineteenth Century (Continued) A SHORT LIST OF REFERENCE BOOKS 
Particular attention may be called to the important and valuable publica- 
tions of the John Crerar Library, Chicago, prepared by Aksel G. S. Josephson : 
viz. A List of Books on the History of Science (1911); Supplement to the Same 
(1916); and A List of Books on the History of Industry and Industrial Arts 
(1915). 
A. General 
Abelson, Paul. The Seven Liberal Arts. A study in 
medieval culture. 
Arago, F. E. Biographies of Distinguished Scientific 
Men. 
Avebury. (See Lubbock.) 
Bacon, Francis. The Advancement of Learning. 
Bacon, Roger. (See Little.) 
Bury, J. B. A History of Freedom of Thought. 
Dannemann, F. Die Naturwissenschaften in ihrer Ent- 
wickelung und in ihrem Zusammen- 
hange. 4 vols. (1910-1913.) 
Aus der Werkstatt grosser Forscher. 
Draper, J. W. History of the Conflict between Religion 
and Science. 
History of the Intellectual Development 
of Europe. 
Griffiths, A. B. Biographies of Scientific Men. 
Henderson, L. J. The Order of Nature. 
Holland, R. S. Historic Inventions. 
Hume, M. The Spanish People. 
Huxley, T. H. Advance of Science in the Last Half 
Century. 
Isis. Revue consacree a Vhistoire de la science. 
(March 1913-June 1914.) Edited by 
G. Sarton. 
459 460 
A SHORT HISTORY OF SCIENCE 
Ker, W. P. The Dark Ages. 
Lee, Sidney. Great Englishmen of the Sixteenth Cen- 
tury. 
Libby, W. Introduction to the History of Science. 
Little, A. G. Roger Bacon, Commemoration Essays. 
Lubbock, J. Fifty Years of Science. 
Merz, J. T. A History of European Thought in the 
Nineteenth Century. 
Nineteenth Century, The. A review of progress during the past one 
hundred years in the chief departments 
of human activity. 
Ostwald, W. F. Grosse Manner. 
Ostwald, W. F., Editor. Klassiker der Exacten Wissenschaften (con- 
tains many volumes of historical in- 
terest). 
Pearson, K. The Grammar of Science. 
Phillips, L. March. In the Desert. 
Picard, Emile. La science moderne et son etat actuel. 
Putnam, G. H. Books and their Makers during the 
Middle Ages. 
Rashdall, H. Universities of Europe in the Middle Ages. 
Rolt-Wheeler, F. W. The Science-History of the Universe. 
Routledge, R. A Popular History of Science. 
Sarton, G. (See p. 459.) 
Schaff, P. The Renaissance. 
Smiles, Samuel. Lives of the Engineers. 
Symonds, J. A. Renaissance in Italy. The Revival of 
Learning. 
Thomson, J. A. Progress of Science in the Century. 
Vernon-Harcourt, L. F. Achievements in Engineering during the 
Last Half Century. 
Wallace, A. R. The Wonderful Century. 
Walsh, J. J. Catholic Churchmen in Science. 
The Popes and Science. 
The Thirteenth, Greatest of Centuries. 
Whewell, W. History of the Inductive Sciences, from the 
Earliest to the Present Times. (1837.) 
The Philosophy of the Inductive Sciences, 
founded upon their history. (1847.) A SHORT LIST OF REFERENCE BOOKS 
461 
Whewell, W. History of Scientific Ideas. Being the 
first part of the philosophy of the induc- 
tive sciences. (1858.) 
White, A. D. A History of the Warfare of Science with 
Theology in Christendom. 
Williams, H. S. A History of Science. 
Nineteenth Century Science. 
B. Ancient Science 
Allman, G. J. Greek Geometry from Thales to Euclid. 
Apollonius of Perga. Treatise on Conic Sections, (c. 220 b.c.) 
Aristotle. On the Parts of Animals; on Generation, 
etc. 
Baikie, James. Sea Kings of Crete. 
Breasted, J. H. Ancient Times: a history of the early 
world. 
Butcher, S. H. Some Aspects of the Greek Genius. 
Davidson, Thomas. Aristotle and Ancient Educational Ideals. 
Euclid. Elements, (c. 300 b.c.) 
Fairbanks, A. First Philosophers of Greece. 
Freeman, K. E. Schools of Hellas. 
Frontinus. The Waterworks of Rome. (c. 100 a.d.) 
Galen. On the Natural Faculties, (c. 180 a.d.) 
Gibbon, Edward. Decline and Fall of the Roman Empire. 
Gomperz, Theodor. Greek Thinkers: a history of ancient 
philosophy. 
Gow, James. A Short History of Greek Mathematics. 
Grote, George. Plato and the Other Companions of 
Socrates. 
Hawes, C. H. and H. Crete the Forerunner of Greece. 
Heath, Thomas L. Apollonius of Perga. 
Archimedes. 
Aristarchus of Samos. 
The Thirteen Books of Euclid’s Elements. 
Diophantus of Alexandria. 
Jastrow, M. The Civilization of Babylonia and Assyria. 
Lewes, G. H. Aristotle; a Chapter in the History of 
Science. 462 
A SHORT HISTORY OF SCIENCE 
Lubbock, John. Prehistoric Times. 
Lucretius. On the Nature of Things, (c. 90 b.c.) 
Mahaffy, J. P. What have the Greeks done for Modern 
Civilization ? 
Alexander’s Empire. 
Osborn, H. F. Men of the Old Stone Age. 
Pliny. Natural History, (c. 70 a.d.) 
Ptolemy. Almagest, (c. 150 a.d.) 
Seneca. Physical Science in the Time of Nero. 
(c. 30 A.D.) 
Strabo. Geography, (c. 10 a.d.) 
Tannery, Paul. La Geometrie Grecque. 
Walden, J. W. H. The Universities of Ancient Greece. 
C. Mathematical Science 
(Mathematics, Astronomy, Mechanics) 
Arthur, James. Time and its Measurement. 
Ball, R. S. Great Astronomers. 
Ball, W. W. R. A Primer of the History of Mathematics. 
A Short Account of the History of Mathe- 
matics. 
Berry, Arthur. A Short History of Astronomy. 
Bibliotheca Mathematica. Zeitschrift fur Geschichte der mathemati- 
schen Wissenschaften. 
Brewster, David. Martyrs of Science. 
Memoirs of Sir Isaac Newton. 
Bryant, W. W. A History of Astronomy. 
Cajori, Florian. A History of Elementary Mathematics. 
A History of Mathematics. 
A History of the Logarithmic Slide-rule 
and Allied Instruments. 
The Teaching and History of Mathe- 
matics in the United States. 
Cantor, Moritz. Vorlesungen uber Geschichte der Mathe- 
matik. Vols. I-IV. (1892-1898.) 
Chasles, Michel. Apergu historique sur Vorigine et le de- 
veloppement des methodes en geometric. A SHORT LIST OF REFERENCE BOOKS 
463 
Clerke, A. M. A Popular History of Astronomy during 
the Nineteenth Century. 
The Herschels and Modern Astronomy. 
Modern Cosmogonies. 
de Morgan, Augustus. A Budget of Paradoxes. 
Descartes, Rene. A Discourse on Method, etc. (1637.) 
Dreyer, J. L. E. Tycho Brahe. 
History of Planetary Systems from Thales 
to Kepler. 
Duhem, Pierre. Les origines de la statique. 
Enriques, F. Problems of Science. 
Fink, Karl. A Brief History of Mathematics. 
Forbes, George. History of Astronomy. 
Galileo, G. Dialogues concerning Two New Sciences, 
(1638) translated by H. Crew and A. 
de Salvio. 
Grant, Robert. History of Physical Astronomy, from the 
earliest ages to the middle of the 
nineteenth century. 
Haldane, E. S. Descartes, His Life and Times. 
Hale, G. E. The Study of Stellar Evolution. 
Hobson, E. W. “ Squaring the Circle.” A History of the 
Problem. 
Lockyer, J. N. The Dawn of Astronomy. A Study of 
the Temple Worship and Mythology 
of the Ancient Egyptians. 
Lodge, Oliver. Pioneers of Science. 
Mach, Ernst. The Science of Mechanics. A Critical 
and Historical Exposition of its Prin- 
ciples. 
Miller, G. A. Historical Introduction to Mathematical 
Literature. 
Morley, Henry. Jerome Cardan. 
Napier, John. Tercentenary Memorial Volume. 
Newton, Isaac. Philosophiee naturalis principia mathe- 
matica. (1687.) 
Pierpont, James. The History of Mathematics in the 
Nineteenth Century. 
Poincare, H. Science and Hypothesis. 464 
A SHORT HISTORY OF SCIENCE 
Poincare, H. The Value of Science. 
Reeves, E. Maps and Map Making. Three lectures 
delivered under the auspices of the 
Royal Geographical Society. 
Smith, D. E. Rara Arithmetica. A catalogue of the 
arithmetics written before the year 
MDCI with a description of those in 
the library of George Arthur Plimpton 
of New York. 
History of Modern Mathematics. 
Todhunter, I. History of the Mathematical Theories of 
Attraction and the Figure of the Earth 
from the Time of Newton to that of 
Laplace. 
A History of the Theory of Elasticity and 
of the Strength of Materials, from 
Galileo to Present Time. 
A History of the Mathematical Theory of 
Probability from the Time of Pascal to 
that of Laplace. 
D. Physical and Chemical Science 
Armitage, F. P. A History of Chemistry. 
Arrhenius, S. A. Theories of Chemistry, being lectures 
delivered at the University of Cali- 
. fornia. 
Barus, Carl. The Progress of Physics in the Nineteenth 
Century. 
Bauer, Hugo. A History of Chemistry. 
Berry, A. J. The Atmosphere. 
Boyle, Robert. Sceptical Chymist. (1660.) 
Cajori, Florian. A History of Physics in its Elementary 
Branches, including the Evolution of 
Physical Laboratories. 
Clarke, F. W. The Progress and Development of Chem- 
istry during the Nineteenth Century. 
Fahie, J. J. Galileo, His Life and Work. 465 
A SHORT LIST OF REFERENCE BOOKS 
Gilbert, W. On the Loadstone and Magnetic Bodies, 
and on the Great Magnet, the Earth. 
Glazebrook, R. T. James Clerke Maxwell and Modern 
Physics. 
Gray, A. Lord Kelvin. 
Hales, S. Haemostatics, etc. (1733). 
Hilditch, T. A Concise History of Chemistry. 
Jones, Bence. Life and Letters of Faraday. 
Jones, H. C. A New Era in Chemistry; some of the 
more important developments in 
General Chemistry during the last 
quarter of a century. 
Kelvin. (See Thomson.) 
Lodge, Oliver. Signalling through Space without Wires. 
Being a Description of the Work of 
Hertz and His Successors. 
Mach, Ernst. History and Root of the Principle of the 
Conservation of Energy. 
Mendenhall, T. C. A Century of Electricity. 
Meyer, Ernst. A History of Chemistry from Earliest 
Times to the Present Day. 
Muir, M. M. Pattison. A History of Chemical Theories and Laws. 
The Story of Alchemy and the Beginnings 
of Chemistry. 
O’Reilly, M. F. Makers of Electricity. 
(Potamian). 
Priestley, Joseph. Experiments and Observations on Differ- 
ent Kinds of Air; On the Generation 
of Air from Water, etc. 
Ramsay, William. Essays Biographical and Chemical. 
Ramsay, William. The Gases of the Atmosphere. The 
history of their discovery. 
Redgrove, H. S. Alchemy: Ancient and Modern. 
Roberts, E. Famous Chemists. 
Schorlemmer, Carl. The Rise and Development of Organic 
Chemistry. 
Tait, P. G. Lectures on Some Recent Advances in 
Physical Science. 
Thomson, William. Popular Lectures and Addresses. 466 
A SHORT HISTORY OF SCIENCE 
Thorpe, T. E. Essays in Historical Chemistry. 
History of Chemistry. 
Tyndall, John. Faraday as a Discoverer. 
Venable, F. P. The Development of the Periodic Law. 
“ “ “ A Short History of Chemistry. 
Whetham, W. C. D. The Recent Development of Physical 
Science. 
Whittaker, E. T. A History of the Theories of yEther and 
Electricity from the Age of Descartes 
to the Close of the Nineteenth Century. 
E. Natural Science 
(See also p. 398) 
Agassiz, E. C. Louis Agassiz, His Life and Correspond- 
ence. 
Allbutt, T. C. The Historical Relations of Medicine and 
Surgery to the End of the Sixteenth 
Century. 
Buck, A. H. The Growth of Medicine to the Nine- 
teenth Century. 
Buckley, A. B. A Short History of Natural Science and 
of the Progress of Discovery from the 
Time of the Greeks to the Present Day. 
Clodd, Edward. Pioneers of Evolution from Thales to 
Huxley. 
Darwin, Francis. Life of Charles Darwin. 
Duncan, P. M. Heroes of Science. Botanists, Zoolo- 
gists and Geologists. 
Foster, Michael. Lectures on the History of Physiology 
during the Seventeenth and Eighteenth 
Centuries. 
Garrison, F. H. Introduction to the History of Medicine. 
Geikie, Archibald. The Founders of Geology. 
Goode, G. Brown. The Beginnings of Natural History in 
America. 
Green, E. L. Landmarks of Botanical History. 
Green, J. R. A History of Botany. A SHORT LIST OF REFERENCE BOOKS 
467 
Haddon, A. C. History of Anthropology. 
Harvey, William. An Anatomical Dissertation upon the 
Movement of the Heart and the Blood 
in Animals. (1628). 
Huxley, Leonard. Life and Letters of Thomas Henry 
Huxley. 
Huxley, T. H. Man’s Place in Nature. Essays. 
Advance of Science in Last Half Century. 
Judd, J. W. The Coming of Evolution. 
Lankester, E. R. History and Scope of Zoology. 
Locy, W. A. Biology and Its Makers. 
Lull, R. S. Organic Evolution. 
Osborn, H. F. From the Greeks to Darwin. An outline 
of the development of the evolution 
idea. 
The Origin and Evolution of Life. 
Osier, William. The Growth of Truth as illustrated in the 
discovery of the circulation of the blood. 
Paget, Stephen. Pasteur and After Pasteur. 
Park, Roswell. An Epitome of the History of Medicine. 
Payne, J. F. Thomas Sydenham. 
Power, D’Arcy. William Harvey. 
Romanes, G. J. Darwin and After Darwin. 
Sachs, J History of Botany. 
Thomson, J. A. The Science of Life. An outline of the 
history of biology and its recent 
advances. 
Vallery-Radot, R. Life of Pasteur. 
Woodward, H. B. History of Geology. 
Zittel, K. A. von. History of Geology and Palaeontology to 
the End of the Nineteenth Century. INDEX 
Abacus, 31, 41, 147, 155, 184 
Academies, 269, 301 
Accademia dei Lined, 229, 269 
Acceleration, 246, 326 
Adams, 341 
Agassiz, 373, 385 
Agricola, 226 
Agriculture, 167, 447 
Ahmes Papyrus, 30 
Air-pump, 257, 267 
Albertus Magnus, 177, 179 
Alchemy, 166, 170, 181, 187, 226, 260 
Alcmseon, 56, 63 
Alcuin, 153 
Alexandria, 87 
Algebra, 100,133, 156,162, 179, 232, 240, 
277, 327 
A1 Hazen, 163 
Alkarismi, 162 
Almagest, 127, 193 
Ampere, 354, 363 
Anaesthesia, 444 
Aniline, 446 
Analytic Geometry, 110, 273, 277, 282 
Analytic Method, 71 
Anatomy, 113, 226, 319, 373, 390 
Anaxagoras, 60 
Anaximander, 47 
Anaximenes, 48 
Animatism, 5 
Animism, 5 
Anthropology, 3 
Antiphon, 67 
Antipodes, 150, 180 
Antiquity of Man, 1, 387, 390 
Antiseptic Surgery, 379 
Apollonius, 108-112, 115, 241, 303 
Arabic Numerals, 159, 161, 176, 182, 184 
Arabic Science, 156, 160-171, 176 
Archimedes, 95-107, 111, 112, 115 
Architecture, Roman, 142, 143 
Archytas, 75, 105 
Aristarchus, 116 
Aristillus, 119 
Aristotle, 37, 45, 48, 55, 66, 74, 79-84, 
156, 172, 177, 265 
Arithmetic, Babylonian, 26 
Greek, 40, 50, 125 
Roman, 147 
Arkwright, 441 
Arrhenius, 365 
Art, Geometry in, 230, 234 
Arts, Seven Liberal, 148, 174 
Aryabhata, 157, 160 
Assyria (see Babylonia) 
Astrolabe, 127, 129 
Astrology, 23, 132, 181, 204, 210, 253 
Astronomy, Babylonian, 27 
primitive, 21 
Atmosphere, 256, 260, 307, 315 
Atomism, 61, 348, 361 
Attraction, 293, 348 
Augustine, St., 150 
Avogadro, 362 
Babylonia, 2, 7-11, 26-29, 36, 43, 139 
Bacon, Francis, 81, 203, 270, 312 
Bacon, Roger, 177, 180, 400 
Bacteriology, 380 
Baer, von, 375, 393 
Balance, 258, 308 
Barometer, 256, 267, 302 
Barrow, 289 
Becher, 262 
Bell, 445 
Bergmann, 305 
Berkeley, 298 
Bernard, 376 
Bernoulli, 326 
Bessel, 344 
Bessemer, 446 
Bhaskara, 158 
Biogenesis, 381 
Biology, 315, 371, 392 
Black, 305, 312 
Black Death, 185 
Blood, Circulation of, 255, 412 
Boerhaave, 263 
Boethius, 148, 153, 182 
Bonnet, 320, 372 
Botany, 315, 317, 371, 374 
Boyle, 258, 260-262, 334 
Brahmagupta, 157, 160, 162 
Briggs, 244 
Brown, 393 
Brunelleschi, 234 
Bruno, 254 
Bryson, 67 
469 470 
INDEX 
Buffon, 318, 335, 372, 386 
Bunsen, 353 
Biirgi, 244, 252 
CffiSEr 143 
Calculus, 96, 99, 112, 147, 253, 273, 279, 
282, 296, 324 
Calendar, 21, 29, 38, 143, 181, 242 
Calippus, 78 
Caloric, 265, 312 
Calorimetry, 312 
Candolle, de, 374 
Canopus, Decree of, 108 
Capella, M., 147 
Carbon, 309 
Carbonic Acid Gas, 305, 309 
Cardan, 239 
Carnot, 350 
Catastrophism, 383 
Cavalieri, 278-280 
Cave Animals, 386 
Cavendish, 306, 309 
Cell Theory, 375, 391 
Centrifugal Force, 288 
Chaldeans, 8 
Chambers, 369 
Charlemagne, Schools of, 153 
Chemistry, 226, 260, 304, 360 
Chladni, 311 
Church, Attitude of, 149 
Circle, Squaring of, 28, 32, 67, 98, 
337 
Clausius, 357, 364 
Clepsydra, 27, 40, 127, 148, 163 
Clocks, 188, 248, 266, 287, 328 
Columbus, 181, 192 
Combustion, 260, 262, 307, 309 
Comets, 83, 207, 302 
Comparative Anatomy, 390 
Compass, 164, 187 
Computation, 41, 147, 182, 236, 242 
Computing Machine, 285, 301 
Conic Sections, 76, 95, 108-112, 281 
Conservation of Energy, 289, 348, 349, 
355, 357 
Contagion, 375 
Continuity, 79 
Coordinates, 110, 122, 125, 273, 277 
Copernicus, 55, 194-203, 339, 407 
Cosmogony, 371, 387 
Cosmology, 23, 45, 73, 77 
Cotton Gin, 441 
Coulomb, 354 
Counting, 23-25 
Crete, 16-18 
Crusades, 172 
Crystallography, 364 
Ctesibius, 123 
Cube, Duplication of, 68, 71, 75, 76, 108, 
241 
Cubic Equations, 100, 135, 163, 179, 
239 
Cuneiform Writing, 9 
Cusa, Nicholas of, 192 
Cuvier, 372, 386 
d’Alembert, 328 
Dalton, 361 
Dante, 181 
Dark Ages, 141, 152 
Darwin, 367, 393, 394-396 
Davy, 364 
Decimal Fractions, 252 
Decimal System, 159, 178 
Degrees, 29 
Della Porta, 228, 269 
Democritus, 33, 46, 48, 61, 62, 77 
Desargues, 280 
Descartes, 111, 265, 270, 271, 273- 
280 
Descent of Man, 395 
De Vries, 396 
Dinostratus, 65 
Diophantus, 133-138, 276, 282 
Disease, 264 
Dissipation of Energy, 358 
Dissociation, electrolytic, 365 
Division, 183 
Dufay, 314 
Durer, A., 235 
Dynamics, 248, 293 
Dynamite, 439 
Earth, Motion of, 54, 85, 116, 128, 192, 
195 
Size of, 82, 107, 163, 199 
Eclipses, 10, 28, 30, 43, 55, 121, 129 
Ecliptic, 55, 108, 129 
Economy of Nature, 282, 283, 390 
Egypt, 11, 29-34, 36, 43, 45, 49, 139 
Elasticity, 346 
Electricity, 313, 354, 445 
Electric Lighting, 445 
Electrolysis, 364 
Electromagnetic Theory of Light, 355 
Elements, 45, 47, 52, 59, 74, 83, 261, 
315, 363 
Ellipse, 99, 110, 279, 280 
Embryology, 256, 375 
Empedocles, 59 
Energy, 348, 349, 355, 357, 358 
Engineering, 104, 123, 142, 321, 447 
Epicurus, 84 
Epicycles, 118, 129, 199, 212 
Epigenesis, 375 
Equant, 129, 199 INDEX 
471 
Equations, Theory of, 241, 277, 296 
Equinoxes, Precession of, 120, 129, 165, 
199 
Erasistratus, 113 
Eratosthenes, 107, 122 
Ether, 286, 356 
Euclid, 88-95, 97, 110, 134, 178 
Eudemus, 38, 42 
Eudoxus, 77, 88, 90, 105, 119 
Euler, 283, 326 
Eustachius, 227 
Evolution, 225, 317, 322, 343, 349, 366, 
388, 395 
Excentric Orbits, 117, 129 
Exhaustion, Method of, 77, 112 
Experimentation, 259 
Explosives, 440 
Fabricius, 227, 255 
Falling Bodies, 80, 239, 246 
Fallopius, 227 
Faraday, 354, 364 
Fermat, 110, 282, 301 
Fermentation, 258, 378 
Ferrari, 239 
Fibonacci, 177, 178 
Finger Reckoning, 147, 182 
Fizeau, 352 
Fluids, 249, 289. (See also Hydrostatics) 
Fluxions, 296, 326 
Food Preservation, 446 
Fossils, 56, 228, 316, 317, 385 
Foucault, 352 
Fractions, 30, 135, 163 
Frankland, 362 
Franklin, 270, 314, 329 
Fraunhofer, 353 
Fresnel, 351 
Frontinus, 144 
Function, 243, 324, 327, 336 
Galen, 113, 142, 146 
Galileo, 63, 217-226, 230, 245-253, 267, 
310, 414 
Galvani, 314, 321 
Gas, 258, 305, 321, 443 
Gauss, 323, 337, 340 
Geminus, 38, 85, 97 
Geography, 107, 132, 145, 181 
Geology, 315, 371, 383 
Geometry, Egyptian, 32, 33 
primitive, 25 
Gerbert, 155 
Germ Theory, 378 
Gesner, 228, 386 
Gilbert, 228, 313 
Gioja, 187 
Glaciers, 385 
Goethe, 374 
Gravitation, 214, 291-295, 342 
Gray, 314 
Greek Science, 35, 59, 131, 139 
Green, 345, 354 
Grew, 268 
Guericke, 257, 313 
Gunpowder, 164, 439 
Gutenberg, 189 
Hales, 263, 268, 305, 335 
Haller, 319 
Halley, 296, 302 
Hargreaves, 441 
Harriott, 245 
Harvey, 255, 264, 271, 375, 412 
Hawksbee, 311, 313 
Heat, 249, 312, 350 
Hebrews, 16 
Helmholtz, 311, 323, 356-359, 439, 444 
Helmont, Van, 258 
Heraclides, 85 
Hero, 123 
Herodotus, 16 
Herophilus, 113 
Herschel, 333 
Hertz, 356 
Hicetas, 54, 56 
Hindu Science, 156-160 
Hipparchus, 119-123, 126, 129, 131 
Hippias, 65 
Hippocrates of Chios, 67, 90 
Hippocrates of Cos, 63, 113, 377 
Oath of, 399 
Hooke, 268, 316 
Hour, 28, 40 
Huggins, 353 
Humanism, 185, 191 
Humboldt, von, 393 
Hunter, 319 
Hutton, 317, 383 
Huxley, 269, 366, 372 
Huygens, 266, 286 
Hydrogen, 306, 309 
Hydrostatics, 103, 249, 252 
Hypatia, 138 
Hypsicles, 115 
Impact, 288, 290 
Incommensurables, 89 
India-rubber, 444 
Indivisibles, 278, 297 
Industrial Revolution, 320, 367 
Infinitesimals, 66, 78, 299, 326 
Infinity, 159, 251 
Integration, 280, 289 
Internal Combustion Engines, 446 
Inventions, 123, 189, 320, 396, 438-448 472 
INDEX 
Ionian Philosophers, 42 
Ions, 364 
Irrational Numbers, 52, 101, 290 
Janssen, 354 
Jenner, 322, 422 
Jordanus nemorarius, 179 
Joule, 349, 351, 357 
Jupiter, Satellites of, 221 
Jussieu, de, 318, 374 
Justinian, Edict of, 152 
Kelvin (see Thomson) 
Kepler, 208, 210-217, 222, 245, 279 
Kinetic Theory of Gases, 357 
Kircher, 264, 267 
Kirchhoff, 353 
Koch, 380 
Laboratories, 360 
Lactantius, 149 
Lagrange, 251, 273, 328-330, 336 
Lamarck, 372, 386, 392 
Laplace, 330 
Lavoisier, 308, 360 
Leap-year, 30, 79, 108 
Leeuwenhoek, 264, 268 
Leibnitz, 299 
Lenses, 163, 265, 302, 313 
Leonardo da Vinci, 228, 234, 252 
Leucippus, 61, 62 
Lever, 80, 102, 123 
Leverrier, 341 
Liebig, 260, 360 
Light, 313, 327, 351, 355 
Velocity of, 216, 250, 286, 351 
Limit, 78, 112, 253 
Linnaeus, 318, 372, 374 
Lister, 379 
Lobatchewski, 338 
Lockyer, 354 
Locomotive, 442 
Locus of Equation, 277 
Logarithms, 215, 230, 242 
Lucretius, 144 
Ludolph von Ceulen, 245 
Lyell, 317, 384, 394, 429 
Maclaurin, 326 
Magnetic Charts, 303 
Magnetism, 228, 313, 354 
Malpighi, 256, 268, 319 
Malthus, 390 
Manometer, 266, 268 
Maps, 25, 48, 132, 143, 192, 241, 303 
Margarita philosophica, 235 
Maxwell, 354, 356, 357 
Mayer, 349, 357 
Mayow, 260 
Mechanics, 80, 102, 123, 133, 214,’ 218, 
219, 230, 234, 245-252, 286-289, 
328, 330, 346 
Medicine, 56, 63, 113, 144, 166, 226, 263, 
322 
Menaechmus, 76 
Mendel, 396 
Mendelejeff, 363 
Mercator, 241 
Metallurgy, 226 
Meton, 39 
Micrometer, 267 
Microscope, 267 
compound, 319, 374, 442 
Middle Ages, 151 
Milky Way, 61, 63, 179, 222 
Mineralogy, 226 
Molecules, 361 
Moment of Inertia, 288 
Monge, 282, 335 
Month, 22, 38 
Moorish Science, 156, 165, 176 
Moro, 317 
Mortality Statistics, 303 
Motion, Laws of, 246, 290, 293 
Muller, 376 
Multiplication, 31, 182 
Napier, 242 
Natural History, 146, 227, 315, 370 
Natural Selection, 395 
Natural Theology, 369 
Navigation, 192, 193 
Nebular Hypothesis, 278, 331, 343 
Negative Numbers, 159 
Neptune, 341 
Newcomen, 312, 440 
Newton, 290-301, 420 
Nicomachus, 125 
Nitrogen, 306 
Nitro-glycerine, 439 
Non-Euclidian Geometry, 91, 337 
Number Theory, 41, 50, 72, 89, 125, 282 
Numbers, Imaginary, 340 
Oersted, 354 
Ophthalmoscope, 444 
Optics, 95, 124, 132, 163, 215, 264, 286, 
291, 302 
Organic Chemistry, 263 
Origin of Species, 369, 394 
Oxygen, 260, 307, 309 
Oughtred, 245 
Pacioli, 232 
Palaeontology, 385 
Palissy, 228 INDEX 
473 
Pappus, 38, 110, 132, 276, 277 
Paracelsus, 226 
Parallelogram of Forces, 248, 253 
Parasitology, 378, 380 
Parmenides, 59 
Pascal, 257, 282 
Pasteur, 377-382 
Pathology, 264, 377 
Pendulum, 163, 219, 248, 266, 287 
Periodic Law, 363 
Perspective, 61, 235 
Perturbation, 295, 331, 342 
Petrarch, 186 
Peurbach, 193 
Philolaus, 54 
Phlogiston, 262, 305 
Phoenicians, 13-16, 43 
Photography, 444 
Physical Chemistry, 364 
Physics, Modern, 350 
Physiologus, 175 
Physiology, 319, 376 
Pisano, Leonardo (see Fibonacci) 
Planets, 22 
Plato, 58, 69, 105 
Pliny, 146 
Pneumatics, 123 
Poincare, 339 
Polyhedra, regular, 51, 89, 90, 211 
Poncelet, 357 
Posidonius, 108, 147 
Power, 320, 439 
Priestley, 306, 309 
Prime Numbers, 89, 108 
Principia, 292, 420 
Printing, 189 
Probability, 282, 283, 284, 332 
Proclus, 38, 88, 90, 92, 119 
Projection, 122, 132, 242 
Projective Geometry, 280 
Protoplasm, 375, 391 
Ptolemy, 119, 126, 138, 176 
Pumps, 246, 257 
Pyramids, 29, 32, 44 
Pythagoras, 49, 90, 310 
Quadrants, 163, 206 
Quadratic Equations, 65, 89, 134, 162 
Quadrivium, 50, 148, 174 
Rainbow, 292, 402 
Ramus, 210, 213 
Ray, 316, 369 
Recorde, 202, 236 
Reductio ad absurdum, 65, 67 
Refraction, 132, 163, 181, 215, 265, 266, 
283, 287, 292, 327 
Refrigeration, 445 
Regiomontanus, 191, 193 
Reinhold, 201 
Renaissance, 172, 185 
Retrogressions, 118 
Rheticus, 196, 241 
Romer, 266, 286 
Royal Society, 268, 269 
Rumford, 312, 322, 350, 357 
Rutherford, 306 
Sacrobosco, 180 
St. Hilaire, 372 
Saturn, Rings of, 223, 266, 286 
Sauveur, 311 
Scheele, 308, 334 
Scholasticism, 155, 174 
Seebeck, 354 
Seleucus, 117 
Servetus, 227 
Sewing-machine, 442 
Slide-rule, 245 
Smith, 317 
Snellius, 266 
Socrates, 69 
Sophists, 64 
Spallanzani, 320, 382 
Sound, 250, 310 
Spectroscope, 353 
Spectrum, 266, 353 
Spinning Jenny, 441 
Spirals, 99 
Spontaneous Generation, 381 
Stahl, 262 
Stars, Distance of, 116, 209, 303, 344 
Steamboat, 442 
Steam-engine, 258, 312, 320, 440 
Steel, 446 
Stevinus, 252 
Stifel, 235, 242 
Strabo, 145, 147 
Sun-dial, 28, 29, 39, 45, 134, 148 
Sydenham, 264, 377 
Symbols, 159, 184, 231, 233, 236, 237, 
240, 245, 277, 301 
Tables, astronomical, 128, 165, 176, 193, 
201, 215 
Tartaglia, 237 
Telegraph, 445 
Telephone, 445 
Telescope, 215, 219, 229, 266, 267, 291, 
313 
Thales, 42 
Thesetetus, 70, 88 
Theon, 38, 90 
Theophrastus, 46, 47, 84 
Thermodynamics, 351, 365 
Thermometer, 249, 267 474 
INDEX 
Thomson, W., 278, 341, 356, 358, 362 
Three Bodies, Problem of, 331 
Tides, 117, 181 
Time Measurement, 27, 39, 127, 207, 247 
Timocharis, 119 
Torricelli, 253, 256 
Treviranus, 371, 392 
Trigonometry, 115, 122, 128, 133, 163, 
193, 241 
Trisection of Angle, 29, 65, 241 
Trivium, 148, 174 
Tycho Brahe, 203-209 
Tyndall, 311, 382 
Universities, 175, 184, 321 
Uraniborg, 206 
Uranus, 333, 342 
Vaccination, 322, 425 
Vacuum, 247, 257 
Van t’ Hoff, 364 
Vesalius, 226, 264 
Vieta, 240 
Vitruvius, 103, 143 
Vivisection, 113 
Vlacq, 245 
Volta, 314 
Vortices, 278 
Wall, 314 
Wallace, 367, 395 
Wallis, 269, 289 
Water, Composition of, 309, 316 
Watson, 314 
Watt, 312, 320, 441 
Week, 22 
Weismann, 396 
Werner, 317, 386 
Wheeler, 314 
Wohler, 363, 375, 391 
Wren, 288 
Xenocrates, 77 
Xenophanes, 56 
Year, 21, 38, 120 
Young, 351 
Zeno, 66 
Zero, 40, 159 
Zoology, 315, 317, 371, 372 
Zymology, 378 
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