WMMW^WQ!^®^ Price, including Key and Portfolio, $6.60 Price 28 francs. FULLER'S COMPUTING TELEGRAPH. The proprietor of the Computing Telegraph, in sub- mitting it to the inspection of the people of Great Britain, has the satisfaction of annexing the following testimonials of well-known mathematicians. A NEW COMPUTING TABLE. (From the Liverpool Mercury of Friday, the 27th ult.J We have been much pleased with the inspection of an ! American invention, entitled " Fuller's Computing Scales, and Time Telegraph," which, for the multiplicity of calculations which it embodies,—the accuracy of the results,—the facility and expedition with which they are obtained,—and the neat and commodious form of the I table itself, deserves to stand amongst the first, if not of the greatest service. A few hours cxpeiience will ensure facility, and its use must supersede the " ready reakoners" at present so constantly employed. REV. J. ENGLAND, M.A., Head Master, High Street, Liverpool." To be remembered in using the Telegraph. Let the 1 be placed at the right hand; examine the arrangement of the figures; observe their position, that the two sets of figures are precisely alike, and one-third the space on the circle is found between the one and two, and that all thesubsequent figures gradually approximate until, from 98 to 99, the space becomes very narrow; that this same approximation continues until,from995 to 1 or the very foremost, of all the calculating machines and11000, the space is also very contracted. Although there | ready reckoners of the day. It must, obviously, have > been the result of many years study and intense appli- r cation. It will be found of the greatest practical utility i to merchants, shopkeepers, tradesmen, anc mechanics, ! as well as to members of the learned proiessions, to ) students, and men of science generally. It solves almost { instantaneously an almost endless variety of problems, ' and to show that our estimate of its importance is not . over-rated, and also to render some little service to i science, we may quote the following certificate:— \ From the London Daily Times. | Extract from Prof. A. De Morgan's recommendation. ' " Having examined Mr. Fuller's Circular Sliding > Rule, I can certify that it is an excellent thing of the > kind. It represents a common sliding rule of upwards ! of twenty-six and nearly twenty-seven inches in length. ) A rule can be learned in a minute or two; and a few | hours of perseverance will make any one a tolerable | master of the instrument. i " The neglect of the sliding rule by computers is the '• neglect of a very great advantage. I always use one ! myself when I have several arithmetical processes to I go through at one time; and I find it a great source of [ accuracy, and, of course, a great relief. To know that ' one error cannot possibly amount to so much as a ' farthing in the pound by mechanical means, sets the i computer free to turn his greatest attention to the ! smaller quantities. | " I think Mr. Fuller's instrument deserving of | success, and strongly recommend it. > " University College, ! Jan. 23, 1849." " A. DE MORGAN. ' The annexed is from the Rev. Mr. Hall, Professor of { Mathematics at King's College, London. » " January 30, 1849. ? "I think the calculating table a very ingenious one, i and might be useful to us. I will recommend you to , order one for the College. 5 " To Mr. Cunningham, Secretary." " T. G. HALL. > Extruct from Rev. Mr. Dixon's recommendation. > " Having carefully examined Mr. Palmer's computing ' scale, improved by Mr. Fuller, I have great pleasure in t bearing my testimony to the accuracy of the gradua- J tions, and to the perfection with which it performs the J operations of multiplication and division, either sepa- > ratolv or simultaneously 5 I consider the invention to be founded on an unerring , principle, viz., that of registering the numbers accord- J ing to their logarithmic values calculated on the cir- ! cumference of the circle; and, consequently, from its J unerring correctness, of the utmost importance, and j the greatest possible use to all practical men. , Mr. Fuller's scale has also one great recommenda- ! Hon, that a few hours' study and application are suffi- | cient for gaining a very fair knowledge of its use. It | might also be occasionally used with advantage to im- j press the rules of arithmetic upon the minds of the i young, in which use it would form both an agreeable i exercise, and a good preparation for the rapid calcula- ', tions of the counting-house. | REV. THOMAS DIXON, M. A., ' Late Fellow and Mathematical Lecturer of Jesus College, ) Cambridge, and Head Mathematical Master of the > ^ Liverpool Collegiate Schools. | Liverpool, 23rd October, 1848." | " High School, Mechanics' Institution, I October, 1848. i Having carefully tested Palmer's Computing Scale, as ! improved by Fuller, I can with confidence speak of its I correctness in the particular which constitutes its great ; value—its careful graduation. Formed on principles which have the sanction of rigid demonstration, its worth as a machine for speedy and correct calculation, depends on the minute agreement of its several parts , and combinations. In this respect it is quite equal to I the beat rectangular scales, whilst its form gives beauty | and compactness to accuracy, and enables the operator | to solve his question with ease. In all cases where mere mechanical rapidity is required, it will be found ire 1000 divisions, no two of them are at the same distance from each other. Each figure may be called according as the nature of the problem to be solved may require. For example—161 may be called fS/JL, tB> W. or 16.100, or 161.000, or 1,610,000, and if any mistake be made it must at least be tenfold, which would be seen at a glance. The best method of finding any given number is to observe the following rule:—Bring the two ones even with each other. Should you require 135, look for 13, and between that and 14, you find it; if 695, look for 69, itc , after the same manner as you would look for some vol. in a set of books. A very short practice will enable one to find the number as quick as thought. Let no one allow himself to be disconcerted in the outset, as a few moments thought will prove the simplicity of rhe arrangement. Telegraphic Rule, by which Multiplication and Divi- sion is performed by a single operation. If this rule be correctly understood, all the calculations in the rule of proportion become very simple, and may be performed as readily as the statements can be made. It consists in making the divisor (which is at all times found on the stationary part) the gauge point, as will be seen by the following examples:—Suppose a room is 12 feet square, required the number of yards. Place 12 on the moveable part at 9 on the stationary, then 12 on the stationary part gives 16, the number of yards. If the room be 18 feet each way, then 36 would be the answer. American law allows one passenger for every fourteen superficial feet of deck surface. A ship is iH feet wide, and 165 feet long,—howmany passengers m»y she carry? Set 32 at 14, and at 165 is 377. This produces the same result a3 if 165 be multiplied by 32, and that product be divided by 14. To bring shillings ,5c pence into pounds by one operation. Rule.—Place the shilling and decimal part of the same at 20, which acts as the divisor, and at the multiplicand is the answer in pounds and decimal parts of the pound. Example.—Is. 6d. per yard for 24 yards: place 1 and ■fr at 2, and at 24 is £1. 16s. or 1 JL. To bring pence into shillings. Rule.—Set the pence and parts of pence at 12, which acts as divisor, and at the multiplicand is the answer in shillings and decimal parts of the same. Example.—Paid 3d. per yard for 72 yards: place 3 at 12, and at 72 is the answer 18s. In 240 yards at ljd. per yard, how many shillings ? Set 175 at 12, and at 24 is 35s., the answer. To bring farthings into pence. Rule.—As 4 farthings make Id., set the farthings and parts of the same at 4 : paid 3 farthings each for 16, and place 3 at 4, and at 16 is 12d. the answer. Average of Accounts or Equations of Payments. The computing telegraph will be found invaluable in the above-named work. The time telegraph is of grea^ value to the accountant, whether he work by the follow- ing rule or not. It will be seen that the time to each entry on the book is obtained by a single setting of the time to 365. Exa mple.—The following bill is on three months, and is supposed to be settled, and the note given at the last date, November 17. It will be obvious that a portion of the time of credit is already expired, as the first item was September 23, and the next October 25. The simplest method of getting the average here is the same as that taught by many book-keepers, viz., to make up the interest account at the uniform rate per cent., and find how long that interest will pay the same per cent. on the entire bill. The following example will illustrate the principle, which may be extended to an indefinite number of items:— £ s. d. Time. Interest. September 23.........191 10 8.........55 days.........£1.44 October 25.........104 5 6.........25 „ ..........52 November 17.........122 15 S.........— „........ — Ipay 5 per cent, interest on this bill? The answer is found by placing the 478 at 73, and looking at the amount of interest on the moveable line, the time will be 30 days. Now set the 17th of November on the time table at 365, and by reference to the 30 on the back line is the 18th of Oct., the time to date the note, and, of course, it will become due Jan. 18. Feet in a mile, f English, j 5280, at three feet per step. Required, the steps taken in a mile. Set 3 at 1, and at mile gauge point is 1760, the steps or yards in a mile. This is often useful as in the following Average speed of B. and N. A. Steam Ships. Example.—The British and North American steam- ships average about thirteen days in crossing theAtlantic, which is about three thousandmiles.Required the average feet per second. This requires several changes,but needs only the ordinary care and is done in one minute. Set 3 at 13 and at 1 is 231 per day. 231 per day, how many per hour ? Set 231 at 24, and at 1 is 9^-\ miles per hour, or 9.62 for 60 minutes. How many minutes for 1 mile ? 1 mile on the moveable is at 6T20s;p. A mile is, as the gauge point informs us, 5280. Set this at 6.22, and if she run 5280 feet in 6-f£s minutes, then at 1 is 847 feet for 60 seconds. Set this at 6 or 60, and at 1 is the answer 14 feet 2 inches per second. It must be obvious to all, that, in addition to the mental training, a vast amount of pleasant amusement will be gained, while the arithmetical rules are revived and fixed indelibly in the mind. This has led many persons, after using it for a season, to remark, that while they*found themselves essentially benefited by its use in correcting mistakes, it afforded as pleasant recreation and amusement as any invention of the age. Per-centage Mule for Calculating Dividends or any In- solvent Estate by Decimals. A bankrupt or insolvent debtor has cash on hand £1100, and owes £7100, what per cent, can he pay, and how much will a demand of £8 receive ? Rule.—Place 11 on the moveable at 71 on the sta- tionary, and at 1 on the stationary is ISA. N.B.—All the stationary lines are called demands, and all the moveable lines are to be called dividends. All questions of per centage, whether it be whole num- bers or fractions, are calculated in like manner, whether the sums be pounds, shillings, or pence, dollars or cents. Cubic Feet in Boxes. The present custom for obtaining the precise measure- ment is to multiply the inches and tenths of the inch in thickness by the height, and this product by the length, this being the total of cubic inches must be divided by 1728, which will, of course, cause many figures. By thf Telegraph it is done Instantly; for example,—a tea chest is in thickness 16^, in height 17^, and in length 22^. Place 16.6 at 1, and at 17.8 is 296. Set this 296 at 1728 on the stationary, and at 22.9 is 392, being 3 feet ^j'j, This will apply equally to all the measure- ments of cubical contents. Where feet and inches are given it will only be necessary to observe the following rule, which is the decimal of one inch, the decimal of one penny, or the decimal for any number of 12ths. 478 II 5......._ ......... 1.96 The above bill is £478. lis. 6d., or a fraction over £478). The next question is, how long will £1 and^'j l-12ths or 1 inch or Id. is 81, lOOths, 2-12ths or 2 „ or 2d. is 16 J, 100 „ 3-12ths or 3 „ or 3d. « 25TJ7, 100 „ 4-12ths or * ., or 4d. is 33i, 100 „ 5-12fhs or 5 » or 6d. is 41J, 100 „ 6-l'2ths or 6 „ or 6d. is 50, 100 „ 7-12ths or 7 „ or 7d. is 58L, 100 „ 8-12ths or 8 „ or 8d. is 66f, 100 „ 9-12ths or 9 » or 9d. is 75, 100 „ 10-1'Jths or 10 „ or lOd. is 83^, 100 „ Il-I2ths or •1 » or lid. is 91J, 100 „ 12-12ths or 12 „ or 12d. is 100, 100 „ A few moments' reflection will, with the assistance of the telegraph, enable any person to calculate by feet and decimal parts of the foot. Example:—A box measures 2 feet 1 inch in width, 2 feet 3 inches in breadth, and 2 feet 4 inches in length. 2 feet 1 inch or 2 feet 81 by 2 feet fifa is 4.69; set this at 1, and at 2 feet 4 inches or 2 feet ffe on the stationary is the answer 11 feet. By this rule, wood, timber, and all kinds of merchandise is also measured; and this method will test the accuracy of the former. Amongst the thousands who havepurchasedtheabove- named work, a very large number use it to examine computations made in the ordinary manner. It is uni- versally admitted that the most perfect mathematicians find themselves sometimes in error in setting down the numbers, or in placing the fractions for addition. To Measure Timber A stick 131 by 15, and 32 feet long :—Set 15 at 1, and at 13.5 is 202; set this at 144, and at 32 is 45 feet. Superficial Measure. Set the whole width in inches at 12, and at the entire ^M^^^^MMJ^M^^X^^M^W&M^MiMM Price 22s. 6d. Jfd length is the feet. fQ Required the surface measure of a stick 7} by 6, and Jg 19 long, this is 45 inches wide:—Set this at 12, and at Freight, 15s. per ton, how many shillings for 1200 lbs.? „A Set 15 at 2 or 20, and at 12 is 9s., and in like manner J\jP| for all prices and quantities. J/g Rule for Manufacturers and Mechanics. «- - The speed of drums and pulleys is obtained in the /O following manner:—The moveable part may be called I" ^ the diameter of the pulleys, in feet or inches, and the $-^ fixed part the number of turns they may be driven. Ex- Jig ample:—A 12-inch drum is driven 96 turns per minute. JjS Set the 12 at 96, and by looking at 11,88 is found; and JE? at 9, 72 is found to be the proportions. If a greater yp} speed be required, the size of the drum is at once ob- 3i2 tained as follows :—Required the size of the drum to 5{g run from this 12 inches, running 96 turns per minute, ""g to obtain 128 turns per minute. Set the 12 at 128, and ?E? at 96 (the former point) is the diameter of the drum «ij required, 9 inches; being the same result as would be 3c5 obtained by multiplying 96 by 1-, and dividing that .tr^ product by 128. This rule will apply equally to all ., I other cases, as it performs multiplication and division ^fc> by one process. A drum 14 inches diameter is driven j«5 by one 11 inches, and running 98 turns per minute. 8tj Set 14 at 98, and at 11 is the answer, 77 turns |f^ Required the number of yards of cloth to the lb., the ?tj package weighing 142 lb., and containing 815 yards. jS^ Set 142 at 815, and at 1 (lb.) on the moveable part is j/~ 5|, the answer. N.B. All the figures on the fixed part 3c5 are yards, and those on the other are pounds, as 12 lbs. JfiR 69 yards, &c. £R How many yards of cloth will one loom weave, at 64 JJjifS threads per inch, and throwing 125 threads per minute ? Jig Set 64 at 1, and at 36, the inches in 1 yard, is 2304, «5 the threads in 1 yard. Set 125, the divisor at 1, and !«?> that in 2304 is 18$, the minutes to weave 1 yard. Set «)§ 18} at 1, and at 60 is 3,24, the yards per hour, should gQ the loom not stop. Allow 25 per cent, for the stoppage. flg Set 75 at 1, and at 324 is 2,43, the discount off. Mul- «g tiply this by the running time, 12 hours, and the result *£> is 29J yards. Multiply this by the whole number of cJ»B looms, and the amount is obtained. The rule may be 2690. N.B.—All the fixed lines of numbers are pounds ?«5 of coal, and all the opposite lines are shillings and parts 0\C> of the same. By obtaining the weight of one cubic •/■,. foot of coal, a body of any dimensions may be calculated, •*- and the number of tons given in one minute. '^(S Exchange of the different currencies into Pounds, Shil- 'Jrl lings, and Pence. ■ Example—If 444 cents be equal to 20s., required the value of 19s. Set 444 at 2, and at 19 is 4.22, at 18 is 399 J, and at 17 is 377, and against each number of shil- lings on the stationary part is the answer. If pounds be 1/qI required instead of shillings, place the 444 at the 1, and . ~' on the moveable part are the dollars equal to any num- ;tO ber of pounds and parts of a pound, on the opposite y^ side. The par value of a dollar is 4s. 6d., or £9. is equal to forty dollars. !ft~> The same rule is applicable to all other coins or cur- L^>' rencies. 3p^ If 25 francs are equal to 20s., how many francs for J>£? 12s. ? Set 25 at 20, and at 12 is 16, the answer. j«j If 25 francs are £1., how many francs for £9.? Set -{O 25 at 1, and at 9 is 225. j ! If 25^ francs for £1. then £8. is 204 francs. ?Q> If 3 guilders are equal to 5s., how many shillings for Vj 33 guilders ? Set 3 at 5, and at 33 is 55, the answer. , ., English and French Measures. KrS Set the number of inches in the yard—36, against the '-(jQ number of inches in any French measure, and at any given number of yards English is the French. Example.—Set 36 at 39.3, and at 11 French is 12 the English yard, or at 100 is 109. At l^jd. per yard, what would 32 yards cost ? Set 112| at 12, and at 32 is 3 shillings, the answer. Calculations of salaries. If £100,000. per annum, 3£3 how much per hour ? Set 1 at 365, and at the other 1 7Q is £274. per day ; set 274 at 24, and at 2 is the answer, ,^ 228} the shillings per hour. (t> o '. IMPROVEMENT TO 5 PALMER'S ENDLESS SELF-COMPUTING SCALE AND KEY; IDAJTINO IT TO THE DIFFERENT PROFESSIONS, WITH EXAMILKS AND ILLUSTRATIONS FOR EACH PROFESSION ; AND ALSO TO COLLEGES, ACADEMIES AND SCHOOLS, WITH A TIME TELEGRAPH, MAKING, BY UNITINO THE J WO, A COMPUTING TELEGRAPH. BY JOHN E. FULLER. PALMER'S ENDLESS SELF-COMPUTING SCALE. NEW-YORK: PRINTED FOR THE PUBLISHER 2851. The proprietors of this invaluable work, beg leave to pre- lent the public with the following notice. This Scale (the result of three years' incessant labor) is designed as an assistant in all arithmetical calculations. The simplicity, rapidity, and accuracy of its results, have as- tonished our best mathematicians. It consists of a loga rithmic combination of nnmbera, arranged in two or more circles, one of which is made to revolve within the other; which process constantly changes the relation of the figures to each other, and solves an infinite variety of problems. Its advantages are,— 1st. A complete saving of mental labor ; for, by the use of this Scale, the most intricate calculations are but a pleasurable exercise of the mind. 2d. A great saving of time. Computations requiring from three to four days, are wrought out by this Scale in the incredible short space of one minute. 3d. Complete accuracy. The results of the computations on this Scale, are infallible. Errors are entirely out of the question, except through sheer carelessness. 4th. Mental improvement. By this Scale, a knowledge of the philosophy of numbers, and their relation to each other, is soon obtained. So that, in a little time, many of the common calculations are wrought out by the mere exercise of the mind. Brockport, Feb. 19, 1842 I have carefully examined " The Endless Self-Computing Scale," by Mr. Aaron Pa".mer; and, without hesitation, give it as my opinion, that it will be found a very useful invention. All the problems in arithmetic can be readily solved upon it, and most of them with great expedition, particularly the rules for computing interest for months and days, at any per cent, the Rule of Three, and Fractions. In the apportionment of County, Town, and School Taxes, it will be found almost in- valuable, as it requires to be set but once, to show each man's tax. JULIUS BATES, M. A. Principal of Collegiate Institute Cambridge. Oct. 20, 1843. I have examined Mr. Aaron Palmer's -Endless Self-Com- puting Scale;" it is simple and most ingenious, and I cheer- fully concur in Mr. Julius Bates's judicious recommendations of its utility. BENJAMIN PEIRCE, Perkins Professor of Astronomy and Mathematics in Harvard University. Boston, October 24, 1S43 Mr. Palmer's "Self-Computing Scale" is certainly a very ingenious arrangement of numbers, and it will save a greal amount of time in the hands of those who have computing tfi perform, whatever be the subject of the computation. FREDERICK EMERSON, Author nf the North American Arithmetic I heartily concur in the above recommendation. WILLIAM B. FOWLE. Ijale Teacher of the Female Monitorial School, Boston Boston, October 23, lS4f< Mr. Aaron Palmer, Sir: Your "Self-Computing Scale" appears to me an exceedingly useful invention. I shall be glad to possess one of them, as it will save me much labor, aud I doubt not that many persons will find the same advantage in its use. Respectfully your servant, JOHN S. TYLER, .votr/ry Public and Insurance Broker 6 Boston, October 24, lb 13. I have examined Mr. Aaron Palmer's " Self-Computing Scale j" it strikes me as being a very convenient labor-saving machine, and that it will be highly useful in calculating interest, general average, or dividends on a bankrupt's estate, and for other similar purposes. S. E. SEWALL. Counsellor at Lux NORTHEEN DISTRICT OF NEW YORK. TO WIT: Bl IT Remembered, That on the eleventh day of December, Anno Domini, 1813, JOHN COTTS SMITH, of the said District, has deposited in this Office the title of a Book, the title of which ia in the words follow- ing, to wit: " A Key to the Endless, Self computing Scaje, showing its Application to the different Rules of Arithmetic, Ax. By Aaron Palubk." The right whereof he claims as proprietor. In conformity with an Act of Congress entitled An Act u> amend the several Acts respecting Copy Bights. [A true copyol record.; ANSON LITTLE, Clerk of the District. STSnEOTTPKD BV OEORGE A. CURTIS, ■SW ENGLAND TYPE AND BTEEBOTYPE POUNDEY, BOSTON. RECOMMENDATIONS OF THE ENDLESS SELF-COMPUTING SCALk. Rochester, Jan. 19, 1842. The- " Self-Computing Scale," by A. Palmer, is a very in- genious and interesting instrument for performing most of the operations in arithmetic. The principle is very plain; and the accuracy, and certainty, and rapidity of the results are very striking. C. DEWEY, Principal of Collegiate Institute. Rochester, January 19, 1842. Having particularly examined Mr. Palmer's "Self-Com- puting Scale," I fully concur in the above testimonials of Dr. Dewey. SAMUEL LUCKEY, D. D. Attica, March 5, 1842. From an examination of the " Self-Computing Scale," by Mr. Palmer, I can most cheerfully concur in the above recom- mendations, and hope it may be introduced into our schools and academies. E. B. WALSWORTH, Principal of Attica Academy Buffalo, April 5, 1842. We have examined tne above mentioned Scale, and concur in the certificate of Professor Dewey. W. K. SCOTT, Civ. Eng. R. W. HASKINS, M. A. I have examined "The Endless Self-Computing Scale" cf Mr. Palmer, and with pleasure express my high admiration of it. It is constructed on the only principle acknowledged by scientific men, since the invention of Logarithms, adequate to such purposes. Over all sliding Logarithmic Scales, it possesses a vast superiority, both in facility of use and ac- curacy of result. For this superiority, it is indebted to its circular form. With a diameter of about eight inches, ii a equivalent to a common sliding scale of four leet with its slide of the same length, making when drawn out, a rod of about eight feet in length. It will be seen that its accuracy will be proportionably greater, as a circle can be constructed more exact than such a scale. G. C. WHITLOCK, • Professor of Mathematics and Natural Science in Genessee Wesleyan Seminary. Mr. Aaron Palmer, Sir: I have taken much pleasure in testing the power of your " Self-Computing Scale," by examples from nearly all the arithmetical rules. I am particularly struck with its great facility and accuracy in computing interest, apportioning divi- dends, and performing proportions generally. From the best sxamination I have been able to give it, I think it at once a most simple and wonderful invention; and I am confident, that when perfected, it will come rapidly into extensive public use, and will prove of singular benefit to those having occa- sion to make frequent computations in Bankruptcy, Insol- vency, Insurance, Averages, Taxation, and the like branches of business. AMOS B. MEREILL, 10 Court Street, Boston. THE TIME TELEGRAPH. The Time Telegraph is composed of a beautiful steel plate engraving, neatly executed by G. G. Smith, of Boston, upon the surface of which is arranged in circles four lines or rows of numbers; upon the move- able circle is placed the names of the twelve calendar months, to which is affixed the number of days in each month, 365 making the entire circle; the inner row of numbers found upon the stationary circle, running from 1 to 365, is used for calculating time to come; the outer row of numbers on the stationary circle is reversed, and is used for the purpose of calculating time past. The manner of ascertaining the number of days from any given day in any month, is readily found by simply turning the moveable circle unto the day of the month from which you compute is directly opposite the gauge point affixed at the figures 365 then opposite the day of the month to which you wish to reckon is found the exact number of days required. Upon the stationary circle is also found the weeks, from one to 52; to these are added divisions of 30 days, so that any portion of the year can be brought into months as readily as the fingers of the hand can be reckoned. The Time Telegraph will be found of invaluable benefit in working equation of payments, &c. Entered according to Act of Congress, A.D. 1845, By John E. Fullcr. INTRODUCTION. The undersigned, proprietor of the Copy Right of Palmer's Endless Self-Computing Scale, and hav- ing been engaged in introducing and selling the same for abou< eighteen months past, and become exten- sively acquainted with the wants of the community, has been induced to introduce an improvement for which he has secured a Copyright, both for the Scale anc Key, and is assured that all persons in com- mencing the use of the Scale will be very mucn assisted. The character of the Scale is too well estab- lished to need remarks. Having personally introduced it to about Four Thousand persons ; by very many of 'vhom he has had repeated assurances of their high ap- preciation of its value, he can with confidence refer oth- ers who may wish to possess it, to any of those who may have used it in any of the various rules of Arithmetic. His only desire is that its future patronage shall be pro- portionate to its true merits. JOHN E. FULLER. KEY TO THE SCALE. DESCRIPTION OF THE SCALE. The figures on both parts of the scale, are pre- cisely alike, and may be called whole numbers or parts of numbers, according to the nature of the problem to be solved. The large figure 1 may be called ToW. or -rfoj, or 1\,, or 1, or 10, or 100, or 1000, or 10000, &c., &c. If it be called ^fa,, the large figure 2 will be y^n-, the large 3 will be t^, and so on j and the next sized figures between those large ones, will then be t^, ^f^, rvivv, &c.; and the still smaller ones will be -rinhviK &c. If the large 1 be called 1, then 2 is 2, 3 is 3, &c.; and the next sized figures are tenths, and the third sized ones are hundredths, &c. If the large 1 be called 10, the large 2 is 20, 3 is 30, &c.; and the next sized figures are whole numbers—the first after the 1 is 11, the next 12, the next 13, &c. If the large 1 be 10 called 100,2 is 200, &c.; and the next sized figures then will read 10, 20, 30, ice.; and the smallest sized figures will then be whole numbers. N. B.—Whenever fig. 1 is referred to, it means the large fig. 1 at the diamond—unless otherwise explained. A TABLE OF GAUGE POINTS USED ON THIS SCALE. I., at the diamond, is the gauge point for Multipli cation, Division, &c, &c. A. Area of a Circle. C. Circumference of n. Circle. B. G. Beer Gallons. W. G. Wine Gallons. 15. for months, at 8 per cent. for months, at 7 per cent. 2. for months, at 6 per cent. for days, at 8 per cent. for days, at 7 per cent. for days, at 6 per cent. 107. Compound Int. for years, at 7 per cent. 106. do. do. do. 6 do. 160. for Acres. 144. for Square Timber. 9. Yds. Square. 886. Square and Circle, equal in Area. 707. Inscribed Square. 577. side of Inscribed Cube. 12 87. side of Inscribed Triangle. 689. side of Pentagon, (5 sides.) 5. side of Hexagon, (6 sides.) 437. side of Heptagon, (7 sides.) 383. side of Octagon, (8 sides.) 337. side of Nonagon, (9 sides.) 31. side of Decagon, (10 sides.) 282. side of Undecagon, (11 sides ) 26. side of Dodecagon, (12 sides.) 464. diameter of 3 Inscribed Circles. 416. diameter of 4 Inscribed Circles. 785 . point for Area. 314 . point for Circumference. « f 13 To Perform Multiplication. Rule.—First find the multiplier on the circular. Place it opposite 1, then opposite the multiplicand found on the fixed part, is the product on the circular. Example.—What is the product of 4 by 2 ? Place 2 opposite 1: then opposite 4 is the pro- duct =8. N. B.—Observe, now, that all the numbers and parts of numbers on the fixed part, are multiplied by 2, and their products are directly opposite them on the circular. So of any other multiplier. What is the product of 12 oy 7 ? Place 7 opposite 1: then opposite 12 is 84, the answer. 0;3by 3? Place 3 opposite 1: then opposite 3 is 9, the answer. What is the product of 8 by 2J ? Place 2-5 opposite 1: then opposite 8 is 20, the answer. What is the product of 10 by 5 ? Plr.ce 3 opposite 1 : then opposite 10 is 50, the answer. Here you have to use the same figures both o 14 times, calling them 1 and 5 the first time, and adding a cypher to each the next time. What is the product of 13 by 3 ? Place 3 opposite 1", then opposite 13 (found be- tween the large 1 and 2) is 39, the answer. What is the product of 50 by 4 ? Place 4 opposite 1: now we must call the large 5 50: opposite it is 200, the answer. What is the product of 24 by 3 ? Place 3 opposite 1: then opposite 24 (found be- tween the large 2 and the large 3) is 72, the answer. What is the product of 3 multiplied by -2 (two tenths) ? Now we must call the large 2, two tenths. Place it opposite 1: then opposite 3 is -6, (six tenths,) the answer. Division. Rule.—Find the divisor on the circular. Place it opposite 1: then opposite the dividend, found also on the circular, is the quotient on the fixed part. Example.—2 is in 8, how many times ? Place 2 opposite 1 : then opposite 8 is 4, the answer. 3 is in 12, how many times ? 15 Place 3 opposite 1: then opposite 12 is 4, the •nswer. How many times 4 in 14 ? Place 4 opposite 1: then opposite 14 is 3 and five tenths, (3-5,) the answer. Note.—Whenever a divisor is placed opposite 1, all the numbers and parts of numbers on the circular are divided by it. The quotients are on the fixed part. Example.—Place the divisor 2 opposite 1: now opposite 2 is 1, opposite 12 is 6, opposite 4 is 2, opposite 6 is 3, opposite 14 is 7, opposite 24 is 12, opposite 125 is 62-5, opposite 75 is 37-5, &c. To Multiply by one number and Divide by another BY ONE SIMPLE PROCESS. Rule.—Place the multiplier on the circular oppo- site the divisor: then, opposite the multiplicand is the result. Example.—What is the result of 22 multiplied by 13 and divided by 14 ? Place 13 opposite 14: then opposite 22 is 20-4-|- ihe answer. FRACTIONS. To Change an Improper Fraction to a whole o« mixed Number. Rule.—Place the numerator found on the circular lb- opposite the denominator: then opposite 1 is the answer. Example.—A man spending £ of a dollar per day, in 83 days would spend ^ of a dollar. How much would that be ? Place 83 opposite 6: then opposite 1 is $13 83, the answer. In f of a dollar how many dollars ? Place 8 opposite 4: then opposite 1 is $2, the answer. To reduce a Mixed Number to an Improper Fraction. Rule.—Place the mixed number opposite 1: then opposite the denomination to which you wish it re- duced is the answer. Example.—In 16^ of a dollar, how many 12ths of a dollar ? Place 16^r opposite 1: then opposite 12 is the number of 12ths in 16^-, viz., 197=-ij^-, the answer. To reduce a Fraction to its lowest and all its Terms. Rule.—Place the numerator found on the circular opposite the denominator: then all the numbers standing directly opposite each other, are other terms »f said fraction; and the lowest of said numbers are its lowest terms. 17 Reduce || to its lowest terms. Place 12 opposite 16: now 9 is opposite 12 (•&,) 6 is opposite 8 (f,) and 3 is opposite 4 (f,) the answer. To divide a Fraction by a Whole Number. Rule.—Place the whole number found on the cir- cular opposite 1: then opposite the denominator is a number, which, placed opposite the numerator, is the answer. Example.—If 2 yards of cloth cost § of a dollar, how much is that per yard ? 2 is in § how many times ? Place 2 opposite 1: then opposite 3 is 6. Now place this opposite 2, and it will read f, the answers£. 2 is in | how many times ? Place 2 opposite 1: opposite 8 is 16. This, placed opposite 7, makes fc, the answer. To multiply a Whole Number by a Fraction, or a Fraction by a Whole Number. Rule__Place the numerator found on the circular opposite the denominator: then opposite the whole number is the answer. N. B.—Whenever a numerator is placed opposite a denominator, all the numbers on the circular are that fractional part of the numbers opposite them. 18 Example.—Place 3 opposite 4: this is f. Now the 3 is J of 4; 6 stands opposite 8, being £ of 8; 9 is opposite 12; 12 is opposite 16, Sec., Sec. Now move the circular until 3 is opposite 5: now all the numbers on the circular are $ of those opposite them. Note.—Whenever a numerator is placed opposite a de- nominator, thereby forming a vulgar fraction, the decimal of said vulgar fraction is opposite 1; hence, To reduce Vulgar Fractions to Decimal Fractions. Rule.—Place the numerator found on the circular opposite the denominator: then opposite 1 is the decimal fraction. Example.—What is the decimal of f ? Place 3 opposite 4: now opposite 1 is -75, the answer. What is the decimal of J ? Place 7 opposite 8: opposite 1 is *875. To reduce Decimal Fractions to Vulgar Fractions. Rule.—Place the decimal found on the circular opposite 1: then any two figures standing directly opposite each other is the answer. Example.—What is the vulgar fraction equivalent to the decimal -5 ? id Place 5 opposite 1 iow 1 is opposite 2 = £, the answer. To multiply one Fraction by another. Rule.—Reduce one to decimals: then place the numerator of the other opposite the denominator: then opposite the decimal is the answer in decimals, which, if desired, can be reduced to a vulgar fraction by the preceding rules. to reduce the different currencies to federal Money. Rule.—Place the 1 on the circular, opposite the number of s'hillings and parts of a shilling composing a dollar of the currency to be reduced: then, opposite the given number of shillings is the answer. Example.—Reduce 5 shillings, New York cur- rency, to Federal money. Place 1 (on the circular) opposite 8: ther opposite 5 shillings, is -625, the answer. In 15 shillings, how much ? Opposite 15 is 1875, the answer. In 32 shillings, English currency, how much ? Place 1 (on the circular) opposite 4-5: then oppo- »ite 32, is $7-11, the answer. In 9 shillings, how much ? Opposite 9 is $2, the answer. 20 INTEREST. To compute Interest for Years. Rule.—Place the rate per cent, found on the eir cular, opposite 1: then opposite the principal is the interest. Example.—What is the interest of $50 at 7 per cent. ? Place 7 opposite 1: then opposite 50 is $3-50, the answer. What is the interest on $40 at 6£ per cent. ? Place 6-5 opposite 1: then opposite 40 is $2-60, the answer. To compute Interest for Months. Rule.—Place the principal, (found on the circular,) opposite the gauge point for months at the given per cent.: then opposite the given number of months is the answer. Example.—What is the interest on $50 for three months at 7 per cent. ? Place 50, (found on the circular,) opposite 1714, (the gauge point for months at 7 per cent.,) then opposite 3 months is -875, the answer. What is the interest on $60. for eight months at 6 per cent J 21 Place 60 opposite -2, (the gauge point for months at 6 per cent.,) then opposite 8 months is $2'40, the answer. To compute Interest for Days. Rule.—Place the principal, (found on the circular.) opposite the gauge point for days at the given per cent.: then opposite the number of days is the answer. Example.—What is the interest on S55 for 15 days at 6 per cent ? Place 55 opposite -600, (the gauge point for days at 6 per cent.,) then opposite 15 days is -13 3-4. The Principal and Interest being given, to find the rate per cent. Rule for one Year.—Place the interest opposite the principal: then opposite 1 is the rate per cent. Example.—Received $7-00 for the use of $50-00 for one year; what was the rate per cent. ? Place 7 opposite 50 : then opposite 1 is 14, the an- swer, 14 per cent. Gave $4-00 for the use of $80-00 one year: what was the rate per cent. ? Place 4 opposite 80: then opposite 1 is 5, the an- swer, 5 per cent. 22 Rule for Months.—Place the given interest op- posite the given number of months: then observe the number opposite 12. Now place this number oppo- site the principal: then opposite 1 is the rate per cent. Example.—Paid 25 cents for the use of $500 for 4 months : what was the rate per cent. ? Place 25 opposite 4: then opposite 12 is 75. Now place 75 opposite $5-00: then opposite 1 is 15, (15 per cent.,) the answer. Gave 14 cents for the use of $60-00 one month : what was the per cent. ? Place 14 opposite 1: then opposite 12 is 1-68- Now place 1-68 opposite 60: then opposite 1 is 2-8, (2^ per cent.,) the answer. Rule for Days.—Place the given interest oppo- site the given number of days: then observe the in- terest opposite 365 (the number of days in a year). Place this opposite the principal: then opposite 1 is the rate per cent. Example.—Paid 14 cents for the use of $64-00 29 days: what was the rate per cent. ? Place 14 opposite 29: now opposite 365 is $1-76. Now place 1-76 opposite 64: then opposite 1 is 2-75 (2J per cent.,) the answer. Paid 23 cents for the use of $5000,21 days: what was the rate per cent. ? 23 Place 23 opposite 21: now opposite 365 is 4, Place 4 opposite 50: then opposite 1 is S per cent. the answer. The Rate per cent, and the Interest bbm< given, to find the Principal. Rule for one Year.—Place the per cent, oppo site 1: then opposite the interest is the principal. Example.—At 7 per cent. I paid S3-50 for the us of money 1 year: what was the principal ? Place 7 opposite 1: then opposite 350 is S500C the answer. Rule for Months.—Place the interest opposit the given number of months : then opposite the poir of the given per cent., for months, is the answer. Example.—Gave $2-00 at 7 per cent, for thre months : what was the principal ? Place 2 opposite 3: then opposite 1-714 is SI 14-31 the answer. Rule for Days.—Place the given interest oppi site the given number of days: then opposite tl gauge point for days stands the principal, Example.—At 7 per cent., gave 15 cents for £ days : what was the principal ? Place 15 opposite 20: then opposite 521 (tl gauge point for days at 7 per cent.) is $39-00, tl answer. 24 The Rate per cent., Interest, and Princtal beii> given, to find the Time. Rule.—Place the interest of the given princip for one year opposite 12: then opposite the given i: terest will be the answer in months and decimals ( a month. Or, place the interest of the given print pal for one year opposite 365: then opposite tl given interest will be the time in days. Example.—Gave 87,5 cents at 7 per cent, f $50-00 : how long did I have it ? The interest of $50-00 for one year, is $3-5 Place 3-50 opposite 12: then opposite -875 is tl answer, 3 months. Gave 24 cents at 7 per cent, for the use of $5< how long did I have it ? Place $3-50 opposite 365: then opposite 24 is t answer, 25 days. Compound Interest. Rule.—Place the principal opposite fig. 1: th opposite the rate per cent, added to 100, on the fix part, is the amount for one year. Place this amou opposite fig. 1 : then opposite the same point is t amount for two years. Place this last amount opt site 1: then opposite the same point is the amoti for 3 years, &c. 25 27 29 Example.—What is the compound interest on $5-00 for 5 years at 6 per cent ? Place 5 opposite 1: then opposite 106, (the per cent, added to 100,) is $5-30, the amount for 1 year. Now place $5-30 opposite 1: then opposite 106 is $5-62, the amount for 2 years. Now place $5-62 opposite fig. 1: then opposite 106 is $5-95. the amount for 3 years. Now place $595 opposite fig. 1: then opposite 106 is $6-31, the amount for 4 years. Now place $6-31 opposite fig. 1: then oppo- site 106 is $6-69, the amount for 5 years. LOSS AND GAIN. Bought a hogshead of molasses for $60 : for how much must I sell it to gain 20 per cent. ? Rule.—Place 20 opposite 1: then opposite 60 is what must be added to the original cost to gain said per cent., viz.- 12 : which added to 60-= 72. Bought cloth at $2-50 per yard; but, being damaged, I am willing to sell it so as to lose 12 per cent. How must I sell it per yard ? Place 12 opposite 1: then opposite $2-50 is -30, the amount to be deducted from $2-50, which will leave 2-20, the answer. Bought cloth at 50 cents per yard: sold it for 10 cents advance from cost. What per cent, did I make? 3 26 Place 10 opposite 50: then opposite 1 is 20 per cent., the answer. Another Method.—Place the original cost oppo- site 1: then opposite the rate per cent, added to 100, is the answer. Example.—Bought corn at 50 cents per bushel: at how much must I sell it to gain 20 per cent. ? Place 50 opposite 1: then opposite 120, is 60 cents, the answer. Bought cloth at $2 per yard, and sold it at $3 per yard: what per cent, did I make ? Place 2 opposite 1: then opposite 3 is 150,50 per cent., answer. RULE OF THREE, OR PROPORTION. Rule.—Place the second term opposite the first. then opposite the third term, is the answer. Example.—If 2 yards of cloth cost $4.00, what cost 8 yards ? Place 4 opposite 2: then opposite 8 is 16. Note.—All numbers of yards at that rate, are now on the scale, and may be determined without moving the circular. At | of a dollar per yard, what cost 4 yards ? Place 7 opposite 8: then opposite the given num- ber of yards, is the answer. If 1 ton of hay cost $8-00, what cost 900 pounds ? Place 8 opposite 2000, (the number of lbs. in a ton:) then opposite 900 is the answer; and so of any other number of pounds. FELLOWSHIP. Rule.—Place the whole gain or loss opposite the whole stock: then opposite each man's share of the stock is his share of the gain or loss. Example.—A invested $30, B invested $20, and they gained in trade $12: what is each one's share of the gain ? Place 12 (the whole gain) opposite 50 (the whole stock): then opposite 20 (A's stock) is $4-80; and opposite 30 (B's stock) is $7-20. EVOLUTION. To extract the Square Root. Rule.—Move the given number around until it is opposite the same number which is opposite 1; and that number is the answer sought. Example.—What is the square root of 42 ? Move 42 on the circular around until it comes opposite 6-48. Now 6-48 is opposite 1: hence that is the square root of 42. To extract the Cube Root. Rule.—Move the given number around until it 28 comes opposite a number, the square of which at the same time is opposite 1; and that number is the root sought. Example.—What is the cube root of 27 ? Move 27 around until it comes opposite 3: at that time 9 is opposite 1: hence 3 is the answer. TO APPORTION TAXES. Rule.—Place the whole tax to be raised, found on the circular, opposite the whole valuation: then oppo- site each man's valuation, is his tax. Example.—A tax of $1,500-00 is levied on a val- uation of $200.000-00: what is a man's tax whose valuation is $700-00 ? Place 1500 opposite 200.000: then opposite 700 is $5-25, the answer. School Tax. 1550 days have been sent, and $33*20 tax is to be raised: how much is each man's tax ? Place 33-20 opposite 1550: then opposite the days each man has sent is his tax. A has sent 28 days: his tax is 60 cents. Opposite 70, the number of days B has sent, is his tax, $1-50; and so of every other man's tax, without moving the scale. TO COMPUTE TOLL. What is the toll on 6000 pounds, for 289 miles, at 4 mills per mile per 1000 pounds ? Place the 4 opposite 1000 : opposite 6 is -024 (two cents four mills). Now place this opposite 1: then opposite 289 is $6-936, the answer. TO MEASURE SUPERFICES. Rule 1.—Place the width in inches opposite 12 : then opposite the feet in length, is the answer in feet and tenths of a foot. Example.—Give the contents of a board 6 inches wide, 14 feet long. Place 6 opposite 12 : then opposite 14 (the length), is the answer, 7 feet. Rule 2.—Place the width in feet opposite 1: then opposite the length in feet, is the answer in feet. How many square feet in a floor 20 by 20 ? 20x20=400, the answer. i-------1 How many square feet in a garden 9/ I-------1 by 54 feet ? •^ 96X54=5184 feet, answer. Note.—If one side be inches and the other feet, place the given number of inches opposite the number of inches 3* 30 in a foot, viz. 12 : then opposite the length in feet, will be the answer in feet. If one side be feet and the other rods, the answer will be in rods by placing the feet opposite the number of feet in a rod; &c, &c. In a lot of land 120 rods long and 60 rods wide, how many acres ? Place 60 opposite 160 (the number of rods in an acre): then opposite 120, is 45 acres, the answer. If a board be 8 inches wide, how much in length will make a square foot ? Place the width, 8 inches, opposite 1: then oppo- site 144 (the number of square inches in a foot) is the answer, 18 inches. If a piece of land be 5 rods wide, how many rods in length will make an acre ? Place 5 opposite 1: then opposite 160 (the num ber of rods in an acre) is the answer, 32 rods. Square Yards. How many square yards of carpeting will it require to cover a floor 20 feet long and 14 feet wide ? Place 20 found on the circular opposite 9 (the gauge point for yards square): then opposite 14 on the fixed part is 31 yards, the answer. The Width and Contents given, to find thr Length. Rule-—Place the contents on thr r-.!rc-;!'ir opposite 31 33 35 the width in feet: then opposite 9, on the fixed part, is the length in feet. Example.—I have a room containing 20 square yards: I wish to cover it with a piece of carpeting 2£ feet wide : how many feet in length will it re- quire ? Place 20 on the circular opposite 2-5 (2J): then opposite 9, on the fixed part, is 72 feet, the answer. To measure Land in Chains and Links. Rule.—Place one of the sides in chains and links, opposite 1: then opposite the other side, in chains and links, are the number of acres and parts of an acre. Example.—To find the acres in 7 chains and 50 links by 6 chains and 40 links. Place 750 opposite. 1: then opposite 640 is 4-80 (4rV*ff) acres, the answer. To fina tne acres in 7 chains and 75 links by 9 chains and 64 links. Place 775 opposite 1: then opposite 964 is 7^5 acres, the answer. To find the amount of land in 1 chain and 80 links by 2 chains and 50 links. Place 180 opposite 1: then opposite 250 is "45 (t* 'he answer. products together, the square root of which is the answer. Example.—What is the hypotenuse of a right- ingled triangle, one side of which is 3 feet, the other 4 feet? 3X3=9 and 4X4=16: these two added together, make 25, the square root of which is 5 feet, the answer. A To measure a Triangle. Place half the base opposite 1: then oppo- site the perpendicular height, is the area. Example.—What is the area of a triangle whose base is 32 inches, and perpendicular height 14 inches ? Place 16 (J of 32) opposite 1: then opposite 14 is 224 square inches, the answer. To find the Solid Contents of a Pyramid. Rule.—Multiply the area of the base oj | of the perpendicular height, whether it be d square, triangular, or circular pyramid. Example.—What is the solid contents of a pyra- mid whose base is 4 feet square, and perpendicular height 9 feet ? 4X4=16, the base. Place this opposite 1. Now J of 9 is 3. Opposite 3 is the solid contents, 48 feet. To find the Solid Contents of a Frustrum of* Cone. Rule.—Multiply each diameter by itself sepa. rately, multiply one diameter by the other, add these three products together. Now place the length opposite 382 : then opposite the products thus added is the answer. ______ To find the Circumference of a Circle from j'jj Diameter, or its Diameter from its Circumfet ence. Rule.—Place letter c, (found on the circular,) opposite fig. 1: then the figures on the fixed pari are diameters, and those on the circle are circumfer- ences. Opposite each diameter is its circumference. Example.—What is the circumference of a circle whose diameter is 9 inches ? Place c opposite fig. 1: then opposite 9 is 28-2, (28 inches and 2 tenths,) the answer. To find the Area of a Circle. Rule.—Place the square of the diameter opposite 1: then opposite the letter a is the o Example.—What is the area of a circular garden whose diameter is 11 rods? Place 121 (the square of 11) opposite 1: then opposite letter a is 9503 rods, the answer. 32 To measure Square Timber. Rule.—Place the product of the width by the thickness, opposite 144: then opposite the length is the answer in feet and tenths. Example.—What is the solid contents of a stick 4 inches by 7, and 20 feet long ? 4 X 7=28. Place 28 opposite 144: then opposite the length, 20 feet, is 39 feet, the answer,=3T% feet. What is the solid contents of a stick of timber 18 inches by IS inches, and 13 feet long ? The product of 18 by 18, is 324. Now place 324 opposite 144: then opposite 13 (the length) is 29-3, (29-rV,) the answer. N. B.—If it be desired to have the answer in inches, instead of placing the product of the width by the thickness, opposite 144, place it opposite 1: then opposite the length in inches, will be the solid con- tents in inches. Note.—Any bale, box, or chest may be measured by the preceding rule. To measure a Hypotenuse. ab hypotenuse, bc perpendicular, AC base. 34 A There is a cone whose height is 27 feet, and whose base is 7 feet in diameter • what are its contents ? Place the square of 7 (49) opposite 1: then oppo- site a is the area of the base. J of 27 is 9. Place 9 opposite 1: then opposite the area (386) is the answer, 346£ solid feet. To find the Solid Contents of a Frustrum of a Pyramid. Rule.—To the product of one end by the other, add the sum of the squares of each end. Place this opposite 144. Then opposite J of the length, is the answer. Example.—What are the contents of a stick of timber whose larger end is 12, whose smaller end is 8 inches, and whose length is 30 feet ? The product of one end by the other is 96, the square of 12 is 144, the square of 8 is 64. These, all added, make 96 144 64 304. Place this opposite 144. then opposite 10 (J of the length) is the answer, 21J feet. 36 To find the side of a Square equal in area to any given Circle. Q Rule.—Place '886, found on the circular, opposite fig. 1: then opposite any diameter of a circle upon the fixed part, is the side of a square equal in area, on the circular. Example.—What is the side of a square equal in area to a circle 4 feet in diameter2 Place '886 opposite fig. 1: then opposite 4 is 3*55 feet, the answui. To find the side of the greatest Square that can be inscribed in any give rcle. Rule.—Square each of the sides and add theii Rule.—Place '707, found on the circular, opposite fig. 1 : then opposite any diameter of a circle (found on the fixed part,) is the side of its in- scribed square. Example.—What is the side of an inscribed square equal in area to a circle 45 rods in diameter ? Place '707 opposite fig. 1: then opposite 45, on the fixed part, is 318 rods, the answer. To find the length of one side of the greatest CuU that can be taken from a Globe of a given diam- eter. Rule.—Place 577, found on the circular, opposite fig. 1: then opposite any diameter, on the fixed part, is the length of one side of the greatest cube. 37 39 41 Example. What is the length of the side of the greatest cube that can be taken from a globe 82 inches in diameter? Place 577 (the gauge point for the side of an inscribed cube) opposite fig. 1: then opposite 82, on thr fixed part, is 47-3 (47-j^) inches, the answer. to find the length of the side of the greatest equi- lateral triangle that can be inscribed in a given circle. Rule.—Place 87, found on the circular, opposite fig. 1: then opposite any diameter on the fixed part, is the length of the side of an inscribed triangle And opposite the length of the side of any triangle on the circular, is the diameter required to inscribe it in. Example.—What is the length of one side of the greatest equilateral triangle that can be inscribed in a circle 62 inches in diameter ? Place 87 opposite fig. 1: then opposite 62, on the fixed part, is 54 inches, the answer. What is the least diameter of a circle in which a triangle may be inscribed whose side is 6-5 inches (6|)? Place 87 opposite fig. 1 : then opposite 6-5, on the circular, is 7-48 (7-j*^) inches, the answer. 4 in which may be inscribed an undecagon (eleven- sided figure.) one side of which is 13 inches long ? Place 282 opposite fig. 1: then opposite 13 inches, found on the circular, is 46-1 inches, the answer. To find the greatest diameter of each of three equal circles that can be inscribed within a circle of a given diameter. Rule.—Place, 464 opposite fig. 1 : then op- posite any diameter on the fixed part, is the diam- eter of one of the three inscribed circles. Example.—What is the greatest diameter of each of three circles, that can be inscribed within a circle 25 inches in diameter? Place 464 opposite fig. 1: then opposite 25 on the fixed part, is 11-6 inches, the answer. To find the greatest diameter of four equal circles that, can be inscribed, within another circle of a given diameter. Rule.—Place 416 opposite fig. 1: then opposite any given diameter on the fixed part, is the diameter of each of the four inscribed circles. Example.—What is the greatest diameter of each of four equal circles that can be inscribed in another circle 22 inches in diameter ? Place 416 opposite fig. 1 : then opposite 22, on the fixed part, is 9-15 (9yVo") inches, the answer. In a log 7 inches diameter, 15 feet long ? Answer 4 y&j feet. Note.—If the diameter and length are both given in inches, place the square of the diameter opposite 1728: then opposite the inches in length, is the answer in feet. Note.—A cylinder that is 12 inches in diameter and 12 inches long, and a globe that is 12 inches in diameter, and a cone that is 12 inches high and 12 inches diameter at its base, bear a proportion to each other as 3, 2 and 1. Therefore if you place the contents of any cylinder on the circular opposite to 3 on the fixed part, then opposite 2 on the fixed pait is the contents of an inscribed globe, and opposite fig. 1 is the contents of an inscribed cone. To find how many Solid Feet a Round Stick of Timber will contain, when hewn Square. Rule.—Place double the square of half the diam- eter opposite 144: then opposite the length is the answer. Example.—In a log 28 feet long, 22 inches diam- eter, half the diameter is 11, the square of which is 121. This doubled, is 242. Now place 242 oppo- site 144: then opposite 28 (the length) is 47+the answer. To find how many feet of Boards can be sawn from a Log of given Diameter. Rule —Find the solid contents of the log wb»r 4* 33 To find the length of the side of the greatest figurt that can be inscribed in a given circle. Rule for a Pentagon (5 sides) Place 589. Hexagon 6 " " 5. Heptagon 7 " " 437. Octagon 8 " " 3-83 Nonagon 9 " " 337 Decagon 10 •' " 31 Undecagon 11 " " 282 Dodecagon 12 " " 26 opposite fig. 1: then opposite any given diameter on the fixed part, is the length of the side of the greatest figure that can be inscribed in it. Example 1.—What is the length of one side of the greatest pentagon, or five-sided figure, that can be inscribed in a circle whose diameter is 51 inches ? Place 589 opposite 1: then opposite 51, on the fixed part, is 30 inches, the answer. Example 2.—What is the length of one side of the greatest nonagon (nine-sided figure) that can be ■ascribed in a circle 82 feet in diameter ? Place 337 opposite fig. 1: then opposite 82, found on the fixed part, is 27-6 (27 fij) feet, the answer. Example 3.—What is the least diameter of a circle 40 mo find the Solidity of a Cylinder, or to measurt Round Timber. l|i Rule.—First find the area of the ^^Sb^^EI Dase DV tne ru[e for finding the area of a circle, place that area opposite 144, then oppo- site the length in feet, is the answer in feet and decimals of a foot. Note.—If the diameter be given in feet, place the area opposite 1, instead of placing it opposite 144. Example.—What are the solid contents of a cyl- inder 5 inches in diameter, and 13 feet long? Place 25 'the square of 5) opposite 1 : then oppo- site a is 1-965. Now place 1-965 opposite 144. then opposite 13 (the length) is 1-77 feet, the answer. How many solid feet in a round log 15 inches in diameter, and 14 feet long? Place 225 (the square of 15) opposite 1: then opposite a is 1-77 the area. Now place 1-77 oppo- site 144: then opposite 14 is 17-2 feet, the answer. In a log 12 feet long, 14 inches diameter ? Answer, 12-8 feet. In a log 16 feet long, 11 inches in diameter1 Answer, 10-5 feet. made square, then place 12 opposite the thickness of the board (including the saw-calf:) then opposite the solid contents is the answer in feet. To find the Area of a Globe or Ball. #Rule.—Place the diameter opposite 1: then opposite the circumference is the answer. Example.—How many square inches of leather will cover a ball 3| inches in diameter ? Place 3J opposite 1: then opposite d. is 11, the circumference. Opposite 11 is the area, 38J inches. How many square feet on the surface of a globe 4 feet in diameter ? Place 4 opposite 1: then opposite d. is 12-55 feet, the circumference. Opposite 12-55 is 50-4, the answer. To find the Solid Contents of a Globe or Ball. __ Rule.—First find its area by the preceding ^ rules: then multiply its area by £ of its ^mr diameter. Example.—What are the solid contents of a ball 14 inches in diameter ? Place 14 opposite 1: then opposite d. is 44 inches, the circumference. Opposite 44 is 617, the area. \ of the diameter, is 233 J. Place this opposite 1: then opposite 617 (the area) is 1437 inches, the solid contents. 43 What are the solid contents of a ball 5 inches in diameter ? Place 5 opposite 1: then opposite d. is 15-7 inches, the circumference. Also, opposite 15-7 inches is 78-4 inches, the area. | of 5 is -835. Place this opposite 1: then opposite 784 inches (the area) is 654 inches, the solid contents. There is a ball 20 inches in circumference : what are its solid contents ? Place 20 opposite letter d. Opposite 20 is 127, the area. £ of the diameter is 1-06. Place this opposite 1: then opposite 127 is 1350 inches, the solid contents. To find the Area of an Ellipse. Rule.—Place the product of the trans- verse diameter multiplied by the conjugate diameter opposite 1: then opposite letter a is the answer. Example.—What is the area of an ellipse whose transverse diameter is 12 inches, and conjugate diameter 10 inches ? 10 X 12= 120. Place 120 opposite 1: then oppo- site letter A is 94-25, the area. 45 6-7x67=45,and45X6.7=301-5. Place301-5 opposite 1: then opposite 14 is 42-29 pounds, the answer. A ball 5-54 inches diameter ? Answer, 24 pounds nearly. A ball 32 inches circumference ? Place 32 opposite d: then opposite 1 is the diameter. Now cube the diameter, and place that cube opposite 1: then opposite 14 is 148 pounds, the answer. To find the Weight of a Leaden Ball from its Diameter or Circumference. Rule.—Place the cube of the diameter opposite 1: then opposite 21-5 is the weight. A ball is 6-6 inches in diameter: what is its weight ? Answer, 61-6 pounds. A ball 5-3 inches in diameter ? Answer, 32 pounds nearly. To find the Diameter of an Iron Ball from its Weight. Rule.—Place the weight opposite 1: then oppo- site 7-11 is a product, the cube root of which is its diameter 46 What is the diameter of a 24 pound ball ? Answer, 5*54 inches. To find the Diameter of a Leaden Ball from its Weight. Rule.—Place 14 opposite 3: then opposite the weight is a product, the cube root of which is the answer. A ball 8 pounds in weight is 3*34 inches in diameter. 47 Specific Gravity and Weight of Bodies. Pure Platina oz. 23000 Clay . . . cz. • 2160 Fine Gold . 19400 Brick . . . ■ 2000 Standard Gold 17720 Common Earth 1984 Quicksilver . 13600 Nitre . . . 1900 Lead . . . 11325 Ivory . . . . 1825 Fine Silver . 11091 Brimstone 181(1 Common Silver 10535 Solid Cunpowder 174.5 Copper . . 9000 Sand . . . . 1520 Copper Pence 8915 Coal . . . • 1250 Gun Metal . 8784 Mahogany . 1063 Cast Brass 8000 Boxwood . . . 1030 Steel . . . 7850 Sea Water . . 1030 Iron . . . 7645|Common Water 1000 Cast Iron . . 7425 Oak . . . Q9H Tin ... 7320 Gunpowd'r shook close937 Crystal Glass 3150 " in a loose heap 836 Granite . . 3000 Ash ... 800 White Lead . 3160 Maple . . . 755 Marble . . 2700 Beech . . . 700 Hard Stone . 2700 Elm . . . 600 Green Glass . 2600|Fir .... 550 Flint . • • 2570JCork . . . 240 Common Stone 2520|Air at a mean state 1$ Note.—The several sorts of wood are supposed to be dry. Also, as a cubic foot of water weighs just 1000 ounces, the numbers in this table express, not only the specific gravities of the several bodies, but also the weight of a cubic foot of each, in avoirdupois ounces; and there- fore the weight of any other quantity, or the quantity of any other weight, may be found, as in the next two proposi- tions. 48 To find the Magnitude of any Body from its Weight. 44 Gauging Casks. To find the Mean Diameter of a Cask. Rule.—Add § of the difference between the head and bung diameter to the head diameter. This re- duces the cask to a cylinder. Then multiply the square of the mean diameter by the length. Place the product opposite 1: then opposite bg is the num- ber of beer gallons, and under wr. is thp number, of wine gallons. Example.—There is a cask whose head arameter is 25 inches, bung diameter 31 inches, and whose length is 36 inches : how many beer gallons and how many wine gallons does it contain ? 6 is the difference between 25 and 31. $ of 6 is 4. This, added to 25, makes 29 inches, the mean diameter. The square of 29 is 841. Place this opposite 1: then opposite 36 is 302-f-. Place this last opposite 1: then opposite bg is 85 gallons, and opposite wg is 103 gallons, the answer. To find the Weight of an Iron Ball, from its Diameter. Rule.—Place the cube of the diameter opposite I: then opposite 14 is the weight. Example.—What is the weight of an iron ball 6*7 inches in diameter ? Rule.—Place the weight of the material in ounces under its specific gravity: then opposite 1728 is its magnitude in cubic inches; and opposite 1 is the answer in cubic feet. Example.— How many cubic inches of gunpowder are there in one pound weight, shaken close ? Place 16 (the number of ounces in a pound) oppo- site 937 : then opposite 1728 is its content or magni- tude, 29£ inches. How many cubic inches are there in 3 pounds of cast brass ? Place 48 (the number of ounces in 3 pounds) oppo- site 8000 : then opposite 1728 is the answer, 103-5. To find the Weight of a Body from its Magnitude. Rule.—Place the contents of the body opposite 1728 : then opposite its specific gravity is its weight in ounces. How many ounces avoirdupois in 864 cubic inches of sand ? Place 864 opposite 1728: then opposite 1520 (the specific gravity of sand) is 760 ounces, the answer. 49 51 TABLES OF SQUARES AND CUBES. 53 Measure, fyt- 5,280 feet in a mile. 63,360 inches in a mile. 190.0S0 barley-corns in a mile. 32,000 ounces make one ton. 43,560 square feet in an acre. 4,S40 square yards in an acre. 32 gills in one wine-gallon. 7-22 cubic inches in a gill. 28-875 cubic inches in a pint. 57-75 cubic inches in a quart. 2,1504-f- cubic inches in a bushel. 12444 cubic feet in a bushel. 3,600 seconds in an hour. S6.400 seconds in a day of twenty-four hour* 31,557,600 seconds in a year. 1.72S cubic inches in a foot. 128 feet make one cord of wood. TABLE9 OF SQUARES AND CUB ES; Number. Bquere. t*.. Number. W]..are. Cube. To facilitate the Mensuration of the Surjuies ««« 201 40401 8120601 251 63001 15813251 Solidities of Bodies. 203 40804 824241)3 352 63504 16003008 203 204 41209 41616 6365427 8489664 253 254 64009 64516 16194277 16387064 ■ S,u«e. ube bel- Square. Unbe. 205 206 42025 42436 8615125 B7418I6 255 356 65025 65536 16581375 16777216 l 1 1 50 2500 125000 2 4 8 51 2601 132651 207 42849 B869743 257 66049 10S74593 3 9 27 52 2704 140608 208 43264 6998912 358 66564 17173512 4 16 64 53 2809 148877 209 43681 9123329 259 67081 17373079 5 25 125 54 2916 157464 210 44100 9261000 260 67600 17576000 6 36 216 55 3025 166375 211 44521 9393931 261 68121 17779581 7 49 343 56 3136 175616 212 44944 9528128 262 6.-644 17984728 8 64 512 57 3249 185193 213 45369 9663597 263 69169 18191447 9 Bl 729 58 3364 195112 214 45796 9800344 264 69696 18399744 10 100 1000 59 3481 205379 215 46225 993-375 265 70225 18609625 11 121 1331 60 3600 216000 216 46656 10077696 266 70756 I6821096 ia 144 1728 61 3721 226981 217 47089 10218313 267 71269 19034163 13 169 2197 62 3844 238328 218 47524 10360232 268 71824 19248832 ! 14 196 2744 63 3969 250047 219 47961 10503459 2 9 72361 19465109 l 15 235 3375 64 4096 262144 220 48400 1U648000 270 72900 191.83000 ; 16 256 4096 65 4225 274B25 221 46841 10793861 271 73441 19902511 17 289 4913 66 4356 287496 222 49284 10941048 272 73984 B012-1648 18 324 5832 67 4489 300763 223 49729 110 9567 273 74559 20346417 « 19 361 6859 68 4624 314432 224 50176 11239424 274 75076 20570824 | 80 400 eooo 69 4761 328509 225 50625 11390625 275 7.5C25 20796875 ! 21 441 9261 70 4900 343000 226 51076 11513176 276 76176 21024576 ] 92 4S4 10648 71 5041 357911 227 51529 11697083 277 76729 21253033 23 529 12167 72 5184 373248 228 51984 11852352 278 77284 21484952 | 24 576 13824 73 5329 389017 229 52441 121)01*989 279 77841 21717639 | 25 625 15625 74 5476 405224 230 55900 1216700;) 280 78400 31952000 i 26 676 W576 75 5625 421875 231 53361 12326391 281 78961 22188041 j S7 729 19683 76 5776 438976 232 53824 1V487HM 282 79524 22425768 1 28 784 21952 77 5929 456533 233 54289 126(9337 263 Mint 9 22605187 ! 29 841 24389 78 6084 474552 234 51756 12812904 284 80(151) 229 6304 30 900 27000 79 6241 493039 235 55225 12977875 285 81225 23 43125 j 31 961 29791 80 6400 512000 236 55696 13144256 2*6 81796 2339365(1 32 1024 32768 81 6561 531441 237 56169 13312153 287 82369 23639903 33 1089 35937 82 6724 551368 238 56644 13481272 288 82914 £'887879 ' 34 1156 39304 83 6889 571787 239 57121 13651919 | 269 835> 1 24137509 1 35 1225 42875 84 7056 592704 240 57600 13824000 | 290 8-11011 36 1296 46656 85 7225 614125 241 58081 13997521 291 84681 37 1369 50653 86 7396 636056 242 58564 14172488 292 85264 38 1444 54=172 87 7569 658503 343 59049 14348907 ' 293 6.iS4!) 33 1521 59319 88 7744 681472 244 59536 14526784 294 86436 251! ■_■:.0 Comparative Value and Weight of Different Ktndi of Fire Wood, assuming as a standard the Shell Bark Hickory. 52 TABLES of squares and CUBES. Lbs. in a Cord. Compar. Val. P ets. Shell-Bark Hickory Button Wood Maple Black Birch White Birch White Beech White Ash f'oimnon Walnut i'itch Pine White Pine i, vnbardy Poplar Apple Tree White Oak Biick Oak Scrub Oait Spanish Oak Yellow Oak Red Oak White Elm Swamp Whortleberry N0TE.—It is estimated that a cord of wood contains, when green, 1443 pounds of water, equal to 1 hogshead »nd 2 barrels of water. 4469 100 7 4C 2391 52 3 95 2668 54 4 00 3115 63 4 67 2369 1- .3 56 3236 65 4 81 3420 77 5 70 4241 95 7 03 1904 43 3 IS 1868 42 3 11 1774 40 2 96 3115 70 5 1 = 3821 81 6 00 3102 66 4 89 3337 73 5 40 2449 52 3 85 2919 60 4 44 3254 69 5 11 2592 58 4 29 3361 73 5 40 Number. Sq..«re. Cbe. N„„be, Square. Cube. 970S99 150 23500 3375000 10000 1000000 151 32801 3442951 1030301 152 33104 3511808 1061208 153 33409 3581577 10609 1092727 154 23716 3653264 1124864 155 24025 1157625 156 24336 3796416 1191016 157 24649 3869893 1225043 158 24964 3944312 1259712 i 159 25281 4019679 11881 1295029 160 25600 4096000 1331000 161 25921 4173281 12321 1367631 163 26244 4251528 1 12544 1404923 163 26569 4330747 12769 1442897 164 26896 4410944 12996 1481544 165 27225 4492125 13225 1520875 166 27556 4574296 13456 1560896 167 27889 4657463 13689 1601613 168 28224 4741632 13984 1643032 169 28561 4826809 14161 1685159 170 38900 4913000 1728000 171 29241 1771561 172 29584 1815841 173 29929 5177717 1860867 174 30276 15376 1906624 175 30625 1953125 176 30976 15876 2000376 177 31329 2048383 178 31684 3097152 179 32041 2146689 180 32400 2197000 181 32761 2248091 182 33124 17421 2399968 183 33489 2352637 184 33856 17956 2406104 185 34225 2460375 186 34596 2515456 187 34969 18769 3571353 188 35344 2628072 189 35721 19321 2685619 190 36100 2744000 191 36481 £803221 192 36864 20164 2863288 193 37249 20449 2924207 194 37636 2985984 195 38025 3048625 196 38416 21316 3112136 197 38809 3176523 198 39204 • 21904 3241792 199 39601 149 22201 3307949 200 40000 54 THE STEAM-ENGINE. The power of the steam-engine is measured by that of the horse. A horse-power, as fixed by Watt, ia equal to 33,000 lb. avoirdupois, raised one foot high per minute; and one day's work of a horse, is this power, acting through eight hours. The pressure of our atmosphere is reckoned as equal to that of thirty perpendicular inches of mercury; or 14-701 b. per square inch, or 11-55lb. per circular inch. To find the, Horse's power of an Engine, according to the Rule given by Mr. Watt. From the Diameter of the cylinder in inches, sub- tract 1, square the remainder, multiply the square by the velocity of the piston in feet per minute, and di- vide the product by 5640. The quotient will be the number required. COJNDhNSING ENGINES. Proportion of the Cylinder.—The best proportion is when the length is twice the diameter ; because the cooling surface is then least, in proporton to the con- tent of steam. Proportion of the Air-Pump and Condenser.—In double condensing engines, these are made, by Boul ton and Watt's rule, each to measure one eighth th« content of the cylinder. 55 57 59 Velocity of the Piston toproduce the best effect.—In engines working the steam expansively, 100 times the square root of the length of the stroke in feet, is the best velocity in feet per minute. In engines not working expansively, 103 times the square root of the length of the stroke in feet, is the best velocity in feet per minute. To find the quantity of Water required for Steam tml Injection.—Multiply the area of the cylinder in (eel, by half the velocity in feet for single, and by the whole velocity in feet for double engines. Add l-10th for cooling and waste ; and this, divided by 1497 (at the common pressure on the valve of 21b. per circular inch), wil give the quantity of water required for steam per minute. The quantity of water for injection should be 21 times that required for steam. The diameter of the injection-pipe should be l-36th part of that of the cylinder. The valves should be as large as practicable. The boiler should be capable of evaporating abou 12 gallons per hour for each horse power. NON-CONDENSING, OR HIGH PRESSURE ENGINES. The length of the cylinder should be at least twice i's diameter. The velocity of the piston, in feet per minute, should be 103 times the square root of the length of the stroke MARINE ENGINES. The construction and arrangement of the Marine Steam Engine necessarily differ from that of the ordi- nary condensing Engine, on account of the peculiar form of the floating structure in which it is placed, and of the absence of that solid support which can be ob- tained for Engines on land. The importance of et fecting economy of room and weight on board a steam vessel, has led to the adoption of various methods of communicating motion to the paddle wheels; and vertical, oscillating, and other varieties of Engine have been introduced, with more or less success; but the more general form is that of the beam or lever Engine, the position of the beam being reversed on being plac- ed on each side of the bottom of the cylinder. The arrangement of the condenser, air-pump, &c, is also necessarily accommodated to the space in which the machinery is required to be fixed. The following Dimensions are given by Mr. Rus- sell, for the Cylinders of Marine Engines of various power: For 10 horse power, 20 inches diameter, 2 ft. 0 in. stroke. .. 20 .. 11 .. 2ft.6in. .. .. 30 .. 32 .. 3 ft. 2 in. .. .. 40 .. 35 .. 3a.6in. .. .. 50 .. 40 .. 4 ft. 0 in, .. = I Duration t of Kxper |T. t ilCwisump- Mean Pre* ti.niu pound* sure in At- 1'ivoir.lupuis mospheres. ICnnniunptinn prrliour, in pound.1*. .■o....!w...r.|e"rf-|2Jrl 43h 15iii|14f2.7|f>387.1 13.82 33h30m|l!l"-'■'- "ni.59 3.5 .191i 5tnm!l4ti9.0 , - i».2.'l Q.S 2.57 2.55 W,t« (▼■porn rdbyl ro.it, |fw*| ?«' ."fl.ytf lli)3.!l|5.66 " 5-U6 331.7 5.61 2.73 1.24 | 15.22 |210.7|5.32 Friction of Steam-engines. The difference in loss of power by friction, between beam and direct action engines is found by experiment to be so trifling, as to be unnecessary to be taken into account in estimating their relative advantages. The amount of pressure upon the piston, expended in each kind of engine in overcoming friction appears, on an average, to be not more than about 1 lb. to the square inch, in well-constructed engiucs. Steam-engines for Cotton and Paper Mills. For Cotton Mills.—The best steam-engines for cot- ton-mills are the double-acting, working the steam expansively. The most advantageous mean pressure in the piston with low pressure steam is 51b per circu- lar inch, ami each circular inch will suffice to drive three spindles of cotton yarn twist with the machinery. For mule yarn, add 15 to the number of the yarn, and multiply the sum by -26 ; the product will be the number of spindles for each circular inch of piston. Or, one horsepower will drive 100 spindles with cotton yarn, and machinery. And for mule yarn, add 56 in feet j or 100 times, if the steam is worked expan- sively. The area of the cylinder should be, to tlie area of the steam-passages, as 4800 is to the velocity of the piston, found as above. Form and Direction of Steam-pipes.—Enlargements in steam-pipes succeeded by contractions, always re- tard the velocity of the steam—more or less according to the nature of the contraction—and the like effect is produced by bends and angles in the pipes. These should therefore be made as straight, and their internal surface as uniform and free from inequalities as may be practicable. The following proportions of velocity, from Mr. Tredgold, will exemplify this :— The velocity of motion that would result from the direct unretarded action of the column of fluid which produces it, being unity .... 1000 or 8 The velocity through an aperture in a thin plate by the same pressure is .625 or 5 Through a tube from two to three diame- ters in length, projecting outwards .813 or 6.5 Through a tube of the same length, pro- jecting inwards - • - .681 or 5.45 Through a conical tube, or mouth-piece, of the form of the contracted vein .983 or 7.9 58 For60 horse power, 43 inches diameter, 4 ft. 3 in. stroke. .. 70 46 4 ft. 6 in. .. 80 49 4ft. 9in. .. 90 52 5 ft. 0 in. ..100 55 5 ft. 6 in. ..125 - 59 6 ft. 0 in. ..150 62 6 ft. 3in. ..175 66 6 ft. 6 in. ..200 70 7ft.0in. .250 76 7 ft. 6 in. ..300 82 8ft.0in. ..350 87 8 ft. 6 in. ..400 92 9 ft. 2 in. ..500 100 10 ft. 0 in. Economy of Steam-jackets. The following Table presents the results of three experiments made in France to ascertain the economy of steam-jackets to the cylinders of Engines, in the consumption of fuel. In the 1st, the steam first enter- ed the jacket round the cylinder, and passed from thence into the cylinder. In the 2nd, the steam enter- ed the cylinder directly, without passing into the jacket. In the 3rd, the steam entered both the cylinder and jacket directly, by means of separate communications between them and the boiler. The result shows an increase in the consumption of fuel of nearly five- sevenths, in the second experiment, over that in the first. 60 !5 to the number of the yarn, and multiply by 3 ; tne product will be the number of spindles for each horse- power. One horse-power will work 12 power-looms, with the preparatory machinery.—Brunlon. For Paper Mills.—A beating machine requires about 7 horse-powe r. The new paper machines require from 2 to 2 1-2 horse-power) 3 1-2 horse-power will prepare 1 ton old rope per week, working ten hours per lay.—Fenwick. Steam-power required to drive various lands of Ma. chinery. A series of experiments instiuted by .Mr. Davison, at Messrs. Truman and Co.'s Brewery, to ascertain the power required to drive various kinds of machinery, gave the following results : 1st. That an engine which indicated 50 horses power wnen fully loaded, showed, after the load and the whole of the machinery were thrown off, 5 horses, or one. tenth of the whole power. 2nd. 190 feet of horizontal, and 180 feet of upright shafting, with 34 bearings, whose superficial area was 3300 square inches, together with 11 pair of spur and bevel wheels, varying from 2 feet to 9 feet in diameter, reqi-te, d a power equal to 7.65 horses. 3rd. A set of three-throw pumps, 6 inches in diame- ter, pumping 120 barrels per hour, d a height of 165 feet,=4.7 horses. 61 By the usual mode of calculation (viz., 33,000 lbs. lifted one foot high per minute), it would appear that there was, in this case, friction to the extent of 13 per cent. 4th. A similar set of three-throw pumps, 6 inches in diameter, pumping 160 barrels per hour, to a height of 140 feet,=6.2 horses. By the same mode of calculation as before, there was here friction to the amount of 15 per cent. 5th. A set of three-throw pumps, 5 inches in di- ameter, raising SO barrels per hour, to a height of 54 feet,=l horse. By calculation as before, the friction amounted to 12 1-2 per cent. 6th. A set of three-throw "starting" pumps, pump. ing 250 barrels of beer per hour, to a height of 48 feet, =4.37 horses. By calculation as before, the friction amounted to 15 1-2 per cent. 7th. Two pair of iron rollers and an elevator, grinding and raising 40 quarters of malt per hour=8.5 horses. 9th. An ale-mashing machine, made by Haigh, of Dublin ; mashing at the time, 100 quarters of malt,= 5.63 horses. 9th. Two porter-mashing machines, made by More- land, mashing at the time, 250 quarters of malt,=10.8 horses. 6.1 Lowoy cooling in the cylinder and pipes .... 0-160 Loss by friction of the piston and waste .... 1-250 Force required to expel the steam through the passages - 0-070 Force required to open and close the valves, raise the injection water, and overcome the friction of the axes - . . 0-630 Loss by the steam being cut off be- fore the end of the stroke - 1-000 Power required to work the air-pump 0-500 Amount of deductions ----- Effective pressure 3-680 6-320 Pressure and Density of Steam. The following formula has been given by Mr. Wm. Pole for calculating the pressure and density of steam for engines working expansively, which is stated to produce a very near approximation to the truth; the mean error being only .0062 lb. per square inch : Let P be the total pressure of the steam in lbs. per square inch, and V its relative volume, compared with that of its constituent water. 24250 24250 Then P=------, or V =-----rlus65. V-65 P 65 To prevent Incrustation in boilers.—The intioduc- tion of potatoes andother vegetable substances will, in a great degree, prevent incrustation on the bottom and sides of a steam boiler, and animal substances, such as refuse skins, will accomplish it still more effect- ually. Iron Cement for joining the Flanches of Iron Pipes, SfC.—Take of Sal Ammoniac, 2 ounces; Flowers of Sulphur, 1 ounce; clean cast-iron Borings or Filings, 16 ounces : mix them well in a mortar, and keep them dry. When required for use, take one part of this powder, and twenty parts of clean iron borings or fil- ings, mix them thoroughly in a mortar, make the mix- ture into a stiff paste with a little water, and apply it between the joints, and screw them together. A little fine grindstone sand added, improves the cement. A mixture of white paint with red lead, spread on can- vas or woollen, and placed between the joints, is best adapted for joints that require to be often separated. For Copper, a cement is used, composed of powder- ed quick lime, mixed to a proper consistence with serum of blood, or white of egg—and used immediate ly it is made. THE MECHANICAL POWERS. Power is compounded of the weight and expansive force. of a moving body multiplied into its velocity. The power of a body which weighs 40 lbs., and 62 10th. 95 feet of horizontal Archimedes screw, 15 inches diameter, and an elevator, conveying 40 quar- ters of malt per hour, to a height of 65 feet,=3.13 horses. Mr. Tredgold's Estimate of the Distribution and Expen- diture of the Steam in an Engine. IN A NON-CONDENSING ENGINE. Let the pressure on the boiler be 10 000 Force required to produce motion of the steam in the cylinder will be 0-069 Loss by cooling in the cylinder and pipes.....0-160 Loss by friction of piston and waste 2-000 Force required to expel the steam into the atmosphere - - 0-069 Force expended in opening the valvs, and Iriction of the various parts 0-622 Loss by the steam being cut off be- fore the end of the stroke - 1 -000 Amount of deductions — Effective pressure IN A CONDENSING ENGINE. Let the pressure on the boiler be Force lequired to produce motion of the steam in the cylinder - 0-070 3-920 6080 10000 54 This formula is applicable, with little risk of error, to engines working with from 5 lbs. to 65 lbs. per square inch. TABLE Of the Pressure on a square and circular Inch, respec tively, excited by the elastic force of Steam at various degrees of Temperature, with the Height of the col- umn of Mercury it will support. 1. 1'RESSURE ON A SQUARE INCH, i 2. PRESSURE ON A riRrm .» tfr.n 3 ^ lid Propor pressure on Inches ofll Si Mercury = £ -* .upport- U'sS'M ed. \\gu. l| |f Propor. pressure on Inches o.' f-~" lis! inch in Iba. ig§ incli in lbs. support ed 220 21 1.963 5.15 222 2J 3.183 6.56 222 3 2.356 6.18 224 3 3.819 7.87 223 H 2.749 7.21 226 34 4.456 9 n 225 4 3.141 8.24 228 4 5.093 (..5 227 4J 3.534 9.27 230 4J 5.729 l> t 223 0 3.927 10.3 232 5 6.366 13.1 230 5.J 4.320 11.3 234 51 7.002 14.4 231 6 4.712 12.3 236 6 7.639 15.7 233 6i 5.105 13.4 236 fi* 8.276 17.0 234 7 5.498 14.4 238 7 8.912 18.3 235 74 5.890 15.4 239 7* 9.549 19.7 236 8 6.283 16.5 241 8 10.18 21.0 237 84 6.676 17.5 242 84 10.82 22.3 239 9 7.068 18.5 244 9 11.45 23.6 240 91 7.461 19.6 245 31 12.09 24.9 241 10 7.854 20.6 247 10 12.73 26.2 242 101 8.247 21.6 248 10* 13.36 27.5 243 11 8.639 22.6 250 11 14.00 28.9 244 111 9.032 23.7 251 1U 14.64 30.1 215 12 9.424 24.7 252 12 15.27 31.5 252 15 11.78 30.9 259 15 19.09 39.3 261 20 15.71 1 41.2 270 20 25.46 52.5 269 35 19.63 51.5 278 25 31.83 65.6 276 30 23.56 61.8 287 30 38.19 78.7 283 35 27.19 72.1 294 35 41.56 91.8 289 40 31.41 82.4 300 40 50.92 105 294 45 35.34 92.7 305 45 57.20 118 300 50 39.27 103 309 50 63 66 131 66 moves with the velocity of 50 feet in a second, is the same as that of another body which weighs 80 lbs., and moves with the velocity of 25 feet in a second ; for the products of the respective neights and veloci- ties are the same. 40 multiplied by 50-200 ; and 80 by 25-2000 Power cannot be increased by mechanical means. Power is applied to mechanical purposes by the lever, wheel and axle, pulley, inclined plane, wedge, and the screw, which are the simple elements of all machines. The whole theory of these elements consists simply, in causing the weight which is to be raised, to pass through a greater or a less space than the power which raises it; for, as power is compounded of the weight or mass of a moving body multiplied into its velocity, a weight passing through a certain space may be made to raise, through a less space, a weight heavier than itself. Power is gained at the expense of space, by the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. LEVER. Case 1.— When the fulcrum oftb" lever is between the power and the weight. Rule.—Divide the weight to be raised by the pow- er to be applied; the quotient will give the difference 67 69 71 of leverage necessary to support the weight in equili- brio. Hence, a small addition either of leverage or weight will cause the power to preponderate. Example 1.—A ball weighing 3 tons, is to be rais- ed by 4 men, who can exert a force of 12cwt., requi- red the proportionate length of lever ? 60 3 tons = 60 cwt.; and — = 5. 12 In this example, the proportionate lengths of the ,Jever to maintain the weight in equilibrio, are as 5 to 1. If, therefore, an additional pound be added to the power, the power side of the lever will preponderate, and the weight will be raised. But, although the ball is raised by a force of only one-fifth of its weight, no power is gained, for the weight passes through only one-fifth of the space. The products, therefore, aris- ing from the multiplication of the respective weights and velocities are the same. Example 2.—A weight of 1 ton is to be raised with a lever 8 feet in length, by a man who can exert, for a short time, a force of rather more than 4 cwt.: re- quired at what part of the lever the fulcrum must be placed ? 20 cwt. ------= 5 ; that is, the weight is to the power as 5 -V cwt. [to 1 : therefore, 40 multiplied by 12 furthest from the weight; and------------—=24 20 feet cwt. on the support nearest to the weight. wheel and axle. Rule.—As the radius of the wheel is to the radius of the axle, so is the effect to the power. Example.—A weight of 50 lbs. is exerted on the periphery of a wheel whose.radius is 10 feet; requir- ed the weight raised at the extremity of a cord wound round the axle, the radius being 20 inches. 50 lbs. multipliedvby 10 ft.; by 12 inches. 20 inches. 300 lbs. [Ans. PULLEY. Rule.—Divide the weight to be raised by twice the number of pulleys in the lower block; the quotient will give the power necessary to raise the weight. Example.—What power is required to raise 600 lbs., when the lower block contains six pulleys? 600 .—.-----------= 50 lbs., Ans. 6 multiplied by 2 INCLINED PLANE. Rule.—As the length of the plane is to its height, so is the weight to the power. The base of the triangle is the circumference of the cylinder; its height, the distance between two con- secutive cords or threads; and the hypothenuse forms the spiral cord or inclined plane. p Rule.—To the square of the circumference of the screw, add the square of the distance between two threads; and extract the square root of the sum. This will give the length of the inclined plane ; its height is the distance between two consecutive cords or threads. When a winch or lever is applied to turn the screw, the power of the screw is as the circle described by the handle of the winch, or lever, to the interval or distance between the spirals. Velocity is gained at the expense of power by the lever, and the wheel and axle. LEVER. Case.— When the weight to be raised is at one end of the lever, the fulcrum at the other, and the power is ap- plied between litem. Rule.—As the distance between the power and the fulcrum is to the length of the lever, so is the weight to the power. Example.—The length of the lever being eight feet, and the weight at its extremity 60 lbs., requited the power to be applied six feet from the fulcrum to raise it? As 6 : 8 :: 60 : 80 lbs., Ans. 68 ■= 1 foot and a third from the weight. 5 multiplied by 1 Example 3.—A weight of 40 pounds is placed one foot from the fulcrum of a lever; required the power to raise the same when the length of the lever on the other side of the fulcrum is five feet ? 40 multiplied by 1 ---------------= 9 lbs., Ans. 5 Case 2.— When the fulcrum is at one extremity a, fcV lever, and the power at the otlier. Rule.—As the distance between the power and the fulcrum is to the distance between the weight and the fulcrum, so is the effect to the power. Example 1.—Required the power necessary to raise 120 lbs., when the weight is placed six feet from the power, and two feet from the fulcrum ? As 3 : 2 :: 120 : 30 lbs., Ans. Example 2.—A beam, 20 feet in length, and sup ported at both ends, bears a weight of two tons at the distance of eight feet from one end: required the weight on each support ? 40 cwt. multiplied by 8 ft. - = 16 cwt. on the suppor 20 feet 70 Example.—Required the power necessary to raise 540 lbs. up an inclined plane, five feet long and two feet high. As 5: 2:: 540 : 216 lbs., Ans. WEDGE. Case 1.— When two bodies are forced from one an- other by means of a wedge, in a direction parallel to Us back. Rule.—As the length of the wedge is to half its back or head, so is the resistance to the power. Example.—The breadth of the back or head of the wedge being three inches, and the length of either of its inclined sides 10 inches, required the power neces- sary to separate two substances with a force of 150 lbs. As 10 : 11-2 :: 150 : 22 1-2 lbs., Ans. Case 2.— When only one of the bodies is moveable. Rule.—As the length of the wedge is to its back or head, so is the resistance to the power. Example.—The breadth, length, and force, the same as in the last example. As 10: 3 :: 150 : 45 lbs., Ans. SCREW. The screw is an inclined plane, and we may sup- pose it to be generated by wrapping a triangle, or an inclined plane, round the circumference of a cylinder. 72 N.B. Any other example may be computed br reversing any of the foregoing operations. -ixllY. >k ****•*. o*» ^..ei' ^c i/', '■%. LI 3:7?. AI «*« "igton 'u 0> The Numbers on this Scale are arranged according to their Logarithmic Value; and occupy the same relation to each other in space that they do in Talne|p To find the mount of a per centage on any given sum. | 1 the rate on the moving circle; place it opposite 1; found to contain two sets of numbers running from 1 to 1000, then opposite the sum found on the fixed circle in the answer, ^making 3 revolutions around the scale. The large size figures Required the per cent, protit §> j) Directum for using the Scale. - r,nnd m comain two sets of numbers running from 1 to 1000, then opposite I It on molasses bought at 28 cents per ^making l"ha"i»t.™«nd coiVuining'ali" the numbers "under 10-lhe gallon and sold at 33. Place 33 opposite 28, and opposite 1 is 1.18. "" ze from 10 to 100—t'le 3d size to 1000— running in full mini- $1.18 for $lis 18 per cent advance. to 300, and fnm 300 to 500 every other number with decimal Interest on any Sutnof Monty is computed at any Rate per cent., Warks intervening and rxm 500 to 1000 with »»»>*",»»* ""ft By observing gauge points with . on the margin-they run jnato as before. Observe all the divisions areJr. '"JJSTlf/e, cent, and are marked with theVords, Days. lOOths. The large 1 etc be ken a; 'he right h«"d. »"'«" °f Pcr <-, -Months, Per Ct. To accommodate banks and others, It ecessity removed, as tl keeps the figures in the best position lor^ reckoned M0 'day8 ^ year Any sum of m011eVj al any rau, per rent., for any lengih of time—Examples : By placing the 1 opposi*. Sit will give 1-4 of any number; 1 $04 for 55 days—Place 04 opposite 6, and opposite 55 Is 86 cts pposite 3 gives 1-3 ; and i opposite 5, 3-5lhs. ---- ----$3:100 for one day—place 33 opi»site b, and opposite I is 55 cents. NOTE-On this Scale, all answers in multiplication and pro Makcjhis M.000, and the answer is 5.50^ Add »notJwr_ciphcrand| ln „„„ ra||()i ,„„„ ,„ SJ8 mllM now mU(.n ©portion of numbers are found on the movable circle, and all an ^swers in division upon the fixed circle. To perform M*IHplicat\ Role—Place the Multiplier founii on the Movable Circle oppo- To get the Tonnage of a Ship. Rule—Multiply the length of the ship deducting 3-5 tne breadth of the beam, by the breadth of the beam; set the product oppo- site 05, ihcoopposite the depth in the hold is the answer. A ship Is 120 feet long, 25 feet beam, one half of which, 12 1-2, is the depth In the hold. Set 25 opposite 1, and opposite 105 is 262 1-2—set this opposite 95, and opposite 12 1-2 is the answer, 346t tons. ^r- Navigators, Masters, and Officers of Vessels, Wilt And the Scale of invalnable importance, in nearly all the different calculations required in a ship's reckoning. The follow- ing examples are sufficient to illustrate the fuct. If a ship run 0 1-2 mites per hour for 24 hours—place 9.5 op- posite 1, and opposite 24 is 228 miles. If she make 1-4 leeway Set 1.25 opposite 1, Speed of Drum*. QV% Role—Multiply the diameter of the drum by its number ofjgpi revolutions, and divide the product by diameter of the driver. The£*' quotient will be the revolutions driven. The drum 8 nches. in;i-.. I king 100 revolutions per minute— the driver is5 incben how irni^ » = revolutions will it make t Set 8 opposite 5, opposii I is 160, Ho; I ole—Place the Multiplier lounu on ine raovaoij I'""" "i't~-iopposite the large 1; ,___the large figure 1. then opposite the multiplicand found on the^'^ j8(beinte%at afixed part, is the answer on the Revolving Circle. 3 Example—What Is the product of 4 multiplied by 2! Place 2 Sfound on the movable circle opposite figure 1 on the fixed circle, 3then opposite 4 on the fixed ia 8 on the revolving circle. S Note I. All numbers and parts of numbers on the fixed circle ~are now multiplied by 2, and the answers stand opposite to them jjon the movable circle. ~£ Note n. If In multiplying large numbers you cannot determine gwhat is the last or unit figure in the answer, look opposite the latte ^figure on the fixed circle which is the same as the unit or last ar- s 33H.OOO, and then the answer is $55. Cents or small frac-j ,nd opriMl^ 228 js 28 ,_.j mj|cs ins may be cast separately and added. if ,ne currem 8et a s„jp ou, of her cour«e 5-8ths of To compute Interest for Years. hour, how many miles in 15 1-2 hours ? Set 5 opposite 6, and ojh Rolb—Find the rate per cent, on the movable circle; place it uosile'J t"2's 9.70 then opposite the principal found on the fixed! Teachers of Navigation have recommended the r-cale and given j the most undoubted assurance that the mo*t difficult parts offUic. , „ , „ . , work will be readilv done on the Scale. See rule for getlhl'-'the Place»1-2 opposite To find the Interest for Days and Months, at any rate per cent. apportionment of Whalemen's voyages. Rule—Find the rate per cent, on the movable circle; place it oppo- . , „„ , _. '. site the I on the fixed circle ; then opposite 305 found on the movaJ Required the amount due a Seaman of a Whale Ship, rcle is the gauge point for days, and opposite 12 on the samel Drawing one 165th of a cargo of 72.000 gn Is. Place 165 opposite 1 circle is the gauge point for months, at that rate per cent. If the answer required is Bank Interest, or 30 days to the month .. in your multiplicand, and the nnit or last figure on the mova- e circle standing against it, will be the answer. Example—What is the product or 234 multiplrsd by 8? Place found on the movable circle opposite figure 1. Then opposite 14, on the fixed circle, is 187t something. Now to determine the nit or last figure, look opposite the large 4 on the fixed circle, (4 sing the nnit figure of the multiplicand ) and the unit or last flg- re on the movable circle against it, is the answer, viz. 2: mak ing the answer 1872. Division. Find the Divisor on the Movable Circle; place it opposite 1 or the fixed circle ; then opposite the dividend found on the movable irele is the answer on the fixed, in whole numbers and tenths of he Divisor. To MuUiply by one number and Divide by another by one tisnpli process. Role—Place the multiplier found on the movable c irele opposite the divisor found on the fixed circle, then opposite the.multiplicand found on the fixed circle, is the answer ou the movable circle. then opposite 360 is the gauge point for days. To Compute Interest for Months, Rule—Place the principal (found on the movable circle) opposite the gauge point for months at the given per cent.; then opposite the number of months found on the fixed circle is the answer. Discount. Place the amount opposite 1, and opposite the sum to be com- puted, on the outside, is the answer. Required the discount on $150 at 5per cent. Place 95 opposite 1, and opposite 150 is 142.5 What is 9 1-2 per cent, advance on $2001 Place 109.5 oppositt 1, and opposite 200 is 219. This rule will be found to apply equal ly well iq all kinds of articles, as well as money. Equation of Payments by Casting Interest on each Sum. Example—$155 Jan. 1 to June 15—105 days—$4.26 Interest. $168 Feb. 28 to June 15—105 days—$2.99 $145 May 1 to June 15— 45 days—$ 1.09-^ $50 June 15. Total $518. Interest $8.34 Set $518 opposite 6 per cent, and opposite 8 34 on the outside' jj «,,,(„. the number of days equal lo the interest, 96 1-2 days. If the credit is 4 months, 23 1-2 days will make it equal Equation of Payments. Role—Multiply the first sum by the second and divide by the amount of both. $300 on the 1st of April ad $400 — and opposite 72 is 437 gallons. Multiply this by 36 cents per gal Ion, the price paid for the oil, and the result is $157. Ifthe voy- age has been of 17 months, how much would that be per month 1 Place 157 opposite 17, and opposite 1 is the answer, $9.25, Price of Freight. Required the price of lfiSOIbs. freight at $1.75 per ton. Place 1.75 opposite 20, and then opposite 1680 is 1.47 Superficies. Required the number of feet of boards to cover a house 27 feet high and 40 by 54 feet square. Place 27 opposite 1, and opposite 188 [the feet round the snme] is 5060 ft. Ans. If the rafters art 24 ft. or 3*5ths the width of the house, then place 54 opposite 1 arid opposite 48 is 2592. Ifthe gable ends are 40 by 14, then plact SpChair of 40] opposite 1, and opposite 14 is 280. To bring the! BfSJt^nd last into square yard: If a 12 Inch drum makes 100 revolutions, how large Is the drnmj make 200' Set 12opposite 200, then opposite 1 Is o, the ansvi or. an 8 inch pulley makes 250 revolutions—how large shall it tu,- inake 1001 Set 8 opposite I, then opposite 850 Is the ans. 20m Required the speed ol a 5 inch pulley, if the 10 Inch throw tei rns per minute. Place 5 opposite 65, and opposite lu is 130. '; Speed of Pulleys. l If one Inch pulley throw 84 turns, h.iw many 2 Inch 1 Dace ir opposite 84, and opposite 1 is 42. If one inch throw 84, bow; le Pcr many 8 inch! Set 8 opposite 84 and opposite 1 Is 105. (; ■ - . To get the number of Cogs for any size Wheel. A wheal 18 feel diameter will require how many top at 6 I-2fc to the fool 1 Place C in circle opposite 1, and opposite lb is 55 1-2 , ace f> 1-2 opposite 1 and opposite 56 1-2 ia 367. In a wheel l9inchos diameter, how many teeth of 5-8 inch pitchy from centee of one to centre of the next tooth t Place C in circlets opposite 1, and opposite 19 is 59 3-4. Then set 5 opposite 8, ancfc, noslle59 3-1 is 95 3-4. fc Geering of Wheels. Multiply the number of teeth in the driver by its number of rev- olutions, and divide the product by the number of teeth In the driven—the quotient is the number of revolutions of the driven.}-! Example—the driver has 80 teeth, making 30 revolutions ; thei: drive, has 40 teeth: how many revolutions will It make) Setjy ~ opposite 40, and opposite 30 is 60, the snswer. ^ Geering of Wheels. A Wheel Kith 740 teeth runs upon one with 84.3; Role—Place 84 opposite 1 and opposite 740 is the answer, 88. A wheel with 18 teeth runs upon this, and this goes into 84=4.66.! This may be carried to any extent. Speed of Locomotive Engines. How many revolutions would the driving wheel of an englni make in running from Greenport to Brooklyn, the height of wheelj I 188 is 562- square yards. X- . , 9, and opposite 14 is 31t, t^fteTjuired the square yards of carpeting to let 27 opposite 9, and opposileiheing 6 1-2 feel, and the distance being 96 miles T Place C in clr- In gable end, set 20 opposite'eleopposite 1, and opposite 6 1-2 is 20.4-place this 20.4 opporlie 1, and opposite 5280, the feet per mile, is 258, the answer for one ___4 opposite 7, and opposite 3 is 17.2. April 1 /, Ans Should $300 also be added on the 1st of July, making the sun i-il$l0i0' Place 3 opposite 10, and opposite 7 is-21 days. If $400 more be added July 23d, how much more lime required Place 10 opposite 14, and opposite 10 is 28 1-2. These days added together give the time. Insurance. Required the premium on $7,000 at 3-8 per cent. Place 3 oppo j ls "165," iiieprTce." If for yards, place 20 opposite 9, and the yards. To .Measure Plank. A plank 13 1-2 wide, 3 1-2 thick, and 15 1-2 long: what Is the contents 1 Place 3 1-2 opposite 1, then opposite 13 1-2 is 47.2— set this 47.2 opposite 12, and opposite 15 l-2is 61ft. board The Price, at $27 per 1000—Set 27 opposite 1, then opposite 61 l*creuc,inFroctions-aneasyioay,ogetakno«,Udge.fthcSc*^ ! 1 placed at 2 gives a half; at 3, a third; at 4, a quarter; and so f2on The numbers placed as they would be written, will give: he Sdeslred result. If cloth cost $5.25 per yard, how much for 5-8ths lof a yard ! Place 5 opposite 8, and opposite 9.25 is i.si. To Reduce a Fraction to its Lowest and All its Terms. I Role—Place the numeratorfound on the movable circle oppos the denominator found on the fixed circle; then all the numbers Standing directly opposite each other, are other terms ot saiairac- ion, and the lowest of said numbers are its-lowest terms. To Multiply a Whole Number by a Fraction, or a Fraction by a Whole Number. RoLE-Place the nnmerator found on the movable circle opposite .he denominator found on the fixed circle; then opposite the whole number on the fixed circle is the answer. J To Reduce Vulgar Fractions to Decimal Fractions. g RoiE-Placethennmeratorfoundonthemovable ?ircteooposite &h£denominator found on the fixed circle; then opposite 1 found on Sthe fixed circle, is the answer, or decimal fraction. f. To Reduce Decimal Fractions to Vulgar Fractions. % ROLE-Piace the decimal found on the movable circle opposite1 |hen any two figures standing directly opposite each other is the ans Q Tin Ware. % To get the cubic inches and Gallon, in a Coffee-pot 5 inches D; Samaterat the top and 8 inches at the bottom-eet half the square J |7e top and 8 bottom being 44 i-2 at ■ reduce the corner- by look «i„„ at Area of Circle, being 35 mean diamater. Multiply tnis py fet iYhcta.Tand the result is 490 inches Cubic Div.de hi.hy §231 the inches in a Gallon and at 490 is the Answer 2 Gallon 12.100. 5 itufe/or Price of Meta\s. If a oennvweight of gold is worth $ 1.05, how much for 14 pen wVXS Vla» 109'opposite 1, and opposite 14 is 14.7, the ans sitp ft and opposite 7 on the outside is 26 .-» 3 M per cent is assessed upon the premium notes of a Mutual, To get the Price of Lumber. s Co Rule—place 3 1-4 opposite 1, and opposite any amount 13.50 per 1000 for 800 feet—Place 13.50 opposite 1, opposite 800 on the outside is the sssessment. is the price, 10.80. Rule for reducing the different Currencies to Dollars,Cents, and Mills\ Land Measure. Place any sum of foreign currency opposite any equal sum of] How many acres in a piece of land 356 by 444 feet ? Place 356 Federal money. If $4.44 be fl.howmuch for £10 lOs.l Placeatl, then at 444 is the number of feet. 158.00—divide this by 4 44 opposite 1 and opposite 10.5 is 46.70.- If 5 francs be 94 cts. 43,560, and the area Is 3.63.100 opposite the number of feet ' how much is 68 francs 1 Place 5 opposite 94,'opposite 68 is 12.80, the answer. 0B_P,ace;helon.h.m„,ab,edrcleop^ curS^betffi^ M^.l IJZ ^^ added on the fixed circle, is the answer. Hoia many feet of Boardsin a Box, The sides and ends of which are 18 inches wide, and 64 inches 18 inches \ 'nyweight! Place 105 opposite Weight of Water. „ a cubic foot of water weigh 62 1-2 P.^V7JL00748°ooDo1sOiIe much will a barrel or 31 1-2 gallons weigh' Place 748 opposite 12 1-2, and opposite 31 1-2 is 263 lb. the answer. To aicertoin the Rate per cent, one sum bears to another. , RoLE-Placethe less sum found on the moving circle opposite 'thf larger f lien opposite 1 on the fixed cirele is the answer. Barter or Exchange. V 67 cents will boy a bushel of corn wd 73 cents will buy o«,-he7oCfen£."bow m'any bushels of « &£%*£ of "»'" ^Place 67 opposite T^am^posite WjsKW theanswer. Measure of Boxes. Place the product of the width by the thickness opposite 1728, °<1and opposite the breadth in inches is the answer in cubic feeL- '- Example, 16 by 19 and 22-place 16 at 1, opposite 19 is 304- place this opposite 1728, and the answer is 3 7-8 feet. Required the Cubic Inches in a Cylinder. 12 inches diameter and 12 Inches long—place 144, the square of 12 oooosite 1, (this is the surface of a foot including corners,) look opposite to Area of circle near 8, (the comers off,)-place this 113 opposite 1, and the answer is opposite 12: 13.56. Rule of Three, or Proportion. If a barrel of flour cost $5 25, what cost 28 lbs.! Pla^e 555 op- posite 196 (the pounds in a barrel,) and opposite 28 ls 75 cents. Required the Cubic feet in a Block of Granite. A block 3 feet wide, 2 12 feet thick, and 6 feet long. Set 3 oppo sitt 1, and opposite 2 1 -8 is 7.50-ttas by 6 teet length isi«Jfeel 12 feet of duincy granite weigh a ton-place 12 opposite 20 anc opposite 45 is the weight, 7500 dounds, or 3 3-4 tons. Rale to Measure Grain by its Weight. ,„« the actual Weieht found on the movable circle opposite the weight required byslatute. Then opposite the number of running bushels found on the fixed circle, is the number of lawful bushels. -1 . «™ .™,i„Jmile—place this 258 opposite 1, and opposite 96 ls the answer—: ,imw|uiicu liic eijuaie j-urus ui carpeting iv wivcr S nOOr twelve,-, mq J feet square. Place 12 opposite 9, and opposite 12 is the answer,ii,''DB' ,. ...... „ j ? 1 To get the number of tons o] Coal m quantity lym%\n a Body, p To measure the outside of a house in feet or yards. \ Multiply the feet Into cubic measure, and divide by the cubicj; . t» ■ „- , ^, ^ .,,;,,. .. , l j.t j , _,. feet in one ton. Requ red the tons of coal In a parcel 7 feet deepj A hmise is 8, feet high, and 40 by 54 in breadth and length.- ." d M f *M b s4 ls 576 th„ by 7 „ 4032. If 36 6M&. Place 27 opposite 1, then opposite 188 is 50,6. wejeh 2000i '6 int0 4032_ g^ 36 oppOSite i, and opposite 4032 is| the 1st of Togetsquare yards, set 27 opposite 9, then 188 is the yards. 'l!2 lorn ired the p--- bushels. Place 11 opposite 36, opposite 1 is 30.5—and for 7 1 bushels is 2.29. Price and Weight of Coal. If coal be $5.25 per ton gross weight, how much for 20001bs.! Place 5.25 opposite 2240, then opposite 20 is 469. Observe the lbsv of coal gross weight are on the outside, and the price at $5.25 lag opposite, inside. § If the coal be reckoned at 2000 for 5.25, then set that sum 5.25J opposite 2000, and you have the result in net pounds To Measure Wood. Place the height of the pile found on the movable circle oppostl on the fixed circle; then opposite the length found on the fixedi _ rcleis the number of feet and lOthsof a foot, which If diYidrJ by 6 will give the cords. Another Rule to measure any load or range of Wood. Rule—Place the product of the height by .he width opposite 16, snd opposite to the length is the answer in wood nwasure. Required the contents of a load of wood 3 1-2 feer wide, 7 1-2 fl high, and 7 1-2 feet long. Place 3.5 opposite to l,snd opposite 5 is 26 1-4; place this opposite to 16, and opposite 7.3 is the-^ answer, 12.31. a The Price of the same Is found by placing the pries per cordg opposite 8. Required the price of 12.3 at $4.50 per cord. Placeg, 4.50 opposite 6, and opposite 12.3 is 6.91, the answer. To Measure and find the Weight of a Cast Iron Shaft. An Iron shaft is 12 inches diameter and 23 feet long: how man cubic feet will it contain, and what is the weight at 7465 ounces t a cubic foot on being reduced to pounds ? 7465 divided by 16 isg 466 lbs lo the Bquare foot. The square of 12 cuts opposite 144, ice 144 opimsite 1, snd opposite area of circle is 113—plsce Ibii 144 and opposite 23, the length, Is the cubic feet, 18t! now placeg 466 opiwsite I, and opposiie 18, the cubic feet, is 8400lbs=4 lonsg 400 pounds, the answer. If 3.57 cubic inches make a pound, then, 484 pounds make a cubic foot, the shaft would then weigh 872510S, or 4 tons 725 The latter is the mual standard for cast Iron, To find the capacity of a Cistern. A cistern is 10 feet diameter and 14 feet deep. Place 120 [> inches) opposite 1, and opposite 120 is 144—place this opposite Number of Threads in a yard of Cloth. if cotton cloth have 50 threads to one inch, how many thread; to Iheyard? Set 50 opposite I, then opposite 36 is 1,800. ln 50 yards how many threads ? 50 by 1100 is 94.000. 50 yards for 12 hours is how many for one hour ? Place 5 opposite 12, opposite 1 Is 4.17-. Row many Brick are required to lay a Wall. 21 1-2 to the cubic foot—the wall 36 feel high, 60 long, and 15 Inches thick. Place 30 opposite I. and opposite 60 is 2160 liply this by setting 125 at I, and opposiie 216 is 27,00—multiply this by 21 1-2, the brick in a foot, and the answer is 58.000. To get the Cubic feet in a Cistern, and change the same to Gallons Required the number of cubic feet in a cistern measuring 6 ft or 72 inches in diameter and 6 11. or 72 inches deep. Place 72 al 1 opposite 72 is 518. Set this at 1. and opposite area circle is 407— this opposite 144, and opposite the depth.6 feet is 170—multi- ply this by 7.48, and the answer is 1270 gallons. See same resullknen opposite 168 the inches in 14 feet is found,_242• Place thlsg 72 by 72, and that product bv 72. and that by 34-or at gauge point at 1. then at WineGallon gauge poin is Ihe answer, 82-5. Dlvldeg for Wine Gallons, gives 1270. Rule for Dry Goods Merchants in taking Account of Stock. 17 yards of calico al 38 cents—place 17 si 1, and at 38 is 6.46 • 1.2broHilcloth at $4.75--41.37 1-2 ^i^J jAiJjAao »ij . >^J. by 31 1-2 and you have the barrels, 2611. To get the Cubic Yards of earth in o»y Cellar. •ay 24 feet square and 7 feet deep. 24 by 24 is 5760—this by 403.00—divide by27 and the squa 149, the answe t J ■••:♦*$ 'At .Mi ', ..as ^: »m *s, (V K t v .--/V-X' 1