IN MEMORIAM 
 FLOR1AN CAJORI 
 
<U# 
 
, * 
 
IMPROVED ABACUS: 
 
 AN EXPLANATORY TREATISE 
 
 THEORY AND PRACTICE 
 
 ARITHMETIC AND MENSURATION, 
 
 BY THOMAS RAINEY. 
 
 ENERGY IS THE PRICE OF SUCCESS. 
 
 CINCINNATI: 
 
 PUBLISHED BY E. D. TRUMAN 
 
 111, MAIN STREET. 
 
 1849. 
 
RECOMMENDATIONS. 
 
 From the Cincinnati Evening Nonpareil. 
 
 " We have received a copy of ' Rainey's Improved Abacus,' a treatise on Arithmetical 
 calculations, and take pleasure in recommending it as a work deserving the patronage of 
 business men desiring a knowledge of a plain, concise, and practical system. It simpli- 
 fies and shortens calculations bv canceling or expunging the numbers, and is of incalcula- 
 ble benefit to those whose business requires of them a simplified method of calculation." 
 
 From the Cincinnati Daily Evening Times. 
 
 ' Messrs. E. I). Truman & Co. have just issued a Treatise on Arithmetic by the can- 
 celing method. The work is by Prof. Rainey, and is well prepared for popular instruction. 
 All questions in numbers are of either Multiplication or Division, and this is the base of 
 the system. We can truly say, the canceling method is a superior one for all ordinary 
 calculations, and we trust it will be generally cultivated. Mr. Rainey has given to the 
 public his analysis in cheap form, and they are under great obligations therefor." 
 
 From the Cincinnati Great West. 
 
 "THE IMPROVED ABACUS: an explanatory treatise on the Theory and Practice of 
 Arithmetic and Mensuration, in which the general principles involved in practical calcu- 
 lations are thoroughly elucidated and illustrated by numerous analytic and abbreviated 
 examples. By Thomas Rainey. Energy is the price of success." 
 
 " This is the title of a neat volume from the press of Messrs. Truman & Co. The design 
 of the work is to give to practical men and advanced students a thorough theoretical and 
 practical knowledge of all such calculations as arc necessary in commercial and mechani- 
 cal transactions; and this, too, on common sense principles ; ' for,' says the author, ' rules 
 without reasons are ridiculous, and insulting' to the inquiring mind.' That a work ot 
 this character has long been needed, no practical man can deny ; and every advanced 
 student in the science will hail its appearance with delight. 
 
 " Knowing practically, the many deficiencies of the old systems used, we cannot too 
 highly recommend this excellent treatise." 
 
 From Prof. WM. McGoomN, Sidney, O. 
 
 " I have examined a small work published by Prof. T. Rainey, on Interest, Profit and 
 Loss, Proportions, Discounts, Insurance, Stocks, Commission, Mensuration, etc., and am 
 much pleased with it. It is just such a work as the learner needs ; it supersedes the ne- 
 cessity of a teacher. The roles are concise and practical. 
 
 WiM. MC'GOOKIN." 
 
 From Prof. PRATT, University of Missouri. 
 
 " Having attended your course of lectures, in Arithmetic, etc., you will allow me to ex- 
 press to yon my entire conviction that your system is peculiarly adapted to the simplifica- 
 tion of Arithmetical calculations generally, and to the abbreviation of most intricate 
 computations. Very respectfully, &c., GEO. C. PRATT." 
 
 " I fully concur with Prof. PRATT. WM. T. DAVIS, 
 
 Columbia High School." 
 
 From H. F. WEST, of Indianapolis. 
 
 *****" The unquestionable accuracy, as well as the facility of Prof. 
 Rainey's process of solving problems, commends itself to the plain common sense of 
 
 every "observer, as it is so happily adapted to the practical purposes of life The 
 
 ready manner by which the results of intricate questions were obtained, correctly too, 
 showed that Prof. Rainey had happily adapted the principles of his improvements to all 
 those perplexing rules that fill up our school Arithmetics. * * * Every 
 
 business man, and every young man that intends being one, should not be unacquainted 
 with this important attainment in the science of numbers." 
 
 From Prof. McWiLLiAMS, Springfeld, O., late Presiding Officer of Ohio 
 
 School Convention. 
 
 * * * * " He lectures at present to our school at large, and from this test, I find 
 him to be what he professes. * * Do not fail, if practicable at all, to try his 
 
 system ; and the farther the better." 
 
From the Cincinnati Gazette. 
 
 . . . . " The author commences by showing that if a quantity is to be multiplied and divi- 
 ded by the same number, it will remain unchanged in value ; and consequently neither op- 
 eration need be performed : Or, if a quantity is to be multiplied by a given number 
 and divided by another number half as great, we may in the first instance, multiply by half 
 the number, and neglect the division altogether, and the result will be the same that it would 
 have been, had we performed both operations, and gone through with twice the amount 
 of labor. The author does not confine himself to the simple examples we have given, 
 but shows that this great principle that PlHJOsing forces _ojLftqu.al strength will destroy 
 each otlie_tr-runs through the whole systeMf, and^ thal^it may be used to" immense advan- 
 tage under all of its rules ; in fact, it is the main pillar upon which the author has reared 
 his beautiful system ..... What next strikes the reader of this instructive work, is, 
 that the author permits no opportunity of imparting instruction to pass by him unimproved. 
 The terms of art which he uses are all clearly and beautifully explained ; and the mechan- 
 ical and tedious methods of instruction adopted by many of the old writers, are ex- 
 posed." 
 
 From the Tennessee [Nashville] Organ Rev. JOHN P. CAMPBELL, Editor. 
 
 NEW AND USEFUL BOOK' FOR EVERYBODY. 
 
 RAINEY'S IMPROVED ABACUS : AN EXPLANATORY TREATISE ON THE THEORY 
 AND PRACTICE OF ARITHMETIC AND MENSURATION. 
 
 THE work above noticed, was written with direct reference to the wants of all classes 
 of business men. The system is new, very short, and beautiful. 
 
 Experience teaches the business man that the old system requires too many figures : that 
 the rules are too tedious and arbitrary : that in too .many instances results are found by the 
 arbitrary and incomprehensible arrangements of the books ; and that if he is unable to re- 
 tain or apply such rules, he must fall back on common sense, and construct his own rules. 
 Now, all rules, without reasons, we consider, ridiculous and insulting to the intelligent and 
 inquiring mind : but with reasons presented first, and afterward calculations to illustrate 
 their application and use, it becomes an easy matter for any person of common mind to 
 make his own rules. Hence, every rule in Arithmetic and Mensuration is fully, clearly, 
 and satisfactorily explained ; while a great number of examples, such as occur in every- 
 day business, are wrought out; thus enabling the practical man to find in the book the 
 method of making calculations similar to any that he may find necessary. 
 
 The practical calculator will find in this work a system of calculations extraordinarily 
 short, simple, and satisfactory; and explained in PLAIN, FAMILIAR LANGUAGE, 
 free from difficult terms, which too often obscure the sense, without satisfying the 
 judgment. 
 
 No department of calculations is omitted in this work, which would be of interest to 
 business men and students, of any trade or profession. 
 
 MERCHANTS and COMMERCIAL men will find everything relating to their voca- 
 tions, and a great variety of new and interesting matter, not before introduced into any 
 work of this kind. We invite particular attention to the system of Interest, Discount, 
 Profits and Losses, the several varieties of Commission, Insurance, etc. ; the combina- 
 tion of several different statements in one statement ; and to the Tables for Banking, 
 Equation, etc. 
 
 MECHANICS are invited to examine the great variety of work pertaining to MA- 
 CHINERY ; the MECHANICAL POWERS ; the TABLES of AREAS, etc. ; WEIGHTS of 
 METALS, etc. ; Circles, Cylinders, Globes, Balls, etc. ; Contents and Weights of Solid 
 .Bodies, and all of the Superficial and Solid Measurements necessary in any department 
 of mechanics. 
 
 The attention of SCIENTIFIC men, is directed to the new and beautiful system of 
 PROPORTIONS, CAUSE and EFFECT ; the Philosophy of the general Method of Statement 
 throughout the whole work : COMBINATION of STATEMENTS ; COMPLEX FRACTIONS ; the 
 general and easy method of disposing of Fractions, etc., etc. 
 
 It has been the constant aim of the author to combine UTILITY, BREVITY, and SIMPLI- 
 CITY ; to use such language in all explanations, as could be easily understood ; and to 
 present to the public a real improvement. 
 
 All such persons as wish to learn a short, certain, and easy method of making general 
 business calculations, without the assistance of a Teacher, will find in this work every- 
 thing that is calculated to assist them in this desirable undertaking. ASSISTANCE TO THE 
 PRIVATE STUDENT is a peculiar feature of this treatise. 
 
 Several hundred COIN PLATES arc added, of whose utility it 'is unnecessary to say 
 anything to those concerned. 
 
 "It is, without doubt, a very useful book." Cincinnati Daily Commercial. 
 
RECOMMENDATIONS. 
 
 From the Wiyne County Whig. 
 
 " We again call the attention of our readers to the appointment of Prof. Rainey, to 
 lecture at the Methodist Church, on this Wednesday evening. We had the pleasure of 
 attending: the lectures delivered in our place by this gentleman last week. Under ordinary 
 dircnmBtanees, the subject of Mathematics is exceedingly dry and uninteresting. To the 
 lectures of Mr. Rainey, we listened with most intense interest, as did all who heard him. 
 jives to the sul.jrct a novelty and interest we have never before witnessed. His new 
 mode of calculation is the shortest, most rapid, and simple of any mode now in use. We 
 
 entertain the very liiirlu^t opinion of the practical utility of his plan, and commend it to 
 the favorable notice of students and business men. The following communication was 
 sent us for publication. We indorse readily and fully the commendations it contains. It 
 comes from men well known as worthy of every confidence, and well qualified to judge of 
 what they speak. 
 
 " MR, EDITOR : Prof. Rainey, who has been lecturing in our place some days, on his 
 short and beautiful system of practical calculations, intends visiting your town. We 
 take pleasure in commending this matter to your attention, because we feel convinced 
 that he will greatly benefit all who go to hear him. In his lectures in this place, he has 
 proved his system of calculations to be the shortest and the most simple, as well as the 
 most expeditious that we have ever seen. His operations at the blackboard convince 
 every thinking man of brevity and precision, certainty, simplicity, and final satisfaction ; 
 while they excite equal wonder and pleasure for their philosophical beauty and grandeur. 
 He throws away, instead of using figures, and works all fractional questions with ease and 
 simplicity. 
 
 " Nothing that we have ever seen can equal his calculations in Simple Interest. Instead 
 of a dozen different rules for different rates per cent., he has one simple rule, short and 
 easily understood, for working questions of any conceivable principal, time, and rate per 
 cent. This every practical man knows to be a great convenience. Prof. Rainey has 
 with him a large work in which his system is thoroughly developed ; treating of every va- 
 riety of calculation that can possibly occur among business men, either in Commercial and 
 Mechanical Arithmetic, or in Mensuration. This is an excellent work, and the first we 
 have seen that thoroughly explained these operations. He (jives his reasons for a thing first; 
 then deduces the rule. Hence, his rules appeal to Common sense. His work in these rules, 
 as well as others, cannot fail to excite admiration and attention wherever intelligent men 
 will take time to investigate. As he has but a short time to stay in your place, we think 
 it well to call the attention of your citizens to the subject, that they may secure the op- 
 portunity, and not regret when too late. Let all your people hear him, and not fail to se- 
 cure his books. 
 
 [Signed ] JAMES M. POE, \ 
 
 CHARLES FISKE, > Teachers, Richmond, la. 
 WILLIAM AUSTIN, ; 
 
 DR. JOHN PRICHET,) n , .,, T 
 REV. E. MCCHORD,! Centreville, la." 
 
 * * * * " RESOLVED, 11. That Prof. Rainey's book is just such a work as 
 the learner needs ; because every principle and operation is so thoroughly explained and 
 illustrated, that by its investigation, the ordinary reader will be enabled to comprehend 
 and practice the system. 
 
 " RESOLVED, 12. That his system of Interest, as taught in his work, is of itself, worth 
 more than his charge for both instruction and his books, and being short and easily under- 
 stood, is pre-eminently adapted to the counting-house. 
 
 [Signed by] P. Y. Wilson, Finley Bigger, John W. Barber, Dr. W. Frame, J. A. 
 Kendall, Dr. A. Norris, and others of Rushville, la ; by John C. Osborn, Thomas Kirby, 
 G. B. Holland, Dr, Andrews, Esquire Swarr, George B. Norris, and others, of Muncie, la ; 
 and by Win. F. Kelso, James McMeans, and twenty others, of Newcastle, la." 
 
 From the Cincinnati Evening Dispatch', 
 
 also, 
 
 From the Cincinnati Daily Enquirer. 
 
 " RAINEY'S IMPROVED ABACUS. ... It purports to be An explanatory Treatise 
 on the Theory and Practice of Arithmetic and Mensuration ;' and from a hasty examina- 
 tion it appears to be of great practical use in making calculations in a great variety of 
 business relations. It is recommended in high terms by teachers and others competent to 
 judge, who have examined it." 
 
HAINEY'S IMPROVED ABACUS; 
 
 AN EXPLANATORY TREATISE ON THE 
 
 THEORY AND PEACTICE 
 
 OF 
 
 AEITHMETIC AND MENSURATION: 
 
 IN WHICH THE GENERAL PRINCIPLES INVOLVED IN PRAC- 
 TICAL CALCULATIONS ARE THOROUGHLY ELUCI- 
 DATED AND ILLUSTRATED BY NUMEROUS 
 ANALYTIC AND ABBREVIATED 
 EXAMPLES. 
 
 BY THOMAS R A I 1ST E Y . 
 
 ENERGY IS TH!i P1UC1? OF SUCCESS. 
 
 CINCINNATI: 
 E. D. TRUMAN, PUBLISHER. 
 
 1849 
 
NOTE TO TEACHEKS. 
 
 The Teacher will observe that this work is devoted exclusively to the de- 
 velopment and illustration of the principles of numbers, with the introduc- 
 tion of such a number, and variety of examples, only, as subserve this pur- 
 pose. It is deemed the privilege of the teacher, to present such examples 
 for practice and test, as may best accord with his judgment, and in the 
 highest degree develop the capacities of the learner. 
 
 This arrangement presupposes the plan of instructing classes at the 
 blackboard, by familiar illustrations, and the occasional test of each pupil's 
 progress, in the presence of the whole class. A blackboard isindispensa~ 
 ble to every good school. 
 
 In Mensuration, every figure explained in the text should be carefully 
 drawn on the blackboard; that the twofold purpose of illustration and 
 draining might be subserved at the same time. 
 
 It has been deemed useless to treat of the Elements of Arithmetic, af 
 this department of numbers is generally studied in a separate book; and 
 as a considerable number of good elementary works now claim the patron- 
 age of the public. 
 
 Entered according to Act of Congress, in the year 1849, 
 
 By THOMAS RAINEY, 
 In the Clerk's Office of the District Court, for the District of Ohio. 
 
 C. MORGAN & Co., Stereotypers. 
 MORGAN & OVEREND, Printers. 
 
PREFACE. 
 
 IT would be both absurd and arrogant for an author, at this 
 day, to present a new text-book on Arithmetic, according to 
 the old and long cherished standards. A large number of ac- 
 complished mathematicians and experienced teachers, as well 
 in our own country as in Europe, have plied their talents and 
 energies to this subject with peculiar ability and success; pre- 
 senting works, perfect in their order and arrangements, and 
 sufficiently intelligible and satisfactory, so far as the old sys- 
 tems are concerned, for all practical purposes. 
 
 _ Although Arithmetic has, until recently, been neglected by 
 scienlific^ttfiii, yet the onward progress of latter-day^improve- 
 
 "ments has necessitated a corresponding advance in this sci- 
 ence; until a place is now conceded it among the rational and 
 explicable sciences, instead of among the mere handicraft arts, 
 as hitherto. This advance in the principles of the science, 
 has involved new issues. It is found that the ordinary system . 
 of statements is too mechanical) circumstantial, and uncertain] 
 that the method of statement, within itself, precludes that ra- 
 tional analysis, and thorough demonstration of principles neces- 
 sary to the proper appreciation and use of any science; and 
 that a sum of labor is performed, intne practical reduction " 
 of calculations, which is not only unnecessary, but which di- ; - 
 verts the mind from the proper issues involved in the pro- 
 blem, and leads it into a labyrinth of doubts and obscurities. 
 It is to the remedy of these palpable defects, that our labors 
 are addressed; and to the presentation of a rational, satisfac- 
 tory, simple, unique and brief system of calculations, such as 
 demanded by the practice of ordinary business transactions. 
 
 It is likewise designed to furnish the private student with an 
 easy and certain guide to proficiency in numbers, on such 
 
 (iii) 
 
IV PREFACE. 
 
 principles, and in sucli accordance with common sense, as will 
 appear to him reasonable and convincing. Hence, while the 
 theory of each department of numbers treated, is clearly elu- 
 cidated, the practice is illustrated and demonstrated by familiar 
 examples, occurring in business; and, too, by a method so 
 short and general in its application, as to admit of demonstra- 
 tion more conclusive and pointed, than when encumbered by 
 the masses of unnecessary multiplication, division, etc., which 
 always attend the solution by the ordinary method. 
 
 Throughout the whole treatise, the reader will observe that 
 much importance is attached to the comprehension and proper 
 application of the principles of Proportion; as constituting, 
 chiefly, the basis of all those operations which follow the ele- 
 mentary rules. In accordance with this view, all of the 
 statements in this work, except those of addition and sub- 
 traction, are made by Simple, Compound, or Concatenated 
 Proportion; hence, they are unique, and may be easily 
 remembered. - 
 
 The beautiful theory of Cause and Effect is thoroughly 
 discussed, and applied to Compound Proportion. The theory 
 of Inverse Proportion will, as dependent on Cause and Effect, 
 be found quite different from any hitherto presented. 
 
 Much attention has been given to Mensuration, because of 
 its practical utility, and the constant necessity of the appli- 
 cation of its principles in active life. 
 
 Contrary to the custom of many authors, we have excluded 
 from this treatise all of those sub-divisions of numbers whose 
 explanation depends on algebraic principles; such as the Po- 
 sitions, Alligation, the Progressions, Permutation, Cube Rootj 
 etc.; none of which offer any reward for the arduous labor 
 lost in the impossible task of their attainment in Arithmetic. 
 Jf one-half the time devoted to these principles, in their arbitrary 
 form in arithmetic, were given to the study of Algebra, the pupil 
 would not only learn a great portion of that beautiful science, 
 but would thus secure the only key to the principles involved in 
 these rules. 
 
 It has been a prime object, first, to discuss principles, and 
 
PREFACE. 
 
 then deduce practical directions; for rules, wUJiout reasons, are ^\ /v /, 
 ridiculous, and insulting to the inquiring mind. I W */ 
 
 No secondary principle has been used in the elucidation or 
 illustration of one that is primary; nor has any princi- 
 ple been anticipated; but each, used in its natural sequence, 
 has been made the basis of a yet higher principle, in such 
 manner, as to cultivate the reasoning powers of the learner, with- \ 
 out embarrassing them. 
 
 Jechnical phraseology has been avoided, as far as consistent 
 with the requirements of such a treatise; a-s likewise, puzzles, 
 and all giddy theorizing on trivial and unimportant topics, 
 which should be beneath the dignity of a scientific man, al- 
 though well calculated to please the fancies of a vacant mind: 
 for he who would be a useful man, must be a practical man; 
 and the less acquainted with fascinating chimeras, the better he 
 is adapted to his great purpose. 
 
 It is not claimed for this system that cancelation can be 
 availed in every solution; but that a great majority of prac- 
 tical questions can be much abbreviated by it; the excellency 
 claimed for the system, is, that while it abbreviates the work, 
 the statement is so simple, so philosophical, and the result, so I 
 inevitable, that no intelligent individual can fail practicing its t 
 principles, whether the arbitrary rules be remembered or not 
 
 We shall endeavor to deal mildly with those who, being 
 bound to the old system, as their hobby, cannot, or will not, \ . 
 epen their eyes to the evident advance of modern improve- Uu / 
 ments; and, therefore, submit our labors to the investigation * 
 of those candid and intelligent minds, which are not shackled // / 
 down to such usages of the past, as are endeared more by^ /, 
 habit, than by any rational merit / / 
 
 T. RAINEY. 
 
 ^Cincinnati, July, 1849. 
 
INDEX. 
 
 PAGE. 
 
 Theory of Cancelation, 5 
 
 Rule for Cancelation, 7 
 
 Common Fractions, 8 
 
 Multiplication of Fractions, 9 
 
 Division of Fractions, 10 
 
 Rule for Fractions, 11 
 
 Complex Fractions, 11 
 
 Reduction of, 12 
 
 Multiplication of, 14 
 
 Division of, 15 
 
 Ratio and Proportion, 47 
 
 Definition of Proportion, and 
 new method of Statement 
 
 and Solution, 48 
 
 Analysis in Proportion, .... 50 
 
 Contrast of old and new systems, 51 
 Obscurity of the ordinary 
 
 form of Statement, 55 
 
 Statements and Solutions,.. 57 
 Ex. in Complex Fractions,. . 61 
 Rule for Direct Proportion,. . 62 
 
 Simple Interest, 16 
 
 4 Theory of, 17 
 
 Rule for all Rates, 25 
 
 Interest at 6 per cent., 28 
 
 Illustration of Theory ,. .... 32 
 Rule for 6 per cent., 35 
 
 Interest at 7 per cent., 36 
 
 Rule for 7 per cent., 39 
 
 Interest at 8 per cent., 37 
 
 Interest at 12 per cent., 38 
 
 Rule for 8 and 12 per cent.,. . 39 
 
 Rate and Forfeiture Table in Int. 39 
 
 Law of making and transferring 
 
 Notes,.! 37 
 
 Lapse of time between two Notes, 39 
 
 Partial Payments, 40 
 
 Ohio, Indiana, and Ken- 
 tucky Rule, 40 
 
 Calculation illustrating Rule,... 42 
 U. S. Supreme Court rule, . . 45 
 Commercial or Vermont, and 
 Connecticut rule, 46 
 
 Discount: Theory, 95 
 
 Rule for Discount, 99 
 
 True and False Discount,.. 100 
 
 To find the Face of a Note, to 
 diaw a Specific Sum, ...... 101 
 
 (vi) 
 
 PAGE. 
 
 Equation of Payments, ...... t .. 171 
 
 Rule for Equation, 173 
 
 Time Table for Banking and 
 
 Equation, 278 
 
 Profit and Loss, 62 
 
 Discussion of General The- 
 ory, 63 
 
 Five Varieties of, 64 
 
 Var. 1 To find the selling 
 price at a given per cent. 
 
 gain or loss, 66 
 
 Var. 2 To find the rate per 
 centum profit or loss on 
 
 sales, 72 
 
 Var.3 To find the Cost Price 
 after gaining or losing a 
 
 given per cent., 75 
 
 Var. 4 Compound Profit and 
 
 Loss, 81 
 
 Var. 5 Combination of state- 
 ments, 85 
 
 Examples in Combination, . . 88 
 Miscellaneous examples,... 92 
 Rule for Combining several Oper- 
 ations in one Statement, . 94 
 
 Commission, 102 
 
 Two methods of deducting, . 105 
 
 Brokerage and Stocks, 105 
 
 Statements combined, 107 
 
 Rule for combined Broker- 
 age, Stocks, etc.,... 107 
 
 Insurance, 108 
 
 Four Varieties, 109 
 
 Discussion of propriety of In- 
 surance, Ill 
 
 Var. 1, and Rule, 113 
 
 Combination of Statements, 115 
 
 Var. 2, and Rule, 116 
 
 Var.3, and Rule, 117 
 
 Var. 4, and Rule, 118 
 
 Life Insurance, 119 
 
 Tolls and Rule, 120 
 
 Compound Proportion, 123 
 
 Theory of Cause and Effect: 
 all animate things Causes,126 
 
 Causes of Time, 126 
 
 Geometrical Extent a Cause,127 
 Capital aCause 127 
 
INDEX. 
 
 vii 
 
 PAGE. 
 
 Classification of Causes in the 
 
 statement, 129 
 
 The Causes and Effects,the four 
 terms of a geometrical pro- 
 portion, 131 
 
 Ratio among Causes, 132 
 
 Examples and Proofs, 134 
 
 Com. Prop, in Fractions, 138 
 
 Passive Causes, 141 
 
 To find Principal, in Interest,. ... 142 
 
 To find the Time, 143 
 
 To find the Rate, 143 
 
 Causes of Capacity, 145 
 
 To find the Side of a cubical 
 figure when two of the sides 
 
 are given, 145 
 
 Relative Contents of Hollow 
 
 bodies, 147 
 
 Contents and Sides of Cribs,. .148 
 Contents and Sides of Boxes, 
 
 etc., in bushels and gallons,. 149 
 Size and relative Weight of Me- 
 tallic bodies, 150 
 
 Novelty in Contraction, 152 
 
 Com. Prop, by single statement,153 
 
 Rule for Com. Proportion, 154 
 
 Simple Inverse Proportion, .... 155 
 
 Theory of statement, 156 
 
 Insufficiency of com'n method,. 159 
 Inv. Proportion in Fractions, . . 159 
 
 Calculations in Machinery, 162 
 
 To find number of Revolutions, 162 
 
 To find Size of Wheel, 163 
 
 To find the number of Teeth or 
 
 Diameter of Wheel, 164 
 
 Concluding remarks on the 
 
 Proportions, 166 
 
 Rule for Inverse Proportion, . . 167 
 Conjoined Proportion, or Chain 
 
 rule, 168 
 
 Theory of, 168 
 
 Exchange of Moneys, 170 
 
 Rule for Conjoined Prop 171 
 
 Fellowship Simple, 174 
 
 Fellowship in Fractions, 176 
 
 General Average, 177 
 
 Fellowship Compound, 179 
 
 Rule for the Fellowships and 
 
 General Average, 181 
 
 Barter and Commerc'l Exchange,]82 
 
 Barter by Reduction, 184 
 
 Origin of State Currencies,. ..185 
 Combinations of statements,.. 186 
 Rule for Barter, Commercial 
 Exchange, and Reduction,.. 187 
 
 Duties, and Tare and Tret, 188 
 
 Law relating to, 189 
 
 Definitions in Tare and Tret,. .189 
 
 PAGE. 
 
 Specific Duties, 190 
 
 Ad Valorem Dut ies, 191 
 
 Law relating to, 192 
 
 Tare and Tret proper, 193 
 
 Combination of statements, . .195 
 
 Rule for Duties & Tare & Tret,196 
 
 Commercial Exchange proper, ..197 
 
 By combination, 197 
 
 Rule for Commerc'l Exchange^" 
 
 Decimal Fractions, \ _ 
 
 Theory of Decimals, 200 ) 
 
 Addition of Decimals, 203 
 
 Subtraction of Decimals, 204 
 
 Multiplication of Decimals, . . . 204 
 
 By Contraction, 205 
 
 Division of Decimals, 208 
 
 By Contraction, 209 
 
 To reduce Decimal to common 
 
 Fractions, .210 
 
 To reduce com. to Dec. Frao.211 
 To reduce Denominate numbers 
 
 to Decimals, 212 
 
 Mensuration, or Practical Geom- 
 etry, 213 
 
 Theory and general remarks,. .213 
 
 Measurem't of Wood and Bark, 216 
 
 Combination of statements, . .217 
 
 Measurement of Lumber, 219 
 
 Combination of statements, . .220 
 
 Cubic Measurement, 221 
 
 By Combination, '221 
 
 Masonry, 223 
 
 Plasterers' and Pavers' Work,.. 224 
 
 By Combination, 225 
 
 Carpenters' and Joiners' Work,. .226 
 
 Cribs, Boxes and Bodies, 227 
 
 Wine, Beer and Dry Measure,227 
 
 Combination of rules, 228 
 
 To find the Side of a Crib, Body, 
 
 Box, etc. Rule, ...228 
 
 Tonnage of Vessels, 229 
 
 Weight of water per cubic foot 
 
 and inch, 229 
 
 Cylinders, Spherical and Conical 
 
 inches, Sea-water, etc 229 
 
 U. States and English Standards 
 
 of Liquid Measure, 230 
 
 U. States and English Standards 
 
 of Dry Measure, 230 
 
 The Winchester Bushel, 230 
 
 The Connecticut Bushel, 230 
 
 Government rule for Tonnage. 230 
 
 Carpenters' Rule, 232 
 
 Superficial Geometry, 232 
 
 Variety of Figures, Definitions 
 
 and Derivations, 233 
 
 To find the Contents of a Rectan- 
 gle, 235 
 
viii 
 
 INDEX. 
 
 PAGE. 
 
 Contents of a Parallelogram and 
 
 Rhomboid, 235 
 
 Triangles, 236 
 
 Polygons, 238 
 
 The Circle, 239 
 
 Definitions and Derivations, ....240 
 
 Quadrature of the Circle, 241 
 
 History and difficulty of, 241 
 
 Inscribed and Circumscribed 
 
 Polygons, 242 
 
 Ratio between the Diameter 
 
 and Circumference, 242 
 
 To find the Circumference of a 
 
 Circle, 243 
 
 Area of the Circle, 244 
 
 The Ellipse, 247 
 
 Circumference of, 247 
 
 Area of,.. 247 
 
 Contents of a Square Inscribed 
 
 in a Circle, 248 
 
 Solidity of a Log when Squared,. 249 
 
 Side of an Inscribed Square, 249 
 
 To find the largest Square that a 
 
 Round Stick will make, 259 
 
 The Side of a Square given to 
 find the Diameter of a Cir- 
 cumscribed Circle, .250 
 
 To find the Diameter of a Circle 
 circumscribed about a Square, 
 or how large a Round stick 
 must be to make a Square of 
 
 given Side, 25 1 
 
 To find the Side of a Square of 
 Area equal to that of a given 
 
 Circle, 251 
 
 To find Diameter and Circumfer- 
 ence of same, . 251 
 
 Measurement of Cisterns, 252 
 
 Square and Circular, 253 
 
 Conical and Pyramidal, 254 
 
 To find the Side or Diameter of 
 
 a Cistern, 255 
 
 Table of Cisterns, 256 
 
 Solid bodies; definitions and de- 
 rivations, 257 
 
 The Cylinder, 258 
 
 Superficial Contents, 258 
 
 Solid Contents, 259 
 
 Contents of Boilers, 260 
 
 Table of Contents of, 269. 
 
 PAGE. 
 
 Boilers and hollow Cylinders,. 261 
 
 Air Pressure, 261 
 
 Cones and Pyramids, 262 
 
 Convex Surface, 262 
 
 Solidity of, 262 
 
 Frustum of, 863 
 
 The Sphere or Globe, 264 
 
 Surface of,... .....264 
 
 Solidity of, 264 
 
 Globe and Cylinder compared,265 
 
 The Spheroid, 265 
 
 Solidity of, 265 
 
 Proportions among Lines, Areas 
 
 and Solidities, 265 
 
 To ascertain the Weight of 
 
 Globes, 265 
 
 Weight of iron Cylinders and 
 
 Globes given, 26^ 
 
 Gauging Casks, 2Rt 
 
 Mechanical Powers, 26? 
 
 The Lever, 26^ 
 
 The Wheel and Axle, 269 
 
 The Inclined Plane, 269 
 
 The Wedge, 270 
 
 The Screw, 270 
 
 Square Root, 271 
 
 Currency, 274 
 
 Customhouse value of Foreign 
 
 Coins, 274 
 
 Table of Moneys of Account 
 
 at Congress valuation, ....275 
 Jewish Standard Weights and 
 
 Measures, 277 
 
 Jewish Standard of Money, and 
 
 Table of Value, 278 
 
 Table of Areas of Valves, Circles, 
 
 etc., , 279 
 
 Table of weight of Square Rolled 
 
 Iron, 280 
 
 Table of weight of Round Rolled 
 
 Iron, 281 
 
 Table of weight of different bo- 
 dies of Cast Iron, 281 
 
 Table of weight of Flat Bar Iron,282 
 Table ofweig. of Cast Iron Pipes 
 and Cylinders from 1 to 30 in. 
 
 diameter, 284 
 
 Fac simile Coin Plates represent- 
 ing the coins used by all na- 
 tions, 285 to 316 
 
KAINE Y" S 
 
 IMPROVED ABACUS. 
 
 CANCELATION. 
 
 ALL arithmetical computations are effected 
 by increase and decrease, which depend in their 
 relations, on the converse operations of Multi- 
 plication and Division. The latter are but 
 . abbreviated methods of adding and subtracting. 
 Increase and decrease are the results of the 
 relative difference between different numbers 
 and quantities of the same thing : hence their 
 result depends, in all reckoning, on the great 
 principles of Ratio and Proportion. Therefore, 
 by Proportion, as the rationale of statement, and 
 Multiplication and Division as the mechanical 
 media of reducing such statements to their 
 results, we have in a few words, an epitome 
 of all arithmetic. 
 
 As by this, Multiplication and Division are 
 presented as the leading operations of reckon- 
 ing, we may profitably spend some little time, 
 in ascertaining a more expeditious method of 
 determining products and quotients, than by 
 the old, tedious, and circumlocutoiy formulae 
 of the books. 
 
 When 7 is multiplied by 3, the product is 21 : 
 this product divided by another 3, gives 7 again; 
 
6 RAINEY'S IMPROVED ABACUS. 
 
 the 7 is not changed by the Multiplication and 
 Division : it may, therefore, be inferred, that, 
 when any number is both multiplied and divided 
 by any other number, the former remains un- 
 changed. Hence, such multipliers and divisors 
 may be dropped, as useless, and the numbers 
 canceled. If the 7 be multiplied by 5, and the 
 product divided by 10, the result will be 3i, or 
 i of the 7 ; because the 5 has only one half 
 the capacity in elevating, that the 10 has in 
 depressing : consequently, the 7 is affected 
 twice as much by Division as by Multiplication. 
 We may therefore divide 10 by 5, and place 
 the quotient 2, on the side of the 10, which 
 shows the relation between these numbers; 
 the one, increasing by multiplication, on the 
 right ; the other, decreasing by division, on the 
 left. Again : 
 
 Two numbers, as 12 and 16, sustain to each 
 other a relation, that may be expressed by 
 smaller numbers. They may be reduced to 
 such smaller numbers, by extracting a factor 
 or figure which has been instrumental in pro- 
 ducing the numbers in each case. Thus, 12 is 
 composed of 4 times 3, while 16 is composed 
 of 4 times 4. Here the same factor which has 
 been used in making each number, the 4, may 
 be extracted in each case, leaving 3 in the 12, 
 and 4 in the 16 : which shows that \ f are equal 
 to | ; or, that i-f make J. In this case, as in 
 all other cases of factors, the number sup- 
 posed, must not be written down; and must 
 be contained without a remainder, in some 
 number on both the right and left of the line. 
 Such numbers on the two sides of the line, as 
 
CANCELING. 7 
 
 are divided by the supposed factor, may be 
 canceled, and the other constituent factor must 
 be set on the side of the canceled number, 
 from which taken.* 
 
 When there are numbers on the right and 
 left, terminated by ciphers, these ciphers may 
 be stricken off in equal numbers, as so many 
 factors of 10. In | we cancel the two ciphers, 
 and leave f ; which is equivalent to extracting 
 10 in each case. No cipher can be canceled 
 which has a significant digit on its right ; for its 
 value is qualified by such digit. Numbers on 
 the left of ciphers may be canceled with other 
 numbers on the opposite side of the line ; as 
 such numbers are but factors co-operating with 
 the ciphers on the right, to constitute their 
 sum. Ciphers thus isolated indicate 10. From 
 these considerations we conclude, that, to can- 
 cel numbers, after they have been arranged 
 on the two sides of the vertical line, 
 
 1st. Cancel all equal numbers on the two sides 
 of the line : 
 
 2d. Cancel ciphers in equal numbers : 
 
 3d. Divide, leaving no remainder, from one 
 side of the line into the other, and, vice versa, 
 placing the quotient on the side of the larger : 
 
 4th. Extract all possible factors from any two 
 numbers occupying different sides of the line, 
 and leave the other constituents of such numbers, 
 on the side of each. 
 
 A few examples in the multiplication of 
 fractions are given, merely to illustrate the 
 
 * Cancel, is from the French cancelkr, which signifies 
 literally, to cross a writing. 
 
8 RAINEY'S IMPROVED ABACUS. 
 
 application of the foregoing directions. Multi- 
 ply | of f of T V of 2, by of Lf of li of fi of 5. 
 Here, as in the multiplication of all fractions, 
 we place all the Numerators on the right, and all 
 the Denominators on the left. We have not 
 sufficient space to give the theory of these 
 statements, in this little treatise, which, it is 
 designed to devote more particularly to opera- 
 tions in practical business ; leaving all the work 
 preparatory to this, to lectures at the black- 
 board, or to the elementary works of others.* 
 
 The following remarks may be proper just 
 here : 
 
 The upper part of a fraction is called the 
 Numerator ; and the lower part, the Denomina- 
 tor. The Denominator, from de, concerning, 
 and nomen, a name, shows the name of the 
 fraction, or the number of parts into which the 
 unit or whole thing, is divided. The Nu- 
 merator, from numerus, number, shows the 
 number of parts, of the size indicated by the 
 Denominator, taken. A whole number is con- 
 sidered a numerator, whose denominator would 
 be 1. Mixed numbers, such as 4i, 3|, &c., 
 before placed on the line, must be reduced to 
 improper fractions. This is done by multiply- 
 ing the whole number by the denominator of the 
 appended fraction, and adding in the numerator. 
 The denominator of the number is again used, 
 as the denominator of the improper fraction. 
 Thus, in 4i, twice 4 make 8 3 and 1 makes |. 
 In 3^, five times 3 are 15, and 1 is '/. Hence, 
 
 * Day and Thomson's Practical and Higher Arithme- 
 tics. 
 
MULTIPLICATION OF FRACTIONS. 
 
 I 
 
 8 
 
 t 
 
 8Ans. 
 
 9 and 16 are the numerators, although they 
 are larger respectively, than their den ominators. 
 
 Fours equal : 5 into 10, twice ; 
 this 2 equals 2 opposite : 8 into 
 24, 3 times, while 3 times 3 on 
 the right make 9, which goes 
 into 18 on the left twice : now 
 twice 6 on the left equals 12 on 
 the right : 5 into 35 seven times, 
 and 7 into 14 twice; this 2 into 
 16 eight times, on the left: we 
 have remaining on the left 8, 
 and on the right 5 ; making 
 f , Ans. 
 
 After canceling as far as practicable on the 
 two sides of the line, multiply continuously 
 together all the terms on the right, for a new 
 numerator, and all the terms on the left, for a 
 new denominator. If nothing remains on the 
 right, one is understood. If the number on the 
 right be smaller than the number on the left, 
 the answer is a fraction ; of which, in all 
 cases, the right is the numerator ; but, if the 
 number on the left be smaller, the right must 
 be divided by the left : in such case, the 
 answer will be a mixed number. 
 
 What will 7i Ibs. iron come to at 2| cts. per 
 
 lb.? Here 7i make 
 
 1_5 
 
 and 
 
 2 1 make l -j 
 
 We place the numerators on the 
 right, and the denominators on the ft 
 left. Five into 15, three times ; and 1 18 cts. 
 2 into 12 six times ; while six times 
 3, the only remaining numbers, make 18 cts., 
 the answer. 
 
10 RAINEY'S IMPROVED ABACUS. 
 
 Multiply, -jf of a yard of cloth, by }f of a 
 dollar per yard, thus, 
 
 _ 3 Here, we must suppose some 
 3 factor, before the numbers can be 
 
 20 
 
 20 
 
 reduced. Let us take 5, which 
 goes into 25 five times, and 
 into 15 three times ; again : 4 into 
 16 four times, and into 12 three times; we 
 have 5 and 4 as factors, neither of which has 
 been placed down ; only their quotients. Now 
 3 times 3 on the right, and 4 times 5 on the 
 left, make ^ of a dollar. 
 
 What will 2f yards of gambroon come to 
 at 1.20 cents per yard? 
 
 Eight is contained in 120, 15 
 times ; and this number multiplied 
 by 21 on the same side, gives $3,15 
 cents for the answer. 
 
 #0-15 
 
 3,15 
 
 If T 8 5 of a farm are divided among 4 heirs, 
 how much will each get? 
 
 In this instance the dividend, T 8 j, must be 
 placed on the right of the line, and 4, the di- 
 visor, on the left; thus, 
 
 A fractional number occupies the 
 
 15 
 
 * 2 
 
 right or left of the line, when its nu- 
 merator is on the right or left. Let 
 15j 2 the numerator be located first; then, 
 15 I the denominator is merely placed op- 
 posite. If I direct the pupil to place 
 T 9 T on the left of the line, he must place the 9 
 on the left only, and the 17 on the right. The 
 numerator always indicates the locality and 
 value of the fraction. 
 
 Divide J of f of 20, by T V of f of '/ of \ j 
 of 4i of v- 
 
COMPLEX FRACTIONS. 
 
 11 
 
 The numerators of the divi- 
 dend are placed on the right, and 
 those of the divisor on the left, 
 with all the denominators oppo- 
 site their respective numerators. 
 
 Four into 20 five times, and 5 
 equals 5 on the left; 9 into 18 
 twice, and twice 2 on the right 
 make 4, which goes into 12 on 
 the left 3 times; 5 into 25 five 
 times, and this 5 again into 10 
 on the right twice ; 8 into 40 five 
 
 7 
 A 
 
 3-/W 
 t-tt 
 
 5-40 
 
 105 
 
 16 
 
 105 
 
 times ; 3 into 9 three times ; 3 equals 3 ; 2 and 8 
 remain on the right, and 5, 3, and 7 on the left, 
 which multiplied separately give T W 
 
 From the foregoing we may deduce the fol- 
 lowing directions : 
 
 To multiply fractions ; place all of the numera- 
 tors, both of the multiplicand and multiplier, on 
 the right of the vertical line, and all the denomina- 
 tors on the left. 
 
 To divide fractions; place the numerators of 
 the dividend on the right, and those of the divisor 
 on the left, with the respective denominators of 
 each opposite their numerators. 
 
 COMPLEX FRACTIONS. 
 
 (It is not designed in this short treatise to devote ranch space to either 
 the theory or practice of fractions ; as it is believed that there are very 
 many elementary works accessible, which do entire justice to this depart- 
 ment of numbers. We shall merely introduce a page or two on complex 
 fractions,, for the consideration of teachers, that we may present a short 
 and simple method of using them, which is not found in the books.) 
 
 Division of ordinary fractions leads to the 
 consideration of those that are complex. The 
 
12 RAINEY'S IMPROVED ABACUS. 
 
 doctrine may be advanced, that to increase the 
 terms of a fraction, so as not to change its value, 
 multiply both the numerator and denominator by 
 the same number. 
 
 If | be multiplied by 2, it is made f . Now, 
 
 3_L 
 if instead of f , the fraction were-j-, we might 
 
 double the numerator 3i, by multiplying by 
 the special denominator 2, and adding in the 
 special numerator 1 ; thus making J of the 
 numerator. 
 
 But if this numerator is thus increased by 
 the 2, the denominator should be likewise : 
 hence the 4 is multiplied by the same 2, mak- 
 ing 8, by which we have t, which is equivalent 
 
 3 1 
 
 to -p. Both terms of the fraction are in- 
 4 
 
 creased by the same number, the denominator 
 of the fraction annexed to the numerator. 
 After reducing the numerator, it may be ex- 
 
 7 
 
 pressed thus : T -r, showing that the two de- 
 " 
 
 nominators have been thrown together, and 
 may, consequently, be combined in multipli- 
 cation. 
 
 It appears thus, that if we would reduce 
 complex to simple fractions, and at the same 
 time to the lowest term, we should multiply 
 each term of the fraction by the denominator or de- 
 nominators of the fraction or fractions annexed 
 to the numerator or denominator, or both, adding 
 in at each separate multiplication, the given nume- 
 rator or numerators. For example, in the frac- 
 
COMPLEX FRACTIONS. 13 
 
 tion o^,we multiply first by the denominator 
 
 of the numerator, 2. Twice 3 are 6, and one, 
 the numerator, added, makes 7, or J: now 
 we multiply the denominator 2|, by the 
 same 2; twice 2 are 4, and twice i are f, 
 making 4|. This 4f must be placed under 
 
 the 7, thus, j^. We have now reduced the 
 
 complex fraction in the numerator by multi- 
 plying it and the whole denominator below, by 
 the denominator 2 above, and proceed to that 
 of the denominator. Taking the new fraction 
 4| to work on, we say 3 times 4 are 12 and 2 
 are 14, for a new denominator, and 3 times 7 
 are 21, for a new numerator; thus, f-J. This 
 reduced to its lowest term, by dividing the nu- 
 merator by the denominator, gives 14 for the 
 answer. 
 
 It is seen, therefore, that in each case, it is 
 necessary to reduce both the numerator and 
 the denominator to a mixed number. After 
 this is done, knowing that the denominator 
 of a fraction is the divisor, we may deduce the 
 following rule: 
 
 To reduce complex to simple fractions, of the 
 lowest term : Reduce both terms to an improper 
 fraction, and place the numerator of the numera- 
 tor on the right, and the numerator of the denomi- 
 nator on the left, with their respective denominators 
 opposite. 
 
 Let us reduce in this way ^. The nume- 
 rator 34 is first reduced to J and placed, 
 2 
 
14 RAINEY'S IMPROVED ABACUS. 
 
 the numerator on the right of the line, and 
 the denominator on the left; thus, 
 
 Next reduce the denominator 2J to an 
 improper fraction, and divide by it, plac- 
 ing the 7, or numerator, as in other 
 cases of division, on the left. Seven equals 7 ; 
 and 2 is contained in 3 one and a half times, 
 which is the answer, as above. 
 
 By this process, the complex is not only re- 
 duced to a simple fraction, but to the lowest 
 term of that fraction ; all of the factors being 
 excluded in the canceling: while it is plain 
 and intelligible, and at once proves itself ne- 
 cessarily correct to all who take the trouble to 
 know that the numerator of a fraction is the 
 dividend, and the denominator the divisor. 
 
 In the same manner that other fractions are 
 multiplied, we multiply these, by placing on 
 the right the numerators, both in the multipli- 
 cand and multiplier. Hence, 
 
 To multiply complex fractions and reduce them 
 to their lowest term : Place the numerators of the 
 numerators, both of the multiplicand and mul- 
 tiplier, on the right; the numerators of the denomi- 
 nators on the left ; and all respective denominators 
 opposite. 
 
 Multiply ^ by j|. In the first place 8j. 
 
 make \ 5 , which are placed on the right, while 
 4, the denominator, is placed with the 3 on the 
 left. Under this are placed the 4|, or y , the 
 other numerator; and f, the denominator, is 
 placed on the left; thus, 
 
COMPLEX FRACTIONS. 
 
 15 
 
 Four times 6 on the left equal 24 on 
 the right. The answer is 8J. 
 
 25 
 
 f 
 
 ~3 
 
 1 6. 
 
 Again: Multiply fby~. Thus, J is divided 
 
 4" 9" 
 
 by f , and this is multiplied by f and divided 
 by*. 
 
 Three is contained in 6 twice, and $ f 
 
 twice 4 equals 8 on the left. Seven 
 
 equals 7, while 5 is contained in 9 f $ fi 
 
 If times. 5 I 9 
 
 Division of complex fractions may be per- 
 formed as in the division of other fractions, 
 by inverting or placing on the left of the line 
 the numerator of the divisor. Hence, 
 
 To divide one complex fraction by another'. 
 Place the dividend on the right and the divisor on 
 the left. 
 
 4 2 
 Divide ~ by ^|. Here we place 4| on 
 
 right, and 7i on the left ; and dividing by % 
 place it on the left, while its denominator 
 is placed opposite. 
 
 4] 
 
 It appears that all of the numbers frfr 
 equal: hence, one is understood on the 
 right for the quotient. 
 
 the 
 
 .11 
 
 It is foreign to our purpose to fill this little 
 volume with examples for the reader to solve ; 
 nence, after pointing out clearly the principles 
 
16 
 
 governing statements and solutions, we shall 
 give him the privilege of working such ques- 
 tions as his experience and business may sug- 
 gest ; a few examples, however, will be intro- 
 duced, to familiarize him with canceling and 
 using figures by this system. 
 
 Multiply 3J Ibs. cheese, by 8J cents per Ib. 
 
 Multiply J of T \ of 20, by y of f of T 8 T of 
 JL of f of 10i of 180. 
 
 3-^ 2 
 
 Multiply - 1 of 40 by -, and divide by f 
 
 of T V 
 
 Divide i by i, and multiply it by 4 of 20. 
 
 The examples above are deemed sufficient 
 to enable the learner to proceed with ease, 
 after the questions are stated, without a teacher. 
 If the numbers on the two sides of the line 
 are such as cannot be canceled, they must be 
 multiplied and divided, and the result will be 
 the same. The vertical line will be explained 
 in Simple Proportion. 
 
 SIMPLE INTEREST.* 
 
 $5- The following method of casting interest is designed for every rate 
 percent. Many of the questions wrought under this head, can be wrought 
 with less than half the number of figures here required, by the short 6 per 
 cent., and other methods which will follow. 
 
 SIMPLE INTEREST is an allowance made for 
 the use of money, and is different in different 
 countries. It is supposed that, as in one year 
 the products of the soil, on which all other 
 profits are primarily based, yield their regular 
 
 * I am indebted to the Rev. William McGookin, of Ohio, for many impor- 
 tant suggestions on simple interest. 
 
THEORY OF INTEREST. 17 
 
 increase, so it is reasonable that a borrower 
 of money should be required to refund the 
 amount borrowed, annually. Hence, the unit 
 of time for which money is lent, is one year ; 
 called per annum or by the year, from the Latin 
 words, per by, and annum a year. The 
 charge of a specific price or bonus for the use 
 of money, requires that a sum be established, 
 as a general sum, on which, in one year, the 
 specified, legal interest shall be charged; so 
 that a larger sum would receive proportionally 
 more, and a smaller sum proportionally less 
 interest ; while likewise a greater or less length 
 of time than one year, would make the inter- 
 est proportionally more or less. This specified 
 sum is 100; a number chosen because easily 
 used in dividing ; being the round product of 
 two decimates. The 100 is called per centum, 
 from per by, and centum, a hundred. Hence 
 per centum per annum means, by the hundred, 
 by the year. Cent, is a contraction of centum. 
 Most of the States of our Union have estab- 
 lished 6 per cent, per annum, as the legal rate 
 of interest. Now, it is manifest that if per 
 centum or 100, in per annum or 1 year, that 
 is, if 100 dollars in one year, gain 6 dollars 
 interest, a larger or smaller number of dollars 
 will gain more or less than 6 dollars, in one 
 year ; while likewise, $100 in a greater or less 
 time than one year, would gain more or less 
 than 6 dollars interest. If $100 in one year gain 
 6 interest, 200 would gain twice 6, or 12; and 
 if 200 in one year gain 12, in two years it would 
 gain twice twelve ; or in one half of .a year 
 the half of 12, or 6. So we perceive that the 
 
18 RAINEY'S IMPROVED ABACUS. 
 
 interest on a sum of money for a given period 
 of time depends on the relation that such sum 
 bears to 100, as well as the relation of the 
 time to 1 year. These relations are properly 
 ascertained by Compound Proportion ; but we 
 will here present them in the form of two 
 connected simple proportions. What is the 
 interest on $50 for six months, at 6 per cent. ? 
 that is, if $100 in 12 months gain $6 interest, 
 what interest will $50 gain in 6 months ? 
 Now we place the demand, $50 and 6 months, 
 on the right of a line, and the terms of the 
 same name, 100 dollars and 12 months, as the 
 supposition, opposite these, on the left. When 
 we place dollars opposite dollars thus, it is to 
 get the ratio between the two numbers of 
 dollars : the same is the case with the months, 
 which are placed opposite. The ratio between 
 100 and 50 is i ; and between 12 months and 
 6 months \ ; so that the two ratios multiplied 
 together, that is \ times i, make \ ; hence, the 
 $6 interest multiplied by this i, is reduced to 
 li, which is the result ; thus, 
 
 The ratio of these numbers 
 compared, is obtained as by a 
 fulcrum and scale. Hence, 50 is 
 contained in 100 twice, and 6 in 
 12 twice, and this 2 in 6 three 
 times, and 2 on the left in 3 on the right li 
 times, which is \\ dollars, the answer. Suppose 
 the time above, instesd of being 6 months, were 
 6 days ; then we cc aid not place one year 
 opposite it in the form of 12 months ; but only 
 in the form of 360 days : for, to obtain the 
 ratio of time that 6 days bear to a year, we 
 
 3 
 
PRACTICE OF INTEREST. 19 
 
 must compare days with days, not with months : 
 hence if the time were 6 days, we would place 
 360 days, which make a year, opposite; or 
 which is the same thing, 30 days opposite, to 
 make a month, and 12 months to make a year : 
 these two numbers make 360, the number of 
 days in an interest year. Let us find the 
 interest for 6 days ; thus, 
 
 2 ,100 
 
 20 
 
 Jl^ 
 
 20 
 
 We must place on the left op- 
 posite the specified time, 1 year, 
 12 months, or 360 days, just as 
 the time specified, may be in years, 
 months or days. Here again, 50 
 into 100 twice : 6 times 6 on the 
 right equal 36 on the left ; leaving 
 the cipher or 10 to be multiplied into the 2, 
 making the denominator 20. One being un- 
 derstood on the right, the answer is ^V f a 
 dollar; equal to 5 cents. 
 
 We see then, that 100 and the time, either 
 in years, months or days, go to the left ; while 
 we place the sum on which the interest is 
 to be obtained, the time, and the rate, all on 
 the right. 
 
 Now if the rate be 6, or 10, or any other per 
 cent., it is so many hundredths ; 100 being the 
 denominator, while 6 or 10, &c., is the nume- 
 rator. This denominator is composed of two 
 decimals. Decimal is from the Latin word 
 decent, which means ten ; hence, the decimal is 
 the tenth part of a unit. Two of these deci- 
 mals, or the tenth times one tenth, make one 
 one-hundredth : hence, any two decimal factors, 
 express so many hundredths. It is necessary 
 in expressing a decimal fraction, that there 
 
20 RAINEY'S IMPROVED ABACUS. 
 
 be one figure less in the numerator than in 
 the denominator: and the denominator 100, 
 containing 3 figures, we always consider that 
 there are but two in the numerator or rate. 
 Then, if we consider the rate per cent, two 
 decimals, every thing multiplied into it must be 
 made 100 times smaller : that is, if dollars be 
 multiplied by the rate, they become hundredths 
 of dollars, or cents : and if cents be thus multi- 
 plied by the rate, or two decimals, they become 
 hundredths of cents. Now, in the first question 
 wrought, let us make this rate two decimal 
 factors, by dropping the 100 at the left. It is 
 unnecessary here to place a cipher at the left 
 of the 6, to show that the rate is composed of 
 two decimals : this will be understood. 
 
 Certainly, when we multiply the 
 $50 by this rate per cent., 6, it is 
 made 100 times smaller than dol- 
 lars, and becomes cents ; so that 
 the answer is 150 cents. We 
 will therefore cut off two figures at the right 
 of the result for cents. Suppose again, we 
 consider this 50 cents, instead of 50 dollars: 
 then the cents being multiplied by the rate, 
 become hundredths of cents ; so that in the 
 answer, we cut off two for hundredths of cents, 
 two more, if we have them, for cents, while 
 the remaining figures at the left, are dollars. 
 But having only 3 figures, the answer is 1 cent 
 and 50 hundredths; or li cents. Let us get 
 the interest on $60, for 317 days, at 6 per cent. 
 We call the $60 here, as we call the sum in 
 all other cases, the PRINCIPAL the 317 days, 
 the TIME, and the 6 per cent, per annum, the 
 
 50 
 
INTEREST IN FRACTIONS. 21 
 
 RATE. We place these numbers on the right, 
 thus, 
 
 and as the time is in days, we 00 
 
 place 30 and 12 opposite, or 360, 0317 
 
 which is the same thing ; and # A# r 
 again dispense with the 100. If $|3,17 
 
 we dispense with this 100 on the | 
 left, the answer will be 100 times smaller than 
 the principal; and this being dollars, the an- 
 swer will be cents. Cyphers equal : 6 into 
 12 twice : and twice 3 on the left, equal 6 on 
 the right. We have 317 left, and conclude, 
 that the answer is 3 dollars and 17 cents. If 
 the principal were 60 cents, the answer would 
 be 3 cents and 17 hundredths. From this we 
 conclude, that, When the Principal is dollars, 
 the answer is cents; and when the principal is 
 cents , the answer is hundredths of cents. 
 
 What is the interest on $80, for 9 months, 
 at 7 per cent. ? Here, we place the P. T. & R. 
 as before, on the right , and place 12 only, on 
 the left; because the time is in months, or T 9 ^ 
 of a year. 
 
 We use the factor 4, which 
 goes into 12 three times, and into 
 8 twice : 3 into 9 three times : 
 now 3X2X7X10 make 420, or 
 $4 and 20 cents. Were this 
 
 $0 2 
 X fi-& ft 3 
 
 $|4,20 
 
 time 9 days, we would place 30 with 12 on 
 the left. Were the principal 80 cents, the 
 answer would be four cents and T \\. 
 
 What is the interest on 37 1 cents for 18 
 days at 74 per cent. ? The principal and rate 
 being mixed numbers, must be reduced to im- 
 proper fractions ; and the numerators placed 
 
22 RAINEY'S IMPROVED ABACUS. 
 
 on the right, with their denominators on the 
 left. The numerator of a fraction in all cases, 
 occupies the same place that otherwise the whole 
 number would : while the denominator is invaria- 
 bly placed opposite. Hence again, the time being 
 days, we place 30 and 12 on the left. The 
 principal is cents ; \ 5 cents ; hence, the answer 
 will be hundredths ; and as such, we will strike 
 off two numbers for hundredths, and two for 
 cents. 
 
 2 75 j Fifteen into 30, twice, and 
 
 2 #0 / l$_ 3 2 into 18, 9 times : The fac- 
 4 & tor 3 into 9 three, and into 
 
 12, four times; 4X2X2 are 
 16 on the left, and 3X75 on 
 the right, are 225 ; which di- 
 vided by 1 6 gives 1 4 T l . The 
 answer is no dollars, no cents, and 14 T J hun- 
 dredths cents. Such examples as this are 
 scarcely of any practical value, and only show 
 the full extent of the theory and practice of 
 interest by this system. 
 
 What is the interest on $600,60 cents, for 3J 
 years, at 4i per cent.? We make the 3 J years, 
 y , and place the 10 on the right, and 3 on the 
 left, and divide by nothing, except the denom- 
 inator. All that we divide by the numbers 
 12, and 12 and 30 for, is to reduce the time to 
 years ; hence, when the time is already in 
 years, division by any number becomes unne- 
 cessary, except by such denominators, as from 
 mixed numbers, may fall on the left ; which 
 is the case with 3 and 2 in this example. 
 Here, the 4i make | per cent. 
 
INTEREST FOR BROKEN TIME. 23 
 
 Two into 10 five, and 3 into 
 9 three times: now 3X5X600,60 
 make 900900. We cut off two for 
 hundredths, and two for cents : 
 
 600,60 
 
 405 
 ' 3 
 
 $|90,09,00 
 
 hence, the answer $90,09 cents, 
 and no hundredths. This answer is in hun- 
 dredths, because the principal is in cents. 
 
 What is the interest on $600, for 3 years, 6 
 months and 20 days, at 6 per cent. ? Here it 
 is necessary to reduce all the years to months, 
 and add in the given months ; and likewise re- 
 duce the days to the fractional part of a month, 
 and add such fraction to the months. In three 
 years there are 36 months, and 6 more added, 
 make 42 months. Now, 20 days are f %- of a 
 month, which, canceling the tw r o ciphers, makes 
 . The time, therefore, is 42f months, which 
 make -f- months. We place this 128 on the 
 right, and 3 on the left. The time now being 
 in months, we divide by 12 only. 
 
 Six into 12 twice, and twice 000 
 
 3 on the left, equal 6 on the % 128 
 
 right: we consequently draw # 40 
 
 down the 128, and annex the f|128,00 
 
 two ciphers, making the answer 
 $128,00. Suppose the time had been 1 year, 
 1 month, and 10 days. One year and 1 month 
 make 13 months : 10 days are 1^ or i of a 
 month : consequently the time is 13 J, or \ 
 months. Here, 40 should go to the right, and 3 
 to the left, with 12. Again : Suppose the time 
 6 months and 15 days : These 15 days are i.f 
 or \ month ; so that the time is 64, or l -/ 
 months. Here, again, 12 should be placed on 
 the left. Suppose the time 2 years, 9 months 
 
24 RAINEY'S IMPROVED ABACUS. 
 
 and 25 days: the 25 days make thus, |f, equal 
 to | of a month ; and two years and 9 months, 
 make 33 months, which, with the f annexed, 
 is 33 1 or 2 f 3 months. Suppose the time 27 
 days : this would be f J, or T \ of a month, to be 
 annexed to all the months. Were it 28 days, 
 it would be ||, or 1 j of a month. Nine days 
 would be JL, or T %. of a month : so would 8 
 days be -J^, or T 4 j of a month. Suppose the 
 time were three months and 29 days. Most 
 business men would call this 30 days : but to be 
 accurate, we would multiply the months by 30 
 and add in the 29. Thus, the whole time would 
 be reduced to 119 days : and we would conse- 
 quently divide by 30 and 12. When the days 
 make a number that cannot be reduced to the 
 fraction of a month, to secure entire accuracy, 
 the years must be reduced to months, and all 
 the given months added in ; then, these months 
 must be reduced to days, and the given days 
 added in. 
 
 What is the interest on $50 for 1 year, 3 
 months and 5 days, at 6 per cent.? One year 
 and 3 months make 15 months: 5 days are 1 
 of a month: hence 151, or y months, is the 
 time. We divide by 12 only. 
 
 Sixes equal: we have 12 on the 
 91 left, and on the right 50X91 which 
 makes 4550. This divided by 12 
 gives for answer $3,791 cents. 
 
 From the foregoing principles and 
 operations, we are justified in ma- 
 
 12|4550 
 
 king the following 
 
RULE FOR ALL INTEREST. 25 
 
 SUMMARY OF DIRECTIONS, 
 
 For working Interest of any conceivable Prin- 
 cipal, Time, and Rate. 
 
 Place the Principal, Time, and Rate, on the 
 right of the vertical line ; and if the time is days, 
 place 30 and 12 on the left: if the time is months, 
 place 12 only, on the left: and if the time is 
 years, place nothing on the left. 
 
 If the Principal, Time, or Rate is a mixed 
 number, reduce it to an improper fraction, and 
 place the numerator on the right, with the denom- 
 inator on the left. 
 
 When the Principal is dollars, the answer is 
 cents: in such*' case, two figures must be cut off 
 for cents : when the Principal is cents, the an- 
 swer is hundred ths of cents: here, cut off two fig- 
 ures, commencing at the right, for hundredths, 
 two more for cents, and the remainder at the left 
 is dollars. The figures thus cut off for cents, 
 hundredths, tyc., must be whole numbers; while 
 any existing fraction will be only a fractional 
 part of such cents or hundredths. 
 
 When the time is months and days, or years, 
 months and days, reduce the years to months, and 
 add in all the given months : then reduce the days 
 to the fractional part of a month, and annex this 
 fraction to the whole number of months: reduce 
 all to an improper fraction, and place the numerator 
 on the right, and \he denominator on the left. In 
 such case, divide by 12 only. If the time cannot 
 be reduced to the fractional part of a month, re- 
 duce tfie whole time, years, months and. days, to 
 days, and divide by 30 and 12. 
 
26 RAINEY'S IMPROVED ABACUS. 
 
 If 4ht time is years and months, reduce the 
 months to the fractional part of a year : add to 
 the years : reduce all to an improper fraction, and 
 divide by the denominator only. 
 
 If the answer to a question be $80, 20 cents 
 and 38|f hundredths, and it were written thus, 
 80.20.38^1, it would be wrong to cut off eith- 
 er the 18 or 29 by itself, or both together, for 
 the denomination of hundredths; for they make 
 only the i| part of one one hundredth part of 
 a cent. Hence, to strike off cents or hun- 
 dredths of cents, place the separatrix between 
 integral numbers only. 
 
 The use of 360 days to the year, may be by 
 some thought singular; but it grows out of the 
 standard commercial usage, 3 days to the 
 month, and 12 months to the year. The busi- 
 ness year, the civilized world over, is called 
 360 days. If, however, any wish to use the 
 365, they can easily do so by substituting 365 
 on the left, for 30 and 12. The difference in 
 the result is only T * part. In banks, the in- 
 terest is always reckoned for three days more 
 than the time specified by the borrower, which 
 are called days of grace, or days given the bor- 
 rower to allow for any accidents or exigencies 
 which may prevent the money being funded 
 at the close of the discounting period. Grace 
 means gift. Really the three days are not days 
 of grace ; for interest is reckoned on them as 
 part of the general discounting time. Banks, 
 too, charge more than the legal rate of inter- 
 est ; on what principle of ethics, however, I 
 have never been able to learn. If the rate be 
 6 per cent., the note for one year, is given for 
 
INTEREST FOR MONTHS AND YEARS. 27 
 
 $100; and the interest on $100, which is $6, is 
 deducted from the money when issued to the 
 borrower; so that he gets only $94. Now 
 here, he pays $6, not for the use of 100, which 
 would be equitable, at 6 per cent., but for 94. 
 The interest on $94, the sum that the borrow- 
 er receives, at 6 per cent, will not be $6; so 
 that- he loses clear the difference between the 
 $6, and the sum of interest that 94 would gain. 
 This difference is quite important in heavy 
 transactions. 
 
 It is frequently impossible to cancel in ques- 
 tions of interest ; when this is the case, all the 
 numbers on the right must be multiplied to- 
 gether for a dividend, and all on the left for a 
 divisor : after which the former must be divi- 
 ded by the latter. Some would ask, " what 
 benefit in working interest in this way, if at 
 times, it is necessary to multiply and divide, as 
 in the old system ?" It is because all ques- 
 tions, whatever be the principal, time or rate, 
 can be wrought by this one, simple rule, with- 
 out a separate rule for every varying per cent.; 
 and which rule itself, is based on a principle 
 so remote, as seldom to be seen by the ordi- 
 nary arithmetician : because the statement 
 can be easily made and understood by any or- 
 dinary mind ; because the work is unique and 
 systematic; and because in most cases the work 
 can be greatly abbreviated by 
 canceling. We give but two 
 
 other examples, and these 3 ^ 
 
 without the work. What is ~~3'65000 
 
 the interest on $800, for 2 years, 
 
 8 months and 15 days, at 10 per |. b ' 
 
 65 
 10 
 
28 RAINEY'S IMPROVED ABACUS. 
 
 cent.? Here then, the time makes -V- months. 
 
 30 
 12 
 
 10,000 
 90 
 
 What is the interest on $10,000 
 for 90 days, at i of 1 per cent.? 
 This time may be placed on the line 
 as 90 days 3 months, or i of a 
 year. In the first case, we would 
 
 divide by 30 and 12 ; in the second by 12 only, 
 
 and in the last by nothing. 
 
 6,25 
 
 INTEREST AT SIX PER CENT. 
 
 Interest at 6 per cent, has long since been 
 reckoned by dividing by 60, when the' time was 
 days, or when months, multiplying the princi- 
 pal by half their number. This process is 
 very easy, when the number of months is 
 even, and the half can be found without ma- 
 king it necessary to multiply the principal by 
 a mixed number ; but when the time is an odd 
 number, or has years, months and days ; or 
 when the principal is a mixed number, it is 
 difficult, by the process ordinarily pursued, to 
 use the fractions, and ascertain the precise 
 result. Hence, the time is generally made too 
 great or too little, and many fractions are 
 thrown away ; whereas by the use of the ver- 
 tical line, and the consequent advantage of 
 placing the numerators and denominators on 
 its two sides, this difficulty is entirely obvia- 
 ted ; so that if the numbers on the two sides 
 cannot be canceled, they can at least be mul- 
 tiplied and divided, as by the old method. By 
 this method, therefore, a clear gain is made of 
 
INTEREST AT SIX PER CENT. 29 
 
 all numbers that can be canceled, as well as 
 greater ease and perspicuity in the statement. 
 We find by the first method presented in this 
 work, that the interest on one dollar, for 60 
 days, at 6 per cent., is one cent. It is likewise 
 the same for 2 months, thus : 
 
 |1 cent. |1 cent. 
 
 In the former case, the time being 60 days, 
 30 and 12 are used on the left; in the latter, 
 the time being 2 months, 12 only is used. The 
 result is the same. 
 
 If one dollar, as above, give 1 cent interest 
 in 60 days, it will in 6, which is the tenth part 
 of 60 days, give the tenth part of a cent, or 
 one mill. The fact is therefore established, 
 that 
 
 One dollar in 6 days, at 6 per cent., will gain 
 ONE MILL interest ; and ONE DOLLAR in 2 months 
 at 6 per cent., will gain ONE CENT interest. Hence, 
 we are justified in making the following 
 statement in Proportion: If 1 dollar in 6 days 
 on the left, give one mill interest, last on the 
 right, how many mills will any other number 
 of dollars and days give on the right? Or, 
 if one dollar, in 2 months on the left, give one 
 cent interest, last on the right, how many 
 cents will any other number of dollars and 
 months give, on the right ? 
 
 What is the interest on $40 for 240 days? 
 Here, the principal and time are placed for a 
 
30 RAINEY'S IMPROVED ABACUS. 
 
 demand, on the right, 1 dollar and 6 days, for 
 the same name on the left, and 1 mill, last on 
 the right, for the denomination of the answer: 
 Thus. The answer is consequently, 
 mills ; hence, one figure must be cut 
 | j off for mills, and all at the left of it 
 
 jj-gQ-Q- are cents and dollars. The answer 
 is one dollar, 60 cents and no mills. 
 The ones are placed on the two sides of 
 the line, merely to indicate the proportion in 
 the statement. They are unnecessary in 
 practice, and may be dropped in the statement 
 and calculation. 
 
 What is the interest on 20 cents 
 
 01794 
 
 3|7940 
 
 for 794 days ? Two is contained in 
 six three times, which we can di- 
 vide by, no farther, and consequent- 
 ly bring down on the left. The 
 794 are multiplied by 10, by merely appending 
 the cipher. In this case, the principal is 
 cents ; hence, the answer is 1000 times smaller 
 than if it were dollars, and is consequently 
 thousandths of cents. We strike off one fig- 
 ure for thousandths, two more for hundredths 
 of cents, and the remaining figure is cents ; 
 hence the answer, 2 cents, 94 hundredths, and 
 6 thousandths. The business man every- 
 where, would call this 3 cents. By this it is 
 seen that, when the principal is dollars, the an- 
 swer is mills, and when the principal is cents, the 
 answer is thousandths of cents. 
 
 It is quite preferable that the answers 
 should be in dollars, cents, and hundredths of 
 cents. To effect this, when the time is days, 
 and we divide by 6, as in the foregoing, we 
 
INTEREST AT SIX PER CENT. 31 
 
 may cut off and throw away at the right of 
 the answer, one figure, for mills or thousandths 
 of cents, as useless; and the answer will be 
 cents or hundredths, according to the denom- 
 ination of the principal. It must be remem- 
 bered, that this figure is thrown away at the 
 right of the answer, only when the time is 
 
 MO - 3401 
 
 What is the in- ^g_3i 102 03 
 terest on 680,20 cts. -%p -- qrnrzrtt A 
 for 93 days? $10,54,310 Ans. 
 
 Here, the factor 3, is contained in 6 twice, 
 and in 93 thirty- one times ; this 2 in 6802 is 
 contained 3401 times. We multiply the lat- 
 ter number by 31, by using the 3 only, not set- 
 ting it down, but placing its product one move 
 to the left of the unit's place, and adding the 
 two numbers. To this sum we annex the ci- 
 pher. One figure, the cipher, is thrown off, 
 which leaves the answer 10 dollars, 54 cents, 
 and 31 hundredths. The long method of mul- 
 tiplying by 31 would be quite as easy for 
 most persons as that use'd above. Or the two 
 original numbers 680,20 and 93 could be mul- 
 tiplied, and their product divided by the 6, pro- 
 ducing the same result. 
 
 It may be remarked here, that whenever it 
 is necessary to multiply by the numbers 21, 
 31, 41, 51, 61, 71, 81, 91, the left figure only, 
 may be multiplied by, and its product removed 
 one figure to the left of the unit's place, and 
 the two numbers added. Likewise to multi- 
 ply by 13, 14, 15, 16, 17, 18 and 19, use the 
 right hand figure only, placing the product one 
 move to the right, and add as before. In 
 
32 
 
 RAINEY'S IMPROVED ABACUS. 
 
 11,750 
 
 87,5 
 
 neither of the cases do we write the number 
 multiplied by. 
 
 What is the interest on 87^ cents, for 120 
 days ? In this instance we may place the 
 principal on the line as a mixed 
 number -J-, as in the annexed ex- 
 ample, or we may make the i cents, 
 5 decimals, and place it down as 87,5, 
 thus, 
 
 The result will be the same, ex- 
 cept that, in the latter case, one 
 figure must be cut off for the deci- 
 1 1,75, 00 mals; then the figures remaining 
 may be treated as usually. In the first ques- 
 tion, 6 times 2 equal 12 on the right; in the 
 2d, 6 into 12 twice, and twice 87,5 are 1750, 
 which with the 10 annexed, becomes 17500. 
 The result is the same, after cutting off the 
 decimal in the latter case, and afterwards one 
 figure in each case for the thousandths. Hence, 
 the answer, one cent and 75 hundredths. 
 
 What is the interest on 287,37i cents, for 
 80 days ? We give the i cent here the deci- 
 mal expression, .5, and will throw away one 
 figure in the answer for it. 
 
 Here multiply by 8, annex 
 the cipher, and divide by 6. The 
 answer is 3 dollars, 83 cents, 
 16 hundredths, &c., &c. 
 
 What is the interest on $200, 
 for 15 days? Ans. Fifty cents. 
 
 Jrt) 
 
 22990000 
 
 3,83,16,66^ 
 
 00 
 
 ,500 
 
SIX PER CENT. INTEREST. 
 
 33 
 
 What is the interest on $360, for Jgf " 
 
 97 days ? 
 
 $|5,820 
 
 We now come to questions in which the 
 time is months, or months and days, or years, 
 months and days. In this case 2 is used on 
 the left, because in 2 months $1 gains 1 cent 
 interest; the answer will be in cents and hun- 
 dredths, without throwing off' one figure. 
 
 What is the interest on $200, for 7 
 months ? Here we place principal and 
 time on the right, and 2 on the left ; 
 the answer is $7, and no cents. 
 
 Interest on $387,20 cents, for 10 
 months? Two into 10 five times, 
 and five times 387,20 are 19 dol- 
 lars 36 cents, and no hundredths, 
 answer 
 
 387,20 
 
 19,36,OC 
 
 What is the interest on $47, for 5 
 months? In this instance, we multi- 
 ply and divide only. 
 
 47 
 
 |235 
 
 What is the interest on $480, 
 93| cents, for 1 month ? Three- 
 fourths are made .75, in the form 
 of two decimals : hence, in the 
 
 480,93,75 
 1 
 
 2,40,46,87^ 
 
 answer, two figures are cut off for decimals, 
 two for hundredths, and two for cents. 
 
 What is the interest on 12J cents, 
 for 20 months ? This principal is 
 reduced to halves : answer, 1 cent 
 and 25 hundredths ; or li cents. 
 3 
 
 25 
 
34 
 
 RAINEY'S IMPROVED ABACUS. 
 
 13 
 
 $|2,60 
 
 $ ^002 
 11 
 
 $|22,00 
 
 What is the interest on $80, for 6 
 months and 15 days ? 15 days being 
 i of a month, the time is 6i, or \ 3 - 
 months. Hence the statement. 
 
 What is the interest on $1200, for 
 
 3 months and 20 days ? Twenty 
 
 20 
 days are |E, equal to f of a month : 
 
 hence the time 3f, or -U- months. In these 
 instances, the denominators are placed on the 
 left with the 2 months. 
 
 What is the interest on $1000, for 1 1 
 months and 12 days? Twelve days 
 are 1J> equal to | of a month: the 
 
 ^iK 1 ? nn 3 o * JL 5 
 
 b7 > UL time, consequently, is 11|, or -y- 
 months. Here, 5 times 2 on the left, equal 10, 
 or cipher on the right. The answer is $57,00. 
 What is the interest on $1500, for three 
 years, 8 months and 25 days ? Three years 
 and 8 months make 44 months : and 25 days 
 are J of a month, which makes the time 44 1 
 months. This reduced to an improper fraction, 
 in |-2. months : thus, 
 
 Twice six on the left, are 12, 
 which goes into 1500, one hun- 
 I dred and twenty five times : and 
 $|336,25 I 269X125=$336,25, the answer. 
 What is the interest on 80 dollars, for 4 
 years, 10 months and 9 days ? 
 Four years and 10 months, 
 make 58 months, and 9 days 
 make y\ of a month, which 
 
 ,4^00 ^5 
 
 269 
 
 40583 
 
 H| 23,32 Ans. 
 
 are 58 T \, equal to - 5 T \ 3 - months. 
 
 In cases where the time is an even number 
 of years, it is only necessary to multiply 
 
RULE FOR SIX PER CENT. INTEREST. 85 
 
 together principal, time and rate, and cut off 2 
 or 4 figures for cents, as indicated by the de- 
 nomination of the principal. 
 
 SUMMARY OF DIRECTIONS. 
 
 When the Time is DAYS, place the Principal and 
 Time on the right, and 6 on the left. If the 
 Principal is dollars, the answer is MILLS; here, 
 cut off one figure at the right, for mills, and two 
 more for cents : if the Principal is cents, the an- 
 swer* is thousandths of cents; here, cut off one 
 figure for thousandths, two for hundredths, and 
 two more for cents. Or, cut off one figure at the 
 right in cither case, and the answer will be cents, 
 or hundredths, according to the denomination of 
 the principal. 
 
 When the Time is MONTHS, place Principal 
 and Time on the right, and 2 on the left. If 
 the Principal is dollars, the answer is cents ; rf 
 cents, the answer is hundredths of cents. 
 
 When the Time is YEARS, place Principal, 
 Time and Rate on the right, and multiply contin- 
 uously. The answer is cents or hundredths, ac- 
 cording to Hie denomination of the principal. 
 
 When the Time is years and months, or years, 
 months and days, reduce the years to months, and 
 add all the given months: then reduce i/te days to 
 the fractional part of a month, if practicable ; 
 annex this fraction to the months : reduce all to 
 an improper fraction, and place the numerator 
 en the right, and denominator on the left. 
 
36 RAINEY'S IMPROVED ABACUS. 
 
 If the days cannot be reduced to the fractional 
 part of a month, reduce the whole time, years, 
 months, and days, or months and days, to days, 
 and divide by 6, as in other cases. 
 
 When the principal is a mixed number, reduce 
 it to an improper fraction, and place the numera- 
 tor on ike right, and denominator on the left : Or, 
 express the fraction in decimals, and cut off as 
 many figures on the right of the answer for deci- 
 mals, as indicated by the number of decimals in the 
 principal. 
 
 INTEREST AT SEVEN AND EIGHT PER CENT. 
 
 The first method given in this book for reck- 
 oning at any rate per cent., is quite applica- 
 ble to 7 and 8 per cent. Yet we may use 
 a method which requires fewer figures, in con- 
 nection with the 6 per cent, method just noticed. 
 
 In states where the rate of interest is 7 and 
 8 per cent., it will be found very convenient to 
 ascertain the interest first at 6 per cent., by 
 the short rule given, and then add | more for 
 7 per cent., or i for 8 per cent. One per cent, 
 is the i of 6 per cent.: hence we may divide 
 the interest at 6 per cent, by 6, and thus get , 
 which will be added to the former interest. 
 So two per cent, over 6, making 8, is of 6: 
 hence the interest at 6 may be divided by 3, 
 and the quotient added to the former interest. 
 
 When the rate is 12 per cent., instead of di- 
 viding by 6 as in 6 per cent, calculations, 3 
 only may be placed on the left of the line, and 
 the answer divided off as in 6 per cent, calcu- 
 lations ; because at this rate per cent, one cfol- 
 
SEVEN AND EIGHT PER CENT. INTEREST. 37 
 
 lar in three days gives one mill interest. A few 
 examples are given for illustration. 
 
 What is the interest on $350 for 160 days at 
 7 per cent.? We work as in 6 percent.; thus, 
 
 350 
 
 30 i00 8 
 
 28000 
 
 9,33,3 
 
 $ 10,88,8i 
 
 Placing 6 on the left, at 6 per 
 cent, the answer would be 
 $9,33i cents. We now place 6 
 at the left of this answer, which 
 is equivalent to multiplying the 
 answer by |, not using the nu- 
 merator on the right, and divide 
 the first answer, setting the quo- 
 tient beneath. This added gives for the an- 
 swer, $10,88 cents and 8i mills. 
 
 Again : What is the interest on $20 for 8 
 months at 7 per cent.? 
 
 Here, 2 is used on the left, the time 
 being months, and the answer 80 cents 
 is obtained. One-sixth or 13J added 
 gives 93J cents. One advantage in 
 this method is that the numbers to be 
 divided by, on the left, 
 
 20 
 
 80 
 
 ,931 
 
 on the leit, are never 
 
 large, and the fraction lost by the last division, 
 is of no practical importance. 
 
 What is the interest on $397,20 for 40 days 
 
 at 7 per cent.? 
 
 The answer in this instance 
 is $3 and nearly 9 cents. 
 
 What is the interest on 
 8 per cent? 
 
 ^,20662 
 40 
 
 264800 
 441331 
 
 $ 3,08,93,3i 
 for 371 days at 
 
38 
 
 RAINEY'S IMPROVED ABACUS. 
 
 6 
 
 ^03 
 371 
 
 11130 
 3710 
 
 ; 14,84,0 
 
 Again : 
 days at 8 
 
 In this example, after finding the 
 interest at, 6 per cent., it is divided 
 by 3, which is equivalent to multi- 
 plying by i, and the quotient is ad- 
 ded, making 14 dollars and 84 cents, 
 answer. 
 
 What is the interest on $399 for 20 
 per cent.? 
 
 1330 
 
 The answer is one dollar, 77 
 cents, and 3J mills. 
 
 What is the interest on $487,20 for 4 months 
 at 8 per cent.? 
 
 487,20 
 
 ^ ^ 2 The time being months, we di- 
 
 97440 vide first by 2, and afterward, the 
 
 32480 answer thus obtained by 3. 
 
 $ 12,99,20 
 
 What is the interest on $1500 for 211 days 
 at 12 per cent.? 
 ii00 _ 5 | We here divide by 3 only, be- 
 
 #2 
 
 211 
 
 cause, as said above, 1 dollar in 3 
 days at 12 per cent, gives 1 mill in- 
 $[105,50,0 terest. Hence, one figure is cut 
 off in the answer for mills. 
 
 When the time is months and the rate 12 
 per cent., the principal and time may simply 
 be multiplied together; because 1 dollar in 1 
 month at 12 per cent., gives 1 cent interest. Hence 
 nothing more is necessary than multiplication, 
 as 1 placed on the left of the line could serve 
 
RATES OF INTEREST. 39 
 
 no other purpose than to show the nature of 
 the statement by proportion. 
 
 What is the interest on $873 for 7 months 
 at 12 per cent.? 
 
 The answer i*in this, as in all other 
 cases of months, obtained in cents, 61 
 dollars and 11 cents. 
 
 873 
 
 _7 
 
 1,11 
 
 DIRECTIONS FOR SEVEN, EIGHT, AND TWELVE PER 
 CENTUM. 
 
 State as in cases of 6 per cent.: if the rate is 
 7 per cent., divide the answer by 6, and add the 
 quotient: if the rate is 8 per cent., divide the an- 
 swer by 3, and add the quotient. 
 
 If the rate is 12 per cent., and the time days, 
 place principal and time on the right and 3 on the 
 left: cut off one figure at the right, and the an- 
 swer will be cents or hundredths of cents. 
 
 When the time is in months, multiply principal 
 and time together, and the answer will be in cents 
 or hundredth of cents. 
 
 When the time is years, multiply principal, 
 time, and rate together; and the answer will be 
 cents, or hundredths of cents. 
 
 RATE AND FORFEITURE TABLE. 
 
 RATES OF INTEREST legalized in the several states of our union, and in 
 foreign countries, with the PENALTIES for USURY.* 
 
 FTATES. RATES.! PENALTIES FOR USURY. 
 
 Maine, 6 per cent. Forfeit of entire debt. 
 
 New Hampshire, 6 " " Three times the usury. 
 
 Vermont, 6 " '* Usury recoverable with costs. 
 
 Massachusetts, 6 " " Three times the usury forfeited. 
 
 * On all dues to the United States, 6 per cent, is charged, ever; where the 
 states legalize higher rates. 
 
 t If the rate is not mentioned in the note, interest may be collected at the 
 rate established by the law s of the state in which the transaction occurs. 
 
40 
 
 RAINEY S IMPROVED ABACUS. 
 
 RATE AND FORFEITURE TABLE. 
 
 STATES. 
 R. Island, 
 
 RATES. 
 
 6 per cent. 
 
 PENALTIES FOR USURY. 
 Forfeit of interest and usury. 
 
 New York, 
 
 7 
 
 cc cc 
 
 
 cc cc cc cc ' 
 
 
 New Jersey, 
 
 6 
 
 cc cc 
 
 
 cc cc cc cc 
 
 
 Pennsylvania, 
 
 6 
 
 cc cc 
 
 
 cc cc cc^ cc 
 
 
 Delaware, 
 
 6 
 
 cc cc 
 
 
 cc cc cc cc 
 
 
 Maryland, 
 
 6 
 
 cc cc 
 
 (1) 
 
 Such contracts void. 
 
 
 Virginia, 
 
 6 
 
 cc cc 
 
 
 Forfeit twice the usury. 
 
 
 North Carolina, 
 
 6 
 
 cc cc 
 
 
 cc cc cc 
 
 
 South Carolina, 
 
 7 
 
 cc cc 
 
 Forfeit interest, usury, and costs. 
 
 Georgia, 
 
 8 
 
 cc cc 
 
 
 Forfeit three times the usury. 
 
 
 Alabama, 
 
 8 
 
 cc cc 
 
 
 Forfeit usury and interest. 
 
 
 Mississippi, 
 Louisiana, 
 
 8 
 5 
 
 cc cc 
 
 (2) 
 (3) 
 
 Forfeit of usury and costs. 
 All such void. 
 
 
 Tennessee, 
 
 6 
 
 cc cc 
 
 
 cc cc cc 
 
 
 Kentucky, 
 
 6 
 
 cc cc 
 
 
 Costs and usury recoverable. 
 
 
 Ohio, ' 
 
 6 
 
 cc cc 
 
 
 All such void. 
 
 
 Indiana, 
 
 6 
 
 cc c 
 
 
 Forfeit of twice the usury. 
 
 
 Illinois, 
 
 6 
 
 cc cc 
 
 (4) 
 
 Forfeit interest and three tirm 
 
 BS usury 
 
 Missouri, 
 
 6 
 
 CC Ct 
 
 (5) 
 
 Forfeit interest and usury. 
 
 
 Michigan, 
 Arkansas, 
 Florida, 
 Wisconsin, 
 
 7 
 6 
 8 
 
 7 
 
 cc cc 
 
 (6) 
 
 (7) 
 
 Forfeit one-fourth debt and us 
 Whole usury forfeited. 
 Forfeit interest and usury. 
 Forfeit three times usury. 
 
 mry 
 
 Iowa, 
 
 7 
 
 cc cc 
 
 (8) 
 
 cc cc cc cc 
 
 
 Texas, 
 
 10 
 
 cc cc 
 
 
 All such void. 
 
 
 Dist. Columbia, 
 
 6 
 
 cc cc 
 
 
 cc cc cc 
 
 
 England, 
 
 5 
 
 cc cc 
 
 
 Forfeit three times the debt. 
 
 
 France, 
 
 5 
 
 cc cc 
 
 
 
 
 Ireland, 
 
 6 
 
 cc cc 
 
 
 
 
 Canada, 
 
 6 
 
 cc cc 
 
 
 
 
 Nova Scotia, 
 
 6 
 
 cc cc 
 
 
 
 
 W. Indies, 
 
 8 
 
 cc cc 
 
 
 
 
 Constantinople, 
 
 30 
 
 cc cc 
 
 
 
 
 MAKING AND TRANSFERRING NOTES. 
 
 In closing the article on interest, it may be well to give a few practical 
 directions on making and transferring notes. Much litigation arises from 
 inattention to the following requirements of law : 
 
 1. A promissory note is an instrument of writing in which the promisor 
 or maker pledges the payment of money or property to a second person, 
 at or before a specified time, in consideration of equivalent value received. 
 
 2. The sum of money or property for which the promisor gives the note, 
 is called the " face of the note," and after being expressed in figures at the 
 beginning of the note, should be written in words in the body of the same 
 
 3. The words " value received " should always be found in the body of 
 
 (1) Contracts in tobacco may be as high as 8 per cent. 
 
 (2) By contract as high as 10 per cent. 
 
 (3) By agreement as high as 10; bank interest 6 per cent. 
 
 (4) By agreement as high as 12. 
 
 (5) By agreement as high as 10. 
 
 (6) Any rate not above 10 per cent. 
 
 (7) By contract as hig'h as 12. 
 
 (8) By agreement as high as 12 per cent. 
 
LAWS REGULATING NOTES. 41 
 
 a note, as the laws require that money shall be paid only for a " con- 
 sideration " or equivalent. Without this, notes are said to be invalid or 
 worthless. 
 
 4. The individual who makes the note is called the drawer or giver ; the 
 person to whom it is given is called the payee ; and the person who has it 
 in legal possession, is called the holder. 
 
 5. The payee of a note may sell or transfer it to a third person, if it be 
 written payable " to order," or " to bearer ;" and this third person or holder 
 may sue and collect as if he had been the original payee. Such a note is 
 called negotiable, because it can be traded by the payee, and made paya- 
 ble to such person as he may order. 
 
 6. The law requires the holder of a negotiable note to indorse it, or write 
 his name on the back, if he wish to sell it ; provided that such note be 
 transferable, or " payable to order." If the holder is unable to collect the 
 note of the drawer, then the indotser is responsible, and can be made to 
 pay it. The holder of a note which is made payable "to bearer," can 
 transfer without indorsing it ; and is, in this event, not liable for it. In the 
 transfer of such notes there is said to be no recourse. Thus, if a bank- 
 note is not indorsed by the individual who pays it, he is no longer liable to 
 lose it, if it prove worthless. 
 
 7. A note made payable to a particular individual without the words 
 payable " to order," or "to bearer," cannot be negotiated or traded to an- 
 other ; nor can another individual, except in the name of the payee, col- 
 lect it. 
 
 8. A note specifying no time for payment, must be paid on demand. 
 Such are generally called " due-bills." 
 
 9. Conventional usage, and in some instances law, has established, that 
 a note shall not be collected until three days after it is due ; and interest 
 is calculated on these three days, as on the rest of the time. A note be- 
 coming due on Sunday, with these three days included, should be paid on 
 Saturday. Three days thus given, is to allow for all exigencies ; and are 
 called " days of grace," because they are given gratuitously. Grace means 
 gift. 
 
 10. Maturity of a note is the time specified for its payment. If a note 
 is not paid at maturity that has been transferred, the holder must legally 
 notify the indorser of the fact, or the indorser is released from his liability. 
 
 11. Interest cannot be charged on a note paid at maturity, without it 
 has been specified ; the words, " with interest," being inserted. But 
 when a note not containing these words, is not paid until after maturity, 
 the rate of interest legal in the state can be charged from maturity. 
 
 12. Notes bearing interest without the rate being inserted, bear the legal 
 interest of the state in which drawn. Any agreement for a rate of inter- 
 est less than the legal interest, must ba specified, or legal interest can be 
 collected. 
 
 13. Notes for any commodity of merchandise cannot be negotiated ; and 
 if payment is not made at the time specified, the holder may recover the 
 amount in money. In such cases the three days grace are not allowed. 
 
 14. When two or more persons give a note conjointly, the payee or 
 holder may collect it of either or any of them. 
 
 PAYMENT ON NOTES. To compute interest on notes, we must ascertain 
 the time which elapses between the period when interest commences, and 
 
42 RAINEY'S IMPROVED ABACUS. 
 
 that on which the payment is made, by subtracting the former from the lat- 
 ter date,* which is done thus: 
 
 What is the interest on a note for $500, dated May 20, 1843, and paid 
 January 19, 1845 ? 
 
 After arranging the former time tinder 
 the ' latter thus, if the number of days in 
 the lower line is larger than that in the 
 
 yrs. mos. days. 
 1845 " 1 " 19 
 1843 " 5 " 20 
 
 Time, 1 " 7 " 29 
 
 upper, 30 days must be added to the upper 
 line, and the subtraction made from the 
 whole number above, and the remainder 
 set under the days. One is carried to the 
 
 act uuuci me uuys. wiie is uarncu LU me 
 
 lower line of months. If this number of 
 months is larger than that above, 12 must be added above and the subtrac- 
 tion continued as before. It will be observed here, that the months are 
 placed down according to the order they occupy in the year. May is the 
 5th month; hence we use 5 as the number ; so is 1 used lor January, it be- 
 ing the first month. 
 
 PARTIAL PAYMENTS. 
 
 Below, we give the Ohio rule for casting the 
 interest on partial payments, which is the 
 method used in Indiana, Kentucky, and most 
 of the states of the union. 
 
 The rule, and the calculation to illustrate 
 its application, are extracted from Swan's 
 Treatise. 
 
 " The Ohio rule for calculating partial pay- 
 ments, is as follows : Where payments exceed- 
 ing the interest are made after the debt is due : 
 In such case interest should be calculated on 
 the debt up to the time of payment, and the 
 principal and interest then added together, 
 and the payment subtracted from the total. 
 Subsequent interest should be computed on 
 the balance of principal thus found to be due. 
 
 "Where the payment is less than the inter- 
 est due, the surplus of interest must not be 
 added to the principal ; but interest continues 
 on the former principal, the same as if no 
 
 * The day on which a note is dated, and that on which it becomes due, 
 hould not both be reckoned. The former is excluded among business men. 
 
PARTIAL PAYMENTS. 43 
 
 payment had been made, until the period 
 when the payments added together, exceed the 
 interest due; and then the surplus of pay- 
 ments is to be applied towards discharging the 
 interest. For instance, upon a note for $100, 
 payable in one year with interest, if a pay- 
 ment of 10 dollars is made at the end of two 
 years, and 10 dollars at the end of four years, 
 and 19 dollars at the expiration of six years ; 
 here interest on the whole amount of the note 
 should be calculated up to the time of the 
 payment of the 19 dollars, and then the sever- 
 al payments should be added together, and 
 deducted from the amount of all the principal 
 and interest; the balance would be the amount 
 due, and upon which interest should be after- 
 wards computed. 
 
 The following calculations will illustrate 
 the rule in the text: A., by his note, dated 
 Jan. 1st, 1840, promises to pay to B. 1000 dol- 
 lars, in 6 months from date, with interest from 
 the date. On this note are the following en- 
 dorsements : Received, April 1, 1840, 24 dol- 
 lars; Aug. 1, 1840, 4 dollars ; Dec. 1, 1840, 6 
 dollars; Feb. 1, 1841, 60 dollars ; July], 1841, 
 40 dollars; June 1, 1844,300 dollars ; Sept. 1, 
 1844, 12 dollars ; Jan. 1, 1845, 15 dollars ; Oct. 
 1, 1845, 50 dollars; and the judgment is to be 
 entered Dec. 1, 1850. 
 
44 RAINEY'S IMPROVED ABACUS. 
 
 CALCULATION. 
 
 The principal sum carrying interest from January 1, 
 
 1840, $1000 00 
 Interest to April 1, 1840, 3 months, 15 00 
 
 Amount, 1015 00 
 Paid April 1, 1840, a sum exceeding the interest, 24 00 
 
 Remainder for a new principal, 991 00 
 
 Interest on- 991 from April 1, 1840, to February 1, 
 
 1841, 10 months, 49 55 
 
 Amount, 1040 55 
 
 Paid August 1, 1840, a sum less than the in- 
 terest due, $4 00 
 Paid December 1, 1840, do. do. 6 00 
 Paid February 1, 1841, do. greater do. 60 00 
 
 70 00 
 
 Remainder for a new principal, 970 55 
 Interest on 970 55 from February 1, 1841, to July 1, 
 
 1841, 5 months, 24 26 
 
 Amount, ~ 994 81 
 
 Paid July 1, 1841, a sum exceeding the interest, 40 00 
 
 Remainder for a new principal, 954 8 1 
 Interest on 954 81 from July 1, 1841, to June 1, 
 
 1844, 2 years 11 months, 167 00 
 
 Amount, 1121 81 
 
 Paid June 1, 1844, a sum exceeding the interest, 300 00 
 
 Remainder for new principal, 821 81 
 Interest on $821 81 from June 1, 1844, to October 1, 
 
 1845, 1 year and 4 months, 65 75 
 
 Amount, " 887 56 
 
 Paid September 1, 1844, a sum less than the in- 
 terest due, $12 00 
 
 Paid January 1, 1845, do. do. 1500 
 
 Paid October 1, 1845, do. greater, with the two- 
 last payments, than the interest then due, 50 00 
 
 77 00 
 
 Remainder for new principal, 810 56 
 
 Interest on $810 56, from October 1, 1845, to Decem- 
 ber 1, 1850, the time when judgment is to be en- 
 tered, 5 years and 2 months, 251 30 
 
 Judgment rendered for the amount, $1061 86 
 
PARTIAL PAYMENTS. 45 
 
 The following is the rule of the Supreme Court of the 
 United States, as given by Chancellor Kent, Johnson's Chan- 
 cery Reports, vol. 1st, page 17; and is adopted by most of the 
 States of the Union, among which are Massachusetts and 
 New York: 
 
 SUPREME COURT RULE. 
 
 I. " The rule for casting interest, when partial pay- 
 ments have been made, is to apply the payment, in the 
 first place, to the discharge of the interest then due. 
 
 II. " If the payment exceeds the interest, the surplus 
 goes toward discharging the principal ; and the subse- 
 quent interest is to be computed on the balance of the 
 principal remaining due. 
 
 III. " If the payment be less than the interest, the 
 surplus of interest must not be taken to augment the 
 principal ; but interest continues en the former principal 
 until the period when the payments, taken together, exceed 
 the interest due, and then the surplus is to be applied 
 toward discharging the principal ; and interest is to be 
 computed on the balance as aforesaid" 
 
 A. gave to B. his note for 12,000 dollars; at the expiration 
 of three months he paid 2,000 dollars ; in three months more 
 6,000, and at the expiration of three months more, 3,000 dol- 
 lars: what did he pay to B. when the note was taken up at 
 the close of the year, the note being made on the 1st day of 
 January ? We reckon these payments by the Supreme Court 
 Rule. 
 
 Principal $12,000 
 
 Interest on the whole for three months - - - ,180 
 Amount of principal and interest ----- 12,180 
 
 First payment to be deducted ------- 2,000 
 
 Balance due after first payment ------ 10,180 
 
 Interest from 1st to 2d payment, 3 months - ,152.70 
 
 Amount to be reduced by 2d payment - - - 10,332.70 
 Second payment to be deducted ------ 6,000. 
 
 Balance due after 2d payment 4,332.70 
 
 Interest from 2d to 3d payment, 3 months - 64.99.05 
 
 Amount to be reduced by 3d payment - - - 4,397.69.05 
 
 Third payment to be deducted 3,000. 
 
 Balance due after 3d payment - ------ 1,397.69.05 
 
 Int. from 3d pay't till settlement, 3 months _ 20.96.53.57$ 
 
 Balance due on settlement 1,418.65.58.57$ 
 
 4 
 
46 RAINEY'S IMPROVED ABACUS. 
 
 The following is called the Commercial Rule, and is adopted 
 by Vermont: 
 
 COMMERCIAL OR VERMONT RULE. 
 
 Find the amount of the whole debt until the time of 
 settlement ; then find the amount of each payment from 
 the time of payment until the time of settlement ; add 
 these, and subtract the sum from the former amount : 
 the remainder will be the sum due. 
 
 For all payments made within one year, this rule is iden- 
 tical with that of Connecticut, which follows : 
 
 CONNECTICUT RULE. 
 
 I. " Compute tlie interest on the principal to the time 
 of the first payment ; if that be one year or more from 
 the time the interest commenced, add it to the principal, 
 and deduct the payment from the sum total. If there be 
 after payments made, compute the interest on the balance 
 due to the next payment, and then deduct the payment 
 as above; and in like manner from one payment to 
 another, until all the payments are absorbed ; provided 
 the time between one payment and another be one year 
 or more 
 
 II. "If any payments be made before one year's 
 interest has accrued, then compute the interest on the 
 principal sum due on, the obligation, for one year, add it 
 to the principal, and compute the interest on the sum, 
 paid, from the time it was paid up to the end of the 
 year ; add it to the sum paid, and deduct that sum from 
 tJie principal and interest added as above. 
 
 III. " If a year extends beyond the time of payment, 
 then find the amount of the principal remaining unpaid 
 up to the time of settlement, likeioise the amount of the 
 indorsements from the time they were paid to the time of 
 settlement, and deduct the sum of these several amounts 
 from the amount of the principal. 
 
 11 If any payments be made of ct sum less than the 
 interest arisen at the time of such payment, no interest is 
 to be computed, but only on the principal sum for any 
 period.' 9 KIRBY'S REPORTS. 
 
PHILOSOPHY OF RATIO. 47 
 
 Ignorance of the correct method of calcula- 
 ting interest on partial payments, is the cause 
 of much litigation. Hence, it behoves men 
 to remember, that interest should be reckoned 
 till the time of the first payment, and added 
 to the principal, and the payment deducted ; 
 provided the payment is greater than the in- 
 terest that has accrued. But if the interest is 
 greater than the payment, the payment must 
 be set aside, and the interest reckoned to 
 another payment ; or continuously from one 
 payment to another, till the sum of the pay- 
 ments shall exceed the sum of the interest 
 accrued. Then the several sums of interest 
 should be added to the principal, and the sum 
 of the several payments deducted. The re- 
 mainder will be a new principal on which in- 
 terest runs till the next current payment, or 
 till the debt^is paid, or judgment rendered. 
 This method will stand in the courts, of the 
 great majority of the States in the Union. 
 
 RATIO AND PROPORTION. 
 
 That the principles of subsequent statements 
 may be understood, we introduce a few re- 
 marks on Ratio and Proportion. The princi- 
 ples and various ramifications of this subdi- 
 vision of numbers, could not be well developed 
 and elucidated, in less space than is occupied 
 by this entire volume. Hence, we attempt 
 only its outline. 
 
 Ratio is the relation that exists between different 
 
48 RAINEY'S IMPROVED ABACUS. 
 
 quantities and numbers of similar things ; and is 
 ^ always expressed by an abstract number. Ten 
 apples are twice as many as five apples : hence, 
 the ratio or relation is 2. We have here two 
 different numbers of the same object : but it 
 could not be said that, ten apples were twice 
 as many as five biscuits ; because there is no 
 relation between one apple and one biscuit : 
 neither can one be said to be larger than the 
 other. Again, 15 yards of cloth are five times 
 as many as 3 yards of cloth : here the ratio is 5. 
 Twenty gallons of melasses are 4 times as 
 many as 5 gallons : but 20 gallons of melasses 
 are not 4 times as many as 5 gallons of rum. 
 Six bushels \vheat are one fifth times as 
 many as 30 bushels : that is, the ratio is \. 
 Hence, the fact assumed in the outset is proven, 
 that Ratio is the relation between similar things. 
 __J^^xatios_make a proportion ; that is, as 
 many times greater or less as & second thing is 
 than the first, so many times greater or less is a 
 fourth than the third. These relations are 
 ascertained, by first ascertaining the ratio that 
 exists between the two numbers or quantities 
 in the proposition, which are alike. For in- 
 stance : If 4 yards of cloth cost $5, what 
 will 12 yards of cloth cost? Here the ratio 
 between the 4 and 12 yards must be obtained 
 first. This is done by using a vertical line, 
 which is considered the fulcrum of a scale or 
 balance, thus, 
 
 The 12 yards are placed on the right, 
 
 and the 4 yards opposite, on the left. 
 
 Hence, as in the two ends of the scale, 
 
 we compare their value. Four is contained 
 
SIMPLE PROPORTION DIRECT. 49 
 
 in 12 three times : hence, the ratio is 3. Now, 
 the four yards on the left, are equal in value 
 to 5 dollars ; which to balance the 4 yards, 
 must be placed, likewise, last on the right. 
 The ratio being an abstract number, and being 
 on the right with the dollars, may be multiplied 
 into the dollars : hence, 3 times $5 are 15 
 dollars ; not 15 bushels, or 15 of any thing else, 
 except the denomination into which the ab- 
 stract ratio is multiplied. 
 
 From this, the philosophy of the statement 
 seems to require, that the price or denomina- 
 tion of the answer, be placed last, on the right, 
 where it can be multiplied by the ratio between 
 the two quantities above, and increased or 
 decreased accordingly, as the ratio is an integer 
 or fraction. From this, tao^tojce^j^ju^ the_ 
 theory of equIIi5ghH^ 
 
 extent, equals fKe~price^or nameTof answer, 
 must be placed opposite, or on the left : then the 
 number which is to be compared with this 
 supposed quantity on the left, must be placed 
 on the right, directly opposite the term on the 
 left, for the purpose of ascertaining the ratio. 
 When stated thus, it is not essential that the 
 ratio be actually obtained, by dividing one upper 
 term into the other; for the position indicates 
 the ratio, and the numbers may be canceled by 
 using one of the numbers with either of the 
 other two. 
 
 In applying this statement to questions in 
 Profit and Loss, the conditions of the question 
 must be strictly noticed : cost price, compared 
 with cost price; par per cent., with par per 
 cent. ; reduced with reduced, and advanced 
 3 
 
50 RAINEY'S IMPROVED ABACUS. 
 
 with advanced per centum. By this means, 
 and the directions following, the statements 
 made in Profit and Loss, become rational and 
 easy. 
 
 In questions where any of the terms are 
 fractional, or mixed numbers, such mixed num- 
 bers may be reduced to improper fractions, 
 and their numerators located in such position 
 on the line, as otherwise the whole number 
 would occupy ; with all respective denomina- 
 tors opposite. 
 
 If 7i yards cloth are worth $20, what will 
 18J yards come to? The term of demand, or 
 which is connected with the demand, is 18-J 
 yards. This must be placed on the right of the 
 line first, in the form of \ 5 . The numerator 
 75 will consequently occupy the right, thus, 
 
 The 7i belongs to the left, and 
 being made y , we place the nume-* 
 
 05 
 
 9 1 50 
 
 rator 15 on the left : now the term 
 in which the answer is required, 
 $20, is placed last on the right : so 
 
 that being multiplied by the ratio between the 
 two quantities above, it will be made more or 
 less. The question has now lost its fractional 
 form ; and being easily stated, can be more 
 easily wrought. Thus, if the terms were such 
 that they could not be canceled, they might be 
 multiplied on each side ; and the right divided 
 by the left, with very little trouble. 
 
 If i of 6 yards cost $3, what will i of 20 
 yards cost ? Here, i of 20 is the demand, and 
 is placed on the right, by its numerators, 1 
 and 20 : \ of 6 is the supposition of the same 
 
SIMPLE PROPORTION DIRECT. 51 
 
 name, and is placed opposite, by its numerators : 
 the price, $3, is placed last on the right; thus, 
 
 This may be proven by saying, 
 if i of 20 cost 7i, what will i of 6 
 cost? J of 6 is the demand; i of 
 
 20 the same name, and $7i the 2 $ 
 
 term of answer, all of which are 
 located accordingly, by their nu- 
 merators; thus, 
 
 3 
 1 
 
 20 
 
 The result is three dollars, and the 
 question is proven. 
 
 This department of numbers is called " the 
 rule of three, 51 because three terms are given 
 and used in a statement to find a fourth. 
 When this fourth term is ascertained, the pro- 
 position is complete ; so, also, is the proportion. 
 
 The rule of three having three terms given 
 to find a fourth, we conclude that, two of these 
 being either means or extremes, we may divide 
 either couplet by the remaining mean or ex- 
 treme, and find such mean or extreme as con- 
 stitutes the fourth term. 
 
 It is a doctrine in proportion, that the product 
 of the two means equals the product of the two ex- 
 tremes. If this is true, it is evident that the 
 product of either the means or extremes, is 
 the result of the multiplication of the two 
 terms together as factors. 
 
 Suppose the extremes be multiplied and the 
 product divided by one of the means; it is evi- 
 dent that the other mean will be obtained, and 
 
52 RAINEY'S IMPROVED ABACUS. 
 
 that if it be multiplied with the first mean, their 
 product will equal that of the extremes. Or, 
 if the product of the means be divided by one 
 of the extremes, the other extreme will be ob- 
 tained, which multiplied with the given ex- 
 treme will yield a product equal to that of the 
 means. 
 
 In the proportion 2 : 4 : : 8 : 16 (two to four 
 as eight to sixteen), we multiply 8 by 4, mak- 
 ing 32; this divided by 2, one of the extremes, 
 gives 16, the other extreme: or, divided by 16 r 
 gives the extreme 2. If, again, the two ex- 
 tremes, 2 and 16, be multiplied, and the pro- 
 duct, 32, divided by 4, one of the means, the 
 other mean, 8, is obtained : or, if this product 
 be divided by 8, the other mean, 4, is obtained. 
 This law will be found necessary in the expla- 
 nation of Compound Proportion, by cause and 
 effect. 
 
 It will require care and attention in the 
 learner to distinguish the demand at times, 
 the language of questions being frequently so 
 transposed as to present seemingly new issues 
 to those not acquainted with, or accustomed 
 to, analyzing questions before stating them. 
 
 The general nature of every question must 
 be understood before much can be done by 
 any process of solution. And this is the only 
 service that analysis can render, the elucidation 
 of the general bearings of the problem. 
 
 We have always been opposed to the me- 
 chanical routine by which persons have gone 
 on from one question to another, and wrought 
 some, merely because others were wrought so. 
 So long as this method of solving questions 
 
SIMPLE PROPORTION DIRECT. 53 
 
 by a rule, whose principles may at first have 
 been understood, is practiced among those 
 seeking a first knowledge of the science, we 
 may not look for such a knowledge of the sub- 
 ject as will enable them to apply the princi- 
 ples to the great every-day transactions of 
 life. 
 
 We shall now solve a variety of problems 
 and prove them ; that after the path is pointed 
 out, the pupil may pursue it with some degree 
 of pleasure. 
 
 If 10 yards of cloth .cost $16, what will 15 
 yards cost? 
 
 The demand, 15 yards, is placed 
 on the right, and the term of the 
 same name, 10 yards, opposite; 
 while the price, 16 dollars, is 
 
 24 
 
 placed last on the right, where it can through 
 multiplication, be increased by the ratio of the 
 two numbers of yards. We use the factor 5 
 into 10 and 15. The answer is 24 dollars. 
 
 We prove this by saying, if 15 yards cost 
 24 dollars, what will 10 cost? 
 
 Thus it is proven that 10 
 yards cost the 16 dollars assumed 
 
 in the outset. ft 1A 
 
 jfjUO 
 
 We may now state yet differently, and say, 
 if $16 buy 10 yards, how many yards will 
 $24 buy? thus, 
 
 Here 24 dollars become the 
 demand, and 16 the same name; 
 while 10 yards is the denomina- 
 tion of answer, and is conse- 
 
 15 yds 
 
 quently placed on the right. The ratio is ob- 
 
54 RAINEY'S IMPROVED ABACUS. 
 
 tained between the dollars, and multiplied into 
 the 10 yards, increasing them to 15 yards. 
 
 Again: if $24 buy 15 yards, 
 
 how many will 16 buy? 
 10yds. 
 
 The first form of the question above would 
 be stated and solved thus by the old method: 
 10 : 15 : : 16 : x. The second and third 
 terms, 15 and 16, would be multiplied togethei 
 and their product divided by the first; thus, 
 
 10 : 15 :: 16 
 15 
 
 80 
 16 
 
 10)240 
 
 24 dolls. 
 
 By this mode of pro- 
 ceeding, the figures are 
 accumulated to a large 
 number by first multiply- 
 ing up, and then dividing 
 down again; whereas both 
 operations might be dis- 
 pensed with, and the term 
 of answer multiplied simply by the ratio be- 
 tween the 10 and 15. 
 
 It appears singular that any intelligent au- 
 thor should teach that it was necessary in such 
 a question as the one just wrought, to mul- 
 tiply the 16 dollars by the 15 yards, which is 
 within itself impossible ; for such an operation 
 has no meaning. Fifteen yards times 16 dol- 
 lars will make neither 240 yards nor 240 dol- 
 lars. It may be urged that it is multiplying 
 the dollars by the abstract terms of a fraction, 
 which represents the ratio. This may be ad- 
 mitted in the case of those prime numbers 
 whose ratio cannot be reduced to a single ex- 
 pression. Even then we would prefer express- 
 ing it by a positive fraction, in a fraction's 
 
SIMPLE PROPORTION DIRECT. 55 
 
 form, or by a mixed number, either of which 
 would indicate how many times the dollars, or 
 other denomination of answer, was to be 
 taken. For when the other process is pursued, 
 the whole idea of ratio between the number 
 multiplied and divided by, is lost; and the 
 pupil multiplies and divides just because the 
 rule so teaches him. But tell him to get first 
 the ratio between two numbers or quanti- 
 ties, and then to multiply the denomination in 
 which the answer is required, by it, thus in- 
 creasing or decreasing it, and he understands 
 what he is doing. He knows that if the ratio 
 be larger than 1, he will have an answer as 
 much larger than the last term as this ratio 
 is greater; and if the ratio be smaller than 
 unity, his answer will be smaller than the 
 term to be multiplied by such ratio. But that 
 the second and third shall be multiplied to- 
 gether to a product, but to be divided down 
 again by the first, is an absurdity that in its 
 very form excludes the idea that we should 
 keep constantly before us, that of ratio. It is 
 not that the boy could not understand ratio, 
 but that the mode of applying it had so little 
 affinity to the principle, that when the appli- 
 cation was commenced in multiplying and di- 
 viding, he found himself in a new field with- 
 out any discern able directions to his point of 
 destination. Such absurd regulations give the 
 young an idea that nothing can or must be at- 
 tempted beyond the comprehension of mere 
 mechanical landmarks; while if he were told 
 that he must get the ratio between the two 
 numbers, and increase or decrease another by 
 
56 
 
 RAINEY'S IMPROVED ABACUS. 
 
 it, the mist would flow from his eyes, and de- 
 velop the sunlight of unclouded truth. 
 
 If 5 sheets of paper make 150 pages of a 
 book, how many sheets are required to make 
 800 pages? 
 ' i 800 
 
 It requires 26| sheets. 
 
 26f 
 
 If 26f sheets make 800 pages, how many 
 will 5 sheets make ? 
 
 In this instance the demand 
 is 5 sheets, and the same name 
 26| sheets ; the latter is placed on 
 the left after being reduced to 
 150 pages. 8_o. That is, the numerator is 
 placed on the left, where the term of same 
 name should go, and 3, the demominator on 
 the right. Knowing that 26| sheets make 800 
 copies, and that in the use of this 800 we must 
 find how many five sheets will make, we place 
 the 800 last on the right, that the answer may 
 be in pages. 
 
 If 240 feet of lumber cost 9 dollars, how 
 many feet can be purchased at 18f dollars? 
 
 A fp 25 1 The two terms to be com- 
 
 ~^ 2 pared are dollars: hence 18f 
 
 dollars is the demand. The 
 answer is the number of feet 
 
 !500 
 
 which this sum buys, 500 feet. 
 
 If a chicken that cost 5 cents is sold for 8 
 cents, what is the gain per 100 cents? 
 
 1002 
 
 We know that 5 cents gains 3 
 cents, and inquire, what will 100 
 cents or per centum gain? The an- 
 swer is 60 per cent. 
 
SIMPLE PROPORTION DIRECT. 57 
 
 If a perpendicular wall 80 feet high cast a 
 shadow at noon 60 feet wide, how wide a 
 shadow will a perpendicular church steeple 
 cast, which is 240 feet high ? 
 
 Here the terms to be compared 
 are not the wall and the steeple, 
 w r hich are both perpendicular, but 
 the separate hights of these two 
 
 60 
 
 180 
 
 dissimilar objects. The answer is desired in 
 width or extent; this is consequently placed 
 last on the right, where it can be multiplied by 
 the ratio between the different hights. An- 
 swer, 180 feet wide. 
 
 If a shadow 180 feet wide be cast by a 
 steeple 240 feet high, how high must the 
 steeple or other perpendicular object be that 
 will cast a shadow 60 feet wide? 
 
 Here the widths are com- $ ^01^0 
 pared, and the answer is ob- ^0 8 
 
 tained in hight. gO 
 
 How many gallons of oxygen will be neces- 
 sary to make 720 gallons of water, if 9 gal- 
 lons of water require 8 gallons of oxygen ? 
 
 We compare water with water. 
 
 The answer is 640 gallons oxygen. 
 
 b40 
 
 If nine gallons of water require 1 gallon of 
 hydrogen, how much hydrogen is required to 
 make 720 gallons of water? 
 
 ,W 8 
 The answer is 80 gallons of \ 
 
 hydrogen. 
 
 If after I see the flash of a cannon, I hear- 
 
58 RAJNBY'S IMPROVED ABACUS. 
 
 the report in 4 minutes, how far will it be off, 
 if sound flies at the rate of 1142 feet per 
 second ? 
 
 11240 
 3 1142 
 
 1760 
 
 51 TT 
 
 Four minutes make 240 seconds, 
 which is the demand, while 1 se- 
 cond is the same name, and 1142 
 feet, the distance which sound flies 
 in the one second, is the term of 
 
 19 
 
 16 
 
 171 
 
 answer. We say, how many 
 yards will these feet make, if 3 feet opposite 
 make 1 yard : and, again how many miles 
 will these yards make if 1760 yards opposite 
 make one mile? Thus three distinct propor- 
 tions are combined in one, which produces no 
 inconsiderable economy in the use of figures. 
 If | of a pound of butter costs 4j cents, 
 what will li pounds cost? 
 
 The demand is placed on the 
 right as |, the same name on the 
 left as f, and the price last on right 
 as y. It is seen that the numera- 
 tor of the same name occupies the 
 left, and its denominator the right. 
 None of the numbers in this question can be 
 canceled; yet no sane man will dispute that 
 the question is stated in much better form than 
 by the old rules; for here the pupil sees at a 
 glance what terms must be multiplied togeth- 
 er, and what divided by, from their very loca- 
 tion on the two sides of the line ; which is not 
 the case in the old method. Then, the state- 
 ment being rational and easy, it is far prefer- 
 able, though not a figure can be canceled. 
 
 If 4j pounds of wool cost 30 cent/, what 
 will 18| pounds come to? 
 
SIMPLE PROPORTION DIRECT. 59 
 
 This example may easily be 
 proven by using the answer, in 
 making inquiry with regard to 
 some other portion of the ques- 
 tion. 
 
 "A hare starts 12 rods before a hunter, and 
 ecuds away at the rate of 10 miles an hour: 
 now, if the hunter does not change his place, 
 how far will the hare get before him in 45 
 seconds?" 
 
 The demand is here 45 seconds, which we 
 place on the right : now, that we may reduce 
 the hours to seconds and use them on the left, 
 we place opposite the 45, 60 seconds, which 
 make a minute ; while the one minute is 
 placed on the right last, as the denomination 
 of answer. But that we may continue on, 
 and reduce these minutes to hours, we place 
 opposite this one minute 60 minutes, which 
 make an hour, and which is equivalent to 10 
 miles running of the hare. As this 60 minutes 
 equals the 10 miles running, we must place the 
 latter last on the right; for we wish the an- 
 swer in distance. Now, it is evident that if 
 the question were wrought without going far- 
 ther, the answer would be in miles; but wish- 
 ing it in rods, we place 1 mile opposite 10, and 
 8 furlongs on the right, saying, how many fur- 
 longs will all the miles on the right make, if 
 1 mile makes 8 furlongs ? We say again, 
 how many rods will all of these furlongs 
 make, if 1 furlong opposite the 8, make 40 
 rods ; which being the term of answer, is 
 placed last on the right; thus, 
 
60 
 
 RAINEY'S IMPROVED ABACUS. 
 
 40 
 12 
 
 152 rods. 
 
 The 12 rods which the 
 hare had in the start, added 
 to this, makes the answer, 52 
 rods. 
 
 Again: "If a dog by running 16 miles an 
 hour gain on a hare 6 miles every hour, how 
 long will it take him to overtake her, if she 
 has 52 rods the start?" 
 
 The demand is 52 rods, and 40 rods 
 opposite equal 1 furlong, 8 furlongs 
 equal 1 mile, 6 miles equal 60 minutes 
 of running, and 1 minute equals 60 
 seconds : hence the answer is 97i 
 seconds. 
 
 Again: "A hare starts 12 rods before a 
 greyhound, but is not perceived by him until 
 she has been up 45 seconds; she scuds aw r ay 
 at the rate of 10 miles an hour, and the dog 
 after her at the rate of 16 miles an hour: 
 what space will the dog run before he over- 
 takes her?" 
 
 40 t 52 
 
 6|i 
 
 160 
 
 60 
 
 60 
 60 
 
 195 
 
 1 
 
 16 
 
 40 
 
 What will 97i or 'f 5 seconds be, if 
 60 seconds make 1 minute, and 60 
 minutes equal 16 miles running of 
 the dog, and 1 mile equal 8 furlongs, 
 and 1 furlong equal 40 rods, the de- 
 nomination of the answer? The dog 
 will run 1381 rods 
 
 These questions cannot be fully elucidated 
 
SIMPLE PROPORTION IN FRACTIONS. 
 
 61 
 
 in this little work, and are given merely to in- 
 dicate the capabilities of this beautiful system 
 of statement and solution. 
 
 If of a 
 5 
 
 pound of rice feed 3 men, how 
 
 many will ~ pounds feed? 
 
 We treat complex fractions in 
 the statement as all others, plac- 
 ing the demand on the right, same 
 name on the left, etc. In the de- 
 mand, the numerator of the nume- 
 rator is placed on the right, and 
 the numerator of the denominator 
 on the left, with all respective de- 
 nominators opposite their numerators. 
 
 of 2 yards of muslin 
 
 2A 
 
 30 
 
 If 
 
 11 15 
 
 4f 
 
 be worth 
 
 3 
 
 yards of gold lace, how many yards of gold 
 
 40 
 lace will pay for of 12 yards of muslin ? 
 
 j 402 
 
 4 
 
 Five times 3 equal 15; 3 into 
 9 three times, and 3 times 2 equal 
 6; 5 into 10 twice, and into 25 
 five times. 4X4X2X2X12=768, 
 which divided by 5 gives 153f 
 yards of gold lace for the answer. 
 
 5768 
 
 153J 
 
 All such propositions as these, though not 
 practical in their bearing, will, nevertheless, 
 5 
 
62 RAINEY'S IMPROVED ABACUS. 
 
 afford interesting entertainment to those study- 
 ing for the mere beauty of theory, while it is 
 believed that a sufficient number of examples 
 have been given to meet all practical purposes. 
 We are aware that many authors consider 
 the treatise of Proportion under two heads, as 
 superfluous. It may be superfluous when 
 treated mechanically ; but when cause and ef- 
 fect, as the bases of these principles, are de- 
 veloped, they induce the division on natu- 
 ral, rational, and necessary grounds. 
 
 SUMMARY OF DIRECTIONS FOR DIRECT PROPORTION. 
 
 Ascertain first the term of Demand: 
 Place the Demand Jtrst on the right : 
 Place the term of the Same Name opposite the 
 Demand, on the left: 
 
 Place the term in which the answer is required, 
 last on the right, and the answer will be in the 
 same denomination. 
 
 In all fractional terms, the Numerator must oc- 
 cupy the side of the line ordinarily assigned to the 
 integer. 
 
 PROFIT AND LOSS. 
 
 Profit and Loss, as constituting a depart- 
 ment of the Arithmetic, relate to the various 
 gains and losses of general business transac- 
 tions. They are reckoned in two ways: spe~ 
 
THEORY OF PROFIT AND LOSS. 63 
 
 cificcdly, and at a certain per centum. A specific 
 gain, is where any gross quantity, at a given 
 cost price, is sold at some other price, without 
 reference to any regular sum of profit : as 30 
 galls, of wine, which cost $19, being sold at 
 $31,40; hence, the gain is $12,40, on the cost. 
 
 Gain or loss per cent., is where a certain 
 sum is gained on the hundred ; for instance, a 
 bill of goods costs $50 ; but it is desired to sell 
 it at 30 per cent, profit; consequently, it must 
 be sold at $65 ; the 15 dollars gained, being a 
 systematic profit. 
 
 Some business men adopt a conscientious 
 mode of trading, and are willing to gain such 
 a per cent, profit, as is just and reasonable; 
 while others, of less principle, and more exor- 
 bitant in their exactions, make the ignorance, 
 credulity, or necessity of the purchaser, their 
 only standard; and grasp any advantage that 
 circumstances may afford them. 
 
 Some individuals calculate their profits and 
 losses with reference to time; while others, 
 and indeed the great mass of business men, 
 regard only the simple transaction. That we 
 should reckon with regard to the length of 
 time occurring between the transactions of 
 buying and selling, appears very reasona- 
 ble, when we consider that, all capital, as a 
 medium of gain, has delegated to it, all the 
 productive capacities of active individual effort. 
 
 The general and unavoidable expenses of 
 every establishment, require the influx of a 
 constant gain to preserve the capital stock 
 entire. Hence, the capital stock that a man 
 invests in business, must yield periodically, and 
 
64 RAINEY'S IMPROVED ABACUS. 
 
 at short intervals, a sufficient amount of profit 
 to sustain these. And if the capital, which 
 may be the only gain producing element in a 
 man's business, be permitted to lie during a 
 long interval, between the purchase and sale, 
 the profit on the sale when made, must be 
 larger in a degree corresponding to the length 
 of time thus invested ; or the deficit in meeting 
 the expenses, must convert this seeming gain 
 into a positive loss. 
 
 Let us suppose money worth 10 per cent, at 
 interest. The business man who has 2000 
 dollars may easily lend it, and realize at the 
 end of the year, 200 dollars profit. But he 
 prefers investing it in goods, which he will sell 
 at 10 per cent, profit. If the stock of goods 
 be sold in 6 months, he may invest again, and 
 sell again, in 6 months. Thus he would make 
 400 dollars clear money. But suppose he sell 
 his first stock at the end of the year, at the 10 
 per cent, profit; he will gain 200 dollars. Sup- 
 pose, again, he sells only half, it is manifest 
 that he gains only 100 dollars; or that, if he 
 sell the whole in a year and a half, his gain 
 will be 200 dollars in H years, or 6f percent, 
 per annum. 
 
 Profit and Loss may be divided into five 
 distinct varieties. They are, 
 
 First: To find how an article must be sold 
 to gain or lose a certain per centum; or to find 
 the sum of gain or loss, at a specified per 
 centum. 
 
 Second: To find the rate per cent., profit or 
 loss, when an article is purchased at one price 
 and sold at another. 
 
VARIETIES OF PROFIT AND LOSS. 65 
 
 Third: To find the cost price, when an 
 article has been sold at a certain per cent., 
 gain or loss. 
 
 Fourth: To find the rate per cent., gain or 
 loss, when an article, sold at a certain price, 
 with a specified gain or loss, is advanced or 
 reduced to yet another price. 
 
 Fifth : To find the selling price of an article, 
 whose cost price is affected by commission, 
 premium, discount, loss or drawback, gain or 
 loss per cent., &c. 
 
 All operations coming under any of these five 
 heads in Profit and Loss, depend primarily on 
 the principles of ratio and proportion. Ope- 
 rations in specific profit and loss, depend on 
 addition and subtraction, only. Per centum 
 being the great acknowledged basis in the rule, 
 everything is in a ratio, greater or less than 
 100. The arbitrary rules in all of the old 
 books, on this subject, have tended to make 
 this simple and beautiful department of Arith- 
 metic, complex; and even, in many cases, 
 unintelligible; whereas, when the relations 
 and bearings of the proportional principles 
 involved, are demonstrated, we see a harmony 
 and system, a regularity and order, that no 
 other portion of the science can excel. Calcu- 
 lations in profit and loss are, however, of a 
 more apparently abstract nature, that in the 
 general calculations of ordinary business con- 
 cerns. Hence, it becomes necessary, that we 
 make nicer distinctions between the terms, and 
 their specific names and qualities, than where 
 material objects, such as yards, Ibs., &c., are 
 concerned. Here, we have to compare cost 
 
66 RAINEY'S IMPROVED ABACUS. 
 
 price with cost price, selling price with selling 
 price, advanced with advanced, and reduced 
 \vith reduced price. And, too, these distinctions 
 are vital and necessary; for they constitute 
 supposition, demand, and term of answer. 
 Some authors work profit and loss by a system 
 of decimals, which, although correct, yet ob- 
 scures entirely the principles of ratio ; making 
 the statement, as well as the work, depend on 
 a mechanical arrangement. 
 
 VARIETY FIRST. 
 
 If a Ib. of coffee cost 10 cents, for how much 
 must it be sold to gain 20 per cent.? We may 
 remark here, that the cost price of an article 
 is called the "par" price; par, as meaning the 
 equal of something else, or the equal of the 
 price. When, therefore, we say the par of per 
 centum, we mean the 100, without increase or 
 decrease, as its own established rate indicates. 
 When we say the par price of an article, we 
 mean the cost. Thus par price, and par per 
 centum, may be considered the same thing. 
 When 100 is compared with any specific thing, 
 as cents, in the above case, it loses its abstract, 
 and assumes the denominate form of the 
 specific thing with which compared. Hence, 
 in the case above, we may say that the 100 
 becomes 100 cents , because compared with 10 
 cents. Now, the position assumed at first was, 
 that per centum, or 100, gained 20, which, 
 changed from this abstract form to the denom- 
 inate, is 100 cents giving 20 cents, or 100 cents 
 being advanced to 120 cents; for, if 100 cents, 
 
TO FIND SELLING PRICE. 67 
 
 or anything else, gain 20, they must be ad- 
 vanced to 120. 
 
 Now, we know that 100 is the par, or cost, 
 or first value of per centum ; but 120 is the 
 advanced price or value of it. Ten, we know, 
 likewise, is the price of the coffee. Now, our 
 demand, 10 cents, cost price, is placed on the 
 right; 100, or per cent., opposite, on the left; 
 and 120, advanced price, last on the right : con- 
 sequently, the answer would be in the advanced 
 or selling price of the coffee. Therefore, the 
 ratio is obtained between the 10 cents and the 
 100 cents; and the selling price, 120, is multi- 
 plied by it. The ratio is T J 7 ; consequently, T ^ 
 of 120, or 12, is the selling price of the coffee, 
 
 thus : \id\ib id 
 
 We may, instead of placing the |i20 
 
 advanced or selling price last on the T~- 
 
 right, place the gain of 100 there; get 
 the answer, in gain, and add it to the cost 
 price. If 100 gain 20, what will 10 gain ? We 
 add this gain, 2, to the cost price, and i^xxi^x 
 have, as before, 12, selling price. The K^ 
 
 latter process, however, is no more per- 
 
 spicuous, while it is more tedious than ' 
 the former. Hence, it is best to find the sell- 
 ing price, by placing the advanced or reduced 
 per cent., last on the right. Let us now find 
 how we will sell this 1 Ib. to lose 20 per cent. 
 We know that if per cent, loses 20, it 
 will be reduced to 80; so, if 100 cost, 
 be 80, selling price, what will be the 
 Belling price of 10, cost? 
 
 Let us find the result, by the 2d process, as 
 4 
 
68 RAINEY'S IMPROVED ABACUS. 
 
 before ; by ascertaining first, the loss, and then 
 subtracting it from the 10. 
 
 What will 10 lose, if 100 lose 20? 
 We have 2 as before, which subtract- 
 12 ed, leaves 8. 
 
 One thing may be noticed very particularly, 
 in the above ; that the sum of profit or loss on 
 any given sum, as the 100 above, is the same. In 
 this case 10 either gained, or lost 2. This leads 
 us to remark, that a large sum will gain or lose 
 more than a smaller. While 10 either gains 
 or loses 2, 12 will gain or lose more than 2 : 
 20 cents will gain or lose twice as much as 
 10 cents, while 5 cents will gain or lose only 
 one half as much. We make these remarks, 
 because there are some who may not appre- 
 hend the difference between the gains or losses 
 on larger or smaller sums. Such think that 
 if 20 per cent, gain will advance 10 to 12, as 
 a matter of course, 20 per cent, loss will reduce 
 12 to the 10. The difference consists in this, 
 that 10 will gain 2, and advance to 12: but 
 12, losing more than 2, will be reduced to 9|. 
 It will be well for the pupil to bear this differ- 
 ence in mind, as its importance will be seen 
 in the calculations that follow. We might 
 prove the foregoing questions, by asking the 
 cost price, after knowing the advanced price r 
 12 : but as this operation comes under variety 
 3d, we will defer proof till we treat of that 
 variety. It may be proven by variety 2d, by 
 finding the rate per cent. From the foregoing 
 we conclude, that, 
 
 To find hoiv an article must be sold, to gain or 
 
PROFIT AND LOSSVARIETY FIRST. 
 
 69 
 
 10015 
 24225 
 
 lose a given per cent.; place the cost price on the 
 right, for the demand: 100 on the left, for the same 
 name; and 100, increased by the gain per cent, 
 added, or diminished by the loss per cent, subtract- 
 ed, last on the right, for the term of answer. 
 
 Purchased sugar at 5 cents per lb.: how must 
 it be sold to gain 12i per cent.? 
 Here, 5 cents is the demand; 100 the 
 same name; and 112i, the name of 
 answer. The latter is reduced, ma- 
 king ---, and is stated as above. 
 
 Again : how must cloth that cost 
 $7| per yard, be sold to lose 40 per 
 cent.? that is, what will $y be re- 
 duced to, if $100 are reduced to 60? 
 
 If a horse cost $50, how must he 
 be sold to gain 50 per cent. ? Here, 
 the demand is $50, 100 the same 
 name, and 150, the term of answer. 
 
 100 
 
 15 
 60 
 
 $|4i 
 
 150 
 
 |75 
 
 We find that 50 per cent, advances to $75 : 
 or, that it gains $25. 
 
 Let us see if selling another horse 
 that cost $75, at 50 per cent, loss, 
 will bring 75 back to 50. It is per- 
 ceived here, as in a former question, 
 
 100|75 
 150 
 
 1374 
 
 that 50 per cent, loss on $75, brings 75 down 
 to $37i ; for while 50, at 50 per cent., gains 
 or loses 25, 75, at 50 per cent., gains or loses 
 37|. This difference, and the reason for it, 
 are too palpable to need further comment. 
 
 If 8 yards of cloth cost $20, for how much 
 must 200 yards be sold, to gain 20 per cent. ? 
 Here, it is manifest that the statement is made, 
 first by proportion, thus, 
 
70 
 
 RAINEY'S IMPROVED ABACUS. 
 
 8'200 This being the cost of 200 yards, 
 
 1 20 we on anc ^ sa y> wna t will these 
 $500 be advanced to, if 100, or per 
 100 50C cent . ? be advanced to 120? The 
 120 answer is 600. It is wholly unne- 
 $1600 cessary to make two separate state- 
 ments in this case : for, knowing 
 that the answer is involved in the first state- 
 ment, that is, that the number of dollars which 
 the 200 yards would cost, could be found by 
 working the question, we say, what will this 
 supposed answer or sum, be yet farther ad- 
 vanced to, as the demand, if 100 opposite, be 
 made 120 ? thus, 
 
 The $100, or per cent., placed 
 on the left, is placed opposite the 
 $20, which term represents the 
 answer, and which is the demand. 
 
 6 
 
 6 
 
 18 
 
 1 
 
 8 
 
 100 
 
 15 
 
 9 
 
 1200 
 
 275 
 
 $1775, 
 
 If i of | of a farm cost $1200, 
 for how much must 1 of T | of the 
 same be sold, to gain 37i per 
 cent. ? The statement of the 
 question made, by proportion, 
 nothing more is necessary than 
 to place 100 opposite dollars, or 
 involved answer, and 137i, or 
 ~-j on the right. 
 
 I have 300 Ibs. bacon, which cost 5 cents 
 per pound : how must I sell the whole, to gain 
 25 per cent. ? It is necessary in the first place, 
 to find the cost of the whole bacon ; conse- 
 quently, we say, if 1 Ib. cost 5 cents, what 
 will 300 Ibs. cost ? thus, 
 
COMBINATION OF STATEMENT. 71 
 
 Then we say, what will all the cts. ., , , 
 
 5 
 
 125 
 
 paying for the bacon, be advanced 
 to, if 100 be advanced to 125? 
 
 Bought 200 galls, oil, at 60 cents , --, 
 per gallon ; I lost 40 galls, and wish 
 to know how. I must sell the remainder to gain 
 30 per cent, on the whole investment. What 
 will 200 galls, cost if 1 gallon cost 60 cents? 
 Then we know that after deducting 40 gallons 
 loss, we have but 160, which this sum of 
 money, on the right, has paid for. Supposing 
 this on the right to be the price of the 160 
 galls., we say, what will one gallon cost on 
 the right, if 160 opposite, cost these cents : 
 the result would be the advanced price at 
 which 1 gallon of the 160 would be sold; 
 so that the whole sale would bring the money 
 first invested in the 200. We then say, what 
 will this price be yet farther advanced to, if 
 100, or per cent., opposite, be 130? We get 
 the answer in the price of 1 gallon, at 30 per 
 cent, profit, thus, 
 
 160 
 100 
 
 200 
 
 60 
 
 130 
 
 In the case above, all that is ne- 
 cessary, is to suppose that the de- 
 mand does exist on the right ; for we i ,^ , 
 know that this demand is but the ' 
 result of another proportion, preceding, which 
 we could easily ascertain by making the sepa- 
 rate statement. The demand, 1 gallon, is 
 merely supposed ; for it is wholly unnecessary 
 to assign a place to a unit, which, so far as the 
 work is concerned, is useless. 
 
72 RAINEY'S IMPROVED ABACUS. 
 
 SECOND VARIETY. 
 
 If I buy a Ib. of butter for 10 cents, and sell 
 it for 15 cents, what do I gain per cent. ? The 
 demand here is, what will per centum, or 100 
 gain, if 10 cents opposite, gain 5? The ques- 
 tion is not what will 100 be advanced to, if 10 
 be advanced to 15; but to know how much 
 100 will gain. Stating accordingly, the answer 
 will be the gain of 100, or the rate per cent, 
 profit. It is supposed, not unfrequ ently , that 
 the per cent, must be calculated on the selling 
 price, which, in the case cited, would present 
 this absurdity: if 15 cents gain nothing, what 
 will 100 cents, or per cent., gain? We know 
 that the gain has been effected by the use of 
 the 10 cents; and if the 10 cents have gained 
 5 others, certainly in the same ratio, 100, or 
 per cent., which are 10 times as many, will 
 gain 10 times as much as 5, which is 50 per 
 cent. We could .find this rate per cent, profit 
 in the following way: if 10 be advanced to 
 15, what will 100 be advanced to? In this 
 case, the cost price, 10, is added to the gain, 5, 
 to make the advanced price ; therefore, the par 
 per centum, 100, will be added to the gain of 
 the 100, to make advanced per cent. Hence, 
 the necessity, if the question be wrought thus, 
 of subtracting 100 from the answer. This 
 mode of finding per cent, is, however, neither 
 direct nor natural. Here, it is seen, that as we 
 would subtract the par, 10, to leave the gain, 
 
RATE PER CENT. PROFIT AND LOSS 
 
 BO we would subtract par per cent, 
 to leave the gain per cent. This is 
 presented only for its theory, which 
 will be found applicable in variety 
 fourth. Fifty per cent, answer. 
 
 in 
 
 inn 
 
 
 
 - 
 I 50 
 
 40 400 
 2 
 
 20 
 
 We now revert to variety first, and knowing 
 that the one Ib. of coffee, which cost 10 cents, 
 was sold at 12, to gain 20 per cent., we will 
 see if this is correct. When purchased at 10 
 and sold at 12, the gain was 2 : now, 
 if 10 cents gain 2, what will 100 cents 
 gain? Twenty, which is 20 per cent., 
 is the answer. 
 
 Suppose the question above were thus : if 
 10 cents lose 2, what is the loss per cent.? It 
 is evident that it would be precisely the same 
 operation, and that 20 per cent, would be the 
 result. Again : If I buy raisins at 7| cents 
 per Ib., and sell them at 10 cents, what do I 
 gain per cent.? The first thing to do in all 
 such cases is, evidently, to find the gain or loss 
 on the cost, and say, if the cost has given this 
 gain, what will per centum give? Now, 7-J- 
 cents gain 2i ; the question is stated 
 accordingly; disposing of fractions 
 in the usual manner. The result, 33 J 
 per cent., is evidently correct, if we 
 consider that 2i being i of 7i, the 
 answer should likewise be J of 100. 
 
 If a horse is bought for 40 dollars, 
 and sold for 80, what is the gain per 
 cent. ? If 40 gain 40, per cent, will 
 gain 100. The result is 100 per 
 centum. 
 
 15 
 
 100 
 
 100 
 
 
RAINCY S IMPROVED ABACUS. 
 
 From the foregoing \ve conclude that, To 
 find the rate per cent., profit or loss, when an 
 article is purchased at one price and sold at 
 another; 
 
 Ascertain the gain or loss on tfie cost price, by 
 subtraction: make 100 the demand: the cost price, 
 the same name; and the gain or loss, the name of 
 answer, last on the rigM. The answer will be the 
 gain, or loss, per cent. 
 
 ^ k^l * g 00 ^ 3 cost 2 400 dollars, 
 and was sold for 3000; what was 
 the gain per cent. ? The 2400 gain- 
 ed 600; hence, the per cent, is % J 
 
 2400 100 
 600 
 
 [25 
 Bought cloth at 4j dollars, and sold it at 4f ; 
 
 how much per cent, was gained or lost ? We 
 find that 45 dollars lose J of a dollar : hence, 
 we inquire, what will 1 00 lose ? The rate of 
 loss is 21-f per cent., thus, 
 
 19 
 8 
 
 100 
 
 100 
 19 
 
 19 
 1850 
 
 844 
 
 75 
 
 |25 
 100 
 
 4 
 425 
 
 be proven by Variety 1st, 
 by asking, to what price will 43 be 
 reduced, to lose 2i-| per cent. ? That 
 
 I is, what will y be reduced to, if 100 
 be reduced to 97 T 7 ^ or ^ff-? Four 
 
 1 and five-eighths is evidently the 
 answer; for this was the price at 
 which it was first sold, at a loss. 
 
 Ribbon that cost 6 cents per yard, 
 is sold at 7^ cents, what is the gain 
 per cent.? 
 
 Cloth that cost 183 cents per yard, 
 is sold at 12i cents: what is the 
 loss per cent.? The 183 lose 6i; 
 therefore, 100 will lose 33:, which 
 is the rate per cent. loss. 
 
PROFIT AND LOSS : COST PRICK. 75 
 
 It must be observed here, that loss, as well 
 as gain, is made on the cost price. 
 
 THIRD VARIETY.* 
 
 Sold a yard of cloth for 250 cents, and 
 thereby gained 50 per cent.; what did it cost?. 
 It is manifest that 50 per cent, was calculated 
 on the cost price, and then added to it, for the 
 selling price ; so that to take 50 per cent, from 
 the selling price, which is considerably larger 
 than the cost price, would be taking off a 
 larger sum than the cost price, at 50 per cent., 
 would give. It is, therefore, necessary to find 
 the cost, and then 50 per cent, reckoned on 
 this and added, would make the selling price. 
 The 250 cents are the advanced price, from 
 which we wish to deduct the per centum that 
 has been added. To do this, we must compare 
 it with the advance value of per cent., or with 
 100, advanced by the rate, added to it. Now 
 the rate 50, when added to 100, makes 150, for 
 the advance or amount of per cent. If this 
 be the advance or amount of per cent., to find 
 the cost or par, we must reduce it to 100. We 
 have, therefore, 150 advance per cent, to com- 
 pare by ratio with 250, advanced price of 
 cloth, and conclude, that if 150 advance, be 
 reduced to 100, par or cost, 250 advance 
 must be reduced in the same ratio, to find its 
 par or cost. Hence, we make 250 the demand, 
 150 the same name, and 100 the term of 
 answer: thus, 
 
76 
 
 RA1NEY ; S IMPROVED ABACUS. 
 
 150 
 
 250 
 100 
 
 In reducing the 150 above, it is 
 not the 150 losing 50 per cent. ; for 
 this would reduce it to 75, instead 
 $|l,66f f 10 : but it is 150 losing the gain, 
 being brought back to the cost or 
 sum, that first gained it. If 100, at 50 per 
 cent., be 150, certainly 100 of it is original 
 value, and 50 gain ; both making the amount 
 150. So, likewise, with the cloth ; 166| is the 
 first value, and 83 J the gain: both together, 
 making the amount or advanced price, 250. 
 Now, by proportion we find, that if 100 would 
 gain $50, 166| would gain 83|. So, it be- 
 comes reasonable, that in reducing an article 
 that has been sold at an advanced, to its cost 
 price, to compare advanced price of the article 
 with advanced per cent. We may now prove 
 this by both preceding Varieties. 
 
 100 
 
 500 
 150 
 
 $12,50 
 
 500 
 
 100 
 
 3 
 
 31250 
 
 |50 
 
 First, \vhat must a yard of cloth 
 which cost 166| 5 be sold for, to gain 
 50 per cent. ? 
 
 Second, If a yard cost 1,66|, and 
 sell for 250 cents, what is the gain 
 per cent.? We know that this ought 
 to be 50. We say then, if 166J 
 gain 83^, what will 100 gain ? 
 
 It is seen here, that when we take 50 per 
 cent, from 250, we reduce it to 1 ,66| ; and that 
 50 per cent, on this 166|, will elevate it to 250 
 again. Some think it singular that it can be 
 done in this case, and not in the case of the 
 first example, in Variety First. The great 
 reason of this difference, is the difference of 
 
PROFIT AND LOSS VARIETY THIRD. 77 
 
 names. In the latter case, we are falling from 
 the amount to the principal : from the advance 
 to the cost, by reducing the amount of per 
 cent., to its par or first value : in the former, 
 the work has no reference to finding the par 
 or cost price ; but merely to laying on or taking 
 off per cent, on larger or smaller sums; as 10 
 and 12 cents : 5 and 20 cents, &c. In the 
 latter case our supposition is the advance per 
 cent. : in the former, it is not advance, but par 
 per cent., or 100 being reduced to some loss 
 price. Advance, par, loss, reduced, &c., be- 
 come, therefore, important distinctions if we 
 would inquire the prime reason of different ope- 
 rations, which are rather apparently the same. 
 Suppose I sell a yard of cloth for 480 cents, 
 and thereby lose 20 per cent. : It is evident 
 that I have sold it for a price, 20 per cent, too 
 small. Now, many would think that we 
 might advance the 480 cents, 20 per cent., 
 and have the cost price ; but this is a mistake : 
 for the 20 per cent, loss, is so much on the 
 cost price, and could not be calculated on the 
 480 ; because it is the reduced price ; and 20 
 per cent, on this reduced price will not make 
 as much as on the cost price. We say there- 
 fore, what will this reduced price, 480, be 
 advanced to for nar, if 80, the reduced value of 
 per cent., be advanced to 100, par ? It is seen 
 here, that we have reduced opposite reduced, 
 and last on the right, par or 100 for the term 
 of answer. We find that the cloth cost 600 
 cents, thus, 
 
78 
 
 RAINEY'S IMPROVED ABACUS. 
 
 80 
 
 480 
 100 
 
 cts.|600 
 
 600 
 
 100 
 
 80 _ 
 
 cts.|480 
 
 600 
 
 100 
 120 
 
 [20 
 
 It was this 600 that lost the 20 
 per cent. ; not the 480 : and if 600 be 
 reduced 20 per cent, by Variety first, 
 the reduced or loss price will be 
 found 480 ; thus, 
 
 This may be proven again, by 
 Variety 2d, thus : If cloth that cost 
 600, is sold for 480, what is the loss 
 per cent. ? We know that it was 
 20 : hence 20 per cent. 
 
 225 
 
 500 
 
 2 
 
 100 
 
 From the foregoing, we conclude, that, 
 To find the cost price, when an article has been 
 sold at a specified per cent., gain or loss, make 
 the selling price the demand ; 100, increased by 
 the gain per cent, added, or diminished by the loss 
 per cent, subtracted, the same name, opposite; 
 and 100, the term of answer, on the right: the 
 answer will be the cost price. 
 
 If a man sell a yard of cloth 
 for $5, and thereby gain 12i per 
 cent., what did it cost him ? What 
 will 500 advanced price, be re- 
 duced to, if 112i advanced per 
 cent., be reduced to 100? 
 
 If I have a yard of cloth that cost 
 4,441, how must I sell it, to gain 
 12i per cent.? Variety 1st. 
 
 If I buy cloth at 4441, and sell 
 it at 500 cents, what do I gain 
 per cent. ? Variety 2d. It is evi- 
 dent that 4441 gain 55 1 cents: 
 then, what will 100 gain ? Here, 
 12 1 is the answer. 
 
 100 
 
 14,441 
 
 4000 
 225 
 
 4000 
 
 |5,00 
 100 
 
 500 
 
PROFIT AND LOSS! VARIETY THIRD. 
 
 79 
 
 If, when wheat is sold at 80 
 cents per bushel, 20 per cent, is 
 lost, what did it cost ? What will 
 the reduced price, 80, be advanced 
 to, if 80 reduced per cent, be ad- 
 vanced to 100, par? 
 
 When wheat is purchased at 
 100 cents, and sold for 80, what 
 is lost per cent.? It is the 100 
 cents here, that lose 20 : hence 20 
 per cent, answer. 
 
 If hemp, sold at $4i per cwt., 
 gains 10 per cent., what did it 
 cost? 
 
 If hemp cost $4,09 T 1 T cents per 
 cwt, for how much will it be sold 
 to gain 10 per cent.? 
 
 Purchased flour at 82,40 per bar- 
 rel, which was 40 per cent, below 
 cost : what was the cost? 
 
 How must flour that cost $4,00 
 be sold to lose 40 per cent.? 
 
 100 
 100 
 
 1104,50 
 100 
 
 4,09 T V 
 
 60 240 
 100 
 
 4,00 
 
 100 400 
 60 
 
 2,40 
 
 Sold a horse for 120 dollars and thereby lost 
 20 per cent., whereas I ought have gained 40 
 per cent.; how much was he sold under his 
 value? It is plain in the first place, that if I 
 lost 20 per cent, in selling him at $120, he 
 must have cost me more than this : conse- 
 quently this is the reduced price. It may be 
 said, therefore, what will this reduced price be 
 
80 
 
 RAINEY'S IMPROVED ABACUS. 
 
 00,100 
 
 140 
 
 advanced to, for cost, if 80, the reduced per 
 cent., be advanced to 100? This will give the 
 cost price of the horse. Now, this cost price 
 must be advanced 40 per cent. Hence, we 
 say, what will this cost price be advanced to, 
 if 100 be advanced to 140? Thus, the two 
 statements are combined, 
 
 We find that the horse should 
 have sold, to comply with these 
 conditions, for 210 dollars; from 
 which, subtracting 120, we have 
 a loss of 90 dollars. We might 
 take the two questions separate- 
 ly; or, as they are combined, we 
 might drop the hundreds for mere convenience, 
 saying 80, reduced price, may be advanced to 
 140, selling or advanced price. 
 
 Suppose, when a horse is sold for $120, 20 
 per cent, is gained, whereas a loss of 40 per 
 cent, might be sustained ; how much is he 
 sold over his value ? 
 
 210 
 120 
 
 90 
 
 60 
 60 
 
 We find that the horse might have 
 been sold for $60, and is consequent- 
 ly sold for sixty too much, in making 
 the price 120. 
 
 Again : In the latter example the 
 100 is suspended on each side, while 
 the statement is still quite as per- 
 spicuous as before. 
 
 Operations in this variety of profit and loss, 
 are quite similar to those of discount. In dis- 
 count the advance made from principal or par, 
 to amount, is based on the losses of a speci- 
 fied time; whereas, in this case, the advance 
 
 60 
 60" 
 
PROFIT AND LOSS: VARIETY FOURTH. 81 
 
 depends on a usage of common consent, 
 whereby an individual is allowed to make a 
 reasonable per cent, in lieu of the accommo- 
 dation extended to those around him. 
 
 FOURTH VARIETY. 
 
 If, in selling a yard of cloth for $5, 1 gain 
 20 per cent., what will be gained if it sell 
 for $6 ? 
 
 In this case we may say, what per cent, wild, 
 be gained by selling at $6, if by selling at $5 
 we gain 20 per cent? Thus, 
 
 It would gain 24 per cent. Now 5 
 advancing to 6 gains 1 dollar ; what 
 per cent is it? or, what will 100 gain 
 if 5 gain one? 
 
 We find that the simple advance 
 
 124 
 5100 
 
 from 5 to 6 is equivalent to 20 per cent. 
 Here the general advance is 24, and 
 the advance from 5 to 6, 20 per cent., 
 which, added, make 44 per cent, as the gain 
 in selling at $6. 
 
 We say above first, what will 6 gain, if 5 
 gain 20: and, again, what will 100 gain, if 5 
 gain 1 ? In the first place 5 gains 20 as a 
 mere addition ; and in the second, it gains 1 
 as a specified per cent. Knowing, then, that 
 we must yet further use the 20 to advance it 
 to a larger amount, and likewise the 100 to get 
 the per cent., we combine the two operations, 
 by adding 100 to the 20 for the purpose of 
 getting the per cent.; suspending one of the 
 fives on the left ; and finally, subtracting this 
 
RAINEY'S IMPROVED ABACUS. 
 
 100 thus added above, which leaves the true 
 answer; thus, 
 
 6 
 1202 
 
 144 
 100 
 
 44 
 
 By thus adding the 20 and 100, 
 we work two propositions, and per- 
 form addition at the same time. It 
 is very seldom necessary to add 
 numbers on the line ; and is done in 
 this instance only to avoid two state- 
 
 ments, and to shorten the work. 
 
 When one price is advanced to another, and 
 a gain made, we subtract 100 from the an- 
 swer, and take the remainder as the per cent.; 
 and when one price is reduced to another, the 
 answer is subtracted from 100, while the re- 
 mainder is, as before, accounted the true per 
 cent. 
 
 To prove this answer correct, we may find 
 the cost of the yard of cloth, knowing that $5 
 is 20 per cent, more than the cost; thus, 
 
 We find by this, that the 
 cost was 4 dollars. Now, if 
 4 dollars, in advancing to 6, 
 gain If dollars, what is the 
 rate per cent, gain ? 
 
 Thus, by variety second, we find 
 that the original cost was 4, and 
 after subtracting this from the sell- 
 ing price, by variety second, find 
 also that the rate is 44 per cent., 
 
 i 
 11 
 
 44 
 as above. 
 
 In selling a watch for $10, I lose 20 per 
 cent ; what per cent, would be lost in selling 
 it at $8? 
 
PROFIT AND LOSS: VARIETY FOURTH. 
 
 83 
 
 08 
 
 64 
 
 Here the selling price, 8, is the de- 
 mand; the first reduced price, 10, the 
 same name; and 100, reduced by the 
 loss per cent, subtracted, the same 
 name. It may be observed here, that if the 
 per cent, is gain, it must be added, and if loss, sub- 
 tracted. Taking 64 above, from 100, we have 
 36 per cent, loss, for the answer. 
 
 We know that if $10 lose 20 per cent., 8 
 will lose 16; arid likewise in descending from 
 10 to 8 we lose 20 per cent.: now 20 and 16= 
 36, as above. 
 
 This may be again proven, thus : What will 
 $10, reduced price, be advanced to, if 80 re- 
 duced per centum be advanced to 100 ? 
 
 Thus the cost price of the watch 
 was $12i ; now, in selling for 8, 12i 
 loses 4i; hence we ask, what will 100 
 lose? 
 
 80*0 
 100 
 
 12* 
 
 The loss in reducing to this price 
 is thus found to be 36 per cent., 
 which proves the work correct in 
 every part. 
 
 If, in selling a pound of tobacco for 30 cents, 
 I lose 10 per cent., what will be lost if it is 
 sold at 25 cents ? 
 
 36 
 
 25 
 003 
 
 The result is 75 cents, which we 
 subtract from 100, leaving 25 per 
 cent, as the answer. 75 
 
 Now, if 30 was 10 per cent, loss, we can 
 easily see what it cost; thus, 
 
 3-4 
 
 The cost was 33 J cents. 
 
 100 
 
84 
 
 RAINEY'S IMPROVED ABACUS. 
 
 If 33J reduced 
 100 lose? 
 
 100 A 00 
 
 to 25, loses 8, what will 
 
 
 
 25_ 
 25~ 
 
 Again the question is proven by 
 finding the per cent, between the cost 
 and selling price. 
 
 3040 
 130 
 
 100 
 
 If, in selling a horse at $30, I gain 30 per 
 cent., what will I gain per cent, by selling him 
 at $40? 
 
 We subtract 100, and have for an- 
 swer 73J per cent. We now find, by 
 variety third, the cost of the horse, 
 which in selling at $30 gained 30 per 
 cent.; saying, what will 30 advanced 
 price be reduced to, if 130 advanced 
 per cent., be reduced to 100? 
 130 30 Now, 23 T ^ being the cost price, we 
 
 100 know that it gains 16}| in advancing 
 to $40. Therefore, by variety second, 
 ^rV we say, if 23 T ^ gain 16}f , what is the 
 rate per cent., or what will 100 gain? 
 
 Again, the work is proven correct 
 by subsequent statements. 
 
 220 
 
 From the foregoing we conclude, that, 
 To find ike rate per cent., gain or loss, when an 
 article, sold at a specified price, with a spe- 
 cified gain or loss per cent., is advanced or re- 
 duced to yet another price : Make the last selling 
 price, the demand; the first selling price the same 
 name', and 100, increased by the gain per cent, 
 added, or reduced by the loss per cent, subtracted^ 
 the term of answer. If a per cent, is gained^ 
 
PROFIT AND LOSS! VARIETY FIFTH 85 
 
 subtract 100 from the answer; if lost, subtract 
 the answer from 100, and in either case the re- 
 mainder will be the true rate per cent. Or, 
 
 Proceed according to directions in variety third, 
 and Jind the original cost ; then find by subtrac- 
 tion the difference between the cost and the selling 
 price; and by variety second, Jind the rate per 
 cent, gain or loss. 
 
 This variety of profit and loss is of but lit- 
 tle practical value, but may serve to awaken 
 close investigation in the mind of the reader. 
 
 FIFTH VARIETY. 
 
 We shall consider under this head questions 
 of a general character, with particular refer- 
 ence to the combination of several separate 
 statements into one general statement, by 
 which a specific answer may be obtained. As 
 all operations, in this or any other department 
 of profit and loss, depend primarily on pro- 
 portion, it will be necessary to make all of the 
 statements occurring in combination, by the 
 general principles indicated in simple ratio. 
 Such statements of concatenated proportions, 
 might be called conjoined proportion, which 
 in the strictest sense is but making the answer 
 of a preceding, the demand of a subsequent pro- 
 portion. This being the case, and all propor- 
 tion depending in solution on multiplication 
 and division, we conclude, that these several 
 multiplications and divisions may be made all 
 together, and at the same time. 
 
 The statement of questions by combination 
 
86 RAINEY'S IMPROVED ABACUS. 
 
 gives free exercise to all the analytic powers 
 of the student's mind, and tends greatly to the 
 cultivation of correct modes both of thinking 
 and reasoning. It will be found necessary, in 
 all such questions, to commence with the com- 
 mencing point, and keep up all of the natural 
 relations of the questions until the term of an- 
 swer is found. 
 
 Sent to Cincinnati for 440 gallons of wine, 
 and paid 87i cents per gallon : paid 2 per 
 cent, commission to my agents; and in ex- 
 changing specie for depreciated Kentucky pa- 
 per, gained 10 per cent- premium: lost 40 gal- 
 lons by leakage : how much must the remain- 
 der be sold for per gallon, to gain 20 per cent, 
 on the prime cost? 
 
 It is evident that we must find what the 440 
 gallons cost, at the price, and that we must 
 then increase this sum 2 per cent, for commis- 
 sion; for the commission is reckoned on the 
 sum of money paid for the wine. We know 
 that this depreciated paper will be received in 
 payment for the wine and commission, quite 
 as well, if necessary, as gold ; and that $1 10 of 
 the paper are worth $100 of the specie. We, 
 therefore, find what sum of specie will pay 
 for this paper, or cost of the wine and com- 
 mission. This, then, would give the amount 
 of specie to be sent to Cincinnati; but losing 
 40 gallons, we divide the whole quantity of 
 specie paying for all the wine, by the number 
 of gallons left, 400, and find the advanced 
 price which each gallon of the reduced lot 
 must sell at, to reproduce the amount expen- 
 ded. We then advance this price 20 per cent., 
 
PROFIT AND LOSS! COMBINATION. 87 
 
 and find the selling price of the wine, per gal- 
 lon. It is a settled matter, that the commis- 
 sion must be paid on the sum invested, wheth- 
 er in specie or paper. It is likewise easily 
 seen, that it is the amount of specie sent to 
 Cincinnati that must be divided by the re- 
 duced quantity of wine ; for the purchaser 
 merely wishes to know the advanced price per 
 gallon that will reinstate him in his expendi- 
 ture, without reference to the 10 per cent, dis- 
 count saved ; for if he were to divide the price 
 paid in paper, he would reindemnify his loss 
 of 40 gallons, and retain also the 10 per cent. 
 But, as it is, he wishes only to make 20 per 
 cent, on the whole transaction, by advancing 
 the specie price of each gallon of the reduced 
 quantity 20 per cent. We consequently make 
 the statement by proportion ; thus, 
 
 What will 440 gallons cost, if 
 1 gallon cost 87|; what will this 
 sum, or price of all the gallons, be 
 advanced to, if 100 be advanced 
 to 102, for amount of both commis- 
 sion and payment? Now, what 
 will all this sum of money in pa- 
 
 100 
 110 
 400 
 100 
 
 440 
 175 
 
 102 
 100 
 120 
 
 1,07, 
 
 per, which pays for the wine, be 
 reduced to, for specie, if 1 10 paper, opposite, be 
 worth 100 of specie, on the right? Then, 
 what will 1 gallon cost, if 200, the remainder 
 after the loss, be worth this last sum in specie? 
 Here, the demand 1, is understood, not ex- 
 pressed. Again : What will this price per 
 gallon on the right, as demand, be advanced 
 to, if 100 opposite, be made 120? We find 
 
88 
 
 RATNEY'S IMPROVED ABACUS. 
 
 that the wine must be sold at 1 dollar and 7 T * y 
 cents per gallon. 
 
 The several successive steps, or proportions 
 in this statement, may be made separately, as 
 follows : 
 
 1 
 
 2 
 
 440 gallons. 
 175 cents price. 
 
 Per ct. 100 
 
 385,00 whole price of wine. 
 102 commission. 
 
 Dig. 110 
 
 392,70 amount with commission. 
 100 specie. 
 
 
 357,00 reduced amount of specie. 
 
 Red. qu. 400 
 
 1 1 gallon demand. 
 357,00 whole price in specie. 
 
 Per. ct. 100 
 
 89J price per gallon, red. qu. 
 120 20 per cent, profit. 
 
 Ans. 
 
 I,07y 7 retail price. 
 
 The concatenation of the statement is here 
 kept up, except in one instance, when 1 be- 
 comes the demand on the right, and the an- 
 swer of the former question, 35700, is the 
 price of the 400 gallons. We might dispense 
 with this 1 ; and use it here, only to show the 
 full proportional relations. Hence, we say, 
 if 400 gallons cost $357,00, what will 1 gallon 
 cost; and get the ratio between the 400 and 
 the 1, and by this ratio, which is ^, multiply 
 the price 35700 cents, and thus decrease it to 
 89 cents. We have shown the absurdity of 
 attempting to multiply together two denomi- 
 nate things : the same reasoning is true with 
 regard to dividing one denominate by another 
 
PROFIT AND LOSS*. VARIETY FIFTH. 
 
 89 
 
 denominate thing, as cents by gallons, or gal- 
 lons by cents. 
 
 Placing the 110 on the left according to dis- 
 count may seem wrong, until we reflect that 
 39270 is the amount of paper which pays for 
 the wine; and which we wish to reduce to 
 gold. This amount of paper must be com- 
 pared with another amount of paper that will 
 equal per centum or 100 in specie; for specie 
 is the par per cent, of exchange ; this amount 
 of paper we know is 110; for the par, 100 
 specie, will pay for this sum of discounted 
 funds. Hence, the operation is one of pure 
 discount. 
 
 Bought 5 hogsheads of sugar, containing 
 each 1200 pounds, at 2% cents per lb., and lose 
 1000 Ibs.: how must I sell the remainder per 
 lb. to gain 6i per cent, on the prime cost? 
 
 The separate statements occur in the follow- 
 ing order: How many pounds will 5 hhds. 
 make, if 1 hhd. make 1200? what will these 
 pounds come to, if 1 pound cost 2i cents? 
 then, what will 1 pound cost, if the 5000 
 pounds, after the loss of 1000, cost the price 
 indicated by the last answer in cents ? what 
 will this advanced price be yet further ad- 
 vanced to, if 100 be advanced to 106; thus, 
 
 We say 5 times 5 on 
 the right make 25, which 
 goes into 100 on the left 
 4 times, etc., etc. In 
 this question 1 again oc- 
 curs as demand, while 
 5000 is the same name, 
 and the preceding an- 
 
 1^003 
 
 1651 
 
90 
 
 RAINEY'S IMPROVED ABACUS. 
 
 swer in cents, the denomination of the an- 
 swer; that when the answer is obtained it may 
 be advanced 6i per cent further. 
 
 Suppose I purchase 90 yards of broadcloth 
 at $5 per yard, on a credit of 1 year; but for 
 ready payment am allowed a discount of 10 
 per cent.: after receiving the cloth I lose 10 
 yards; how must I sell the remainder per 
 yard, to gain 10 per cent, on the prime cost? 
 
 What will 90 yards come to, if 1 yard cost 
 $5? what will this amount, 1 year hence, be 
 reduced to for ready payment, if 110 opposite 
 be reduced to 100? then, if this price pay first 
 for 90 yards, or, after sustaining a loss, for 80 
 yards, what price will pay for 1 yard? what 
 will this price of 1 yard be advanced to, if 100 
 be advanced to 110 for selling price? thus, 
 
 The answer 5f dollars, or $5,62i 
 cents, is correct; because when we 
 deduct from the cost of the 90 yards, 
 10 per cent discount, and divide the 
 remainder by 80 yards, we have 
 the cost price of 1 of the remaining 
 
 190 
 06 
 
 80^00 
 
 54 
 
 yards ; thus, 
 
 1 
 
 00 
 
 110 
 
 5 
 
 80100 
 
 te 
 
 225 
 352 
 
 100 
 44 
 
 180 
 
 10 per ct. 
 
 Now, the difference between this 
 price and 5f , for which it sold, is |f|. 
 We may now prove that in selling at 
 5f , 10 per cent, is gained, by variety 
 second; thus, 
 
 If 5/f gain |f f , what will 100 
 or per cent, gain? This 10 
 per cent, gain is made on the 
 amount of money invested, as 
 has been shown; consequent- 
 
 ly, on the first cost. 
 
,100 120 
 
 PROFIT AND LOSS: COMBINATION. 91 
 
 Purchased cloth at $3 per yard, but being 
 damaged, I was allowed a deduction of 20 per 
 cent.; for what must I sell it to gain 20 per 
 cent.? 
 
 The first statement evidently is, what will 
 $5 be reduced to, if 100 be reduced to 80 : the 
 second, what will this price be advanced to, if 
 100 be advanced to 120; thus, 
 
 To some minds the old diffi- 400 A 
 culty is here again presented, of 
 finding the reduced price at 20 
 per cent, loss, which at 20 per 
 cent, gain, will not produce the 
 former price. We must recollect that when 
 $5 lose 20 per cent., they will be reduced to a 
 number which will not at 20 per cent, gain a 
 sufficient amount to advance to $5 again. 
 Five dollars in losing 20 per cent, are reduced 
 to $4; but $4 to gain 20 per cent would ad- 
 vance to $4,80 cents only. The defect, 20, 
 arises from the $4 being too small to gain at 
 20 per cent, the sum that $5 would gain. The 
 first statement of the question above, if 
 wrought separately, w r ould give $4, the reduced 
 price of the cloth; hence, when we increase 
 this answer 20 per cent., we make it $4j. 
 
 I purchase another piece of cloth, on which 
 the sum made was 20 per cent.; in considera- 
 tion of damages he lets me have it at cost 
 price, or at a discount of 20 per cent.: what 
 should I get for it? 
 
 Five dollars is the advanced price; hence, 
 we will, by variety third, compare with it 120, 
 advanced per centum, and reduce it to cost or 
 par price, by the par, 100, on the right.' Then, 
 
92 RAINEY'S IMPROVED ABACUS. 
 
 by another combined statement, we will ad- 
 vance it 20 per cent, for selling price ; what 
 will the selling price be ? The 20 per cent, 
 advance must be reckoned, like all other pro- 
 fits and losses, on the cost price ; now, 20 per 
 cent, discount has been taken off for the pur- 
 pose of finding cost price ; thus, when the cost 
 price is found, we must, by 20 per cent., ad- 
 vance it again to its former advanced price, 
 $5; thus, 
 
 The difference in these two opera- 
 tions is, that in the latter case we 
 discount from the advanced to find 
 the cost ; whereas in the former, we 
 reduced the cost, by simple loss, to find 
 the reduced price. 
 
 We will now give a few solutions on gene- 
 ral principles. 
 
 A purchases 500 pounds of sugar at 6 cents 
 per lb.; how must he sell it per Ib. to gain $20 
 on the whole lot? 
 
 It is manifest that it is necessary to find, first, 
 the cost of the sugar, which is $30,00. To 
 this we must add 20, making $50, the whole 
 price that the 500 Ibs. must sell for, to gain 
 00 1 I $20. Now, what will 1 pound 
 0,00 cost, if 500 pounds cost $50; 
 
 1A . thus, 
 10 cts. 
 
 A miller sold a quantity of rye at $1 per 
 bushel, and gained 20 per cent.; soon after, he 
 sold of the same to the amount of $37,50, and 
 gained 50 per cent.; how many bushels were 
 there in the last parcel, and at what did he 
 sell it per bushel ? 
 
PROFIT AND LOSS! COMBINATION. 
 
 93 
 
 150 37,50 
 100 
 
 $25 
 
 2,00 
 3 
 
 30 bu. 
 
 We know that Si is the cost price of one 
 bushel, with 20 per cent, profit added : conse- 
 quently, according to variety third, the cost 
 price of the first quantity is 83i cents; thus, 
 
 Now, $37,50 is 50 per cent. 120 
 more than the cost price of the 
 second lot; so, by the same pro- 
 cess, we ascertain that the lot 
 cost $25; thus, 
 
 Now, we say, how many bush- 
 els will $25, the cost of the lot, 
 buy, if 83, cost price, buy 1 bush- 
 el ; thus, 
 
 We find that 30 bushels were 
 sold. Now, if these 30 bushels 
 cost $37,50, what will 1 bushel 
 cost? thus, 
 
 It is found by the following 30 
 statement that he sold 30 bushels 
 at $1,25 cents per bushel; thus, $ 1,25 cts. 
 
 We see here beautifully illustrated, the dis- 
 tinction between cost, advanced, and reduced 
 prices. We find the cost of each; compare 
 the cost of 1 with the cost of the lot, and find 
 the number of bushels in each lot. Then, 
 knowing that these bushels cost the price of 
 the lot, we find the price of one bushel. This 
 is an easy and simple process of reasoning; 
 yet the conditions and relations of such ques- 
 tions are seldom understood, unless nice dis- 
 tinctions are made in the terms. 
 
 A merchant bought a parcel of cloth, at the 
 rate of $1 for 2 yards, of which he sold a cer- 
 
 1 
 37,50 
 
94 RAINEY'S IMPROVED ABACUS. 
 
 tain quantity at the rate of $3 for 5 yards; 
 and then found that he had gained as much 
 as 18 yards cost; how many yards did he sell? 
 We know that the cloth cost 50 cents per 
 yard, and that he sold it for 60 cents; conse- 
 quently, he gained 10 cents per yard. Now, 
 he gained as much in selling a quantity of it 
 as 18 yards cost; which is 900 cents. If, 
 therefore, 10 cents is the gain of 1 yard, of how 
 many yards is 900 cents the gain? Nine hun- 
 dred is the demand; 10 cents the same name, 
 and 1 yard the term of answer; thus, 
 
 90yds, 
 
 We have now sufficiently illustrated all of 
 the practical operations in profit and loss, to 
 enable the careful and reflective student to 
 perfect his knowledge of the subject by exam- 
 ples and experiments of his own, both in the- 
 ory and practice, in all of the usual business 
 transactions of life. 
 
 A great number of questions in business 
 come under the head of variety fifth, which> 
 judiciously arranged and stated, may be easily 
 wrought; and frequently with one-fourth the 
 number of figures required by former methods, 
 and separate statements. 
 
 From the foregoing illustrations, we deduce 
 the following 
 
 SUMMARY OF DIRECTIONS. 
 
 For COMBINATION OF STATEMENTS in Profit and 
 Loss. 
 
DISCOUNT. 95 
 
 I. Place first on the right the quantity of the 
 article and the cost price: 
 
 II. If it is desired to advance or reduce this 
 entire cost, by commission, premium, transporta- 
 tion, drawback, or other consideration, place 100 
 on the left, and 100, increased or reduced by the 
 per cent., on the right. 
 
 III. To effect a discount, place 100, increased 
 by the rate, on the left, and 100 on the right: Or, 
 if profit and loss follow, place 100, increased or 
 reduced by the gain or loss per cent., on the right, 
 in the place of the 100 or par of discount. 
 
 IV. If a specified portion of the article of mer- 
 chandise be lost, and it is desired to know how 
 a unit of the quantity must be sold, subtract the 
 quantity lost, and place the remainder on the left. 
 
 Or, Make the whole statement a concatenation of 
 proportions, and proceed according to the specified 
 directions in the various rules involved. 
 
 DISCOUNT.* 
 
 DISCOUNT is reckoned by two methods ; one 
 true; the other false. The false method is 
 very often used by business men; which is 
 merely to reckon the interest on the amount, 
 
 * Discount is from the French dccompte to count back, and is used 
 synonymously with rebate, which is from rebattre, to strike otV. It implies 
 the striking of a portion from an amount made of separate sums, 
 as is the case in discount, where the amount is composed of the original 
 principal, and the interest supposed to have accrued. .Amount is from 
 monter, to ascend ; which is from the root of the Latin, moiis, a moun- 
 tain. 
 
96 RAINEY'S IMPROVED ABACUS. 
 
 and deduct it for the discount; making the 
 remainder the present worth. By the true 
 method, the amount is reduced to such a sum, 
 as, connected with its own interest for the 
 time, and at the rate, will restore the amount. 
 In other words, the interest on the present 
 worth, which is equal to the Discount, will, if 
 added to the present worth, restore the original 
 amount or debt. Some individuals use both 
 of these methods : the former, if they are pay- 
 ing out money ; the latter, if receiving it. 
 
 The theory of discount is based on the sup- 
 position, that the debt becomes due at a future 
 period, and bears interest from date, or from 
 the specified time when it is to be paid. 
 Therefore, it is necessary to use some standard 
 of present and future value, such as 100, or 
 per centum. One hundred is the value or 
 standard at present time; but 100, with its 
 own interest for the time and rate, added, 
 will be the future value, of such standard. 
 Suppose the time 1 year, and the rate 10 per 
 cent. : the standard of future value will be 
 110. This 110 is both principal and interest, 
 which added, make the amount which is 
 always the future value. In the same man- 
 ner, the debt on which the discount is to be 
 made, is both principal and interest, or 
 amount: hence, the propriety of comparing 
 amount of debt with amount of standard, and 
 by proportion, reducing the amount of debt to 
 its par or present value. In the supposition 
 above, if 110 dollars, one year hence, be 
 reduced to 100 dollars, present time and value, 
 what will any other amount, as $100 debt, be 
 
110 
 
 100 
 100 
 
 THEORY OF DISCOUNT. 97 
 
 reduced to, in the same proportion : that is, 
 if $110 be reduced to $100, for present worth, 
 what will 100 dollars be reduced to for present 
 worth? Here, 100 is the demand; 110 the 
 same name; and 100 the term of answer, or 
 present worth. We state accord- 
 ingly, and the answer will be the 
 sum of money payable at present 
 time. 
 
 By annexing two ciphers, this answer might 
 be obtained in cents. The present worth is 
 901-0. dollars; or 90 dollars, 90 cents, and \. 
 This sum with interest for one year, at 10 
 per cent., will amount to 100 dollars ; which 
 proves the position correct, that the interest on 
 the present worth is equal to the sum of discount. 
 It is necessary to make a distinction between 
 the terms used. The sum is the whole of any- 
 thing, from summum, the whole : as the sum of 
 interest, the sum of present worth, the sum of 
 discount, &c. The amount is the result of two 
 or more sums added, as present worth and 
 discount added, which make the amount of 
 debt. Amount is from the French monter, to 
 ascend. The present worth is the portion of 
 the debt ^remaining, after the discount is 
 deducted. 
 
 It is necessary, in stating this sub-division 
 of numbers, to place amount opposite amount, 
 on the line, that we may ascertain the ratio 
 between such different amounts, and apply it 
 to the par, present worth, or 100, and increase 
 or decrease it accordingly. 
 
 What will be the present worth of 400 dol- 
 lars, 10 years hence, at 6 per cent ? Here, as 
 
98 RAINEY'S IMPROVED ABACUS. 
 
 in all other cases of discount, the interest on 1 
 dollar, for the time, and at the rate, must be 
 ascertained, and added to 100 cents; and the 
 amount placed on the left of the line. One 
 hundred cents, in 10 years, at 6 per cent., 
 amounts to 160 cents; or, 100 dollars, in 10 
 years, at 6 per cent., will amount to 160 dolls. 
 It may be observed here, that the denomina- 
 tion of the amount on the left, is determined 
 by the demand on the left. If the demand is 
 dollars, the amount is the same; and if cents, 
 the amount is cents; the left, or amount, being 
 merely an indenominate standard. 
 
 It is manifest that 400 is the demand; 160 
 the same name; and 100 the term of answer, 
 ,thus, 
 
 40025 
 
 This 250 dolls. , in 10 years, 
 at 6 per cent., will gain 150 
 dollars, which, added to the 
 250, restores the original amount, 400. 
 
 What is the present worth of 824 dollars, 
 due 8 months hence, at 4i per cent.? We 
 ascertain first the interest on 1 dollar, at 4J 
 . per cent., thus, 
 
 #_._i g > | We now add this 3 cents 
 
 ## to 100, making 103, which 
 
 r^ ; we place on the left of the 
 
 3 cts " ' line, thus, 
 
 The answer is 800 dollars, 
 on which, in 8 months, at 4-J 
 per cent., the interest would 
 be 24 dollars. This added 
 makes the amount 824. When it is necessary 
 to ascertain the sum of discount, subtract the 
 present worth from the amount due. 
 
 100 
 
RULE FOR DISCOUNT. 99 
 
 What is the present worth of $20,86,24, 
 due in 18 days, at 6 per cent, discount? We 
 find the amount of 100 cents, 
 thus, 
 
 The amount is 100,3. This 
 number is placed on the left 
 of the line, thus, 100, 3 [20,86,24 
 
 In this instance, we have 3 100 
 
 mills on the left, after the ^- Qnn 
 
 cents, and on the right four >|20,80,0 
 
 hundredths of cents. Having this one decimal 
 more on the right, than on the left, one figure 
 must be cut off, on the right, for hundredths. 
 This might be obviated, by using another 
 cipher on the left, making 3 mills 30 hun- 
 dredths. In such case, the demand and same 
 name, would be of the same denomination. 
 From the foregoing considerations, we are 
 justified in making the following 
 
 DIRECTIONS FOR DISCOUNT. 
 
 To find the present worth of an amount of 
 money at discount, 
 
 Ascertain, by interest, the amount of 100 dollar s^ 
 or cents, for the time, and at the given rate: or 
 ascertain the interest on one dollar for the time 
 and rate, and add it to 100; make the amount of 
 the debt the demand ; the amonnt of 100 the same 
 name ; and 100 the term of answer: the answer 
 will be the present worth. To ascertain tlie Dis- 
 count, subtract the present worth from the amount. 
 If the number of decimal places in the amount on 
 the rig/it is greater than in the amount on the left, 
 cut off such surplus in the answer. Cancel as in 
 
100 RAINEY'S IMPROVED ABACUS. 
 
 other cases ; or, if this be impracticable, multiply 
 and divide. 
 
 It may be well to offer a lew remarks to 
 those who suppose that Discount and Interest 
 are the same; or who think that deducting 
 the interest, is a fair method of discounting. 
 
 The true discount of 100 dollars, for ten 
 years, at 10 per cent., would be 50 dollars. 
 But the discount by the false method, of sub- 
 tracting' the interest, would of itself, be 100 
 dollars ; leaving nothing for present worth. 
 The absurdity of this may be better seen, by 
 taking the discount on the same sum for 20 
 years, at 10 per cent. ; in which case, the dis- 
 count would be 200 dollars: so that, if de- 
 ducted, would leave the holder of the note 100 
 dollars in debt to his creditor, by receiving 
 payment, or making settlement : whereas, by 
 the correct method, the discount never can en- 
 tirely consume the debt; as there must always 
 be a present value. We have no space for 
 descanting farther on the beautiful theory of 
 this sub-division of Arithmetic; and will pro- 
 ceed to the consideration of per cent., under 
 other heads. 
 
 In BANK DISCOUNT, the interest is reckoned 
 on the face of the note, and deducted: the 
 remainder is the present worth, or sum drawn. 
 This is allowing a greater rate per cent, than 
 is specified in the note ; and on what plea, in 
 morals and justice, I am unable to learn. 
 
GIVING NOTES IN BANK. 101 
 
 FACE OF NOTES GIVEN IN BANK. 
 
 It frequently becomes necessary to find the 
 face of a note given in bank, to draw a spe- 
 cific sum of money. Suppose the rate per 
 cent, discount be 6; then $100 face of note 
 will give 94 dollars ready money. Suppose it 
 is desired to draw 4700 dollars. We say, 
 therefore, what will 4700 dollars ready money 
 be advanced to for face of note, if 94 dollars 
 ready money be advanced to 100 dollars, face 
 of note? thus, * 
 
 Now, the interest at 6 per 
 cent., for 1 year, deducted - 
 from this 5000 dollars, will 
 leave 4700 dollars, the sum to be drawn. 
 
 Suppose the rate to be 4 per cent., and it is 
 desired to draw 1800 dollars: we know that 
 96 dollars will be the sum drawn for the face 
 100, and state accordingly; saying, what will 
 1800 be advanced to, if 96 be advanced to 
 
 1.10025 
 
 Here, the factor 12 
 into 96 eight times, 
 and into 180, fifteen 
 times; again, 4 into 8 twice, and into 100, 25 
 times; while 2 on the left into 10 on the right, 
 five times; which, multiplied thus, 5X15X25, 
 makes 1875, the answer. The interest on this 
 r-.im, at 4 per cent., is 75 dollars, which 
 cieducted, leaves 1800, the face of the note. 
 Hence, To find the face of a note given at bank, 
 to draw a specific sum. 
 
 Deduct the interest of 100, /or the time and at 
 
102 RAINEV'S IMPROVED ABACUS. 
 
 the rate, from 100; place the remainder on the 
 left for the supposition, cash drawn; the sum to 
 be drawn in cash, on the right, for the demand; 
 and 100, face of note, last on the right, for the 
 term of answer : the answer will be the face of the 
 note. 
 
 Bankers generally ascertain the face of the 
 note, (though they frequently wish to avoid it 
 altogether) by a species of approximation, by 
 casting interest on the sum, and subsequently 
 on each separate sum of interest, till the result 
 is too small toj3e noticed further. The precise 
 face of the note could never be obtained in 
 this way. 
 
 COMMISSION, BROKERAGE, &c. 
 
 Commission, Insurance, Brokerage, Taxes, 
 &c., are wrought by Proportion, and in their 
 standards of value, are based on per ce?itu?n. 
 Transactions of this kind are generally made 
 without reference to time, that is, the time 
 required for the specified transaction, is con- 
 sidered a unit. 
 
 Commission is a specified sum paid per 100, 
 for the purchase and sale of merchandize, &c. 
 The rate of commission varies from 1 to 20 
 per cent. The sum paid for commission is 
 called bonus, which is the amount of reward 
 for the trouble incurred; from the Latin, bonus, 
 good. 
 
 A. sends to B. 500 dollars worth of books, 
 
THEORY OF COMMISSION. 10$ 
 
 to be sold on commission, and agrees to allow 
 him 2i per cent, commission, what sum does 
 B. receive? Five hundred, , M ^ bM 
 the sum, is the demand; 100, 
 or per cent., the same name; 
 and 2J, the rate bonus, the 
 term of answer. The answer is consequently 
 124 dollars. 
 
 The same name and the term of answer, are 
 conformed in their denomination, to that of 
 the demand : for being only standards, they 
 have no specific names, and may become 
 dollars or cents, as indicated by the name of 
 the demand. 
 
 What is the commission on 750 dollars 
 worth of wheat at 3J per ct.? , ,-^Q 
 
 Here, we suspend the 100 j J 15 
 
 on the left, and cut off 2 | filFv) 
 
 figures for decimals of a | 
 
 dollar in the answer. 28,02-J 
 
 What it the commission on 800 bushels 
 of wheat, worth 60 cents per bushel, at 6^ per 
 
 OP-ttt 
 
 J T ' ' , . . , 100i$00 2 
 
 In this instance, we place I '60 
 
 the number of bushels, and A 25 
 
 the price, on the right, which irscToo 
 
 is equivalent to multiplying 
 them ; and which makes the statement the 
 following : what will all the cents that 800 
 bushels cost, pay for commission, if 100 cents 
 pay 6i cents commission? The 100 might 
 again be suspended on the left, and two more 
 figures cut off in the answer, for hundredths 
 of cents. 
 
 A factor receives 708 dollars and 75 cents, 
 
104 RAINEY'S IMPROVED ABACUS. 
 
 and is required to purchase iron at 45 dollars 
 per ton ; he is to receive 5 per cent, commission 
 on the money paid : how much iron will he 
 purchase ? 
 
 The demand is the amount of money, 708,75 
 cents, and the same name, 100 with the com- 
 mission added, or 105; and the name of 
 answer, 100. This would give the amount to 
 be invested. In this instance it would be im- 
 proper to charge commission on the whole 
 sum of money, that is, to charge commission 
 on the commission received. The question is, 
 what sum must the factor invest in iron, so 
 that the commission on the same, would make 
 such sum amount to 708,75, the original amount 
 of capital. We know that for every 100 dol- 
 lars that he invests, he receives $5 commission : 
 this added makes 105: now, this $105 capital 
 will make 100 investment ; and we say accord- 
 ingly, what must 708,75 capital be reduced 
 to for investment, if 105 capital, be made 100 
 investment money ? thus, 
 
 105 
 45,00 
 
 708,75 
 100 
 
 1 15 tons 
 
 This will give the whole num- 
 ber of cents that may be invest- 
 ed : hence, we say again, how 
 many tons iron will all these 
 cents, in this involved or implied 
 
 answer, buy, if 4500 cents opposite, buy 1 ton ? 
 Here, we combine the two statements in one, 
 and have for the answer, 15 tons. That the 
 commission should be charged on the sum 
 invested only, may be better illustrated by the 
 following contrast : A. sends to B. $100 worth 
 of books to be sold on commission at 25 per 
 cent. : what commission does B. receive? It 
 
BROKERAGE. 1 Q5 
 
 is manifest, that as B. has the trouble of selling 
 the whole lot of books, that part which pays 
 his commission as well as the other, he should 
 receive commission on the entire 100 dollars 
 worth. The commission is consequently, $25. 
 Again : 
 
 Suppose A. send to B. $100 with which to 
 purchase books : what is the commission at 
 25 per cent. ? Here, the commission being 
 ready at hand, the agent has no further trouble 
 than to deduct his commission, and invest the 
 remainder. It would be manifestly unjust for 
 him to charge commission on his commission, 
 with which he had invested no time. We will 
 suppose that if he wished to purchase $100 
 worth of books, he would necessarily send to 
 B. 100 to invest, and 25 to pay commission, 
 or 125 capital to make 100 investment. If 
 then, $125 capital make $100 investment, what 
 investment will $100 capital, make? 
 
 1/100 4 
 1/100-20 
 
 It is thus ascertained that 
 B. must invest $80 and reckon 
 his commission on the same ; 
 which, at 25 per cent., would 
 be $20 ; consuming the entire $100. 
 
 The difference consists in the medium that 
 the factor operated on ; the one, ready capital ; 
 the other, merchandize, which must first be 
 converted. 
 
 BROKERAGE. 
 
 Brokerage is but another form of commis- 
 sion, in which the factor or agent operates in 
 monies, stocks, &c. The rates of brokerage 
 vary from ^ to 10 per cent. 
 
106 RAINEY'S IMPROVED ABACUS. 
 
 Brokers operate in two ways : by keeping 
 a current account with their dealers, in which 
 they charge the various sums of premium due 
 them ; or by deducting the premium from the 
 capital before investment, as in the case of 
 the purchase of books just mentioned. 
 
 What will be the premium or bonus for 
 
 purchasing 300 shares Whitewater canal stock, 
 
 worth $50 per share, at If per cent, premium? 
 
 \Ttfak Here, the number of shares is 
 
 100 50 multiplied by the price per share, 
 
 #j 5 10 placed opposite, and the rate, If, 
 
 j or |- last on the right. We might 
 
 $|250 a g a i n suspend the 100 on the left, 
 and cut off two decimals for it on the right. 
 Either method may be used with ease and 
 safety. 
 
 B. has 200 shares Illinois Canal stock, which 
 he \vishes to sell 20 per cent, below par, and 
 agrees to pay to his broker 7 7 F per cent., for 
 effecting the sale : stock worth $100 per share. 
 The number of shares is here mul- 
 tiplied by the price, and the whole 
 sum of stock reduced 20 per cent., as 
 in profit and loss, which gives the 
 true value of the 200 shares. Then 
 we say, what premium will all of 
 these dollars give, if 100 opposite give T 7 of 
 a dollar? Hence the answer 112 dollars. 
 We might suspend the 100, and the T \, and 
 ascertain the discounted value of the 200 
 shares stock, only. 
 
 If my broker purchase for me 300 shares 
 Railroad stock at 10 per cent, advance, and 
 
 00 
 
 ,100400 
 
 100 
 
 80 
 
RULE FOR COMMISSION. 107 
 
 charge me 1 per cent., brokerage on the sum 
 invested, what will my stock cost me ? 
 
 Here, the advance is added to 
 
 00 
 
 the 100 on the right; whereas, in ,100,100 
 the question above, the 20 per 1,100110 
 cent, was subtracted from the same 101 
 
 number. The one per cent, added $133 330 
 to the 100 on the right, is not to 
 ascertain the premium, but the whole price to 
 which the stock is advanced ; and is identical 
 \vith Variety 1st, in Profit and Loss. 
 
 From the foregoing considerations, we are 
 justified in making the following 
 
 SUMMARY OF DIRECTIONS, 
 
 for calculating bonus and premium on Commission, 
 Brokerage, Stocks, fyc. 
 
 To ascertain bonus, premium, tyc., Place the 
 amount of money, merchandise, stock, fyc. on the 
 right, for the ^demand: 100 on the left, for tJie 
 same name; and the rate, bonus or premium, last 
 on the right, for the term of answer ; the answer 
 will be in the denomination of the amount : or, 
 suspend the 100 on the left, and cut off two figures 
 in the result, for hundrcdths. Again : 
 
 To Jind the sum of money to be invested, after 
 the commission is deducted from a given amount, 
 Proceed as in discount: and make the amount 
 the demand: 100 with the per cent, commission 
 or brokerage added, the same name ; and 100 the 
 term of answer. The answer will be the sum to 
 be invested. 
 
 To ascertain premium, or selling price of stock, 
 when effected by gain or loss on par, 
 
108 RAINEY'S IMPROVED ABACUS. 
 
 Place the number of shares, and the prize per 
 share, on the right: 100 opposite; and 100 in- 
 creased by the gain per cent., or, reduced by the 
 loss per cent., on the right: then, to ascertain the 
 premium, place 100 on the left, and the rate on 
 the right ; or, to ascertain the selling price of the 
 stock, place the 100 on the left, as before, and 100 
 on the right, increased by the gain, or reduced by 
 the loss per cent. The answer will be the price 
 of the stock, including gain or loss, and brokerage. 
 
 INSURANCE. 
 
 Insurance is security against hazard or loss 
 of property on land or sea , and is usually di- 
 vided into two kinds : Fire and Marine. 
 
 Fire Insurance is that which secures build- 
 ings and other property on land : Marine, 
 which is from maris, the sea, is to secure vessels, 
 boats, cargoes, &c., on sea, or on rivers, lakes, 
 &c. The propriety of insurance is found in the 
 fact, that by a single accident an individual 
 may lose his entire property, with little or no 
 hope of recovery : whereas, if he pay a small 
 bonus to an association, his losses may be 
 restored to him, by the association paying him 
 from a large common stock fund ; which fund 
 is kept up by the bonus paid by each indivi- 
 dual who insures. Reason dictates that it is 
 better to pay a small share of our profits for 
 certain safety, than saving it, to be continually 
 subject to lose, not only the small sum so 
 saved, but the entire capital on which the hope 
 of all gain is based. 
 
THEORY OF INSURANCE. 109 
 
 Property is insured in two ways : by corpo- 
 rations, which are legalized associations of in- 
 dividuals, with specific powers and privileges, 
 based on a definite amount of common stock 
 fund : and by individuals, according to private 
 contract. Insurance effected in the latter way, 
 is called, " out-door" insurance ; and is never 
 so safe as the former, except when the insurer 
 is careful not to insure property to a 1 arger 
 amount than his personal property would in- 
 demnify, in case of loss, on part of the assured. 
 
 The instrument of writing given to an indi- 
 vidual as evidence of his insurance, is called a 
 Policy : and the sum paid by such individual 
 for insurance, is called Premium. The latter 
 is always a certain per cent, on the value of 
 the property insured. The agent of the com- 
 pany, or the individual who signs this policy 
 or contract, is called an underwriter. 
 
 The question arises, what per cent, should 
 be paid for the insurance of property? This 
 is always according to circumstances ; as prop- 
 erty is more or less subject to damage. Hence, 
 insurers divide property into hazardous, not 
 hazardous, and extra hazardous ; charging dif- 
 ferent rates, according to the degree of expo- 
 sure. These distinctions refer more particularly 
 to fire insurance on land; as, in marine in- 
 surance, there is but little difference in degree 
 of hazard. 
 
 1st: A stone or brick building situated re- 
 mote from other buildings, is, from its location, 
 and the quality of material of which construct- 
 ed, less liable to destruction by fire, than simi- 
 lar buildings otherwise located ; or buildings 
 8 
 
110 RAINEY'S IMPROVED ABACUS. 
 
 of a different material in the same place. For 
 such, the rate of insurance would be low. 
 
 2d : Buildings in blocks, densely surrounded, 
 where fire may communicate from one to 
 another, and constructed of more destructible 
 materials, being more hazardous, are insured 
 at higher rates. 
 
 3d : Buildings devoted to purposes involving 
 greater danger, such as chemical manufacto- 
 ries, drug establishments, foundries, &c., being 
 greatly exposed, pay yet higher, and extra 
 rates. In some cases of similar nature, it is 
 impossible to effect insurance at any rate. 
 
 The unit of time for insurance on property, 
 is one year. Vessels and their cargoes are 
 usually insured for the voyage. Coasting ves- 
 sels are generally insured by the year ; being 
 less liable to loss than out-sea vessels. Coast- 
 ers are insured at rates varying from 3 to 8 
 per cent. ; the difference being the result of a 
 state of peace or war in the vicinity. Whale 
 ships are insured by the voyage, at from 4 to 
 10 per cent. Insurance on goods, ships, stores, 
 manufactories, chattels, dwellings, &c., varies 
 from i to 2i per cent., per annum, acording to 
 exposure. 
 
 The amount of insurance taken on property, 
 is always less than the property is worth ; and 
 is generally not above f of its assessed value. 
 This leaves still a partial degree of risk on the 
 owner, as well as on the insurance company, 
 which keeps him awake to the safety of his 
 property. But were the whole value insured, 
 or an amount greater than this, it would be an 
 inducement to some, to destroy their property, 
 
THEORY OF INSURANCE. HI 
 
 for the base purpose of converting it into 
 money, or of realizing a small profit on the 
 capital invested in it. Thus, it becomes ne- 
 cessary to insure a definite amount, which 
 entire amount may be recovered, if it is shown 
 that such an amount of property has been lost. 
 But if less than this amount is lost, the whole 
 insurance cannot be recovered ; only a share 
 proportional to the damage sustained. For 
 instance ; if I insure property to the amount 
 of $1000, and lose one half of it, I cam recover 
 but $500 insurance. 
 
 If by fire or other accident, a loss occurs to 
 property, and does not exceed 5 per cent., it 
 is sustained by the owner; otherwise, he 
 might carelessly consign everything to wreck 
 around him, knowing that the company would 
 have to pay for repairs. 
 
 It is supposed by some who have little 
 experience, and less common observation, 
 that if insurance companies can make money 
 by taking risks, individuals certainly can by 
 running risks; but nothing is more illogical. 
 A company with a large common fund, may 
 locate their risks in a great number of places, 
 so that a loss in one place will be more than 
 overbalanced by the gains in another ; for it is 
 scarcely reasonable to suppose that losses will 
 occur simultaneously in a great number of 
 places, and on the particular property insured 
 by a certain company. But suppose an indi- 
 vidual lose all his property in one place : he 
 is not likely to have enough in another to 
 retrieve this loss by equal gains. And suppose 
 he lose all, which not unfrequently occurs, he 
 
112 RAINEY'S IMPROVED ABACUS. 
 
 is prostrate ; his energies tied, whatever they 
 be ; for his gain-producing property or element 
 is gone. Now, as before said, which is better, 
 that he save a little and risk the loss of all, or 
 lose a little, and secure the safety of the 
 balance ? Common sense would pronounce 
 the former insanity, in all cases of considera- 
 ble hazard. 
 
 Let the economical merchant in Upper 
 Missouri invest his whole worth in a stock of 
 goods inSt. Louis, and ship them to Lexington 
 without insurance. The boat is old and pretty 
 well insured, and the owner would be willing 
 to lose her, if by her sinking he would be 
 enabled to purchase a new one; he sinks her: 
 or, a good boat may be shattered on a destruc- 
 tive stump, or burned: his goods are lost; his 
 money gone ; and too, his friends, one by one, 
 or en 7nasse, have disappeared; what resource 
 is left him of the earnings of former years of 
 toil, but the peaceful shades of undisturbed 
 poverty! He was too eager to be rich; 
 whereas, had he insured, he might, on the 
 recovery of his money, have reinstated him- 
 self in business, at the defiance even of 
 accident. 
 
 Insurance, like other divisions of Arithmetic 
 involving per cent., is wrought by Simple Pro- 
 portion. One hundred, or per centum, is the 
 given sum that gives a specified premium. 
 Any other sum, therefore, will, as it is larger 
 or smaller than 100, gain a greater or less sum 
 of premium. For instance : 
 
INSURANCE VARIETY FIRST. 
 
 113 
 
 VARIETY FIRST. 
 
 A building is insured at the valuation, 2000 
 dollars, at li per cent, premium. Now, it is 
 quite evident that the li premium is gained 
 by 100, or it could not be so much per centum. 
 The question may, therefore, be stated in 
 proportion, thus : What premium will insure 
 2000 dollars worth of property, if li dollars 
 insure 100 dollars worth of property ? The de- 
 mand is 2000, the same name 
 100, and the term of answer 
 li premium. The answer 
 will be the premium for in- 
 surance in dollars. The premium is $25. 
 
 Again : what is the sum 
 premium for insuring a coast- 
 ing vessel worth 3000 dolls., 
 at 7i per cent. ? 
 
 What premium must be 
 paid on a shipment of goods 
 from New Orleans to Havre, 
 worth 6,280, at 2i per cent.? 
 
 What is the annual insurance 
 
 ,10014000 5 
 
 41 5 
 
 A00I3000-5 
 
 
 on a cotton 
 factory worth 80,000 dollars, at 4 per cent.? 
 
 It is perceived here, that 
 the rate is placed last on the 
 right; consequently, the an- 
 swer is in premium. If the 
 amount insured is dollars, 
 
 0,0002 
 
 $|600 
 
 the answer is 
 dollars ; if cents, the answer is cents ; because 
 if the demand is dollars or cents, we place 
 100 dollars or cents opposite, and the premium 
 must be a given sum on 100, of the same 
 
114 RAINEY'S IMPROVED ABACUS. 
 
 denomination. We might obtain the same 
 result in the cases above, by suspending the 100 
 on the left, and cutting off two figures at the 
 right of the result for decimals of dollars or 
 cents, as the case might be. Thus, in the last 
 example : 
 
 Here, we cut off the two 
 
 ] #0,000 2 
 
 
 decimals of a dollar, which 
 are found on the right as the 
 consequence of suspending 
 the 100 on the left. But when the rate is frac- 
 tional, which is not un frequently the case, we 
 must, to multiply by it with facility, place the 
 denominator on the left of the line. In cases 
 of prime numbers, or of odd cents in the sum, 
 it may be well to drop the 100, and place on 
 the right, the sum and rate only. What is 
 the premium on a whaler for the voyage, 
 worth 9783 dollars ? 
 
 9?83 , The. 100 could not be ea- 
 l sily divided by in this case; 
 hence, it is dropped, and two 
 
 ' figures are stricken off for it 
 
 in the answer. In fact, it may be said to be 
 used for the purpose of keeping up the pro- 
 portional statement, more than anything else. 
 A. wishes to purchase a coasting vessel 
 worth 8000 dollars, and insure it for one year 
 at 7f per cent. : what sum of money will be 
 necessary to pay both purchase and insurance ? 
 This may be done in two ways : first, by find- 
 ing the premium, and adding it to the cost of 
 the vessel; 2d, by saying, if 100, cost price, 
 amount to 107J cost and premium, what will 
 8000 cost, amount to, with premium? 
 
INSURANCE STATEMENTS COMBINED. 115 
 
 We find that the cost of 
 this vessel, with the premium 41431' 
 
 added, amounts to 8620 dol- liaForf" 
 
 lars : hence we infer that the 
 premium is 620 dollars. 
 
 Questions such as the following frequently 
 occur, when it is quite convenient and simple 
 to combine statements. Purchased 18000 Ibs. 
 of cheese at 5i cents per lb., on a credit of 1 
 year; but for ready money I am allowed 10 
 per cent, discount. I ship the cheese to Nash- 
 ville, and pay If per cent, insurance: what 
 will the cheese cost me in Nashville, exclusive 
 
 of freights? 4 11 $000 3 
 
 In this statement, we find 
 the number of cents that the 
 
 400 
 
 100 
 
 305 
 
 $1915,00 
 
 whole quantity of cheese 
 comes to, by multiplying by 
 o or y cents : I then dis- 
 count 10 per cent., by saying, what will all 
 these cents be reduced to for present worth, if 
 110 be reduced to 100? Having now the 
 present cost of the cheese, we say, what will 
 all these cents be advanced to, if 100 be ad- 
 vanced to 101 1 for insurance, and find that my 
 answer is 915,00 dollars. The two figures are 
 cut off for cents, because the price was in 
 cents, and all the operations have been per- 
 formed on cents. Were the freight in this 
 case 10, 15, or any other rate per cent., it 
 might be added to the 101$. making 111$, 
 116$, &c., which would be placed on the line 
 as above. 
 
116 RAINEY'S IMPROVED ABACUS. 
 
 , VARIETY SECOND, 
 
 In insurance, teaches the method of finding 
 the rate per cent, at which an insurance is 
 effected, when the premium and sum are 
 given; as in the following: Paid 60 dollars 
 premium for the insurance of a building, 
 valued at 3000 dollars : what was the rate per 
 cent. ? The question here is, what will be the 
 premium on 100 dollars, or per cent., if on 
 $3000 it is 60 dollars? Hence, per cent., or 
 100, is the demand, 3000 the same name, and 
 60 dollars premium, the term of answer. We 
 wish the answer in premium, thus, 
 
 Jt is found by . this ' that the 
 ra ^ e i nsure d a t is 2 per cent., 
 which, may be proven as 
 
 , 
 I follows: if 100 gains 2 pre- 
 
 mium > what wil1 300 g ain ? 
 The answer is here, 60 
 dollars, the sum of premium 
 first ascertained. This ex- 
 ample, proving the other, is wrought according 
 to Variety 1st, in Insurance. From the nature 
 and statement of the former question, it may 
 be inferred that, 
 
 To Jind the rate per cent, of Insurance, when 
 the value of the property and premium arc gain, 
 Make 100 the demand: the sum insured, the same 
 name; and the premium, the term of answer : the 
 answer will be the rate per cent, premium. 
 
 If a man pay 40 dollars per annum for the 
 insurance of his house, worth 1200 dollars,, 
 what rate per cent, does it cost him? 
 
INSURANCEVARIETY THIRD. 
 
 The rate is 3 J per cent. ; for 
 this is the price of insurance 
 for 100 dolls, worth of property. 
 This question could be proven 
 in the same manner as the one above. 
 
 VARIETY THIRD, 
 
 Is to find the sum insured, when the premium 
 and rate are given. Thus, if I pay 90 dollars 
 premium at 3 per cent., what is the sum 
 insured? It is clear, in this case, that 3 dolls, 
 premium, require 100 dolls, worth of property ; 
 now, therefore, what will 90 dollars premium 
 require? We consequently make the whole 
 premium, the 90 dollars, the demand : the rate 
 of premium, or per cent., the same name ; and 
 $100, value of property insured by the 3 per 
 cent., the term of answer. The answer is 
 found, therefore, in amount of property, thus, 
 
 We find by this, that the - ^ 3 
 
 amount insured is $3000; 
 which may be proven by 
 showing that the premium 
 
 100 
 
 $|3000 
 
 on this sum, at 3 per cent., is 90 dollars. 
 Therefore, 
 
 To find the amount of property insured, when 
 the rate and premium are given, make the whole 
 sum paid for the premium, the demand : the rate 
 per hundred, the same name; and 100 the term 
 of answer. The answer will be the value of the 
 property insured. 
 
 An importer paid $700 premium, on wines 
 imported from Madeira to Cincinnati, which 
 
118 RAINEY'S IMPROVED ABACUS. 
 
 was H per cent, on the sum insured ; how 
 much did he insure ? 
 
 We find that the cargo of wines 
 was worth $56000. This may be 
 100 proven by finding the premium on 
 
 this sum at li per cent., by Variety 
 
 $|56000 lgt) w hich would be $700: or by 
 ascertaining the rate of premium by Variety 2d. 
 
 VARIETY FOURTH, 
 
 Is somewhat different. By it we ascertain 
 what sum must be insured on a specified 
 amount of property, to recover both property 
 and premium in case of loss. 
 
 A. owns a coasting vessel worth $8400, 
 and wishes to insure at 4 per cent., so that if 
 lost, he may recover both the value of the 
 vessel and the premium : what sum must he 
 insure on ? While the rate of premium is 4 
 per cent., nothing is plainer than that on a 
 policy of $100, only $96 worth of property can 
 be insured. Therefore, if $96 worth of prop- 
 erty, be advanced to $100, property and pre- 
 mium, the question arises, what will $8400 
 worth of property be advanced to for both prop- 
 erty and premium ? 8400 is the demand, 96 
 the same name, and 100 the term of answer, 
 thus, 
 
 96i8400 One hundred here, is the amount, 
 100 properly speaking, of the property 
 
 j and premium ; therefore, the answer 
 
 518750 I w jn k e sucn an amount as must be 
 insured on, to secure both. Hence the result, 
 8750. To prove this correct, we will find that 
 the premium on this sum at 4 per cent., is 
 
INSURANCE VARIETY FOURTH. HQ 
 
 $350, which subtracted, leaves the first value 
 of the vessel. Therefore, we conclude, that, 
 
 To find a sum to be insured that will secure 
 both property and premium, in case of loss, make 
 the value of the property the demand : one hun- 
 dred, reduced by the rate per cent., the same name; 
 and 100 the term of answer. The answer will be 
 the value of the property after the premium has 
 been reckoned and deducted. 
 
 Suppose I send to California an adventure 
 of $19000 worth of goods, at an insurance of 
 5 per cent. : what sum must be insured on, 
 that in case of total wreck, I may gain both the 
 original value of the goods and the premium ? 
 
 Here, as in the case above, 
 95 is the sum of property which 
 
 requires for amount of property 
 
 100 2 
 
 $| 20,000 
 
 and premium 100 : hence, we 
 say, what amount of both will 
 $19000 property require ? Thus, it is seen that 
 20000 dollars worth must be insured. It may be 
 proven by finding that the premium on 20,000 
 at 5 per cent., is $1000. This variety of opera- 
 tion is precisely identical with that of finding 
 the face of a note "given at bank, to draw a 
 specific sum. A similar process is used to find 
 the amount to be collected by tax-gatherers, 
 when it is desired to pay both the taxes and 
 the commission on them. 
 
 Life Insurance is reckoned as other insurance; 
 being a stipulated per cent., according to the 
 age and health of the assured. 
 
 To find the amount to be collected on taxes, to 
 pay both tax and commission, proceed as in Variety 
 4th in commission^ as above. 
 
120 RAINEY'S IMPROVED ABACUS. 
 
 TOLLS. 
 
 Tolls are charges made for the transportation 
 of merchandise, &c., by canals, and railroads : 
 and are generally a specific sum per mile, on 
 the 1000 Ibs., or the number of bbls., number 
 of perches, number of feet, &c., &c. What- 
 ever becomes the standard of a given article, 
 as 1 barrel, 1 cord, 100 feet, 1 perch, 1000 
 pounds, &c., must be placed on the left of the 
 line in working the question. For instance ; 
 what will be the tolls on 14000 Ibs. of flour, 20 
 miles, at 14 mills per mile on every 1000 Ibs.? 
 Here, we place down the num- 
 
 1000 
 
 14000 
 
 14 
 
 ber of miles first on the right, and 
 multiply it by the number of Ibs., 
 by placing the latter under the for- 
 mer. The 20 times 14000 would 
 be the whole number of Ibs. to be carried 1 
 mile : now, supposing this answer to be in- 
 volved, we say, what will all of these Ibs. cost 
 on the right, if 1000 Ibs. opposite, cost 14 
 mills ? Here, the answer is mills, because the 
 last on the right is mills. It would be in 
 cents, if the price of the 1000 Ibs. were in 
 cents. Hence, one figure is cut off for mills, 
 and the answer is $3, 92 cents, and no mills. 
 What must be paid for the transportation 
 of 18700 Ibs. bacon, 60 miles, at 15 mills per 
 1000 Ibs. ? 
 
CANAL AND RAILROAD TOLLS. 
 
 Here, we may place the price 
 in mills, and the answer will be 
 mills; or, $16,83, no mills. The 
 same result might be obtained by 
 calling the 15 mills 1$ cents, and 
 placing it down as |, thus, 
 
 Here, the answer is in cents, 
 and we cut off accordingly. It 
 is optional with the operator 
 whether he place the price in 
 mills or cents : but if in the for- 
 
 121 
 
 60 
 
 18700 
 _J_5_ 
 
 $116,83,0 
 
 1000 
 
 18700 
 
 $|16,83 
 
 mer, he must cut off, in all cases, one figure 
 at the right, for mills. 
 
 What are the tolls on 800 bbls. 
 flour, 80 miles, at li cents per 
 barrel ? 
 
 04 
 
 800 
 
 3 
 
 $|960,00 
 
 What tolls on 20000 Ibs. salt, 
 63 miles, at 1 cent per mile ? 
 
 What tolls on 192000 shingles, 
 at 5 mills per 1000, for 20 miles ? 
 The answer is 19 dollars and 
 20 cents. 
 
 What tolls on 1200 cubic feet 
 undressed stone, 40 miles, at 4 
 mills per perch, of 25 cubic feet. 
 
 ,1000 
 
 63 
 
 20,000 
 
 1 
 
 $|12,60 
 
 1000 
 
 
 192000 
 
 1 
 
 $|19,20 
 
 40 
 
 12004 
 
 4 
 
 $|7,68,0 
 
122 RAINEY'S IMPROVED ABACUS. 
 
 |M0* 
 
 1003000 
 
 What are the tolls on 3000 cubic 
 feet of lumber, at H cents per 100 
 feet, for 18 miles ? 
 
 40 
 
 8 
 
 3 
 
 What are the tolls on a pile of wood, 200 
 feet long, 40 feet wide, and 8 feet high, for 60 
 miles, at H cents per 100 feet? 
 
 Here, it is unnecessary to multi- 
 ply together the 3 dimensions of the 
 pile of wood : while the placing 
 of the several numbers on the 
 right, will indicate the same. We 
 1576 00 m ight ca ^ these three dimensions 
 64000 feet of wood, and use, on 
 the right, only 60, 64000, and f , which would 
 produce the same result. 
 
 It frequently becomes necessary to reckon 
 tolls, when it is not convenient to go through 
 with tedious multiplications and divisions : 
 hence, the necessity of combining several op- 
 erations, as above, in one statement. From 
 the foregoing we conclude, that, 
 
 To ascertain tolls on merchandise, Place the 
 distance in miles, and quantity of freight in Ibs., 
 feet, 4* c -> with the price, on the right ; and the 
 standard measure, of the specified article of freight, 
 on the left: that is, if tolls are charged per 1000 
 Ibs., place 1000 on the left: if on 100 feet, place 
 100 on the left, fyc. 
 
ANALYSIS OF COMPOUND PROPORTION. 123 
 
 COMPOUND PROPORTION. 
 
 We have seen that a simple proportion is formed 
 by the combination of two equal ratios ; hence, a Com- 
 pound Proportion is formed of two or more of these 
 simple proportions, instead of a compound and simple, 
 ratio, as is said by some authors. In some cases, as 
 in the following example, the compound proportion is 
 formed of the two combined ratios of the causes, and 
 the one ratio between the effects ; but when there are 
 more terms in the effect than one, it is found necessa- 
 ry to ascertain a number of ratios, as well in the ef- 
 fects, as in the causes; so that in such case, 
 the compound proportion would be composed of two 
 compound ratios, instead of one compound and one 
 simple. 
 
 It may be shown, as in the question following, that 
 by two or more statements in simple proportion, we may 
 easily ascertain the result of a compound question. 
 
 If 4 men in 8 days mow 12 acres of grass, how 
 many acres will 8 men mow in 16 days? The first 
 statement is, as 4 men to 8 men, so are 12 acres to 24 
 acres: the second, as 8 days to 16 days, so 24 acreg 
 to 48 acres Thus, how many acres will 8 men mow, 
 if 4 men mow 12 acres? 
 
 The answer is 24 
 
 12_ 
 24~ 
 
 Again : How many acres will 16 days work give, if 
 8 days work give 24 acres ? 
 
 The answer, 48 acres, is thus obtained by 
 two separate and easy statements. Here 
 it is seen, that in each of the two ratios, we 
 
 24 
 
 48 
 
 have made a proportion; which proportion 
 
 has in each case shown an increase in the number of 
 
124 RAINEY'S IMPROVED ABACUS. 
 
 acres, until this number has become as great as if 
 originally multiplied by the two ratios combined, which 
 make 4; thus, 4 : 8 gives 2, and 8 : 16 gives 2, and 
 twice 2 are 4, the combined ratios. Now the 12 acres 
 multiplied by this ratio 4, becomes 48 acres, as 
 before. 
 
 In compound proportion there are always five terms 
 given to find the sixth, seven to find an eighth, nine to 
 find a tenth, eleven to find a twelfth, thirteen to find 
 a fourteenth, fifteen to find a sixteenth, etc., etc. 
 
 In a compound, as in a simple proportion, the two 
 mean terms are always equal to the two extremes. Be- 
 cause there may be ten, twelve, or more terms in a com- 
 pound proportion question, is no reason that the theory 
 and the governing principles of the question are any 
 the less the doctrine of means and extremes, than if 
 there are four terms only. All of these several terms 
 must be so classified that they can be united in four 
 distinct bodies ; which four bodies become means and 
 extremes of proportion. The great difficulty with 
 most authors on arithmetic has been to systematize 
 this classification, so as to present it as a general 
 truth. This classification depends on the relations of 
 cause and effect ; and although many years since a 
 European writer, Dr. Lardner, discovered that these 
 causes and effects were the great foci in the state- 
 ment of such questions, yet no system of bringing 
 them together, or into their appropriate connection 
 and place, was discovered, by which their use could be 
 availed. It has been so in all sciences. Many bril- 
 liant practical and useful axioms have been discovered, 
 that have lain dormant in neglect, merely because the 
 modus operandi of their application did not accom- 
 pany the discovery, so as to give it efficiency. 
 
 We shall, therefore, consider, 
 
 FIRST The principles involved in Cause and 
 Effect : 
 
THEORY OF CAUSE AND EFFECT. 125 
 
 SECOND The application of these principles; a?id, 
 
 THIRD The form of statement according to the 
 necessities of their relation, so as properly to avail all 
 the benefits of cancelation in their reduction. 
 
 Causes are anything that involve action^ or imply 
 capacity ; and which in their action, or the contents 
 of their capacity, produce effects. Action always 
 commences in a life-giving principle; pursues some 
 regular medium; and invariably shows some effect 
 when it ends. Capacity, or Geometrical extent, in- 
 stead of producing any effect, merely exercises an in- 
 fluence over effects. It produces nothing, because it 
 has no action; this action being always essential to 
 the formation of an object. Geometrical extent per- 
 tains to containing, circumscribing, or consuming ei* 
 fects; and, as such, is not an active, but a passive 
 cause. 
 
 The principles in Cause and Effect pertain strictly 
 to matter ; and, as such, admit of no subdivision. 
 They form simply the two categories of the produc- 
 ing and the produc-ed : the agent and the object. 
 
 The medium through which causes operate to pro- 
 duce effects cannot be real or material; has no parts, 
 and is only imaginary. It is like a ray of light shoot- 
 ing through the aperture of a window of a darkened 
 room, and leaving a brilliant spot upon the wall. The 
 bright spot is the effect ; the sun, the cause ; the space 
 between them being only a straight line, conceivable 
 and imaginary, though without parts. A house that 
 has been built by a man is the effect, while the man 
 is the cause. No relation exists between these but 
 the builder and the built. It may be urged that there 
 are instrumentalities necessary to enable the cause or 
 agent to do this work or produce this effect. A pro- 
 per analysis will show, however, that these instrumen- 
 talities should be merged into their prime causes, and 
 become part of them. It would be said, that in 
 9 
 
12$ RAINEYS IMPROVED ABACUS. 
 
 building the house, tools were the instruments ; but a 
 little reflection will suggest that these are used only 
 to give the hands more efficiency, and facilitate the 
 execution of the work. So the truth is resolved back, 
 that the man is the plastic cause that conforms and 
 molds to his own taste; and the house, the object 
 made. 
 
 Ancient philosophers divided and subdivided these 
 agencies and effects, in actual and material operations, 
 into ten or twelve different categories; but these are 
 both useless and unreasonable, and tend greatly to 
 confusion ; while the practical fact recurs, that all ob- 
 jects in nature are directly, either causes or effects. 
 
 Causes implying action as well as capacity, and 
 causes definitively speaking, being active and endowed 
 with life, we conclude, that 
 
 FIRST All animate things are causes. Hence we 
 say, active causes are men and animals. When we say 
 men, we include the whole human race men, women, 
 and children. These constitute a higher order of 
 causes, being endowed with reason ; and may, there- 
 fore, be called intelligent causes. When we say ani- 
 mals, we refer to every living, moving, creeping, fly- 
 ing thing that Grod has made. Such may be classed 
 among active irrational causes. 
 
 Capacity being considered cause, we conclude that 
 
 SECOND All time is cause ; that is, every conceiv- 
 able subdivision of time, as centuries, generations, 
 years, months, days, hours, minutes, seconds. Time 
 within itself has none of those active powers that can 
 entitle it to the name of an efficient cause; but it 
 seems to be co -efficient, from the fact that it gives ex- 
 tent or capacity in which active and operative causes 
 may produce effects. This- capacity, or sphere of ac- 
 tion, it will be shown, is as essential to the existence 
 of the effect as the cause itself; though in a secondary 
 degree in point of production. 
 
THEORY OF CAUSE AND EFFECT. 127 
 
 Geometrical extent is not unfrequently the bounds 
 determining the extent of matter consumed or used; 
 and hence the occupation of the vacuum or limit, de- 
 pends on its prescribed limits Hence we conclude, 
 that, 
 
 THIRD Geometrical extent is sometimes a cause, 
 and sometimes an effect : a cause when no more osten- 
 sible or efficient cause is found in the question; and 
 an effect when the dimension of something produced 
 by an efficient cause. 
 
 For instance: it is a cause when, as in making 
 cloth, etc., a certain number of yards in length and quar- 
 ters in width consume a given quantity of wool : it is an 
 effect only when the dimension of something produced ; 
 as the length, width, and hight, of a wall, etc., built 
 by a given number of men, etc. In the latter case, it 
 is an effect, because the work is effected by active 
 causes, such as men. As the dimensions of a wall, 
 this extent might be a cause; as in determining the 
 quantity of stone consumed in its construction. 
 
 Anything representing active endeavor may be call- 
 ed a cause : hence we conclude, that, 
 
 FOURTH Capital is a cause, when it produces in- 
 terest ; because it has delegated to it all the productive 
 and cumulative capacities of active individual effort. 
 Capital may be said to be a cause only when it produces 
 interest. There are, however, instances in which a 
 sum of money becomes a cause; but only on similar 
 grounds to that of gaining interest, as the representa- 
 tive of active powers. It is only in transferring ac- 
 tion to the object that the object becomes a cause. 
 When this is done by conventional usage, as in the 
 case of interest, where the laws give this quality to 
 capital, it may be called a relative cause. 
 
 It is a self-evident truth in common sense, that a 
 certain number of separate causes must produce the 
 
128 RAINEY'S IMPROVED ABACUS. 
 
 same number of separate effects : or that the extent 
 of the effect is greater or less, as the cause is greater 
 or less ; and that the quality of the effect depends on 
 the nature of the cause ; or as the change is, in the 
 cause, so must it be in the effect. 
 
 That causes always exist before effects are pro- 
 duced ; and that no effect can be produced without a 
 commensurate cause; and that causes, as causes, 
 always produce effects, are alike self-evident and 
 reasonable. 
 
 The cause occupying one position and the effect an- 
 other, thus, 
 
 CAUSE E FFECT, 
 
 and the space between them being an imaginary line, 
 no one can doubt that they are as directly opposite 
 in their nature and relations as east and west ; right 
 and left. Nor will it be disputed, that if from the 
 cause we pursue a given direction to find the effect, we 
 must, in returning from the effect to the cause, trace 
 back the line by which we first sought the effect. 
 
 We know that the united energies of any number 
 of causes, cannot in a natural operation produce more 
 than one effect at a time : otherwise it would be ne- 
 cessary for them to operate in two directions at the 
 same time, which would be impossible ; for as unity of 
 cause produces one thing, so, whatever produces more 
 than any one thing at the same time, must be more 
 than one train of causes. But, as no object can be 
 encompassed or circumscribed by the direct operation 
 of any one other object, influence, cause, or thing, so 
 no one cause, or agent in the sum of cause, can, unaided 
 by another agent, produce an effect. 
 
 Although men may be considere.d one of the most 
 active species of causes constituting the sum of causa- 
 tion, yet, the 8 men, above would never, of themselves, 
 mow 12 acres of grass ; for, however active they may be 
 
THEORY OF CAUSE AND EFFECT. 129 
 
 as one element of causation, yet their powers must have 
 some sphere of action', and this sphere is time. So 
 that when we combine the energies of the men with 
 the capacity of the time, both together constituting a 
 community of endeavor, are enabled to encompass or 
 effect the desired object. Any one thing by itself as 
 a unit* is inoperative; but when multiplied into some- 
 thing else, cumulates or expands in the ratio of 
 squares. It is this expansion of the combined ener- 
 gies of causes that enables them to encompass effects ; 
 and it is the necessity of this, which prevents any one 
 bare agent in a cause producing any effect by itself. 
 * It may be said that certain chemicals produce pow- 
 erful effects by their lone agency : this, however, can- 
 not prove the position, that any one essential element 
 of cause can produce an effect ; for whenever any chem- 
 ical effect is produced, it is the consequence of the com- 
 bination of different essential elements, within the 
 chemicals, or in the air, which is the medium of 
 operation. Hence, different effects are produced by 
 the combination of different elements; the combined 
 effort being the parent of the inception. 
 
 From the foregoing we may safely conclude, that 
 Causes are 
 
 MEN, 
 ANIMALS, 
 TIME, 
 
 CAPITAL, or 
 MEDIUM : 
 Or, whatever produces an effect. 
 
 Every problem in Compound Proportion, and in 
 Simple Inverse Proportion, is composed of its terms 
 of supposition, and its terms of demand; and every 
 supposition and every demand has in each, causes and 
 effects. 
 
 These causes and effects we will endeavor to clas- 
 sify, so as to form a rational and philosophical state- 
 
130 RAINEY'S IMPROVED ABACUS. 
 
 ment, which will, at the same time that it is clear, be 
 unencumbered with those varied unmanageable depen- 
 dencies of terms, common to the old form of state- 
 ment; while it will be general in its applica- 
 tion, and susceptible of all the abbreviation prac- 
 ticable in cancelation. Some of the old works have 
 indeed canceled some little in this department of arith- 
 metic ; but not to any very useful extent, by reason of 
 having no system of arranging all of the terms so as 
 to assume the form of dividend and divisor. This is 
 the great desideratum, so far as canceling is concern- 
 ed, in all modern- works that pretend to use it, in any 
 other, as well as in this department of numbers. 
 
 The student has to analyze and state a question 
 thoroughly, before he can determine what terms are 
 divisors and what dividends. He can have, therefore, 
 no regular or unique system of statement ; and is 
 at all times necessitated to hunt up dividends and 
 divisors merely as such ; whereas, by this method, he 
 has the well-defined landmarks of cause and effect, 
 and states according to the two great and infallible 
 directions that nature has given him, in all created 
 things. 
 
 In the following problem we will assign to causes 
 and effects their appropriate places. 
 
 If 4 men in 8 days mow 12 acres of grass, how 
 many azres will 8 men mmv in 16 days ? 
 
 We write the supposition first, and then the de- 
 mand, and draw a line under each ; thus, 
 
 men. days, acres. men. days, acres. 
 
 4 ., 8 , 12 8 . 16 .. 
 
 A short line is likewise drawn under the causes both in 
 the supposition and demand. Here 4 men and 8 
 days are the causes in the supposition, and 12 acres 
 the effect : in the demand, 8 men and 16 days are the 
 
APPLICATION OF CAUSE AND EFFECT. 1 fU 
 
 causes, while a cipher or blank indicates the place of 
 acres, which is the term in which the answer is 
 demanded. 
 
 We now draw two vertical lines near each other ; 
 the left for the supposition and the right for the de- 
 mand. On the left side of each line we place 
 and on the right, effects ; thus, 
 
 CAUSE. EFFECT. CAUSE. EFFECT 
 
 Supposition* Demand. 
 
 The left being the line of supposition, we write 
 supposition under it ; and the right being the line of 
 demand, we write demand under it. Now, all of the 
 causes in the supposition are placed on the left side 
 of the left line, and all of the effects on the right. 
 All of the causes in the demand are placed on the left 
 side of the right line, and their effects on the right. 
 It appears in this question that we have no effect in 
 the demand ; consequently, a cipher is placed on the 
 right, to show that it is deficient, and that the answer 
 is required in this term; thus, 
 
 , E It is perceived here, that the cause 
 12 8 1 6 ^ a ^ P ro( ^ uces the l^ acres, is divided 
 
 o -jJ into two parts, men and days; but 
 
 these men and days, are placed together 
 on the same side of the line, that their powers may 
 be combined by multiplication ; thus rendering them 
 unique as a cause. The same is the case with the two 
 -causes on the other line. Therefore, we may multiply 
 the two causes into one., in each separate case. Four 
 times 8 in the first, make 32, combined cause ; and 8 
 times 16 in the second, make 128, a similar combina- 
 tion. Instead of placing, as above, the separate fac~ 
 
 c. 
 
132 RAINEY'S IMPROVED ABACUS. 
 
 tors of each cause on the line, we may place their 
 product; thus, 
 
 32 '12 128 Here it is perceived, that we 
 have three terms of a proportion 
 _ given to find the fourth; that is, 
 the two means 'and one extreme, to find the other 
 extreme. Now, we know that if these two means 
 be multiplied together and divided by the given ex- 
 treme, the other extreme will be found; which, too, 
 will be the required answer. The question, as now 
 stated, appears to be, as the cause, 32, is to the cause 
 128, so is the effect, 12, to the required effect, 48 ; or, 
 as the first cause is to the first effect, so is the second 
 cause to the second or required effect.* 
 
 This proportion may be transposed as other pro- 
 portions, into eight different forms or readings; the 
 causes and effects being the terms. When either the 
 means or extremes are multiplied together, they are the 
 product of one cause and one effect ; and this product 
 is divided by the given cause, to find the required ef- 
 fect ; or by the given effect, to find the required cause. 
 
 But suppose this question be changed and proven, 
 by saying: If 4 men in 8 days mow 12 acres of grass, 
 in how many days.wiB 8 men mow 48 acres? Here, 
 4 men and 8 days are causes in the supposition, and 
 12 acres the effect : 8 men and blank (0) days the 
 causes in the demand, and 48 acres the effect ; thus> 
 
 12 8 
 
 
 48 We p er ceive that it takes both men 
 and days, two causes, in the supposition, 
 to produce the effect, 12 acres ; and in- 
 fer that two similar causes should exist in the de- 
 mand, to produce the one effect, 48 acres. In the de- 
 
 * We do not allude here to the specific ratio between the causes and their 
 effects ; we only refer to their categorical and regular bearings. Strictly 
 speaking, there is no ratio between cause and effect, except that acciden- 
 tal ratio which depends upon the nature ef things, in their unclassified and 
 irregular state. We desire to discard this forced, and irregular, and cir- 
 cumstantial connection or ratio, and place proportion on its true basis,- 
 $i between, things tkat are alike.. 
 
PROPORTION AMONG CAUSES AND EFFECTS. 133 
 
 mand only one of these causes is found ; hence, the 
 cause is deficient; that is, one of the terms of the 
 means is deficient; and a member that is necessary to 
 constitute the causation, producing the effect, 48. 
 Hence, while all of the extremes are multiplied to- 
 gether for a dividend, the product should be divided 
 by all of the remaining means to get the required 
 mean. We know that 16 days is the answer required. 
 Now, the product of the extremes is 4x8x48=1536, 
 which, divided by the product of the means 12x8= 
 96, gives 16, the number of days required. Thus, we 
 might drop any one of the terms, of either the means 
 or extremes, and ascertain it, by dividing the product 
 of the other two, by the remaining terms. 
 
 The deficient term, or that in which the answer is 
 required, is always either a cause or an effect. This 
 deficiency must always be indicated by a cipher. It 
 never falls on either side of the left, or line of 
 supposition ; because the supposition must in all cases 
 be perfect, and have, consequently, no deficiency; but 
 it is always found on either the right or left side of 
 the right line, as the term wanting may be an effect or 
 a cause; for this demand is always deficient; and it 
 is this deficiency which makes it a demand, by asking 
 what will supply it ; while the operation is performed 
 for no other purpose than to supply it. 
 
 To be certain that the blank, or cipher indicating 
 the deficiency, is located correctly, we must consider 
 whether the answer desired is a cause or an effect. 
 If a cause, the blank is placed under the head of 
 cause, on the left side of the line ; if an effect, it must 
 be placed under the head of effect, on the right side 
 of the right line. Or we may count the number of 
 causes on each line, and if they be equal, the defi- 
 ciency is an effect. Again : we may count the num- 
 ber of effects on each line ; if they are not equal, the 
 deficiency is found; if they are equal, the deficiency 
 must be among the causes. 
 
134 RAINEY'S IMPROVED ABACUS. 
 
 After placing all of the terms in the question on 
 the two lines, according to the directions given above, 
 notice whether the blank falls among the mean terms ; 
 that is, between the two lines ; or whether among the 
 extremes, or terms outside the two lines : if it fall 
 among the inner terms, all of the inner terms must be 
 placed on the left side of the vertical line, or line of 
 ratio, on which the question is to be wrought: if 
 among the outer terms, all of the outer terms must be 
 placed on the left of this line. This is based on the 
 principle of dividing by the mean or extreme, as the 
 one or the other may be deficient, to ascertain the 
 deficient term. Being thus stated, the question may 
 be canceled as in other cases. Let us now recur to 
 the first question : 
 
 If 4 men in 8 days mow 12 acres, how many acres 
 will 8 men mow in 16 days? and state it thus, 
 
 Here we inclose the mean terms en- 
 
 12 8 
 16 
 
 
 
 tirely, which are 12, 8, 16, that they 
 ~ may be clearly distinguished from the 
 extreme, which are 4 and 8. The blank falling on the 
 outside, the outer terms become the divisors, and the 
 inner terms the dividend, and are placed on the 
 right; thus, 
 
 ^ 143 I 
 
 Eight equals 8 ; 4 into 12, 3 times, 
 and 3x1^=48 acres, the answer. 
 |48 acres, i 
 
 We now resume the second question, which will 
 prove this correct. 
 
 If 4 men in 8 days mow 12 acres of grass, in how 
 many days will 8 men mow 48 acres? 
 
 Four and 8 are the causes, in the supposition, and 12 
 the effect; 8 and blank the causes, in the demand, and 
 48 acres the effect. After stating the question, thus, 
 
COMPOUND PROPORTION. 
 
 135 
 
 
 4.8 
 
 * s seen that ^ ne blank is among the 
 means; hence, the means, 12 and 8, be- 
 come the divisors, and are accordingly 
 placed on the left ; while 4, 8, and 48, the extremes, are 
 placed on the right of the line, where the dividend ia 
 always placed ; thus, 
 
 14 
 
 m 
 
 16 da. 
 
 ! 
 
 Days being the deficient term, the an- 
 swer is 16 days. 
 
 Again : If 4 men in 8 days mow 12 acres, how 
 many men will be required to mow 48 acres in 16 
 days? 
 
 12 
 
 
 16 
 
 48 
 
 8 men. 
 
 Here, again, the mean terms be- 
 come the divisors. Hence, 8 men, 
 the answer. 
 
 Again: changing the supposition and demand, 
 If 8 men in 16 days mow 48 acres, how many acres 
 will 4 men mow in 8 days ? 
 
 848 
 16 
 
 
 
 4ST-1 
 
 12 acres. 
 
 In this instance, one of the 
 terms of supposition is dropped, 
 which makes it deficient; it becomes, consequently, 
 the demand; while all of the terms of the former 
 demand are supplied; making the demand, in its 
 turn, the supposition. Hence, the reason for placing 
 it on the left line. The answer is 12 acres. 
 
 Again: If 8 men, in 16 days, mow 48 acres of 
 grass, in how many days will 4 men mow 12 acres? 
 thus, 
 
136 
 
 RAINEY'S IMPROVED ABACUS. 
 
 848 
 
 16 
 
 12 
 
 The first question has now been 
 proven in all of its terms ; showing 
 
 40- 
 
 8 da. 
 
 conclusively the similarity between the two causes, and 
 the two effects, as the four terms of a proportion. 
 
 Having presented the foregoing rationale of Com- 
 pound Proportion, we will now state and solve a num- 
 ber of problems involving all of the varieties com- 
 mon to this department ; that in them the reader may 
 have a sure guide to the solution of all similar ques- 
 tions, occurring either in theory or practice. 
 
 If 4 men, in 8 days of 10 hours long, mow 40 
 acres of grass, in how many days of 12 hours long 
 will 15 men mow 60 acres? 
 
 440 
 8 
 10 
 
 1560 
 
 12 
 
 Here, the different lengths of the 
 days are expressed by the hours, as 
 
 A* 4 
 
 2|da. 
 
 causes, in each case. Hours have, therefore, assigned 
 to them the place of cause, as all other causes of time ; 
 and cooperate with the men and days to produce the 
 effect, by extending the time or sphere of execution, 
 and defining its limits. The answer is 2| days. 
 
 If 4 men in 20 days of 8 hours long, build a wall 
 400 feet long, 32 ft. wide and 5 ft. high, in how many 
 days, 15 hours long, will 6 men build another wall, 
 300 ft. long, 80 feet wide, and 30 feet high? 
 
 While 4, 2, and 80 are the causes in the supposi- 
 tion, 400, 32, and, 5 are the dimensions of the effect, 
 which, properly considered, is a unit. In both cases, 
 supposition and demand, these dimensions are made 
 the effect, and we proceed with the work as hereinbe- 
 
COMPOUND PROPORTION IN FRACTIONS. 137 
 
 fore described. The answer being required in days 
 the blank comes under the head of cause, to show it. 
 Hence the mean terms are the divisors, and the an- 
 swer is a term of cause, 80 days. 
 
 20 
 
 8 
 
 400 
 32" 
 
 
 15 
 
 300 
 
 80 
 
 30 
 
 Two ciphers equal two ciphers; 8 
 times 4 equal 32 ; 6 times 5 equal 30 ; 
 3 into 15, 5, and 5 times 4 equal 20; 
 the answer is 80 days. Now, let us I 
 prove this question by using this answer, and drop- 
 ping some other term ; the 80 feet wide, for instance. 
 The question will be as follows : 
 
 If 4 men in 20 days of 8 hours long, build a wall, 
 400 feet long, 32 feet wide, and 5 feet high, how wide 
 will that wall be, the length being 300 feet and the 
 hight 30 feet, which 6 men, in 80 days, 16 hours long, 
 will build ? 
 
 :400 
 
 >32 
 
 85 
 
 2032 
 
 80 
 16 
 
 300 
 
 
 
 30 
 
 Here it is seen that the blank 
 comes under the head of effect, one 
 of the dimensions of the wall, which 
 causes us to divide by the extremes. 
 
 300 
 
 7*00 
 32' 
 
 A 
 
 85ift.w. 
 
 The following question involving fractions may be 
 as easily disposed of as the others. We will simply 
 place the mixed numbers on the two lines of supposi- 
 tion and demand, without reducing them, until they 
 are brought to the line of ratio; when, after being re- 
 duced as in other cases, to improper fractions, the nu- 
 merators will occupy just such position as if they 
 were whole numbers, with their respective denomi- 
 nators opposite them. Thus, the question will be 
 
S IMPROVED 
 
 the I'orm of whole nunibtjrs, avuiuij-p: thi 
 inorous iractioiial difficulties which usually render such 
 solutions too complex to be interesting to the general 
 reader. 
 
 If 5 men, in 7-J- days of 12^ hours long, dig a ditch 
 400 feet long, 4| feet wide, and 6 feet deep, how 
 many men will be required in 20 days, 6 hours 
 long, to dig another ditch 640 feet long, 3f feet wide, 
 and 3i feet wide ? 
 
 500 
 4f 
 
 4 
 
 0640 
 203 
 
 We have placed all of the inner 
 terms on the left, and ah 1 of the 
 outer terms on the right, without 
 observing any particular order as 
 to which should be placed on the 
 line first. 
 
 Men. 
 
 815 
 
 If 6 men in 10 days, 8 hours long, dig a ca- 
 nal 200 yards long, 80 yards wide, and 3 yards deep, 
 in how many days of 12 hours long, will 5 men dig an- 
 other canal 300 yards long, 40 yards wide, and 16 
 yards deep, supposing the difficulty or density of the 
 soil to be, in the latter case to the former, as 5 to 2 ? 
 
 It is palpable, that 2 degrees density of soil must 
 be placed, among the effects of the supposition, to 
 determine the extent or difficulty of the effect to be 
 produced; while, for the same cause, 5 degrees of 
 density are placed among the effects of the demand. 
 The same results could be obtained, if 2i were placed 
 among the effects of the demand, or f among those 
 of the supposition ; in this instance, however, the sym- 
 metry of the terms would be lost, and the statement is 
 made according to the first suggestion; thus, 
 
COMPOUND PROPORTION IN FRACTIONS. 139 
 
 10 
 
 200 
 
 80 
 
 3 
 
 2 
 
 5J300 
 040 
 
 12 
 
 16 
 
 5 
 
 6 
 
 12 
 
 10 
 
 
 
 200 
 
 300 
 
 80 
 
 40 
 
 3 
 
 16 
 
 2 
 
 5 
 
 
 80 da. 
 
 We find in Compound Proportion 
 many conditions that can be appended 
 to questions, which ordinarily render 
 them apparently very complex; but 
 these conditions may be disposed in the most summa- 
 ry and natural manner, by assigning to the terms repre- 
 senting them, their proper places as causes and effects. 
 
 How long will 8100 Ibs. of bread serve a garrison 
 of 100 men, if 900 Ibs. are consumed in 50 days by 
 20 men? If, in this instance as in all others, indi- 
 cates the supposition, which is 20 men, 60 days, and 
 900 Ibs. of bread ; while the demand is 100 men, 
 days, and 8100 Ibs. We state as in other cases; re- 
 membering that whatever the language in which a 
 problem is stated, whether the men eat, or whether 
 the bread is consumed, whether the demand or sup- 
 position be given first, we must transpose the ques- 
 tion, until it is brought before the mind in its proper 
 bearings ; and then state according to common sense. 
 Some would think, at first sight of this question, that 
 because the pounds of bread served the garrison, these 
 pounds must necessarily be the cause, and the garrison 
 the effect. Such must consider, and ask in what 
 term the greater action exists ; whether in the bread, in 
 supplying or being eaten, or in the men, by eating and 
 consuming it. The latter implies action; while the 
 former is only subject to action, and as an object in 
 its passive form, becomes an effect, 
 
 201900 
 
 50' 
 
 100 8100 
 
 
 000 50 
 
 #009 
 
 90 days. 
 
140 
 
 RAINEY'S IMPROVED ABACUS. 
 
 If 60 horses in 20 days consume 15 bushels of oats, 
 how many horses will 600 bushels feed 120 days? 
 
 Horses become the deficient term ; hence we divide 
 by the inner terms, and find that 400 horses will con- 
 sume the 600 bushels in 120 days. 
 
 60 
 20 
 
 15 
 
 01600 
 
 120[ 
 
 0004 
 
 400 
 
 This question, as well as the fore- 
 going, and all that will follow, may be 
 proven in as many different ways as there are differ- 
 ent terms given in the supposition and demand. 
 
 If 400 musquitoes, in 30 nights, 15 hours long, 
 raise on an animal 60,000 lumps, of an inch in diam- 
 eter and i inch high, in how many nights, of 10 hours 
 in length, will 800 musquitoes be required to produce 
 50,000 lumps, -fj of an inch in diameter, and ^V of 
 an inch in hight? 
 
 Here, the number of lumps, their diameter, and 
 hight are the effects, in each case; while the musqui- 
 toes, nights, and hours are the causes. We state 
 accordingly. 
 
 400 
 30 
 15 
 
 60,000 
 
 
 
 800 
 10 
 
 50,000 
 
 800400 
 
 1030 
 60,00015 
 14 
 18 
 50,000 
 
 169 
 201 
 
 In this question numerous other 
 conditions might be added: it 
 might be said that the warmth of 
 the night, in the one case, was to 
 that of the other as a given ratio : 
 also, it might be said, that a cer- 16 m. 
 
 tain ratio of difference existed between the two sub- 
 jects operated on, as to adhesiveness of material, etc. 
 All of these conditions would be legitimate in such 
 questions, at least in theory. 
 
 We now come to the consideration of questions in 
 which the causes that produce effects, instead of being 
 
RELATIVE CAUSES IN PROPORTION. 141 
 
 active, are merely passive; with an efficiency dele- 
 gated, rather by the attendants, and controlling cir- 
 cumstances and customs of the day, than any inherent 
 or vital principle of action. 
 
 Capital and time are causes in the production of 
 interest, not because they have any active powers that 
 would enable them to encompass or accomplish this 
 object, but because public common consent grants that 
 capital may, by representing individual effort, be 
 supposed to do this; and this supposition or permis- 
 sion of the thing, is equivalent to the act or fact, 
 " other things being equal," so far as matters of com- 
 mon consent are concerned. 
 
 If I lend my friend, in his need of money, $200, for 
 20 days, and he agrees to render me a similar accom- 
 modation when necessary; and when my occasion re- 
 quires it, he has only $150, how long must I use this 
 sum to remunerate me for the loan of the $200 ? 
 
 The question appears to be this : If $200 in 20 days, 
 render 1 accommodation, in how many days will $150 
 render 1 accommodation? The 200 and 20 are the 
 causes in the first case, and 1 accommodation the ef- 
 fect; while, in the other, 150 and days are the 
 causes, and, as before, 1 accommodation the effect. 
 State as follows: 
 
 200 
 20 
 
 150 
 
 
 Hence the answer, 
 
 150 200 
 
 20 
 
 26| days. 
 
 The pupil will notice, that in the foregoing propo- 
 sition the effects are equal; that is, the effect in the 
 supposition and demand is the same, or common. We 
 shall have occasion to use this fact in our remarks on 
 Inverse Proportion; for it is easily seen, that if the 
 one effect equals the other, they can both, as terms, be 
 dispensed with : hence, in the statement of the ques- 
 10 
 
142 
 
 RAJNEY S IMPROVED ABACUS. 
 
 tion without them, we would find only three terms 
 given for ascertaining the fourth; and this would 
 make it a simple proportion. 
 
 Let a few other questions be now presented, in 
 which capital is cause, and which will teach the pupil 
 the very important method of finding principal, time, 
 and rate in interest. We will find, first, the interest 
 on $200 for 3 months, at $6 gain on the $100, or 6 
 per cent.; and here, as in other cases, we must have a 
 supposition, which, if not given in the question, must 
 be taken from legally-established custom. We say, 
 therefore, 
 
 If $100 dollars, in 12 months, gain $6 interest, 
 what interest will $200 gain in 3 months? 
 
 The causes in the supposition are $100 and 12 
 months, while $6 interest is the effect; and in the de- 
 mand $200 and 3 months, while the effect is wanting. 
 State as follows : 
 
 10016 
 
 12 
 
 200 
 
 
 
 We find the sixth term $3, which 
 is the effect; showing that, if $100 
 
 8. 
 
 3 int. 
 
 in 1 year gain $6 interest, twice this sum, in 
 time, will gain ^ of 6, or $3 interest. 
 
 the 
 
 TO FIND THE PRINCIPAL IN INTEREST. 
 
 Having ascertained the interest, let us now find the 
 capital that in 3 months would produce this interest. 
 To do this we make the following statement : 
 
 If $100, in 12 months, gain $6 interest, what sum 
 of capital will gain $3 interest in 3 months? thus, 
 
 1006 
 
 12 3 
 
 Our answer is $200 capital, which 
 is correct ; because at the given 
 
 100 
 
 & 
 
 $ 200 prin. 
 
TO FIND PRINCIPAL, TIME, AND RATE. 143 
 
 rate per cent, it has just gained, in the time specified, 
 the $3 interest. Here, it has become necessary to 
 assume the standards of per centum and per annum 
 in interest, for supposition. 
 
 TO FIND THE TIME IN INTEREST. 
 
 Changing the statement, it may be well to find the 
 time in which $200, at 6 per cent., will gain $3 inter- 
 est : and when we say at 6 per cent., we assume that 
 $100, in 12 months, 360 days, or 1 year, gains this 
 $6. Thus, the statement : 
 
 If $100, in 12 months, gain $6 interest, in what 
 time will $200 gain $3 interest ? thus, 
 
 1006 013 
 
 12 
 
 200 
 
 The answer is 3 months time. 3 mos. 
 
 This answer may again be changed so as 
 
 TO FIND THE KATE OF INTEREST. 
 
 It is only necessary now, to complete this series of 
 proofs and proportions, to ascertain the rate at which 
 money is lent, when a certain sum has gained another 
 specific sum of interest. We commenced in the ques- 
 tion, by saying, that the rate per cent, was 6 ; and 
 have used this rate in the statement of the supposi- 
 tion with which it is connected. It is now desired 
 to find this 6 per cent., per annum. When we say 6 
 per cent., per annum, we mean 6 on the one hundred, 
 for or by the one year, or 12 months. It cannot be 
 denied, that this $6 was gained by the $100 in one 
 year, or 12 months. The former supposition, "If 
 100 in 12 gain 6," now becomes the demand; one of 
 
144 
 
 RAINEY'S IMPROVED ABACUS. 
 
 its terms, the 6, being deficient. The former demand, 
 therefore, now becomes the supposition, and is as 
 follows : 
 
 If $200, in 3 months, gain $3 interest, what in- 
 terest will $100, or per centum, gain in 12 months, or 
 per annum? 
 
 The answer will certainly be the sum that the $100 
 in 1 year would gain ; that is, $6, which proves it 6 
 per cent, per annum. 
 
 20013 
 I 
 
 10010 
 
 a 2 
 
 6 per ct. 
 
 We may in a similar manner find the rate at which 
 interest has been gained on a given principal, and for 
 any given time, by saying, if such principal, in such 
 time, gain so much interest, what interest will $100 
 principal gain in 1 year, 12 months, or 360 or 365 
 days, as the case may be ; for, if the time in the sup- 
 position be in years, 1 year must be used in the de- 
 mand: if in months, 12 months in the demand; and 
 if in days, 360 days. For example : 
 
 If $60, in 120 days, gain $1| interest, what is the 
 rate per cent, per annum ; or, what will $100 gain in 
 860 days? Thus, 
 
 60111 
 
 120J 
 
 100 
 
 360 
 
 00 ^00 
 
 ^0 0032 
 
 In the question preceding this, 
 the reader will perceive, that al- 6 per cent, 
 
 though this is a compound statement, yet the suppo- 
 sition and demand are located, as in simple proportion, 
 with the term of answer, $3 interest, last on the right. 
 
 This is not an inverse statement, because the an- 
 swer is required in interest, which is an effect. 
 
 Again : Suppose I have a debt of $40 to pay in 80 
 
COMPOUND PROPORTION: CAUSES OF CAPACITY. 145 
 
 days, and have no means of getting the money except 
 by lending to some individual a sum of my capital, 
 large enough to gain this sum of debt, in the given 
 time, at 6 per cent.; how much capital shall I thus 
 put on interest, for the 80 days, to gain the $40 ? The 
 question would be stated, thus, 
 
 If $100, in 360 days, gain $6 interest, what prin- 
 cipal will gain $40 in 80 days ? 
 
 1006 
 360 
 
 
 
 80 
 
 40 
 
 #006 
 
 0100 
 
 405 
 
 It is thus seen, that the sum to be 
 negotiated is $3000, which may be $ 3000 cap. 
 easily proven, by asking the interest, at 6 per cent., 
 on this sum, for 80 days, which would be found $40. 
 
 We next proceed to the consideration of causes of 
 capacity, properly so called. It is known that the di- 
 mensions 8 feet long, 4 feet wide, and 4 feet high 
 make 1 cord. Now, 
 
 If 4 feet high, 4 feet wide, and 8 feet long make 1 
 cord of wood, how many cords will a pile 400 feet long, 
 60 feet wide, and 16 feet high make? 
 
 In each case the dimensions are the causes, and the 
 number of cords which they make, the effect. We 
 state accordingly, 
 
 400 
 60 
 16 
 
 
 
 The answer is 3000 cords of wood. 
 
 460 
 
 3000 cor. 
 
 Now, suppose we wish a pile to be made large 
 enough to make a certain number of cords, where two 
 of its dimensions are given ; for example, 
 
 If 4 feet wide, 4 feet high, and 8 feet long make 
 1 cord, how high must a pile be, whose width is 8 feet 
 and length 40 feet, to make 10 cords? thus, 
 
146 
 
 RAINEY S IMPROVED ABACUS. 
 
 010 
 
 8 
 40 
 
 The pile must be 4 feet high ; | 
 which may be easily proven by sus- 4 feet hi. 
 pending some other dimension, or finding the number 
 of cords that these complete dimensions will make. 
 
 If 10 yards of cloth, J of a yard wide, consume 40 
 Ibs. of wool in the facture, how many pounds will it take 
 to make 20 yards, f of a yard wide V 
 
 We have shown that no one cause, by itself, can pro- 
 duce an effect, until united with some other cause ; we 
 have likewise shown, that no cause, or combination of 
 causes, can produce more than one effect. Now, al- 
 lowing the existence of these truths, it would be ridic- 
 ulous to say that the pounds of wool, in the question 
 above, was the cause of producing so much cloth, 
 so, and so wide. But, to say that the length 
 and width in yards and quarters, disposed of, or con- 
 sumed a given number of pounds, would be both rea- 
 sonable and natural. The want of a little discrimi- 
 nation and reflection here, will involve the student 
 in innumerable absurdities ; while, on the other hand, 
 a proper and intelligent attention to the relations and 
 operations of causes and effects, cannot fail conduct- 
 ing him to proper and palpable results. 
 
 1040 
 
 200 
 
 Our answer, 93i Ibs., is the effect 
 of the capacity of the 20 yards long 
 and wide. These two, as causes of 
 
 10140 
 
 20 
 
 7 
 
 93i 
 
 capacity, consume or appropriate this number of 
 pounds of wool, as the effect. We are apt to fall into 
 the same unheeded absurdity in questions like the 
 following : 
 
COMPOUND PROPORTION : CAUSES OF CAPACITY. 147 
 
 If the transportation of 20 tons of freight, 30 
 miles, cost $60, how many tons can be carried 800 
 miles for $250? 
 
 Here the tons and distance are the causes, and the 
 price the effect: thus, 
 
 2060 
 30 
 
 
 800 
 
 250 
 
 250 
 
 The answer is 3| tons. 3| tons. 
 
 The same result may be obtained, though impro- 
 perly, by saying that the dollars are the causes, and 
 the weight and distance the effect. If all the rela- 
 tions of the more palpable operations of cause and 
 effect have but one tendency, and which reversed, dis- 
 orders the whole, certainly, because in other cases 
 these laws and relations are not so easily perceived, is 
 no reason that they are not founded upon, or governed 
 by, the same fixed and eternaj principles. Nor can 
 these principles be neglected* or perverted, without 
 consequent disorder, at some time, and in some place. 
 
 If $40 pay for hauling 16 cwt. of hemp 28 miles, 
 how many miles can -56 cwt, be hauled for $200 ? 
 
 16140 
 
 28| 
 
 0|200 
 56| 
 
 200 
 
 40 miles. 
 
 It is an easy matter, in any question, to determine 
 what terms have such a connection as to require that 
 their energies be combined ; while it is not less easily 
 determined which term appears the great focus of all 
 these operations. The former are causes, while the 
 latter is an effect. 
 
 If a cistern 8 ft. deep, 20 ft. long, and 3 feet wide, 
 hold 340 barrels, how many barrels will be contained 
 in one 28 ft. deep, 60 ft. long, and 15 ft, wide? 
 
 The barrels are, in each case, the effects. 
 
148 
 
 RAINEY S IMPROVED ABACUS. 
 
 20 
 3 
 
 340 
 
 28 
 60 
 15 
 
 17 
 (7 
 
 15 
 
 It may be observed here, that in 
 all cases, such as the one above, if 17850 bis. 
 
 inches be annexed to, or connected with, the feet, they 
 must be reduced to the fraction of a foot. Three feet 
 4 inches may be reduced by saying, 4 inches are T 4 2 
 or of a foot ; this fraction added, makes 4| or y feet. 
 
 If 2150 inches make 1 bushel, how high must a 
 crib be, to contain 800 bushels, which is 200 inches 
 long and 86 inches wide? 
 
 In this question, it is the geometrical extent of all 
 the sides of the bushel measure, which is the cause ; 
 the 2150 being the product of the three dimensions. 
 The cube root of this number would be one of the 
 sides; and this, placed down three times, on the left 
 side of the line of supposition, would constitute the 
 causes producing the effect, one bushel. As it is un- 
 necessary to find this side an9 use it three times, 
 which, at best, could only reproduce the number 2150, 
 we merely write the 2150, which is the product of 
 these sides, and produces precisely the same result. 
 In place of the two remaining factors constituting 
 2150 y we use two inverted commas; thus, 
 
 2150 
 
 200 
 
 86 
 
 
 
 800 
 
 5 
 
 100 in. hi. 
 
 It is seen, that the crib should be 100 inches high. 
 By this mode of operation, either the contents, or any 
 of the sides of a crib, body, box, etc., may be found. 
 
 The question above may be proven, by asking, how 
 many bushels a crib will contain, which is 200 inches 
 long, 86 inches wide, and 300 inches deep. Again: 
 
 If 2150 inches make 1 bushel,, haw long must a bos 
 
COMPOUND PROPORTION : CAUSES OF CAPACITY; 149 
 
 be, to contain 1 bushel, which is 20 inches wide and 
 15 inches deep? 
 
 21501 
 
 
 
 20 
 15 
 
 We find that it must be 
 
 3 4 
 
 643 
 
 7i in. 
 
 inches long. One bushel is the common effect in both 
 the supposition and demand. We see by this calcu- 
 lation, that a box 7 in. deep, 20 in. long, and 15 in. 
 wide, will contain one bushel. This may be proven, 
 by multiplying the three sides to a continued product. 
 We might, in this instance, give a short, but somewhat 
 mechanical, rule for finding any remaining side of any 
 figure of 6 sides, when the contents were a unit of 
 any given standard, or any number of units of such 
 standard, as 1 bushel, or 80 bushels; 1 gallon, or 20 
 gallons, etc. 
 
 According to the foregoing, we may, in a similar 
 manner, find the. dimensions of a box or cistern that 
 will hold a given number of gallons, if the standard of 
 unity be first assumed, and used as a supposition; 
 thus: 
 
 If 231 inches make a wine gallon, how long, wide, 
 or deep must a box be, to hold any specific number of 
 gallons, if it be of a given width, or depth, etc.? 
 
 Beer gallons may become the standard of measure, 
 by using 282, instead of 231 inches 
 
 If 231 inches make 1 wine gallon, how long must a 
 box be, to contain 100 gallons, which is 15 in. wide, 
 and 11 in. deep? 
 
 231 
 
 
 
 15 
 11 
 
 100 
 
 1002 
 
 140 in. long. 
 
 By this, it is seen, that the box must be 140 inches 
 long. 
 
150 
 
 RAINEY S IMPROVED ABACUS. 
 
 If 2688 inches make 1 dry bushel, how wide must 
 a body be, to hold 100 bushels of coal, which is 120 
 in. long, and 32 in. deep ? 
 
 Here, 2688 is the product of the three sides of the 
 dry bushel ; and, consequently, must be considered as 
 the unit of cause, that produces or contains the 1 
 bushel. We state as in the similar case above. 
 
 2688 
 
 
 
 120 
 32 
 
 100 
 
 100 
 
 70 in. long. 
 
 The width of the body must be 70 inches. 
 
 We know that 1 inch wide, 1 inch thick, and 3 T 8 / F 
 inches long, make 1 pound "of cast iron. Now, these 
 dimensions are the causes in the supposition, and 1 
 pound, the effect. 
 
 We may take the 3 T \\, or, which is equivalent, 3|i 
 solid inches to represent the dimensions of the quan- 
 tity, and say, 
 
 If 3f inches make 1 Ib. of iron, how many Ibs. 
 will a bar make, that is 24 in. long, 4 in. thick, and 6 
 in. wide? 
 
 240 
 
 6 
 
 This 3fi can be. very easily 
 used without the least difficulty in 150 Ib. 
 
 the fraction, as f f , as above. 
 
 The question may now be changed; and having the 
 width and thickness of the bar, let us see how long it 
 must be to weigh 150 pounds. 
 
 By careful attention to the statement and solution 
 of this question, any question arising as to the length, 
 width, or thickness of bars of iron, window-weights, 
 etc., etc., may be easily solved. It frequently be- 
 comes necessary, after the two dimensions and weight 
 are given, to find the other dimension, which is 
 
COMPOUND PROPORTION : CAUSES OF CAPACITY. 151 
 
 often very troublesome to practical men; especial- 
 ly if many fractions are found in the work. The 
 question is, 
 
 If 3f i inches make 1 lb., how long must a bar be, 
 that is 6 in. wide, and 4 in. thick, to make 150 Ibs? 
 
 150 
 
 From this work it appears, that 24 in. long, 
 the bar must be 24 in. long, which is true; because 
 this length was used to find the weight. 
 
 If 3-f.j- inches make 1 lb., 94 inches will make 25 
 Ibs. The product of 96 is composed of 8 inches 
 long, 4 inches wide, and 3 inches thick ; therefore, in 
 stating questions of this nature, it may be said, 
 
 If 96 inches make 25 Ibs ; or, if 8 in. long, 4 wide, 
 and 3 thick make 25 Ibs., how long, or thick, or wide 
 must a bar be, with two dimensions given, to make 
 any specific number of pounds? 
 
 If 8 in. long, 4 in. wide, and 3 in. thick make 25 
 Ibs., how long must a window -weight be, to make 25 
 Ibs., which is 4 in. wide and 1-J-in. thick? 
 
 25 
 
 
 4 
 
 H 
 
 25 
 
 The bar is 16 in. long. Again : 16 in. long 
 
 If 8, 4, and 3 make 25 Ibs., how wide must a bar 
 be, to make 25 Ibs., which is 12 in. long and 1-Jin. 
 thick ? 
 
 25 
 
 
 12 
 
 25 
 
152 
 
 RAINEY S IMPROVED ABACUS. 
 
 This mode of stating such ques- 
 tions can be easily used, if any at- 
 tention be given to the general 
 principles. 
 
 5-j- in. wide. 
 
 All mixed numbers must be reduced to improper 
 fractions, here, as elsewhere. 
 
 We will now solve one or two other questions, and 
 close this article by endeavoring to show how such 
 statements may be made on one line, as in Simple 
 Proportion. 
 
 If 10 men, in 15 days, 8 hours long, compose a book 
 of 16 sheets, 36 pages on a sheet, 25 lines on a page, 
 and 60 letters in a line, in how many days, 10 hours 
 long, will 40 men compose another book, of 48 sheets, 
 32 pages on a sheet 50 lines on a page, and 90 let- 
 ters in a line ? 
 
 10 
 15 
 
 16 
 36 
 25 
 
 60 
 
 4048 
 
 
 1050 
 
 32 
 50 
 90 
 
 $ ,24 
 
 00 
 
 The answer is 24 days. 24 days. 
 
 There is a novelty in the statement and solution of 
 this question, arising from the fact that it can be 
 wrought without- the use of one figure more than is 
 necessary to state it. Instead of canceling, as above, 
 we may have said, 16 into 32, twice ; twice 9 into 36, 
 twice; and twice 4, on the left, equals 8: 10, equals 
 10; cipher, equals cipher; 25 into 50, twice; 2 into 6, 
 three times; 3 into 15, 5 times; five into 10, twice; 
 and 2 into 48, twenty -four times ; the answer, as above. 
 
COMPOUND RROPORTION: SINGLE STATEMENT. 153 
 
 One more example is given, to illustrate the method 
 of stating on one line. 
 
 If 4 men in 16 days, 12 hours long, compose a book 
 of 14 sheets, 24 pages on a sheet, 44 lines on a page, 
 and 40 letters in a line, in how many days, 8 hours 
 long, will 12 men compose a similar work, of 42 sheets, 
 16 pages on a sheet, 48 lints on a page, and 55 let- 
 ters in a line ? 
 
 14 
 24 
 1244 
 40 
 
 16 
 
 12 
 
 
 
 42 
 16 
 
 48 
 55 
 
 Twelve equals 12 ; 8 into 16, twice, 
 and twice 4, on the right, make 8, 
 which goes into 40, five times ; 5 into 
 55, eleven; 11 into 44, four times; 4 24 da. 
 into 16, four times; 4 into 24, 6 times; 6 into 42, 
 seven times ; 7 into 14, twice ; and 2 into 48, twenty- 
 four times ; the answer. 
 
 This question may be stated in the following order : 
 
 In how many days, 8 hours long, will 12 men pro- 
 duce a certain effect, if 4 men produce the same in 16 
 days, 12 hours long? 
 
 The answer is desired in days ; a cause. Hence, 
 this term, 16 days, is placed last on the right for the 
 term of answer; while the causes, which cooperate 
 with it to produce the given effect, are placed on the 
 same side of the line,, where their energies can be 
 multiplied. Now, these causes being necessarily on 
 the right, the causes in the demand must be placed 
 opposite them, on the left. Hence, in all inverse ques- 
 tions, the demand is placed on the left: thus, 
 
 Here, if no effects were to 
 be considered, the answer 
 would certainly be in days : 8 
 days. Thus, we see, that 
 when the answer desired, is a 
 
 Men, 
 Hours, 
 
 12 4 Men, 
 12 Hours, 
 16 Days, 
 
 8 days. 
 
154 RAINEY'S IMPROVED ABACUS. 
 
 cause, the demand, that is, the causes of demand, 
 must be placed on the left ; and those of the supposi- 
 tion, opposite. Now, if causes and effects are always 
 directly opposite in their nature, the effects in the 
 demand, must be placed on the right; and those of 
 the supposition, opposite ; thus, 
 
 Sheets, 1442 Sheets, 
 
 Pages, 2416 Pages, 
 
 Lines, 4448 Lines, 
 
 Letters, 40 55 Letters, 
 
 It is seen here, that the 
 effects occupy the reverse 
 side of the line. This is, 
 both because causes and 
 effects are opposite in their 
 nature ; and because, when 
 an answer, as the one above, is desired in an effect, 
 the proportion is direct, requiring that the demand be 
 placed on the right. Thus, 3 is the answer ; which, 
 multiplied with the 8, gives 24, as in the former work. 
 Each term is placed opposite the term of its own 
 kind, to get the ratio. This order of statement is re- 
 versed, if the answer is required in an effect. In 
 such case, all of the causes, in the demand, are placed 
 on the right, and those in the supposition, opposite ; 
 while all the effects are placed opposite their respec- 
 tive causes. 
 
 From the foregoing, we deduce the following 
 
 SUMMARY OF DIRECTIONS FOR COMPOUND PROPORTION. 
 
 Separate the question into terms of Supposition and 
 Demand : Ascertain which terms, both in the suppo- 
 sition and demand, are Causes, and which are Ef- 
 fects : Draw two vertical lines : Place all of the terms 
 of supposition on the left line ; and all of the terms 
 of demand, on the right line : Place causes on the left 
 side of each line, and effects on the right. 
 
 If the answer be required in a cause, place a ci- 
 pher on the left side of the right line ; if in an effect, 
 place a cipher on the right side of the right line. 
 
INVERSE PROPORTION. 155 
 
 When the cipher falls between the two lines, make 
 all of the inner terms the divisor, and all of the outer 
 terms the dividend : when it falls outside of the two 
 lines, make all of the outer terms the divisor, and all 
 of the inner terms the dividend* The causes are 
 men, animals, time, capital, and medium. 
 
 To state on one line : 
 
 When the answer is desired in a cause, place all 
 of the causes in the demand, on the left ; and all of 
 the causes in the supposition, on the right ; with the 
 respective effects of each, opposite: 
 
 When the answer is desired in an effect, plact all 
 of the causes in the demand, on the, right ; and all of 
 the causes in the supposition^ on the left ; with the re- 
 spective effects of each, opposite. 
 
 SIMPLE PROPORTION INVERSE. 
 
 The consideration of Compound Proportion, leads 
 to that of Simple Inverse Proportion ; the great 
 leading principles of both, being cause and effect. We 
 have seen, from the solution of one or two questions 
 in compound proportion, that the effect in a compound 
 question being the same, or a unit, in both supposi- 
 tion and demand, there are but three remaining 
 terms to use. The effect, being unity in each case, 
 we say, that the two terms are common; and conclude 
 that the problem, as properly enounced, is composed 
 of causes only. It is evident, that if all of the terms 
 are causes, and that as some one of them must be the 
 denomination of the answer, therefore, this answer must, 
 when obtained, be a cause. And here exists the differ - 
 
 * Recollect that the divisor is always placed on the left' side of the line 
 on which the question is wrought, and the dividend, on the right. 
 
156 RAINEY'S IMPROVED ABACUS. 
 
 eace between direct and inverse proportion. From what 
 has been said of the relations of cause and effect, it 
 is evident, that as causes exist before their effects, so 
 the production of these effects is a direct or first ope- 
 ration of the cause ; hence, it is the province of di- 
 rect proportion to ascertain effects. If, again, causes 
 and effects possess an opposite nature, it is reasonable 
 to suppose, that when the cause is demanded as an 
 answer, the course pursued to find it, will be a retro- 
 grade from the effect, to the cause. Hence, the direct 
 course is broken, and reversed ; and the line of opera- 
 tion, which the cause first pursued to find the effect, is 
 retraced from the effect to the cause. This inver- 
 sion, or turning around, is the basis of the theory of 
 Inverse Proportion. The term is derived from the 
 Latin, inverto, to turn back. 
 
 We may always know that a question is inverse, 
 if there are no given terms of effect, whatsoever, above 
 unity. When the effect is not common, the question 
 is necessarily compound; having more than three 
 terms. 
 
 All inverse questions are properly compound propor- 
 tion, and may be solved most intelligibly and easily, 
 by a statement under this head, according to the great 
 principles just treated. Therefore, when they are re- 
 duced to the form of simple proportion, we may rea- 
 sonably expect an entire change in the statement, from 
 questions ordinarily occurring in this department. The 
 following illustration, according to compound propor- 
 tion, may be given : 
 
 How many men will be required to do a piece of 
 work in 15, which can be done in 24 days, by 5 men? 
 
 The enunciation of the question may be changed, 
 thus : 
 
 If 5 men in 25 days do one piece of work, how many 
 men will be required to do one other piece in 15 days? 
 
THEORY OF INVERSE PROPORTION. 157 
 
 The piece of work is the effect, in each case; a 
 common effect -, and is located as follows : 
 
 24 
 
 
 15 
 
 8 men. 
 
 It is seen, that in dividing by the 
 inner terms, the 15 is placed on the 
 left. This term would be the demand in simple pro- 
 portion ; and, it is evident, that it is placed on the left, 
 that, by dividing the product of the other two terms, 
 the fourth term may be found; which, linked with the 
 15, would complete the proportion. It is not, however, 
 placed on the left as a factor only, to find another 
 factor; but that its ratio, with a similar term on the 
 opposite, may be ascertained. The 5 men is the de- 
 nomination of the answer, and must, consequently, be 
 placed last on the right. Now, this 5 men does not 
 equal in value the term of supposition opposite, the 15 
 days, which is the sole connection between the two 
 similar terms in direct proportion; but, as a cause, it 
 is combined with another cause, the 24 days, to pro- 
 duce the common effect, one piece of work. That 
 these two causes may occupy a position where their 
 energies may be multiplied together and cumulated, 
 the 24 days is placed on the right of the line, in the 
 place ordinarily assigned to the demand, and just over 
 its cooperative term, 5 men; for causes producing a 
 common effect, cannot be separated. Now, to obtain 
 the ratio between this 24 days in the supposition and 
 the 15 days in the demand, the 15 must be placed op- 
 posite. Hence, by the necessity of purely philosophi- 
 cal principles, the demand in inverse proportion is in- 
 verted, or placed on the left. This is the only satis- 
 factory reason why the demand changes its place, or 
 is inverted. These laws, necessitating more than one 
 cause to produce an effect, and an actual connection 
 of the various terms employed, so as to elicit their 
 
158 RAINEY'S IMPROVED ABACUS. 
 
 combined energies, in their creative capacity of some 
 effect, have been so clearly elucidated and demonstra- 
 ted in the discussion of first principles, as to leave no 
 doubt, on the part of the reader, that both terms in 
 the supposition must be placed on the right; one, as 
 the denomination of the answer, and the other as the 
 term to be compared with the term of similar name, 
 in the demand, on the left. The two ones may be 
 dropped in the statement. Again : 
 
 If 15 men, in 8 days, do a piece of work, how many 
 men will do the same in 24 days ? thus, 
 
 The demand is placed on the left ; 
 the same name opposite it ; and the 
 ,r term of answer, last on the right. Let 
 
 { us prove this again. 
 If 8 men, in 15 days, do a piece of work, how 
 many days will 5 men be required to do the same ? 
 
 L g 
 
 ** *L g While men are compared in this state- 
 
 ment, days become the term of answer. 
 
 |24days. ! 
 
 It is sometimes difficult to determine what is 
 cause, and what effect; especially, when geometrical 
 extent is a cause. Therefore, for the benefit of those 
 who are not experienced in the designation of causes, 
 we give the following directions for stating such ques- 
 tions correctly ; and which, to some minds, is the only 
 means of determining whether a proportion is direct 
 or inverse: 
 
 If the answer desired should be larger than the 
 same term in the question, phice on the left the smaller 
 of the two terms to be compared by ratio : if the an- 
 swer required be smaller, place on the left, for the de- 
 mand, the larger of the two terms to be compared. 
 This is what is called by the old schoolmen, " more 
 giving less, and less giving more;" which simply 
 
INVERSE PROPORTION IN FRACTIONS. 159 
 
 means, if the one cause or factor in a question be 
 larger, the other must be smaller ; and vice versa. 
 
 Most authors have written about the necessity of 
 inverting terms in inverse ratio, as they call it (which, 
 in truth never existed), as well as about more giving 
 less, and less giving more: but none of them have 
 ever explained satisfactorily the reason for this inver- 
 sion, or why "more gives less, and less more." Nor 
 have any of them so reasoned on the assumption, as 
 to give the pupil even their own vague and indefinite 
 ideas ; much less satisfying his common sense with a 
 rational and demonstrable theory. No doubt that the 
 best arrangements practicable have been made ; for it 
 is wholly impossible to explain the theory on any other 
 principles than those of cause and effect. These are 
 the very embodiments of inverse action ; the one go- 
 ing directly on to create ; the other returning to the 
 creator or cause, from the created effect. The obe- 
 dient pupil has too long followed in the wake of rules 
 that linger along in obscurity and darkness, not only 
 in this but in most of the departments of this beauti- 
 ful science. 
 
 How many yards of cloth, 4 quarters wide, are 
 equal to 40 yards, 5 quarters wide ? 
 
 5 
 
 Here, 4 quarters is the demand, 5 quarters 
 the same name, and 40 yards the term of an- 
 swer. 1 50 
 
 If it take 20 yards of cloth, f of a yard wide, to 
 make a gown, how many yards, wide, will it take to 
 line it? 
 
 The demand, , is placed on the left, by I 7 $ 2 
 its numerator; while the same name, f, is, in 
 the same manner, placed on the right. 
 
 17f 
 
 How many yards of silk serge, 2 yards wide, will 
 
160 
 
 RAINEY'S IMPROVED ABACUS. 
 
 line a coat, that requires 15 yards of cloth, f of a 
 yard wide ? 
 
 Fifteen yards being the denomination 
 of the answer, is placed last on the right. 
 
 5 yds. 
 
 If 5 yards long, and 2 yards wide, line a coat, how 
 long must the piece of cloth be, which is f of a yard 
 wide, to make it? 
 
 t* 
 
 ft p o rpjjg demand is % in this case, while the 
 
 _ 5 same name is 2|-. Again: 
 
 15yds. 
 
 If 15 yards long and f wide make a coat, how wide 
 must 5 yards of serge be, to line it ? 
 
 In this case, the two terms to be com- 
 pared, are in length ; while width is the de- 
 nomination of answer. The serge must be 
 2^ yards wide ? 
 
 If 5 yards long and 2^ wide, make a coat, how wide 
 must 15 yards in length be, to line it ? 
 
 m-i 
 
 43 
 
 t-tt 
 
 4 
 
 
 0-3 
 
 4 
 
 3 
 
 
 4 
 
 The answer is f of a yard wide. 
 
 It is perceived, that these questions are inverse, be- 
 cause all of the terms are causes of capacity. 
 
 Suppose a garrison has provisions to last 4 months, 
 at 20 ounces per day ; how many ounces should they 
 use, for it to last 6 months ? 
 
INVERSE PROPORTION. 
 
 161 
 
 The two different number of months 3 014 
 must here be compared ; 6 months being mQ 
 
 the demand, and 20 ounces the term of "lilTi 
 answer. I 
 
 If, when wheat is worth 60 cents per bushel, the 
 cent loaf weighs 10 ounces, how much ought it to 
 weigh when wheat is worth $1,25 per bushel? 
 
 The prices are compared, as causes, and 
 the weight is the term of answer. 
 
 125 
 
 60 
 10 
 
 4* 
 
 If, when wheat is worth $1,25 cents, the cent loaf 
 weighs 4f ounces, what ought it to weigh when wheat 
 is 60 cents per bushel ? 
 
 60 125 
 
 The answer is 10 ounces. 
 
 24 
 
 10 oz. 
 
 If a bushel of wheat make 40, five cent loaves, 
 how many eight cent loaves will it make ? 
 
 The ratio is obtained between the 8 and 
 5 cents, as the two most active causes con- 
 ducing to the effect. Thus, capital is at 
 times a cause. 
 
 25 
 
 If it require 110 yards of carpeting f of a yard 
 wide, to carpet a room, how many yards, 1-f yards 
 wide, will carpet the same? 
 
 tfaf 2 
 
 The answer is 60 yards. 
 
 60 
 
 If I lend a friend $100, for 40 days, how long ought 
 he to lend me $80, to return the accommodation? 
 
 > 1005 
 The answer must be in days. 
 
 "J50 da. 
 
162 RAINEY'S IMPROVED ABACUS. 
 
 How long must a board be, that is 9 inches wide, 
 to make a square foot? 
 
 The question should be, 
 
 If 12 inches long and 12 inches wide, make 1 foot, 
 what length, with 9 inches in width, will make the 
 same? 
 
 Nine, the width, is evidently the demand, while 12 in 
 width is the same name, and 12 in length is the term 
 of answer; for the answer is required in length. 
 
 $ 4-4 4 Without transferring and working the 
 jLA 4 question anew, we may prove it by 
 simply multiplying 9 and 16, which 
 make 144, the number of square inches 
 in a square foot. 
 
 If a certain pasture supply 1500 cows 75 days, how 
 long will it supply 2000 cows ? 
 
 16 in. 
 
 2000 
 
 1500 
 
 75 
 
 The causes are, in this instance, active 
 causes. 
 
 If a man perform a journey in 24 days, when the 
 days are 12 hours long, how many days will he be re- 
 quired to do the same, when the days are 18 hours 
 long? 
 
 The causes here, are all of the same 
 kind, time ; yet we know that the an- 
 swer must be in days, and place the 
 days, consequently, on the right. 
 We will now present a few questions in the calcula- 
 tion of machinery ; all of which, so far as velocity is 
 concerned, are inverse. 
 
 If a drum, 60 inches in diameter, make 72 revolu- 
 tions per minute, how many revolutions will a pulley 
 connected with it, make, which is 6 inches in diameter ? 
 It is necessary, in this question, to compare the two 
 diameters ; and, as 6 diameter is the demand, we place 
 
CALCULATION OF MACHINERY. 
 
 163 
 
 it on the left, and 60 diameter on the right ; while 72, 
 the number of revolutions, becomes the denomination 
 of the answer. 
 
 The answer is 720 revolutions _ 
 
 720 
 
 Let us now change this question, and find the diam- 
 eter of this pinion, knowing the number of revolu- 
 tions that we wish to make. 
 
 If 72 revolutions require 60 inches in diameter, of 
 what diameter must a pinion foe, to make 720 revo- 
 lutions ? 
 
 NAM 
 
 Thus, the diameter used at the outset 60 
 
 is again found. "~ /T~I 
 
 Again: Making the two diameters of the demand 
 thus found, the supposition, we will endeavor to find 
 #ach term of the former supposition, 72 and 60. 
 
 If 6 inches in diameter nmke 720 revolutions, how 
 many revolutions will 60 inches diameter make ? 
 
 Here, the diameters must be compared. 
 
 720 
 
 72 rev. 
 
 If 720 revolutions require 6 inches diameter, what 
 diameter will give 72 revolutions ? 
 
 Thus, we can find the diameter and 
 revolution of any wheel, the diameter and 
 revolutions of that with which it is con- 
 nected, being given. 
 
 6 
 
 60 in. 
 
 It frequently becomes necessary to Ascertain the 
 revolutions or -diameter of a wheel connected with 
 several others. This is done by what is called " con- 
 joined proportion," which our limited 'space will not 
 permit us to treat here- 
 
10 in. 
 
 200 in. 
 
 164 RAINEY'S IMPROVED ABACUS. 
 
 Suppose a counter wheel in a mill turn 40 times per 
 minute ; how large must a trundle be on a spindle, to 
 make 240 revolutions per minute, the counter wheel 
 being 5 feet in diameter ? 
 
 Two hundred and forty revolutions is 
 the demand, 40 revolutions the same 
 name, and 5 feet, or 60 inches, the diam- 
 eter of the counter wheel, is the term of 
 answer. Hence, the trundle must be 10 inches in 
 diameter. 
 
 How many inches in diameter must a water-wheel 
 be, to make 12 revolutions, if a counter wheel, 60 
 inches in diameter, makes 40 revolutions per minute? 
 
 &4Q These 200 inches make 16| feet, 
 
 which might be easily obtained by substi- 
 tuting 5 feet, for 60 inches, when the an- 
 swer would be in feet. This shows us, 
 that a water-wheel 16-j- feet in diameter, making 12 
 revolutions per minute, working in a crown wheel of 
 equal size, with a counter wheel, 60 inches in diame- 
 ter, making 40 revolutions, must have a trundle 10 
 inches in diameter, to make 240 revolutions per 
 minute. We may go on thus from one wheel to an- 
 other, to any extent, and apply the diameter and rev- 
 olutions of one to another, coming in regular sequence 
 after it, until we ascertain the revolutions and diame- 
 ters in a long chain of machinery. 
 
 Such calculations as the following, are frequently 
 necessary in finding the size of wheels, their " pitch- 
 line," and the number of " cogs," or teeth. They 
 come properly under the head of direct proportion ; 
 but are placed in this connection, that most of the re.- 
 marks of machinery may be found together. The 
 problems are solved as other questions in simple 
 proportion. 
 
 Suppose it be required to make a wheel 1^ inches 
 larger, whose pitch-line ia 12 if inches in diameter t 
 
INVERSE PROPORTION: MACHINERY. 165 
 
 how many cogs will the wheel so enlarged have, if the 
 first wheel had 67 cogs ? 
 
 We first add 1% to 12}f inches, making thus, 12-|- 
 1=13 inches : ^ inch equals T 8 F , and 8-J-15 sixteenths 
 equal f-f , or 1 7 7 F , which added to 13 makes 14y 7 F , the 
 diameter of the pitch -line, after adding the 1^ inches. 
 We now say, as 12 T f is to 14 T 7 g-, so will 67 cogs be 
 to the number required. 
 
 If a wheel 12-J-J- inches in diameter, have 67 cogs, 
 how many cogs will a wheel 14 T 7 g- inches in diameter 
 have; thus, 
 
 231 
 
 16 
 
 67 
 
 The demand is here placed on the right. 16 
 The wheel, thus increased in diameter, will 207 
 have 74|f cogs; and, as the number of 
 cogs must always be even, we will sup- 
 pose that the wheel shall have 75 cogs, 
 being the whole number that nearest expresses the 
 fraction. Now, this will require the pitch-line to be 
 still a little larger. So we say, if 67 cogs be ad- 
 vanced to 75 cogs, what will the pitch-line, 12 T f , be 
 advanced to ? Seventy-five cogs is the demand, 67 
 cogs, the same name, and 12} in diameter, the term 
 of answer. 
 
 67 
 16 
 
 75 
 207 
 
 The answer is a very minute fraction 
 under 14^- inches; and this pitch-line 
 can be made so nearly the right diame- 
 ter, as to step off with the " dividers " 
 75 cogs, without any perceivable fraction. 
 
 Suppose, again, a wheel, that is 18 inches in diam- 
 eter, has 100 cogs; how many cogs must it have to 
 be enlarged 6 inches ? 
 
 Here, 18-|-6=24, the diameter of the required 
 wheel. Now, how many cogs will 24 the desired di- 
 ameter require, if 18 inches, the given diameter, re- 
 quire 100 cogs; thus, 
 
166 RAINEY'S IMPROVED ABACUS. 
 
 18 
 
 24 
 
 100 
 
 1334- 
 
 This gives for the number of cogs in the 
 larger wheel 133-J, which not being an 
 even number, we call 133, that from this 
 
 smaller number of cogs, we may make the 
 new pitch-line proportionally smaller. 
 
 If 100 cogs require 18 inches diameter, what pitch, 
 diameter, will 133 cogs require? 
 
 100 
 
 133 
 
 18 
 
 I"** 
 
 Therefore, to make 133 cogs, which is 
 an even number, the new pitch-line must 
 be 23|J, or 23.94 decimal inches; which, 
 
 too, can be so accurately approximated, as 
 to give 133, in about the desired diameter. 
 
 The article on Inverse Proportion, although short, 
 has, we trust, a sufficient number and variety of ex- 
 amples to enable the judicious and discriminating 
 reader to apply the principles in practice whenever 
 found necessary. 
 
 We now close our remarks on the proportions; and 
 shall hereafter advert to them only to show the philo- 
 sophy of the statements that follow in the various rules 
 of arithmetic and mensuration : as all of these subdivi- 
 sions depend in principle and detail on the proportions ; 
 and are but different branches of them, assuming such 
 apparently different form, as the circumstances of their 
 application may require. We feel confident, that if 
 the reader will carefully review the foregoing pages, 
 with a close attention to principles, rather than prac- 
 tice, he will find no difficulty of perceiving the neces- 
 sary operations to be performed on all questions that 
 will follow in this work; nor in applying the principles 
 that he has learned, in such manner as to conduct to 
 satisfactory results. He cannot expect to do this, by 
 a slight and indifferent course of reading : progress in 
 arithmetical knowledge justifies no such conclusion, if 
 we consult the history of those who have attained 
 great proficiency in its principles. To become tho- 
 roughly acquainted with the science, requires not only 
 
INVERSE PROPORTION: RULE. 167 
 
 a due comprehension of each separate principle, but a 
 knowledge of all their relative bearings; and even 
 with these,, such an acquaintance and familiarity, as 
 to render their constant contemplation, which ever 
 enables us to discern new beauties, rather a pleasure 
 than a task. Nor must we regard cancelation as a 
 short-hand system, merely, that will enable us to ar- 
 rive at results instanter, without that thought neces- 
 sary to the apprehension of great general principles in 
 any science : for the abbreviation of this system, con- 
 sists, not in the shortening of principles, but in their 
 use and concentration. If, moreover, we investigate 
 a proposition for the beauty and grandeur of its prin- 
 ciples and relations, instead of as a forced and neces- 
 sary task, for merely speculative purposes, we will 
 not fail being thoroughly acquainted with it; thus 
 connecting the pleasures of the science, with the neces- 
 sity of the art. 
 
 From the foregoing illustrations, we deduce the 
 following 
 
 SUMMARY OF DIRECTIONS FOR INVERSE PROPORTION. 
 
 Ascertain first the term of Demand : 
 Place this Demand on the left : 
 
 Place the term of the same name of the demand, 
 opposite the demand, on the right: 
 
 Place the term in ichich the answer is required, last 
 on the right. 
 
 If any of the terms are fractional. 
 
 Place the Numerators on the side of the line ordina- 
 rily assigned to the integers, and the Denominator 
 opposite 
 
168 RATNEY'S IMPROVED ABACUS. 
 
 CONJOINED PKOPORTION. 
 
 Conjoined Proportion differs from Simple Propor- 
 tion, only by reason that, instead of one, many differ- 
 ent proportions are found in one question. Hence, 
 all questions coming under this head, may be wrought 
 by continued separate statements in Simple Propor- 
 tion. The demand is placed on the right, and the 
 same name, in the supposition, on the left. This sup- 
 position merely equals in value the term in which the 
 answer is required, and which is placed invariably on 
 the right. Hence, statements in this species of pro- 
 portion are always direct. Now, the term which is of 
 the same name of that just used as denomination of 
 answer, may be placed opposite this denomination of 
 answer, and the new denomination of answer which 
 this second term of supposition, or same name, equals, 
 may be placed last on the right for a new term of an- 
 swer. So, again, this term of answer may be used as 
 demand, with another supposition opposite, and an- 
 other term of answer succeeding, until any number of 
 separate statements are merged into one general state- 
 ment. Hence, the statement will be but a concatena- 
 tion or chain of statements ; from this fact, it has re- 
 ceived the name of " chain rule" Nothing more is 
 necessary, to make this interlinking of statements 
 plain and easy, than to find the term of demand, or 
 that term for which an equivalent term in value, is 
 demanded. It will, therefore, be necessary, to place 
 terms of the same denomination, and such only, oppo- 
 site one another. 
 
 If 5 Ibs. of raisins are worth 81bs. of tamarinds, 
 and 3 Ibs. of tamarinds are worth 18 Ibs. of figs, and 
 20 Ibs. of figs are worth 75 Ibs. of cheese, and 40 Ibs . 
 
CONJOINED PROPORTION. 
 
 169 
 
 of cheese are worth 16 Ibs. of butter, and l&f Ibs. 
 butter are worth 30 yards of calico, and 5 yds. calico 
 are worth 4J- bushels of apples, how many bushels of 
 apples are worth 13^ Ibs. of raisins ? 
 
 It is evident, that the answer must be in bushels of 
 apples, and that the demand is 13^ Ibs. of raisins: 
 hence, we conclude, that if raisins is the demand, 
 raisins must be the same name; and consequently, 
 place opposite the 13f Ibs. raisins, the term of raisins 
 given in the question, which is 5 ; and so on with all 
 of the other terms, making the 8 Ibs. tamarinds, which 
 this 5 Ibs. equals, the term of answer, under the de- 
 mand, and, again, the 3 Ibs. of tamarinds, the same 
 name, opposite, and so on ; thus, 
 
 This may be proven by changing 
 the conditions of the question. 
 Again : 
 
 t 
 
 25 
 
 30 
 
 7128 
 
 If 3 days work of A, are equal to 5 of B, and 4 
 of B to 9 of C, and 10 of C to 6 of D, and 8 of 
 D to 40 of E, and 3 of E to 2 of F, and 3| of F 
 to 18f of G, how many days' work of A will equal 
 25 of G? 
 
 Here, G- is the demand, and certainly G must be 
 the same name ; hence, we state in the retrograde 
 order of the question, until we find A, which was the 
 first term proposed, and which is now the term of an- 
 swer; thus, 
 
170 
 
 RA1NEY S IMPROVED ABACUS. 
 
 3- 
 
 15 
 
 15 
 
 of A. 
 
 The result is T 8 -j of 1 day's work 
 of A. The proportion in these 
 questions is always direct ; because 
 the number on the left merely equals 
 in value some other member on tLe 
 right. Again : 
 
 If $18 U. S. are worth 8 ducats, at Frankfort; 12 
 ducats at Frankfort, 9 pistoles, at Geneva; 50 pistoles 
 at Geneva, 40 rupees, at Bombay; 16 rupees at Bom- 
 bay, 20 rials, at Buenos Ayres ; 15 rials at Buenos 
 Ayres, 6 candarines, at Canton; 4J- candarines at 
 Canton, 8 lepta, at Corinth ; 2 lepta at Corinth, 5 as- 
 pers at Cairo ; and 20 aspers at Cairo, 40 copecks, at 
 St. Petersburgh; how many copecks of St. Peters- 
 burg are worth $120, U. States ? 
 
 The answer is 142f co 
 pecks. 
 
 Dol. 18 
 
 120 Dollars. 
 
 Due. 12 
 
 8 Ducats. 
 
 Pis. 50 
 
 9 Pistoles. 
 
 Bup. 16 
 Ri. 15 
 
 40 Rupees. 
 20 Rials. 
 
 Can. 9 
 
 6 Candarines. 
 
 
 2 
 
 Lep. 2 
 
 As. 20 
 
 8 Lepta. 
 5 Aspers. 
 40 Copecks. 
 
 
 142f 
 
 All questions of a similar nature, in exchange and 
 reduction of monies, may be wrought as the above, 
 and without the least difficulty, when proper attention 
 
EQUATION OF PAYMENTS. 171 
 
 is given to the foregoing, which may be brought into 
 the following 
 
 SUMMARY OP DIRECTIONS FOR CONJOINED PROPORTION. 
 
 Place the demand first on the right: Place the 
 same name opposite ; and place the term of answer, 
 which equals the same name, under the demand, on the 
 right. Again: Place opposite the term of answer, 
 the term of the same name, and the term which equals 
 the same name, under the former term of answer, 
 on the right ; and so on, until all of the terms are 
 placed down. Cancel as in other cases. 
 
 EQUATION OF PAYMENTS. 
 
 It frequently becomes necessary to make several 
 different payments on a note at one time, or to make 
 one payment of several different notes. This is done 
 by liquation of payments, which simply means, to 
 make the time of one payment equal to the average 
 of the several periods. Equation is from the Latin, 
 equa, equal. The following facts may be assumed : 
 
 1, in 3 months, will gain as much as { 
 
 510, in 8 " " " " " < 
 
 >40, in 1 " " " " " 
 
 4, in 20 days " " " " l 
 
 S17, in 11 " " " " " ' 
 
 $, in 1 year " " " " ' 
 
 *19, in 7 " " " " " 
 
 3 in 1 mo 
 8 in 10 
 1 in 40 " 
 
 ^20 in 4 da. 
 
 ;ll in 17 " 
 1 in 8yrs. 
 7 in 19 " 
 
 A merchant owes two notes, payable as follows : one 
 for $10, payable in 8 months, and the other for f 
 payable in 6 months. Now, 
 
172 
 
 RA1NEY 3 IMPROVED ABACUS. 
 
 8 months equals $ 80, for 1 month. 
 40X6 " " $240, for 1 " 
 
 50||320, for 1 
 64 months. 
 
 $50 
 
 Above, each sum is multiplied by its number of 
 months. The original sums are added, making $50, 
 for the equated time ; or, their products in the months, 
 making $320, for 1 month. The latter product is 
 divided by the former, giving the equated time 6f 
 months. The question would be stated thus, by In- 
 verse Proportion : 
 
 If $320 shall be paid in 1 month, in how many 
 months must $50 be paid ? Thus, 
 
 50 320 This is the proper method of arriving at 
 
 1 the proportion in such questions. The an- 
 
 ~~ 7JT" swer is, as before, 6f months ; because 1 
 
 1 J ' month was the last term on the right. 
 
 Again : A owes to B 4 notes, payable as follows : 
 
 ; 25 X in 6 months = 9 150 ] 
 , 75x in 8 " =$ 600 ( 
 200 X in 10 " =$2000 f 1 
 i 60X in 3 " = $ 180 J 
 
 1 month. 
 
 $2930 
 
 360 
 
 2,930 
 
 Ans. 8/g-, equal to 8 months, 4 
 days, and 4 hours. 
 
 I give 2 notes, one for $100, payable in 6 months; 
 the other for $100, payable in 18 months: at what 
 time may both be paid together ? 
 
RULE FOR EdUATION OF PAYMENTS. 173 
 
 $100 X- 6 months = $ 600 ) , 1 ., 
 j$100xl8 < = $1800 ( h * 
 
 $200 #00 #0012 
 
 12 months, answer. 
 
 This method of calculating equation of payments, 
 is not strictly correct ; being based on the supposition 
 that interest and discount are the same ; that is, that 
 the deduction made in advance, is equal to the interest 
 that accrues, after a certain period. Thus, in the 
 question above, $100 are withheld from the creditor 6 
 months, and interest should be charged ; whereas, an- 
 other $100 were paid 6 months before due, and should 
 sustain only a discount. The interest on the 
 money withheld, is greater than the discount on 
 that advanced; hence the difference. This differ- 
 ence is, however, very minute, and of no practical 
 importance. 
 
 When the time is days or years, such days or years 
 must be multiplied into the sum to be paid, as in the 
 cases of months above ; and the equated time will be 
 days or years, as the case may be. When the time is 
 months and days, it must be reduced either to 
 months or days. 
 
 If one of the debts is paid down, or a payment 
 made at the date of the obligation, it will make no 
 product in multiplying into time, and will be used only 
 in finding the sum of the debts or payments. 
 
 From the foregoing, we deduce the following 
 
 DIRECTIONS FOR EQUATION OF PAYMENTS. 
 
 Multiply the sum of each payment by its time, 
 add the several payments: then, add their products ; 
 and divide the sum of the products, by the -sum of the 
 payments. 
 12 
 
174 RAlNEY y S IMPROVED ABACUS r 
 
 The several periods of time must be of the same 
 denomination ; either days, months, or ytars, separate- 
 ly ; and the answer will be the equated time, in days, 
 months, or years, as the ease may be. Or, state as m 
 Inverse Proportion. 
 
 FELLOWSHIP, SIMPLE AND COMPOUND 
 
 INCLUDING PARTNERSHIP, GENERAL AVERAGE, BANK- 
 RUPTCIES, ETC, 
 
 In Simple Fellowship or Partnership, General Ave- 
 rage, Bankruptcy, etc., two or more individuals, on- 
 different sums of money, gain or lose some general sum ; 
 when, it is desired to know each individual gain oir 
 
 loss. 
 
 In Compound Fellowship, not only the sums of cap- 
 ital are different, but are invested for different peri- 
 ods of time, etc. From this fact it takes the name of 
 Compound Fellowship. 
 
 A, B, and C invest $1000 in a cargo of wheat : 
 A puts in $200, B $300, and C $500 ; and agree to 
 share the profits and losses, in proportion to the capi- 
 tal severally invested. They gain $600 ; what is each 
 man's share? 
 
 A's share $200 
 B's " 300 
 
 C's " 500 
 
 $1000. Sum of gain $600. 
 
 Now, fhe whole sum, 1000, gains the whole sum, 
 600 ; therefore, we may ask, what share of 600 each 
 
SIMPLE FELLOWSHIP. 
 
 175 
 
 individual share gains. The statements, according to 
 proportion, are as follows : 
 
 If 1000 gain 600, what will 200 gain? 
 u a 300 " 
 500 " 
 
 000200 
 
 $120 
 
 300 
 600 
 
 500 
 600 
 
 300 
 
 A's share of gain 120 
 B's " " " 180 
 O's " ' " 300 
 
 Whole sum gained, $600 Proof. 
 
 A father divides $1800 among three sons, in the 
 following proportion : A, 1 share ; B, twice as much 
 as A ; and C, three times as much as B ; what is the 
 share of each? 
 
 Let us set A's share down as a unit ; thus, 1 
 B's share, as twice this, 2 
 
 C's share, as three times the latter, 6 
 
 Sum of the shares, 9 
 
 Thus, there are 9 shares ; and A has ; B, f , and 
 C, : hence, their shares would be as 9 to 1 ; 9 to 2 ; 
 and 9 to 6 ; thus, 
 
 0^002 #,i#00 - 2 
 
 2 
 
 200 
 
 400 
 
 6 
 
 1200 
 
 Or, the questions might be stated, thus, 
 What will 1 share be, if 9 shares be 1800? What 
 will 2 shares be ? What wiU 6 shares be, etc.? 200 
 _j_400+1200=$1800, the estate ; which proves the 
 work correct. 
 
176 
 
 RAINEY S IMPROVED ABACUS. 
 
 A owes to B $400; 
 " " " C $600; 
 " " " D $500; 
 " " " E $800; 
 " " " F $200 ; and pays only 
 much does each 
 
 $2500 
 R600: how 
 
 lose? 
 
 $2500 
 
 $900 
 
 The whole sum of the credits loses $900 : hence, the 
 following statements: 
 
 00 
 
 400 00 
 9004 
 
 600 ?00 
 900 4 
 
 500 
 9004 
 
 $ 
 
 144 B 
 
 00 
 
 g 
 
 V 
 
 800 
 9004 
 
 216 C. 
 
 n<M> 
 
 to 
 
 
 
 200 
 900- 
 
 180 D. 
 1 
 
 144+216+180+288+72=900 dollars, the whole 
 Bum lost. 
 
 A man left to his four sons $60, to be divided in 
 
 the proportion of -J-, 
 
 one get ? 
 
 i of 60=20 
 | 60=15 
 i " 60=12 
 i- 60=10 
 
 , i; how much does each 
 
 57 
 
 20 
 60 
 
 is 57 : 
 
 60 
 
 :: 20 : 
 
 X or 21/ T 
 
 " 57 : 
 
 60 
 
 : : 15 : X or 15^5. 
 
 " 57 : 
 
 60 
 
 :: 12 : Xorl2-f4 
 
 " 57 : 
 
 60 
 
 :: 10 : 
 
 XorlOfl 
 
 
 
 
 
 $60 
 
 
 5715 
 
 
 57 
 
 12 
 
 57 
 
 10 
 
 60 
 
 8 
 
 
 60 
 
 
 60 
 
 'l ZA. 
 
 
 1On n 
 
 1H3 
 
 I 4 -**? [ Av i7 
 
 The several sums, added, make the original $60. 
 
GENERAL AVERAGE IN FELLOWSHIP. 177 
 
 The $57 was the deficient sum of capital; and 20, 
 15, 12, and 10, the deficient individual sums. Then, if 
 57, deficient, be advanced to 60, full sum, what will 
 any of the other deficient sums be advanced to, for 
 full sum or share ? 
 
 Railroad, canal, bank, and other stock dividends, or 
 assessments, may be calculated as the first and second 
 examples given. 
 
 GENERAL AVERAGE. 
 
 It frequently becomes necessary, at sea, to throw 
 overboard a large portion of the cargo, to secure safety 
 in time of storm. The property thus sacrificed may 
 belong to one individual, although it is thus thrown 
 overboard for the benefit of the whole. Hence, the 
 whole cargo should sustain a loss proportionate to the 
 value of each individual interest. Now, it becomes 
 necessary to assess the loss according to value : this 
 is called General Average ; while the property ejected 
 is called jettison, from the French, jetter, to throw. 
 All such losses are sustained by the ship, the cargo, 
 and the freight, according to the value of each, which 
 is called pro rata, or in proportion. 
 
 Ordinary losses, such as wear and damage, or sacri- 
 fice made for the safety of the ship alone, must be 
 borne by the owners of the vessel; losses made for 
 the safety of any particular portion of the cargo, must 
 be charged to the individual so losing,andarenot to be 
 brought into the general average. 
 
 The property lost must be reckoned, as well as that 
 saved. 
 
 The cargo is valued at the price it would bring at 
 the place of destination, after deducting storage and 
 all necessary charges. 
 
 One-third is generally deducted from the freight, for 
 
178 RAINEY'S IMPROVED ABACUS. 
 
 wages, pilotage, etc., etc.; in New York, one-half is 
 allowed. 
 
 One- third of the cost of repairs, on masts, spars, 
 rigging, etc., is deducted before the average is made; 
 thus making the valuation of the old. two-thirds of 
 the new. This is done on the principle of insuring 
 only two-thirds of the value of property on land; or to 
 allow for damage. And here, the danger must be 
 imminent, or the general average will not be allowed. 
 
 All necessary charges must be deducted from each 
 individual's interest before the average is made. 
 Example : 
 
 The ship, John Adams, from Havre to Boston, had 
 on board a cargo estimated at $60,000. Of this A 
 owned $20,000; B, $30,000; and C, $10,000. The 
 
 Eoss amount of freight and passage money was 
 .2,000. The ship was worth $50,000; and $800 
 d been paid for insurance. The ship, being in great 
 distress, the master threw overboard $7,800 worth 
 of goods, and cut away her masts, rigging, and an- 
 chors. In port, it cost $3,000 for repairs; what was 
 the loss of each owner, both of ship and cargo. 
 
 Ship valued at .... $50,000 
 
 Premium deducted, . . . 800 $49,200 
 
 Cargo worth, ' 60,000 
 
 Freight and passage, . . .12,000 
 
 One-third deducted for wages, 4,000 8,000 
 
 Sum of individual interests, . . . . $117,200 
 
 Goods thrown overboard, 7,800 
 
 Cost of new masts, spars, etc., $3,000 
 
 One-third deducted for wear, 1,000 2,000 
 
 Com. on repairs, 20 
 
 Port duties and other expenses, ... 180 
 
 Sum of loss, ... . . $10,000 
 
COMPOUND FELLOWSHIP. 
 
 179 
 
 As 117,200 : 20,000 :: 
 As " : 30,000 :: 
 As " : 10,000 :: 
 As : 49,200 :: 
 As : 8,000:: 
 
 PROOF. Whole loss, 
 
 10,000 .: $1706.48 A's loss 
 : 2559.73 B's 
 : 853,24 C's " 
 : 4197.95 Ship's 1. 
 : 682.60 Frt's. L 
 
 $10,000.00 as ab've. 
 
 COMPOUND FELLOWSHIP. 
 
 A, B, and purchase a pasture to be used by them 
 jointly, for $50, in which A keeps 80 oxen 3 months; 
 B, 100 oxen 2 months; and C, 160 oxen 1 month: 
 what part of the cost must each pay ? 
 
 It is not necessary here that the time be expressed 
 .as months ; for it is simply a unit ; and 3, 2, and 1 
 show the ratios of consumption of grass, rather thaa 
 the time^ as the $50 pay alike for 1 month, 1 year, 
 or 10 years. The multiplication of the time and oxea 
 together, shows -merely how many oxen, in each case, 
 remain a common length of time in the pasture ; for, 
 after being thus multiplied, one lot is supposed to be 
 in the pasture as long as the other; and the difference 
 of consumption and price is the effect of the different 
 numbers of oxen thus grazing. 
 
 A's 80x^ months -= 240 oxen for the time. 
 B's 100x2 " = 200 " " " 
 C's 160x1 " = 160 " " " 
 
 The whole = 600 " -" 
 
 Now, if 600 oxen cost $50, what will 240, 200, and 
 160 cost, respectively; thus, 
 
 600 
 
 240 
 50 
 
 600 
 
 20 A's. 
 PJEIOOF. 
 
 200 
 50 
 
 eoo 
 
 16| B's. 
 
 160 
 50 
 
180 
 
 RAINEY'S IMPROVED ABACUS. 
 
 A and B companied; A put in $2,000, January 
 1 ; but B put in June 1 ; what sum did he put in, to 
 have an equal share of the profits with A ? 
 
 A has 2,000 employed 12 months ; it is, therefore, 
 desired to know how much capital belonging to B will, 
 in 7 months, make equal profits with the 2,000, 12 
 months. Hence, the question is one of inverse pro-- 
 portion, and is wrought thus, 
 
 12 
 
 2000 
 
 The demand is on the left. 
 
 $ 3428A 
 
 " In an adventure, A put in $12,000 for 4 months; 
 then adding $8,000, he continued the whole 2 months ; 
 B put in $25,000, and after 3 months took out $10,000, 
 and continued the rest 3 months longer; C put m 
 $35,000 for 2 months ; then, withdrawing f of his 
 stock, continued the remainder 4 months longer : they 
 gained $6,000 ; what was the share of each ? 
 
 = 88,000. 
 
 A'sj 
 A's! 
 
 $12,000x4 BIOS. 
 20,000x2 " 
 
 = 48,000, 
 = 40,000, 
 
 B'sj 
 B's< 
 
 $25,000X3 " 
 |15,OOOX3 " 
 
 = 75,000, 
 = 45,000, 
 
 C's $35,000x2 " 
 C's $25,000x4 " 
 
 Whole sum, 
 
 = 70,000, 
 = 100,000, 
 
 $378,000 
 
 If 378,000 gain 6,000, what will 88,000, A's, gain? 
 If 378,000 " " " " 120,000, B's, " 
 If 378,000 " " " " 170,000, C's, " 
 
 A gains 
 B " 
 
 $1,396.83, ) 
 
 $1,904.76, \ =6,000, PROOF, 
 
 $2,698.41, ) 
 
RULE FOR THE FELLOWSHIPS. 181 
 
 From the examples given, we deduce the following 
 
 SUMMARY OP DIRECTIONS FOR SIMPLE FELLOWSHIP. 
 
 Add the several sums: make the whole sum the 
 supposition ; each separate sum, the demand ; and the 
 whole gain or loss, the term of answer : the answer 
 will, in each case 9 be the gain or loss on the individual 
 sums. 
 
 PROOF: Add the several answers, and the whole 
 sum will equal the whole gain or loss. 
 
 To make General Average, 
 
 FIRST : Ascertain the value of the cargo ; the sum 
 of the freight-Nil, passage, etc., minus one-half, or 
 one-third, as customary : the sum total will be the 
 general value : 
 
 SECOND : Ascertain the sum of loss, goods throivn 
 overboard ; cost of new masts, spars, rigging, etc. y 
 minus one-third; commission on repairs; port duties, 
 and other expenses : add these, and the sum total will 
 be the loss. Then, make each individual loss the de- 
 mand; the sum of all the combined interests, the same 
 name ; and the whole sum of loss, the term of answer. 
 
 PROOF : The several answers, added, will equal the 
 whole loss. 
 
 COMPOUND FELLOWSHIP. 
 
 Multiply each sum of capital by the time invested; 
 add the products ; make, each separate product the 
 demand; the sum of the products, the same name; 
 and the whole gain or loss, the term of answer. 
 
 PROOF : Add the several specific sums, and they will 
 equal the whole sum of gain or loss. 
 
 When a part of an investment is deducted, or an 
 additional sum paid in, make the remainder, or 
 
182 RAINEY'S IMPROVED ABACUS. 
 
 the increased sum, as the c-tse may be, a new invest- 
 ment for the time that it runs ; and multiply by the 
 time, as before. 
 
 When tzoo or more investments, made by one indi- 
 vidual, are multiplied thus, the separate products may 
 be added into one individual product, before being 
 placed on the line. 
 
 BARTER, DUTIES, AND COMMERCIAL EX- 
 CHANGE. 
 
 Barter * is the exchange of one article of specified 
 value, for an equivalent value in something else. 
 Hence, articles in Barter, or Commercial Exchange, 
 are considered in the ratio of their separate values or 
 quantities. 
 
 How many bushels of wheat at 50 cents per bushel, 
 will pay for 200 bushels of corn, at 30 cents per 
 bushel? 
 
 The demand here, is, what will 200x30 cents buy, 
 if 50 cents, opposite, buy 1 bushel? Thus 
 
 The answer is 120 bushels of wheat. 
 The 30 and 200 are here multiplied, 
 j2Q , merely to show that the demand is the 
 
 price of the whole of the corn ; other- 
 wise the proportion would not be recognized. It 
 should be stated with the terms to be multiplied, one 
 jvhove the other; thus, 
 
 * Barter is from the Spanish, baratar, from the Latin root, verto, to turn 
 or exchange. This change implies equality in the articles, or the prices of 
 the articles, exchanged. From its derivation, it hears a striking resem- 
 blance to the word proportion. 
 
BARTER. 
 
 183 
 
 One hundred and twenty, as before. 
 
 200 
 5030_ 
 
 120 
 
 How many bushels of corn, at 30 cents per bushel, 
 will pay for 120 bushels of wheat, at 50 cents per 
 bushel? 
 
 The answer is 200 bushels of corn. 
 
 50 
 
 |200 
 
 If 200 bushels of corn, at 30 cents per bushel, pay 
 for 120 bushels of wheat, what is the price of the 
 wheat ? 
 
 rWp 00 
 
 05 
 
 The answer is 50 cents. 
 
 50 cts. 
 
 How many pounds of butter, at 12^ cents per lb., 
 will pay for 1&| Ibs. of bacon, at 10 cents per lb.? 
 
 Here, the demand is 18f times 10 
 cents ; the same name 12-J- cents ; and 
 the term of answer, 1 pound of butter : 
 hence, 15 Ibs. of butter, the answer. 
 
 15 Ibs. 
 
 How many pounds of sugar, at 3-J- cents per lb., 
 will pay for 20 bbls. of rum, 35 galls, to the barrel, 
 worth 90 cents per gallon ? 
 
 The answer is 18,000 pounds of sugar. 
 
 20 
 
 tl> 
 90 
 
 2 
 
 l> 5 
 
 18,000 
 If 18,000 Ibs. of sugar pay for 20 bbls. of ram 
 
184 
 
 RAINEY'S IMPROVED ABACUS. 
 
 35 gallons to the barrel, and worth 90 cents per gal- 
 lon, what is the sugar worth per lb.? 
 
 What will 1 lb. of sugar cost, 
 if 18,000 Ibs. cost 20x35x90 
 cents ? 
 
 If 18,000 Ibs. sugar, at 3 cts. per lb., pay for 20 
 bbls. of rum, worth 90 cents per gallon, howrnany gal- 
 lons are there per barrel ? 
 
 > 5 
 
 It is seen in this and the preceding 
 examples, that all of the constituents 
 of the given commodity are placed on 
 or I! the right, and the remaining constitu- 
 
 ents, of the commodity about which 
 the inquiry is made, on the left. If there are four 
 constituents on the right for the perfect supposition, 
 and three out of four given, for an equivalent in barter, 
 the three given, must be placed on the left, and the 
 other factor or constituent will be found. It is ob- 
 served that the price is placed on the left, to find its 
 associated quantity ; or, the quantity is placed on the 
 left, to find its associated price. 
 
 BARTER BY REDUCTION 
 
 All reduction, ascending and descending, is a spe- 
 cies of concatenated or conjoined proportion. Where 
 articles are of different denominations, or are paid for 
 in prices of different denominations, it is very con- 
 venient to make these reductions on the line, at the 
 same time that the general question is wrought. And, 
 if treated proportionally, the statement becomes lucid, 
 and the work interesting. 
 
 How many yards of cloth, at $1,25 per yard, will 
 pay for 6 tons of iron, and 5 pence per lb., in the cur- 
 
BARTER BY REDUCTION. 
 
 185 
 
 Ton. 1 
 Cwt. 1 
 Lb. 1 
 D. 12 
 Sh. 6 
 Cts. 125 
 
 6 Tons. 
 20 Cwt. 
 100 Lbs. 
 5 Pence. 
 1 Shil. 
 100 Cents. 
 1 Yard. 
 
 A ns. 
 
 666f yds. 
 
 rency* of New England, which is 6 shillings to the 
 dollar? 
 
 What will 6 tons make, if 1 
 ton make 20 cwt.; what will this 
 make, if 1 cwt. make 100 Ibs.; 
 what will all of these Ibs. come 
 to, if 1 Ib. cost 5 pence; how 
 many shillings will these pence 
 make, if 12 pence, opposite, 
 make 1 shilling; how many 
 cents will all of these shillings 
 make, if 6 shillings, New England, opposite, make 100 
 cents; how many yards will all these cents buy, if 
 125 cents buy 1 yard? Hence, the answer is in 
 yards, 666-|; the number that will pay for the 6 tons. 
 
 Let us now prove it, by asking, how many tons will 
 pay for 666-| yards. The latter number becomes the 
 demand; thus, o 
 
 Yd. 12000 Yds. 
 Cts. 100125 Cts. 
 Sh. 16 Sh. 
 D. 512 D. 
 Lbs. 1001 Lb. 
 Cwt. 201 Cwt. 
 Ton. 
 
 The ones arc wholly unne- 
 cessary in the solution; and 
 are given merely to indicate 
 the ratio 
 
 Ans.\6 tons. 
 
 beautiful decimal system. Sterling money derived its name from Ester 
 ling,vfho first made the coin. The dollar mark is a combination of U. S, 
 
 nnrl \vn nrirrinallv \vrittpn TT S Oft OO. Pt<v 
 
 and was originally written U. S. 20.00, etc 
 
186 
 
 RA1NEY S IMPROVED ABACUS. 
 
 We may prove it, again, by finding the cost of 1 Ib. 
 of iron, which was 5 d.; hence, the 5 d. is the demand : 
 D. 125 D. 
 Sh. 6 1 Sh. 
 Cts. 125100 Cts. 
 Yds. 2000 1 Yd. 
 
 3 
 Ton. 1 6 Tons. 
 
 20 Cwt. 
 
 100 Lbs. 
 
 Denominations of the 
 same kind are placed oppo- 
 site each other. 
 
 Ans. 1 Ib. iron. 
 This may be proven, by demanding cost of 1 Ib. iron : 
 
 This might be proven by 
 several other processes; but 
 further proof is unnecessary. 
 
 Ans. 5 pence. 
 
 How many galls, cordial at $180 per gal , will pay 
 for 10 T. wine, at 6 d. per pt., S. Carolina currency ? 
 10 
 
 Lbs. 100 
 
 1 Lb. 
 
 Cwt. 20 
 
 1 Cwt. 
 
 Tons. 6 
 
 1 Ton. 
 
 3 
 
 
 Yd. 1 
 
 2000 Yds. 
 
 Cts. 100 
 
 125 Cts. 
 
 Sh. 1 
 
 6 Sh. 
 
 
 12 D. 
 
 4^030.00 
 
 Ans. 1200 gallons. 
 
 Tun. 1 
 
 10 Tuns. 
 
 Pipe. 1 
 
 2 P. 
 
 Hhd. 1 
 
 2 Hhds. 
 
 Gal. 1 
 
 63 Gals. 
 
 Qt. 1 
 
 4 Qts. 
 
 Pt. 1 
 
 2 Pts. 
 
 D. 12 
 
 6 D. 
 
 Sh. 20 
 
 1 Sh. 
 
 . 7 
 
 1 . 
 
 Cts. 1.80 
 
 30.00 Cts. 
 
 
 1 Gal. 
 
 Ans. 
 
 1200 gallons. 
 
BARTER AND COMMERCIAL EXCHANGE. 187 
 
 The 7, above equal $30, or 3,000 cents, South 
 Carolina currency. 
 
 In this question, the demand is 10 tuns; the same 
 name, 1 tun : and the term of answer, the next lower 
 denomination, 2 pipes. Thus, the answer of every 
 preceding statement, is made the demand of a subse- 
 quent question. 
 
 This question may be proven in several ways, as the 
 one preceding. 
 
 How many barrels of wheat, at 50 cts. per bushel, 
 will pay for 12 tons of iron, at 4 pence per lb., New 
 York currency ? 
 
 1 
 1 
 1 
 
 1-6 
 
 W 
 &fi 
 100 
 
 A 
 
 /Ljl 
 
 W 
 * 
 
 yk 
 1 
 4004 
 1 
 1 
 
 Ans. 
 
 400 
 
 Ton. 1 
 
 12 Tons. 
 
 Cwt. 1 
 
 20 Cwt. 
 
 Lb. 1 
 
 100 Lbs. 
 
 D. 12 
 
 4 D. 
 
 Sh. 8 
 
 1 Sh. 
 
 Cts. 50 
 
 100 Cts. 
 
 Bu. 5 
 
 1 Bush. 
 
 
 1 Barrel. 
 
 Ans. 
 
 400 barrels. 
 
 In New York, 8 shillings make 1 dollar ; hence, the 
 8 on the left of the line. 
 
 From the foregoing, we deduce the following 
 
 DIRECTIONS FOR BARTER, COMMERCIAL EXCHANGE, ETC. 
 
 Place all of the terms of the commodity whose 
 quantity and value are given, on the right ; and all 
 of the terms of the commodity whose quantity or value 
 is required, on the left : the answer will be the quan- 
 tity or value, as the case may be. 
 
 FOR DENOMINATE NUMBERS: Place the demand 
 first on the right : place a unit of the same name, or 
 the quantity specified of the same name, opposite : 
 place the number which this unit makes in reduction, 
 
188 RAINEY'S IMPROVED ABACUS. 
 
 or such other quantity of something else as the given 
 quantity may equal, on the right : place, again, oppo- 
 site this last term, on the left, the unit or other quan- 
 tity of the same name, and proceed as before ; mak- 
 ing the answer of each preceding, the demand of a 
 succeeding question, ad infinitum. 
 
 DUTIES, TARE AND TRET, ETC. 
 
 We shall bring together and consider in one article, 
 both Duties and Tare and Tret, because of their al- 
 most entire identity, and the similarity of operations 
 necessary to reduce them ; both depending alike on 
 per centage, as well as losses of tare, tret, leakage, etc. 
 
 Commercial duties are sums of money paid to a 
 government, for the privilege of importing foreign 
 goods. The general law imposing these duties, and 
 determining their rate and extent, is called a tariff. 
 Tariffs are different in different countries, varying 
 in their extent according to the necessities which 
 cause their creation. In some countries, a charge 
 is made on goods, produce, etc., for the ostensible 
 purpose of creating revenue to discharge the expenses 
 of government; in others, for the purpose of pro- 
 tecting manufactures, or agriculture, by preventing 
 the importation of foreign manufactures or agricultu- 
 ral products. The manufacturing facilities of Great 
 Britain being such that she can, not only compete 
 with, but defy the whole world, she lays a tariff more 
 particularly on the products of the soil, both for the 
 promotion of industry, with the full development of 
 her natural resources, and the establishment, among 
 
 Tret is from the Latin, tritiis, from tero, to wear ; Tare is French, and is 
 from the same root as the Italian tarare, to abate. 
 
DUTIES, AND TARE AND TRET. 180 
 
 her laboring population, of a long- cherished system of 
 almost arbitrary tenantry. 
 
 In our own country, to the contrary, a tariff 
 is imposed for the avowed purpose of making the 
 premium on foreign privileges, pay the expenses of 
 our government. Such a tariff, varies in its exac- 
 tions, according to the financial exigencies of the 
 government. 
 
 We have had a tariff in our country for quite a dif- 
 ferent purpose ; that of fostering her domestic manu- 
 factures. This was necessarily high enough to pre- 
 vent a large foreign importation, by rendering the 
 profits accruing, too small for consideration. 
 
 Our federal constitution expressly forbids any tariff 
 between the several states of this union. 
 
 1. In every port in the United States, where vessels are permitted to 
 enter, a customhouse is established by the government, at which the du- 
 ties assessed on goods, are paid. The officer who examines the cargo, 
 determines the duties, etc., is called a custom-house officer. All importa- 
 tions, whether by our own, or foreign vessels, are alike subject to these 
 duties. 
 
 2. Duties are divided into specific and ad valorem. 
 
 3. A specific duty is a certain sum charged on some specified quantity, as 
 a cwt., gallon, yard, ton, foot, etc. This kind of duty is levied without re- 
 ference to price. 
 
 4. An ad valorem duty is a certain per cent, on the price of the article. 
 This phrase is from the Latin, ad, according to, and valorem, value ; ac- 
 cording to value. 
 
 5. When specified duties are received, certain allowances are made for 
 tare, tret or draft, leakage, etc. These allowances are sometimes a speci- 
 fied per cent, on the original quantity ; and sometimes a specified deduc- 
 tion on the cwt., cask, box, etc.; and are deducted before the duty is re- 
 ceived ; otherwise duty would be reckoned, alike on the quantity wasted, 
 and that saved. 
 
 6. Tare is an allowance made for the box, bag, crate, etc., which con- 
 tains the article ; and is deducted, either specifically or at a certain per 
 cent. It has been customary to allow 12 Ibs. on the 112, or old cwt. 
 
 7. Tret is a certain allowance on the weight after tare is deducted, for 
 waste, grease, dust, and other extraneous substances, and is usually 4 Ibs. 
 on the 104. 
 
 8. Leakage is an allowance, usually of 2 per cent., on liquids, such as 
 liquors, oils, chemicals, etc.. for waste. 
 
 9. Gross weight is the weight before any deductions are mad. 
 
 13 
 
190 
 
 RA1NEY S IMPROVED ABACUS, 
 
 10. Suttle is the weight after apart of the deductions are made ; as the 
 Weight, after deducting tare, from which tret is to be deducted. 
 
 11. Net weight is what remains, after all deductions are made j and i 
 the basis on which duties are reckoned. 
 
 12. The rates of less are different on different articles, according to th 
 law regulating such deductions. Losses of this kind are sometimes allow- 
 ed in individual transactions, in groceries, etc. 
 
 SPECIFIC DUTIES, 
 
 What is the specific duty on 1900 gallons of me- 
 lasses, at 9 cents per gallon, allowing 2 per cent, for 
 leakage ? 
 
 The first thing to be done, is to find the number of 
 gallons, after deducting the leakage. What will 1900 
 be reduced to, if 100 be reduced to 98 ; thus, 
 
 Here, 1900 gross is the demand, and 
 100 gross, the same name ; while 98, net, 
 is the term of answer. We multiply 
 the net number of gallons by 9 cents, 
 the specific duty, and find the duty, 
 $167,58, cutting off two figures, for 
 cents. We should have placed 1 oppo- 
 site, and 9 under the 98, saying, what will all of the 
 net come to, if 1 gallon, net, be 9 cents? 
 
 What is the duty on 20 hhds. whisky, at 10 cents 
 per gallon, allowing 2 per cent, for leakage ? 
 
 The hhds. are reduced to gallons, as 
 in other cases 
 
 At 6 cents per lb., what is the specific duty on 170 
 kegs of tobacco, weighing each 125 Ibs, allowing 6 
 Ibs. per hundred for tare? 
 
 w 
 
 1900 
 
 98 
 
 
 1862 
 9 
 
 $ 
 
 167.58 
 
 1 
 
 100 
 
 20 
 63 
 
 98 
 
 $ 
 
 123.48 
 
AD VALOREM DUTIES. 
 
 191 
 
 When tare is so much per box, 1 170 
 
 bag, etc., we multiply by the num- ,100 fL&p 25 
 ber of boxes, bags, or whatever 1 94 
 
 they be : find the whole tare, and $ 3 
 
 after subtracting this from the I HQS 50 
 
 whole gross, multiply the net re- 
 mainder by the specific rate ; thus, 
 
 What is the duty, at 44 cents per lb., on 80 boxes 
 tobacco, weighing each 100 Ibs., allowing tare at 20 
 cents per box ? 
 
 Here, the tare would be, at 20 Ibs. per box, on 80 
 boxes, 20x80=1600 Ibs. Now, 8000- 1600=6400 
 Ibs., net. This net weight is multiplied by the spe- 
 cific rate ; thus, 
 
 We find that the duty is $288,00. 
 Thus, it is seen, that we can use the 
 line to advantage, even in specific 
 duties. 
 
 0^0032 
 9 
 
 288,00 
 
 DIRECTIONS FOR SPECIFIC DUTIES. 
 
 Find the net weight, and multiply this by the duty. 
 To find the net weight, make the gross sum the de- 
 mand : 100 gross the same name : and 100, di- 
 minished by the tare, tret, or leakage, the term of an- 
 swer : the answer will be the net weight. Place un- 
 der the term representing net weight, the duty ; cut 
 off two figures in the answer, when the duty is in 
 cents, and the answer will be in dollars and cents. 
 
 AD VALOREM DUTIES. 
 
 It may be again observed, that ad valorum duties 
 are reckoned on the cost price of the article; hence, 
 no deductions of tare, tret, etc., are made, as in spe- 
 cific duties. The duty is reckoned on the invoice of 
 goods, in the same manner that premium is reckoned 
 on insurance. 
 
192 
 
 RATNEY'S IMPROVED ABACUS. 
 
 1. An invoice is a bill giving in all the goods, and setting forth the price 
 of each article. 
 
 2. To prevent deception, this bill must be verified, .or sworn to, by the 
 owners, or one of the owners of the goods, to the effect, that the invoice 
 presents a true statement of cost prices ; and that no discount, drawback, 
 or bounty has been named, that has not been actually allowed. 
 
 3. This oath shall be administered by a consul, commercial agent, or 
 other duly authorized officer, in the country where the goods are pur- 
 chased ; which fact shall be duly certified by such consul or agent. 
 
 4. The officer thus acting, who commits any fraud, in making the in- 
 voice too small, etc., is liable to heavy penalties. Our government keeps a 
 consul, or other commercial agent, in every important part of the world, 
 for this purpose, except where they are forbidden admittance by the laws 
 of the country, as has, until recently, been the case in China. See Laws of 
 the United States. 
 
 What is the ad valorem duty, at 20 per cent., on a 
 lot of boots and shoes, worth $40,000? 
 
 The question is, what duty will $40,000 give, if 
 $100 give $20 duty? thus, 
 
 /100 40,000 
 20 
 
 $ 8,000 
 
 We find that the duty is $8,000. 
 
 A lot of Italian silks cost $9,100 ; wnat is the ad 
 valorem duty on them, at 43 per cent.? thus, 
 
 400 9,100 
 43 
 
 $3,913 
 
 The answer is $3,913. 
 
 What is the duty on an invoice of socks, which cost 
 B,963, at 25 per cent.? thus, 
 
 4-400 
 
 3963 
 
 990f 
 
 We might suspend the 100 on 
 the left, and cut off two figures for 
 decimals of a dollar, on the right. 
 
 When the rate is the aliquot part of a dollar, we may very much abridge 
 the operation by suspending the 100 on the left, and multiplying by such 
 fractional part of a dollar, as the rate makes, as above : 25 per cent, is the 
 % of 100 ; hence, we may multiply the 3963 by , which is equivalent to 
 simply dividing it by 4. 
 
 At 22 per cent., what is the duty on $71,000 worth 
 of crockery- ware ? 
 
COMMERCIAL TARE AND TRET. 
 
 193 
 
 This answer is obtained in dollars. 
 
 #071; 
 
 22 
 
 $ 15,620 
 
 What is the duty on 20 tons of wool, at 30 cents 
 per lb., the duty being 60 per cent.? 
 
 The first point is to know how much 1 20 
 the wool comes to in cents ; hence, the 1 20 
 reduction descending, found in the 
 statement. After being reduced to 
 pounds, it is multiplied by the price 
 per lb.; 100 is placed opposite ; and the 
 rate, last on the right. 
 
 30 
 60 
 
 $ 7200.00 
 
 We will now consider tare and tret in a different 
 form. In ordinary cases coming under this head, 12 
 Ibs., on the 112, are deducted for tare. This 112 Ibs. 
 was the old standard for 1 cwt.; but is not now used. 
 It has been customary to deduct 4 Ibs. per 104 for 
 tret, after the deduction of the tare. This custom 
 has pretty much gone out of use. The 12 Ibs. de- 
 ducted for tare, should be deducted from 112; not 
 from 100, as is dishonestly done by some dealers. 
 Deducting 12 from the 100, instead of 112, is simi- 
 lar in principle to the deduction of false discount. 
 
 In this, as in many of the departments of arithme- 
 tic, too much attention is given to questions merely 
 theoretical, in which denominate numbers are used, 
 such as have no existence in business. We never hear 
 anything of Ibs., qrs., cwt., etc., in business. Hence, 
 the impropriety of introducing such questions into 
 a practical treatise. 
 
 It is to be regretted that our whole system of de- 
 nominate numbers, with the exception of Federal 
 Money and Avoirdupois Weights, is so varied and 
 irregular ; while the decimal substitute could be made 
 with great ease and palpable advantage. 
 
194 
 
 RAINEY 7 S IMPROVED ABACUS. 
 
 We subjoin a few questions involving commercial 
 tare and tret : 
 
 What will 1 ton, gross weight, of hemp come to, 
 at $3,62-| cents per gross cwt.? 
 
 We here use 2240 Ibs. as the gross 
 ton. The price per cwt., or per 112 Ibs., 
 g ross ? being in cents, the answer is cents. 
 
 725 
 
 50 
 
 Or, what will 20 cwt. cost, if I cwt. cost $2,62|? 
 thus, 
 
 120 
 
 725 
 
 The answer is $72,50, as before. 
 
 $ 72,50 
 
 What will 8 tons tobacco come to, at 10 cents per 
 It)., allowing 10 Ibs. per 100 for tare? 
 
 8 
 
 20 
 90 
 10 
 
 The answer is $1440,00. 
 
 $ 1440,00 
 
 What would be the duty on the same at 37-J- per 
 cent.? 
 
 8 
 
 90 
 
 |10 
 
 75 
 
 $ 540,00 
 
 Above, 1 ton equals 20 cwt., and 
 1 cwt. equals 90 Ibs., net ; while 1 Ib. 
 net equals 10 cents, and 100 cts., give 
 y cents duty. 
 
 I have 1500 Ibs. of cotton, worth 5 d. sterling per 
 Ib., net; and wish to know the duty, if I allow 15 
 Ibs. per 100 for tare, and 4 Ibs. per 100 for tret, the 
 duty being 40 per cent.? 
 
 We reduce the 5 d. to Federal money, by using l 
 
COMBINATIONS IN TARE AND TRET. 
 
 195 
 
 on the left, which equals $4,84 on the right; or by 
 saying that 20 shillings equal $4,84, 
 
 WIMW-* 
 
 This combination is quite 
 simple, and enables us to 
 arrive at the result with 
 very few figures. 
 
 5 -/ 
 
 l^-9f- 
 100484 
 
 ,$ 49,36f 
 
 Bought 200 hogsheads of sugar, on a credit of 12 
 months, weighing each 1500 Ibs., gross, for which I 
 pay 2-j- cents per lb., net, the deduction for tare being 
 5 per cent.; how must I sell it, go as to realize 80 per 
 cent profit, being allowed 5 per cent, discount? 
 
 Here, instead of saying 
 in the discount, that 105 
 is reduced to 100, and using 
 another 100 on the left, from 
 which to advance to 80 per 
 ent., we say that 105 dis- 
 count is advanced to 180 
 profit. 
 
 1 
 
 jt00 400 5 
 1 #13 
 
 180 
 
 11700,00 
 
 What would the sugar be sold for, in Havre, after 
 deducting 20 per cent, for duty, from the prime net 
 value ? 
 
 400 
 
 i 
 
 00 
 
 4 005 
 
 t 
 
 80 
 180 
 
 Hhd. 1 
 G. 100 
 
 Net. 1 
 2 
 Dis. 105 
 
 M. D.I 00 
 
 200 Hhds. 
 1500 Lbs., grosa. 
 
 91 Net. 
 5 Ots. per lb. 
 
 80 Minus duty. 
 180 Profit. 
 
 Am. ft 
 
 9360,00 
 
 Above, we find first the number of Ibs,, then 
 
196 RAINEY'S IMPROVED ABACUS. 
 
 duct the tare ; find the cost of the whole at the price 
 per lb.; deduct the discount, and instead of reducing 
 the 105 to 100, merely reduce it to 80, which is allow- 
 ing for the 20 per cent. duty. It is not necessary to 
 use the two one hundreds which come between the 105 
 and 80, in which 105 is reduced to 100, and 100 
 again to 80. The duty being now deducted, and giv- 
 ing the net cost, we advance 80 per cent.,, and find the 
 selling price, which is $9,360. This answer might be 
 obtained in francs, by placing 93 cents opposite the 
 f cents, and 5 francs on the right under the 2^-. Al- 
 though the discount would then be made on the 
 francs, the question would be the same ; as the value 
 would not be changed. 
 
 DIRECTIONS FOR AIX VALOREM DUTIES, TARE AND 
 TRET, ETC. 
 
 To find the ad valorem duty, make the whole sum 
 the demand ; 100, the same name ; and the rate duty, 
 the term of answer. The answer will be the duty in 
 dollars, or dollars and cents, according to the denomi- 
 nation of the sum. 
 
 Or, Multiply the whale sum by the rate duty, and 
 strike off two or more figures for cents r as the case 
 may be. 
 
 FOR TARE ANI> TRET : Place the sum on the right ; 
 the standard on the left ; and the standard, reduced by 
 the tare, on the right. The answer will be the net 
 weight. 
 
 FOR DUTY: Place 100 on the left, and the rate 
 duty on the right : the answer will be the duty in the. 
 denomination of the sum. 
 
 To MAKE DISCOUNTS, PROFITS, LOSSES, ETC.: Place 
 10.0, increased by the rate discount, on the left ; and 
 
COMMERCIAL EXCHANGE. 
 
 197 
 
 100, on the right: also, 100 on the left, and 100, in- 
 creased by the gain per cent., or reduced by the loss 
 per cent., on the right. If there be discount, and profit 
 or loss, both in the same question, suspend the two one 
 hundreds. 
 
 COMMERCIAL EXCHANGE. 
 
 Operations in this department of numbers are iden- 
 tical with those of Conjoined Proportion; hence, the 
 statements are simple and easy, the demand being 
 placed on the right, the same name on the left, and its 
 equivalent on the right, as the term of answer. 
 Again, this term of answer becomes a new demand, 
 while a term of the same name is placed opposite, and 
 its equivalent in value again on the right. This order 
 is, however, frequently interrupted, by the introduc- 
 tion of discounts, gains and losses, reductions, etc.; 
 yet such may be easily interwoven and combined in 
 the statement by the reflecting student. 
 
 Bought 2,200 Ibs., gross weight, of wool, and was 
 allowed a deduction of 5 Ibs. per 105 for tret; I paid 
 for net weight 3s. 6 d. per lb., New York currency, 
 and having a credit of 12 months, was allowed a dis- 
 count of 10 per cent, for ready money; for how much 
 did I afterward sell the whole to gain 20 per cent, on 
 my investment-? 
 
 100 p 
 
 
 G. 105 
 Net. 1 
 2 
 Sh. 8 
 Dis. 110 
 Par. 100 
 
 2200 Gross. 
 100 Net. 
 7 Shillings. 
 
 1 Dollar. 
 100 Par. 
 120 Profit. 
 
 !P 
 
 1,000 
 
 Above, what will 2,200 gross be reduced to, if 105 
 
I ( J8 
 
 RAINEY'S IMPROVED ABACUS. 
 
 gross = 100 net; and 1 Ib. net = f shillings; and 8 
 shillings = $1 ; and $110, in discount, is reduced to 
 $100 par; and $100 par, opposite, is advanced to 
 $1*20 , 20 per cent, profit ? In the solution, the two 
 100s, par of discount, and par of profit, were left out ; 
 and 110, discount, simply advanced to 120, profit. 
 
 Bought 1,500 Ibs. of butter, at 7 d. 2 far. per Ib., New 
 Jersey currency, and was allowed 30 Ibs. per 100 for 
 firkin, and 4 Ibs. per 100 for impurities or tret, t had 
 a credit of 1 year, at 5 per cent, interest, but paying 
 the cash, was allowed 5 per cent, discount. I imme- 
 diately sold the same so as to realize 60 per cent, 
 on my money invested : what did I get for the butter, 
 in Federal money 
 
 ttW 
 
 160 
 
 G. 100 
 
 1500 
 
 Gross. 
 
 Sut. 100 
 
 70 
 
 Suttle. 
 
 Net. 1 
 
 96 
 
 Net. 
 
 2 
 
 15 
 
 Pence. 
 
 D. 12 
 
 
 
 Sh. 20 
 
 1 
 
 Shilling. 
 
 's. 3 
 
 1 
 
 
 
 Dis. 105 
 
 8 
 
 Dollars. 
 
 Par. 100 
 
 100 
 
 Par. 
 
 
 160 
 
 Profit. 
 
 $ 
 
 128 
 
 Ans. 
 
 A merchant has 20,000 Ibs. of cotton, which he can 
 sell a t 4 d. per Ib., New England currency. Fail- 
 ing to find a purchaser, he gives to A, in barter, 4-J- 
 Ibs. cotton for 15 Ibs. of butter: he then barters 
 with B, giving him 40 Ibs. of butter for 3 yards of 
 gambroon : again, he barters all of his gambroon with 
 C, giving him 2-J yards gambroon for 2 yards broad- 
 cloth : now, he barters his broadcloth with D, giving 
 him 1} yards for 12 yards linen ; and to E he gives 
 30 yards linen for 8 cwt. sugar : he now barters his* 
 sugar with F, giving 3 cwt. of sugar for 50 gallons of 
 
COMMERCIAL EXCHANGE. 
 
 190 
 
 melasses : after this, he gives G 4-^ galls, melasses for 
 1^ galls, of rum: he gives to H 400 gallons of rum 
 for 3 horses: and, finally, to J he gives 2 horses for 
 120 sheep: he sells his sheep at $1,80 cts. per head; 
 how much is he gainer or loser by trading, instead of 
 taking the original offer for his cotton? 
 
 0^0,000 
 
 Lbs. 9 
 
 20,000 Lbs. 
 
 
 2 
 
 B. 40 
 
 15 Butter. 
 
 Ga. 5 
 
 3 Gambroon. 
 
 
 2 
 
 C. 6 
 
 2 Yds. cloth. 
 
 
 5 
 
 L. 30 
 
 12 Linen. 
 
 Su. 3 
 M. 9 
 
 8 Cwt. sugar. 
 50 Galls, mel. 
 
 
 2 
 
 2 
 
 3 Galls, rum. 
 
 R. 400 
 
 
 H. 2 
 
 3 Horses. 
 
 S. 1 
 
 120 Sheep. 
 180 Cents. 
 
 $ 
 
 48000,00 
 
 1 
 
 3 /W 
 
 20 
 
 3 
 
 #0000 
 
 4 
 1 
 
 1 
 
 
 10,00 
 
 Ans. $ 
 
 1111,111 
 
 050 
 
 ,140 
 
 Ans. $48000,00 
 
 1111,111 
 
 Ans. $46888,88| 
 
 In the last calculation, 3 equal $10 ; hence, the an- 
 swer above; which, subtracted from the amount ob- 
 tained in exchange, leaves for the gain of the merchant 
 in trading $46,888,88f cents. The answer of the * 
 
200 RAINEY'S IMPROVED ABACUS. 
 
 first question is cents, because the price of one sheep, 
 last on the line, is cents. 
 
 From the foregoing we deduce the following 
 
 DIRECTIONS FOR COMMERCIAL EXCHANGE. 
 
 Make the gross quantity the demand; the specific 
 quantity the same name; and the specific quantity of 
 the article which it equals, the term of answer : repeat 
 the process, and continue the concatenation of state- 
 merit, until tht last term of answer is placed on the 
 right ; and the answer will be in the denomination of 
 such last term. 
 
 When tare, tret, or other per cent., is to be deducted, 
 place the standard, whatever it be, on the left ; place 
 the same standard, reduced by the deduction, etc., on 
 the right, for suttle, net weight, etc. 
 
 When a discount is to be deducted, place the 
 amount, 100 and rate, added, on the left; and 
 100, on the right, for present worth: 
 
 When a given per cent, is to be gained or lost, 
 place 100 on the left, and 100, increased by the gain, 
 or reduced by the loss per cent., on the right ; and the 
 answer will be the advanced or reduced price. 
 
 DECIMAL FRACTIONS. 
 
 Before commencing the article on Mensuration, it may be well to give 
 some few general remarks on decimal fractions, that the pupil may be pre- 
 pared to use them intelligibly, as they constantly occur in this department 
 of numbers. 
 
 Decimal fractions being of little service to the ordinary arithmetician, 
 except in Multiplication, Division, Addition, and Subtraction, we shall 
 give only such an outline of their nature and relations as will meet the 
 wants of the practical calculator. Consequently, the reader will not look 
 for an elaborate explanation of abbreviations in decimal calculations; of 
 circulating decimals ; or even of the four divisions mentioned. 
 
 Units are divided into regular and irregular 
 
THEORY OF DECIMAL FRACTIONS. 201 
 
 fractions. When divided into 3ds, 4ths, 9ths, IGths, 
 etc., they are called irregular or common fractions; 
 having such denominator as indicated by the number 
 of equal parts into which the unit is divided. 
 
 In decimal fractions the unit is divided into ten 
 equal parts ; while, again, one of the latter is divided 
 into 10 parts, making tenths, hundredths, thousandths, 
 etc. Hence, the name decimal, from the Latin, 
 decem, ten. 
 
 If units increase in a tenfold ratio, from right to 
 left, certainly, from left to right, they decrease again 
 in the same ratio. Now, continuing this decrease, 
 from the units place to the right, it is palpable, that 
 numbers decrease in each successive order, in the ratio 
 of .J-, y i-, T oVo- etc., without limit. Hence, the 
 first figure to the right of units, is ten times smaller 
 than units; the second one hundred times smaller; 
 the third, one thousand times smaller, and so on. 
 
 The point ( , ) is placed between whole numbers and 
 decimals to distinguish them ; and is called the sepa- 
 ratrix or decimal point.. The denominators of the de- 
 cimals .3, .4, and .07, would be T 3 , T 4 , T /^. Hence, if 
 the decimal numerator belong to an order of decimals 
 below tenths, a sufficient number of ciphers must be 
 prefixed to such numerator, to supply the place of the 
 vacant orders. In the case of yf^, above, it is neces- 
 sary, in showing that the 7 occupies the hundred's 
 place, to place a cipher before it, to fill the tenth's place. 
 If the 7 were y^o^, ^ ree ciphers would be placed at 
 the left for this purpose, and would be written, thus, 
 .0007, with the cipher prefixed as far as the order of 
 tens. From this we see, that 
 
 Tke denominator of any decimal fraction is a unity 
 with as many ciphers annexed, as there may be figures 
 in the numerator. 
 
 This is reasonable when we reflect that the denomi- 
 
202 RAINEY'S IMPROVED ABACUS. 
 
 nator of each separate figure in the numerator, is ten. 
 Consequently, 
 
 There must be one figure less in the numerator, 
 than in the denominator. 
 
 If it be necessary to express three, seven hundred 
 thousandths, we know that as there are six figures in 
 the denominator, there must be five in the numerator ; 
 and, as the 3 belongs to the order of hundreds of 
 thousandths, the five vacant orders must be filled by 
 figures that express no value ; thus, .00003. Hence, 
 
 Ciphers prefixed to decimal digits, have no active 
 value, and serve only to show the order occupied by 
 such digits. 
 
 The orders of decimals and their names may be 
 seen in the following 
 
 DECIMAL TABLE. 
 
 J -2 o 
 
 1 .3 sM 11 1 1 . 1 1 
 
 5 * * I is I ^ * * 
 
 'g tl "a o a Is 5 c'cs e la 3 all 
 
 ^ Q> & M *> B i 3 73 _3 *- W 
 
 5347.457923865497329 
 
 Here, at the left of the separatrix, we have 7 units, 4 tens, 3 hundreds, 
 etc.; while, at the right of it, we have 4 tenths, 5 hundredths, 7 thou- 
 sandths, etc.; or .45 hundredths; .457 thousandths, etc. Again, we pass 
 by or suspend the 4 and 5, and say that the 792, being of the order of hun- 
 dred thousandths, taken together, may be expressed, thus, .00792; the two 
 ciphers prefixed, showing the order to which the 792 belong. If these two 
 ciphers were left out, the 792 would be so many thousandths only. 
 
 Three, decimally expressed, would be .3, or -fa; 
 but, with a cipher prefixed, it would be .03, or T . 
 Hence, 
 
 Each move of a decimal one place from the deci- 
 mal point or unit, decreases its value ten tines. 
 
ADDITION OF DECIMALS. 203 
 
 Ciphers annexed to decimals do not change their 
 value; as the significant decimals occupy still the 
 same order or value in relation to the unit's place. 
 
 In decimal fractions, the denominator is never written 
 or expressed ; and is only understood. A great ad- 
 vantage in the use of decimals is, that instead of mul- 
 tiplying or dividing by the denominator, as many 
 figures may be cut off, as there are tens in the 
 denominator. 
 
 We give a few examples in the 
 
 ADDITION OF DECIMALS. 
 
 Add 318.972; 4.38; 62.7895; and 3412.013; 
 thus, 
 
 318.972 
 
 4.38 
 62.7895 
 3412.013 
 
 3798.1545 
 
 All of the units in the whole numbers 
 are written in a column; 8, 4, 2, and 2. 
 At the right of this, the decimals are 
 written, tens under tens, hundreds under 
 hundreds, etc., each order under its sep- 
 arate column, and under a similar order. 
 We add, as in other cases, beginning at the right, and 
 carrying all that may be over nine to the next figure 
 at the left, both in the decimals and the whole num- 
 bers. After this, the separatrix is placed in the sum, 
 in its own column, under the similar separatrices in 
 the sums above. Therefore, 
 
 TO ADD DECIMALS: 
 
 Place down the several whole numbers and deci- 
 mals, units under units; tenths under tenths; hun- 
 dredths under hundredths t etc.: add as in whole num- 
 bers, and place the separatrix of the sum under the 
 separatrices above. 
 
204 RAINEY'S IMPROVED ABACUS. 
 
 SUBTRACTION OF DECIMALS. 
 
 Subtraction in decimals is performed as in case of 
 whole numbers. Let the smaller number be written 
 under the larger; units under units; tenths under 
 tenths ; hundredths under hundredths, etc. 
 
 From 972.3856 subtract 298.534; thus, 
 
 972.3856 
 298.534 
 
 673.8516 i n / 10 o i i 
 
 1 9 from 12, 3, and so on ; borrowing as in 
 
 the subtraction of integers. Hence, 
 
 Nothing being under the 6 at the right 
 hand, we say, from 6 leaves six : 4 from 
 5, one ; 3 from 8, five ; 5 from 13, eight ; 
 
 TO SUBTRACT DECIMALS: 
 
 Place the smaller of the two numbers under the 
 larger ; units under units ; tenths under tenths; hun- 
 dredths under hundredths, etc.: subtract as in whole 
 numbers, and locate the separatrix, as in addition of 
 decimals. 
 
 If there be a larger number of decimals in the lower 
 than in the upper number, ciphers may be annexed, ad 
 infinitum, to the decimal in the upper number. We 
 have seen before, that ciphers thus added, do not 
 change the value of the number of decimals to which 
 they are appended. 
 
 MULTIPLICATION OF DECIMALS. 
 
 Multiply .46 by .5 
 
 We proceed as in ordinary multiplication ; thus, 
 
 .46 ' If a whole number be multiplied by a deci- 
 .5 
 
 .230 
 
 mal, the answer will be a whole number and a 
 decimal combined ; if a decimal be multiplied 
 by a decimal, the product, according to the 
 laws of multiplication, must be decimals only; for 
 decimal factors cannot produce integers ; nor can in- 
 tegers produce decimals ; that is, the product must be 
 
MULTIPLICATION OF DECIMALS. 205 
 
 of the denomination of the multiplicand and the mul- 
 tiplier. Hence, as both of these, in the question 
 above, are decimals, the product must be decimals ; 
 and as there are no integers, there can be no integers 
 in the result. Therefore, 
 
 As many figures must be cut off for decimals^ as 
 there are decimal factors, or places in both the mul- 
 tiplicand and multiplier. 
 
 As ciphers, appended to decimals, have no value, the 
 cipher in the result above may be dropped, and the de- 
 cimal called .23 hundredths, instead 230 thousandths, 
 which is equivalent, as before. 
 
 Multiply 275.437 yards of cloth by 3.07 dollars per 
 yard; thus, 
 
 Here, both the multiplicand and mul- 
 tiplier have both integers and decimals : 
 hence, there are both integers and deci- 
 mals in the answer. Cutting off five 
 places for decimals, the answer is 
 $845.59159 ; or 59 cents and 159 thou- 
 sandths of a cent. Hence,* 
 
 275.437 
 3.07 
 
 1928059 
 826311 
 
 845^69159 
 
 *SHORT METHOD OF MULTIPLYING DECIMALS. 
 
 In cases where the multiplier is 10, 100, 1000, 10000, etc., 
 the decimal point may be removed as many figures to the right, 
 as there are ciphers in the multiplicand. Thus: Multiply 
 198.7486 by 100. Thus, 19874.86, Ans. 
 
 Decimals below the 4th and 5th orders, are so small as to be 
 of very little value : hence, when the decimal places are very 
 numerous in the multiplicand and multiplier, or either, and it 
 is not desired to extend the calcu\ation beyond 4, 5, or 6 deci- 
 mal places, it becomes necessary to resort to a method of find- 
 ing the product without multiplying to whole number of de- 
 cimals. This is done as follows: We assume the following, 
 in ordinary multiplication: 
 
 The left hand figure of the multiplier may be multiplied by 
 
 14 
 
RAINEY'S IMPROVED ABACITST. 
 
 TO MULTIPLY DECIMAL FRACTIONS. 
 
 Proceed as in the multiplication of whole numbers / 
 cutting off as many figures in the product for deci- 
 mals^ as there may be decimal factors or places, both 
 in the multiplicand and multiplier. 
 
 When there is a larger number of decimal places 
 
 first, If tens are still placed under tens; .hundreds under 
 hundreds, etc.; thus, 
 
 1284 
 2475 
 
 2568 
 5136 
 8988 
 6420 
 
 3177900 
 
 1284 
 2475 
 
 2568 
 5136 
 8988 
 6420 
 
 8177900 
 
 Hence, as above, commencing with the left hand figure ID 
 the multiplier, and causing the scale to descend te the right, 
 produces the same result, as multiplying first by the unit's 
 place and descending to the left. 
 
 Now, it is evident, that we may multiply in cases of deci- 
 mals in the same way, and by carrying the multiplication to 
 a certain number of orders to the right, to get the product of 
 the orders so used, throw away all useless and minute mul- 
 tipliers. 
 
 Let us multiply 2.8724 by .37854; thus, 
 
 All of the figures to the right of 
 the vertical line, are useless, as there 
 are five places of decimals on the 
 left of it, being as many decimals as 
 desired in the product. 
 
 In placing down the entire pro- 
 ducts, the figures on the right of the 
 line serve to show what numbers are 
 carried to the first place of decimals 
 retained on the left. 
 
 2.872'4 
 .3785 4 
 
 86172 
 20106 
 2297 
 143 
 11 
 
 8 
 92 
 620 
 4896 
 
 1.0873 1 
 
 8296 Ans. 
 
MULTIPLICATION OF DECIMAL FRACTIONS. 207 
 
 in the multiplicand and multiplier, than in the pro- 
 duct, prefix ciphers to the product until the defi- 
 ciency is supplied. 
 
 We may now show how to get the figures on the left of 
 the line, without having to make those on the right. 
 
 We multiply the first right-hand figure of 
 the multiplicand, by the left-hand figure 
 of the multiplier, placing the product un- 
 der the figure thus multiplied. We next 
 multiply by the second decimal multiplier, 
 7. Multiplying this into 4 would necessa- 
 rily cause the product to be placed one 
 move to the right of the former product, 
 2; and, as this is unnecessary, we multi- 
 ply the 7 into the second figure from the 
 multiplicand, 2, carrying to the product 
 
 2.872'4 
 .3785 4 
 
 ~86172 
 
 20107 
 
 2298 
 
 143 
 
 11 
 
 1.08731 Ans. 
 
 the nearest number of decimals which this 7 and the 
 suspended 4 would make. Seven times 4 making 28, 
 nearly 3 decimals, we say, 7 multiplied by 2 equals 14, and 3 
 added, makes 17. Hence, the 7 is placed under the 2, at the 
 right, and 1 carried to the product of 7 into 7, which makes 
 49, making it 50, and so on. We multiply again by the next 
 decimal multiplier, 8, suspending both the 4 and 2, at the 
 right of the multiplicand, and carry its product into the 7 
 above, the nearest number of decimals that the 2 and 4 make. 
 Thus, 8 times 7 are 56: now, 8 times the former 2 are 16, 
 and 8 times 4 are 32 ; carrying 3 from 32, to 16, makes 19; 
 very nearly 2 decimals: hence, 2 added to 56 make 58; the 8 
 being placed under the right-hand column, and the 5 carried 
 as before. In like manner we proceed next with the 5 and 4. 
 After multiplying 8 by 5, we carry 3; because 5 into 7, the 
 suspended order, makes 35, which we call only 3 decimals, 
 although it is three and a half, allowing this half over 
 3, for the deficit in previous numbers, where the number of 
 decimals carried, was rather too large. Thus, the numbers 
 become pretty well balanced. 
 
 Above, we must always cast off one figure less in the mul- 
 tiplicand than the number of decimals which we wish to re- 
 tain: for the first decimal figure of the multiplier, when mul- 
 tiplied by, would otherwise give one factor too many; and 
 consequently, one place of decimals too many in the product. 
 The first multiplier will, however, be multiplied into the re- 
 jected figure, and the nearest number of decimals in the pro- 
 
In this division, the decimal on the left, 
 equals or neutralizes one decimal on the 
 
 .208 RAINEY'S IMPROVED ABACUS. 
 
 DIVISION OF DECIMAL FRACTIONS. 
 
 How many gallons of melasses, at .4 of a dollar per 
 gallon, can be bought for .96 of a dollar ? 
 
 Here, .4 equals T \ ; and .96 equals T 9 6 o We now 
 divide the latter common fraction by the former; 
 thus, 
 
 $ 24 | The answer is in units of gallons. 
 Now, instead of dividing 24 by this 
 10 on the left, it is quite as easy to 
 cut off one figure at the right of the 
 24; thus, 2.4, and the 4 is understood as T \, or a de- 
 cimal. Hence, it may be divided decimally ; thus, 
 
 .4) .96 
 
 24 
 
 right : hence, the remaining decimal becomes 
 
 a unit. 
 
 In multiplying decimals, the product must have as 
 many decimals as there are decimals in both the mul- 
 tiplicand and multiplier. The dividend is always 
 equal to the product of the divisor and quotient; 
 hence there must be as many decimals in the divisor 
 and quotient, taken together, as there are in the divi- 
 dend. Therefore, 
 
 To locate the decimal point, ascertain the differ- 
 ence between the number oj decimals in the divisor 
 and the dividend ; the remainder will be the number 
 of decimals to be cut off, in the quotient. If there are 
 not as many decimal places in the quotient as the dif- 
 ference^ prefix ciphers to the quotient, until the num- 
 
 duct, will be carried to the product of the first figure of the 
 multiplicand. The rejected figure is in the column of the 
 last decimal of the answer. Hence, count to the left from 
 this, and locate the decimal point accordingly. 
 
 This subject cannot be treated elaborately here; as it be- 
 longs to Elementary Arithmetic. 
 
DIVISION OF DECIMALS. 
 
 209 
 
 ber in the, quotient equals the number in the dif- 
 ference. 
 Divide .00954 by 3.08. 
 
 3.08).00954(.003 
 924 
 
 ~30 
 
 The answer is 3 thousandths. 
 By annexing ciphers to the 30, the 
 division may be continued, and the 
 quotients placed at the right of 
 
 the .003. 
 
 However far the division may be carried in this example, it will not ter- 
 minate. It is called a circulating decimal. The division has been con- 
 tinned on to fourteen places, giving the following, 
 
 .00309740292207792 
 
 When the number of decimals in the dividend and divisor, is the same, 
 the quotient is a whole number. When there are not as many decimals 
 in the dividend as in the divisor, annex ciphers to the former until they 
 are equal. 
 
 It is generally very useless to carry decimal calculations further than 
 four or five figures 
 
 TO DIVIDE DECIMAL FRACTIONS.* 
 
 Proceed as in the division of whole numbers ; and 
 
 ^CONTRACTION IN THE DIVISION OF TRACTIONS. 
 
 Division of decimals may be very much abridged when 
 there is a very large number of decimal places in the divisor, 
 as in the following example. 
 
 Divide 4.3125 by 3.2364, retaining four decimal places in 
 the answer. 
 
 Common Method. Contraction. 
 
 3.2364)4.3125(1.3324 
 32364 
 
 10761 
 9709 
 
 1052 
 971 
 
 81 
 
 _65 
 
 16 
 
 3.2364)4.3125 
 32364 
 
 (1.3324 
 
 10761 
 9709 
 
 
 2 
 
 1051 
 970 
 
 80 
 92 
 
 80 
 64 
 
 880 
 728 
 
 16 
 12 
 
 1520 
 9456. 
 
 3 
 
 2064 
 
210 RAINEY'S IMPROVED ABACUS. 
 
 cut off, in the quotient, a number of figures equal to 
 the excess of the decimals in the dividend, compared 
 with those in the divisor. If the number of decimals 
 in the quotient be too small, prefix ciphers until the 
 number equals the excess, above named. 
 
 Reduce the decimal .225 to an equivalent common 
 fraction of the lowest term. 
 
 Subscribing the denominator, .225 becomes fWo- 
 This reduced to its lowest term is \. Hence, 
 
 TO REDUCE DECIMALS TO COMMON FRACTIONS. 
 
 Cancel the decimal point, and place the denomina- 
 tor below the. given decimal; reduce the fraction to its 
 lowest term, and the answer will be an equivalent 
 common fraction, in its lowest term. 
 
 The first figure of the quotient is 1. Now, instead of an- 
 nexing a cipher to each remainder, and thus multiplying it 
 successively by 10, we reject at each separate division the 
 right-hand figure of the divisor, which is equivalent to divi- 
 ding it successively by 10. In multiplying the last unrejected 
 figure in the divisor, 6, by the second quotient figure, 3, mak- 
 ing 18, we carry 1, which is the nearest decimal that the pro- 
 duct of the 3 and the rejected 4, will make. The decimal 
 accession, from the rejected figures of the divisor is consid- 
 ered hi each subsequent multiplication and division, until the 
 4 required decimal orders are found for the quotient. In 
 multiplying the 3 in the divisor, by the third 3 of the quo- 
 tient, the product is increased to 11 by the accession from the 
 last two rejec4ed figures, 3 times 6 making 18, and 3 times 4 
 making 12, the sum of which is 20, or 2 decimals. 
 
 If the divisor has more figures than the number required 
 in the quotient, including integers and decimals, take as many 
 on the left of the divisor as required in the quotient, and di- 
 vide by them, as in other cases. 
 
 If the number in the divisor be smaller than that required 
 in the quotient, divide as ordinarily, until the deficiency is 
 filled; after which, contract as before. 
 
 When the divisor is 10, 10-0, 1000, etc., remove the separa- 
 trix to the left, as many places as there are such ciphers; and 
 the division will be performed without further reckoning. 
 
COMMON FRACTIONS REDUCED TO DECIMALS. 211 
 
 Keduce f to an equivalent decimal fraction. 
 
 We multiply the numerator and the denominator, 
 each, by 100; because the product of two tens into 
 the numerator, is the smallest number that can be di- 
 vided by the denominator, 4. This makes f -J-, which 
 divided by the original denominator, 4, gives --- 
 This T 7 o%- = .75; for always dividing the product of 
 the numerator and denominator into tens, hundreds, 
 tc., by the same denominator thus multiplied, it is 
 evident that the denominator must always be composed 
 of tens. .Since, therefore, these tens, hundreds, etc., 
 in the denominator of a decimal, are useless, we avoid 
 the process of getting them, and simply annex ciphers 
 to the numerator of the common fraction, and divide 
 by the denominator, until an exact result is obtained, 
 or as many decimal places as requisite ; Thus, 
 
 The division here terminates in two places ^ 
 of annexed ciphers. Again: ,75 
 
 Reduce 4 to a decimal, 
 
 Here the divisor terminates in one place of | ' 
 annexed ciphers. ,i 
 
 Beduce '-|* to a decimal ; thus 
 
 8)200000000000000000000 
 
 666666666666666666664- 
 
 TO REDUCE A COMMON TO A DECIMAL FRACTION. 
 
 Append ciphers to the numerator, and divide by the 
 Denominator , until the denominator terminates, or 
 
 *Tbis is called a repeating decimal ; showing that although the process 
 anight be repeated ad infinitival, yet the true result would never be ob- 
 tained. Hence, although we get nearer to the true answer at every step, 
 yet, we would never get it entirely, although the division were continued 
 forever. In such cases the division need not be carried further than from 
 four to six places, as in the seventh place one of the sixes would be only 
 ^. ___ &. ___ a very minute and scarcely conceivable common fraction, 
 
 1 (' 0' J 
 
 iT.h? /'/MA- iu;irk is appended to aho.w that it is still .imperfect. 
 
212 RAINEY'S IMPROVED ABACUS. 
 
 until a sufficient number of decimals is obtained. Cut 
 off, for decimals, in the quotient, a number of places 
 equal to the number of ciphers annexed. 
 
 When the figures in the quotient are not equal to 
 the number of ciphers annexed, prefix ciphers to the 
 quotient, until the deficiency is supplied. 
 
 The method of pointing off above, will appear reasonable, when we re- 
 flect, that every cipher annexed to the numerator, multiplies it by 10; 
 hence, after it is divided by the denominator, the quotient will be ten 
 times too large, and should, consequently be divided again by 10. This is 
 done most easily, by cutting off one figure toward the left. The same rea- 
 soning is true as regards annexing two, three, or more ciphers, and increas- 
 ing in the multiplication, by 100, 1000, etc., necessitating a division of the 
 result by the same numbers. Hence, the propriety off striking off a num- 
 ber of figures in the result, for decimals, equal to the number of ciphers, 
 appended. 
 
 To reduce compound numbers to decimals, Reduce the denom- 
 inate number to a fraction of the denomination required, and 
 this fraction, to a decimal. 
 
 Reduce 5 shillings 3 pence to the decimal of a pound. 
 5s. =60 d; and 60+3=63 d: now, 1=240 d: henceJLSL 
 of a pound must now be reduced to a decimal, thus, 
 
 24)63000(.2625 
 48 
 
 150 
 144 
 
 60 
 
 48 
 
 If we divide by the 240, there would be 
 a cipher in the unit's place; but dropping 
 the cipher in the divisor, we have no unit 
 in the quotient, and place the decimal point 
 at the left of the answer, which is .2625 
 
 decimals of a pound. 
 120 
 120 
 
 Reduce 15 minutes, 30 seconds to the decimal of an hour. 
 !5X'60==>900-)-30=930 seconds: now, 1 hour contains 
 3600 seconds: hence, reduce -^^g- to a decimal; thus, 
 
 36)9300QOO(.258S33 
 
 The result is .258333. This is a repeating decimal; hence> 
 the calculation is discontinued at six places. 
 
 All other redactions in denominate numbers may be mada 
 as these, 
 
DEFINITIONS IN GEOMETRY. 213 
 
 MENSURATION, 
 
 OR 
 
 PRACTICAL GEOMETRY.* 
 
 MENSURATION is that department of the 
 science of numbers, which treats of the meas- 
 urement of lines, superfices, solids, &c., and 
 is derived from measura, measure. 
 
 The general principles and laws regulating 
 this department of the science, are derived 
 from Geometry. 
 
 Geometry is the science of magnitude, in all 
 its various forms and relations ; and is divided 
 into practical and theoretical. The latter 
 treats of those portions which are so complex 
 as to require symbols and the higher mathe- 
 matical formulae for their illustration : the 
 former treats of such portions only, as depend 
 on the simple relations of numbers, as mani- 
 fested through proportion. 
 
 Geometry is from 7*7, the earth, and /if?po*>, 
 measure, and primarily signified the measure- 
 ment of the earth. 
 
 Many of the laws of Geometry are demon- 
 strated by formulae that the ordinary reader 
 
 * It may be remarked, while treating of superficial measurement, 
 that Abacus is a Latin word which means flat, fn the primitive 
 ages, ail calculations were made by the Oriental nations on boards 
 covered with dust, on which lines and signs could be easily 
 traced. From the word abak, signifying dust, the Greeks 
 deduced their word a@*g. This Abacus used by the Romans, and 
 Abax by the Greeks, was a large board with transverse lines 
 drawn on it, on which calculations were made by sundry move- 
 ments of pebbles, or calculi. Hence, the derivation of our En- 
 glish word calculate, from cakulo, which is from calculus, a pebble. 
 The latter word is from the Syriac kalkai, gravel. 
 ft 
 
214 RAINEY : S IMPROVED ABACUS. 
 
 would not comprehend ; they can, however, be 
 made quite as intelligible by the manifest 
 relations and deductions of common sense, 
 without the exercise of which, all formulae 
 become mere mechanical arrangements ; being 
 neither appreciated nor understood. Too 
 many writers endeavor to teach Mensuration 
 by the introduction of Geometrical signs and 
 reasonings ; thereby endeavoring to teach a 
 primary, by the rules of a secondary science. 
 Hence, the reason of so many failures in this 
 study ; and hence, the mechanical patch-work 
 by which many practical men make such 
 calculations. 
 
 Practical Geometry is divided into superficial 
 and solid. Superficial, which is from superficies, 
 the surface, the outside, fyc., relates to the meas- 
 urement of surface, which has extent merely, 
 without bulk ; and has two sides given to find the 
 contents : Solid Geometry relates to the measure- 
 ment of bodies or magnitudes, which have 
 length, breadth, and thickness. This species of 
 measurement is generally called cubic. A cubic 
 foot of timber, is a foot Io7ig, a foot wide, and 
 a foot thick, or 12 inches in everyway: hence, 
 when these 3 twelves are multipled continu- 
 ously, they make 1728, the number of cubic 
 inches in a cubic foot.* 
 
 Cubic is from the Latin cubicus, from cubus, 
 a die. Hence, cubic is a congregation of par- 
 ticles, forming a solid mass of six equal sides. 
 
 According to the laws of Multiplication, 
 concrete objects cannot be multiplied together ; 
 
 * This solid foot, or 1728 cubic inches, weighs 1000 ounus 
 rain water. 
 
THEORY OF MENSURATION. 215 
 
 nor can concrete objects of different denomi- 
 nations be multiplied. Feet multiplied by 
 feet, through ratio, will give feet; but feet 
 multiplied by inches, will give neither feet nor 
 inches. Ten feet long and 12 inches wide 
 will give neither 120 feet nor 120 inches. 
 Hence, when the denominations are dissimilar ', 
 such reductions must be instituted as will make the 
 terms alike. Above, if we divide the width 12, 
 by the number of inches in a foot, we find that 
 the width is 1 foot; now, the length and width 
 being in feet, we conclude that there are 10 
 superficial feet. 
 
 Ten feet in length and 10 in width give 100 
 feet superfice : this multiplied by 10 feet in 
 height, will give 1000 feet solidity. 
 
 These similar dimensions , length, height h, and 
 width, multiplied together, give the cubic or solid 
 contents of the figure, in the denomination of the 
 dimensions : as a crib, a box, a wall, a boat, a 
 cistern, &c. 
 
 Let the learner keep these truths before his 
 mind, and but few difficulties will present 
 themselves in ordinary measurements. When- 
 ever mathematical rules are introduced, they 
 must be received by the student on authority, 
 as it would be impossible, In a treatise on 
 numbers, to develop their principles. 
 
 It may be remarked here, that too much 
 time is generally spent on algebraic and 
 mathematical solutions, while the learner pro- 
 poses to study arithmetic. The introduction 
 of such questions and rules, is an oversight in 
 too many authors : for the student thus wastes 
 his time in pursuing the work of mathematics, 
 
216 RAINEY'S IMPROVED ABACUS. 
 
 which is impossible in arithmetic; while it 
 should be devoted to numbers only ; for noth- 
 ing: else than numbers can be learned in 
 arithmetic. 
 
 WOOD AND BARK. 
 
 WOOD and BARK are generally measured by 
 the cord, which is a pile 4 feet wide, 4 feet 
 high, and 8 feet long. The word cord is 
 derived from the Welsh cord, signifying a twist, 
 relating to a rope : hence, the cord, or rope, 
 with which the ancients were accustomed to 
 measure a pile of wood, gave 128 solid feet, 
 which these dimensions, 4, 4 and 8 make, 
 when multiplied. A pile of wood contains 
 more or less than a cord, when it has more or 
 less than 128 solid feet. Hence, wood is 
 measured by proportion. We may multiply 
 together the dimensions of the pile, and com- 
 pare the whole number of feet with 128; or 
 we may compare the several separate dimen- 
 sions with 4, 4, and 8. The latter is preferable. 
 
 How many cords of wood in a pile 120 feet 
 long, 20 feet wide, and 2 feet high? 
 
 g.^rfx ir i Here, we place the several 
 
 2 
 
 4 05 
 
 |37-J- cords. 
 
 dimensions of the pile, on the 
 right, and the dimensions of 
 a cord opposite these, on the 
 
 left ; and say, what will all 
 these feet on the right make, if 8, 4, and 4, 
 on the left, make 1 cord, last on the right, 
 it is unnecessary to place the 1 on the right, 
 as it will not assist in the calculation. 
 
WOOD AND BARK MEASURE. 
 
 217 
 
 How many cords in a pile 200 feet long, 3? 
 feet wide, and 16 feet high? 
 
 Here, we say 4 times 4 on 
 the left, equal 16 on the right. 
 The answer is eighty-seven 
 and a half cords. 
 
 cords. 
 
 7 
 
 32|175 
 
 51 1 cds. 
 
 How many cords in a pile of bark 20 feet 
 long, 3 feet 4 inches high, and 10i feet wide? 
 
 In this instance, 4 inches 
 are of a foot, making the 
 height 3J or y> feet. The 
 numerator of this, as well as j 
 of the *y , is placed on the 
 right, and the denominator 
 opposite. The answer is 
 5i| cords. 
 
 What will a load of wood 8 feet long, 2 feet 
 6 inches high, and 3 feet 4 inches wide, come 
 to, at 1 dollar and 80 cents per cord. 
 
 Again, the inches are made 
 the fractional part of a foot, 
 and added to the given feet 
 in each case ; while the mixed 
 number is placed on the line 
 in the form of an improper 
 fraction. We know that the 
 8, 4, and 4, on the left, make 
 
 5 
 
 /1 05 
 
 41375 
 
 93|cents. 
 
 one cord, or that these 128 feet are worth the 
 price, 180 cents; then the price is placed last 
 on the right in the place of the one cord, and 
 the answer must be the price of the whole 
 pile of wood at 180 cents per cord. This is 
 nothing more than simple proportion. Twice 
 
218 
 
 RAINEY'S IMPROVED ABACUS. 
 
 3 on the left, goes into 18 on the right three 
 times. The answer is 93 J cents. This method 
 is quite preferable to ascertaining the quantity, 
 which may be fractional, and multiplying it by 
 the price as a separate operation. 
 
 How many cords in a pile 
 of wood 10 feet long, 3 feet 
 wide, and 7 feet high? and 
 what will the same come to, 
 at 240 cents per cord? 
 
 There are 1|1 cords wood. 
 
 We will now state both in 
 one, thus. The two was used 
 in 4 on the left, and in 10 on 
 
 64J105 
 
 9 | 3,93f 
 
 2|75 
 
 \r4JA. o r* 11JL * ULL ILL 
 
 _itt=tl 6 the right. 
 
 What will a pile of wood 
 40 feet long, 3 feet high, and 
 20 feet wide, come to, at 2 
 dollars per cord? 
 
 The price is dollars in this 
 case, and the answer is in 
 dollars ; 37 dollars arid 50 
 cents. 
 
 From the foregoing, we conclude that, 
 To ascertain the number of cords in a pile or 
 load of wood or bark, place all of the dimensions 
 on the right in feet, and 4, 4, and 8, or 128, on 
 the left. If there are inches in any of the dimen- 
 sions, they must be reduced to the fraction of a 
 foot, and added to the feet, and treated as other 
 improper fractions . 
 
 If the answer is desired in the price of the 
 whole quantity of wood, place the price of one cord 
 last on the right, in dollars or cents, and the 
 
BOARD MEASURE. 219 
 
 answer will be the price of the whole, in dollars or 
 cents. 
 
 LUMBER MEASURE. 
 
 Under this head may be classed superficial 
 board measure, and the measurement of solid 
 timber. We have only two general dimensions 
 in board measure; length and width. The 
 thickness is generally considered a unit ; inch 
 measure being the standard. Anything under 
 one inch is not noticed ; but all above an inch 
 in thickness, as two inches, three inches, &c., 
 is called two, three, &c., thicknesses of lumber. 
 If a piece of lumber 20 feet long, 16 inches 
 wide, and 8 inches thick, be measured, the 
 thickness is called eight planks.* 
 
 The first thing to be done with such a ques- 
 tion as the one above, is to reduce the width, 
 which is in inches, to feet, that width and 
 length in feet may be multiplied together for 
 the superficial contents. This would after- 
 wards be multiplied by the 8 thicknesses, giving 
 8 times as many feet as in the one piece. All 
 of this may be done in the same operation ; thus, 
 
 It will be observed here, that 
 the length, width, and thickness 
 are all placed on the right of 
 the line, and 12 only, on the 
 left, to reduce the width, 16 
 inches, to feet. Hence, the an- ' |213 ft. 
 
 swer is 213J feet. 
 
 * In America the words board and plank, are variously 
 used to denote the same thing. This is incorrect. While 
 a board is a thin piece of timber, a plank is a thick and 
 heavy piece. The word is from the Dutch plank, or the 
 Danish planke, a thick board. Hence the difference. 
 
RAINEY'S IMPROVED ABACUS. 
 
 8|135 
 
 161 ft. 
 
 When it is necessary to get the cubic con- 
 tents, we place another 12 on the left, to reduce 
 the thickness in inches to feet, that having all 
 three dimensions in feet, the product may be 
 feet. 
 
 How many feet of lumber in a board 18 
 feet long, 7i inches wide, and li inches thick? 
 Here, the mixed numbers are 
 reduced to improper fractions, 
 as in all other cases. No other 
 statement is necessary in board 
 measure, than such as will ad- 
 mit of the several dimensions 
 being multiplied together. Now 
 the width being generally in inches, and the 
 thickness often fractional, it is quite conve- 
 nient to throw the numbers on the line, and 
 the standard which reduces the inches to feet, 
 with the denominators, on the left. 
 
 <15 How many feet in a board 7i 
 
 $ ^ feet long, 8 inches wide, and 4i 
 
 3 inches thick ? When the length 
 ~, , is a mixed number, as in this in- 
 ^ ' stance, it must be reduced to an 
 improper fraction, as in other cases. 
 
 What will a pile of planks containing 120 
 pieces, 16 feet long, 15 inches wide, and 6 
 inches thick, come to, at 37i cents per 100 ft. ? 
 In this instance, we mul- 
 tiply by 120, the number of 
 pieces, and ascertain the 
 number of feet in the whole 
 pile : the question is then 
 proportional, and by com- 
 bination of statement, we 
 
 | 54,00 
 
MENSURATION. 
 
 221 
 
 say, what will all of these feet come to, on the 
 right, if 100 feet opposite, cost 37i, or y cents ? 
 The answer is 5400, the number of dollars and 
 cents, which pay for the whole. 
 
 12 
 
 16 
 
 100 
 275 
 
 16 
 
 15 
 
 6 
 
 120 
 
 13 
 
 Suppose in the case above the 
 timber will lose T \ of an inch in 
 sawing. We say, what will the 
 whole quantity of lumber be re- 
 duced to, if y be reduced to i-f ? 
 
 $|43,87i 
 
 What will 4 pieces of timber come to, at 
 $2i per 100 ft. which are 10, 20, 18 and 12 feet 
 long respectively, and 16 inches wide, and 3 
 inches thick ? 
 
 10 
 20 
 18 
 12 
 
 60 entire length. 
 
 In this instance, it is necessary to add the 
 several lengths, and place their sum on the 
 right. Had there been 10, or any other num- 
 ber of pieces in each pile, 10 or such number 
 would be placed on the right, once, and only 
 once : for the question, by getting the sum of 
 the lengths, was changed into this, how many 
 feet in a piece 60 feet long, 16 inches wide, 
 and 3 inches thick ? Hence, ten times the 
 number in each case, would be ten times the 
 60 feet. 
 
 What will 10 piles lumber, with 40 pieces 
 lo 
 
222 
 
 KAINEY'S IMPROVED ABACUS. 
 
 in a pile, come to, at $H per 100 feet, the 
 plank being 18 inches wide, 3J inches thick, 
 and 20, 16, 17, 19, 23, 10, 7, 12, 6, and 20 feet 
 long? The sum of the lengths is 150 feet: 
 hence, we place it on the right, thus, 
 
 2, 
 
 * 
 
 4 
 
 150 
 
 3 
 15 
 
 8|3375 
 
 j 3 
 
 Here, 40 planks in a pile, is 
 placed down once for the whole 
 lot, considering that the lot is 
 now 150 feet long. The answer 
 is in dollars, because the price 
 was dollars. 
 
 What will 80 pieces of lumber, 
 8 feet long, 9 inches wide, and 
 2i inches thick, come to, at 60 
 cents per hundred ? 
 
 We deem the examples given, 
 sufficient for the measurement 
 of lumber, as there is but very 
 little difficulty in the statement. 
 
 How many cubic feet in a stick of timber 
 30 feet long, 8 inches thick, and 10 inches wide ? 
 In this example it is ne- 
 cessary to divide both the 
 width and the thickness by 
 12, to reduce them to feet, 
 that by multiplying all the di- 
 mensions in feet, the product 
 may be solid feet. Hence, 
 
 60 
 
 jlGfft. 
 
 To measure lumber, Place the length in feet, 
 the width in inches, and the thickness, in incJics, 
 on the right , and 12 on the left. 
 
 To ascertain the number of feet in the whole 
 pile, when of the same dimensions, place the number 
 
MASONRY. 223 
 
 of picas, likewise on tlw right. If the answer is 
 desired in dollars, or dollars and cents, place 100 
 on the left, and the price per hundred, on the right. 
 To lose a fraction for saw-cut, subtract the 
 fraction lost from such a number of parts of the 
 same she as would constitute a unit, place the 
 remainder on the right, and the number making a 
 unit, on the left. 
 
 MASONRY. 
 
 Masonry, as a department of measurements, 
 may properly be classed with cubic timber 
 measure. Stone work is measured by the 
 perch, which is generally 25 solid feet, or 16 J 
 feet long, \\ feet wide, and 1 foot high. A 
 solid perch in masonry, is a mass 162 feet in 
 every way. The word perch is derived from 
 the French perche, which signifies sharp, extend- 
 ing, fyc., as a pole or rod for measurements. 
 Hence the name is derived from the limits 
 which define it. 
 
 How many perches of masonry in a wall 80 
 feet long, 15 feet high, and 2i feet thick? 
 
 Here, we make 25 solid , , k\$Q 4 
 
 feet a perch, saying, what 
 will all of the feet in the wall 
 
 3 
 
 make, if 25 opposite make 1 r^r 
 
 perch. Hence 120 perches. 
 
 We may easily ascertain the price for a 
 piece of work at the same time that the quan- 
 tity is obtained, by placing the price per perch 
 last on the right. 
 
224 RAINEY'S IMPROVED ABACUS. 
 
 ^ o ~ What will it cost to put up 
 
 a wall 200 feet long, 6 feet 3 
 inches high, and 3f ft. thick, 
 at 120 cents per perch of 25 
 
 15 
 
 The price being in cents, 2 
 figures are cut off at the right of the answer, 
 for cents. 
 
 How much will it cost to wall a cellar, at 
 $1,60 cents per perch, 20 feet square, and 7i 
 feet deep, with a wall li feet thick? 
 
 / , 27 It is evident that the two 
 
 end walls are each 3 feet 
 shorter than those of the 
 sides : hence, the entire length 
 
 4 
 
 |1,60 
 
 ^h I CO QO 
 
 place this, with the height 
 and thickness, on the right, and the denomi- 
 nators on the left. We use the factor 5 on the 
 two sides of the line. Hence, 
 
 To ascertain the number of perches in a piece 
 of stone work, place the length, height, and width, 
 in feet, on the right, and 25, or whatsoever stand- 
 ard is acknowledged, on the left: the answer will 
 be solid perches. If the cost is desired, place the 
 price per perch last on the right, and the answer 
 will be tJie cost of the entire work, in dollars or 
 cents. 
 
 PLASTERERS', PAVERS', AND BUILDERS' WORK. 
 
 Plasterers and Pavers calculate their work 
 by the square yard, or 9 square feet : Builders 
 reckon by the square, which is 100 square 
 
PLASTERERS' AND PAVERS' WORK. 
 
 225 
 
 110 
 
 1 80 yards. 
 
 
 
 18 
 4 2 
 
 
 36 
 80 
 
 feet, in weather-boarding, ceiling, framing, 
 shingling, &c. 
 
 How many square yards of plastering in a 
 room 18 feet square, and 10 feet high. We 
 place the side, 18, on the right, and 4, which 
 will give all the sides. 
 
 Having the dimensions in 
 feet on the right, we place 9 
 feet, which make a square 
 yard, on the left : the answer 
 is the sum of the four sides, 
 80 yards. We now place 18 
 on the right of another line 
 twice, and ascertain the num- 
 ber of yards overhead, by the 
 same process, which is 36. 
 The two added, make the 
 number of yards in the room, 116. We might 
 have ascertained the cost of the whole, quite 
 as easily, by placing the price in each state- 
 ment, last on the right. 
 
 What will the plastering of a room come to, 
 which is 15 by 20 feet, and 12 feet high, at 
 22i cents per yard? 
 
 The 4 sides make 70 feet 
 around, which we place with 
 the height and price, on the 
 right. The cost of the sides is 
 $21,00. Again, we place 15 
 and 20 on the right, with the 
 price, and 9 on the left. This 
 makes the plastering over- 
 head come to 7 dollars and 
 50 cents, which, added to 
 the sum above, makes the 
 
 70 
 
 2 
 
 21,00 
 7,50 
 
 $|28,50 
 
 t 5 
 
 $|7,50 
 
226 
 
 RAINEY'S IMPROVED ABACUS. 
 
 30025 
 
 75 
 
 $|56,25 
 
 cost of the room amount to 28 dollars and 
 50 cents. 
 
 How much will it cost to lay a 
 pavement 300 feet long, and 4 
 feet 6 inches wide, at 37i cents 
 per square yard ? 
 
 This question is identical with 
 those just wrought in plastering. Hence, 
 
 To ascertain the number of yards of plastering, or 
 paving, place the whole length of the walls, or pave, 
 with the width or height in feet, on the right, and 
 9 on tJie left: if the answer is wished in money, 
 place the price per square yard, last on the right : 
 the answer will be the price of the whole. 
 
 How many squares of weather-boarding on 
 a building 50 by 40 feet, 21 feet high? What 
 will the same come to, at $1,50 per square? 
 
 The whole length of the build- 
 ing, or sum of the sides, is placed 
 on the right, with the price, and 
 i 100, the number of feet in a 
 
 JP 4O, /U I .-l -i n 
 
 1 square, on the left. 
 
 What does it cost to shingle a roof 80 feet 
 long, and 20 feet from the eaves to the cone, at 
 87i cents per square ? 
 
 The roof is 80 by 40 feet, which 
 dimensions are placed on the 
 right, with the price. Thus, by 
 proportion, what will all of these 
 feet come to, on the right, if 100 
 feet, opposite, cost 87i cents? The answer 
 is $28. Hence, 
 
 To ascertain the cost of weather-boarding, shingling, 
 framing, flooring, <fyc., place the length and width in feet, on 
 the right ; ] 00 on tlie left ; and the price per square, last on 
 the right. 
 
 180 
 
 21 
 
 1,50 
 
 m 
 
 40-2 
 176 
 
CORN, COAL, AND LIQUID MEASURE. 227 
 
 CRIBS, BOXES, AND BODIES. 
 
 The standard of measurements of this kind, is gener- 
 ally inches; the number of inches making a bushel, a 
 gallon, &c. 
 
 A compact bushel, as wheat, shelled corn, salt, <fyc. 9 contains 
 2150i cubic inches, which may be expressed and used deci- 
 mally, in the form of 2150,2. 
 
 A dry busJiel, as potatoes, apples, coal, <$-c., contains 2688 
 cubic inches. 
 
 A wine gallon contains 23 1 : a beer gallon, 282 cubic inches. 
 
 A solid foot contains 1728 cubic inches. 
 
 If the length, width, and height of a body, crib, or box, 
 are placed on the right of the line., their product will be 
 the number of cubic inches in such body, crib, &c. The 
 question then is, by proportion, what will all these inches 
 on the right give, if 2150 opposite, give 1 bushel ; or, if 
 2688 give one dry bushel; or, 231 give 1 wine gallon; or, 
 if 282 give 1 beer gallon, &c. 1 Hence, after these 
 dimensions are placed on the right, it is only necessary 
 to place the number making a unit of the desired 
 measure, on the left. If, in measuring *coal, the body is 
 wider at the top than at the bottom, take the mean, width, 
 by measuring half way between the top and bottom. 
 
 The 2150 inches used, make an even bushel; hence, 
 when the corn is in the ear, place 10 on the left : when 
 in the husk, place 20 on the left, and the answer will be 
 in barrels. This allows 5 bushels of shelled corn, 10 of 
 unshelled, and 20 in the husk, for a barrel. 
 
 When it is necessary to ascertain the whole price, the 
 price per bushel, &c., may be placed last on the right. 
 
 160 
 
 How many bushels corn in a crib 
 160 inches long, 86 inches wide, and 
 90 inches high 1 
 
 TKe factor 5 is hero used in the 
 10, and 215. 
 
 How many bushels in a crib 120 
 inches long, 45 wide, and 80 high "? 
 
 We frequently find it impracticable I |80 
 to cancel in these calculations. This, !" [8640~ 
 however, is of but little moment. j -- 
 
228 
 
 RAINEY'S IMPROVED ABACUS. 
 
 100 
 
 
 
 $|100,00 
 
 100-15 
 W 
 
 $|4,50 
 
 1 1000 galls/ 
 
 47 
 
 What will a crib of corn come 
 to, at 21^ cents per bushel, 200 
 inches long, 100 wide, and 50 
 deep] 
 
 This combination of state- 
 ment is quite simple and easy. 
 
 What will a load of coal cost, 
 at three and a half cents per 
 bushel, which measures 120 in. 
 long, 48 inches wide, and 60 
 inches high ? 
 
 The price, 4 dollars and 50 
 cents, pays for the whole load. 
 
 How many gallons water in a 
 tan vat, 70 inches long, 60 inches 
 wide, and 55 inches deep ? 
 
 The answer is 1000 gallons, in 
 measure. 
 
 How much will a vat of beer 
 come to, at 50 cents per gallon, 
 the vat being 120 inches long, 
 40 inches deep, and 20 inches 
 wide ! 
 
 The answer is 170 dollars, 21 
 170,211^ cents, and a fraction. 
 
 To find the contents of a crib, body, or box, in compact 
 bushels, place all the dimensions on the right, in inches, and 
 on the left 2150. If the answer is wished in barrels, place 
 5 on the left for shelled corn, 10 for corn in the ear, and 20 
 for corn in the husk. 
 
 When the answer is desired in dry bushels, place 2688, 
 instead of 2 1 50, on the left. In either case, place the price, on 
 the right, last, and the answer will be the price of the whole. 
 
 To ascertain the number of gallons, place the dimensions, 
 as above, on the right, and for wine gallons 231, or for beer 
 gallons 282, on the left. The price per gallon may be placed 
 last on the right ; the answer will be the price of the whole. 
 
 When the contents of any crib, box, or body, and two of 
 the sides are given to ascertain the other side, place the con- 
 tents and the standard of unity, or the number of inches which 
 make a unit of the contents, on the right, and the two given 
 dimensions on the left : the answer will be the required side. 
 
 402 
 
 50 
 
 1 800000 
 
TONNAGE OF VESSELS. 
 
 229 
 
 A body is 120 inches long, and 86 inches wide ; 
 how high must it be to hold 180 bushels of coalV 
 
 It is seen here, that the 2 #0 #0^224 
 body must be 112 in. high; or # #0 j/L$0 p # 
 
 9 ft. 4 in. Again, ~ r~^ ~ 
 
 112 inches. 
 
 A crib is 215 inches long, and 100 inches wide ; 
 how high must it be to hold 1500 bushels of corn? 
 
 Thus, the crib must be 150 in., 
 equal to 12| feet high. 
 
 1500 
 
 150 in. 
 
 TONNAGE OF VESSELS. 
 
 In ascertaining tonnage, it is necessary to ascertain 
 as nearly as practicable, the number of cubic feet of 
 water displaced by the vessel. This is done by mul- 
 tiplying together the length, width, and depth of the 
 vessel, which gives the number of cubic feet con- 
 tained in the hull. Now, it is a law of hydrostatics, 
 that "if a body floats on a fluid, it displaces as much, 
 of the fluid, as is equal to its own weight" Nor 
 does it make any difference what the shape of such 
 body be ; a quantity of water equal to its own weight 
 must be displaced. Hence, in ascertaining the weight 
 that any hollow square will sustain in water, it is ne- 
 cessary, first, to ascertain the weight of water such 
 square would contain; making all due allowance for 
 weight of vessel, room for safety, etc., etc. A cubic 
 foot of water weighs 1000 ounces avoirdupois,* or 
 62i Ibs.; hence, 95, the number of cubic feet allowed 
 
 *The cubic foot of distilled water weighs about 1000 oz. 
 avoirdupois, or very nearly 62J Ibs., at 40 temperature; at 
 GO , which is generally used, It weighs only 62,353 Ibs.,, less 
 than 1000 oz. The foot weighs 911.458 oz. troy, or .5274 oz 
 per cubic inch. The cubical foot equals 2200 cylindrical, 
 3300 spherical, or (F600 conical inches. A cylindrical foot of 
 
230 RAINEY'S IMPROVED ABACUS. 
 
 for 1 ton, or 2240 Ibs. of freight, will weigh 5937|lbs. 
 avoirdupois. This allows nearly three tiroes the 
 weight of the freight in water, to the ton. 
 
 There are two methods used in reckoning tonnage, 
 the carpenters', and the government rule. 
 
 The following is the government rule for measuring 
 single or double-decked vessels : 
 
 " If the vessel be double-decked, take the length thereof, from the fore 
 part of the main stem, to the after part of the stern post, above the upper 
 deck; the breadth thereof at the broadest part above the main wales, half 
 of which breadth shall be accounted the depth of such vessel; and then 
 deduct from the length three-fifths of the breadth, multiply the remainder 
 by the breadth, and the product by the depth, and divide thi's product by 
 
 water weighs 49.1 Ibs., avoirdupois; a cylindrical inch, .02642, 
 and a cubic inch, .03617 Ibs., avoirdupois. 
 
 Seawater weighs 1.03 times distilled water, which is the 
 standard of weight: hence, 1 cubic foot of seawater weighs 
 1030 oz., avoirdupois. f 
 
 Nineteen cubic inches distilled water, temperature 50 
 Fahr., weigh 10 oz. troy, according to act of parliament, 1825. 
 
 It may be observed here, that the standard of liquid measure 
 MI the United States, is the wine gallon, containing 231 inches, 
 equal to 8.339 Ibs avoirdupois, or 58372.1754 grains distilled 
 water. 
 
 The English imperial standard gallon is 10 Ibs., avoirdupois, 
 distilled water, at 62 Fahr., and 30 inches barometer: and is 
 about equal to 277.274 inches. It is about equal to one 
 and one-fifth, or 1.2 gall., wine measure of the U. States. 
 
 In Great Britain, the imperial bushel weighs 80 Ibs., avoirdu- 
 pois, distilled water, at 62 Fahr., and 30 in. of the barometer. 
 It is a vertical cylinder, 18,789 inches in diameter, and 8 inches 
 deep; and contains 2218.192 cubic inches. This standard is 
 adopted in New York. 
 
 The United States standard of dry measure is the Winches- 
 ter bushel, containing 77.627413 Ibs., avoirdupois, distilled 
 water, maximum density, and weighed in air at 30 in. barom- 
 eter. It contains 2150.42 cubic inches nearly, although 
 2150.2 is more used. 
 
 In Connecticut, 2198 cubic inches make a bushel. The 
 measure used, varies in the different states: hence, the pro- 
 priety of knowing the standard of the state in which the cal- 
 culation is made. It should be the same in all of the states, 
 as is our currency. 
 
TONNAGE OF VESSELS. 
 
 231 
 
 95, the quotient whereof shall be deemed the true contents or tonnage 
 of such ship or vessel: and if such ship or vessel be single-decked, take 
 the length and breadth, as above directed, deduct from the said length 
 three-fifths of the breadth, and take the depth from the under side of the 
 deck plank, to the ceiling of the hold; then multiply and divide as afore- 
 said, and the quotient shall be deemed the tonnage." 
 
 The foregoing rule may be used to get the dimen- 
 sions ; after this, the dimensions may be placed on the 
 right of the line, to find the entire number of feet in 
 the boat, and 95 on the left. The proportion will be, 
 as 95 to the continued product of these dimensions, so 
 will 1 ton, which the 95 feet equals, be to the whole 
 number of tons in the vessel. 
 
 What is the government tonnage of a single-decked 
 vessel, 115 ft. keel, 25 ft. beam, and 10 ft. hold? * 
 
 Here, f of the width, which is 25 feet, equals 15 
 feet, to be subtracted from the length. This leaves 
 the length 100 feet. The dimensions are placed on 
 the right, and 95 on the left ; thus, 
 
 In this instance, the answer is 
 263 T 3 g- tons. In carpenters' mea- 
 sure, the | of the width would not be 
 taken off, but would be calculated, 
 thus, 
 
 Here, the answer is 302|f tons. 
 
 100 
 
 10 
 
 5000 
 
 9__0 115 
 
 10 
 
 5750 
 
 302|f 
 
 How many tons in a double-decked vessel, 268 ft. 
 long, 30 ft. wide, and 15 ft. deep ? 
 
 Here J of the width, 18ft., are subtracted from the 
 
 *The keel of a vessel is the main bottom in length: the beam is the 
 greatest width from side to side of the hull; and the hold, the 'depth from 
 the main deck to the bottom of the hull. 
 
232 
 
 RAlffEY g IMPROVED ABACUS. 
 
 length, leaving 250 ft. long. The depth being ^ of 
 the width, is 15 ft.: we state accordingly, 
 :250 
 30 
 
 From this calculation, the ton- 
 nage is 1184 T 4 . 
 
 22500 
 
 What is the tonnage, according to carpenters' mea- 
 sure, of a steamboat, 300 ft. keel, 10ft. hold, and 38 
 ft. beam? 
 
 We here place all of the dimensions on the right, 
 and divide by 95 ; thus, 
 
 300 The factor, 5, is contained in 10 
 
 twice, and in 95, nineteen times : 19 
 into 38, twice, and 2x2x300= 
 1200 tons, the answer. Hence, 
 
 Tons. 
 
 1200 
 
 TO ASCERTAIN TONNAGE. 
 
 Ascertain frst the dimensions of the vessel, according to gov- 
 ernment, or carpenters' rule: place the dimensions on the right 
 of the line, and 95 on the left: the answer will be the number of 
 tons. 
 
 To ascertain any given dimension, when two of the dimen- 
 sions and the tonnage are given, place the tonnage and 95 on the 
 right, and the two given dimensions in feet, on the left: the an- 
 swer will be the required dimension in feet. 
 
 SUPERFICIAL GEOMETRY. 
 
 VARIETY OF FIGURES, DEFINITIONS, ETC., ETO. 
 
 A line is the shortest possible distance between two 
 points. A line is supposed to be straight; when 
 curved, the word, curved, is mentioned. 
 
 Two straight lines are parallel, when equally dis- 
 
DEFINITION OF TERMS IN GEOMETRY. 233 
 
 tant at all parts ; and when they cannot come togeth- 
 er, however far extended or produced.* 
 
 The point of intersection between two lines, is 
 called an angle. ,f 
 
 An angle of 90 degrees, or ^ of the circumference 
 of a circle, is called a right angle : an angle of less 
 than 90 degrees, is called an acute, or sharp angle. 
 
 An angle of more than 90 degrees, is called an ob- 
 tuse, or blount angle. 
 
 A figure with four equal angles, and four equal 
 sides, is called a square.^. 
 
 A figure with four equal angles, and unequal sides, 
 is called a rectangle.\\ 
 
 A figure with four parallel unequal sides, and two 
 acute, and two obtuse angles, is called a parallelo- 
 gram^ or rhomboid.** 
 
 A figure with four equal sides, and two obtuse and 
 two acute angles, is called a rhomboid; sometimes, a 
 lozenge. ,ft 
 
 * The word produced, in this instance, bears its literal signi- 
 fication, lead out, from the Latin, pro and duco. 
 
 t Angle is a French word, which is from the Latin, angu- 
 lus, a corner. 
 
 J Square is derived from the French, quarre, which is ori- 
 ginally from the Latin, quatuor,four. 
 
 || Rectangle is from the Latin, rectus, right, and angulus, an 
 angle. 
 
 .Parallelogram is derived from the Greek, 9rt/>atXAxof, or 
 parallelos, opposite the one to the other, and y^ppa., or gram- 
 ma, a character or figure. 
 
 ** Rhomboid is from the Greek, /Joyu/So?, or rorribos, a rhomb, 
 and wJoff, or eidos, form. Rhomb is from the Latin rhombus, 
 a whirl, or constantly varying square : in fabulous history, a 
 varying or rolling instrument by which witches were said to 
 bring down the moon from heaven. 
 
 ft Lozenge is from the Gr. AOOC or loxos, oblique,, and yuvk*. 
 or gonia y a corner 
 
234 RAINEY'S IMPROVED ABACUS. 
 
 A figure with four unequal angles, and two parallel 
 sides, is called a trapezoid* 
 
 A figure having three or more equal sides, is called 
 a polygon.^ 
 
 The lowest polygon, that of three sides, is called a 
 triangle ;J that of four sides, a quadrilateral; of five 
 sides, a pentagon; of six sides, a hexagon; of seven 
 sides, a heptagon; of eight sides, an octagon; of 
 nine, a nonagon; of ten, a decagon; of eleven, an 
 undecagon ; and of twelve, a dodecagon. 
 
 There are four principal triangles : the equilateral, 
 the isosceles, the scaline, and the rectangle triangle. 
 
 A triangle which has three sides is called equilate 
 ral :\\ one which has two of its sides equal, is called 
 isosceles : one which has three unequal sides, is called 
 scaline :^[ and that which has a right angle, is called a 
 rigM angled, or rectangle triangle. 
 
 The longer arm of the right-angled triangle is called 
 the basefoom the Lat. basis, the bottom or foundation : 
 the shorter arm is called the side ; and the side opposite 
 the right angle, the hypotenuse, from the participle of 
 
 * Trapezoid is from the Gr. rpstTrt^tov, or trapezion, a small 
 table, and s^To?, or eidos,form. 
 
 f Polygon is from the Gr. TTOMZ, many, and ywta, an angle. 
 
 t Triangle is from the Lat. triangulum, from tres, three, and 
 angulus, a corner: hence, a figure with three angles. 
 
 Quadrilateral is from the Lat. quatnor,four, and latus, side; 
 having four sides : pentagon, from the Gr. WSVTS, or pente, 
 jive, and yu>vi, or gonia, a corner. Hexegon, heptagon, octo- 
 gon, nonagon, decagon, undecagon, and dodecagon are com- 
 pounded by prefixing to the yavtx, or gonia, t%, or ex, six; sirra., 
 or epta, seven; x.ra), or okto, eight; Lat. nonus, nine; JMA, or 
 deca, ten; Lat. undecim, eleven; JWSKA, or dodeka, twelve, etc. 
 
 || Equilateral is derived from the Lat. aquus, equal, and late- 
 ralis, from latus, a side: equal sided. 
 
 Isosceles is derived from the Gr. ires or isos f equal, and 
 CTMSXO? or skelos, a leg: hence it has two equal legs. 
 
 If Scaline is from the Gr. O-X.X.XMK, or skalenos, that totters or 
 hangs over to one side, obliquely. 
 
GEOMETRICAL FIGURES. 235 
 
 the Greek verb wroTwovo-et 9 or upoteinousa, extending 
 under, or from corner to corner. 
 
 Any of the sides of an equilateral and scaline tri- 
 angle may be called the base. The base of an isos- 
 celes triangle is the short side. 
 
 The upper point where the two sides of an isosce- 
 les, or other triangle, meet, is called the vertex of the 
 triangle; by some, the apex.* 
 
 The theory of determining angles, and the lengths 
 of the sides of triangles, constitutes the science of 
 Trigonometry ; and cannot be properly treated in arith- 
 metic. We have before seen that, 
 
 To find the contents of a square or rectangle, multiply the 
 two sides together. 
 
 A parallelogram is equal in contents to a square or 
 rectangle, when its base and verticalf hight are equal 
 to the base and side of such square or rectangle 
 
 A rhomboid or lozenge is equal to a square of the 
 game base and altitude. Hence, 
 
 To find the contents of a parallelogram, or rhomboid, multi- 
 ply the base by the vertical hight. 
 
 To find the contents of a trapezoid, multiply half the sum of 
 the two parallel sides, by the vertical distance between them. 
 
 How many yards of plastering in a wall 30 feet 
 square ? 
 
 We divide by 9 feet, which make 1 square 
 yard. 100 
 
 * These two words are frequently used synonymously. 
 Some apply apex to triangles, and vertex to cones, because of 
 the primary signification of vertex, from the Lat. vertex, a 
 point, which is from verto, to turn. The plural of vertex, is 
 vertices; and of apex, apices. 
 
 fThe vertical hight of any quadrilateral or four-sided 
 figure, is a line dropped from the upper plane or vertex, per- 
 pendicular to the base. Hence, the side of a parallelogram 
 must not be multiplied into the base for the contents, as thte 
 is too long, but the side arising from a line dropped at right 
 angles with the base. 
 
236 
 
 RAINEY'S IMPROVED ABACUS. 
 
 How many acres are there in the road from Cincin- 
 nati to Dayton, which is 64 miles long, and 4 rods 
 wide? 
 
 64 
 
 Ans. 
 
 512 A. 
 
 Here, we ask how many rods 64 
 miles will make, if 1 mile make 8 
 furlongs, and. 1 furlong make 40 
 rods ; then, multiplying by 4 rods in 
 width, we say, how many acres will 
 all these rods make, if 160 rods 
 make 1 acre? Ans. 512 acres. 
 
 The first question relates to a square, or equal 
 rhomboid; the second, to a rectangle, or equal paral- 
 lelogram. 
 
 The two parallel sides of a trapezoid are 40 and 60 
 rods, and the distance between them 80 rods : how 
 many acres are there ? 
 i 025 
 #0 The answer is 25 acres. 
 
 Ans. 25 acres. 
 
 TRIANGLES. 
 
 Every triangle is half of a square, rectangle, paral- 
 lelogram, or rhomboid of similar base and altitude. 
 To prove this, let us, on the hypotenuse, or longest 
 side, of any given triangle, erect another triangle, with 
 the side and base parallel and equal, each, to the side 
 and base of the triangle. The figure formed will be 
 a square, rectangle, parallelogram, or rhomboid, which 
 proves the position correct. Hence, 
 
 To find the contents of any triangle, multiply half 
 of the base by the whole vertical hight, or the whole 
 base by half of the vertical hight, as may be most 
 convenient. 
 
 How many acres of land in a rectangle triangle, of 
 240 rods base, and 120 rods side ? 
 
THEORY OF TRIANGLES. 
 
 237 
 
 Ans. 
 
 90 A. 
 
 Here, we place down the whole ' fi.$0 fififi 3 
 base and side, and divide by 2, # 1#0 3 
 
 which is both easy and simple ; 
 while, likewise, we have the advan- 
 tage of dividing by such denominate numbers as are 
 necessary to reduce to a given denomination. 
 
 How many square feet in an isosceles triangle, 
 of 16f ft. base, and 37^ ft. vertical hight? 
 
 505 
 
 c\ 
 
 Here, 2 is thrown on the left; and we 
 divide by the denominators, 3 and 2. 
 Hence, the answer, 312^ square feet. 
 
 625 
 
 If a line dropped from the vertex of a scaline tri- 
 angle, fall outside of the triangle, the line of the base 
 must be produced until it meets the vertical line. 
 
 The square* of the hypotenuse of a right-angled 
 triangle, is equal to the sum of the squares of the 
 base and side. 
 
 For example ; the base of a right-angled triangle is 
 8, the side 6, and the hypotenuse 10. Now, 8x8= 
 64; and 6><6=36; and 36+64=100; hence, the 
 hypotenuse is 10x10=100, which is the sum of the 
 squares of the base and side. A rectangle, whose 
 sides are 4, 3, and, 5, shows the same equality; thus, 
 4X4=16; 3x3=9; and 16+9=25: now, 5x5 
 =25, which proves again that the sum of the squares 
 of the two sides equals the square of the hypotenuse. 
 
 Frem the foregoing, it follows, that 
 
 To find the hypotenuse of a right-angled triangle^ 
 
 *The square of any number, is that number multiplied into itself; 49 is 
 tbe square of 7- When it is desired to square a number, 2 is written over 
 it, thu, 72 
 
 1G 
 
288 RAINEY'S IMPROVED ABACUS. 
 
 extract the square root of the sum of the squares of 
 the base and side.* 
 
 To find the base, when the hypotenuse and side are 
 given, subtract the square of the side from the square 
 of the hypotenuse ; extract the square root of the re- 
 mainder, and the answer will be the base. 
 
 To find the side, when the hypotenuse and base are 
 given, subtract the square of the base from the square 
 of the hypotenuse ; extract the square root of the re- 
 mainder, and the answer will be the required side. 
 
 The hypotenuse of a right-angled triangle is 10 ft., 
 and the side 6 ft.; what is the base ? 
 
 The square of the hypotenuse is 10x10=100; 
 and of the side 6x6=36: now, 10036=64, and 
 the square root of 64, is 8, which is the required base. 
 
 POLYGONS. 
 
 If lines be drawn from the angles of the polygon, 
 to the center, it is manifest that the polygon will be 
 divided into isosceles triangles ; and if the contents 
 of one of these be multiplied by the whole number of 
 angles thus made, the answer will be the contents of 
 the polygon. Hence, 
 
 To find the contents of a regular polygon, multiply 
 half of the diameter of the polygon, vertical to one 
 of the sides, by half of one of the sides; and the 
 product by the number of sides. 
 
 Or, Place the shortest semi-diameter*, and one oj 
 the sides, on the right of the line, and 2 on the left. 
 
 * Carpenters frequently use this method of finding the length of braces, 
 where the distance from the angle, or lower end of the post, to the extreme 
 end of the mortice, is given. 
 
 W!,en the base and side of a rectangle triangle are of the same length, 
 th* 1 hypotenuse may be found by multiplying the base or side by 1.4142. 
 This number is the square root of twice 3J41592, which is the ratio of the 
 circumference to the circle. 
 
 t Semi-diameter means half -diameter, from the Latin semi, half. Ra- 
 dius is used to denote the same thing in circles. 
 
SPHERICAL MEASUREMENTS. 239 
 
 THE CIRCLE.* 
 
 Squares are always used as the units of superficial 
 measurement, by reason of their sides and angles co- 
 inciding, and leaving no intervening space between 
 their limits. The unit assumed is generally a square 
 inch, a square foot, a square yard, a square mile, etc. 
 
 A circle is a plain figure, bounded by a line called 
 the circumference,^ which is, at all parts, equally dis- 
 tant from a point within, called the centre ; hence, 
 
 The circumference of a circle is a line drawn at all 
 parts equally distant from the centre. 
 
 The diameter % of a circle is a straight line drawn 
 from the opposite sides of the circumference, through 
 the centre, dividing the circle into two equal parts. 
 
 The periphery \\ of a circle is its circumference; the 
 two words being used synonymously, at pleasure. 
 
 The radius^ of a circle is a line drawn from the 
 
 * Circle is derived from the Latin circus, a round ring, or 
 limit; or from the Gr. M/MOS or kirkos, a falcon, that, in flying, 
 describes circles ; or from the Arabic kara, to go round. 
 Many individuals confound circle with circumference; where- 
 as, while the latter merely describes the limits, the former is 
 the space included in such limits. 
 
 t Circumference is from the Lat. circum, around, and/ere/i- 
 tia, from fero, to bear. 
 
 Centre is a French word, from a Gr. noun, signifying a 
 goad or point, which is from the root KWWM or kenteo, to prick. 
 
 t Diameter is from the Gr. Jin or dia, through, or through the 
 middle, and jutrpsv or mttron, to measure. 
 
 || Periphery is from iryt or peri, around, about, and pg/>a> or 
 fthero, to bear: hence, it is identical with the circumference. 
 The word perimeter is sometimes used in the same sense, but 
 improperly. It relates particularly to the measurement or ex- 
 tent of circumferences, from peri and metron, to measure 
 around. 
 
 Radius is a Lat. word, from radio, to sJuoot beams of light, 
 etc. The use of this term in geometry, originates from the 
 fact, that when a large number of radii are drawn in a circle, 
 the circle resembles the sun darting his rays in every direc- 
 tion from the center. 
 
240 RAINEY'S IMPROVED ABACUS. 
 
 center to the circumference, or half the diameter ; two 
 or more of these lines are called radii. 
 
 The perimeter* of a circle, or other figure, is the ex- 
 tent of its circumference or bounds. 
 
 The arecfi of a circle, or other figure, is the surface 
 or space contained within the limits of the circumfe- 
 rence or perimeter. 
 
 A circle is said to be inscribed J in a polygon, when 
 the line of the circumference, cuts the sides of the 
 polygon. 
 
 A polygon is inscribed in a circle, when its angles 
 coincide with the circumference. 
 
 A polygon is circumscribed|| about a circle, when 
 its sides coincide with the circumference. 
 
 A semicircle^ is a half circle, described by cutting 
 the circumference of a circle by a right line drawn 
 through its diameter. 
 
 An ellipse** is an oblong, circular figure, having 
 two axes; the minor, a transverse, and the major, a 
 longitudinal line, each drawn through the center, and 
 on either of which, the figure may be supposed to 
 revolve. 
 
 * Perimeter is from the Gr. mp or peri, around, about, par- 
 ticularly around the space described from a center, and jmtrpsv 
 or metron, to measure. 
 
 f Area is a Latin word, which means space within given 
 bounds. Dr. Webster thinks it is from the Chaldee word 
 ariga, a bed; or from a Hebrew word which signifies to 
 stretch or spread The plural of area is area; this is, however, 
 seldom used; being substituted by the anglicised word, areas. 
 
 t Inscribe is from the Latin inscribo, to write within. 
 
 II Circumscribe is from circum, around, and scribo, to write 
 or draw. 
 
 IP Semicircle is from the Lat. semi, half and circulus, a circle. 
 
 ** Ellipse is from the Gr. root &.KWTOO or dleipo, to pass by 
 or reject. 
 
QUADRATURE OF THE CIRCLE. 241 
 
 The circumference of a circle is divided* into 360 
 equal parts, called degrees ; each of these degrees, into 
 60 parts, called minutes ;f and each of these minutes 
 into 60 parts, called seconds. The degree, is marked, 
 thus ( ) ; the minute, thus ( ' ) ; the second, thus 
 ( " ). Sometimes 30 degrees are said to make 1 sign, 
 marked (5); and 12 signs, 1 circle, marked (c). 
 
 QUADRATURE OF THE CIRCLE. 
 
 The circle, from the varied nature of its uses and 
 application, is one of the most interesting, and yet 
 perplexing, figures in geometry. The comparison of 
 circles and squares; the difficulty of determining the 
 ratio of the circumference to the diameter ; the pro- 
 blem of ascertaining the precise area ; and the diffi- 
 culty of a continued application of its principles to 
 the measurement of solid bodies, in the form of 
 spheres, cones, etc., have, in all ages, rendered its 
 study peculiarly interesting to mathematicians. 
 
 We shall consider, first, the relation of circumfe- 
 rence and diameter; next, the area; and after this, 
 apply these principles to a great variety of practical 
 measurements. 
 
 The difficult and impossible problem of the quad- 
 rature* of the circle, is the determination of the area 
 of a circle, whose diameter is equal to that of a given 
 square, or of a circle inscribed in a square. The in- 
 vestigation of this problem commenced with Archime- 
 des, a Grecian. The first step to be taken, was evi- 
 
 * The division of the circle into 360 parts, originated from 
 the division of the year by the ancients, into 360 days. The 
 12 signs represent the 12 months. 
 
 f Minute is derived from the Lat. minutum, a small part: 
 second, from secundus, the second, or second order of minutes. 
 
 t Quadrature is from the Lat. quadratura, squaring, from 
 quatuor ', four ; reducing the circle to a similar area of four 
 equal sides. 
 
242 RAINEY'S IMPROVED ABACUS. 
 
 dently to ascertain the ratio of the circumference to 
 the diameter. And here, the investigation must be 
 commenced, by every geometer. It is impossible to 
 give the process and reasoning used to ascertain 
 this, in a treatise on arithmetic ; we will, however, in- 
 dicate the process, and avail the benefits, without fur- 
 ther investigation. Neither the exact ratio of the 
 circumference to the diameter, nor the exact area of a 
 circle, can ever be ascertained ; and both have long 
 since been abandoned, as impossible, by all good math- 
 ematicians. 
 
 The method used, is to inscribe and circumscribe 
 the circle with regular polygons ; then, to increase the 
 number of the sides of botli of these, to such an ex- 
 tent, that they seem to merge into a common line ; 
 and although the sides can never entirely coincide, yet 
 they so far coincide, as to give almost entire accuracy, 
 to all practical operations. The reason why they can- 
 not coincide is, that a curved and a straight line can 
 never become one line, however small the degree of 
 curvature. The line of the circle is always found 
 between these two polygons ; and when the sides are 
 increased to a very large number, the human eye, as- 
 sisted by the microscope, is unable to see more than 
 one line in the three. Archimides carried the number 
 of sides to 32768, and thus secured the ratio to seven 
 decimal places. He obtained the ratio 3^--- and 3|3-, 
 which, reduced to an improper fraction, gave 2 T 2 , or 
 the circumference to the circle, as 22 to 7. These 
 numbers may be used for all ordinary and rough pur- 
 poses ; but are far from being accurate, when com- 
 pared with the ratio afterward obtained by Metius, a 
 German, who carried it to 17 places of decimals, giv- 
 ing the ratio of 355 to 113. Yan Ceulen, a Dutch 
 mathematician, carried it yet much further, and ascer- 
 tained that if the circle was 1, the circumference 
 would be greater than 3.14159265358579323846264- 
 
CIRCUMFERENCE AND DIAMETER. 
 
 338327950288, and less than 3.14159265358579323- 
 846264338327950289, demonstrating the coincidence 
 of the two polygons to a fraction less than one nonil- 
 lionth ; a difference too small to be adequately concei- 
 ved. The upper number represents the inscribed, and 
 the lower, the circumscribed polygon, between which we 
 may vainly seek the line of the circle. Later math- 
 ematicians have carried the calculation as far as 140 
 places of decimals. 
 
 The relation of the circumference, may, from the 
 above, be safely set down as 3.141592, This gives 
 the exact ratio to 5 places of decimals, leaving a frac- 
 tion as small as TO - OO'O"O ^ or a ^ practical purposes, 
 3.1416 may be used, changing the 5th decimal, 9, into 
 6, in the 4th order. 
 
 What is the circumference of a circle, whose diam- 
 eter is 31 feet? 
 
 3.1416x31=97.3836 Ans. 
 
 Here, we multiply by the decimals, and consequent- 
 ly cast off four decimals in the result. Hence, 
 
 To find the circumference of a circle, when the di- 
 ameter is given^ multiply the diameter by 3.1416, and 
 cut off four places of decimals in the -answer: or, for 
 entire accuracy,, multiply by 3.1415926, and cut off 
 seven places for decimals at the right. 
 
 It is desired to place 40 sentinels around a camp, 
 which is 1 mile in diameter ; how far will they be 
 apart ? 
 
 [0-44 
 
 3.1416 
 
 138.2304 
 
 We reduce* the mile to yards, and 
 get the answer in yards ; they will be 
 placed over 138 yards apart. 
 
 To find the diameter of a circle, when the circum- 
 ferejice is given, divide the circumference by 3.1416. 
 
 Or, multiply the circumference by 7, and divide 
 %22. 
 
 Qr 9 multiply the circumference by .3183L 
 
244 RAINEY'S IMPROVED ABACUS. 
 
 The circumference of a circle is 1 multiplied by 
 3.1416; hence, the diameter is 1 divided by 3.1416, 
 equal to .31831, which multiplied into the circumfe- 
 rence gives the diameter. 
 
 The circumference of a lot is 500 yards, and it is 
 desired to plant 10 trees on the line of its diameter; 
 how far apart will they be ? 
 
 The trees will be placed nearly 16 
 feet apart. 
 
 .31831 
 
 15.91550 
 
 It will be found far more convenient to multiply by 
 this, than to divide by the other decimal. 
 
 THE AREA Of THE CIRCLE. 
 
 In treating of polygons, it has been shown, that to> 
 ascertain the area, multiply the half radius by the en- 
 tire perimeter. It has since been shown, that the cir- 
 cumference or perimeter of a circle includes a vast 
 number of polygons. Hence y 
 
 To find the area of a circle, when the circumference 
 and diameter are given, multiply the circumference 
 by half the radius : 
 
 Or y Place the circumference and diameter on the 
 right of the line, and 4 on the left. 
 
 A circle is 10 feet in diameter, and 31.416 ft. m 
 circumference ; what is the area ? 
 
 One fourth of the diameter, equal to the semi- 
 radius, is 2-J feet, and 31.416x2^=78.540; thus, 
 
 #44^015708 
 
 5 The answer is 78,54 feeL 
 
 T8J4ft r 
 
AREA OF THE CIRCLE. 245 
 
 Let us suppose a square with a circle of equal diam- 
 eter inscribed : the diameter of the circle is equal to 
 the diameter of the square : ^ of the perimeter of a 
 square or circle is the side of such square or circle, 
 which multiplied into itself will produce the area de- 
 noted by the perimeter. The square of the diameter 
 indicates the area of the square ; and 3.1416, the area 
 of the circle : now, as ^ the perimeter of the square 
 is equal to the side of the square, so ^ the perimeter 
 of the circle, .7854, is equal to the side of the circle. 
 Hence, if the square of the diameter be multiplied by 
 J of the circumference of the circle, or .7854, the re- 
 sult will be the contents of the circle. For this rea- 
 son, geometers take the of 3. 141592=. 7854, and 
 multiply it into the square of the diameter, when the 
 circumference is given. Hence, the circle is .7854 of 
 a circumscribed square. That is, if the square is 1, 
 the circle is .7854; or, if the square contains 10,000 
 parts, the circle contains 7,854 parts. Hence, 
 
 To find the, area of a circle, when the diameter 
 only is given, Multiply the squared of the diameter 
 by .7854, and cut off four places in the answer for deci- 
 mals. For greater accuracy, multiply by .785398, 
 and cut off six places : 
 
 Or, Place the diameter and 11 on the right , and 
 14 on the left of the line. 
 
 What is the area of a circle whose diameter is 18 
 inches ? 
 
 Here, 18xl8x.7854=254.4696. The answer is 
 254^ inches, nearly. 
 
 How many acres in a circular field 40 rods in di- 
 ameter ? 
 
 *The square of the diameter , is the diameter multiplied into 
 
 itself. 
 
246 
 
 RAINEY'S IMPROVED ABACUS. 
 
 .7854 
 
 _ 
 
 The field contains nearly 8 acres. We 
 divide by 160, because this number of 
 I square rods equals an acre. 
 
 What is the area of a circle 
 ameter? 
 
 of an inch in di- 
 
 1 
 1 
 
 .7854 
 
 1.19635 
 
 The answer is very nearly 
 square inch. 
 
 The cylinder of a steam engine is 15 inches in di- 
 ameter ; what is the area in inches and in feet ? 
 
 15xl5x.7854=176 
 .715 inches. 
 
 Ans. 
 
 98175 
 
 1.7854 
 
 1.22718+ in ft. 
 
 How many acres of land in a tract 4000 rods in di- 
 ameter ? 
 
 Here, 160 rods opposite, instead 
 of making 1 acre, as would be the 
 case in a square tract, make in the 
 
 circular tract only .7854 of an acre; 
 
 . ,. i 
 
 in stating, we say, how many acres 
 
 will 4000x4000 rods make, if 160 rods, make .7854 
 of an acre? The answer is 78,540 acres. We may 
 ask, 
 
 What will the above land cost at 62i cents per 
 acre? 
 
 We may make this among the other statements, in- 
 stead of separately ; thus, 
 
THE ELLIPSE. 
 
 247 
 
 In this instance, 1 acre is 
 placed opposite .7854 of an 
 acre, and the price which the 1 
 acre equals last on the right. 
 
 125 
 
 49087.500000 
 
 How many inches in a valve 11 J in diameter ? 
 
 895 
 4^95 
 
 Although very few figures 
 can be canceled in this ques- 
 tion, yet there is a decided ad- 
 vantage in locating the frac- 
 tions on the vertical line. 
 
 Ans. 
 
 110.7536+ 
 
 To ascertain the area of a circle, when the circum- 
 ference only is given, multiply the square of the cir- 
 cumference by the square of 7, and the product by 
 .7854; and divide by the square of 22. 
 
 What is the area of a circle 33 ft. in circumference. 
 
 This is an easy and simple method 
 of arriving at the result, by one 
 statement. The answer is 86-| feet. 
 See Table of Circles and Areas. 
 
 
 33 
 33 
 
 22 
 
 7 
 
 22 
 
 7 
 
 
 .7854 
 
 Ans. 
 
 86.59085 
 
 TO FIND THE CIRCUMFERENCE OF AN ELLIPSE. 
 
 Multiply the square root of half of the sum of the 
 two axes squared, by 3.1416; the product will be the 
 circumference. 
 
 The axes are the longest diameters from side to side, 
 and from end to end. 
 
 TO FIND THE AREA OF AN ELLIPSE. 
 
 Multiply the product of the two diameters by 
 
 .7854. 
 
248 RAINEY'S IMPROVED ABACUS. 
 
 TO FIND THE CONTENTS OF A SQUARE INSCRIBED IN A 
 CIRCLE. 
 
 We shall endeavor to prove that such a square is 
 one half of another square circumscribed about a cir- 
 cle; thus, 
 
 Let us draw a square of a given side, and inscribe 
 in it a circle ; then, in the middle of the sides of 
 the square, where the circumference of the circle cuts 
 the square, let us locate the four angles of a square 
 inscribed in the circle. Thus, each side of this in- 
 scribed square will be the hypotenuse of a right-angled 
 and equal- sided triangle. Now, from the opposite an- 
 gles of the inscribed square, draw two diagonals. It 
 will now be perceived, that the inscribed square con- 
 tains four equal triangles, and the circumscribed 
 square, eight : hence, the inscribed square is equal to 
 half of the circumscribed square. 
 
 It is manifest, that if we square the semiradius of 
 this circle, it will equal two of these triangles ; hence, 
 twice the semiradius, will equal four of them, which 
 are equal to the inscribed square, or half of the cir- 
 cumscribed square. 
 
 What is the area of a square inscribed in a circle, 
 10 feet in diameter? 
 
 10 Here, we place the square of the diam- 
 
 $ 4.0 5 eter on the right, and 2 on the left. This 
 
 " KA fl~ is identical with the other operation, in which 
 50 ft. ,, . -,. j ,,' 
 
 the semiradius would be used ; thus, semi- 
 radius, 5; this squared and multiplied by 2, thus, 
 5X>X2=50 f e et, the answer, as before. Thus, by 
 the second operation, we find it troublesome to find 
 the semiradius, then square it, and afterward multiply 
 it ; while the other process is quite simple and easy. 
 
 How many cubic feet in a log 3 feet in diameter, 
 and 40 ft. long? 
 
INSCRIBED AND CIRCUMSCRIBED SQUARES. 249 
 
 Here, the perimeter is simply multi- 
 plied by the length and divided by 2, 
 because the inscribed is one half of the 
 circumscribed square. We ask again, 
 
 2 
 
 100 
 
 $',30.00 
 
 180 feet. 
 
 What will the same come to at $20 per 100 ft.? 
 
 3 
 3 
 
 Here, the statements are combined, and 
 the answer is obtained in dollars. 
 
 '~$36 
 
 How much will a log cost, at 33^- cents per foot, 
 which is 18 inches in diameter, and 80ft. long? 
 
 In this instance, two twelves are JL& 4-$ 3 
 placed on the left, because the two di- 
 mensions on the right are inches, and 
 must be reduced to feet. The 2 is 
 placed on the left as usual ; while 33-J-, 
 the price of 1 foot, is placed on the 
 right; hence, 
 
 To find the contents of a square inscribed in a cir- 
 cle, place the square of the diameter on the right, and 
 2 on the left. 
 
 To ascertain the solid contents, place the length on 
 the right likewise. 
 
 To find the price of the whole, place the price last 
 on the right) and the quantity which it equals, on the 
 left. 
 
 TO FIND THE SIDE OF A SQUARE INSCRIBED IN A 
 CIRCLE. 
 
 By reverting to the figure just used, we find that 
 the side of the inscribed square is the hypotenuse of a 
 right-angled and equal-sided triangle; .hence, the 
 square root of the sum of the squares of these two sides, 
 
250 RAINEY'S IMPROVED ABACUS. 
 
 or, which is the same thing, the square root of twice 
 the square of the radius, is equal to the hypotenuse, 
 or side of the inscribed square. This being trouble- 
 some to attain, and knowing that the smaller square is 
 half of the larger, we extract the square root of 
 3.141592, which is .707106, and multiply it into the 
 diameter of the circle for the side of the inscribed 
 square: likewise, we extract the square root of the 
 product of 1 divided by 3.1416, into .707106, which 
 is .22508 ; this multiplied into the circumference will 
 give the side. 
 
 How large a square can be hewn from a round 
 stick of timber, 20 inches in diameter? 
 
 Thus, 20 X- 7071=14.142. The side is 14 inches 
 and a fraction. 
 
 What is the side of a square that can be sawed 
 from a log 60 inches in circumference ? 
 
 60 X- 22508 13.5048 inches, Ans. We may caU 
 the last decimal .2251, instead of .2250, etc. Hence, 
 
 To find the side of a square inscribed in a circle, 
 multiply the diameter by .7071, or the circumference 
 by .2251. 
 
 THE SIDE OF A SQUARE GIVEN, TO FIND THE DIAME- 
 TER OF THE CIRCUMSCRIBED CIRCLE. 
 
 By reverting to the same figure, as above, we find 
 that the diagonal of the inscribed square, is the hy- 
 potenuse of a right-angled and equal-sided triangle, 
 whose sides are the sides of the square. Hence, the 
 square root of the sum of the squares of these two sides 
 would be the diagonal, hypotenuse, or diameter. This 
 being tedious, we find a decimal, which multiplied into 
 the side, will give the diagonal of the square, or di- 
 ameter of the circle. This decimal is 1.4142. Like- 
 wise, the side multiplied by the decimal 4.443 will 
 give the circumference of the circumscribed circle. 
 
INSCRIBED AND CIRCUMSCRIBED SOTARES. 251 
 
 The former of these numbers is the square root of 
 twice 3.141592 ; and the latter, the square root of twice 
 this number, multiplied by itself. 
 
 How large must a tree be, in diameter, to square 12 
 inches ? 
 
 12 X 1.4142=16.9704 inches, Ans. 
 
 How large is the circumference of a tree, or circle, 
 around a beam or square, whose sides are 20 inches ? 
 thus, 
 
 20x4.443 -88.86 inches, Ans. 
 
 The sides of a sill must be 12-J- inches ; how large 
 must the tree be, in diameter, from which it is sawed? 
 
 By this statement, we avoid #25 
 the difficulty of using the frac- 
 tions. Hence, 
 
 Ans. 17. 5525 inches 
 
 To find the diameter or circumference of a circle 
 circumscribed about a square whose sides are given ; 
 or, to find the diameter of a tree, that will square 
 a given size, multiply the side of the square by 1.4142 
 for the, diameter ; and by 4.443 for the circumference. 
 or girt. 
 
 Jt is frequently necessary to find the side of a square, whose 
 area is equal to that of a given circle; or the diameter or cir- 
 cumference of a circle whose area is equal to that of a given 
 square. As it belongs to geometry to demonstrate the vari- 
 ous relative proportions of figures, we shall pursue this course 
 no further than is demanded by a common-sense view of the 
 subject, and give a few numbers without tracing their origin. 
 
 To find the side of a square whose area is equal to 
 the area of a given circle, multiply the diameter of 
 the circle by .8862, or the circumference by .2821. 
 
 To find the diameter or circumference of a circle, 
 whose area is equal to the area of a given square, 
 multiply the side of the square by 1.128 for the di- 
 ameter, and by 3.545, for the circumference. 
 
 These rules will be found useful to mechanics and 
 other business men generally, and are giyen in such 
 form as to be easily understood and used. 
 
252 RATNEY'S IMPROVED ABACUS. 
 
 APPLICATION OF THE CIRCLE TO CISTERNS.* 
 
 Cisterns are large vaults made underground to hold 
 rainwater. They are of various shapes, square, 
 round, and conical or pyramidal. A conical cistern is 
 round, but of different diameters at bottom and top. 
 A pyramidal is square, and wider at bottom or top. 
 
 All that is necessary in finding the contents of cis- 
 terns, is to ascertain the number of gallons in a solid 
 foot, and multiply the number of feet by it. This is 
 done, by dividing the number of inches in a cubic, or 
 cylindric foot, by the number of inches in a wine or 
 beer gallon. 
 
 A cubic foot contains 7.48 wine gallons. 
 
 " " 6.127 beer 
 
 " cylindric " " 5.875 wine 
 
 4.812 beer 
 
 The numbers 5.875, and 4.812 were obtained by 
 multiplying 1728, the number of cubic inches in a cu- 
 bic foot, by .7854, to reduce them to the number of 
 inches in a circular or cylindric foot;' the product, 
 then divided by 231 and 282, respectively, gave the 
 numbers, as above. Thus, we see the continued ap- 
 plication of the decimal .7854. 
 
 Were this process not pursued, and the number of 
 gallons in a foot obtained, it would be necessary to find 
 the number of cubic inches in a cistern, and multiply 
 by .7854, when round, and divide by the number of 
 inches in a gallon. 
 
 We may dispense with the use of the decimal, par- 
 tially, and multiply by two other numbers in the form 
 of a common fraction. This is done, by multiplying 
 the decimal by some number that will terminate in ci- 
 phers, and dividing by the same. For instance: 
 
 * The root of cistern is the Lat. cista, a box, whence cistcrna, 
 a vault for rainwater. 
 
MEASUREMENT OF CISTERNS. 253 
 
 7.48x2=14.96. Here, 14 is the whole number, and 
 the .9, or T \, at the right of it being very nearly a 
 unit, we carry it to 14 and call the multiplier 15. We 
 divide this by 2, on the left of the vertical line, as 
 well as multiply by it, to make the other number 15. 
 
 A square cistern contains 100 cubic feet ; how many 
 wine gallons does it contain? 
 
 Here 100x7.48=748.00 gallons. By the other 
 process, we place the square of the side on the right, 
 multiply by 15, and on the left, divide by 2 ; thus, 
 
 It is seen here, that the answers 
 vary only 2 gallons in 750. Hence, 
 it is sufficiently accurate for ordinary 
 purposes. When strict accuracy is ' Z77[ i7~ 
 desired, the decimal may be used. I Ans '\" g alls - 
 Therefore, 
 
 To find the contents of cisterns which are square, 
 place the square of the side, and depth in feet ) with 
 15. on the right, and 2 on the left, for wine gallons; 
 or multiply the number of cubic feet in the cistern by 
 7.48. For beer gallons, place 49 on the right and 8 
 on the left, or multiply by 6.127. For circular cis- 
 terns, 47 on the right and 8 on the left, or multiply by 
 5.875, for wine gallons: and for beer gallons, 24 on 
 the right, and 5 on the left ; or multiply by 4.812. 
 
 For conical or pyramidal cisterns, add the areas of 
 the two ends : multiply the two areas, and extract the 
 square root: add this to the sum of the two areas 
 above : place this sum and the depth on the right, and 
 8 on the left : place likewise, on the right and left, the 
 numbers representing gallons, in the measure desired, 
 and, for square or circular cisterns, as the caw 
 may be 
 
 It may be remarked, that wine measure, or 231 cu- 
 bic inches to the gallon, is generally used as the stan- 
 dard in the United States. 
 17 
 
254 
 
 RA1NEY S IMPROVED ABACUS. 
 
 How many wine gallons in a circular cistern 12 ft. 
 in diameter and 20 feet deep? 
 
 12 
 
 4$ 3 
 
 4 #|47 
 
 The same answer might be ob- 
 tained by suspending the 47 and 
 8, and placing 5.875 on the right. 
 
 Ans] 1692 galls. ' 
 
 How many hogsheads of water m a circular cistern, 
 30 feet in diameter, and 21 ft. deep, beer measure ? 
 
 #0 6 In this instance, the answer 
 
 is reduced to hogsheads, by 
 placing 63, the number of gal- 
 lons in a hogshead, on the left. 
 
 How many beer gallons in a circular cistern of con- 
 ical shape, which is 9 ft. in diameter at the top, 7, at 
 the bottom, and 9 ft. deep? 
 
 9x9=81 area of upper end. 
 
 7x7=49 " " lower 
 
 
 
 Ans. 
 
 24 
 
 1440 hhds. 
 
 i 
 
 5 
 
 193 
 
 0-3 
 24 
 
 5 
 
 13896 
 
 Ans. 
 
 27791 
 
 We first square each of the diame- 
 ters, and add them ; next, multiply to- 
 gether these squares, and extract the 
 square root ; we add this root, 63, to 
 the sum of the two squares above, 130, 
 making 193 : this 193, mean area, is 
 placed on the line, with the depth of the cistern, while 
 3 is placed on the left. This is equivalent to mul- 
 tiplying the mean area by -J of the depth. Lastly, we 
 place 24 on the right and 5 on the left, which numbers 
 represent both the number of gallons in a solid foot, 
 and the deduction for the quadrature of the circle. 
 Any conical or pyramidal cistern may be measured in 
 the same way. It is wholly unnecessary above to get 
 
DIMENSIONS AND CONTENTS OF CISTERNS. 255 
 
 first, the circular area by multiplying by .7854 ; as 
 this can be done quite as easily, by the 24 and 5, or 
 by the number which represents the gallons and quad- 
 rature, 4.812 ; for circles and squares are to each other 
 as the squares of their diameters. 
 
 Required the depth of a rectangular cistern, to hold 
 5400 gallons, which is 8 ft. wide, and 10 ft. long. 
 
 We here place the contents of the cistern on the 
 right of the line, and the two given dimensions, with 
 the fraction expressing the number of gallons in a foot, 
 on the left. The fraction in this case is y , the mea- 
 sure being wine gallons : thus, 
 
 Were this beer gallons, we 
 would place 4 9 on the left. If 
 the cistern were circular, we would 
 place on the left 4 7 , or 2 -/, as the 
 case might be, with the square of 
 
 15 
 
 .Ans. 
 
 00135 
 
 9 feet deep. 
 
 the diameter, to find the depth, or the depth, to find 
 the square of the diameter. As the answer, in the 
 latter case, would be the square of the diameter, it 
 would be necessary to extract the square root for the 
 diameter of the cistern, Example : 
 
 What is the diameter of a circular cistern which 
 contains 960 gallons, beer measure, the depth being 
 8 feet? 
 
 The result is 25 : now, the 
 square root of this, 5, is the di- 
 ameter of the cistern. Hence, 
 
 When two dimensions and the contents of a cistern 
 are given to find the other dimension, for rectangular 
 cisterns, place the contents on the right> and the two 
 given sides, with the fraction representing the num- 
 ber of gallons in a solid foot, on the left : the answer 
 will be the required side. 
 
 For circular cisterns, place the contents on the 
 right, and the square of the diameter and fraction, on 
 
256 
 
 RA1NEY S IMPROVED ABACUS. 
 
 the left, for the depth ; or the depth and fraction, for 
 the square of the diameter ; in the latter case, extract 
 the square root, and the answer will be the required 
 diameter. 
 
 TABLE, 
 
 Giving the capacity of Square and Circular Cisterns, 
 Wells, etc., 1 foot deep, in wine and beer gallons. 
 
 Diam., 
 or side 
 in feet. 
 
 Circular Cisterns. 
 
 Square Cisterns. 
 
 The table of cis- 
 terns of different 
 figures, and of dif- 
 
 Wi. gall. 
 
 B'rgall. 
 
 Wi. galls. 
 
 B'r galls. 
 
 3 
 
 52.875 
 
 43.2 
 
 67.5 
 
 55.125 
 
 ferent kinds of 
 
 3V 
 
 71.96 
 
 58.8 
 
 91.875 
 
 75.031 
 
 measure, is given 
 
 4 
 
 994. 
 
 76.8 
 
 120. 
 
 98. 
 
 for the purpose of 
 
 
 118.969 
 
 146.875 
 
 97.2 
 120. 
 
 151.875 
 
 187.5 
 
 124.031 
 153.125 
 
 convenience. The 
 calculations are 
 
 5V 
 
 177.719 
 
 145.2 
 
 226.875 
 
 185.906 
 
 made for cisterns 1 
 
 6 2 
 
 211.5 
 
 172.8 
 
 270. 
 
 220.5 
 
 foot deep; there- 
 
 gi/ 
 
 248.219 
 
 202.8 
 
 316.875 
 
 258.781 
 
 fore, nothing more 
 
 7 
 
 287.875 
 
 235.2 
 
 367.5 
 
 300.125 
 
 is necessary in as- 
 
 7v 
 
 330,469 
 
 270. 
 
 421.875 
 
 344.531 
 
 certaining the con- 
 
 
 
 376. 
 
 307.2 
 
 480. 
 
 392. 
 
 tents of a given 
 
 sy 
 
 424.437 
 
 346.0 
 
 541.875 
 
 442.531 
 
 cistern than to find 
 
 9 
 
 475.875 
 
 388.8 
 
 607.5 
 
 496.125 
 
 the contents of a 
 
 9V 
 
 530.219 
 
 433.2 
 
 676.875 
 
 552.781 
 
 given diameter in 
 
 10 
 
 587.5 
 
 480. 
 
 750. 
 
 612.5 
 
 the table, and mul- 
 
 !!* 
 
 647.719 
 
 710.875 
 
 529.2 
 
 580.8 
 
 828.875 
 907.5 
 
 675.281 
 741.125 
 
 tiply it by the re- 
 quired depth. For 
 
 iji/ 
 
 776.969 
 
 634.8 
 
 991.875 
 
 810.031 
 
 example: it is de- 
 
 12 2 
 13 
 
 846. 
 992.875 
 
 691.2 
 811.2 
 
 1080. 
 1267.5 
 
 882. 
 1035.125 
 
 sired to know how 
 much a circular cis- 
 
 14 
 15 
 
 1151.5 
 1321.875 
 
 940.8 
 1080. 
 
 1475. 
 
 1687.5 
 
 1200.5 
 1378.125 
 
 tern 20 ft. in diam. 
 and 30 ft. deep, will 
 
 16 
 
 1504. 
 
 1228.8 
 
 1920. 
 
 1568. 
 
 contain, in wine 
 
 17 
 
 18 
 
 1697.875 
 1903.5 
 
 1387.2,2167.5 
 1555.22430. 
 
 1770.125 
 1984.5 
 
 gallons. We refer 
 to the table and find 
 
 19 
 
 2120.875 
 
 1732.8 
 
 2707.5 
 
 2211.125 
 
 that such a cistern 
 
 20 
 
 2350. 
 
 1920. 
 
 3000. 
 
 2450. 
 
 1 foot deep will 
 
 21 
 
 2590.875 
 
 2116.8 
 
 3307.5 
 
 2701.125 
 
 contain 2350 galls., 
 
 22 
 
 2843.5 
 
 2323.2 
 
 3630. 
 
 2964.5 
 
 and infer that 30 
 
 23 
 
 3107.875 
 
 2539.2 
 
 3967.5 
 
 3240.125 
 
 times this or 70500, 
 
 24 
 
 3384. 
 
 2768 
 
 4320. 
 
 3528. 
 
 will be the con- 
 
 25 
 
 3671.87513000. 
 
 4687.5 
 
 3828.125 
 
 tents of the re- 
 
DEFINITIONS IN SPHERICAL GEOMETRY. 257 
 CYLINDERS, CONES, SPHERES, ETC. 
 
 A cylinder* is a round solid body, of uniform diam- 
 eter, whose two ends or bases are at right angles with 
 the sides. 
 
 A cone'f is a round solid body, tapering in a direct 
 line from the periphery of the base to a point at the 
 top, called the vertex. 
 
 A pyramid^ is a square body, tapering in a direct 
 line from the periphery of the base to the vertex. 
 
 A frustu7Ji\\ of a cone, or pyramid, is the part re- 
 maining after a portion of the top is cut off by a plane 
 parallel to the base. 
 
 The convex^ surface of a cylinder, cone, pyramid, 
 or frustum, is the curved surface, exclusive of the 
 ends, or bases. The entire surface, includes the area 
 of the ends, or bases. 
 
 A sphere^ is a solid, round body, with a curved 
 surface, all parts of which are equally distant from the 
 center within ; and is generated by the revolution of a 
 semicircle about its own side. 
 
 A spheroid** is an oblong sphere, whose diameter 
 across is less that the diameter of the opposite ends, 
 and is formed by the revolution of an ellipse about 
 either of its axes. The extremes of the longer diam- 
 eter are called the major axis, and of the shorter, the 
 
 * Cylinder is from the Gr. root KVXI& or kulio, to roll. 
 
 t Cone is from the Welsh con, that which shoots a point. 
 
 J Pyramid originates from a Gr. word whose origin is ?rvp 
 or pur, a flame, from its resemblance in shape to a blaz- 
 ing fire. 
 
 || Frustum is a Lat. word, which means a broken piece. 
 
 Convex is from the Lat. convexus, to bend down on every 
 side, as the heavens; the opposite of concave, from cavus, a 
 hollow. 
 
 If Sphere is from the Lat. sphcera, a round ball. . 
 
 ** Spheroid is from the Gr. o-v&ip* or sphaira, a round ball, 
 and ttcfo? or eidos,form; globe-like. 
 
258 RAINEY'S IMPROVED ABACUS. 
 
 minor axis, from the Latin, major and minor, greater 
 and less. 
 
 The axes* of a body are points on which it is sup- 
 posed to revolve. 
 
 The diameter of a sphere is a line drawn through the 
 extreme opposite surfaces. The radius, a line drawn 
 from the center to any part of the surface. 
 
 A section, as of a cone, or other body, is a portion 
 cut off; from the Lat. root seco 9 to cut ojf.f 
 
 THE CYLINDER. 
 
 To find the convex surface of a cylinder, multiply 
 the circumference by the length: for entire contents, 
 add to this twice the area of the base, or the area of 
 the two ends. 
 
 It may be observed that the convex surface of a cylinder is 
 a rectangular figure, supposing it to be rolled out in 
 the form of a plane. Hence, the length multiplied by the 
 width, will give the superficial contents, as in other cases of 
 rectangles. 
 
 What is the convex surface of a cylinder, 18|- in. 
 in circumference, and 8 ft. long? 
 
 6 4 
 
 75 
 
 O 1 j_ 
 
 24 feet. 
 
 Having the length in feet, we di- 
 vide by 12, to reduce the width to 
 the same dimension, that the an- 
 swer may be in feet. 
 
 What is the entire surface of a cylinder 30 inches 
 in circumference, and 40 inches long ? 
 
 * Axes is the plural of axis, which is from ct%u>v or axon, a 
 table at Athens on which the laws were written, and which 
 revolved on centers or axes. This word is from <iyu> or ago, 
 to guide or direct movements-, hence, the direction of the mo- 
 tion of a body. 
 
 fThis is the origin of that beautiful, yet very difficult de- 
 partment of geometry, called Conic Sections, whose demon- 
 stration requires the closest and most perspicuous reasoning. 
 
MEASUREMENT OF THE CYLINDER. 
 
 259 
 
 2 
 
 5 
 
 
 2 
 
 5 
 
 
 22 
 
 7 
 
 
 22 
 
 7 
 
 
 
 .7854 
 
 
 
 2 
 
 
 
 .9939 
 
 area of both ends. 
 
 
 8.3333 
 
 convex area. 
 
 12 
 
 feet. 
 
 Ans. 9.3272 entire surface. 
 
 In the work above, the difficulty is presented of 
 finding the diameter of a circle, then squaring it, and 
 multiplying by .7854, for the area, and 2, for the two 
 ends, which would make two statements necessary. 
 Hence, to avoid this, we multiply together the square 
 of the circumference, the square of 7, the decimal 
 .7854, and 2, representing the two ends, and divide by 
 the square of 22, and the denominators found in the 
 circumference. 
 
 To find the solid contents of a cylinder, place the 
 square of the diameter, the length, and .7854 on the 
 right, and such denominate numbers as may be neces- 
 sary, on the left: or, instead of .7854, place 11 on the 
 right, and 14 on the left. 
 
 What is the solid contents of a cylinder 30 inches 
 in diameter, and 16 feet long? 
 
 Here the two twelves are 
 placed on the left to reduce the 
 square of the diameter to feet, 
 that all of the dimensions may 
 be in feet. Four places of de- 
 cimals are cut off in the answer. 
 
 Ans. 
 
 .7854 
 
 78.5400 ft. 
 
 How many solid feet in a cylinder 10^ feet in di- 
 ameter and 12 ft. long? 
 
260 
 
 RAINEY'S IMPROVED ABACUS. 
 
 21 
 
 ll 
 
 22079 
 
 The 11 and 14 are used in this 
 calculation. These are not suffi- 
 ciently accurate in cases where 
 great precision is desired. 
 
 CONTENTS OF BOILERS. 
 
 To find the contents of boilers, place the square of 
 the diameter and length in feet, on the right with 47, 
 and 8 on the left : the answer will be the contents in 
 wine gallons. Ascertain the contents of the fiues in 
 the same ivay, and subtract these from the contents 
 above. 
 
 For beer gallons, instead of 47 and 8, use 24 
 and 5. 
 
 Or, Multiply the length in feet, by the contents of 
 the given diameter, as found in the subjoined table, 
 and strike off as many figures in the answer for deci- 
 mals as there are decimal places in the multiplier. 
 
 How many wine gallons in a boiler 39 inches in di- 
 ameter and 40 feet long, having two flues, each 9 
 inches in diameter? 
 43 
 
 47 
 
 Boiler. 
 
 Flues. 264.375 
 
 13 
 
 47 
 
 2482.1875 
 264.375 
 
 Ans. 2217.8125 gallons. 
 
 In this example we make a separate calculation for the 
 flues, which are % of a foot in diameter; and as there are 
 two of them, we multiply by 2, instead of working the ques- 
 tion twice. The answer is 2218 gallons, nearly, wine mea- 
 sure. The same process may be pursued for beer gallons. 
 
 The subjoined table will be found very convenient for prac- 
 tical men* 
 
AIR PRESSURE, CONES, PYRAMIDS, ETC. 261 
 
 TABLE 
 
 Of contents of Boilers 1 foot in length, in wine gallons, from 3 
 to 38 inches in diameter. 
 
 Dla. 
 
 Cent's. 
 
 Dia. 
 
 Cent's. 
 
 Di. 
 
 Cent's. 
 
 Di. 
 
 Cent's. 
 
 Di. 
 
 Cent's. 
 
 3 
 
 .367 
 
 7^ 
 
 2-293 
 
 12 
 
 5.875 
 
 21 
 
 17.992 
 
 30 
 
 36.719 
 
 3*X 
 
 .499 
 
 8 
 
 2.611 
 
 13 
 
 6.895 
 
 22 
 
 19.739 
 
 31 
 
 39.207 
 
 4 
 
 .652 
 
 W 
 
 2.947 
 
 14 
 
 7.997 
 
 23 
 
 21.582 
 
 32 
 
 41.777 
 
 4X 
 
 .826 
 
 9 
 
 3.304 
 
 15 
 
 9.179 
 
 24 
 
 23.5 
 
 33 
 
 44.421 
 
 5 
 
 1.02 
 
 9# 
 
 3.682 
 
 16 
 
 10.444 
 
 25 
 
 25.499 
 
 34 
 
 47.163 
 
 5^ 
 
 1.234 
 
 10 ' 
 
 4,079 
 
 17 
 
 11.788 
 
 26 
 
 27.666 
 
 35 
 
 49.978 
 
 6 
 
 1.469 
 
 IQiX 
 
 4.498 
 
 18 
 
 13.219 
 
 27 
 
 29.308 
 
 36 
 
 52.875 
 
 6K 
 
 1.724 
 
 11 
 
 4.937 
 
 19 
 
 14.719 
 
 28 
 
 31.985 
 
 37 
 
 55.853 
 
 7 
 
 2. 
 
 UK 
 
 5.396 
 
 20 
 
 16.319 
 
 29 
 
 34.311 
 
 38 
 
 59. 
 
 AIR PRESSURE. 
 
 To find the air pressure on piston heads and other circular 
 areas, place the square of the diameter and 165 on the right, and 
 14 on the left. 
 
 This rule is based on the supposition that the pressure of 
 air, per superficial inch, is 15 lbs.,and that LI is the ratio of 
 the circle to the square: hence, instead of using the 15 and 11, 
 separately, we make them one number, such as may be easi- 
 ly remembered, by multiplying them together. The pressure 
 may be quite accurately obtained by multiplying by 59 and 
 dividing by 5; though, for entire accuracy, it is best to use 1.5 
 and the decimal .7854, which, multiplied into the square of the 
 diameter, will give the true result 
 
 What is the air pressure on a piston head 10-j- 
 inches in diameter ? 
 
 The answer is nearly 1300 
 pounds. 
 
 Ans. 
 
 21 
 
 i 3 
 
 165 
 
 10395 
 
 12S9| Ibs. 
 
 What is the air pressure on a piston head 35 inches 
 in diameter, in tons ? 
 
262 
 
 RAINEY'S IMPROVED ABACUS. 
 
 22 400/tf 7 
 2-20 ttl 
 
 32 
 
 AnS. 
 
 231 
 
 The answer is 7^ tons, 
 equal to 7 tons, and 437-| Ibs., 
 avoirdupois. 
 
 CONES AND PYRAMIDS. 
 
 To find the convex surface of a cone or pyramid, multiply the 
 circumference or perimeter of the base by the slant hight, and di- 
 vide by 2. 
 
 The reason for dividing by 2, is that the convex surface of 
 a cone or pyramid is just % the convex surface of a cylinder 
 of similar base and altitude. 
 
 The circumference of a cone is 90 inches, and the 
 slant hight 10 feet ; how many feet in its surface ? 
 
 J003 
 
 Twelve is used on the left, to re- 
 duce the circumference to feet. 
 
 105 
 
 150 
 
 Ans. 37| ft. 
 
 To find the solidity of a cone or pyramid, multiply the square 
 of the diameter, the vertical hight, and .7854 together, and divide, 
 by 3, and such other numbers as may be necessary to reduce the 
 dimensions to the came denomination. Cut off four figures for 
 decimals in the answer. 
 
 The reason for dividing the product of the dimensions by 
 3, is, that a solid cone is one-third of a cylinder of similar 
 base and altitude. The same thing is true of a pyramid. 
 
 How many solid feet in a cone 10 feet in diameter, 
 and 30 feet high? 
 
 10 The answer is 785 solid feet, 
 
 X 10 nearly. We might use the 11 and 14, 
 
 instead of .7854, which would give 
 .7854 the same answer within a very small 
 
 fraction. 
 
 Ans. 
 
 785.4000 
 
 To find the convex surface of the frustum of a cone or pyramid, 
 
SOLIDITY OF THE FRUSTUM OF A CONE. 263 
 
 place the sum of the circumferences, or perimeters, of the two 
 ends, with the slant hight on the right, and 2 on the left; for the 
 entire surface, add to the result, the sum of the areas of the two 
 ends. 
 
 A cone, superficially measured, may be easily proven to 
 be a triangle; hence, the necessity of dividing the product of 
 its two sides, as a rectangle, by 2, to get the triangle. Like- 
 wise, the superfice of a frustum may be divided into two tri- 
 angles: hence, the necessity of the sum of the circumfe- 
 rences. 
 
 To find the solidity of the frustum of a cone or py- 
 ramid. To the sum of the areas of the two ends, add 
 the square root of the product of these areas; place 
 this sum, with the vertical hight, on the right of the 
 line, and 3 on the left. 
 
 Thus we obtain the mean and two end areas, which are the 
 bases of thrfce right cones: these three cones are equal to the 
 solid frustum. 
 
 How many solid feet in the frustum of a cone 4| ft. 
 high, and 6 ft. in diameter at the lower, and 5 ft. at 
 the upper end ? 
 
 6x6=36 X- 7854=28.27 area of lower base. 
 5 X 5=25 X- 7854=19.63 " upper " 
 
 47.9 sum of bases. 
 28.27x19-63=^/554.94=23.55. 
 
 Now, 47.9+23.55=71.45, and this multiplied by 
 the hight, and divided by 3, gives the answer ; thus, 
 
 In the work above, we ascer- # fiji.jif; 1429 
 
 tained, first, the two areas, mul- 
 tiplied them, and extracted the 
 square root of their product, 
 which root was 23.55. This, added to the sum of the 
 two areas, 47.9, gave 71.45, to be multiplied by -J- of 
 the hight. It is almost impossible to explain such 
 operations as this satisfactorily by arithmetic ; yet their 
 utility in practical affairs necessitates their intro- 
 duction. 
 
 Ans. 
 
 114.32 
 
264 RAINEY'S IMPROVED ABACUS. 
 
 THE SPHERE OR GLOBE. 
 
 To find the surface, of a sphere or globe, Multiply 
 
 the, circumference by the diameter : 
 
 Or, Multiply the square of the diameter by 3.1416 : 
 Or, Multiply the square of the circumference by 
 
 .3183. 
 
 The decimal .3183 is obtained by dividing 1 by 3.14159. 
 
 What is the surface of a sphere, whose diameter is 
 3 feet? thus, 
 
 3x3=9, the square of the diameter; and this 9x 
 3.1416=28.2744, or 28 superficial feet, the answer. 
 
 To find the solidity of a sphere, 
 
 Multiply the cube* of the diameter by the decimal 
 5236 : 
 
 Or, Multiply the square of the diameter, .7854, 
 of the diameter, and 4 together: 
 
 Or, Multiply the square of the diameter by 3.1416, 
 and the product by of the diameter. 
 
 The decimal .5236 here used, is one-sixth of 3.1416. In- 
 deed, all of the decimals used as multipliers in anything per- 
 taining to circular figures, can be traced back to the number 
 representing the ratio of the circumference of a circle to its 
 diameter. 
 
 A globe is 10 inches in diameter; how many solid 
 inches does it contain? 
 
 10 
 
 10 X 10 X 10=1000 X.5236r= 
 523.6 cubic inches, Ans.; or, 
 
 10xlOX.7854xlfX4=523.6; or, 35 
 
 Ans. 
 
 10 
 3.1416 
 
 523.6000 
 
 Any of the processes above may be used with en- 
 tire safety. 
 
 * The cube of a number is that number multiplied into it- 
 self twice; the cube of 4 is 64. 
 
THE SPHEROID, WEIGHT OF GLOBES, ETC. 265 
 
 A globe is 30 inches in diameter ; how many solid 
 feet does it contain ? 
 
 It is frequently quite convenient 
 to state on the line, where reduc- 
 tion is to be performed, or any 
 fractional numbers used. The 
 question may be stated thus, 
 
 4 /i 
 4^ 
 12 
 
 tfO 
 
 
 3.1416 
 
 
 5 
 
 Ans. 
 
 8.18125 
 
 e are 
 
 2 
 
 5 
 
 imen- 
 
 2 
 
 5 
 
 must 
 
 12 
 
 3.1416 
 
 
 
 5 
 
 The two dimensions in the square are 
 reduced to feet, while the third dim en - 
 sion, in the cube, is still inches, and must 
 be reduced by 12. 
 
 It may be remarked, that when a cylinder is circumscribed 
 about a sphere, whose length is equal to its own diameter, or 
 that of the sphere, the relation of the surface of the sphere 
 to the entire surface of the cylinder is as 2 to 3; and that the 
 relation of their solidities is the same. Hence, an easy method 
 of finding the solidity of a sphere, is to take two-thirds the 
 solidity of a cylinder, of diameter and length equal to tho 
 diameter of the sphere. 
 
 THE SPHEROID. 
 
 To find the solidity of a spheroid, 
 
 Multiply the square of the shorter or minor axis by 
 the longer or major axis, and the product by .5236. 
 
 A spheroid is 20 by 30 inches; what is its solidity? 
 20X20X30X. 5236 6283.2 cubic inches. Ans. 
 
 Lines are to each other as their linear extent: 
 
 Areas are to each other as their squares : and 
 
 Solidities are to each othtr as their cubes. There- 
 fore, 
 
 To ascertain the weight of a globe, when the weight 
 of a globe of similar material is given, Place the 
 cube of the diameter of the globe whose weight is re- 
 quired, on the right, and the cube of the globe whose 
 diameter is given, on the left, and the weight of the 
 given globe, on the right. 
 
266 RAINEY'S IMPROVED ABACUS. 
 
 The solidity of a globe, whose diameter is one inch, is 
 .52359, which, for practical purposes, is called .5236^. 
 
 A globe of wrought iron, 1 foot in diameter, weighs 254.8 
 Ibs., and of cast iron, 242. The weight of bar iron being 1, 
 the weight of cast iron is .95, of steel, 1.02, copper, 1J.6, 
 brass, 1.09, and lead, 1.48. 
 
 A cubic foot of rolled iron weighs 486.65 Ibs., avoirdupois; 
 a cylindric foot, 382.2 Ibs. Hence, a cubic inch weighs 
 .28166 of alb.; a cylindric inch, .22116; now, taking two 
 thirds of the latter, shows that a spherical inch of rolled iron 
 weighs .14744, and of cast iron, .14006 of a Ib. A cubic inch 
 of cast iron weighs .26757 of a Ib.; hence, 3.84 cubic inches 
 of cast iron weigh 1.02 Ib.; 3.84 cubic inches are generally 
 allowed for 1 Ib. 
 
 If a cast iron globe, 12 inches in diameter, weighs 
 242 Ibs., how much will a globe of the same metal 
 weigh, which is 15 inches in diameter? 
 
 Here, the cube of the di- 
 ameter is placed on the line in 
 inches, in each case ; and as the 
 cube of 12 equals 242, the 
 cube of 15 must equal 473, 
 etc. 
 
 Ans. 
 
 GAUGING.* 
 
 Gauging is the measurement of casks or barrels ; 
 and is subject to no specific rules, from the fact that a 
 cask is not identical with any regular geometrical 
 figure; hence, no certain directions can be given to 
 measure all of the various shapes which casks assume. 
 They are generally considered the two equal frustums 
 of a cone, with greater or less lateral curvature. 
 It is not necessary to enter into an investigation of 
 the principles on which we found the following 
 
 * Gauge is from the French, jauge, a measuring rod. 
 
PHILOSOPHICAL CALCULATIONS. 267 
 
 DIRECTIONS FOR GAUGING CASKS. 
 
 Place the sum of twice the square of the bulge di- 
 ameter and once the square of the head diameter, with 
 the length, on the right ; and, on the left, 882 for 
 wine gallons, and 1077 for beer gallons: Or, Multi- 
 ply the square of the mean diameter by the length, 
 and the product by .0034 for wine, and .0028 for 
 beer gallons. 
 
 To ascertain the mean diameter between the bung and the 
 head, where the stave is greatly curved, add to the head diam- 
 eter .7 of the difference between the head and bung diame- 
 ters; when moderately curved, .55; and when very slightly 
 curved, .5. 
 
 How many wine gallons in a cask 49 inches long, 
 30 inches bung, and 21 inches head diameter? thus, 
 
 30x30x2=1800 
 21x21 = 441 
 
 2241 
 
 6 
 
 747 
 
 Ans. 124^ galls. 
 
 The measurement of casks by calculation is of but 
 little utility, as it is now mostly done by a rod, with 
 computations already made, in tabular form. 
 
 MECHANICAL POWERS. 
 
 The remarks on the mechanical powers will be very 
 limited, as this subject belongs legitimately to natural 
 philosophy. Yet we may give such general directions 
 as will enable the student in philosophy to make his 
 calculations with greater facility, than by the old 
 method. 
 
 The mechanical powers are six : the lever, the in- 
 clined plane, the wheel and axle, the pulley, the 
 screw, and the wedge. 
 18 
 
268 RAINEY'S IMPROVED ABACUS. 
 
 Several of these are, however, the same powers, 
 which receive their name from the nature of their ap- 
 plication; as there are, strictly speaking, but two 
 powers, the lever and inclined plane. The wheel and 
 axle, and pulley are revolving levers ; the screw, a 
 revolving inclined plane ; and the wedge, a compound 
 inclined plane. 
 
 The fulcrum* of a lever is the point or pivot on 
 which the lever rests: the arm is the distance be- 
 tween this rest and the power or weight. 
 
 When the two arms of a Lever and the power are given, to find 
 the weight that will equiponderate, proceed as in Inverse Propor- 
 tion: Or, Place the length of the arm for which the weight is re- 
 quired, on the left, and the other arm and the given weight on the 
 right: the answer will be the required weight. 
 
 A lever 20 feet long rests on a fulcrum 5 feet from 
 one end ; on the short end is a weight of 3000 Ibs.r 
 what weight attached to the other end will equipon- 
 derate ? 
 
 teooo 
 
 1000 Ibs. 
 
 This may be proven by finding 
 the length of one of the arms. 
 
 If the arm of a lever 15 feet long, with 1000 Ibs. 
 attached, equiponderate 3000 Ibs., how long is the arm 
 to which the latter weight is attached ? 
 
 1000 Here, 3000 feet is the demand, 1000 
 
 ]Lp 5 the same name, while the term of an- 
 
 ~T~~ r f swer is 15 feet. The 1000 Ibs. and the 
 1 15 cooperate to produce the common 
 
 effect, equiponderance. Hence, they are causes in 
 
 proportion. 
 
 When a weight is sustained between two props or fulcra, pro- 
 ceed by Inverse Proportion; making the entire length of the lever 
 the demand, the sJiort arm the same name, when the weight on the 
 
 * Fulcrum is a Latin word which means a prop, or brace. 
 
MECHANICAL POWERS. 
 
 269 
 
 prop of the long arm is required, and vice versa, and the whole 
 weight the term of answer. 
 
 Two men, A and B, carry a burden on a lever 30 
 feet long, placed 10 feet from A; what is the weight 
 sustained by B ? 
 
 30 
 
 20 
 400 
 
 30 
 
 10 
 
 400 
 
 Ans. 266f A's. Ans. 1.33 Ibs. B's. 
 
 The weight to the answer is inversely as the arm to 
 the whole lever. These two results added, make 400 
 Ibs., the whole weight. 
 
 The diameter of the wheel, the diameter of the axle, and the 
 power given, tofnd the weight: Proceed as in Inverse Propor 
 tion, etc. 
 
 The diameter of a wheel is 21 ft., and that of the 
 axle 8 inches ; what weight attached to the axle will 
 balance 140 Ibs. attached to the periphery of the 
 wheel ? 
 
 We here make the 8 inches f of a 
 foot, and placing it on the left, di- 100 
 vide likewise by 100 which reduces 
 the answer to cwts.: hence, 50| cwt. 
 The radius of a wheel is the long 
 arm; the radius of the axle, the Ans. 
 short arm. The weight is the power. 
 
 10 
 
 21 
 
 440 7 
 
 504 
 
 50| 
 
 This question 
 
 may be proven by finding the power attached to the 
 wheel, or by finding the radius of either the wheel 
 or axle. The ingenious pupil may experiment at 
 pleasure. 
 
 The length, the Jdglit, and the power of an inclined plane given, 
 tofnd the weight: Make the length the demand in direct propor- 
 tion; the hight, the same name; and the power, the term of an- 
 swer: the answer will be the weight. 
 
 To find the power, make the hight the demand, etc. 
 
 An inclined plane is 72 ft. long, and 8 ft. high ; 
 what weight will 781 Ibs. power sustain? 
 
270 
 
 RAUTEY'S IMPROVES ABACUS. 
 
 7gj The demand, 72, is placed on the 
 
 right. Again: 
 
 Ans. 7029 Ibs, 
 
 What power will sustain 7029 Ibs. weight on an in- 
 clined plane 72ft. long, and 8 ft. high? 
 
 We now get the power assumed 
 in- the first question, for the answer. 
 
 The side of a wedge, the thickness of the head, and the power 
 given, to ascertain the force: Make the length the demand; the 
 thickness, the same name ; and the power, the term of answer. 
 
 The dimensions and resistance given, to fnd the power: Make 
 the thickness the demand; the length the same name; and the re- 
 sistance the term of answer. 
 
 The length of a wedge is 40 inches, the head 8 
 inches, and the power 300 Ibs. what is the force? 
 
 This may be proven by finding the 
 power, the force being the resistance ; 
 thus, 
 
 Ans. 
 
 40 
 
 300 
 
 Ans. 
 
 1500 Ibs. 
 
 : 2 
 
 1500 
 
 300 Ibs. 
 
 These statements are made by di- 
 rect proportion. 
 
 The distance between the threads of a screw, the length of lever, 
 and power given, to ascertain the weight: Make the circumference 
 whose radius is the lever, the demand; the distance between the 
 threads, the same name; and the power the term of answer. 
 
 The weight given, to ascertain the power: Make the distance 
 between the threads, the demand; the circumference, as above, the 
 same name; and the power, the term of answer. 
 
 The distance, parallel to the center, of a screw be- 
 tween its threads, is 2-J- inches ; the length of the 
 lever, 56-J- inches, and the power, 9 tons ; what pres- 
 sure will it give ? 
 
SQUARE ROOT. 
 
 271 
 
 Here, 355 is the circumference of 
 twice the radius, 56 . 
 
 5355 
 
 2 
 9 
 
 Ans. 
 
 1278 tons. 
 
 Let us now find the power, the pressure, 1278 tons, 
 being ascertained ; thus, 
 
 We merely reverse the supposition 
 and demand, after finding weight, force, 
 pressure, etc., to find the power. 
 
 We trust enough has been said to render the appli- 
 cation of arithmetic to philosophy, plain and simple, so 
 far as the mechanical powers are concerned. 
 
 355 
 
 2 
 
 5 
 
 1278 
 
 Ans. 
 
 9 tons. 
 
 SQUARE EOOT. 
 
 The extraction of the square root, and all the other 
 roots, depends on principles which it is extremely dif- 
 ficult to explain satisfactorily, in arithmetic. The 
 roots belong properly to Algebra ; and the only ex- 
 cuse for the notice of square root here, is, that we 
 very frequently require it in practical affairs. Cube 
 root, to the contrary, is very seldom needed, except by 
 scientific men, such as have thoroughly studied all of 
 the principles of algebra. I have never, in the course 
 of my life, found occasion for extracting the cube root 
 once, for practical purposes. Hence, the propriety of 
 excluding it from this treatise ; as likewise all of those 
 subdivisions of numbers whose explanation depends 
 on algebraic principles ; such as the Positions, Alliga- 
 tion, the Progressions, Permutation, etc., etc.; none of 
 which offer any reward for the arduous labor lost in the 
 impossible task of their attainment in arithmetic. If 
 
272 RAINEY'S IMPROVED ABACUS. 
 
 one-half the time devoted to these principles in their 
 arbitrary form in arithmetic, were devoted to the study 
 of algebra, the pupil would learn a great portion of 
 that beautiful science, and thus secure the only key to 
 the principles involved in these rules. 
 
 The sign ,J placed before a number, indicates that 
 the square root is to be extracted. By placing 3, 4, 
 5, etc., over it, we understand that the cube, fourth, 
 or fifth root is to be extracted ; thus, 
 
 6=r4; and 4x3=J144. 
 
 SUMMARY OF DIRECTIONS. 
 
 I. Separate the number into columns or periods of two figures 
 each,, by placing a period ( . ) over the unifs figure, and over 
 every second figure from this to the left, and in decimals, over 
 every second figure toward the right. 
 
 II. Find the greatest square number in the first period to the 
 left, and place the root, or an equal factor of such square number, 
 for the quotient, at the right of the whole number: subtract the 
 square of this root, or quotient figure, from the first period, and 
 to the right of the remainder bring down the two figures of the 
 next period, for a new dividend. 
 
 III. Double the root or quotient figure obtained, and place it 
 to the left of the new dividend, for a new partial divisor: ascer- 
 tain how many times it is contained in the new dividend, ex- 
 clusive of the right-hand figure of such dividend, and place the 
 quotient to the right of the first quotient or root: place this 
 quotient, or second figure of the root, likewise to the right of 
 the partial divisor, which was used in obtaining it: multiply 
 the whole number thus found as the divisor, by the figure thus 
 appended, or the last figure in the root: subtract the product, and 
 bring down the next period, for a new dividend. 
 
 IV. Proceed with this period as the one preceding, and thus 
 continue the operation, until the roots of all the periods are ex- 
 tracted. 
 
 V. If there be a remainder after all the periods are thus used, 
 two ciphers may be added at a. time, and the operation continued 
 to any desired number of decimal places. 
 
 VI. The work is correct, if the root multiplied by itself, gives 
 a product equal to the original number. 
 
SQUARE ROOT. 273 
 
 What is the square root of 729? 
 
 It this example, we make 7 the first, 
 and 29 the second period : 2 squared, 
 equal to 4, makes the largest number 
 
 2)729(37 
 4 
 
 that can be extracted from 7 ; for 3 I 47^090 
 squared, would give 9, a number too | 399 
 large. We place this 2 to the right, 
 subtract its square, leaving 3, and bring 
 
 down the 20: we now square the 2, placing the square 
 4, on the left of the new dividend, 829 ; we divide 
 the 4 into the first two figures of the dividend, 32, 
 and find that it would go eight times, and conclude 
 that 8 must be placed to the right of the divisor, 4 ; 
 but, when we multiply the 48 thus found by the root 
 8, we find that it makes 384, a number that cannot 
 be subtracted from 329. Hence, we conclude that 
 the partial divisor, 4, must go into 32 only seven 
 times, making allowance for the one that will be car- 
 ried from the 9 which is rejected, and place the 7 in 
 in the root, and also at the right of the divisor 4, 
 making 47. Now, this 47, multiplied by the root, 7, 
 makes 329, which subtracted leaves nothing. We, 
 therefore, conclude that the root is 27, and prove it 
 by finding that 27 X27=729. Again : 
 
 What is the side of a square field that contains 
 42025 square roods ? 
 
 We first divide this number into periods ; thus, 
 
 The first root is 2, and its pro- 
 
 2)42025(205 
 4 
 
 405)2025 
 2025 
 
 duct, 4, subtracted leaves nothing. 
 
 20 being the next period, we know 
 
 that 4, which is the root doubled, 
 
 would be contained 5 times ; but, 
 
 as 45 multiplied by 5 could not be 
 
 subtracted from 20, we say that 
 
 this period gives no root figure, and supply its place 
 
 by a cipher in the root, and also a cipher at the right 
 
 of the partial divisor, 4. Now, this partial divisor, 
 
 
 
274 
 
 RAINEY'S IMPROVED ABACUS. 
 
 40, is contained in the 202, five times : hence, \ve 
 place 5 in the root, and 5 at the right of 40, and mul- 
 tiply the divisor thus found by the 5, making 2025. 
 Hence, the root is 205. This multiplied by itself 
 gives the original number. 
 
 In decimals appended to a whole number, the ex- 
 traction is effected as in whole numbers. The only 
 tiling to be observed, is to place the decimal point 
 after the last root figure in the whole numbers, and 
 all of the remaining figures will be decimals. 
 
 To ascertain the square root of a vulgar fraction, reduce ike 
 fraction to its lowest term, and extract the roots of the nume- 
 rator and denominator, separately. 
 
 What is the square root of y 8 
 
 81 = 9 J9=3 
 
 9 )777 7^ ; now > ~ 
 
 144=lb 
 
 the roots of 
 
 the fraction, and reduce these to the lowest term; 
 
 /81 = 9 9 =3 
 
 thus, - ; and Ans. 
 7144=12' 12=4 
 
 A number whose root cannot be exactly ascer- 
 tained, is called a surd. 
 
 CURRENCY. 
 
 Value of foreign Gold and Silver Coins, according to Custom- 
 House usage. 
 
 Guinea, English, gold, $5.00 
 
 Crown, " silver, 1.12 
 
 Shilling, .23 
 
 Bank token, English, silver, .25 
 
 Florin, of Basle, silver, .41 
 
 Moidore, Brazil, gold, 4.80 
 
 Livre, of Catalonia, silver, .53K 
 
 Florence Livre, silver, .15 
 
 Louis d'or, French, gold, 4.56 
 
 Crown. " silver, 1.06 
 
 40 Francs, " gold, 7.66 
 
 5 Francs, " silver, .93 
 
 Geneva Livre, silver, .21 
 
 10 Thalers, German, gold, 7.80 
 
 10 Pauls, Italy, silver, .97 
 
 Jamaica Pound, nominal, 3.00 
 
 Leghorn Dollar, silver, $0.90 
 
 Sen do, of Malta, " .40 
 
 Doubloon, of Mexico, gold, 15.60 
 
 Livre, of Neufchatel, silver, .26 % 
 
 Half Joe, Portugal, gold, 8.53 
 
 Florin, Prussia, silver, .22% 
 
 Imperial, Russia, gold, 7.83. 
 
 Rix Dollar, Rhenish, silver, .60% 
 
 " " Saxony, " .69 
 
 Pistole, Spanish, gold, 3.97 
 
 Rial, silver, .12> 
 
 Cross Pistareen, " .16 
 
 Other Pistareens, " 18 
 
 Swiss Livre, " .27 
 
 Crown, of Tuscany, silver, 1.05 
 
 Piaster, Turkish, " .05 
 
FOREIGN COINS AND MONIES OF ACCOUNT. 275 
 Monies oj Account and Coins^made current by act of Congress.* 
 
 Pound Sterling, G, Britain, $4.84 
 Do., Canada and N. Sc., 4.00 
 Do., N. Bran, and N. Found., 4.00 
 Franc, of France and Belgium, .186 
 Livre Tournois, France, ..185 
 Florin, Netherlands, .40 
 Do., Southern Ger. States, .40 
 Guilder, of Netherlands, .40 
 Real Vellon, Spain, .05 
 Do., Plate, .10 
 Milree, of Portugal, 1.12 
 Milree. of Azores, .83% 
 Marc Banco of Hamburg, .35 
 Thaler, or Rix Dollar, Prussia, 
 and Nor. Ger. States, .69 
 
 Rix Dollar of Bremen, $0.78% 
 Specie Dollar of Denmark, 1.05 
 Do.., Sweden and Norway, 1.06 
 Rouble, Russia, silver, .75 
 Florin, of Austria, .485 
 Lira or Lambardo, Venetian 
 kingdom, .16 
 Lira, of Tuscany, . .16 
 Lira, of Sardinia, .186 
 Ducat, Naples, .80 
 Ounce, of Sicily, 2.40 
 Livres, Leghorn, .16 
 Tael, of China, 1.48 
 Rupee, Company, .445 
 Pagoda, India, 1.84 
 
 Foreign Monies of Account, giving the value of the unit, accord- 
 ing to custom, in dollars and cents.^ 
 
 Brazil. 1000 Rees =-1 Milree = in Federal money to $0.828 
 
 The silver coin, 1200 Rees = 994 
 
 Bremen. 5 Schwares =1 Grote; 72 Grotes =1 Rix 
 
 Dollar, silver, .787 
 
 Belgium. 100 cents =1 Guilder or Florin; 1 Guilder, 
 
 (silver),. ,. . 40 
 
 Bencoolen. 8 Satellers 1 Soocoo; 4 Soocoos=l dol~ 
 
 lar or rial, .... 1.10 
 
 Austria. 60 Kreutzers 1 Florin; 1 Florin, silver = .485 
 British India. 12 Pice =1 Anna; 16 Annas =1 Co 
 
 Rupee, silver, ....... . . . . .445 
 
 In Bengal, Madras, and Bombay, the current silver 
 
 Rupee = 444 
 
 Buenos Ayres. 8 Rials =1 dollar, common currency, 
 
 (fluctuating), 93 
 
 Canton. 10 Cash 1 Candarine; 10 Candarines =1 
 
 Mace; 10 Mace=l Tael, ] .48 
 
 (The Cash, composed of Copper and lead, is said to 
 
 be the only money coined by the Chinese.) 
 Cape of Good Hope. 6 Stivers =1 Schilling; 8 sen- = 
 
 1 Rix dollar, 313 
 
 Ceylon. 4 Pice=l Fanam; 12 Fanarns = 1 Rix dol., 40 
 
 Cuba. 8 Rials, plate, =1 dollar; 1 dollar, 1.00 
 
 Columbia, Ecuador, Venezuela, and New Grenada. 8 
 
 Rials =1 dollar; 1 dollar, fluctuating, 1.00 
 
 * Monies of Account are not represented by coin, and are used for fa- 
 cility in reckoning, only, as mills in our country. To ve/ify the tables 
 above, see Laws of the United States. 
 
 .tSee Encyclopaedia JBritaunica, and McCulloctfs Commercial Dictionary 
 
276 RAINEY'S IMPROVED ABACUS. 
 
 Chili. 8 Rials =1 dollar; 1 dollar, silver, = $1.00 
 
 Denmark. 12 Pfennigs =1 skilling; 16 Sk. =1 Marc; 
 
 6 Marcs =1 Rigsbank, or 1 Rix dollar, silver, 52 
 
 Egypt-* Aspers =1 Para; 40 Paras =1 Piaster, sil., .048 
 Hamburg. 12 Pfenings =1 Schilling or Sol; 16 Schil. 
 =1 Marc Lubs;* 3 Marcs =1 Rix dollar, Current 
 
 Marc, silver, 28 
 
 Marc Banco, 35 
 
 Holland. 100 Cents =1 Florin, or Guilder: 1 Florin, 
 
 silver, 40 
 
 Greece. 100 Lepta =1 Drachme; 1 drachme, silver,. . .166 
 Great Britain and France. See tables above. 
 Japan. 10 Candarines ==1 Mace; 10 Mace ==1 Tael, .75 
 Malta. -20 Granif =1 Taro; 12 Tari =1 Scudo; 2> 
 
 Scudi =1 Pezza, 1.00 
 
 Java.- 100 Cents =1 Florin; 1 Florin, as in Nether- 
 lands, 40 
 
 Mauritius. In accounts of state, 100 Cts. =1 dol. = .968 
 Manilla. -34 Maravedies =1 Rial; 8 Rials =1 dollar, 
 
 Spanish, 1.00 
 
 Milan. 12 Denari =1 soldo; 20 Soldi =1 Lira, 20 
 
 Mexico. -8 Rials =1 dollar, 1 dollar, 1.00 
 
 Montevideo. 100 Centesimi =1 Rial; 8 Rial =1 dol., .833 
 Naples. 10 Grani =1 Carlino; 10 Carlini =1 Ducat, 
 
 silver, 80 
 
 Netherlands. Throughout the whole kingdom, ac- 
 counts are kept in Florins or Guilders, and cents, as 
 per law of 1815. See Holland. 
 
 New South Wales. Accounts are kept in Sterling 
 Money, only. 
 
 Norway 4 120 Skillings ==1 Rix dollar, silver, 1.06 
 
 Papal States. 10 Bajocchi =1 Paolo; 10 Paoli =1 
 
 Scudo or Crown, 1.00 
 
 Peru. 8 Rials =1 dollar, silver, 1.00 
 
 Portugal 400 Rees =1 Cruzado;^ 1000 Rees =--1 Mil- 
 ree or Crown, 1.12 
 
 * Lubs indicates that it is the money of the city Lubec; the common coin 
 is the marc currency; the marc banco represents the certificates of deposit 
 of bullion, jewelry, etc., in the bank of Hamburg. Invoices and accounts 
 are frequently made in Flemish pounds, shillings, and pence, which are 
 subdivided as sterling money. The Flemish pound is equal ta7% marcs 
 banco. 
 
 t Grani is the plural of grano; tari, plural of taro; scudi, of scudo; lire, 
 of lira; pezze of pezza; soldi, of soldo; carlini,+f>f carlinoj bajocchi, of 
 bajoccho; and paoli, of paolo. 
 
 t Norway has no gold coin of her owu. 
 
 Cruzadi is plural of cruzado; groschcn, of grosch; centesimi of ctnti- 
 lima; lire piecole t of lira pictola; soldi di pezza> of soldo di pezza* 
 
JEWISH WEIGHTS AND MEASURES. 277 
 
 Prussia. 12 Pfennigs =1 Grosch, silver; 30 Groschen 
 =1 Thaler, or dollar, $0.69 
 
 Russia. 100 Copecks =1 Rouble, silver, 78 
 
 Accounts were kept in paper Roubles previous to 
 1840, 3j^ of which were equal to 1 silver Rouble. 
 
 Sardinia. 100 Centesimi =1 Rira; 1 Lira=l Franc, 
 French, 186 
 
 Sweden. 12 Rundstycks -=1 Skilling; 48 Skilling =1 
 Rix dollar, specie, 1.06 
 
 Sicily. 20 Grani=l Taro; 30 Tari==l Oncia, gold, 2.40 
 
 Spain. 2 Maravedies=l Quinto; 16 Quintos=l Rial 
 of old plate:* 20 Rials vellon =1 Span, dollar, 1.00 
 
 St. Domingo. 100 Centimes =1 dollar: 1 dollar, 33) 
 
 Tuscany. 12 Denari di Pezza =1 Soldo di Pezza; 2 
 Soldi di Pezza =1 Pezza of 8 Rials; 1 Pezza, silver, .90 
 
 Turkey. 3 Aspers =1 Para; 40 Paras =1 Piaster, 
 varying, 05 
 
 Venice. 100 centesimi =1 Lira; 1 Lira =1 Franc, Fr., .186 
 Accounts were once kept in ducats, lire, etc. 12 
 Denari =1 Soldo; 20 Soldi =1 Lira Piccola; 6j. 
 Lire piccole =1 Ducat current; 8 Lire pic. =1 Ducat 
 effective; the Lira piccola is worth, 096 
 
 West Indies, British. Pounds, shillings, pence, etc., as 
 in England; the value varies in the different islands, 
 and is in all of them below that of England. 
 
 Jewish or Scripture, Standard Weights and Measures.^ 
 
 WEIGHTS OF MONEY. 60 Shekels =1 Maneh ; 50 Maneh 
 =1 Talent; or 113 Ibs., 10 oz., 1 clwt., 10 grs., Troy. 
 
 LONG MEASURE. 4 Digits ==1 Palm: 3 Palms ^=1 Span; 2 
 Spans =1 Cubit; 4 Cubits =1 Fathom; 2 Fathoms =1 Ara- 
 bian Pole; 10 Poles =1 Schoenus, which is the measuring 
 line, and is equal to 144 feet, 11 inches. 
 
 ITINERARY MEASURE. 400 Cubits =1 Stadium; 5 Stadia 
 =1 Sabbath-day's Journey; 10 Stadia =1 Eastern Mile; 3 
 Eastern Miles =1 Parasang; 8 Parasangs =1 Day's Journey, 
 or 33| English Miles. 
 
 DRY MEASURE. 20 Grachal =1 Cab; 1 A Cabs=l Gomor; 
 3> Gomor =1 Seah; 3 Seahs =1 Ephah; 5 Ephahs =1 Le- 
 leeh; 2 Leteeh =1 Comer, or 2 Bushels, 1 pint, English. 
 
 LIQUID MEASURE. lj/g Caph ==1 Log; 4 Logs =1 Cab; 
 3 Cabs =1 Hin; 2 Hins =1 Seah; 3 Seahs =1 Bath or 
 
 * Although rial of old plate is not a coin, yet it is the denomination in 
 which exchanges and invoices are reckoned. 
 ISce Kelly's Universal Cambist. 
 
278 
 
 RA1NEY S IMPROVED ABACUS. 
 
 Ephah; 10 Ephah s =1 Chomer, Homer, or Corus, which 
 equals 75 gallons, 5 pints, English Measure. 
 
 1 Talents=113 Ibs., 10 oz., 1 
 
 dwt., 10 grs; or 655714 grs. 
 
 1 Maneh ==13114.28 grs.; or, 
 
 27.3214 oz.; or, 
 
 2.27678 Ibs. T. 
 
 1 Shekel = 218.57133 grs. 
 
 1 Schoenus=145ftll in.; or, 
 1 " =1751. inches. 
 1 Pole = 175.1 " 
 1 Fathom = 87.55 ", or, 
 
 7 ft. 3% inches. 
 1 Cubit =~ 21. 8875 inches. 
 1 Span 10.9437 
 1 Palm = 3.6477 " 
 1 Digit .9119 
 
 1 Day's Journey =33i Eng- 
 lish miles; or 58374.3216 yds. 
 1 Parasang =7296.7902 
 
 1 Eas. Mile =2432.2634 yards. 
 1 Stadium 243.2263 
 1 Sab. D. J .=1216.1315 
 
 1 Comer =2 bu. 1 pt., Eng.; 
 or 16.125 galls.; or 129. pts. 
 1 Leteeh =64.5 pints. 
 1 Ephah =12.9 
 1 Seah 4.3 
 1 Gomor = 1.29 
 
 1 Cab = .7166 
 1 Grachal = .0358 
 
 1 Chomer =15 galls., 5 pts., 
 
 English; or 605 pints. 
 1 Ephah =60.5 pints. 
 1 Seah =21.166 
 1 Kin =10.083 
 1 Cab = 3.361 
 1 Log = .8402 
 1 Caph = .6301 
 
 1 Talent, silver =$1589.61; of gold, =$25415.27. 1 Ma- 
 neh, silver =$31.79; gold, =$508.22. 1 Shekel, =$0.529; 
 gold, =$847; all 24 carets fine, allowing no alloy. 
 
 Time Table, for Banking and Equation, giving the number of 
 days from any given date in one month, to the same date in 
 any other month. 
 
 A.D. 
 
 1849. 
 
 a 
 
 53 
 
 1 
 
 9 
 
 < 
 
 ri 
 ~, 
 
 a> 
 
 c 
 
 o 
 
 1 II 
 
 III 
 
 Jan. 
 
 365 
 
 31 
 
 59 
 
 90 
 
 120 
 
 151 
 
 181 
 
 212 243 
 
 273 
 
 304 334 
 
 Feb. 
 
 334 
 
 365 
 
 28 
 
 59 
 
 89 
 
 120 
 
 150 
 
 181 212 
 
 242 
 
 272 303 
 
 Mar. 
 
 306 
 
 337 
 
 365 
 
 31 
 
 61 
 
 91 
 
 122 
 
 153 184 
 
 214 
 
 245 275 
 
 Ap'l. 
 
 275 
 
 306 
 
 334 
 
 365 
 
 30 
 
 61 
 
 91 
 
 122 153 
 
 183 
 
 214 244 
 
 May. 
 
 245 
 
 276 
 
 304 
 
 335 
 
 365 
 
 31 
 
 61 
 
 92 123 
 
 153 
 
 184 214 
 
 June. 
 
 214 
 
 245 
 
 273 304 1334 365 
 
 30 
 
 61 i 92 
 
 122 
 
 153 183 
 
 July. 
 
 184 
 
 215 
 
 243 |274 |3Q4 
 
 335 
 
 365 
 
 31 ! 62 
 
 92 
 
 123 153 
 
 Aug. 
 
 153 
 
 184 
 
 212 243 273 
 
 304 334 
 
 365 31 
 
 61 
 
 92 122 
 
 Sept. 
 
 122 
 
 153 
 
 181 212 -242 
 
 273 303 
 
 334 365 
 
 30 
 
 61 91 
 
 Oct. 
 
 92 
 
 123 
 
 151 182 212 243 273 
 
 304 335 
 
 365 
 
 31 61 
 
 Nov. 
 Dec. 
 
 61 
 31 
 
 92 
 62 
 
 120 151 181 
 90 11211151 
 
 212 242 273 304 334 365 30 
 182 i ! 212 1243 274 1304 1335 365 
 
TABLE OF DIAMETERS AND AREAS. 
 
 279 
 
 The number of days expiring between any two periods may be very 
 easily ascertained in the foregoing table, by ascertaining the time between 
 the first and second dates, and adding or subtracting the overplus, or 
 deficit, minus 1. For example: 
 
 How long does a note run, dated January 4, and payable December 14? 
 
 In the table above, from Jan. 4, to Dec. 4, is 334 days; and 9 days more, 
 added, excluding the latter date, make 343 days, the time that the note 
 runs. 
 
 How long does a note run from December 4, to January 41 
 
 In the left-hand column we find December, and opposite it, under the 
 head Jan., we find 31 days, the time. The month of the first date must be 
 sought in the column at the left. 
 
 For leap-year, one must be added to the number of days, when the month 
 of February comes within the two dates. 
 
 TABLE 
 
 Of Diameters and Areas of Circles. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 lin. 
 
 .7854 
 
 IOK 
 
 82.516 
 
 19 K 
 
 298.648 
 
 28% 
 
 649.182 
 
 38 
 
 1134.11 
 
 IX 
 
 1.2271 
 
 10 ^ 
 
 86.590 
 
 19% 
 
 306.355 
 
 29 
 
 660.521 
 
 38X 
 
 1149.08 
 
 \y z 
 
 1.7671 
 
 10? 
 
 90.762 
 
 20 
 
 314.160 
 
 29^ 
 
 671.958 
 
 38^ 
 
 1164.15 
 
 1% 
 
 2.4052 
 
 11 
 
 95.035 
 
 20^ 
 
 322.063 
 
 29 y z 
 
 683.494 
 
 38% 
 
 1179.32 
 
 
 3.1416 
 
 ivx 
 
 99.402 
 
 20 y* 
 
 330.064 
 
 29% 
 
 695,128 
 
 39 
 
 1194.59 
 
 2K 
 
 3.9760 
 
 11 y' 
 
 103.869 
 
 20% 
 
 338.163 
 
 30 
 
 706.860 
 
 39K 
 
 1209.95 
 
 2> 
 
 4.9087 
 
 ft% 
 
 108.434 
 
 21 
 
 346.361 
 
 30% 
 
 718.690 
 
 39^ 
 
 1225.42 
 
 
 5.9395 
 
 12 
 
 113.097 
 
 21K 
 
 35-1.657 
 
 30^ 
 
 730.618 
 
 39% 
 
 1240.98 
 
 3 * 
 
 7.0686 
 
 12^ 
 
 117.859 
 
 21 y z 
 
 363.051 
 
 30% 
 
 744.644 
 
 40 
 
 1256.64 
 
 3K 
 
 8.2957 
 
 % 
 
 122.718 
 
 aig 
 
 371.543 
 
 31 
 
 754.769 
 
 40^ 
 
 1272.39 
 
 3> 
 
 9.6211 
 
 !2% 
 
 127.676 
 
 22 
 
 380.133 
 
 31K 
 
 766.992 
 
 40 K 
 
 1288.25 
 
 3% 
 
 11.044 
 
 13 
 
 132.732 
 
 22K 
 
 388.822 
 
 31X 
 
 779.313 
 
 40% 
 
 1304.20 
 
 4 
 
 12.566 
 
 !3K 
 
 137.886 
 
 &X 
 
 397.608 
 
 31% 
 
 791.732 
 
 41 
 
 1320.25 
 
 4K 
 
 14.186 
 
 13 y* 
 
 143.139 
 
 22% 
 
 406.193 
 
 32 
 
 804.249 
 
 41X 
 
 1336.40 
 
 4% 
 
 15.904 
 
 13% 
 
 148.489 
 
 23 
 
 415.476 
 
 32^ 
 
 816.865 
 
 41 K 
 
 1352.65 
 
 4% 
 
 17.720 
 
 14 
 
 153.938 
 
 23>4: 
 
 424.557 
 
 32 y z 
 
 829.578 
 
 41% 
 
 1369.00 
 
 5 
 
 19.635 
 
 14^ 
 
 159.485 
 
 23 y z 
 
 433.731 
 
 32% 
 
 842.390 
 
 42 
 
 1385.44 
 
 5% 
 
 21.647 
 
 !4^ 
 
 165.130 
 
 23% 
 
 443.004 
 
 33 
 
 855.30 
 
 42X 
 
 1401.98 
 
 5>a 
 
 23.758 
 
 m 
 
 170.873 
 
 24 
 
 452.390 
 
 33^ 
 
 868.30 
 
 42^ 
 
 1418.62 
 
 4! 
 
 25.967 
 
 15 
 
 176.715 
 
 24^ 
 
 #1.864 
 
 33^ 
 
 881.41 
 
 42% 
 
 1436.36 
 
 6 
 
 28.274 
 
 15^ 
 
 182.654 
 
 24K 
 
 471.436 
 
 33% 
 
 894.61 
 
 43 
 
 1452.20 
 
 6X 
 
 30.679 
 
 % 
 
 188.692 
 
 24% 
 
 481.106 
 
 34 
 
 907.92 
 
 43X 
 
 1469.13 
 
 6K 
 
 33.183 
 
 U% 
 
 194.828 
 
 25 
 
 490.875 
 
 34X 
 
 921.32 
 
 43 y z 
 
 1486.17 
 
 6% 
 
 35.784 
 
 16 
 
 201.062 
 
 25K 
 
 500.741 
 
 **y* 
 
 934.82 
 
 43% 
 
 1503.30 
 
 7 
 
 38.484 
 
 16>^ i 207.394 
 
 25 K 
 
 510.706 
 
 34% 
 
 948.41 
 
 44 
 
 1520.53 
 
 7K 
 
 41.282 
 
 16 >/ 
 
 213.825 
 
 25% 
 
 520.769 
 
 35 
 
 962.11 
 
 44X 
 
 1537.86 
 
 7 K 
 
 44.178 
 
 16% 
 
 220.353 
 
 26 
 
 530.930 
 
 35^ 
 
 975.90 
 
 44 y z 
 
 1555.28 
 
 TX 
 
 47.173 
 
 17 
 
 2*5.980 
 
 26^ 
 
 541.189 
 
 85 
 
 989.80 
 
 44% 
 
 1572.81 
 
 8 
 
 50.265 
 
 ft* 
 
 233.705 
 
 26K 
 
 551.547 
 
 35% 
 
 1003.70 
 
 45 
 
 1590.43 
 
 8K 
 
 53.456 
 
 17$ 
 
 240.528 
 
 26% 
 
 5ty.00'3 
 
 36 
 
 1017.87 
 
 45K 
 
 1608.15 
 
 8^ 
 
 56.745 
 
 ITX 
 
 247.450 
 
 27 
 
 572.556 
 
 36M 
 
 1032.06 
 
 45}^ 
 
 1625.97 
 
 8% 
 
 60.132 
 
 18 
 
 254.469 
 
 27X 
 
 5S3.208 
 
 36 >/ 
 
 1046.30 
 
 45% 
 
 1643.89 
 
 9 
 
 63.617 
 
 18X 
 
 261.587 
 
 27 >^ 
 
 593.958 
 
 36% 
 
 1060.73 
 
 46 
 
 1661.90 
 
 9X 
 
 67.200 
 
 18 K 
 
 268.803 
 
 27% 
 
 604.807 
 
 37 
 
 1075.21 
 
 46X 
 
 1680.01 
 
 9K 
 
 70.882 
 
 18% 
 
 276.117 
 
 28 
 
 615.753 
 
 37^ 
 
 1089.79 
 
 46 >^ 
 
 1698.23 
 
 9% 
 
 74.662 
 
 19 
 
 383.888 
 
 28^ 
 
 626.798 
 
 37 > 
 
 1104.46 
 
 46% 
 
 1716.54 
 
 10 
 
 78.540 
 
 19X 
 
 291.039 
 
 28^ 
 
 637.941 
 
 37% 
 
 1119.24 
 
 47 
 
 1734.94 
 
280 
 
 RAINEY'S IMPROVED ABACUS. 
 
 TAB LE C&nt inued. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 Dia. 
 
 Area. 
 
 Lia. 
 
 Area. 
 
 47% 
 
 1753.45 
 
 57% 
 
 2619.35 
 
 68K 
 
 3658.44 
 
 85 y z 
 
 5741.47 
 
 9 8 
 
 73.391 
 
 47% 
 
 1772.05 
 
 58 
 
 2642.08 
 
 68% 
 
 8685.29 
 
 86 
 
 5808.81 
 
 9 9 
 
 74.662 
 
 47% 
 
 1790.76 
 
 58% 
 
 2664.91 
 
 68% 
 
 3712.24 
 
 86K 
 
 5876.55 
 
 9 10 
 
 75.943 
 
 48 
 
 1809.56 
 
 58 % 
 
 2637.83 
 
 69 
 
 3739.28 
 
 87 
 
 5944.69 
 
 9 11 
 
 77.236 
 
 48% 
 
 1828.46 
 
 58% 
 
 2710.85 
 
 69% 
 
 3766.43 
 
 87> a x 
 
 6013.21 
 
 10 
 
 78.540 
 
 48% 
 
 1847.45 
 
 59 
 
 2733.97 
 
 09% 
 
 3-793.67 
 
 88 
 
 6082.13 
 
 10 1 
 
 79.854 
 
 48% 
 
 1868.55 
 
 59^ 
 
 2757.19 
 
 69% 
 
 3821.02 
 
 88 K 
 
 6151.44 
 
 10 2 
 
 81.179 
 
 49 
 
 1885.74 
 
 59% 
 
 2780.51 
 
 70 
 
 3848.46 
 
 89 
 
 6221.15 
 
 10 3 
 
 82.516 
 
 49K 
 
 1905.03 
 
 69% 
 
 2803.92 
 
 1Q 1 4 
 
 3875.99 
 
 89% 
 
 6291.25 
 
 10 4 
 
 83.862 
 
 49% 
 
 : 1924.42 
 
 60 
 
 2827.44 
 
 70% 
 
 3903.63 
 
 90 
 
 6361.74 
 
 10 5 
 
 85.221 
 
 49% 
 
 1943.91 
 
 60^ 
 
 2S51.05 
 
 70% 
 
 3931.36 
 
 90% 
 
 6432.62 
 
 10 6 
 
 86.590 
 
 60 
 
 1963.50 
 
 60% 
 
 2874.76 
 
 71 
 
 3959.20 
 
 91 
 
 6503.89 
 
 10 7 
 
 87.969 
 
 50% 
 
 1983.18 
 
 60% 
 
 2898.59 
 
 71# 
 
 3987.13 
 
 91 % 
 
 6573.56 
 
 10 8 
 
 89.360 
 
 50 % 
 
 2002.96 
 
 61 
 
 2922.47 
 
 71% 
 
 4015.16 
 
 92 
 
 6647.62 
 
 10 9 
 
 90.762 
 
 50% 
 
 2022.84 
 
 61M 
 
 2946.47 
 
 71% 
 
 4043.28 
 
 92% 
 
 6720.07 
 
 10 10 
 
 92.174 
 
 51 
 
 2042.82 
 
 6i % 
 
 2970.57 
 
 72 
 
 4071.51 
 
 93 
 
 6792.92 
 
 10 11 
 
 93.598 
 
 51% 
 
 2062.90 
 
 61% 
 
 2994.77 
 
 72% 
 
 4128.25 
 
 93% 
 
 6866.16 
 
 11 
 
 95.033 
 
 51% 
 
 2083.07 
 
 62 
 
 3019.07 
 
 73 
 
 4185.39 
 
 94 
 
 6939.79 
 
 11 1 
 
 96.478 
 
 51% 
 
 2103.35 
 
 62)^ 
 
 3043.47 
 
 73% 
 
 4242.92f 
 
 94% 
 
 7013.81 
 
 11 2 
 
 97.934 
 
 52 
 
 2123.72 
 
 62% 
 
 3067.96 
 
 74 
 
 4300.85 
 
 95 
 
 7088.23 
 
 11 3 
 
 99.402 
 
 52K 
 
 2144.19 
 
 62% 
 
 3092.56 
 
 74K 
 
 4359.1'') 
 
 95% 
 
 7163.04 
 
 11 4 
 
 100.879 
 
 52% 
 
 2164.75 
 
 63 
 
 3117.25 
 
 75 
 
 4417.87 
 
 ft. in. 
 
 feet. 
 
 11 5 
 
 102.368 
 
 52% 
 
 2185.42 
 
 63X 
 
 3142.04 
 
 75 y* 
 
 4476.97 
 
 8 
 
 50.265 
 
 11 6 
 
 103.869 
 
 53 
 
 2206.18 
 
 63% 
 
 31G6.92 
 
 76 
 
 4536.47 
 
 8- 1 
 
 51.317 
 
 11 7 
 
 105.379 
 
 53% 
 
 2227.05 
 
 63% 
 
 3191.91' 
 
 76 
 
 4596.35 
 
 8 2 
 
 52.381 
 
 11 8 
 
 106.901 
 
 53% 
 
 2248.01 
 
 64 
 
 3216.99 
 
 77 
 
 4656.63 
 
 8 3 
 
 53.456 
 
 11 9 
 
 108.434 
 
 53^ 
 
 2269.06 
 
 64# 
 
 3242.17 
 
 77% 
 
 4717.30 
 
 8 4 
 
 54.541 
 
 11 10 
 
 109.977 
 
 54 
 
 2290.22 
 
 64% 
 
 3267.46 
 
 78 
 
 4778.37 
 
 8 5 
 
 55.637 
 
 11 11 
 
 111.531 
 
 54% 
 
 2311.48 
 
 64% 
 
 3292.83 
 
 78> 
 
 4839.83 
 
 8 6 
 
 56.745 
 
 12 
 
 113.097 
 
 54% 
 
 2332.83 
 
 65 
 
 3318.31 
 
 79 
 
 4901.68 
 
 8 7 
 
 57.862 
 
 13 
 
 132.732 
 
 54% 
 
 2354.28 
 
 65^ 
 
 3343.88 
 
 79% 
 
 4963.92 
 
 8 8 
 
 58.992 
 
 14 
 
 153.938 
 
 55 
 
 2375.83 
 
 65 % 
 
 3369.56 
 
 80 
 
 5026.56 
 
 8 9 
 
 60.132 
 
 15 
 
 176.715 
 
 55K 
 
 2397.48 
 
 65% 
 
 3395.33 
 
 80 % 
 
 5089.58 
 
 8 10 
 
 61.282 
 
 16 
 
 201.062 
 
 55% 
 
 2419.22 
 
 66 
 
 3421.20 
 
 81 
 
 5153.00 
 
 8 11 
 
 62.444 
 
 17 
 
 226.980 
 
 55% 
 
 2441.07 
 
 66K 
 
 3447.16 
 
 8L& 
 
 5216.83 
 
 9 
 
 63.617 
 
 18 
 
 254.469 
 
 56 
 
 2463.0r 
 
 66 % 
 
 3473.23 
 
 82 
 
 5281.02 
 
 9 1 
 
 64.800 
 
 19 
 
 283.529 
 
 56K 
 
 2485.05 ; 
 
 66% 
 
 3499.39 
 
 82 y, 
 
 5345.0-3 
 
 9 2 
 
 65.995 
 
 20 
 
 314.160 
 
 56% 
 
 2507.19 
 
 67 
 
 3525.66 
 
 8S 
 
 5410.62 
 
 9 3 
 
 67.200 
 
 21 
 
 346.361 
 
 56% 
 
 2529.42 
 
 67^ 
 
 3552.01 
 
 83^ 
 
 5476.00 
 
 9 4 
 
 68.416 
 
 22 
 
 380.133 
 
 57 
 
 2551.76 
 
 67% 
 
 3578.47 : 
 
 84 
 
 5541.78 
 
 9 5 
 
 69.644 
 
 23 
 
 415.476 
 
 57K 
 
 2574.19 
 
 67% 
 
 3605.03 
 
 84K 
 
 5607.90 
 
 9 6! 70.833 
 
 24 
 
 452.390 
 
 57% 
 
 2596.72 
 
 68 
 
 3631.68 
 
 85 
 
 5674.51 
 
 9 71 72.130 
 
 25 
 
 490.87* 
 
 Weight of a Lineal Foot of Square Rolled Iron, in lbs.,from 
 to 12 inches square. 
 
 Size, 
 
 Wei't, 
 
 Size, Wei't, 
 
 Size, 
 
 Wei't, 
 
 Size, 
 
 Wei't, 
 
 Size, 
 
 Wei't, 
 
 in in. 
 
 inlbs. 
 
 in in. 
 
 in Ibs. 
 
 in in. 
 
 in Ihs. 
 
 in in. 
 
 inlbs. 
 
 in in. 
 
 in Ibs. 
 
 K 
 
 .211 
 
 % 
 
 2.588 
 
 1 
 
 7.604 
 
 2% 
 
 15.263 
 
 2% 
 
 25.560 
 
 % 
 
 .475 
 
 1 
 
 3.380 
 
 1% 
 
 8.926 
 
 2% 
 
 17.112 
 
 2% 
 
 27.939 
 
 
 .845 
 
 1% 
 
 4.278 
 
 1% 
 
 10.352 
 
 2% 
 
 19.066 
 
 3 
 
 30.416 
 
 i 
 
 1.320 
 
 1% 
 
 5.280 
 
 1% 
 
 11.883 
 
 2% 
 
 21.120 
 
 3% 
 
 33.010 
 
 % 
 
 1.901 
 
 lg 
 
 6.390 
 
 2 
 
 13.520 
 
 2% 
 
 33.292 
 
 3K 
 
 35.704 
 
WEIGHT OF DIFFERENT BODIES OF IRON. 
 
 281 
 
 TABLE Continued. 
 
 Size, 
 in in. 
 
 Wei't, 
 in Ibs. 
 
 Size,, 
 in in. 
 
 Wei't, 
 in Ibs. 
 
 Size, 
 in in. 
 
 Wei't, 
 in Ibs. 
 
 Size, 
 in in. 
 
 Wei't, 
 in Ibs. 
 
 Size, 
 in in. 
 
 Wei't, 
 in Ibs. 
 
 3K 
 
 86.503 
 
 4% 
 
 72.305 
 
 5^ 
 
 111.756 
 
 7& 
 
 203.024 
 
 10 
 
 337.920 
 
 B 
 
 41.408 
 
 4% 
 
 76.264 
 
 5K 
 
 116.671 
 
 8 
 
 216.336 
 
 10K 
 
 355.136 
 
 3% 
 
 44.418 
 
 4% 
 
 80.333 
 
 6 
 
 121.664 
 
 8K 
 
 230.068 
 
 10 K 
 
 372.672 
 
 3% 
 
 47.534 
 
 5 
 
 84.480 
 
 6K 
 
 132.040 
 
 8K 
 
 244.220 
 
 10% 
 
 390.628 
 
 3% 
 
 50.756 
 
 5K 
 
 88.784 
 
 6K 
 
 142.816 
 
 8% 
 
 258.800 
 
 11 
 
 408.960 
 
 4 
 
 54.084 
 
 5K 
 
 93.168 
 
 6% 
 
 154.012 
 
 9 
 
 273.792 
 
 UK 
 
 427.812 
 
 4K 
 
 57.517 
 
 5% 
 
 97.657 
 
 7 
 
 165.632 
 
 9K 
 
 289.220 
 
 UK 
 
 447.024 
 
 4K 
 
 61.055 
 
 5K 
 
 102.240 
 
 7K 
 
 177.672 
 
 9K 
 
 305.056 
 
 11% 
 
 466.684 
 
 4K 
 
 64.700 
 
 5% 
 
 106.953 
 
 7K 
 
 190.136 
 
 9& 
 
 321.332 
 
 12 
 
 486.656 
 
 4K 
 
 68.448 
 
 
 
 
 
 
 
 
 
 Weight of Round Rolled Iron, 1 foot long, and from % to 12 
 inches in diameter. 
 
 - 
 
 Dia., 
 
 Wei't, 
 
 Uia-, 
 
 Wei't, 
 
 Uia., 
 
 Wei't, 
 
 Dia., 
 
 Wei't, 
 
 Dia., 
 
 Wei't, 
 
 in in. 
 
 in Ibs. 
 
 in in. 
 
 in Ibs. 
 
 in in. 
 
 in Ibs. 
 
 in in. 
 
 in Ibs. 
 
 in in. 
 
 in VSs. 
 
 K 
 
 .165 
 
 2K 
 
 11.988 
 
 8# 
 
 39.864 
 
 5% 
 
 84.001 
 
 8% 
 
 203.269 
 
 % 
 
 .373 
 
 2K 
 
 13.440 
 
 4 
 
 42.464 
 
 5?4 
 
 87.776 
 
 9 
 
 215.040 
 
 % 
 
 .663 
 
 a% 
 
 14.975 
 
 4X 
 
 45.174 
 
 5% 
 
 91.634 
 
 9K 
 
 227.152 
 
 % 
 
 1.043 
 
 2K 
 
 16.688 
 
 4V 
 
 47.952 
 
 6 
 
 95.552 
 
 9K 
 
 239.600 
 
 % 
 
 1.493 
 
 a* 
 
 18.293 
 
 4?< 
 
 50.815 
 
 ex 
 
 103.704 
 
 93^ 
 
 252.376 
 
 % 
 
 2.032 
 
 2% 
 
 20.076 
 
 4K 
 
 53.760 
 
 6# 
 
 112.160 
 
 10 
 
 266.288 
 
 1 
 
 2.654 
 
 ^ 
 
 21.944 
 
 4% 
 
 56.788 
 
 6%r 
 
 120.960 
 
 10K 
 
 278.924 
 
 IK 
 
 3.360 
 
 3 
 
 23.888 
 
 4# 
 
 69.900 
 
 7 
 
 130.048 
 
 10K 
 
 292.688 
 
 IK 
 
 4.172 
 
 *K 
 
 -25.926 
 
 4% 
 
 63.094 
 
 ? 
 
 139.544 
 
 10% 
 
 306.800 
 
 IK 
 
 5.019 
 
 3X 
 
 28.040 
 
 5 
 
 66.752 
 
 K 
 
 149.328 
 
 11 
 
 321.216 
 
 IK 
 
 5.972 
 
 3% 
 
 30.240 
 
 5K 
 
 69.731 
 
 7% 
 
 159.456 
 
 UK 
 
 336.004 
 
 1% 
 
 7.010 
 
 3K 
 
 32.512 
 
 5# 
 
 73.172 
 
 8 
 
 169.856 
 
 UK 
 
 351.104 
 
 IK 
 
 8.128 
 
 3% 
 
 34.886 
 
 5'< 
 
 76.700 
 
 8* 
 
 180.696 
 
 u% 
 
 3b6.536 
 
 IK 
 
 9.333 
 
 8* 
 
 37.332 
 
 5^ 
 
 80.304 
 
 8^ 
 
 191.808 
 
 12 
 
 382.208 
 
 2 
 
 10.616 
 
 
 
 
 
 
 
 
 
 Weight, in Ibs., of different bodies of Cast Iron, 1 foot in length, 
 and from % to 12 inches diameter or side. 
 
 Side, 
 ordi. 
 
 Squa. 
 
 Hex a- 
 gon. 
 
 Octa- 
 gon. 
 
 Circle. 
 
 Side, 
 or di. 
 
 Squa. 
 
 Hex a- 
 gon. 
 
 Octa- 
 gon. 
 
 Circle. 
 
 iru-b. 
 
 
 
 
 
 inch. 
 
 
 
 
 
 y, 
 
 .781 
 
 .675 
 
 .650 
 
 .612 
 
 3K 
 
 33.009 
 
 28.565 
 
 27.475 
 
 25.921 
 
 X 
 
 1.75(5 
 
 1.528 
 
 1.471 
 
 1.387 
 
 8 
 
 38.281 
 
 33.131 
 
 31.818 
 
 30.065 
 
 1 
 
 3.125 
 
 2.703 
 
 2.603 
 
 2.454 
 
 3% 
 
 43.943 
 
 38.031 
 
 36.581 
 
 31.512 
 
 IV 
 
 4.881 
 
 4.225 4.065 
 
 3.854 
 
 4 
 
 50.000 
 
 43.271 
 
 41.621 
 
 39.268 
 
 1U 
 
 7.031 
 
 6.085 ; 5.856 
 
 5.521 
 
 4X 
 
 56.443 
 
 48.353 
 
 46.990 
 
 44.331 
 
 1 
 
 9.568 
 
 8.281 ' 7.971 
 
 7.515 
 
 *K 
 
 63,281 
 
 5-1.768 
 
 52.681 
 
 49.700 
 
 2 
 
 12.520 
 
 10.815, 10.412 
 
 9.815 
 
 4fc 
 
 70.506 
 
 61.021 
 
 58.696 
 
 55.375 
 
 2V 
 
 15.818 
 
 13.990 i 13.168 
 
 12.425 
 
 5 
 
 78.125 
 
 67.515 
 
 65.040 
 
 61.359 
 
 2^ 
 
 19.531 
 
 16.900, 16.256 
 
 15.337 
 
 5J4 
 
 86.131 
 
 74.549 
 
 71.701 
 
 .67.709 
 
 2^ 
 
 23.631 
 
 20.450 19.671 
 
 18.559 
 
 5V, 
 
 94.531 
 
 81.815 
 
 78.696 
 
 74.243 
 
 3 
 
 28.125 
 
 24.340 23.412 
 
 22.087 
 
 &% 
 
 103.318 
 
 89.421 
 
 86.015 
 
 81.126 
 
282 
 
 RAINEY S IMPROVED ABACUS. 
 
 TABLE Continued. 
 
 Side, 
 or di. 
 
 Squa. 
 
 Hexa- 
 gon. 
 
 Octa- 
 gon. 
 
 Circle. 
 
 Side, 
 or di. 
 
 Squa. 
 
 Hexa- 
 gon. 
 
 Octa- 
 gon. 
 
 Circle. 
 
 inch. 
 
 
 
 
 
 inch. 
 
 
 
 
 
 6 
 
 112.500 
 
 97.368 
 
 93.656 
 
 88.354 
 
 9K 
 
 266.781 
 
 231.418 
 
 222.600 
 
 210.800 
 
 6K 
 
 122.058 
 
 105.640 
 
 101.621 
 
 95.871 
 
 9> 
 
 282.031 
 
 244.100 
 
 234.793 
 
 221.506 
 
 sy* 
 
 132.031 
 
 114.271 
 
 109.948 1 103.696 
 
 &2 
 
 296.968 
 
 257.105 
 
 247.315 
 
 233.318 
 
 6% 
 
 142.381 
 
 123.231 
 
 118.534 
 
 111.825 
 
 10 
 
 3-12.500 
 
 270.471 
 
 260.163 
 
 245.437 
 
 7 
 
 153.125 
 
 132.528 
 
 127.478 
 
 120.372 
 
 10K 
 
 328.318 
 
 284.159 
 
 273.341 
 
 257.859 
 
 7K 
 
 161.256 
 
 142.162 i 136.743 
 
 128.986 
 
 10^ 
 
 344.531 
 
 298.193 
 
 286.828 
 
 270.593 
 
 
 175.781 
 
 152.037 ! 146.337 
 
 138.056 
 
 IOM 
 
 351.131 
 
 312.559 
 
 300.645 
 
 283.633 
 
 7% 
 
 187.693 
 
 162.449 156.259 
 
 147.415 
 
 11 
 
 378.125 
 
 327.268 
 
 314.796 
 
 296.978 
 
 8 
 
 200.000 
 
 173.099 
 
 166.503 
 
 157.078 
 
 11 ^ 
 
 393.216 
 
 342.315 
 
 329.268 
 
 310.631 
 
 w 
 
 212.693 
 
 184.087 177.071 
 
 167.049 
 
 U> 
 
 410.281 
 
 357.693 
 
 344.062 
 
 324.587 
 
 *y 2 
 
 225.781 
 
 195.412 187.365 
 
 177.328 
 
 HM 
 
 429.023 
 
 373.325 
 
 359.187 
 
 338.856 
 
 8% 
 
 239.256 
 
 207.078 199.127 
 
 187.912 
 
 12 
 
 450.000 
 
 389.475 
 
 374.613 
 
 353.428 
 
 9 
 
 253.125 
 
 219.078 210.721 
 
 199.203 
 
 
 
 
 
 i 
 
 Weight of a lineal foot of Flat Bar Iron, in lbs.,from % to 
 inches in width, and from % to 5 indies in thickness. 
 
 c 
 
 c. 
 
 i 
 
 
 
 
 
 J 
 
 c I c 
 
 * 
 
 t e 
 
 c 
 
 g 
 
 c 
 
 .S 
 
 ^ 
 
 g 
 
 e 
 
 c 
 
 c 
 
 a 
 
 fl 
 
 5 J I 
 
 a 
 
 c 
 
 2 
 
 5 
 
 c 
 
 ' .S 
 
 [c 
 
 i 
 
 _^ 
 
 g 
 
 i 
 
 g 
 
 g" 
 
 1 
 
 \ ^ 
 H 
 
 .j 
 
 
 
 H 
 
 g 
 
 
 
 _H 
 
 g 
 
 ** 
 
 TJ 
 
 0.211 
 
 
 5,/ 
 
 2.375 
 
 
 % 
 
 4.435 
 
 
 1 S '8 
 
 9.610 
 
 
 1% 
 
 12.673 
 
 
 K 
 
 0.422 
 
 
 % 
 
 2.850 
 
 
 1 * 
 
 5.069 
 
 /8 
 
 ^ 
 
 0.792 
 
 2Ji 
 
 3>8 
 
 0.89S 
 
 
 ?-i 
 
 0.634 
 
 
 /8 
 
 3.326 
 
 
 ji/ 
 
 5.703 
 
 
 34 
 
 1.584 
 
 
 K 
 
 1.795 
 
 ?8 
 
 >a' 
 
 0.264 
 
 
 1 
 
 3.802 
 
 IK 
 
 6.337 
 
 
 
 2.376 
 
 
 x 8 
 
 2.693 
 
 
 
 0.528 
 
 IK 
 
 3s 
 
 0.528 
 
 
 6.970 
 
 
 3^ 
 
 3.168 
 
 
 
 3.591 
 
 
 X 
 
 0.792 
 
 
 K 
 
 1.056 
 
 [5^1 1^ 
 
 0.686 
 
 
 /8 
 
 3.960 
 
 
 1: /8 
 
 4.488 
 
 
 1;, 
 
 1.056 
 
 
 X 
 
 1.584 
 
 
 K 
 
 1.372 
 
 
 ;-i 
 
 4.752 
 
 
 x4 
 
 5.386 
 
 & 
 
 X 
 
 0.316 
 
 
 
 2.112 
 
 
 % 
 
 2.059 
 
 
 j s 
 
 5.544 
 
 
 /8 
 
 6.283 
 
 
 X 
 
 0.633 
 
 
 /8 
 
 2.640 
 
 
 i/ 
 
 2.746 
 
 
 1 
 
 6.336 
 
 
 1 
 
 7.181 
 
 
 
 0.950 
 
 
 k 
 
 3.168 
 
 
 % 
 
 3.432 
 
 
 1 1 B 
 
 7.123 
 
 
 1>8 
 
 8.079 
 
 
 V 
 
 1.265 
 
 
 
 3.696 
 
 
 x^ 
 
 4.119 
 
 IK 
 
 7.921 
 
 
 
 8.977 
 
 
 ^ 
 
 1.584 
 
 
 [ X 
 
 4.224 
 
 
 T/ 
 
 4.8051 
 
 1% 
 
 8.713 j 
 
 
 1/8 
 
 9.874 
 
 7 '8 
 
 l,g 
 
 0.3'39 
 
 
 MX 
 
 4.752 
 
 
 \ 
 
 5.492 
 
 1 1'^ 
 
 9.505 
 
 
 lx/ 
 
 10.772 
 
 
 X 
 
 0.738 
 
 ?: s 
 
 x 
 
 0.580 
 
 
 
 6.178 ; 
 
 ] sj 
 
 10.297 
 
 
 1/8 
 
 11 670 
 
 
 X 
 
 1.108 
 
 
 
 1.161 
 
 
 IK 
 
 6.864 
 
 1 3 'i 
 
 11.089; 
 
 
 1^4 
 
 12.507 
 
 
 
 1.477 
 
 
 
 
 1.742 
 
 
 l/'s 
 
 7.551 2 
 
 3 8 
 
 0.845 I 
 
 
 1% 
 
 13.465 
 
 
 fa 
 
 1.846 
 
 
 K' 
 
 2.325 
 
 
 1 1/ 
 
 8.237 
 
 
 3i 
 
 1.689! 
 
 
 2 
 
 14.362 
 
 
 m 
 
 2.217 
 
 
 
 2.904 
 
 X 
 
 // 
 
 0.739 
 
 
 X 
 
 2.534 
 
 - 1 i 
 
 "K 
 
 0.950 
 
 1 
 
 ]& 
 
 0.422 
 
 
 3 i 
 
 3.484 
 
 
 3i 
 
 1.479 
 
 
 X 
 
 3.379 ! 
 
 
 K 
 
 1.900 
 
 
 14 
 
 0.845 
 
 
 % 
 
 4.065 
 
 
 ^ 
 
 2.218 
 
 
 
 4.224 i 
 
 
 % 
 
 2.851 
 
 
 % 
 
 1.267 
 
 
 
 4.646 
 
 
 K 
 
 2.957 ! 
 
 
 x4 
 
 5.069 
 
 
 ^2 
 
 3.802 
 
 
 
 1.690 
 
 
 i/ 
 
 5.227 
 
 
 * 
 
 3.696 , 
 
 
 J. g 
 
 5.914 | 
 
 
 % 
 
 4.752 
 
 
 /8 
 
 2.112 
 
 
 ik 
 
 5.808 
 
 
 
 4.435; 
 
 
 [ 
 
 6.758 
 
 
 % 
 
 5.703 
 
 
 ; ! .i 
 
 2.534 
 
 IK x 
 
 0.633 
 
 
 7/ 8 
 
 5.178 
 
 
 (X 
 
 7.604 
 
 
 % 
 
 6.653 
 
 
 7/ 
 
 2.956 
 
 
 K 
 
 1.266 
 
 
 1 
 
 5.914 
 
 
 
 8.448 
 
 
 1 
 
 7.604 
 
 1 "o 
 
 i,/ 
 
 0.475 
 
 
 
 1.900 
 
 
 
 6.653 
 
 
 [% 
 
 9.294 
 
 
 IX 
 
 8.554 
 
 
 K 
 
 0.950 
 
 
 i> 
 
 2.535 
 
 
 H^ 
 
 7.393 
 
 
 IX 10-138 
 
 
 IK 
 
 9.505 
 
 
 
 1.425 
 
 
 Sg 
 
 3.168 
 
 
 
 8.132 ' 
 
 
 1% 10.983 
 
 
 1% 
 
 0.455 
 
 X 
 
 1.901 
 
 
 % 
 
 3.802 
 
 
 ik 
 
 8.871 
 
 1% 111.828 
 
 
 1.406 
 
WEIGHTS OF FLAT BAR IRON. 
 
 283 
 
 TABLE Continued. 
 
 .2 
 
 c 
 
 
 
 c 
 
 a 
 
 & 
 
 .2 2 
 
 | 
 
 .S 
 
 c 
 
 & 
 
 c 
 
 .S 
 
 
 
 .2 
 
 c 
 
 
 a 
 
 .2^ 
 
 .2 
 
 s i i 
 
 .2 
 
 d 
 
 c 
 
 a 
 
 e 
 
 c 
 
 C 
 
 i 
 
 g 
 
 JL 
 
 g 
 
 i 
 
 g 
 
 fl 
 
 g 
 
 ji 
 
 H 
 
 g_ 
 
 i 
 
 I 
 
 g 
 
 
 N 
 
 12.356 
 
 
 P' 
 
 12.199 
 
 2M 
 
 26.719 
 
 
 m 
 
 19.221 
 
 
 iX. 
 
 17.953 
 
 
 ''i 
 
 13.307 
 
 
 L K 
 
 13.308 
 
 
 K 
 
 1.267 
 
 
 
 20.699 
 
 
 
 21.544 
 
 
 /8 
 
 14.257 
 
 
 
 14.417 
 
 
 X 
 
 2.535 
 
 
 1% 
 
 22.178 
 
 
 L% 
 
 25.135 
 
 
 
 15.208 
 
 
 1 '":{ 
 
 15.526 
 
 
 
 3.802 
 
 
 2 
 
 23.656 
 
 
 ! 
 
 28.725 
 
 
 ^K 
 
 16.158 
 
 
 \\- 
 
 16.635 
 
 
 jy 
 
 5.069 
 
 
 2X 
 
 26.613 
 
 
 
 32.316 
 
 1% 
 
 X 
 
 1.003 
 
 
 2 
 
 17.744 
 
 
 k< 
 
 6.337 
 
 
 2 ' . 
 
 29.570 
 
 
 2 1 / 
 
 35.907 
 
 
 
 2.006 
 
 
 2 1 - 
 
 18.853 
 
 
 X 
 
 7.604 
 
 
 ix 
 
 32.527 
 
 
 iy 
 
 39.497 
 
 
 /8 
 
 3.009 
 
 
 JU 
 
 19.962 
 
 
 % 
 
 8.871 
 
 
 3 
 
 35.485 
 
 
 3 
 
 43.088 
 
 
 K 
 
 4.013 
 
 
 2 :i .. 
 
 21.071 
 
 
 1 
 
 10.138 
 
 
 3X 
 
 38.441 
 
 
 3X 
 
 46.679 
 
 
 ; S 8 
 
 5.016 
 
 
 2' 1 '.', 
 
 22.180 
 
 
 I lx 3 
 
 11.406 
 
 3% 
 
 K 
 
 1.584 
 
 
 
 50.269 
 
 
 
 6.019 
 
 2% 
 
 K 
 
 1.162 
 
 
 IX 
 
 12.673 
 
 
 X 
 
 3.168 
 
 
 3V 
 
 63.860 
 
 
 % 
 
 7.022 
 
 
 X 
 
 2.323 
 
 
 
 13940 
 
 
 % 
 
 4.752 
 
 
 4 
 
 57.450 
 
 
 
 8.025 
 
 
 % 
 
 3.485 
 
 
 IK 
 
 15.208 
 
 
 K 
 
 6.336 
 
 <%. 
 
 X 
 
 3.802 
 
 
 K 
 
 9.028 
 
 
 K 
 
 4.647 
 
 
 1 5 .. 
 
 16.475 
 
 
 % 
 
 7.921 
 
 
 
 7.604 
 
 
 
 10.032 
 
 
 / 
 
 5.808 
 
 
 1?4 
 
 17.742 
 
 
 M 
 
 9.505 
 
 
 /I 
 
 11.406 
 
 
 L % 
 
 11.035 
 
 
 % 
 
 6.970 
 
 
 1% 
 
 19.010 
 
 
 /6 
 
 11.089 
 
 
 1 
 
 15.208 
 
 
 'K 
 
 12.038 
 
 
 % 
 
 8.132 
 
 
 2 
 
 20.27 r 
 
 
 1 
 
 12.673 
 
 
 IX 
 
 19.010 
 
 
 [; 
 
 13.042 
 
 
 1 
 
 9.294 
 
 
 2X 
 
 22.811 
 
 
 IK 
 
 14.257 
 
 
 
 22.812 
 
 
 [% 
 
 14.046 
 
 
 IK 
 
 10.455 
 
 
 2 \; z 
 
 25.346 
 
 
 IX 
 
 15.841 
 
 
 i ' ; r 
 
 26.614 
 
 
 1% 
 
 15.048 
 
 
 IX 
 
 11.617 
 
 
 2/4 
 
 27.881 
 
 
 1% 
 
 17.425 
 
 
 2 
 
 30.415 
 
 
 2 
 
 16.051 
 
 
 1.3.;, 
 
 12.779 
 
 3X 
 
 i, 8 
 
 1.373 
 
 
 IK 
 
 19.009 
 
 
 2K 
 
 34.217 
 
 
 2/8 
 
 17.054 
 
 
 IK 
 
 13.940 
 
 
 X 
 
 2.746 
 
 
 ^/8 
 
 20.594 
 
 
 
 38.019 
 
 
 w '4 
 
 18.057 
 
 
 1-TjT 
 
 15.102 
 
 
 / 3< ! 
 
 4.119 
 
 
 1 ;*, 4 
 
 22.178 
 
 
 L' '' ^ 
 
 41.820 
 
 2K 
 
 K 
 
 1.056 
 
 
 i ; V 
 
 16.264 
 
 
 4jj 
 
 5.492 
 
 
 ]% 
 
 23.762 
 
 
 3 
 
 45.623 
 
 
 X 
 
 2.112 
 
 
 1% 
 
 17.425 
 
 
 ?i 
 
 6.865 
 
 
 2 
 
 25.346 
 
 
 8X 
 
 49.425 
 
 
 / 
 
 3.168 
 
 
 2 
 
 18.587 
 
 
 /-'t 
 
 8.237 
 
 
 ax 
 
 28.514 
 
 
 
 53.226 
 
 
 ^ 
 
 4.224 
 
 
 2X 
 
 19.749 
 
 
 / 
 
 9.611 
 
 
 
 31.682 
 
 
 '"' ;i '-4 
 
 57.028 
 
 
 K 
 
 5.280 
 
 
 2 '4 
 
 20.910 
 
 
 1 
 
 10.983 
 
 
 2X 
 
 34.85 r 
 
 
 4 
 
 60.830 
 
 
 X 
 
 6.336 
 
 
 2% 
 
 22.0724 
 
 
 1^8 
 
 12.356 
 
 
 ar 
 
 38.019 
 
 
 4X 
 
 64.632 
 
 
 
 7.392 
 
 
 
 23.234 
 
 
 l' 1 1 
 
 13.730 
 
 
 3 ^ 
 
 41.187 
 
 4M 
 
 X 
 
 4.013 
 
 
 l /8 
 
 8.448 
 
 
 v..,, 
 
 24.395 
 
 
 1 '-',.. 
 
 15.102 
 
 
 
 44.355 
 
 
 
 8.026 
 
 
 IK 
 
 9.504 
 
 2% 
 
 K 
 
 1.215 
 
 I IK 
 
 16.47 
 
 1 
 
 2 
 
 1.690 
 
 
 X 
 
 12.039 
 
 
 IX 
 
 10.560 
 
 
 X 
 
 2.429 
 
 
 1 ' f] 
 
 17.84c 
 
 
 X 
 
 3.380 
 
 
 1 
 
 16.052 
 
 
 1/^3 
 
 11.616 
 
 
 % 
 
 3.64- 
 
 
 ] :'. 
 
 19.22 
 
 
 
 6.759 
 
 
 IX 
 
 20.066 
 
 
 IK 
 
 12.672 
 
 
 
 4.85^- 
 
 
 IJ'H 
 
 20.59 
 
 
 X 
 
 10.138 
 
 
 IX 
 
 24.079 
 
 
 
 13.728 
 
 
 % 
 
 6.07:. 
 
 
 2 
 
 21.96 
 
 
 1 
 
 13.518 
 
 
 1 ';. j 
 
 28.093 
 
 
 i ! 
 
 14.781 
 
 
 s v 
 
 7.287 
 
 
 2X 
 
 24.71 
 
 
 ^ 
 
 16.897 
 
 
 2 ' 
 
 32.105 
 
 
 iji 
 
 15.840 
 
 
 % 
 
 8.502 
 
 
 "-'/: 
 
 27.45 
 
 
 
 20.27 
 
 
 -X 
 
 36.118 
 
 
 2 
 
 16.896 
 
 
 1 
 
 9.7K 
 
 
 2% 
 
 30.20 
 
 
 1 l\i 
 
 23.65 
 
 
 2 A , 
 
 40.131 
 
 
 2.'; 
 
 17.952 
 
 
 IX 
 
 10.931 
 
 
 3 
 
 32.950 
 
 
 2 
 
 27.03 
 
 
 2X 
 
 44.144 
 
 
 ^''1 
 
 19.00h 
 
 
 iV,' 
 
 12.145 
 
 ''> l . 
 
 K 
 
 1.471 
 
 
 2X 
 
 30.41 
 
 
 3 
 
 48.157 
 
 2# 
 
 ^ 
 
 20.06 
 l.lOf 
 
 
 
 13.360 
 14.574 
 
 
 X 
 
 2.95 
 4.436 
 
 
 2>^ ! 33.79 
 2% 1 37.17 
 
 
 3 
 
 52.170 
 56.184 
 
 
 X 
 
 2.218 
 
 
 ifl 
 
 15.789 
 
 
 /" 
 
 5.91 
 
 
 3 
 
 40.55 
 
 
 ' ; "' 
 
 60.197 
 
 
 /i 
 
 3.327 
 
 
 i ?( 
 
 17.003 
 
 
 /i" 
 
 7.39 
 
 
 HX 
 
 43.93 
 
 
 4 
 
 64.210 
 
 
 * 
 
 4.4,% 
 
 
 i 7 ,. 
 
 18.218 
 
 
 x-i 
 
 8.87 
 
 
 :;>, 
 
 47.31 
 
 
 4):i 
 
 68.223 
 
 
 
 5.545 
 
 
 2 
 
 19.43i, 
 
 
 r/ 
 
 10.350 
 
 
 
 50.69 
 
 
 11. 
 
 72.235 
 
 
 X 
 
 6.654 
 
 
 
 20.647 
 
 
 1 
 
 11.82 
 
 *x 
 
 I B 
 
 1.79 
 
 5 
 
 X 
 
 4.224 
 
 
 % 
 
 7.763 
 
 
 Likj 
 
 21.861 
 
 
 1 i, a 
 
 13.30 
 
 
 X 
 
 3.59 
 
 
 
 8.449 
 
 
 i 
 
 8.872 
 
 
 - );; i; 
 
 23.076 
 
 
 1 ^ 
 
 14.78, 
 
 
 k 
 
 7.18 
 
 
 /i 
 
 12.673 
 
 
 IK 
 
 9.981 
 
 
 Jl., 
 
 2-1.200 
 
 
 1 -i j, 
 
 16.26 
 
 
 
 10.772 
 
 
 1 
 
 16.897 
 
 
 IX 
 
 11.090 
 
 
 88 
 
 25.50, 
 
 
 1^-. 
 
 17.74 
 
 
 l" 1 
 
 14.364 
 
 
 IX 
 
 21.122 
 
284 
 
 RA1NEY S IMPROVED ABACUS. 
 
 Weight of a lineal foot of Cast Iron Pipes, or Cylinders, in Ibs., 
 the thickness and bore being given. 
 
 c 
 
 S.S 
 
 .2 
 
 8.S 
 
 s c 
 
 2^ 
 
 s^ 
 
 2.S 
 
 d ^ 
 
 J2 
 
 S.S 
 
 J.s 
 
 5 
 
 D '" 
 
 tt.S 
 
 3 g 
 
 11 
 
 * 
 
 11 
 
 j 
 
 ?LS 
 
 11 
 
 IS 
 
 ! 
 
 r 
 
 il 
 
 1 
 
 X 
 
 3.06 
 
 
 * 
 
 43.68 
 
 
 % 
 
 77.36 
 
 17 X 
 
 x 
 
 88.23 
 
 
 % 
 
 5.05 
 
 
 % 
 
 53.30 
 
 
 % 
 
 93.70 
 
 
 % 
 
 111.06 
 
 IX 
 
 X 
 
 3.67 
 
 
 
 63.18 
 
 
 % 
 
 110.48 
 
 
 % 
 
 134.16 
 
 
 % 
 
 6. 
 
 7 
 
 & 
 
 36.66 
 
 
 1 
 
 127.42 
 
 
 % 
 
 157.59 
 
 IX 
 
 K 
 
 6.89 
 
 
 % 
 
 46.80 
 
 12 X 
 
 X 
 
 63.70 
 
 
 i 
 
 181.33 
 
 
 X 
 
 9.80 
 
 
 % 
 
 56.96 
 
 
 % 
 
 80.40 
 
 18 
 
 % 
 
 114.10 
 
 1% 
 
 % 
 
 7.80 
 
 
 % 
 
 67.60 
 
 
 % 
 
 97.40 
 
 
 X 
 
 137.84 
 
 
 % 
 
 11.04 
 
 
 i 
 
 78.39 
 
 
 % 
 
 114.72 
 
 
 % 
 
 161.90 
 
 2 
 
 % 
 
 8.74 
 
 7 n ., 
 
 % 
 
 39.22 
 
 
 1 
 
 132.35 
 
 
 i 
 
 186.24 
 
 
 X 
 
 12.23 
 
 
 % 
 
 49.92 
 
 13 
 
 X 
 
 66.14 
 
 19 
 
 % 
 
 120.24 
 
 2X 
 
 % 
 
 9.65 
 
 
 K 
 
 60.48 
 
 
 % 
 
 83.46 
 
 
 % 
 
 145.20 
 
 
 X 
 
 13.48 
 
 
 % 
 
 71.76 
 
 
 % 
 
 101.08 
 
 
 % 
 
 170.47 
 
 2X 
 
 % 
 
 10.57 
 
 
 1 
 
 83.28 
 
 
 % 
 
 118.97 
 
 
 i 
 
 195.92 
 
 
 % 
 
 14.66 
 
 8 
 
 X 
 
 41.64 
 
 
 1 
 
 137.28 
 
 20 
 
 % 
 
 126.33 
 
 
 % 
 
 19.05 
 
 
 % 
 
 52.68 
 
 13 X 
 
 X 
 
 68.64 
 
 
 % 
 
 152.53 
 
 2% 
 
 % 
 
 11.54 
 
 
 % 
 
 64.27 
 
 
 % 
 
 86.55 
 
 
 % 
 
 179.02 
 
 
 X 
 
 15.91 
 
 
 % 
 
 76.12 
 
 
 X 
 
 104.76 
 
 
 i 
 
 205.80 
 
 
 % 
 
 20.59 
 
 
 1 
 
 88.20 
 
 
 % 
 
 123.30 
 
 21 
 
 K 
 
 132.50 
 
 3 
 
 % 
 
 12.28 
 
 8X 
 
 X 
 
 44.11 
 
 
 1 
 
 142.16* 
 
 
 X 
 
 159.84 
 
 
 K 
 
 17.15 
 
 
 % 
 
 56.16 
 
 14 
 
 X 
 
 71.07 
 
 
 % 
 
 187.60 
 
 
 % 
 
 22.15 
 
 
 K 
 
 68. 
 
 
 % 
 
 89.61 
 
 
 i 
 
 215.52 
 
 
 X 
 
 27.56 
 
 
 % 
 
 80.50 
 
 
 % 
 
 108.46 
 
 22 
 
 % 
 
 138.60 
 
 3X 
 
 i/ 
 
 18.40 
 
 
 1 
 
 93.28 
 
 
 /'8 
 
 127.60 
 
 
 X 
 
 167.24 
 
 
 fc 
 
 23.72 
 
 9 
 
 X 
 
 46.50 
 
 
 1 
 
 147.03 
 
 
 
 
 196.46 
 
 
 X 
 
 29.64 
 
 
 % 
 
 58.92 
 
 ' 4 >a 
 
 X 
 
 73.72 
 
 
 i 
 
 225.38 
 
 3X 
 
 X 
 
 19.66 
 
 
 % 
 
 71.71 
 
 
 % 
 
 92.66 
 
 i3 
 
 % 
 
 144.77 
 
 
 X 5 8 
 
 25.27 
 
 
 % 
 
 84.70 
 
 
 X 
 
 112.10 
 
 
 % 
 
 174.62 
 
 
 X 
 
 31.20 
 
 
 1 
 
 97.98 
 
 
 % 
 
 131.86 
 
 
 K 
 
 204.78 
 
 3M 
 
 X 
 
 20.90 
 
 9X 
 
 X 
 
 48.98 
 
 
 1 
 
 151.92 
 
 
 1 
 
 235.28 
 
 
 x 5 ** 
 
 26.83 
 
 
 P 
 
 62.02 
 
 15 
 
 X 
 
 75.96 
 
 24 
 
 % 
 
 150.85 
 
 
 % 
 
 33.07 
 
 
 & 
 
 75.32 
 
 
 % 
 
 95.72 
 
 
 X 
 
 181.92 
 
 4 
 
 x 
 
 22.05 
 
 
 % 
 
 88.98 
 
 
 X 
 
 115.78 
 
 
 % 
 
 213.28 
 
 
 % 
 
 28.28 
 
 
 1 
 
 102.90 
 
 
 % 
 
 136.15 
 
 
 i 
 
 245.08 
 
 
 X 
 
 34.94 
 
 10 
 
 X 
 
 51.46 
 
 
 1 
 
 156.82 
 
 
 % 
 
 156.97 
 
 4X 
 
 X 
 
 23.35 
 
 
 % 
 
 65.08 
 
 15 K 
 
 1^ 
 
 78.40 
 
 
 K 
 
 180.28 
 
 
 /'8 
 
 29.85 
 
 
 % 
 
 78.99 
 
 
 t/ 
 
 98.78 
 
 
 % 
 
 221.94 
 
 
 X 
 
 36.73 
 
 
 % 
 
 93.24 
 
 
 
 119.48 
 
 
 i 
 
 25-1.86 
 
 4X 
 
 X 
 
 24.49 
 
 
 1 
 
 108.84 
 
 
 J? 
 
 140.40 
 
 16 
 
 % 
 
 196.62 
 
 
 / 5/ 8 
 
 31.40 
 
 10 % 
 
 X 
 
 53.88 
 
 
 1 
 
 161.82 
 
 
 % 
 
 230.56 
 
 
 % 
 
 38.58 
 
 
 % 
 
 68.14 
 
 16 
 
 X 
 
 80.87 
 
 
 i 
 
 264.66 
 
 4% 
 
 X 
 
 25.70 
 
 
 X 
 
 82.68 
 
 
 % 
 
 101.82 
 
 27 
 
 % 
 
 204.04 
 
 
 % 
 
 32.91 
 
 
 % 
 
 97.44 
 
 
 K 
 
 123.14 
 
 
 % 
 
 239.08 
 
 
 X 
 
 40.43 
 
 
 1 
 
 112.68 
 
 
 % 
 
 144.76 
 
 
 1 
 
 274.56 
 
 5 
 
 X 
 
 26.94 
 
 11 
 
 X 
 
 56.34 
 
 
 1 
 
 166.60 
 
 28 
 
 M 
 
 211.32 
 
 
 % 
 
 34.34 
 
 
 
 71.19 
 
 ie y z 
 
 X 
 
 83.20 
 
 
 % 
 
 247.62 
 
 
 % 
 
 42.28 
 
 
 M 
 
 86.40 
 
 
 % 
 
 104.82 
 
 
 1 
 
 284.28 
 
 SK 
 
 X 
 
 29.40 
 
 
 % 
 
 101.83 
 
 
 X 
 
 126.79 
 
 29 
 
 X 
 
 218.70 
 
 
 % 
 
 37.44 
 
 
 1 
 
 117.60 
 
 
 % 
 
 149.02 
 
 
 % 
 
 256.20 
 
 
 % 
 
 45.94 
 
 ix 
 
 X 
 
 58.82 
 
 
 1 
 
 171.60 
 
 
 i 
 
 294.02 
 
 
 X 
 
 31.82 
 
 
 % 
 
 74.28 
 
 17 
 
 K 
 
 85.73 
 
 80 
 
 % 
 
 226.20 
 
 
 % 
 
 40.56 
 
 
 K 90.06 
 
 
 % 
 
 107.96 
 
 
 % 
 
 264.79 
 
 
 X 
 
 49.60 
 
 
 % 106.14 
 
 
 X 
 
 130.48 
 
 
 i 
 
 303.86 
 
 
 % 
 
 58.96 
 
 
 1 122.62 
 
 
 % 
 
 153.30 
 
 
 X 
 
 343.20 
 
 ^ 
 
 X 
 
 34.32 
 
 12 
 
 V* 61.26 
 
 
 i 
 
 176.58 
 
 
 
 
GOLD COIN. United States, English and French. 285 
 
 4 Eagle ri7951._$5.25. Eagle.- $10.50. % Eagle. $5,00. 
 
 Eagle $2,50. Sovereign.- $4,85 
 
286 GOLD COIN. American, English and French. 
 
 40 Francs. $7,60. Double Sovereign. $9,60. Eagle. $10,00. 
 
 Bechtler pieces. $2.37. 20 Francs. $3 83 Sover'n. $2,42- 
 
 American Gold Dollar. 16th Doubloons.- .$1. #1.20. 
 
 Sovereign. $4,85. Double Louis d'or. $8,50. Sovereign.-^$45. 
 
GOLD Go. English, French and South American. 287 
 
 Sovcreign.-$4,85. t Do U bl oou .-S3.75. 
 
 4DoabIooB.-f7.7S. 
 
288 GOLD COIN. French and South American. 
 
 Doubloo i. $15,55-60. Doubloon. $15,50. 
 
 20 Francs. $3.83. 4 Douoioon.- $4. Sovereign. $4.85 
 
GOLD COIN. South American and German. 289 
 
 ^840 
 
 Doub. $3,-0. Doubloon. -$7.75. Double Frecl.d'or. $7.80. 
 
 Moidore [Brazil] $4,80. $ Joe. $8.50. 
 
 $1,96 
 
290 GOLD COIN, S. American, Portuguese, Spanish, German. 
 
 Doubloon. $15,75-16. 
 
 Doubloon. $15,55-60. 
 
 Escudo. $1,90. Moidore. iOc. . 24 cents. 
 
 Thai $1,94. 
 
 Doubloon. $15.55. 
 
 10 Thaler.- -$780 
 
 10 Thaler.-$7.80. 
 
 10Thaler.-|7.80. 
 
 40 Lire.-$7.66 
 
GOLD COIN. Portuguese, German and Italian. 291 
 
 One Mohur. $6.7 10 Tlmlet.-f 7.80 
 
 1-16 Dou. lOcts. 10th Moidore. 50 cents. 1-16 Doub. 90cts. 
 
 10 Thalers $7,80. 1 ) Timie.s $7 80. 10 Thalers $7,80. 
 
 Fred.d'or. 3.90. 
 
 5 Thale7.~ |3.90. 
 
 -"l*^ 
 5 Tlialers $3,10 
 
292 
 
 GOLD COIN. German and Italian. 
 
 Double Ducat $4.40 lOScudo $10 Soverain. $6.50. 
 
 ^ 
 
 20 Lire -$3 80 10 Lire. 1.90 5 Gilder. $2 Ducat. $2.20. 
 
 80 Lire. $ 1 5.32. 5 Thaler $3.90. 
 
 20 Lire. $ : 3.80. \ Joe. 1.75. Fvquni. $2.20. Ducat .^2.20, 
 
 Ducat. 12.20 
 
 20 Lire. 83.80 
 
 DUCUL $2.20 
 
GOLD COIN. German and ltd. SILVER COIN. U. #293 
 
 Gold Crown -$5,72. > Imperial $3,90. 5 Roubles. $3,90. 
 
 10 Guilders $3,95. 20 L ire . $3,82. 2 Christian u'or $7,80. 
 
 U* States Silver Coins. 
 
 I dime, 5 cents. 
 
 U. States One Dollar. 
 
 One Dime, 10 cte. 
 
SILVER COIN. United States 
 
 Quarter Dollar, 25 cents, % Dollar. 25 cents. * Dollar, 25 cents 
 
SILVER COIN. United States and English. 295 
 
 Dollar. 25 cents. Dime. 10 cents. 7 cent 
 
 Shilling 20 cents. ^ Dollar 24 cents. Shilling, 20 cents. 
 
SILVER COIN. English. 
 
 25 cents. 1 Shilling 18 cents. Shilling. 20 cents. 
 
 I Shilling. 19 cents. 18 cents. 1 Shilling. 18 cents 
 
SILVER COIN. English. 
 
 1 Rupeo. 40 cents. U Dollar. lOcts. 
 
 I H ISH 
 
 **SgnJ!^ 
 11 cents Six-pence, 10 cents. 5 Cent3 . 
 
SILVER COIN. English and Spanish. 
 
 _ _ 
 
 Dollar. 10 cts } Pistareen. 8 cts- Real. 10 cents. 
 
VB 17420 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY