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Las diagrammes suivants illuatrent la mAthode. rrata ;o lelure, 1 A 3 32X 1 2 3 1 2 3 4 5 6 m-^' m \j^, (^t^_ -f?!: !>'*?"■ l^.i.-*^ ^•?^^^^*.v.„., S^' n'V",';'.., >«--.• .^,,.y. •■■l < ^' i^V^'i^i^ ■■^ /, />*;;,.*'■ -■'^'f ■ .•,<^v ,A^ 'i "* ' '>»* fc ;'/ -. •¥ * '.«. -4V f ' .;■■■.. -■:\ ARITHMETIC FOR HIGH SCHOOLS AND COLLEGIATE INSTITUTES BY I J. C. GLASH/VN, OTTAWA. * ©otronto : ROSE PUBLISHING COMPANY 1890. ^a^^mv;: J^"'-^'.'*i>l"\';Va»TVjJsVX\';';!^i5ai^.\'.. ...i**"- ^=.r-- ■;■» ■ . -T 'I i.t^:^-'^- }'.•■#•',:.'' ^iy'h.. . MH^iittiMmtm Entered acc Hunter, Rose & Company, Toronto. PREFACE. The following work was prepared for the use of pupils in High Schools and Collegiate Institutes. As all pupils in these schools are required to possess, before admission thereto, a sufficient knowledge of arithmetic to enable them to solve easy problems such as those in Kxer- cise I and IV pp. 45 to 55 and 75 to 85 of the present work, the author has taken for granted the possession of such knowledge by those who will use this book. In other words, he has sought to supplement and to continue without any unnecessary repetition the course of arith- metic begun in the Public School Arithmetic. Furthermore, as the book is not intended for private study but for class-instruction under the guidance and with the intelligent assistance of competent mathe- matical masters, the Author has endeavored to avoid encroaching on the province proper to the instructor and has in general given oidy the main outlines of proofs and investigations leaving it to the teaclier to till in the details and to supply preliminary illustrations. The work consists of three distinct parts. The first part, forming chapters i to iv, treats of Notation and Computation ; the second part, chapter v, treats of Mensuration or Metrical Geometry ; and the third part, chapters vi, vii and viii, deals with Commercial Arithmetic. Chapter I treats of numbers and " notation and of units of measure- ment. The student will already be well enough acquainted with Arabic and Roman notation and with various compound systems to be able to use them more or less freely, but to know a subject is one thing, to know it in words, i. e., to know it so clearly and distinctly as to be able to put that knowledge into words, is quite another thing j — this chapter will it is hoped, help the student to put into words his knowledge of arithmetical notation and of our ordinary units of value, mass, space a,nd time. Chapter I with §§ 42, 62 and 63 of Chapter II and §§ 119 and 120 of Chapter IV lay a foundation on which may be built the theory of num- bers and the rationale of the art of calculation. True the ' ' Funda- mental Theorems " of chapters ii and iv are, strictly speaking, PosttUates defining and determining the particular kinds of addition, multiplica- tion and involution here considered, but this is a distinction which only those who have advanced some way in their studies can understand, and the history of mathematics teaches that ths method of presenting the subject here adopted is the easiest and the best for beginners. The greater part of Chapter II consists of descriptions of come of the methods of computation employed by experts. The proper place for these is a manual on the art of teaching, but as they are not to be found in their proper place and as many of the pupils in our High Schools purpose becoming teachers, it has been thought better to insert IV PREFACE. theso descriptions in the present work than to leave it a matter of chance whether teacliers shall know and practice any other than the traditional school-room methods. Here as elsewhere throughout the book, the Author makes no pretension to originality ; he has selected for descrip- tion the best methods and processes with which a somewhat extensive acquaintance with the literature of elementary mathematics has made him acquainted. Approximation is a part of arithmetic which has until lately been adequately discussed in only the higher classes of text-books, being, one might be led to conclude from their neglect of it, an unknown subject to the writers of the average school-book. But the great practical import- ance of the subject is at length compelling its fuller recognition in school- work, and it will receive more and more att-ention in proportion as arithmetical instructicoi ceases to be impractical and as teach- ers become better acquainted with the requirements of the count- ing-house, the workshop and the laboratory. In Chapter III, two methods of approximation are described ; the first. Approximation by Continued Fractions; the second. Approximation by Abridgment of Decimal Computations. The former of these takes precedence in historical order and also on account of its theoretical simplicity and of the wide range of subjects to which it is applicable, — from the purely speculative questions of Farcy's series and the partitions of numbers to the laboratory problem of determining the formula of an organic com- pound from its percentage composition ; — but the latter method is superior in facility of adaptation to all ordinary computations. Approximation by continued fractions was well known to the ancient Greek and Indian arithmeticians, so much so that in the oldest of their writings now extant it is introduced abruptly and used without explan- ation as an elementary subject with which their readers are assumed to to be already familiar. The whole theory of the subject is contained in the single theorem, — lies in value between — and _, being greater k a -I- h b~+"k than one and less than the other, and the immediate corollary therefrom, — —f "^^ and — are in order of magnitude^ b mb 4- nk k a, b, hf Jc, m and n denoting (absolute) numbers. But in the calculation and use of continued fractions, no proof is needed of the theorem in the general form in which it is here stated, its truth being tested in each separate instance of its application. Hence no reference to the general theorem is required in Chap. Ill and no such reference is made therein. In the arrangement of the factors in the contracted multiplication due to Oughtred and known by his name, the figures of the multiplier are written in reverse order, but the arrangement adopted in Example 1, p. 68 obviates the awkwardness of this reversal and is as simple as Oughtred 's in every other respect. Teachers who prefer to discuss abridged computation before convergents and those who prefer to omit all discussion of the latter will find tnat the method of treatment'which has been adopted will permit of their following their preference. Those PllEFACE. who seek for a fuller treatment of the theory of contracted calculations will find it in the Arithmetics of Munn, Cox, Sang, Serret and Beynac, in Ruchonnet's Elements do Calcul Approximatif, Lionntt's Approxima- tions Numeriques and Vieille's Theorie Generale des Approximations Nuineriques. Chapter IV contains an elementary discussion of Involution, Evolution and Logarithms. Special attention has been given to ' Horner's Method ' of Involution and Evolut'on not only on account of its simplicity, it being merely an extension of the ordinary rule of * Reduction , but also because of its power and generality as a process and of its great and varied utility. The chief value of logarithms, at least in elementary mathematics, lies in their usefulness as an instrument of calculation, buw the surest way to enable pupils to remember how to use tables of logarithms is to require them to c( mpute a portion of such a table considered as a table of exponents to base 10. Teachers who pr' fer to have their pupils learn at first the mere mechanical use of the tables and defer the theory of logarithms until logarithmic series is reached will omit §§ 114 to 122 and 124 to 136 and Exercises X to XV, but it might be well if they should note that the developmert of the theory of logarithms preceded that of the logarithmic and exponential series, preceded even the invention of generrized exponents, and that no large table of logarithms was ever computtc. by the immediate use of logarith- mic series. In connection with the subject of chapters iii and iv, the following extract is a sign of t\v ^-""ogress now making in England :^ "The Council noticeii with pleasure, as an example of what may be done by an examining body in the way of encouraging sound mathe- matical teaching, the following "Remarks" in the prospectus of the Technical College, Finsbury, with reference to the Entrance Examina- tion: — 'Ir Arithmetic, marks will bo deducted on those answers in which bad and antiquated methods are used; for example, if the Italian method in division is not foil' wed ; if decimal workings ?i'e not Sroperly contracted ; if remainders are given in fractions instea t f'A in ecimals; if logarithms are not used where their use would savt t'oie. [Logarithm Books containing Four-figure tables, are provided at the examination.] (Candidates) should be able to work square root and cube root by Hor- ner's method. '" Extract from the, Report of the Council of the Associ- tion for the Improvement of Geometrical Teaching ; England, January, 1889. Chapter IV closes the subject of pure calculation with the exception of the short Appendix on pp. 315 to 317, in which the notation of circulating decimals is explained without the usual hidden reference to infinite series and the method of limits. The curious and those who care to spend time on a subject of no practical and of but little speculative importance may consult the Arithmetics of Sangster, Brook-Smith, Cox, Lock, and Sonnenschein and Nesbitt, or the Traite d' Arithmetique et d' Arith- mologie of P. Gallcz. Chapter V consists of a short treatise on elementary metrical geometry. Much of the text, especially in the stereometry consists of proofs of important geometrical theorems which are not to be found in the vi PREFACE. propositions not properly , § 201, p. 237, are given m authorized text-book of geometry. A few belonging to elementary mensuration, e. g. without proofs. The Author bega to acknowledge his obligations in this part of his subject to Die Elemente der Matheniatik of R. Baltzer, the I'lanimetrie and Stereometrie of F. Reidt in Hchloemilch's Handbuch der Mathe- matik, the Traite de Geometric Klementaire of Rouche and Comberousse and the Theoremes et Problemes do Geometric Elementaire of M. Eugene Catalan. Chapters VI, VII and VIII complete the course of elementary com- mercial arithmetic begun in the Public School Arithmetic. In selecting (juestions for Exercises I and IV, it was assumed that pupils could solve by the so-called Unitary Method, the simpler problems in commercial arithmetic, including simple interest and discount ; but that method, however excellent as a me*'e answer-obtaining process, has an almost irresistible tendency to withdraw the attention of those who make ex- clusive use of it, from the general principles upon which all methods are founded. Numerous easy problems have therefore been proposed in the earlier exercises of Chap. VII, which it is hoped the teacher will take advantage of to endeavor to raise his pupils at this stage of their studies from infantile dependence on the unitary crawling-on-hands-and-knees method, and leaa them to make direct application of general principles including that widest of all principles, Tne Substitution of Equivalents For further information concerning promissory notes and bills of ex change, teachers should consult the Bills of Exchange Act of 1890, Data for an unlimited number of problems on stocks, bonds and deben tures will be found in the Stock Exchange Year-Book by Th. Skinner S.'udents who wish to advance to the higher questions on interest, annul ties and life insurance, will find an elal)orate discussion of these subjects in the Institute of Actuaries' Text-Book ; Part I, Interest, including Annuities-Certain, by Wm^ Sutton ; Part II, Life Contingencies, in- cluding Life Annuities and Assurances, by Geo. King ; to these two volumes may be added Ackland and Hardy's Graduated Exercises and Examples. The Author desires to express his general indebtedness to the writings of De Morgan, Duhamel, Grassmann, Schlegel and Houel. He also takes Sleasure in acknowledging his special indebtedness to Professor R. R. loCHRANE, of Wesley University, Winnipeg, and Mr. Robert Gill, Manager of the Ottawa Branch of the Canadian Bank of Commercj, for much valuable advice and assistance, and for many practical problems, and he tenders these gentlemen his grateful thanks for their kindness. CONTENTS. PAGE Numbers and Notation : Numeration - - - 1 Notation 6 Prime Units - - . 9 Tables of Values, Weights and Measures - - - 10 Metric System of Weights and Measures - - - 17 The Four Elementary Operations : Addition and S«btracti,■ !«■ Vlll CONTENTS. Mensuration : t>AOB Circle and Ellipse,— Area ..-..- 226 Cylinder, Cone and Sphere, — Area .... 233 " " •• Volume - - 241 Proportional and Irregular Distribution . - - . 261 Partnership 266 Percentage : Introduction - - 269 Profit and Loss 272 Insurance 276 Commission and Brokerage ..... 279 Disccunt 282 Promissory Notes and Inland Bills of Exchange - - 284 Interest - 289 " Simple Interest 290 Averaging Accounts - 293 Partial Payments 295 Compound Interest 297 Stocks and Bonds 302 Exchange, — Foreign 308 Appendix, — Circulating Decimals 315 Tables of Logarithms 318 The following selected course may be taken by candidates preparing for the primary examination and by all who have no special aptitude for mathematics:— Pages 1 to 55, 66 to 73, 75 to 93, problems 1 to 7 «i^ Ex. vi, pp. 97 to 105 omitting last six lines, problems 1 to 12 of Ex. ix, pp. 251 to 298 and 302 to 314, and the portions of chanter v covering the requirements in mensuration required for the primary examin:vtion, . but substituting verification by inspection of models for purely logical demonstration of the geometrical theorems quoted. ^^:^..:.X ARITHMETIC. CHAPTER I. OF NUMBERS AND NOTATION. 1. The simplest expression of a Quantity consists of two factors or conii)onent8. One of tlieso factors is the name of a niagnitudo wliich has been selected as a standard of reference and which is necessarily of the same kind as the (luantity to bo expressed. The other factor expresses how many maj^nitudes, each ecpial to the standard magnitude, nmst be taken to make up the required (piantity. The standard magnitude is termed a Unit and its cofactor, the other component of the expression, is termed the Numerical Value of the (juantity. Hence, — A Unit is any standard of reference employed in counting any collection of objects, or in measuring any magnitude. A Number is that which is applied to a unit to express the comparative magnitude of a quantity of the same kind as the unit. A Number is the direct answer to the question, ' How many V Thus, when it is said that a certain slate is ten inches long, the number ten applied to the unit-length, inch, indicates the magnitude of a certain length, that of the slate, compared with the unit-length, an inch. 2. The number and the unit together indicate the absolute magnitude of the quantity ; the number indicates the relative magnitude, or, as it is termed, the Ratio of the quantity to the Wilt. 3. Numeration is counting, or the expressing of numbers in words. The ordinary system of numeration is the Decimal System (Latin, decern ten), so called because it is based on the number ten. 4. The names of the first group of numbers in regular succession are one, two, three, four, five, six, seven, eight, nine. Other AKlTIIMKriC. miinlK'i-nainos aro ton, Imndrod, tlioiisfuid, million, billion, trillion. Ms 1 ! I «lrill <|iiae, as before we counted by singh- units, we get the nund»ers ivu, twt^nty {hirii-ti(i, twain-ten), thirty {Ihtrc-tiij, thri>e-ten), forty (four-ten), ninety (nine-ten). The names of the nvnnbers between ton and twenty are, in order, eleven (<'n(//if/o7i from ot, (Mi, one and /(/ten), twelve [lira two and /(/"ten), thirteen (three and ten), fourteen (foin* and ten), nineteen (nine and ten). Tlu> nan\ea of the numbers between twenty and thirty, thirty and forty are formed by |)lacing the names of the nund)ers one, two, three, nine, in order after twenty, thirty ninety. 7. The number hundred applied to any unit denotes a »piantity which consists of ten ten-iuiits. Coxmting now by a Innidred units at a tii.rj, as before wo counted by single units, we get the numbers one hundred, two hundred, niiu> hundred. The names of the innubers between one hundred and two innidred, two hundred and three hundred, are formed by placing the names of tlui nundiers from inw to ninety-nine in I'egular succession after one liundnMl, two hmulred, nine hundred. k hu NUMHERS AND NOTATION. s !i quantity 8. Tho iMiiiibor tli(iUHanr/-.) 20 shillings = 1 pound, (£.) 1, 2 and 3 farthings aro denoted by |d., ^d., and |d. respectively. Sterling Money is tho money of account employed in Great Britain ant: Ireland. The prime luiit is the pound which is tho value of the coin named a sovereign. The sovereign is coined of standard gold which is composed of 11 parts o{ pure g(>ld to 1 part TAIILKS OK VALirES, WKIlJIITS AND MKA^lIUE.S. 11 of nll»)y. IH(>!) stjuulanl Hovoroigus woigh 480 ounoim Troy of 480 grains ouch, lloiico h Hoveruign nhould coiitfiin 12.'{ '27447 gruiiiH of .standiinl gold of \\ Huoiiohh, lmt;v "romody," orallowiiiicofororror is i)cnnittolatinum- iridium, called tho Dominit)n standard, deposited in tho Dopart'nont of Inland Revenue at Ottawa. Tho weight of this standard is declared to bo " r)9{)0'{)8387 grains when it is weighed in air at tho temperature of (12 ilegrees of Fahrenheit's thermometer, tho ban miotor being at 30 inches," and 7,0^ yard, formerly used in cloth measure. TJtc V'ord iji now obsolete as a tenn of meamiremcnt. The hand = 4 inches, used in measuring the height of horses. The/(^/iom — 6 feet and ) ,, ., Thecable-lemjth = 120hxthouxs, |"«^^^l ^^X ^^^l""- The rod, pole, or perch =6h yards, used in measuring l.ind, hut not by surveyors. The fu rlong = 220 yards = J mile. The hague, not a fixed length, but in England commonly = 3 miles. The geographical or nantical mile, called also a minute of mean latitude, is i^J^jyiy of the earth's semicircumference from pole to pole. Its length is 6,077 feet, but for rough approximations it is taken as = 6,000 feet = 1,000 fathoms. The Paris foot ■■= 12 -79 inches. The French perch = 18 Paris feet =-6 'SOS yards. The arpent, or " acre " = 180 Paris feet =64 yards nearly. The three measures last-named are used under authority of the Dominion Weights and Measures Act for measuring lands in certain parts of the Province of Quebec. Disttmces less than a mile are often stilted in that Province in " acres." The prime unit, or standard of length, is the distance in a straight line between the centres of two gold plugs or pins in a certain bronze bar deposited in the Department of Inland Revenue at Ottawa, measured when the bar is at a temperature of 61 "91 degrees of Fahrenheit's thermometer. il TABLES OF VAI.UES, WEIGHTS AND MEASURES. 13 ing, divided chftin. for special >th nxoasure. ' horses. ig l;vnd, but ily = 3 miles. ito of mean pole to pole. it is taken as nearly. lority of the Ids in certain a mile are [istance in a |or pins in a Lnd Revenue ire of 61-91 Surface Meajsure. 144 square inches (b{\. in.) = l square foot, (sq. ft.) 9 sijuaro foot = 1 square yard, (sq. yd. ) 4,840 square yards =1 acre, (A.) G40 acres =1 square mile, (sq. mi.) 10,000 square links = 1 square chain. 10 square chains = 1 acre. Square links and square chains are used by land-surveyors in describing land-areas ; in calculations they are written as decimals of an acre. In old deeds and descriptions of property the square rod i)ole or perch i=30i sq. yd., and the rood=^ acre are sometimes used, but these terms are now practically obsolete. The prime unit or standard of surface measurement is the square yard, that is, a square surface whose sides are each one yard in length. Hence the prime unit of surface measurement is derived I from, a/)id is determined by, the prims ^mit of linear measurement. Cubic, or Volume Measure. 1,728 cubic inches (c. in.)=l cubic foot, (c. ft.) 27 cubic feet =1 cubic yard, (c. yd.) Firewood and rough stone are measured by the cord of 128 cubic I feet, which is equal to a pile of the material 8 feet long, 4 feet j wide, and 4 feet high. The cord is not a statutory measure, that [is, it is not defined by statute. The prime unit of volume measure is the cubic yard, that is, a [cube whose edges are each one yard in length. Hence the prime unit of volume measurement is derived from, amd is determined by, fhe prime unit of linear measurement. Measure of Capacity. 2 pints (pt.) = l quart, (qt.) 4 quarts =1 gallon, (gal.) 2 gallons =1 peck, (pk.) 4 pecks =1 bushel, (bu.) I gill is one-quarter of a pint ; it is not defined by statute, but is used in the second schedule to the Dominion Weights iasures Act of 1879. w^ ; i ' 1 14 AUITHMETI(\ Tlu piicity «»f cisternH, reBorvoirs juultholikoiHoftoii cxprossed in Ixirrels (bbl.) of 31^ jrjillons each, or in JuMjshccuts (hhd.) of fU^ gullons i!)ic]i. Tlio ]vu' ter, and with the barometer at thirty inches. The weigh 1 cu^ foot of water nuder those conditions is 62*356 lb., consequently a gallon contains or is equal to, 277 '118 cubic inches. The iTiipcrial gallon was formerly declared by statute to be of 277"274 cubic inches c^pa<'ly, wMch is the volume of 10 lb. of pure water at 67'S°V., but this part of the stati .eof weights and measures n.iS been repealed. A cuoic ti.oi, of pure v. iter at 52 ° F. weighs 62*4 lb. =998 "4 oz., and thij is the weight usually adopted in calculations aiming at a ii' 11 oxpro»»i!«l hhd.) of fK^ 10(1 not by 10 following . m It). as, vor ps, ips, )ns fiO ft). m \h. 70 lb. 480 pounds. \ot in Caniuliv. CO, and a few (fl- 5) (tt- 3) (O.) is tho gallon water weighed ^ air at tlio , „!■ >+er, and cu- foot of Dntly a gallon of 277-274 cubic at 67-5°I^.bu' lied. lb. =908-4 055., IS aiming at iv 'T ULEH OF VALUES, WEKiUTS AND MEASURES. 16 high dogreo of accuracy, Vmt wJn^ro groat accuracy is not required, ()2-5 lb. = 1,000 oz. is taken as tlie voight of w.dv.r per cuLic foot. This approximation is close enough for ordinary i)ur[)oae8, the more so as natural waters conbiin mineral matter in solution and ctjUHOiiuontly are somewhat denser than pure water. Measures of Time. <»0 . oads (sec.) — ! miimto (min.) 'JO miiiutes =1 hour (hi*-) '^ i'4 hours =1 day (da.) 7 days =1 week (wk.) '{()."< days =1 common year - - . - (yr.) 3(M{ days =1 leai> year Till! calendar year is divided into twelve parts called months; jsoven o{ these consist of .'Jl days each, four consist of JJO days each, land one (February) consists of 28 days — in leap years of 29 days. |Tlie lengths of the several months may bo remembered from the following rhymes: — Thirty days have September, Ai)ril, Juno, and November ; February has twenty-eight alone. All the rest have thirty-one ; But leap year coming once in four, Gives February one day more. Tho civil day begins and ends at 12 o'clock midnight. A.M. ^enotes time before noon ; M., at noon ; P.M., after noon. The prime unit of time is the day, or, strictly speaking, the lean solar day. -^ solar day is tho time -interval between two kiccessive transits of tho sun's centre over the meridian ; l)ut as |iese intervals are of unequal length, we take the mean or average all the solar days lu the year, and t«» this mefvn solar day we give, ordinary speech, the name day. A year is tho peritx* of tho earth's revolution about tho sun, torn some deteruunato positit)n back again to the same position. tho starting }hha\\ ho the vernal equinox, the interval is called a jpiciil year auv^ lias been ^ und to ctjupist of 365*242216 mean |lar days = 365 da. 5 hr. 4» min. 47| sec. The tropical year il T^ h 1 1 1 I il ! ! , I !■ li «il! !M m ^ I I ! ' I- I 16 ARITHMETIC. determines the recurrence of the seasons, and of all the important phenomena of vegetation and life depending thereon, but to adopt it as the civil or calendar year, the year of ordinary business affairs, would involve having one part of a day belonging to one year and the remainder of the day belonging to the following year. This partition of a day is avoided by having civil years of two different lengths, the one of 365 days which is less than a tropical year, the other, called bissextile or lecap year, of 366 days, which is greater than a tropical year. Now 400 tropical years WvHild be greater than 400 years of 365 days each by -242216 da. x 400 = 96-8864 da., or nearly 97 days, hence if every 400 years consist of 303 years of 365 days each and 97 years of 366 days each, the average civil year will be i)ractically of the length of a tropical yet^r, and the seasons will recur at the same times by the calendar. This is accomplished by making every year whose date-number is exactly divisible by 4, n leap-year, except in the case of the years whose dates are even hundreds, the date-numbers of these must be exactly divisible by : 400. Thus the years 1880, 1884, 1888 were leap-years ; 1881, 1882, 1886, 1887 were not leap-years ; 1600 was and 2000 will be a leap- ; year ; 1800 was not and 1900 will not be a leap-year. Neither the period of the earth's revolution about the sun nor ; the period of its rotation on its axis is absolutely constant. Tlie latter is lengthening by the ^^^j part of itself per hundred years. Angular Measure. 60 seconds (") 60 minutes 90 degrees 4 quadrants, or ) 360 degrees j = 1 minute - (') ~1 degree (°) = 1 quadrant or right angle. = 1 circle or whole circuit. The prime Unit of angular measure is one complete revolution. Angles less than seconds are expressed as decimals of a second. Angles are always measured in practice by Angular Measure, but in | many theoretical investigations another system of measurement, called Circular Measure, is adopted. '^ *-* METRIC SYSTEM OF WEIGHTS AND MEASURES. 17 THE METRIC SYSTEM OP WEIGHTS AND MEASURES. 28. The French or Metric System of Weights and Measures which is a decimal system, or system based on ten as the common I scale of relation among each set of units of the same kind, is used in scientific treatises. Its use is permissive in Canada, the Britisli Islands, and tht United States, and it has been adopted absolutely as ! the sole system throughout great part of Europe and South America. 29. The prime units in this system and their ratios to the prime I units of the Dominion or Imperial system are Length Metre =1 09362311 yards. Area Are =119 601150 square yards. Volume or \ j • . „ / = -00130798582 cubic yards. Capacity j ^"'^^ \ = '22021444 gallons. Weight and Mass. Gramme = 15 -43234874 grains. 30. The fundamental unit of this system is the metre v'hich was intended to be the ten-millionth, (-000,000,1), part of a quadrant jf latitude, i. e. , of the distance of the pole of the earth from the jquator, measured at the level of the sea. In this respect the [egal metre is not quite exact, but this is of no consequence as )ractically the length of the metre is fixed in each country adopting Ihe metric system, by means of Standard Metres marked on metal [ods, just as the Standard Yard is determined. The original of lese rods is the French Standard Metre,* a platinum rod ^posited in the state archives at Paris. From the metre are derived the are, the litre and the gramme. Phe fire is equal to 100 square metres ; the litre to the -001 of a ibic metre ; and the gramme to the -000,001 of the weight in vacuo a cubic metre of distilled water at its temperature of greatest bnsity. In measuring wood, a Stebe = 1 cubic metre = 1,000 litres used, and in weighing heavy articles a Millier or Metric Ton = ^000,000 grammes is employed. 31. The names of the auxiliary units in this system are formed attaching certain prefixes to the names metre, are, litre and imme respectively ; thus : — ihe Canadian Standard Metre is dehiied by statute as equal to 1-0939* ^ndard yards. It therefore differs appreciably from the French Standard ttrc which is equal to 1-09362311 standard yards, the difference amounting rather more than a yard in two miles. Ih 'IN' it !ni Ml ill: IM-' t,! Pi 111 il 'i! !lii:! !" ir' i-f 18 ARITHMETIC. micro — "^ niilli - centi — deci - tleka - hecto — kilo- luyria - mega— J \- ^ motr3 are litre gruiume^ r2s 0) -13 o t m .a I OP = - CO c- - U) Z - «i i/i - ^ = ^ E - 1-1 , read is (jreater than, thus y>| ; < , read is less thuu, thus j-j < 4 ; .'., read therefore, consequently, hence; '. ', read because, siivce, thus *. • S > | and ^ <4 • 1 > y.* nm 1 CHAPTER II. THE FOUR ELEMENTARY OPERATIONS. ADDITION AND SUBTRACTION. 37. Addition is the operation t)f finding tlint (juantity wliich iiiade u]) as a whole of two or more given quantities as its parts. Tlie (quantities to be added together are called addends, — or lldciida. The result of the addition is termed the sum of the addends. Since the sum is the whole of which the addends are the parts, — Addends and sum mud aJl he quantities of the same ki^id, i. e., ley 111 ist all have the same unit. 1 38. The sign of addition is + , read jdus, meaning increased by. he sign + placed before any quantity indicates that the quantity i an addend. Thus 8 + 3 is read ' eight plus three ' and denotes [at 3 is to be added to 8. In like manner 24 + 9 + 5 is read |\venty-four plus nine plus five,' and denotes that 9 is to be added 24 and then 5 added to the sum. [The sum of any number of given quantities is expressed by [iting the quantities in a row in the order in which they are to be Ided, with the sign + between every adjacent i)air. 39. Subtraction is the operation of finding the part of a ven quantity which remains after a given part of the quantity Ls been taken away. The quantity from which a part is to be taken away is called the Unuend. j Tlie part of the minuend which is to be taken away is called the ibtrahend. jThe result of the subtraction is called the remainder and also le difference between the minuend and the subtrahend. Since tlie minuend is the whole of which the subtrahend and the Iniainder are the parts, — \Mlnuend, subtrahend and remainder must all have the same wilt ; Kl, — // the subtrahend be added to the remai')uler the sum xmll be minuend. I I ; ' 1 1 lib ,| 1 I II Hi, ij :; is ll 22 ARITHMETIC. 40. Tho sign of subtraction is — , read minvs, meaning diminishi by. Tho sign - placed before any quantity indicates that tho (luantitj is a subtrahend. Thus 8 - 3 is read ' eight minus three ' and denotJ that 3 is to be subtracted from 8. In like manner 24 — J) — 5 is reil * twenty-four minus nine minus five ' antl denotes that 9 is to subtracted from 24 and then 5 subtracted from the remainder. 24 + 0-5 denotes that 5 is to be subtracted from 24 + 9 whj 24-9 + 5 denotes that 5 is to be added to 24-9. 41. An expression consisting t)f a succession of addends a J subtrahends, such as 8 + 6 - 3 + 6 - 3 - 4, is called an aggregate, The several pa^ta, the addends and the subtr.ahends, as 8, +ij -3, +6, —3, —4, are called the terms of the aggregate ; and The quantity which results from collecting the terms by performii,^ the indicated additions and subtractions is called the total or sii of the aggregate. 42. The Fundamental Theorems of Addition and Subtractii^ are ; — I. If equals he added to equals, the i wholes are equal. II. If equals be sHbtra4^ted from equals, the remainders are eqiKii III. The sum of tiro addends u-'dl be the same whether the secoui be added to the first or the first be added to the second. IV. Addinq to an addend adds an eqvxd quantity to the siim. V. Subtracting from an addend subtracts an equal quantity fm the sum. VI. Adding to the minuend culds an equal qiiantity to the remaiwh VII. Subtracting from the minuend subtracts an equal quaidi^ from the remaiiulev. VIII. Adding to the subtrahend subtracts an equal quantity fnm the remainder. IX. Subtracting from the subtrahoul adds an^ equal quantit\j the remainder. X. Adding zero to any quantity leaves the quantity imchanged. Theorms III to IX may be stated in a single theorem, thus,— 1 Ghanging the order of collecting Ihe teryns of any aggregate, (/' not change the total or s)()n of tJie aggregate. 43. To prove any calculation is to perform another calculatii that will test or put to proof the correctness of the results . the first calculation. ADDITION AND SUBTRACTION. 23 n and Subtractiii Hid quantity fni equal quantitii 44. The simplest and best way to prove a result in addition is to repeat the addition, adding downwards the columns that were added upwards on the first addition and upwards the columns that were then added downwards. 45. In the additions of tabulated numbers which are to be added both vertically and horizontally the agreement of the grand total of the row of partial sums with the grand total of the columns of partial sums is, in general, a sufficient tetv of mere correctness, but if a mistake has been made, it is not enough to detect its existence, the mistake must be located in the partial sums and there corrected. This location and correction is often greatly facilitated by what is known as Computers' Addition. In this method the sum of each columu, is set daunt, separately, the right hand figure of each partial sum being jdaced under the column from which it is derived, and the other figures in their order diagonally downwards to the left. These partial sums are then added together to obtain the sum. By this arrangement the addition of any colunm can be tested independently of that of the preceding column, no knowledge of the * carried ' number being required. Thus if it be knowTi that an error has been committed in the addition of the hundreds, it can be discovered and corrected without adding the tens to ascertain the ' carriage. ' In this example, the sum of the first column is 38. The 8 is placed under the first column and the 3 under the second column but in the line next below that of the 8. The sum of the second column is 69. The 9 is placed under the second column immediately on the left of the 8 and above the 3 of 38, and the 6 is placed on the left of the 3. The sum of the third column is 42. The 2 is placed under the third column immediately on the left of the 9 of 69 and the 4 diagonally below to the left. The sum of the fourth column is 57, of which the 7 is written beneath the fourth column obtained, and the 5 is placed diag(mally below it to the left, partial sums are now added to obtain the sum, 61928. Examplt • 8784 27 3295 19 2133 9 8594 26 6272 17 9585 27 7986 30 9286 25 5993 26 7298 6 5463 20 61928 from which it was to the left. These i. ipii , ii ' ! !| lit !:'!!', I' i 'H ii u 24 ARITHMETIC. 46. Since in this method the columns are added independently, the result may be tested by adding together the digits :'n each horizontal row as shown in the example. The total of these sums, -in the example, 20G, — should be ilio same as the total of the coluiuu-sums, — in the example, '68, 09, 42 and 67, — treated as a row o • nmtually independent numbers. 47. Some computers prefer to arrange thotigures of the colunui- sums fi >m right diagonally upwards to left and to add in the carried number J as is done in the ordinary method. Taking the preceding examj)'e, the lowest addend and the result would by this arrangement appear as in the margin, the ; ; ; ; upper addends being here omitted merely to save space. The column-sums would be 38, 72, 49 and 01. The of 01 the last column-sum, is not written in the carriage-line but is placed at once in the sum-line. 48. Computers' Subtraction. The best way to perform subtraction is the method based on the fundamental theorem that the sum of the subtrahend and the remainder is ecjual to the minuend. Example. From 436,840 take 269,784. It is recjuired to find what number added to 269,784 will make 436,840. Write the subtrahend under the minuend so d'^fiftdf 6993 473 01928 269784 170002 that tlie figures of the same decinud order in oacli shivll be in the same vertical column as in the margin. To 4, the right-handed figure of the subtrahend, 2 must be added to make up the right-hand figure of the minuend ; put down this 2 as the right-hand figure of the remainder. The 8 (ten) of tlie subtrahend cannot be mode up to the 4 (ten) of the minuend, St) niake it up to 14 (ten), this requires that 6 (ten) be added ; put down this (> (ten) in the remj'inder. To the 7 (hundred) of the subtrahend add 1 (hundred) carried from the 14 (ten), thus making it 8 (hundred), and (hundred) is required to make this 8 (hundred) up to the 8 (hundred) of the minuend ; put (hundred) in the remainder. To the 9 (thousand) of the subtrahend add 6 (thousand) to make up 15 (thousand) which will give the 5 (thousand) of the minuend ; and put down this (thousand) in the remainder. ADDITION AND SUBTRACTION. 25 784 will make To thf> 5 (ten thousand) of the subtrahend add l(ten thousand) from the 15 (thousand) already made up and then add 7 (ten thousand) more to make up 13 (ten thousand) hi the minuend, putting down this 7 (ten thousand) in the remainder. To the 2 (hundred thousand) of the subtrahend carry 1 (hundred thousand) from tlu; i;i (ten tiiousand) and add 1 (hundred thousand) more to make up the 4 (hundred thousand) of the minuend, putting this 1 (lunidred thousand) in the remainder. Fancy you are doing addition with the sum at the top of the columns of addends and work thus setting down, as you prcmounce them, the figures here printed in thick-faced type ; — 4 and 2, six ; 8 and 6, fourteen ; 8 and O, eight ; 9 and 6, fifteen ; and 7 thirteen ; 3 and 1 four. After a little practice the minuend-sums need not be pronounced. The actual character of the process will perhaps be better comprehended by working a few examples with the subtrahend written as the lower of two addends, and the minuend written as their sum, the problem being to find the other i'-('(W"2 addend. Arrange the i)receding exam])le in this 259784 way, (see margin), and repeat the working given 4.'^ir«4.<' above. 49. This method is nearly always adopted in " making change" and so lends itself to calculations involving both additions and subtractions that it is almost universally employed by professional computers, and is generally known as Computers' Subtraction. Example. From 95G4 take 1357 + 498 -f- 197f) + 83 -f 3758. Arrange the subtrahends in column under the minuend addends are arranged in addition ; — see margin. Add the subtrahends together and ' make u}) ' to the minuend, setting down the ' making up ' number. Tluis 1st. Column; 11, 17, 25, 32 & 2 ; 34 ; cnrnj'6: 2nd. " 8, 10, 23, 32, 37 c^j 9 ; 4«) ; " 4: 3rd. " 11, 20, 24, 27 it 8; 35; " 3: 4th. " G, 7, 8 it 1 ; 9. — 50. To prove any subtraction add the subtrahend to the remainder, the sum should be the same as the minuend. IS 9664 1357 498 1970 83 3758 1892 r t ! i I III Pi Ml hi ; :|!:ii| ■ :« 111 li 26 ARITHMETIC. MULTIPLICATION AND DIVISION. 51. Tho simplest oxpression of a tiuantity consists of two components, one naming tho unit, the t)ther stating the number of I such units in tho quantity. But since the unit is a magnitude itf may itself be considered as a (piantity and expressed in terms ofi another unit which relative to it is called a primary unit. Thus the expression of a given quantity ma\j consist of threk; comjwnents, one naming a primary unit, a second statinxj the] NUMBER of these primary vnits composing a standard qnanUty oil derived unit, and a third stating tue number of these derived \ units in the given quantity. 52. The number of primary units in such a quantity h called tho product of tho number of primary units in the derived unit multiplied by the number of derived units in the quantity. Thus 35 marbles is the same quantity as 5 counts of 7 marbles | each, therefore 35 is the product of 7 multiplied by 5. In this caso tho primary unit is a marble and the derived unit is a count of 7 marbles. Again ^ yd. is the same quantity as f of a yd., hence i is thoj product of I multiplied by §. In this case the primary unit is a yard and the derived unit is | of a yard. 53. Multiplication is the operation of finding the product oi two numbers ; in other words. Multiplication is the process of finding the number of units of a given kind in a quantity which contains a given number of standard quantities each consisting of a given number of units of the given kind. The numbers to be nmltiplied together are called the factors of tho product. The factor which is to be multiplied by the other is called tho multiplicand. The factor by which the other is to be multiplied is called the multiplier. 64. A boy who has to read 18 pages of 38 lines each wishes to know how many lines he has to read. Here it is required to find the iiumber of lines in the quantity 18 pages-of-38-lines, a quantity whose unit, a page-of-38-lines, is express ^d in terms of the primary MULTIPLICATION AND DIVISION. 27 g the product of the factors (if er is called tlio d is called the 10 priniHiy M unit, a line. The required number may bo found by counting, or by addition, or by nmltiplication. In this cuso the product, that «)f .'i8 and 18, my 4. So .'i() X .'} : 4 uonotes tlmt 'M is to be multiplied ])y IJ and the product divided by 4, wliilo 3(5 : 3 x 4 denotes that 3(5 is to be divided ])y 3 and tho (juotient multiplied by 4. 60. In an expression c«>ntainin^ a sueceasion of multipliers and divJHors, the operations are to bo performed in order from left to right. Thus, 9 X 5-^3 X 6-7- 10-r4 = 45-^-3 X 6-4-10-^4-15 X 0-f-10-.-4 = 90-1-10^4 = 9^4-2^. Compare this with 9 + 5-3 + 6—10—4 = 14—3 + 0—10—4 = 11 -t- (J— 10— 4 = 17—10— 4=7— 4 = 3. In an aggregate whoso terms contain multipliers and divisors, *lie miiUipllcdfiojis (Oid the, dirlHioiiH are to be perfonned before the roduet will he ::ero. Theorems XIII to XIX may be stated in a single theorem, thus : — If an expression contain a succession of multipliers and divisois, changing the order of the mxdtipliers and the divisors does icot change the value of the expression. Example. lO^Sx 12-^3 = 10-T-5-^3x 12 = 10-f3xl2-:-5 = 10x12h-5-t-3=8. 63. The Fundamental Theorems connecting the operations of addition and subtraction with the operations of multiplication and division are, — XXII. Multiplying the several terms of an aggregate by any number midtiplies the aggregate by that number. XXIII. Dividing the several terms of an aggregate by any number divides the aggregate hy tJmt numher. 64. Scholars* Multiplication. Multiplications in which both multiplier and multiplicand require many digits to express them are generally best made by means of a table of midtiples of the multiplicand. This table may be formed by successive additions of the multiplicand written on a slip of paper to be moved down the column of multiples as the successive additions are made. The multiples should extend from the first to the tenth, the last testing the accuracy of the work ; and, for convenience of reference, straight lines should bo drawn undei? the first, fifth and ninth multiples. MULTIPLICATION AND DIVISION. 31 plies the jyroduct ic product by the mnltiplies the 'es the quotient ides the qiiotient lies the quotient Example. Find the product of 74,853,169 and 2968457. Multiple Table. 2068457 2968457 . ^Jb«457 74853169 1 '2 3 4 _6 6 7 8 _9 10 5936914 8905371 11873828 14842285 17810742 20779199 23747656 26716113 26716113 17810742 2968457 8905371 14842285 23747(556 11873828 20779199 29684570 222198413490233 65. Scholars' Division. A table of multiples of the divisor may be employed in the case of division in which both divisor and dividend require many digits to express them. Example. Divide 2808332109244 by 58679. Multiple Table. J 2 3 4 li 7 8 lo 58679 117358 17(5037 234716 293395 352074 410753 469432 528111 586790 47 859236 58679)2808332109244 234716 461172 410753 504191 469432^ 347590 293395 541959 528111 138482 117358 211244 176037^ "352074 352074 66. Computers' Multiplication. In multiplying by a number re<[uiring several digits to express it, we may set down each partial product as it is calculated, and then sum the whole of them ; or, as each partial product after the first is calculated, we may add to it the sum of all the previously calculated partial products. il ! ! li \ if Si! £ ;: t! I m !l!i: ' I! il I: * ill ilil. 32 ARITHMETIC. Bi 67 ^Hmuli 66437 ^Hiiiulti 3862967 Hiif tii( 396059 ^Hii. 378127 ^■ilie 545745 ^piiuilti 167448 ^Bdiilits 298929 ^1 481388 ^^W'111ll>!4 217449898679 Example. Multiply 66437 by 3862967. The first line of products is simply 7 times the multiplicand. The next line is formed thus : — 6 times 7 and 5. the tens of the first lino of products, = 47. Write the 7 beneath the 5 added in and carry 4. 6 times 3 and 4 carried =^ 22. Write 2 on the left of the 7 last written and carry 2. 6 times 4 and 2 carried and 5 from the first line of products ==31. Write 1 on the left of the 2 last written and carry 3. 6 times 6 and 3 carried and 9 from the first line of products = 48. Write 8 on the left of the 1 last written and carry 4, 6 times 5 and 4 carried and 3 from the first line of products = 37. Write 37 on the left of the 8 last written. The partial product thus formed with the 9 brought from the line above is 378127^ which is 67 times 56437, the multiplicand. The third line of partial products is formed by multiplying the multiplicand 56437 by 9 (hundred) and adding in successively the proper digits of the second partial product, thus : — 9 times 7 and 2 from the second partial product = 65. Write 5 beneath the 2 added in and carry 6. 9 times 3 and 6 carried and 1 from the second partial product =34. Write 4 on the left of the 5 last written and carry 3. 9 times 4 and 3 carried and 8 from the second partial product =47. Write 7 on the left of the 4 last written and carry 4. Proceeding in this way we obtain as third partial product 6467457.0 (the 79 being brought down from the lines above) which is 967 times 56437. In like manner, multiplying by 2 (thousand) and adding in the third partial product we obtain 2967 times 66437. Next multiplying by 5(ten thousand), then by 8 (hundred thousand) and finally by 3 (million), each time adding in the last-obtained partial product, we obtain 21744S898679 which is the product of 56437 multiplied by 3862967. The six figures on the right in this finul product, viz. 898679, are the right hand figures of the six preceding partial products, MULTIPLICATION AND DIVISION. S3 dding in the 67. Computers' Division. In computers' multiplication the [)r()duct is built up liy successive additions of multiples of the iiultiplicand, these multiples being determined by the several digits )f the nuiltiplier. In computers' division this process is reversed ; Hie dividend is broken up or resolved by successive subtractions of multiples of the divisor, these multiples determining the several kligits of tlie (juotient. ^Kcamplr 1. Divide 217,440,808,579 by 5(),437. 3852007 Here 50437 is contained* 3 (million) times in 217440 (million). Write 3 in till' (|Ui)tient and proceed to obtain the ['remainder' by computers' subtraction, thus : — 50437)217440808570 48138V35144 21490()290 107483148 U0086628696 8x9 = 72000+2 = 800 = 72 =36000 36872. 592403 4140821 4140821 7 490 4900 456742713 5397. MULTIPLICATION AND 1)1 VIS ')N. 37 U l + 5+0=f| f 72856340 i - 07285034 65570706 the inultij)lieaii(lj U|t eacli liguro ti'| lining from twd.i ten on both tluj . take 3 tiniesl . . . times tlicl .... take :^(l| 1 tinu'si ]>liera ex])ressor'J :;s otlier than n. iiethod, see the] 400 times. 390 times. i})]ier constitute I the nmltii)lioi| xiv. If the multiplier is seen to be the product of two or more iiuall factors, multiply the given multiplicand by any one of these liicttirs, nuiltiply the product so formed by a second factor, this Lc'Diid product by a third factor, and rfo continue till all tlie factors (lave ])een used. The tinal ])roduct is the product retpiired. XV. To vlivide by 5, multiply by 2 and divide the product by 10. |;5x2-10.) xvi. To divide by 25, nudti^jly by 4 and divide the i)roduct !iy l(K). (25x4 = l(X>.) wii. To divide by 125, multiply by 8 and divide i;he product .y 1()00. (125x8 = 1000.) xviii. To divide by 75, or 175, or 225, or 275, , nudtiply |by 4 and divide the product by 300, or 700, or 000, or 1100, . . . . , as the case may be. xix. To divide 375, o'- 875, or 1375, multiply by 8 land divide the product by 300C, or 7000, or IKHK), #.s the I case may be. XX. If the divisor is seen to be th product of two or more [ factors each less than 13, divide by these factors 'in succession,' I the Hnal (piotient is the ([uotient required. Example. 8765348-r462. •102=6x7x11. 6 7 11 8765348_ U60891|[_ 208«98^[j_ 1897¥j^.^ 9 / 3 7 490 4900 i 6397. Tlie -r might have been written }^ in which case ^5 would have become .Vj and ^jf.;^ become .V:f j. These are the forms in which ^he fractions would have ai)peared had the divisor 462 been resolved ir.io 2 x 3 x 7 X 11 and these four factors used as successive divisors. xxi. Since 100 = 99x1 + 1 2(K) = 99x2 + 2 3m) = 99x3 + 3 325 = 300 + 25 = 99 x 3 + 3 + 2{. 894 = 800 + 94 = 99 x 8 + 8 + 94 &c. = &c. therefore when any number is divided by 99 the cemainder increased if necessary, by 99 or a multiple of 99, exceeds the reuxaiuder ( I t 1! ''i I ill ^'1' ii !li in' II' ;, 5 US AIUTHMETIC. wlirn tli"t number is divided l)y KM), by the t|uotieiit wlion KXHs] the divisor. iSuccusaivo Hpi)licati(iiiH of this leads to a convenient nietliocl ot dividin;^ any number by 1)9. Thus : — 297081)- 25>7«(H) + 89 = 2970x99 + 2!>70 + 89 = 297(5 x99 + 29(M> + 7(» + 89 = 297(5 X 99 + 29 x 99 + 29 + 7(5 + 89 = 3005x99+194 = X05x99+l(M) + 94 = 3005x99 + 1x99 + 1 + 94 = 3(K)(>x 99 + 95 .-. 297689 + 99 = 300()|;5. This may be arrantjed for woiknif; as follows :— 297(5 i 89 29 i 7(5 ! 2J) ' 1 j i)4 1 1 Quot. =300(5 I 95 = Rem. Similarly for any niunber expressed by 9's only. 70. Tests of Exact Divisibility. The following tests of exact divisibility are often useful in a search for the factors of ;i luunber. i. A numberiscxactly divisible by 2 if its right-hand figure is zero or a inunber exactly divisible by 2. ii. A number is exactly divisible by 4 if its two right-hand figures are zeros or express a number exactly divisible by 4. Kramplc^. 173528 is exactly divisible by 4, for 28 is exactly divisible by 4 ; but 319378 is not a nudtiple of 4, for 78 is not exactly divisible by 4. iii. A numl^er is exactly divisible by 8 if its three right-hand figure? are zeros oi express a nund)er exactly divisible by 8. E.vaDtple.t. 536 is a nudtii)le of 8, therefore 139753(5 is exactly divisible by 8 ; but 356 is not a multiple of 8, consequently 4()7935t) is n(jt exactly divisible by 8. iv. A number is exactly divisible by 5, 25, 125, if the number expresse d by the right-hand figure or the two, three, right-hand figures is exactly divisible by 5, 25, 125, kiX. MULTIPLICATION AND DIVISION. :v.) figure is zero V. A miinbor is exnctly divisible by * if the 'sum of its digits i» ex.ictly divisible by .'{. vi. V nuiaber is exactly divisible by \) if the sum of its digits if oxiictly divisible by !). Examples. Test whetlier 18(5.37500 and 7385(;21 are divisible by 1). 1 + 8 + 6 + 3 + 7 + 5 + + 9-45 = 9x5, .-. 18037569 is exactly divisible by 9. 7 + 3 + 8 + 5 + 6 + 2 + 1 = 32 = 9x3 + 5, .-. 7385621 ic not exactly (li\'isible by 9. vii. A number is exactly divisible by 6 if it is exactly divisible by both 2 and 3. viii. A number is exactly divisible by 12 if it is exactly divisible by both 4 and 3. i.\. A lunuber is exactly divisible by 11 if the difl'erence between tlie sum of its 1st, 3rd, 5th, 7th, S:c. figures and tlie sum of its 2nd, 4th, 0th, 8th, &c. figures is 'mvo or a number exactly divisible by 11. Examples. Test whether 729583024 and 457983021 are exactly divisible by 11. 4 + + 8 + 9 + 7 = 34; 2 + 3+5 + 2 = 12; 34-12 = 22=11x2; . •. 729583024 is exactly divisible by 11. 1 + + 8 + 7 + 4 = 20; 2 + 3 + 9+5 = 19; 20-19 = 7; .-. 457983021 is not exactly divisible by 11. There are no easily applied tests for exact divisibility oy 7 ftnd by l.'{, l)ut in the case of very large numbers the following may be ap])lied. X. P(nnt off the number into periods of three figures each, beginning on the right ; if the difterence between the sum of the 1st., 3rd., 5th., &c. periods and the sum of the 2nd., 4th., (>th., i^c. ])eriods is zero or is exactly divisible by 7, by 11, or by 13, the number is exactly divisible by 7, by 11, or by 13, as the case may be. Example. Test 0,570,353,093 for 7, 11, and 13 as factors. 093 + 570 - 353 -0 = 910 = 10 x7i< 13; . •. 7 and 13 are factors but 11 is not a factor. xi. Any number less than 1000 will be exactly divisible bj 7 if the sum of the ones figure, thrice the tens figure and twice the hundreds figure be exactly divisible by 7. f!^ i I ;] i'l 40 ARITHMETIC. Exiunylcs. Is 7 jv factor of 023 iiiul of ($85 '? 3 + H + 12 = 21-7 X 3, . •. 7 is H factor of 023. 5 + 24 + 12 = 41-7x5 + (), .-. 7 is not ji factor of f)8o. xii. If a number is exactly divisible by each of two numbers luime to each other, it is exactly divisible by their product ; and conversely, xiii. If a number is exactly divisible by the product of t\v<» numbers, it is exactly divisible by each of the luunbers. 71. Theorems xii and xiii follow inuuediately from Theorem XIX, § 62. The truth of the other theorems may bo shown as follows : — i. 2 is a measure of 10 ; . •. 2 is a measure of every multiple of 10 ; , •. 2 is a measure of the part of any number consisting of tlu; tens, hundreds, thousands, &,c.\ . •. in testing any number for exact divisibility by 2, the tens, hundreds, thousands. Arc. may be neglected as certainly nudtiples. ii. 4 is a measure of 100 ; .'. 4 is a measure of every multiple of 100; . •. 4 is a measure of the i)art of any number consisting of the hundreds, thousands, ten-thousands, tfcc. ; . •. in testing any number for exact divisibility by 4, the hundreds, thousands, ten-thousands, &c., may be neglected as certainly multiples. iii. 8 is a measure of 1000 ; . •. 8 a measure of every multiple of 1000 ; 8 is a measure of the part of any number consisting of the thousands, ten-thousands, hundred-thousands, »S:c. ; in testing any immber for exact divisibility by 8, the thousands, ten-thousands, hundred-tliousjinds, ♦fee. , may be neglected as certainly nmltiples. iv. The demonstration is similar to those for 2, 4 and 8. V. and vi. Both 3 and 9 are mea^iures of 9, 99, 999, 9999, *fec., that is, of 10-1, 100-1, 1000-1, 10000-1, &c. . •. if from any number there be deducted the ones and 1 from each 10, 1 from each 100, 1 from each 1000, &c., the remainders from iuJL MITI.TIPLICATION AND DIVISION. 41 iisisting of tlu! sisting of the isting of the ilu- tons, tl»»! lunitlivds, tlu) thoiisjvnds, Ac, constitute n iuunl)er which will be u iiiulti[>lo of t) iiiul therefore also of '.) and which may he neghjctetl in testing tho number for exact divisibility by cither J> or 3. There will then remain ti» bo tested the total of tilt' deductions; these were tho ones, tho iniiiilxr of tens, tho iiiniilicr t>i hundreds, the inotihir (^i thousands, Arc, and thereforo their total is sim])ly the sum <>f tho digits of tho given nund)er. J. U: Is 7H54() exactly divisible by either 1) or ',i '^ 7(H»(K) -7 times 10 fc b 8 r)(H)=5 u 100-5 u 1>1) ki a 40 = 4 n 10 = 4 (( «.) k b 4 (5-0 i' 1 = () (( ii (> Adding, 78546= a nudtiple of and 30 .">() is exactly divisible by 3 but not by 0, . ", 7854() is exactiy divisible by 3 l)ut not by 5). It should bo noticed that tho theorem really prtn'cd is :- The rcmainilcr in diviili'iKj amj iivmher h is llic tatiiif. . IM 'I ! Kviimfili . Is M7r»r)431» lixiiclly (liviHii)lo hy 11. H(MKMMM) 8 timort l(MMMMM)--8 times 01KMMM> iiii -7 1(MKKM) = 7 r>(KKM> 5 l(MMX) 5 (KMN) () l(H)0-=<} 4(H)^ 4 100 = 4 30- -3 10 = 3 »- = {) 1 = » Adtling, 87r)= a inultiplo of 11 nnd 2«5 less Hi 2(i U)SH KU^ 10 wliicli is not exactly divisible by 11 . •. 87r>04.')0 is nut exactly divisible by 11. Hero also it slioidd bo noticed that the Theorem really |)rf)ved is, The. irnKiiitilfi' hi, diridiinj any wnnlhr Inj 11 in (he satncoHtlir reituiliiifn' in (firUlnuj the )i\(mhvr olttdincd Jni .sh/>/ )•(»(■/ //(o Irsa tlmii i>, will In; tho riiii.iiiitUa' in dividing,' tho given nunihor l»y ; if it bo 5), tho ivimiindor will l»o /or«>. Exuwplc, I. Ciist tho nines out of 7.'Wr)«;S)42. 7 + .'? + H + r» -f <; -f 4 + 2 -- ^5, .'I + 5 --- 8, rniKtiiulr.i: Instead of julding all tho diyits-toj^etlufr iind cHHtiuj,' tho ninoH out of tho sum, tho nines may })0 cast out of tho ])artial nums as fast as they rise above 8. Adoptinji^ this nietiiod tlio proeodin;^ oxfimplo would ai»i>eai' 7 + ;5-10, (1+0-1), 1 + 8- !);r) + (5^= 11, (1 + 1-2), 2 + 4 + 2 = 8. Wording ; ten, oiif, nine, eleven, tint, six, eight. Example '.i. C;ist tho nines out of nr)87fM>Hr){)4. «, K;, (7), 14, (5), 11, (2), 10, (1), (J, 10, 1 vnmuiukr. The api)lications of tho operation of casting out tho nines dei>end iqiiiu two the»)renis; — A. 77h! .sioji of tiro humhcrH ha.t tin' .s«w«! retnaindii' to !> os tha M'/d, . B. Tilt' ]>roili>et of tiro inimhers /lax tlw, soma remahidci' to nn the jiroihii-t of their revi(ti)i(lcrK to t>. All numbers may be regarded as multiples of + their remainders to!). On adding or nudtiplying these numboi's, all the nuiltii)les • if S) will yield nudtiples of 9 and all those will disappear in casting out tho nines ; tlio result will therefore be the same as if tlie munbers had boon reduced from tho first to their remainders to lies at once 73. Proofs of Multiplication. INIultiplication may bo jiroved, (1.) l)y repeating tho calculation with nuiltiidier and nudtiplicand interchanged. (2.) By dividing the i)roduct by the nudtiplicand ; tho (quotient sliould be eiiual to the midtiplier. (.">. ) By casting tho nines « )ut < if tho nudtiplier and the nudtiplicand, then multiplying tho remainders together and casting tho nines out of their product ; the remainder thus obtained should be tho same as the remainder from casting the nines out of the product of multiplier and multiplicand. i| ! 44 ARITHMETIC. 4- In urrjinging tho sovoral reinaimlurs it is usual to write the reuiainder from tho uuiltiplicaiul ou tlio lel't-hand of an ol)U(iui! cross, tho rouiaindur from the multiplier on the right-hand of tho cross, the remainder from the product helow tlio cross and the remainder from the product of tlie remainders above the cross, Krdmple J. Aj)ply tho test of casting out the nines to 29()8457 X 748r)31()l) - 2221U84i:}41M )2i]3. ('.St c i; <>4. ) Mult lid. h^ Multr. 5x7 = 35-9x3 + 8. Product. Example :.\ Prove 5(5437 x 38529G7 = 217440808570 by casting out the nines. (iSee ^ 66.) ^' 74. Of these three proofs the second possesses the advantage of locating any errors that may be detected but it doubles the labor t)f calculation. The third proof is by far the easiest of a^jplication but it is subject to the serious disadvantage of not pointiiTg out an error of 9 or a nndtiple of 9 in the product. Thus if has been written for 9 or 9 for 0, if a partial product has been set down in the wrong jilace, if one or more noughts have been inserted or omitted in any of the products, if tAV(j ligures have been interchanged or, generally, if one tigiu-e set down is as nuich too great as another is too small, casting out the nines will fail to declare tho presence of error, for in each case the lemainder to will remain unaffected by the error. 75. Proofs of Division. Division may be proved, - (1.) By repeating the calculation Avith the integral part of the cpiotient for a divisor. (2.) By multiplying the divisor by the cv)mplete (piotient ; or, as it is generally stated, by nudtiplying th.e divisor by (the integral part of) the quotient and adding the remainder to the product ; the result should be e([Uiil to the dividend. (3.) By casting tho nines out of the divisor, the integral part of tho quotient and the 'remainder' in the division, nudtiplying the ■RBRS MULTIPLICATION" AND DIVISION. 45 '''^ hy castin-. first two of these reniaiiulcrs together and adding the third to their pi()(kict and casting the nines out of this sum ; the remainder to 9 thus obtained shouhl he the same as the remainder from castini' the nines out of tlie dividend. Example. Prove 38938G5378 -^ 1 71) - 2175:34 37 ] 5 ;} 1 )y casting ( )ut Ihe nines. (See ^ G8.) J>ivkor ^^ 2 Quut. 8 X 5 + 2 = 42 - 9 X 4 + 6. Dividend. The proof of division hy casting out the nines labors under disadvantages corresjumding to those to Avhioh the jiroof of iuulti])lieation hy casting out the nines is subject. EXERCISE I. MISCELLANEOUS PROBLEMS. l>;irt of Uie 1. By what number must 2000 be divided that the quotient and the ' remainder ' may be the same as the quotient and the ' remainder ' in the division of 101 by 11 ? 2. What number ccmtains 13*75 as often as 18"27 contains "0693 { 3. If a strip of car])et 27 in. wide and 50 yd. long make a roll weighing 1351b., what area could be covered by 4 T. of such carj)et ? 4. A rectangular block of granite measures 7' I'x 2' 4"x 1' 3" ; what must bo the length of another rectangular block 2' 1" x 1\ (i), if it is to weigh the same as the first block, (ii), if it is to have the same surface-area as the first lilock, (a) exclusive of end-surfaces, (6) inclusive of end-surfaces '{ 5. A boat's crew rowed a distance of 4 mi. 8(X) yd. in 3(5 min. 45 sec. What was the speed per hour ? What was the average time per mile ? 6. A man who owns l^ of a mill, sells ? of his share ; what fraction of the mill does he still own ? Had he sold ?? of the mill, what fraction of the mill would he still have owned 1 40 AHITHMETIC. 7. A grocer drew off 4 gal. from a full barrel of vinegar and filled the barrel up with w^ter. Next day he drew off 4 gal. of the mixture and then filled up the barrel with Avater. On the third day ho drew off 4 gal. of the mixture and filled up the barrel with water. If the barrel held just 32 gallons, how many gallons of the A'inegar originally contained in the barrel remained in it after the third drawing off? 8. (r can do as much work in 4 days as H can do in 5 days, or as much in 5 days as M can do in 9 days. The three undertake a contract and G and H work together on it for 18 days, then M takes (r's place and H and M work together on it for 26 days and thus finish the contract. How long would it have taken G Avorking all the time alone to have executed the contract ? 9. A grijcer buys two kinds of tea, one kind at 23ct. per lb. , the other kind at 35ct. per lb., and mixes them in the proportion of 51b. of the cheaper to 3 lb. of the dearer kind. At what price jier pound (an integral number of cents) must he sell the mixture to gain at laud 30% on the buying price ? 10. Find the hiterest c)n $794-35 for 188 days at 5%. r' till' 11. The product of 25 and 25 is 625. By how much must this product be increased to obtain the product of 26 and 25 ? By how much must the product of 26 and 25 be diminished to obtain the product of 26 and 24 'i Hence by how much must the prer hour, at what hour of the day v.ill he arrive at the end of his journey ? How much sooner will this be than would have been the hour of his arrival had he not quickened his pace 1 48 ARITHMETIC. 1! i|i; 25. Herbert's age is juafc § of M.iud's. Four years ago, his father, who is now 3G years old, Avas just 5^ times as old as Herbert tlien was. How old is Maud ? 26. A ina" ays out ^-i of his incr me for rent and |(\j for taxes. What fracti( f his income do these two sums form ? If the tw(t sinus amour >gether to $220-90, Avhat must be the amount of the uian's income i Oil. Trt'o blocks of exactly Am sauie size, the one of birch, the other of willow, weighed 4541b. and 241) '7 lb. respectively. The block of birch floated in water with only ^j- of its volume immersed. Ht)W nnich of the volume of the willow-block Avould bo immersed were the bh ick to float in water ? 28. ^1 i-i and C can do a piece of work in 10 days, all three working together. The three undertake the job and work on it for 4 days, then C leaves off work, but A and ]> continue and finish the piece of work in 10 da3^s. Ir A could have dime the whole Avork by himself in 30 days, in what time could B, and in what tiuie could C have done it ? 29. A tradesman sold ?j of a ceitain lot of goods at a loss of 10"', at what per cent, advanc^i on the cost must he sell tlie reuiainder of the lot in order to gain 20% on the whole ? 30. To what sum would ^87 "(58 amount in 97 days (^ (>i% interest '^ 31. Express the foUoAving distances in kilometres : — (i), From Montreal to Toronto, 333 miles ; (ii), from Toronto to Hamilton, 38*72 miles ; (iii), from Tt)ronto to Stratford, 88*34 miles; (iv), from Hamilton to London, 7o"90 miles; (v), from Stratford to London, 32*68 miles; (vi), froux Montreal to London via Hauiilt(m ; (vii), from Montreal to Loudon ri"(.K)6 ft. ^r^nmeter, if its |eiit;tli is (i) o in a 200 yd. race that they may run a dead heat ? 39. (((). vl's age is greater than ii's by 12 yr. which is 2b /^ of A'ti ugo. Determine JB's age. (/<). il-Ts ago which is 69 yr. is greater than y» age by 15/ of Xs a2"78 interest in 123 days ? 41. What number is the same part of 95*9 that 18*27 is of 29, and Avhat number is the same multiple of "119 that 57057 is of 42. The standard of fineness of British gold coins is -^ of alloy, and 480 oz. Troy of standard gold is coined into 1869 sovereigns equal in value to $4'86§ each. Find the value of (i) an oz. Troy, (ii) an oz. avoirdupois, of pure gold. 43. Find the area of the outer surface of a cylindrical stove-drum 16" in diameter and 24" in height, deducting two circles, the pi})e-holes, of 7 " diameter each. 44. The depth ol water in a rectangular cistern of 3' by 2' 9" horizontal section increases at the rate of 5' 4" in 12 min. What is tlie rate of inflow in gallons per minute ? 45. A horse trotted a mile in 2 min. 12 sec. Taking his stride ■'it 1(1 ft., how many times per second did his feet touch the ground ? 50 A11ITH,METIC. . I : -I 'I : /''!•! 40. Tho muiiicipftl ratos hoing reduced from 19|| mills to 171 mills on tho $1, my taxes are lowered liy .$4 'OS. For how much am I assessed ? 47. A boat's crow that can row at the rate of 2H4 yd. per min. in still water, rowcc miles down a stream in 10 min. Find tlie velocity of the st :.a. 48. Sold 19 yd. of silk @ $1'86 a yard, thus gaining the cost price of 12 yd. Find the cost price per yard. 49. A's age which is 49 yr. is less than i^'s age by 12|% of J is age, and jB's age is less than Cs age by 12i% of ("s age. What is Ca age ? 50. The interest on $270-25 for 93 days is $4-82 ; to what sum would $725 amount in 125 days at the same rate 1 51. The sum of two numbers is 1()() and one exceeds tho other by 28*62. What fraction is the small« r of the larger number ? 52. Find the area of a circular field enclosed by a ring fence 440 yd. long. 53. A circular pond 17' 6" in diameter and 5' deep is to be filled by means of a pipe which discharges 100 gal. per min. How long will it tak s to fill the pond ? 54. A train runs the first 120 miles of a trip of 280 rniles at a speed of 32 miles per hour. At what speed must the remainder of the trip be run, if the whole trip is to be accomplished in 8 hours ? 55. If during the day I pay out h, then I, next j\r, and lastly y^.- of the money I had in the morning, what fraction of it have I left ? If the sum left amounts to $1 '54 what sum had I at first ? 56. A train 220 ft. in length is running at the rate of 25 mi. per hour. How long will it take to pass another train 330 ft. long if the second train be (i), standing on a parallel track ; (ii), moving in the opposite direction at the rate of 15 mi. per hour ; (iii), moving in the same direction at the rate of 15 mi. per hour ? 57. A has $480 which is less than what B has by 20% of what B has, and the sum B has is greater than what (* has by 20% of what C has. What sum does G possess ? 58. A has more money than B by 10% of J5's money. By what per cent, of ^'s money is jB's money less than A'al tlio a coniii conn 61 EXERCISES. 51 aiiiing tlie cost ; t<) what sum 59. At wliafc rati! of interest winilcl $*J)7^'4r) iwiumut to $30(> in 1^45 (lay a ? 60. Tlio owner of n house offered an agent .i?5(X) connnission if tlio ai^'ont coukl sell the house for ^10,500. What rate i)er cent, coinmission was the own r offering? Had the owner offered 5% coiuiiiission, what would have heen the coniniission on $10,500? 61. xV sum of nioncy was divided between -I and B, A i-eceiving S'} for every $4 received hy B, and it was found that .1 had received $12 '(JO less than double of what B had received. How imich did each receive ? 6*2, The area oi Eurojio is 3,823,400 scj. lui. and its average tk\ation above the level of the sea is 974 ft. Find the vohnno in cubic miles of the portion of Europe above sea-level. 63. If a cubic foot of gold weigh 1208 lb. , what must be the thickness of a gold ribbon 1^ in. wide and 10 ft. long, weighing 4S() grains? (AvolrdupoLs iVeight.) 64. If the telegraph poles beside a certain railway are placed at intervals of 50 yd., at what speed is a train running which traverses two of these intervals in seven seconds? 65. In a certain subscription list ^ of the number of subscripticms are for $5 each, ^ are for $4 each, i are for $2 each, J are for $1 eacli, and the remaining subscriptions, amounting to $10"50, are for jOct. each. Find the whole numbjr of subscribers and the total amount of their subscriptions. QQ. A train 80 yd. long crossed a bridge 140 yd. long in 22| sec. Find the average speed of the train while crossing. 67. Find the gain per $100 on a cargo of raw-sugar boi-ght at $53 per ton of 2240 lb., refined at a cost of $1-35 per 100 lb. of refined sugar and sold at 6|ct. per lb. , if TSb. of raw sugar yields .") 11). of refined sugar. 68< A and B insure their houses against fire and A has to pay 87 "50 more than B who pays $28*75. Find the value of their houses, the rate of insurance being g %, and express the value of />"s house as a decimal of the value of ^'.s house. 69. In w^hat time would the interest on $182 '50 amount to $5 at 5X? ^ I no AlUTHMETIC. 70. A lujiii })(>ught 50 sliares in a c<>in[))iiiy fit $40 per Hlmro. Nuxt yciir (ho jnico was ^45 ])i!r hIuh'o but cficli year ihcroaftcr tlioro was a fall of $4 per share. Each year fr(»iii tho dato t>f his piirchaso ho sold out 10 shares and found at tho end of live years that including his dividends with tho amounts realized by tho sales of his shares ho had neither gained nor lost. What dividend per share did tho company pay ? 1 ir 71. Prove that if 12 bo added to the product of tho lirst 11 integers, 13 will bo a factor (oy.s each, giving the same number of marbles to each l)oy in a class. Among the boys in the first class he distributeil half the marbles; among those of the second class, .'. of them; among those of the third class, J of them ; and among those of the fi.urth class, the remaining marbles which aUowed the Itoysjust (11. 1' ai)iece. H(.w many did each boy in the other three classes receive and how many marbles wt^ro there altogether? 77. A train 54 yd. h>ng ruiniing at the rate of lil mi. per hr., jiassed another train 78 yd. long running on a pai-allel track ; the t\v(» trains completely clearing each other (i) in 6-4 sec. from the time of meeting, (ii) in 22 "5 sec. from the time of the former train overtaking the latter. Find the speed of the slower train ? 78. A house assessed at ^22CX) was rented for ^23 a month, the tenant to pay taxes and ws'ter-ratcs. The taxes were 174 niills on the }?i and the water-rates were $6 per (piarter year. How i.uich altogether did the tenant pay per year for the house. If the jjioperty had cost the landlord $250(), what rate percent. j)er year was he receiving on his investment ? 79. In what time would $143 amount to $150 at 7 % interest ? 80. The manufacturer of an article makes a profit of 25 %, the wholesale dealer makes a profit of 20 %, and the retail dealer makes a profit of 30 %. What is the cost to the manufacturer of an article that retails at $15 'OO. 81. Prove that if tiie number of integei's less than 9 and prime to it be nudtiplied by the number of integers less than 10 and prime to it, the pi'oduct will be the number of integers less than 144 ( = 16 X 9) and prime to it. 82. How many times must a man walk round a rectangular play-gr(jund 1G5 ft. by 132 ft. in order to travel 4| miles ? 83. How many cubic feet of air wdl a rectangular room 27' 8" X 18' 3" X 12' 4", contain, and how much will the air in the room weigh if a cubic foot of the air weigh 505 grains ? 84. A can run a mile in 4 min. 50 sec. , B cari run a mile in 5 min. 2'^) sec. If A give B 27 yd. start, in what distance will lie overtake him ? 54 ARITHMETIC. 85. Make out mid rocuii>t for K. Dewur it Son the fctllowiii'^' Hccount : - K. Duvvur it Son of Striitford sold to Edwin Roosor on ii'xl Sept., 1885, 28 lb. Furnaco Cement 0" 20 ct., 7 ft. of 12-in. Uot-uir Pil)e ("I 4() ct., 14 lengths of 8-in. Snutkc-pipo (<" 18 ct., 1 Chinniey Riui,', 25ct., 1 8-in. Ell»o\v, 50 ct., 2 8-in Rings for Lnwsou Regulator C"^,'J5ct., 21 1| -in. I{«)lts (^'>2ct., 8 2-in. Bolts (^ .'Jet., ir> 2^-in. Bolts (o> 4».t. A man and an assistant from K. Dewar iV Son's worked 49 liours cleaning and repairing K. Reesor's furnace, rate for the two together, 35 ct. per hour. E. Reesor paid !ii<15 on this account on the I3rd Oct. and the halunce on the 28th Nov.. 1885. 86. A man sold i of his farm, then ?. of the remainder, tluii ^ of what remained, then I of what still remained, and he then found that he hiul sold altogether 72 acres more than ho had remaining. How many acres had he at tirst ? 87. ( runs the lirst half mile at a speed of 3tK)yd. j)er min. and the second half mile at a sjjcciI of 340 yd. per min. Which wins the race and by how many yards ? 89. What principal woidd at 7% interest amount to 8450 in 213 days ? 90. An agent receives 87850 to be invested. What sum shouid he invest if he pay $12'30 ex})enses and charge 1 J"/ connuissmn on the amount of the investment i 91. Two wheels in gear with one another have 30 and 128 teeth respectively ; 1k)w many revolutions will the smaller wheel make while the larger revolves G75 tinu^s ? If two marked teeth, one on each wheel, are in contact at a certain moment, how many revoluticms will each wheel make before the same two teeth are in contact again 1 r:XERCISES. 55 92. Find t'lio volume in cu. in. of 101b. of (n) load, (h) cast- iroii, (<') iimrblo, (d) brickwork/*'^ oak, (f) l)irch, if a ciiliic fool of load vvx'igh 712 1b., of cast-iron 444 11)., of niarblo 172 ll>., of l»iickwork 112 11)., of oak 54 lb., and of ])irch 444 lb. 93. Taking thu woiglit of a cubic foot of water to bo 91)7 "7 <»/•., what weight of water would lill a rectangular bath 36' 0" by 13' 3" l.y .V 7 A"? 94. Make out an invoice of the following, suj)plying names and dates : — .1. B. bought of C. D. 15 doz. First Readers Pt. I (Jh $(1-20, 18 do/. First Readers Pt. II @ $180, 27 doz. Second Readers (<'^.ii<38e values ap})roach so near the value of any given fraction that it is imi)08sible to insert between the given fraction and any of the fractions found another fraction intermediate in value but with terms less than those of the fractions found. The fractions which fullil this condition are teimed Convergents to the given fraction. 77. The following examples exhibit the simplest method of computing the com ergents to a given fraction. Example 1. Find the convergents to ^*A. They are 111 BO- g'j .-jr. .48 1' 2' 3» 1«> 5i»» 7 1' IIM' l.">5' i' 7' 1U' 1^' 4i!' T9r» ;J5-.J' **'^* The method of calculation is as follows : — ■ Write J and below it 7 as initials. From these initials treated as if they vrere fractions form another fraction with the sum of their numerators as its numerator and the 0+1 1 sum of their denominators as its denominator; fXo" 1 • This newly formed fraction, \, being greater than the given fraction j'g^, write it in the upper line, the line of J. 1 I' M^jy-p* f m . ! - jJ^^lJtJ CONVEHGENT FKACTIONS. 57 From I and ^ form u fraction with tho Bum of their numerators HH its numerator and the Hum of tlieir denontinatoi-H an its denominator ; ,-£-, =17 in the upper line. This k being greater than ,*A, write it 2 1 , « , , - . 1+0 1 *rom - and , form tho fraction rr7-r=—. i > |^,^,» . '. write J in the upper lino, 10 , . ,. , . 1 + } From --and \,- form the intermediate fraction ., , . = -, ti 1 '^~r* ♦ i < iV:,) • "• write | in the h»wer lino. From and form the intermediate fraction :.",.■ 4 t> 4"i '' 7 '' 1^^-.' • "• write s in the lower line. „ 2 1 . . . -'+1 ."> From -^ and -. - form the intermediate fraction ;=~T~., = Trk» 4 o i -}-•> 10 j''„ < j'^A, . •. write j'\, in the lower line. .31"'. . ^ . .'i+ 1 4 From :,,, and-,, form the intermediate fraction ... , ., —.„. 10 «> 10+ i> IJ u; ^~ A^-.» • '• write ,*., in tho lower line. ^41 . 4-fl r. From :,., and ., form the intermediate fraction 777-;-.,= ,,.. lo o lo-f-t> lb 5 I'ff ^ A*^-.' • "• write /n in tho upi^er line. 4 ." . . .. . . 5 + 4 9 From :, . and - , form tho intermodiate fraction TT^'ii. >-»«,• ■jiu ^ A*:.' • "• write .VIJ in tho upjjor line. 9 "^ 4 " . " . . 9 + 4 13 From -^ and i,- form the intermediate fraction rt' -]3 < Afr,, . ■. write ]i! in tho lower line. ^^ ,9 ' , . " . , . 13+0 22 From ^-j and ^^ form the intermodiate fraction JolLQ'i""!" i\ >-iV5) •"• write ='■[ in the upper line. Continuing this process wo arrive at length at the given fraction {*A which may he placed in eiihtr line. If tho calculation l»e continued beyond this, the succeeding convergonts will be all less or all greater than tho given fraction accoiding as the latter was 58 AHITHMKTIO. writton in lowor lino. In tlui preocdinjj; list, wo hjvvo i)ln,co obtain 1 1 ] 1 I 1 ;i II 0' I' J' I 1 ■p •>' I (^ ' ) 1 ' r. ^ 1 1 'J 7 ' I .■!' ."• !l> Conipntation. o+i 1 i-fo 1 1+0 1 1+0 1 140 1 l-rO~l' 1+1 L> 1+0 1 1+1 2 5+1"" (>• (5+ 5 "Ml' 2+1 a 11 +r." n + 2 -*+i ;{ a+i •i+i 10 o 5 + a H + n H)+ii 27" 27+1(5 4;r 4;!+n>" J")!r 11+ a 14 14+11 25 55)+ 1(5' 75" 75+50~ia4- Tho oonvorgents following y-.f, aro oniittod. Horo tho Principal Ctnivorgonts aro V, J,, IP I a ant 1 I Kx rawpic ;> h Find convorgonts to Conijniting as boforo wo obtain 4 7 10 p -J' ;! ' 4:1 4n 1 p p l:t .•I ;t;( ' ' I 0' ^imputation. 0+1 1 1+12 1+0" J • J+0 1 2+1 _ a i+o" 1 (^ONVKHOKNT KIIACTIONS. 59 noxt. two .'{^_i._4 4+:{ 7 7+.'i 10 1+0 1 ■ 1-f V 2' 2+1" a' 10+;} i;{ i:{+i() 2a 2:{+io Ha aa+10 43 a + 1 4 • 4 + a 7 ■ 7 + a "" lo" io+'a " la' Tlio Priiicipul (^uiivcirf^ontH are '^ iind '.j*. 79. From ilu! iiK!t.li^- K** > T'^- ■i%>-\> 4 8, >aii_-4H v.g 1 nuH 2 155-^3 ifio-^iff 155'^ ait isrt^ri ir.r.'^^^M and 1 ,-. 5 1 4 4h:. 1 .-. -. .4H. 1 r. 5 > •IH.. 10 -^ir.r, ■^>^^.-]h S:c. 80. From these two fiuidamental laws, four otliers follow aa immediate con8e 4- but <'/:5^., Avith terms less than those rincii>(d rimvcnii'iit to it is less than the difi'erence between the given fraction and any fraction with terms smaller than those of the principal convergent. 81. The Corolhiry to the Second Law applies to ])rincipal convergents only and distinguishes them from internuuliate i^ 60 ARITHMETIC. convergents, it being iiftted that the principal convergents to any given fraction are alternately greater and less than the given fraction. 82. The Fourth Law holds for principal convergents liut does not necessarily hold for intermediate convergents. Cases occur in which a convergent in one line differs less from the given fraction than does a succeeding and therefore larger-termed intermediate fraction in ^/(t; oZ/iw line. Thus, in Example. 1 page 50, r^^, -j> i - ^, so als<> iVk - ? > i - m ^ndt^ - m>^h-h' Hence b.)th ^ and » are inferior to \ and ^\ is inferior to j*rr, if we consider th(3se fractions solely as apj^roximations to -^X, regardless of whether they are approximations in excess or in defect. This Fourth Law, therefore, marks out the Principal Convergents to any large-termed fraction as, in general, the best small-termed substitutes for such large-termed fraction, in approximate calculations. It is consequently important to have an ex])editious method of calculating the principal convergents to any given fraction. Such a method is exhibited in the following exami)les. Example 1. Find the principal convergents to -^^. A. Divide both terms of the given fraction by the numerator. 48 155 Now3<34-fi, 1 —L 155 -r 48 3 + }| 1 48 '•"•'3 "155- B. Divide both terms of J J l)y the numerator. Now 4 • 1 11 48^ 4 + A, 1 '4+t^' 1 48^11 1 3 + i < 3 -f- 4+ A 4+iV 1. e. 48_ i55' 3 1- 4+ A (i) 3 + ] 48 rr)5' (ii) CONVERGENT FRACTIONS. 61 C. Divide both teiina of /^ uy the numerator. 11 ll-^4~2 + |' 1 3 + - Now 2<2+|, '^ + 1 TT >::r- 2 2 + 1' 4 + . 2 + 1 > — 3 + 4 + ^ 3 + - 48 I. e > Tirz: 4+-. 1- 3 + 2 + 1 155' 4+; D. Divide both terms of -| by tlie numerator. 3 1 1^ 4 ~4+3~r+ .1' Now l: il I 6': I !! '■ Wo thus find that ;^^^.- is less than ., but yrroatcr than 3+ n is less than - - 1 )ut iireater tluui (i) .t (ii) (iii^ & (iv) 3+ ;J- 4+i t/ ■'" — •4+ 2+{ and IS cqu id to (v) :j+- 4+- 2+- i-t-I 83. That these fractions are the principal oonvergents to ^'.^ may be shown thus ; — Reducing all to simple fractions they are /., /.j, -J-',, jv!, /A. 1°. -i-!L — Lii = __L_ 135 ll! 15 5X41;* . ■. the terms of all fractions < /A but>]i; are greater thaii tiie terms of y^A ; i^/s is the principal convergent to itself, . '. Yi i^ ^^^^ principal convergent next preceding ^rf.^. 2°. " JL > -11. > A« and Jl - li.' =__1_., 2 « 1 5 5 4 •«> i; I) 4 1.' li 9 X 4 1: . '. the terms of all fractions > j^'^- but< J', are greater than the terms of \^ ; Y'i is a principal convergent to y*A, . '. oHy is the prinei])al convergent next preceding \-L 3". Similarly it may be proved that j'^r and \ are tlie other principal convergents to -^A . 84. The operations A, B, C and D may be summarized as follows; — Divide 155 by 48; divide 48 by 11, the remainder in the preceding division ; divide 11, the first remainder, by 4, the second remainder ; divide 4, the" second remainder, l)y li, the third remainder ; divide 3, the third remainder, by 1, the fourth remainder. Now this is nothing else than the series of operations for finding the G. C. M. of the two numbers 48 and 155. Arranging; the u m CONVERGENT FllACHONS. 63 (v) IS work iis in the rubllc School Arithmetic, pa^o 100, it appears thus ; — 3 4 2 1 .3 Quotients. 155 144 48 44 11 8 4 3 3 3 1 11 4 3 1 1 The couvcrgenta may now 1)0 written down from the line of (quotients, thus ;— 3, 4, 2, 1, . 3. , 1 f 1 1 3+1 1 ' 3+ 4+i 5+- 3+- 4+ '-^+1 4+ 2+ 1 l+,l The simple fractious eciuivalent to tlicse convergeuts may be calculated by the ordinary method of reducing complex fractions to simple forms, or otherwise thus ; — Quotients. '^^Icidation. Convergents. i: . , . . (Initial. ) (i) (ii) (iii) (iv) (v) » 1 . 3 1+0x3 1 0+ 1x3" 3' 4 0+ 1x4 4 1+ 3x4""l»* 2 1+ 4x2 9 3 + 13x2 29' 1 4+ 9x1 13 13 + 29x1 42' 3 9 + 13x3- 48 29 + 42x3 155 85. Limits for the errors arising from the substitution of ^, -^, &c. for T^ may be obtained as follows ; — (il AUITHMKTK!. Hi is I ft:. I.T .1 1ft ft i:» ;iy i;ii i, c. llui orror arisiii;' fi-nm tlio ust) 4 .'■ iov ,'A is Ichs than 4 . I H_ i,'i " f ft ft ■IM ;ix ; ,1 •J 11 i:i i;» i.ix'Jit /. « . tlio oiTor nrisiiig from lu' luf -j'.j f'*!' /A is less !li;iii Siiiiiljuly it uuiv hn shown (Iiaf^ — ^ — is ji siincii«i!' limit »'i' i;rrur ! :l\'Jjr in tlu> snl)stihition of ,}\ for ,■*. •-' (' X I 1 1ft; - ' is the error in the suhstitntion of ]■}, f 4 .: \ 1 ft ft I 'J or Examplr J. Find the prinoipjil eonver«j;ents to f^'.\ 17H 2 ;{ .i;{ :{,'{r»!>;{ 2:{Ji.- iom;> 2;{478 202;{0 51744 ;{7i L'HO 280 27:{ 5>i |7 101 15 1 .{248 1 .'{71 28f* !<] ' 7 (^^uotients, 1, •> [\ 8, 1, Converyeisls, j\, ',', ',, ','■ ' I'oi ' s ,s;i' (1 .'i 1* '.\ * Limits of en 'r, . ' , l\;i . I 1 o\,s;i' 1 _ >s;i ^ '.t;i (J. C^ M.oJ" teriMs. •: ft a iii'jv t T'.t» .'l.'ift.l ■J ri'.f B6. Tf the sj;i\ rn fraction he improper, reiluee it to a mixi d nuriher and nse iiu» integral jvirt of tlie mixed numher na nmu- citor in plaee of l) in tlie initial */. F,ra,i:)^h' .>'. Find jv series of eonvergents to .'{•14155)2<)5 wliieh is iV]>i)roxini;ilely the ratio of the eireumferenee of a circle to its diameter, ,, ., approxim.ately the measure of the circumference in terms oi tho 0 .•{055 5)5)114855 885145 88205K) HllO 885145 5a0781«) 442.5725 882090 :{055 27105) 24440 2(>(>5)0 24440 Quotients, Convei'gents, J, 15. 1. > T Uy m- Hence the circumference of a circle is longer than o tliamctera of the circle, is shorter th an diameters, is longer than diameters and again is shorter than '^f s' diameters. The lin respecti limit>s t)f error are 1, ively. •M0t5 1 fl X 1 1 s and 1 1 iax;iv>(iftO "■*''■" '."' "7";:-^r:*"--''- --"y.' CONVEROENT FRACTIONS. 05 i:j. 'U :\ J t! ■> 87 From this oxamplo ifc is ovidonfc that those rtmrrrffenis irhich hmmiiatdy precede large qiMtknts are the heat approximathms to V nploy as mbstitntesfor exact valnrs. Example 4. Find u sorius of convorgcnfc comparisDiis of tho a ,,ro = 39-370432 in. and tho yard = 3« in. The quotients of 30/39-370432 aro 1, 10, 1, 2, 7, 2, 1, 5 ; hihI the corresponding convorgonts, omitting initials, are „ 1, H, \h 'ih nh ni iU, mi Hence 10 m. < 11 yd. but ] 1 m. > 12 yd. ; 32 m. <35 yd. but 235 ni. >257 yd. ; &c. EXERCISE II. Find tho principal convorgonts to ; — 1. a If «. a. VA- 7. s. Ul N. /i. If?- 0. 5. VVH^ 10. ,1^.050 1 () ;j It • 11. 1-4142. 16. •0498756. 11 1 ;t 1 12. 1-73205. ly. •2439. 13. 2-44940. i§. 1-41844. iVWm- 14. •43589. 19. 2-71828. mm- 15. -55744. 30. 2-302585. Find a series of convergent comparisons of : — ai. The kilometre = 1093 •0)2311 yd. and tlio mile = 17r)0 yd. 33. The hectare and tho acre. 23. The kilogramme and tho pound. 34. The millier and the ton. 3 5. The kilolitre and the cubic yard. 36. The Iftre and tho quart. 37. The Canadian standard metre ^39-382 in. and the French standard metre* 39 -37043 in. 3§. The earth's polar diameter = 41708954 ft. and its longest equatorial diameter =41863258 ft. 39. The tenacity of stool and tho tenacity of copper wire the former being ^JJ times the latter. 30. The excess of the mean solar year of 365 da. 5 hr. 48 m. 47 "46 sec. over the ordinary civil year of 365 da. , and one day. Hence show that if there were 8 leap-years in every 33 years, this system would not be wrong by so much as 1 day in 4224 years, and compare this with the Gregorian system of 97 leap years in every 400 years. t I i*;- •■ b.;=3;-'- 4 <■«»• GO ARITHMETIC. APPROXIMATE CALCULATIONS. !' »■ ► ■■ * - I- ■ 88. Tho greater part of the labor of computation in calculations in which fractions occur arises in general from the several fractions having ditieront denominators. For example, if two or more fractions are to bo added together, they nmst all bo brought to the same denominator, if one fraction is to be divided by one or more others all t)f different denominators, the terms of the quotient are in most cases much larger than the terms of the dividend. The labor of computation may be lessened by using convergents instead of exact values ; it may often bo lessened and the calculations may always bo simplified by replacing the fractions by approximately e([ual decimal numbers. If wo adopt either of these ways of lessoning the labor of ctMnputation, we deliberately incur an error in calculation which we know will give a result sufficiently near tho truth for all practical purposes. 89. In calculations concerning quantities which presuppose measurements, it should bo remembered that these measurements cannot bo made with absolute accuracy. In the measurements of evory-ilay life we are satisfied if wo do not err by mt)re than one part in a thousand ; in the most careful scientific work it is rarely jiossible to reduce the error below one part in a million. The results of calculations based on such measurements are necessarily att'ected by the errors of measurement and it is therefore a mere waste of time and labor to carry any calculation beyond the degree of accuracy with which measurements can be made. It is moreover misleading, for the results then present an appearance of exactness where exactness does not and cannot exist. 90. The first significant figure in any number is tho first digit, — the first figure other than zero, — on the left of the number. Exainples. In 980*61 min., is the first significant figure and in •000122 da., 1 is the first significant figure. 91. A number is said to be correct to two, three, foiu', significant ligures if it doos not f.S. 67 degree ami in [t \V(»uUl fourth, Example 1. If it ia said tlmb the Uuigth of a cerbiinline is '.id in. correct to /»v> Higniticaut figures, it in meant that the actual length is between Ii'85 in. and ii'Oo in. If the lengtli is given as 3"f)4 in. correct to Ihrcc significant figures, it is meant that tlie actual length lies lietween ,'J*935 in. and a -945 in. If the length is said to 1 ,; 3*037 in. correct to ftnir significant figures, the actual length may be any between 3"y3(>5 in. and 3 -9375 in. Examples. If the lengtli of the greatest equatorial diameter of the earth bo given as 41,852,0(H) ft. and tlie length of the polar diameter as 41,710,000 ft. , correct in l>oth cases to j^ re significant figures, it is meant that the actual length of that particular equatorial diameter is not less than 41,851,500 ft. but is less than 41,852,500 ft., and that the actual length of the polar diameter is n(»t less than 41,709,600 ft. but is less than 41,710,500 ft. 92. The degree of any approximation is measured by the fraction which the totjil error ia of the exact value, i. e., by the ughly comiwirod by comparing together the significant figures known to bo correct in each case. Thus if tho first three significant figures are known to bo correct the a})proximation is about ten times as close as it would bo if only the first two were known to be correct. " Correct to six significoiitj h- 1^:^ 68 AtllTHMETtC. figures " incnns im approximation .ihout KXK) times as closo as that of *' correct to throe sigiiificant figures." 94. In expressing mixed numbers and fractions by approximately equal decimal numbers, it is in general sufficient if the calculations are correct to four or at most to seven significant figures. Beytnul seven fujiires we very seldom need go. So also if one approximate number is to bo multiplied by another or to bo divided by another, the result need not be calculated to a greater number of significjvnt figures than are cr)rrect in the given numbers. Example 1. Find the product of ()78 -233 multiplied by 47 0583 correct to six significant figures. Uncontracted Form. 678-233 1 47-0583 27120-32 4747 -(53 1 ()10-40 07 33 01 165 5.42 5864 •20 34609 Contracted Form. Qn-'m 47-9583 27120-32 4747-63 . . (a). 610-41 . . ('') 33-91 . . ('•) 5-42 . . {d) •20 . . (e). 32526-9 32526-90 I 16839 We begin by multiplying by 4, the first significant figure in the multiplier. The product contains 7 significant figures ; this is one more than the number required to be correct, but we retain all seven that we may determine the * carriage ' to the sixth significant figure when adding together the partial products. We contract the subsequently calculated partial products thus ; — (a). Strike the right hand 3 from the multiplicand and multiply by 7, carrying 2 from the 3x7 struck out. (6). Strike the second 3 from the already contracted multiplicand, and multii>ly by 9 carrying 3 from the 3x0 struck out. (c). Strike 2 from the multiplicand as contracted in Q)) and multiply by 5 carrying 1 from 2x5 struck out. {d). Strike 8 from the multiplicand as contracted in {c) and multiply by 8 carrying 6 from the 8x8 struck out. (e). Strike 7 from the multiplicand as contracted in {d) and multiply by 3 carrying 2 from the 7 x 3 Btruck out. §li /■• APPROXIMATE CALCULATIONS. 69 Tho sum <»f tho right-haiul liguroa <»f tho purtial products is 9. This would bo tho sovouth signiticHut liguro o( tho product, ])ut a& tho product is to be correct to only six Biguificimt figures, wo change 9 t«) tho nearest multiple of 10 which in this case is 10 itself. Wo n(jw complete tho jvddition of tho partial products as in tho ordinaiy uncontracted form. The approximation in lino (' ...^ Aw'^ . ■■i^.'"l/"-^..'■,*^^■■4 ■>.■■„ 70 ■I - l! ■i" -■■ AUITHMETIC. Uiicuiitnwtod Form. Col itnictcd F(inii. 47-i»58:j (57823. 47 0583 (5782:j:{),'{252(;h 27121)32 J>3252(51)OM 27121)32 531)758 4747(53 1 531)768 4747(53 .... (a). (}4!M>4 O-O m04ofc)ro I ■()83i) douotoH tlmt tlio (|U(»tiont 47 '0583 is too (♦')» ('0 '"ih /, piigo 08. Computers' Contracted Form. 3252(51K)!»- 539758 04996 3954 563 21 1 47-9583 Examphi Jf. Find tho weight (in Imperial tons of 22401b. each) of tho carbon in tho carbonic acid gas in tho atmosphere resting on a s "5 " " ^ 9' 5 " " = 5' Tho accented tigures are those of 595 ■()(). (See §09, Cme xii.) 06. In tho j)roceding calculation, the solo influence of the 27 cents in the principal is the addition to the annKal interest, of the 2 cents 'carried' inline (a). Even this small increment disappears from the interest for 93 days, $1'51 being practically the interest on 179 for 93 days at 7.^%. The omission from the principal or the addition to it of any nundier of cents less than 50, will not in general change by more than one cent the computed amount of the interest for a short-term loan, but the retention of the cents in the calculation will consideral)ly increase the labor of computation. For this reason, business men compute »m tho nearest number of dollars, when reckoning slu)rt-term interest and when determining tho eijuuted time of an account. (See Public- School Arithmetic, p. 1G8.) EXERCISE III. 1. Find the sum of 143-035472, 29 680037, 089173, 4-99870 and 2923-937958, correct to 4 decimal places. 2. Find the valueof379-2805()+29-68043+r)-8409207 -44 -398042 - 3 ■7984(mi+ -2308592 - 300-790797, correct to significant figures. 3. Find the product of 478*593 and 3-14159 correct to 3 dechual places. 4. Find the value of 427*803 x -00749 correct to 5 decimal places. 6. Find the value of 3-1410 x 3-1410 x 3-1410 to the nearest i nteger. 6. Find the i)roduct of 2 9957323 and -4342946 correct to decimal places. 7. Fhid tho value of 5-70.37825 x 4342945 correct to decimal places. 8. Find the value of 3141592()5 x m x 995 x 9998 x 99992 X -999997, correct to (J decimal places. ■■4- <.- ■,('' places, loarest lecimal lecinial EXERCISES. 73 9. Find tho valuo of 2-7182818 x -8 x -992 x -9993 x 99998 X -999993 X 9999994 correct to 7 deciraal places. 10. Find the valuo of 2 3025851 x '9 x 97 x -996 x 99995 X 999997 ctn-rect to 7 decimal i)laces. 11. Find the product of 1 -(XK^m x 1 -004 and 99898 x 99898 correct to 7 decimal places. i2. Find the valuo of 1(KHX)127 x 999987, correct to 8 significant figures. 13. Find the value of 16-934x16-934x10-934 correct to 5 significant figures. 14. Find the value of 4 8784x4-8784x4-8784 correct to 5 significant figures. 15. Find the valuo of 9-0708324 x 9 0708324 x 9-0708324 X 9;0708324, correct to (> significant figures. 16. Find the value of 2 0188223x2-0188223x2-0188223 X 2-0188223 x 2-0188223 x 2*0188223 correct to 6 significant figures. Find tho values of the following (quotients, correct to 6 significant figures : — 17. 1(K)^ 1-414214. 18. 25000^3-141593. 19. 07-2-64575. 20. 1 -95 -T- 139 -6424. 23. 1-^3-14159265. 24. 1^ 43429448. 25. 11 -f- 2 -22398 ^2 -22398. 26. 4517 -^ 16 -5304 -7- 16 -5304. 21. -6931472-^2-302585. 27. 19 5 -=-2 236068 ^6 -244998. 22. 1 -098612 -^ 2 -302585. Find the values of the following, t<» 5 significant figures : — Oft 1 4--^ 4- — - -i 1 ^ I ^ I- *&c. ^tJ. -r ^ ^1x2^1x2x3^1x2x3x4^1x2x3x4x5^ ,1,1 1,1 1 , „ OQ 1 — h&C. ^^- 1^1x2 1x2x3^1x2x3x4 1x2x3x4x5^ 11 i_ I \ 1 , ^^- 1 "'"l X Trf X 3 x 5"^1 X 3 x 5 X 7 i x 3"x 5 x 7 x SV ^'^' 1111111 31. -J + 2 +-4 + H ■^'l6+32"^64+ *^'^- 1 Jl^ I ^4-1 _l_l4.1-L 32. 1 + 1?-^ 9 "^27~'"Hr 243"*" *^- 11111 ^ 1 "^ 5""^25"^125"^625"^ *^'^- i . J 'I . i 74 AUJTHMETIC. 34. Prove lli.it. tho Huawci- (,<» i)r(>l)h!in 21) in (liu rociprocjil to 5 .sii^iiiticiuit lit^iiri's of tlio jinswor to ]))'ol)l(Mn 28. 35. KxprosH yd. in iiiutrcis correct to 4 Hignificaut. li»j;iiros. 36. H]x[)ri!H.s J7<><) metres in ytirds correct to 4 Higniliciint ligures. 37. ExjjresH 4840 8(|. yd. in centiares correct to 4 aignilicaiifc tigiires. 38. Express 4840 centiaros in sq. yd. correct to 4 sij^nificant figures. 39. Express 1(H) acres ii) hcctai'es correct to 4 significant figures. 40. M.qtresa KM) hectares in acres correct to 4 sigriificant ligures. 41. Express (JOO litres in gallons cori'cct to 4 significant figures. 42. Express ]ii2 gallons in litres correct to 4 significant ligures. 43. Tlie mean distance of the moon from the earth is '.'38800 miles ; express this in kilonu>tres to 4 significant figures. 44. The mean distance of the sun from the earth is 91,430,(KH) miles ; express this in kilometres to 4 significant figures. 45. The mean distance of Saturn from the sun is 872,140,000 miles, and of the earth from the sun {)l,430,(K)O m'''es; form a series of convei'gent com])arisons of thesis distances. 46. Form a series of c(mvergeut comparisons of 34()"019 da. and 25)"r)30(5 da., and hence show that 19 times the former period is nearly ecpial to 22.'> times the latter. Express these products in terms of a year of 305 •25 days. 47. Taking the length of the sidereal year as 3fi5 •25630 days and that t)f the liniar month as 29 •53059 days find a series of convergent c(miparisons of the lunar month and the sidereal year. 48. Mars revolves ahout the sun in 08(5 "9797 days and the earth revolves about the sini in 3f)5^25()4 days ; find a series of convergent ccmiparisons of the length of the Martian year with that of the earth. 49. -hipiter rot.ites on its axis once every 9 hr. 55 min. 20 sec, and the earth once every 23 hr. 50 min. 4 sec. ; luid a series of convergent comparisons of these thues of rotation. 50. Mercury revolves about the sun in 87*9093 da. at a mean distance oi 35,392,000 miles ; and the earth revolves about the sun in 305^2504 da. at a mean distance of 91,430,000 miles. Find c(mvergent comparisons of the speed of Mercury and the earth in tlieir orbits. oil; $: the n with the ai 9. )§225 all fro earth iirth. ace, 103 of luean sun Find th in MISCELLANEOUS PROULEMS. 75 EXERCISE IV. MISCELLANEOUS PROBLEMP.. 1 . The mercury in a barometer rose •121 in. , 'O/Ii in. and "019 in. in three successive days, it fell •(>54 in. and '065 in. durin«^ the two following days, rose "OSS in. on the sixth day and foil "028 in. 50 kiL )gr.vmmes of silvor ' tTe extracted. What percentage of the ore was silver ? Fxpress the weight of the ore in Imperial tons and the weight of the silver in Troy ounces, and employing these t!xprossions of the weights, recalculate the percentage which the silvor constitutes of the ore. 8. A. B. bought goods amounting to $74(50 subject to 25 and 5 off, $3730 subject to 30 off and $1492 subject to 20 and 10 oil', find the net cost of the goods. Were the invoice-clerk to bill A. B. with goods amounting to $12(182 subject to 30 off, what would be the amount of the error in the net cost of the goods ? 9. Find the equated time of payment of a bill for $748 of which $225 is at 30 days, $245 is at 150 days and the balance is at 90 days all from 3lBt Aug. 1889. ii ,i d 76 AlUTHMETIC. 10. Tho procouds (..f ;i, draft for $«>28-<)(> drawn at DO days, aniountud to $G15*7*J. What waa the rate of discount ? 1^ It it 11. A train is duo at a ccrt'iin sfaition at 42 min. past 2 p.m. The actual times of its arrival at tho station for a ccrtrin week were : — Monday, 2,38 p.m. ; Tuesday, 2,47 p.m. ; Wednesday, JJ,07 p.m. ; Thursday, 2,o9p.m. ; I'riday, 2,42 p.m. ; Saturday, Ji,ll p.m. By how many minutes on an average was the train late that'week, (i) not counting * minutes ahead of time,' (ii) including ' Minutes ahead of time ' in the averaging 'i 12. How often is the circumference of a circle 1' 9" radius con- tained in the diameter of a circle whose circumference is 100 foot? 13. What will be the weight of a rectangular sheet of glass 0' 3i" long by 4' 4i" wide and y'^y in. thick, the glass weighing 103 lb. per cubic foot ? 14. How many days were there from 13th Nov. 1887 to Otli ine 1888 ? Express the interval from noon on the former day t(» noon on the latter day as a fraction of the year 1887 and also as a fraction of tlie year 1888. 15. A watch is set right on Monday at 9,15 a.m. and it gainii 'S^ sec. per hour. On what day and at what hour will it have gained exactly 5 min. and what time will it then indicate 'i What will be tho correct time when the watch indicates 9,15 on the following Monday morning 'i 16. Out of a certain sum of money one-half was spent, then one-third of the remainder, next one-twelfth of what still remained and lastly one-fifteenth of what then remained, leaving 39ct. less than one-half of what was spent. What was the original sum ? 1 7. A man buys milk at 5ct, a quart and having mixed it with water, sells the nn'xture at Get. a (piart. His profits are ecjual to 40% of the c»)st of the milk. How much water is mixed with each quart of milk ? What proportion of the mixture is water 1 18. If an investment of $748:V50 yield a net profit of $483-67, > investment ? If per pr( "y this profit is reinvested along with tlie original iiivv^stmunt, and the whole yield a d profit at the same rate per cent, as the first. second whiit will be tho amount of this second profit ? m-] MISCELLANEOUS PROBLEMS. 77 .. ' X 3' ()" X §"] if a cubic foot of the slate weigh 178 lb. 24. In a certain gold mine, 11 tons of f)ve yielded 7;^ oz. (Troy) of pure gold, what fraction of the ore was gold ? Express the j)roportion of gold to t)re in grammes per metric ton. 25. A clock which gains 9 sec. per 1 hr. 11 min., is set right at 10 a.m. on 1st March, when wi'l it denote correct time again ? 26. After drav^'ing off 15 gal. of the contents of a certain cask and then j\ of what was left, the remainder sold at 5| ct. a pint brought $3*96. How many gallons were there originally in the cask ? 27. A mixture of cofl'ee and chicory in the proi)ortif)n of 8 parts ofc(»tfee to 1 part of chicory is sold at 35 ct. a pound, being an advance of 40% on the cost. The chicory cost 9 ct. a jiound, find the cost of the coflee per pound. 28. A man bought a house and lot for $4750. After spending $1143 on repairs and improvements and paying $128 for taxes and other expenses, he sold the property for $0800. What rate per cent, of profit did his investment yield him ? i! / fhi :,! -> *■■; r. V V ill 15 ?Hli 78 ARITHMETIC. 29. On 18th Juno 1888, a merchant purchased goods amounting per catalogue i)rices m $047*80, subject to 25 and 5 off. lie was allowed 3 months credit after which he was charged interest at 8%. Find the amount of the account on 21st? February 1889. 30. Find tlie difference between the discount taken off a draft for $500 drawn at 90 days and discounted at 7% and the interest on the proceeds for 93 days at 7%. Find the interest for 93 days at 7% on the amount of the discount taken off the draft. 31. A man skated 10 miles in 36 min. 37"? sec. ; what v;,i,s his speed Ja .^ ; ds per minute, in miles per hour, in metres per min., in ki"i"iiuetrt-3 per hour ? 32 '< riircular race-track 24 ft. wide encloses a circle of 50 y working days of 10 hours each ? Had the time of making a revolu- tion btien increased by Jl^j of itself, how many rtnntlutions would the wheel have made in 6 days of lO hours each '( 37. A grocer buys SO lb. of tea ;'.t 21 ct. a lb. auvi mixes it with some dearer tea Ik; has on hand. Selling the mixture for $43-75, this being at the rate of 35ct. a lb., he clears $15*25 on the whole. How many i)ouuds of the higher priced tea did he mi v with the vthev and hf>w nuich per pound did this higher priced tea coat him? 38. The po})ulation of a certain city was 27,41.'i .at the date of tjiking one census and at the time of tjvking the next census the l>opuiati»»n had risen to 44,229 ; find the incre.w«e per cent, correct to 4 significant figures. Express this as an increase per thousand, he n (> .In- .uUl ole. the im? of the 'ect nd, V'> ■•;yi}.,^v'i'^::fi V- MISCELLANEOTTS PllOBLEMS. 70 30. What rate of interest ia equal to 8% discount for one year? 40. On I'Jth Aj)ril 1889, a merchant jjurchased goods amounting per catalogue prices to $1239 "35, subject to 30 and 6 off; terms 3 months credit or 5 off for cash, ^% per month on accounts overdue. Find the amount of this account on 19th Oct. 1889. What would have been the amount had the account been paid on 19th April 1889? What rate of interest will the merchant be paying if ho settle on the 19th Oct. instead of on 19th April ? 41. Find to the nearest 100 sec. and also to the nearest minute the time occupied by light in passing from the sun to the planet Neptune, the velocity of light being 187,200 miles per second and the distance of Neptune from the sun being 2,746,000,000 miles. 42. How many yards of carj)et 27" wide will be required to carpet a room 25' 8" by 15' 8" allowing 9" per width for matching ? How many rolls of waU-papev and how many yards of bordering will be re(juired for the same rtxmi, aUowing on the wall-paper a width of 42" each for 3 windows and 2 doors ? , 43. Find the value of a pile of cordwood 13' 4" long by 3' 9" higli at $^.50 the cord ? 44. Find the weight of a circular copper plate ^ in. thick and 11" in diameter, copper weighing 549 lb. per cubic foot. 45. If an express run at 30 mi. an hour and an accommodation train at 22 niiles an hour, what is a inan's time worth if he wo aid lose 45ct. in triwelling a journey of 270 miles by accommodation instead of by express? 46. Find the number of cubic inches which 10 lb. of (<() water, (?)) hard-coal, (c) silver, (d) oak will occupy if a cubic fo(»t of water weigh (52 lb. <5'8 oz. and if hard-coal be 1*6 times and silver belO'5 times heavier, volume for volume, than water, and if a cubic foot of oak weigh g as much as a cul)ic foot of water, 47. A publisher sells a certiiin book at 78ct. i)er copy. He pays the j)rinter 17Act. , the binder 15et., and for other expenses Oct. <»ii every copy }mnU'd. He als( > }>ays t ho author 12r>ct. on evmy c( >])y sohl. Of one e«lition of 10(K) u>pies he sells 879 and the rest are left on his hands. Does he gain or docii he lose on the transaction ? How much ? At what rate uer cent. ? 4f:m 1 '"'in '■i. j\% ■M!'' ■I I \ i ■ t I '?>::" 80 ARITHMETTC. 48. A did J of a picco of work, B did ^ of the remainder, C did f <»f what was loft undone by B, and D then finished the work. How nuich sliould D get for his work if A receive $7 00 for his? 49. Find the proceeds of the following joint note discounted in St. Thomas on 18th Dec. 1888, at 7^%. i^347,''(p(j. St. Thomas 18th Dec, 1888. Ninety days after date wo jointly and severally promise to pay to the order of Jno. Locke & Co. , Three hundred and forty- seven Y^j^ dollars, at the Standard Bank hero. Value received. Isaac Harper. H. H. Frikdlabnder. 60. What rate of discoinit is equal to 8% interest reckoning (a) for a year, (b) for 93 days, (e) for 63 day's ? 61. The British ship Egeria found a depth of ocean of 4430 fathoms at a certain place oft' the Friendly Islands and the U. ,S. ship Tuscarora found a depth of 4055 fathoms ofF the north-east coast of Japan. What nuist be the pressure per scpiare inch duo to the superincumbent water at these depths, sea-water weighing 64*05 lb. per cubic foot ? Express tho pressure in kilogrammes per scjuare centimetre. 52. What will be tho cost (.£ 1000 yards of side- walk 8 ft. -wide, made of 3 in. plank laid on three lines of cedar stringers, if tho ^)lanks cost $12*00 per M., the cedars 4ict, per running-foot and preparing and laying the sidewalk $3*50 per yard ? 53. Out of a cii'cle 18" in diameter there is cut a circle 13*5' in diameter. Whnt fraction of tho origaial circle is left ? 54. Find the av eight of a cast-iron cylinder 8' in length and 7" in diameter, if a cubic foot of cast-iron weigh 444 lb, 55. A vessel holds 2^.^ qt., how many times can it be filled from a barrel containing 31i gal, of oil ? After filling the vessel as often as possible how much oil will remain in the barrel ? What fraction of a vesselful will this remaining quantity be? 66. If 9 lb. of rice cost as much as 6i lb. of sugar and 10| lb. of sugar cost as much as 1 lb. 10 o/,. nf tea and 1*25 lb. of tea cost as much as 2§ lb. of coffee, find the cost of 100 lb. of coffee if rice is worth 7 ct. a pound. MISCELLANEOUS PllOHLEMS. 81 per 57. Tf a Luiip bum '08 of a pint of oil por hour and (> lamps arc usod ovory iiiglit and .'50 gal. of oil aro consumed from 27th Sept. to 4th Jan. next following, both nights included, how many hours per night aro the lamps alight ? 58. In an examhiation .1 obtained 78% <'f the full number of marks beating /J by !()% of the full iiumlier. If A received 975 marks, how many did />* receive? What percentage of A's number was />"« number? What percentage of i"s mnnber was ^'s number? It was afterwards decided to deduct 20% from the total nundnu- of marks and also from the numbers obtained by A and B, what eftect would this change have on tho answers to the preceding three ((uestions ? 50. Find the ])roceed8 of the following note discounted in Toronto on 17th Oct. 1888 at 7^%, exchange j% reckoned to nearest cent. §211/>j^}j. Hamilton, 12tb Oct., 1888. Three months after date I promise to ])uy to the order of A. J. Wilson it Co., Two hundred and eleven Dollars at the Bank of Commerce here. Value received. Henry Tomlinsox. 60. For how nuich iimst a ninety-day note be drawn to realize 8190 when discountinl at 6'/^ ? II L3-5' id 7" cost lice IS 61. Tf 8 metres of silk cost 70 francs what will be the jmce of 10 yd. at the same rate, reckoning 10 francs ecpial to $1'93 ? 62. If 2 horses are wt)rth as much as 7 oxen and 3 oxen as mucli as 17 sheep, find the value of 5 horses that of sheep being .*^60. 63. Find the price of a rectangular slate blackboard 2.3' 4" hmg by 3' 6" Avide (o! 44 ct. per square foot. 64. Find the Aveight of a cast-iron pipe 7' fi" long and of 5|" external and 4" internal diameter, a cubic foot of cast-iron Aveighing 444 lb. 65. A train is running at the rate of 20 miles per hour and a second train starts after it at the rate of 2'th miles per hf>ur and overtakes it in 3 hr. 2.5 min. Hoav many miles aii hour did the second train gain on tho first ? Hoav far ahead Avas the first train when the second train st'trtf^d I 1 82 AlUTHMETIC. 60. A man who luis had liis wages iiicroascd l>y ./;. is in rocciptof $12"50 por wook. What fvactiuii of itsolf must ho takuu urt' lUl-: weekly huui to reduco liis wages to tlio original rato ? 67. How many })oyH each doing '(') of tlio work of a man must he engaged with 51 men to do in 'JO days as much w<»rk as 28 men could do in 45 days ? 68. The average rainfall at 'J\>ronto is leas than the average riinfall at St. .John, N. B. , hy 45ij'/^ of the latter, and the average rainfall at "Windsor, Out., which is oO in. pur annum, is greater than the average rainfall at Toronto by 8*17o of tlie latter. Find the weight per acre of the average annual rainfall at St. John, N. B. 69. Find the proceeds of the following draft discounted on 15th Feb., 188!) at «7^, exchange ^ 7, : 8701 i^i^-. GuELi'ii, 12th Feb., 1889. Sixty days after date i)ay to the order of Henry INIeadows it Co. . of Belleville, Seve'i hundred and ninety-one ^^^;'^J dollars. Value received. Stuart & Gee. To J. J Newcomb, Believille, 70. The proceeds of a note payable in 3 months from 1st Feb. 1880 and discounted on tiio «)th Feb. 1880, amounted to $847 "18. For what sum was the nt)te drawn '^ 71. How many tiles 6" s([uare would pave a hallway ^ the size of a courtyard which refpiired 03(50 bricks to pave it, at the rate of 82' by 4^" per brick ? Find the length of the courtyard and the width of the hallway given that the length of the hallway and the width of the courtyard are each 42 ft. Gin. 72. Find the weight of 5 miles of steel wire of "147" diameter, the steel weighing 402 11>. per cubic foot. 73. Sound travels at the rate of 1120 ft. per second more slowly than light ; at what distance is a lightning-tlash the thunder < if which is heard 7J j- sec. after the lightning is seen ? 74. ^1 by working on piece-work if as fast again as B is able td earn $2 "00 per day. How much does B earn per day 1 MISCELLANEOUS I'llOHLEMS. 83 cipt-, of lUSt 1)0 1 cuuUl Lvevago IVOVilgO cr tliHU iud tho uu 15th 1889. '8 it Co. Value EE. 1st Feb. 847 -IS- I the size |e rate of land the and the liameter, i-e slowly Lnder of Is able ty decide to "work together and to finish the work in days. Ilow many hours a day must they work ? 78. A }nan travels oOO nules in 12 days travelling 8 hours per day. If he increase his speed by 20 °/^, how many hours per day less than before need ho travel in order to accomi)lish 450 miles in 20 days'^ 79. A market-woman bouglit a certain number of eggs (2? 11 for Oct. and sold them, iiU but 3 which were broken and thrown away, at {> for 11 ct., thus cj aring $'2"03 on tho transaction. How many eggs did sho buy and what rate per cent, of profit did she make 1 80. On 2.'h-d July 188J), Messrs. Ingram, Hughes, Leighton &, Co., of Toronto, take to tho Bank of Connnerce, to be discounted and tho proceeds placed to their credit, drafts as follows : — One at 00 days from date on S. CaSsidy & Co., Paris, for $'372'85 ; one at 1)0 days from date on Th. Moore & '^^)., Owen Sound, for $02{)i>0 ; one atlOdaya from date on Gregg ct Weir, Belleville, for .^125; one at 45 days fronx date on Brock it Eaton, St. Thomas, for $748 "50 ; one at 4 mo. from date on Colby > V jS& I °w Photographic Sciences Corporaiion m v 4^ ^ ' o^ 33 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 ^ ^ ^ .>\ ,.v <;. ,^ ^ 4i, /A lA '/, ^ 84 ARITHMETIC. 82. Find the cost of painting the walls and ceiling of a hall 62' X 34' 6 " X 16' 6" at 27 ct. per square yard, — no deductions for openings. 83. A rectangular box made of boards Ij" thick, measures on the outside 3' 7" by 2' 5" by 1' 10". Find its internal content, (n)tho measurements, including the lid ; (b) the measurements being of the box without the lid. 84. If a boat 36 ft. long travel f of its length at each stroke of the oars, how many strokes will be required in rowing a distance of 2'^l miles ? How many strokes per minute will the rowers re(piire to make in order to row the distance in 26 min. 40 sec. ? 85. If a man earns |- as much as 7 women and a boy earns ^ of -^ of the wages of 2 women, what fraction of a man's wages does a boy earn; the time of earning being in all cases the same ? 86. A town council wa» offered gravel unscreened at $4*50 a cord, screened at $5*50 a cord. Allowing 25ct. as the cost of screening a cord of unscreened gravel ; at what fraction of the unscreened gravel do the above prices estimate the loss by screening ? 87. Divide $40 "71 among 7 men, 16 women and 25 children, so that 5 men may get as nmch as 6 women, and 5 women as much as 6 children. 88. By selling a certain book for 83-96 I would lose 12 % of the cost ; what advance on this proposed selling price would give a profit of 12 % of the cost ? What rate ^er cent, on the proposed selling price would this advance be ? 89. On 28th Aug. 1888, a merchant purchased goods amounting per catalogue prices to $987 "50 subject to 20 and 5 off; terms 3 monf- 3 credit or 5 % off for cash. To what rate of interest is this 5 % off for cash equal ? If the merchant were to discount at 7 % a note drawn at 3 months for the credit amount of the above account, by how much would the proceeds of the note exceed the cash amount of the account ? 90. A merchant buys goods amounting per catalogue prices to $1573-45, subject to 20 and 10 off ; terms 90 days credit or 6% off for cash. For how nmch must the merchant make a note payable in 90 days, that the note discounted at 7 % may realize the cash amount of the above bill ? For how much must the note be drawn to allow ^ % off for exchange ? so 1 as the this %a )unt, ount 3S to off ^able cash rawn MISCELLANEOUS PROBLEMS. 85 91. Four foremen Ay B, C, D, are placed over 260 men. For every 4 men under A there are 5 under 0, for every 9 under B there are 10 under D, and for every 2 under A there are 3 under B. How many are under each 1 92. Find the cost of plastering the walls and ceiling of a room 27' 8" X 13' 4" X 9' 2" ot 22ct. per square yard, there being 3 windows 6' 9" X 4' 3" and 2 doors T 3" x 4' 3". How many cubic feet of i)laster would be required to plaster the room, the average thickness of the plaster being half an inch ? 93. Find the surface-area and the volume of a rectangular block 3' 9"X2' 4"xl' 3". What fraction of the block would be cut away and by what fraction of itself would its surface be diminished were 2" each to be taken off its length, its breadth and its thickness ? 94. How long will it take to travel IS^ kilometres at the rate of 11 '9 miles in 1 hr. 45 min. ? How long will it require to travel 13^ miles at the rate of 11 '9 kilometres in 1 hr. 45 min. ? 96. A line A is half as long again as B and B is one quarter as long again as G. What fraction of the length of A is equal to J of the length of C? 96. A merchant sold i of his stock for | of the cost of the whole stock ; ^ of the remainder at a gain of $80 ; ^ of what still remained for its cost, $150 ; and the rest at a reduction of § of the cost. What was his total gain ? 97. If 7 men, 15 women and 9 boys earn $8701 '40 in a year (313 working-days) and if a woman's earnings are '6 of a man's and a boy's are f of a woman's, what are the weekly earnings of a n-an, of a woman and of a boy respectively ? 98. Goods are sold at a loss of 15 % on the cost. By what per- centage of itself should the selling price be advanced to yield a profit of 15 % on the cost ? 99. What rate of discount is equal to 5% off for cash on a purchase on 90 days credit, reckoning |% for exchange with the discount ? 100. What nmst a merchant charge for goods that cost him $976*50 cash, in order that after giving 6 months cx-edit, thus involving the discount @ 7% of a note drawn at 90 days to yield the cash price of the goods and a renewal note also drawn at 90 days and discounted at 7%, he may obtain a i)rofit of 15% on the cash price paid by him for the g)ods ? « CHAPTER IV. THE THREE HIGHER OPERATIONS. INVOLUTION. 97. An Integral Power of any number is the product or the quotient resulting from successive multiplications or successive divisions by the number, the initial multi^Jicund or initial dividend being in every case one. The power is said to be positive, if it be formed by multiplications; negative, if formed by divisions. In naming positive powers, the term positive is usually omitted. The second positive power of a number is conmionly called the square of the number; the third positive power, its cube ; and the initial 1, neither multiplied nor divided, the zeroth power. The first (positive) power of 5 is 1 x 5 =5 The second power or square of 5 is 1x5x5 = 5x5 = 25 The third power or cube of 5 is 1x5x5x5 =^--25x5 = 125 The fourth power of 5 is 1x5x5x5x5 ^125x5 = 625 The fifth power of 5 is 1 x 5 x 5 x 5 x 5 x 5 = 625 x 5 = 3125 The first negative power of 5 is 1 -i- 5 =} The second negative power of 5 is 1 -f 5 -=- 5 = 5 -r 5 = ^j^^ The third negative power of 5 is l-^5-^5-i-5 = ^t-5 = y^5 The zeroth power of 5 is 1 =1 98. The base of a power is the number used as niultiplier (or as divisor) in forming the power. 99. The Exponent or Index of a power is the number which expresses how often the base occurs as fa-^tor (multiplier or divisor) in forming the power. The figures of ;^n exponent are usually made somewhat smaller than those ttf its base and are placed on the right of the base and a little above it. The sign minus is employed as a negative sign and is written before the exponents of negative powers. Instead of 1 x 5 x 5 or 5 x 5 we write 5- which is read "5 square." Here 5 is the base and 2 is the exponent. Instead of lx7x7x7x7or 7x7x7x7, wo write 7* which is read "7 to the fourth," power being understood after fonvth. In this exjjmple, 7 is the base and 4 is the exponent. INVOLUTION. 87 Similarly 3' which is read " 3 to the beventh " (power), represents 1x3x3x3x3x3x3x3; and 3"% read "3 to the negative fifth," represents l-i-3-r3-T-3-i-3-T-3. In 3^, the base is 3 and the exponent is 7 ; in 3~^ the base is 3 and the exponent is - 5. The exponent 1 is not usually expressed, the first positive power of any number being simply the number itself. Hence ivhen no exponent is expreased, the expo^ient 1 is to he ^ittderstood. ICX). The Degree of a power is the number of successive multiplications (or successive divisions) by the base. The exponent of any power is, therefore, tho Index of the Degree of the power. The greater the number of multiplications, or the less the number of divisions by the base, the higher is the degree of the power ; the fewer the multiplications, oi the more numerous the divisions, the lower is the degree. The phrases * higher power, ' * lower power ' are very frequently used instead of the full phrases * power of higher' degree, ' ' power of lower degree. ' 101. Involution is the operation of raising a given base to a power of given degree. In other words, Involution is the operation of finding the product of a given number of factors each equal to a given number. Example 1. Find the first six positive powers of 1 '4678 correct in each case to five figures. 1 1-4678 1-4678 2 2-9356 68712 3 4-4034 8807 4 6-8712 1027 5 7-3390 117 6 8-8068 10-2746 2 15443 7 2-9366 8 11-7424 14678 9 13-2102 7339 687 59 4 Ist power. 3-16227 . . . 3rd power. 4-4034 14678 8807 294 29 10 ¥-64168 . . . 4th power. 2nd power. 5-8712 88068 5871 147 73 12 () -81291 . . . 5th power. 3 1622 V 3r(l power. 8-8068 1 17424 1468 294 132 1 9-99999 . . . 6th power. •i;;,: 88 ARITHMETIC. Wo liHvu tiiiulo 1*4678 the imiltiplicHiul in onch luultipiiuntiun, bectiuau by ho doing, only a singlo tiiblo of nutltiplus is rutpiirud. (Soo JiJxainple ~^, pugo 09.) Tho coniput4vt.i«»n8 Imvo boon carriotl to Hix iiguros in ordor toonHuru jiccumcy in tho tiftli. Tho six i)owoi'8, ouch corroot to iivo tiguros, tuo 14078, 21544, 31(>2:{, 40410, 0-812J) and 10 rospt jtivoly. Exam})le ;?. Find tho vnbio of , -47 1+ ,-+- t ' 4< •47* 1 X 2 X 'i& 1x2 X ;J X 4"^i X 2 X n X 4 X 5* 1 "^1x2"^ ci>rroct to 4 dociniul pl.voo.s. (Work to 5 dooinmls.) •4T •7.'^ 1 2 *> o 4 5 (> r- 4 8 _47 1)4 1-41 1-88 2:tr) 2-82 :{-2U a -70 4-23 •47 (a) •1105 (b) 173 ((•) 20 (./) 2 (0 10 Wo square '47 and divide by 2 and thus obtain (/>). Wo next multiply '47 by {h), this gives one-half of the cube of '47 ; we divide by 3 and obtjiin (c) the sixth part of the cube. Wo then multiply "47 by (c) and divide by 4 to obtain ( 00 ^*"*1 0') ci>»'»'"^fc to the fourth decimal. S s s th INVOLUTION. 89 EXERCISE V. next ; wo then |y47 lof 1, Writo tho foUuwing pri)ductH hh ptjwors — 1. 2x2x2. ft. IXlXlXl. 9. :i X 3. 6. 2-3 X 2 3 X 2 -3 X 2 •;{. X 2 3. 3. 5x5x5x5. 7. Axixi 4. 10x10x10x10x10. «. |xlx|.x;Jxix^ Writo tho following jjowofh hh jmnUiftH : — 9. 3*. 19. 25 «. 1ft. (.^)'. 10. 12''. 13. 2-6«. 16. (\l)\ 11. 15a. 14. •25«. 17. 2''x 32x5x72. Find tho valuo of : — 18. 2". 9S. •02" 38. ay. 39. (iy. 19. 6^. 99. 1"02«. 90. 5^ 30. 492. 4"' ^•- 5* ai. 4». 31. 4-92. 41. 2'»X3<. 93. 23752. 39. -492. 49. 2^x3'>x5-». 9S. 58733. 33. 23(l«. 43. 22x3-«x7-. 94. 273^. 34- 2-3(}"». 44. 2«x5-»x7Xll''Xl32. 9ft. 27". 3ft. •O230'». 4ft. (72)-'. 90. il->. 36, (|k 3T. (|)«. 46. 5"-^52. 9r. i". 47. 27x3*x5'»-f2^^32-^5". Rosolvo tho following numbers into their prime factors, oxpreBsing tho repetition of a factor by an index : — 48. 2520. 49. 70200. ftO. 1024. ftl. 11308. ft9. 530712. Find tho value, correct to four Hignificaut figures, of: — 11111111 ft3. 5-^-5^ + ' 5^ + 5*+ 50+ 5.) + «. ^ 1 1 1 *^- ■«+l>2 + -6^+-(5^ 1 57"^ 5«"^ 1 ftft. ft6. ft7. 1 1 '9 ~ 9- "^ 1 2 5"" 1 -1. 102 1 3 1 9""^ 9: lx2x3^ro^" Arf ll Airf 1 + -. 3 J 1_ 1_ 9.!- 94+ 95- 1 '1x4^ 1 3x4x5 KM "^1x2 10" ~ 1 1 J__ 111 1 i _1 + 5^ 2''" 7'' 2''^ 9 ^'2»"ll^ yii" 1111111 :i "" :V' "^ 5 ^ :p " '7 ^ 3" "^ 9 "" 90 ARITHMETIC. 1 1 J. 1 JL 1 _L 1 J ^L 1 58. 2 -'3^ 2^'^ 6^ 2'' '7^ 2'"^'J^ 2'-' ll^Y'i 11 1 1 1 1 i_ 1 _L 11111 8 3 8''^ 5 S^' (1 1 ^1 J_ 1 _l_ 1 JL 1 *• ''^llJ " 3 ^ iP"^ 5^^ 3'~ 7 ^ n"'*' J> "^ :5''J 1 _1 1_ 1 J^ + 7-3 X 7»+5 X 7.V ^^ r 1 1 1 1 1 1 1111 5» 1 1 1 1 1 X 70'^ 3 ^70"^"^y9 3 ""y^-^" r 1 _ 1 _i^ 1 j^ 1 j_ I _ _i_ "'• *\ 5 3 "^ 5'''^'5 ^ 5- 7^ 5-/ 235 55)' •7 . 7 .73 .7.-. 62. 1+ ^ +Ix2'^lx2x3''"lx2x3x4"*"lx2x3x4x5 .70 + 1x3x3x4x5x6 .7^ -. + &C. 63. 1--4-+ ' •7^ .3--rTrr.+ *t;. 64. F Ix^ " r>72 X 3 "^ 1x2x3x1 " 1 X 2 X 3x4x17 r^ r 1 i__ i_ __i__ 1 ^l'"" I3I "^3x313+51^315+7x31- J r_i 1 j_] r_i_ _ JL j 1 + '^^ U9 + 3x49^ + 5x49'J + U(Jl + 3x 1(J1=' J / ,49^3x49^"^5x49 65. 2x|23x [;^ + 3^+^p+^^ r i_ 1 1 ^ r_i_ 1 -] ) + 17x [49+3 X 49^+6x495 J +^"^ ll01+3xi0pJ /' 102. Homer's Method. — The simplest and easiest method of raising a given base to a power of given positive integral degi'ee, is that which was adopted in Example 1, p 87. In that system the successive positive integral powers were calculated one after another, all the figures of the base being used in the multiplicand in each multiplication. The calculations may however bo conducted t)n a INVOLUTION. 91 Ac. }• >dof 3, is the |her, each i)U a (lifToroiifc plan. Wo may begin with a single figure of the base, (by pruferenco the first on the left-hand) and having raised this single- digit number to the assigned degree, we may then proceed to build up the recjuired power step by step as wo add figure by figure to the base. This way of computing the positive integral powers of numbers is known as the Method of Differences, and tho best arrangement of tho process, that exhibited in tho following examples, is named Horner's Method. In ordinary cjises of involution, Horner's Method if neither so easy nor so simple as that employed in Example 1, p. 81, but it has the advantage of being applicable to whole classes of problems for which the other method is of little or no use. Example 1. Find tho square of 3472. Mark off into column s the space set apart for the calculation, the number of columns being greater by one than the exponent of the required powei". At the top of the left-hand column write 1, this 1 is to be understood as repeated in every line down this initial column. The other columns contain '>»e actual calculations and may be called the working-columns and v; ubered from the left. At the top of each of these write a zero. This forms the first or initial line of the calculation. Multiply 1 in the initial column by 3, the left-hand digit of the base 3472, and add the product to the zero in the first working- column. Set the result which is 3, in the first working-column. Multiply the 3 just set in the first working-column by the base-digit 3 and adding the product to the in the second working-column, set ihe result which is 9, in the second working-column. Begin again with the initial 1, multiply 1 by the base-digit 3 and add the product to the 3 in the first working-column. Set the result which is 6, in the first working-column. We have now instead of the initial line 1, 0, 0, the new line 1, 6, 9 ; the 6 being double the base-digit 3, and the nine being the square of this 3. Prepare this line for the next step by placing «>j(c 1 3 900 = 302, 60 1 64 680 115600 = 3402, 1 687 12040900 = 34702, 6940 1 6942 12054784 = 34722. 92 ARITHMETIC. zero ftfter the and tiro zeros after the U, thus converting them into 60, the double of 30, and 900 the square of 30. Wo now repeat the syateni of operations just described using 4, the next tigure of the base after 3 instead of 3 and the lino 1, GO, 900 instead of the line 1, 0, 0; thus : — 1 X 44-<>0 = G4, which is to be placed in first working-column. 04x4 + 900 = 1156, to be placed in the second working-column. 1 x4-|-64 = 68 to be placed in the first working-column. We thus obtain a third line of calcuhvtion, 1, (\8, 1166, the 68 being double the base 34 and 1156 being 34'-. Prepare this line for the next step by j)lacing one zero after 68 and tico zeros after 1156, thus converting them into 680-340x2, and 115600 = 340'-. Repeat this system of operations usint; 7, the next figure of the base, as multiplier and 1, 680, 115600 as line of calcuhition, thus : — 1x7+6^== 687, in 1st working-column ; 687 X 7 + 115600 = 120409, in 2nd working-colunni ; 1x7-1-687 = 694, in 1st working-column. This gives as fourth line of calculation, 1, 694, 120400, which, pre])aratory for the next step, is converted into 1, 6940, 12040900. Repeat the first course in this system of operations using as multiplier, 2, the last figure of the base, and as line of calculation, 1, 6940, 12040900. 1 x2-f 0940 = 6942, in 1st working-column. 6942 x2 + 12040900= 12054784, in 2nd working-column. This cojnpletes the calculation of 3472'-^. Example ii. Find the cube of 2574. (a). (6). \d). (0- (?)• (A%). (0. (m). (h). I. II. III. 2 4 8000 4 1200 60 65 1525 15625000 70 187500 750 757 192799 16974593 764 198147(K) 'mM^ = 203 = 2503 = 25703 10 14 198455.56 170,53975224 =25743 (771H) (19876428) (7722) INVOLUTION. 93 Hero wo ftro rocinircd to find a third power, wo must theroforo have three working-columns. In the tirst sob of operations we tako 2, the loft-hand digit of tho base 2574, as multiplier and wo have 1, 0, 0, as initial lino, (a) 1 X 2+0=2, in column I, 2x2+0 = 4, in col. IT. 4x2+0 = 8, incol.JTI. Change 8 to 8lKK). (b) 1x2+2=4, in col. I. 4 X 2+4 = 12, in col. II. Change 12 to 1200. (f) 1 X 2+4 = 6, in col. I. Change (5 to 00. Wo havo now a now line of calculation, 1, (JO, 1200, 8()00. In this lino, 60=20x3, 1200=20'-^ x 3 and 8(KX)-2()''. Repeat tho system of operations starting from this new line of calculation and using as multiplier 5, the second figure of tho base. {<() 1x5 + 60=66. Col. "I. 6^x5 + 1200=1525. Col. II. 1525x5 + 8000= 15625. Col. III. 15625000. (t) 1x5 + 66 = 70. Col. I. 70x5 + 1525 = 1875. Col. II. 187600. (/) 1x5 + 70=75. * Col. I. 750. Wo thus obtain a third lino of calculation, 1, 750, 187600, 16625(XK), in which 760=250x3, 187500=260- x 3, 16625000 =250-'. Repeat tho system of operations, starting from tho third lino of calculation and using tho third figure of the base as multiplier. (if) 1x7 + 750 = 757. Col. I. 757 X 7 + 1875(X) = 192799. Col. II. 192799 X 7 -^ 15625000 = 16974593. Col. III. (/i) 1x7 + 757 = 764. Col. I. 764 X 7 + 192799 = 198147. Col. II. (Jc) 1x7+764=771. Col. I. We thus obtain a fourth line of calculation, 1, 7710, . 19814700, 16974593000, in which 7710 = 2570x3, 19814700 = 25702x3, 16974593000 =2670=^ Startmg with this fourth line of calculation, repeat the first course of the system of operations, employing as multiplier the fourth figure of the base. (0 1x4+7710=7714. Col. I. 7714 X 4 -H 19814700= 19845556. Col. II. 19846556 x 4 -f- 16974593000== 17053975224. Col. III. 94 AIUTMMKTIC. III! iii 11 ! I7'>r>.'H)7»V224 hoing tlio cubo of 2574, wo nood go no farther in tluH HyHtoui of ojiuriitionHunloHH wo wIhIi to j>ro|)iiri) for ui'otlior .•♦top in ndviiiico. This wo havo dono in tho oxiuni)le, having calculatod and recorded (within paronthesuH) tho lines marked (m) and (n) respectively. Examph', 3. Fhid 15848!).'U9Ii'^ (inrroct U^ \) signiticanb iigurea. Tho recpiired power being tho fifth, live working-cohnnns will bo needed. Nino figures aro reipiired tc) l)o correct, tho computation nniat therefore bo carried to at least oleven figures in tho fifth working-column. Tho decimal jioint is omitted as unnocosBary, except in tho lasL working-ooluuni. 1 1 1 1 2 3 4 5(NN)0 3 6 l(MNM) 4 10(H) 50 55 1275 16375 131875 HO 1575 24250 2531250000 65 1900 33760000 • 70 225000 750 • 758 231064 35598512 281(}03809(J im 237192 .37496048 3116(HK)480 i <4 243384 39443120 y 782 245)640 ^ X 790 \ \ \ « \ ^ \ 250 .395431 313182372 ^ ^ \ 396431 .314768096 397431 ( 3976'' 31508618 3978 31540442 3980 > 40'^ 3154404 »» 3154764 \ \ \ \ l-(NMNN) ,1 7 r)9.';754768 ,8 9-9718.^).3425(5 ,4 9-997060.3200 ,8 9-9998992836 ,9 9-9999939264 ,3 9-9999970812 ,1 9-9999999207 ,9 10-0000000152 ,3 Hence 1-584893193" = 10 '00000002, correct to the last figure. In the first set of operations, we begin with 1, 0, 0, 0, 0, as the initial line of calculation and we take as multiplier 1, the left-hand digit of the base, 1-584893193. We obtain therefrom, the new line of calculation 1, 5-0, 10-00, 10*000, ^-0000, 1-00000. »VVOI.UTi()N. 96 ,1 ,s ,4 ,8 .9 ,3 ,1 ,0 ,3 III tliu Huuniul Hi;t of ()[)ui-iiti()iiH, wu hugiii with tluH now linu of c.-ilrulii(iiiii uiul wu i)iku HM iiiiilliplii!!' '5, thu hucoiuI tigitrc of thu biis'j. Wo o})tain thorofroiu hh third lino of ciilculatioii, 1, 7-r»'>, 22-f)0o^), 'XV^fiOono, 2r)'^V2Ci(>0()(), 7-r>l)M7r.f/OfW>r;, in vvliich it iH wtirtliy of iiotico thiit 7-5- IT) X 5 22G() = l-5-xlO, ;j.'{-750-=l-5-»xlO 2r):U25 = l-5ix 5, and 7r)Ua75 = l-5'''. Ill tho third set of oporatioiiH, wo hegiu witli tho lino of calcul'itioii lash obtained and wo tako as luultiplior *(>H, tho third tiguro of tho l)aso. Wo obtain thorefrom as fourth lino of calculati* >n 1, 7'nO<), 24-064000, :«> -443120000, .•{l-J«00«48(V^000, ;> •84(55804708(^00(^0 ; in which it should bo iioticocl that 7-90-1-58 X 6, 24-JHMO = l-58axlO, 39-44.'3120 = l-583xlO, 31-16006480-^1-58* X 5, and 9-8465804768 = l-58«. Tho contracting begins at the figure 4 of the base ; the uncontracted fifth working-column would on passing from 8 to 4 of (lio base, receive an extension of Jive figures, these are all omitted and as a consequence the other working-columns must also be contracted by five figures each. Allowing for their "extensions" this will require the cancelling of the right-hand figure in the fourth working-column, of two figures on the right in the third, of three figures on the right in the second and of four figures on the right in the first working- column. In like manner, on proceeding from 4 of the base to the 8 following it, from 8 to 9, from 9 to 3, from 3 to 1, 74'' l.o oltt^iiii llu) viilut) of ;UJr>H''' hy j^ivin^^ tlio Ihibo 2r>74 tho HUcooHHJvo incrotiiontH 4, HO niul KMK). I 7722 772<{ 77.'«> 77:54 7H14 7H".>4 7J>71 H!>74 :M»r)H''+2r>74'' l!)H7r>42H ioj):{82r)2 2()r)C>a;i72 21IJ)4«U2 i7t>r>:«>7r>224 - 2574 • i7i.*i.'W)(>45r>2 2r>7H» IH778 initial lino I, 0, 0, inii^loyod in finditiji; tho vjiluo v>f 4H77'', uso tho initial lino 1, 0, 0, IK), tlio si^n bofoni I H5 ilonotinj^ (hat tho ilitlorcnco i« to 1)0 takon hotwoon JKJ and tho tnnnlKv jarriod froni tho sooond working-oolnnin to tho third. 1 4 ]<'> 8 ■18(M) 120 128 r.824 i;i(i (>*M2(H» 1440 1447 7(M;i25) 1454 7iir.()7o<> I4<>10 14(117 7i2r):{oio lhMKv.4-877'' H<5 •()(H)()74i ;{.*{. 1J(; 52(MH) ,4 -r)-4()80 ,•8 -•41>8(><)7 ,7 + (MMM»74i;5;! • t i EXERCISE VI. Find tho valuo of : — I. 2-, 2;r-, 2:{r)'-, 2.".57-, 2.'ir)78-, 2;i578j-. a. 4'», A\v\ A[\r\ 4;57r>\ 4;i7r)9"'. a. 12-, 12'> ; 127-', 127'' ; 1278'-, 1278" ; l'>78(5-, lt>78({-«. 4. 51-44!)-, 51-449«, 51 -441)'. 5. 1.%-', •\'MV\ laU', •\'MV\ WW\ «. 205:W1)- 5 •!)•». T. 1 70 -5 ran «. 8. ;M4159^ .'UXi. 9. 8-241 •■' - 8-24r'2 +8241 - 500. {Take 1, - 1, 1,-500 a^ Initial lini.) 10. ll-48-' + ll-48--1554. (Take 1,1, 0, 1554 a.'< initinl line.) EVOLUTION. 97 EVOLUTION. 103. Tho square root of a given number is that number whoso square is tho given num})er. ExamplcH. 4 is tho B. Tho cube ,oot of a given number is that number wh<>8e cube istlie given mnii])or. Kxamphn. 8 is tho cube of 2, . '. 2 is tho cube root of 8 ; 12.5 is tlio cube of 5, . '. 5 is tho cube root of 125 ; 100(3 is tlie cube of 10, . •. 10 is tho cube root of KHK). Tho fourth rooty fifth root, sixth root of a given number is tliat number whose fourth ])owcr, fifth j)ower, sixth power is tho given number. Examples. 81 is tlio fourth power of 3, .'. 3 is the fourth root of 81 ; •000:{2 is tho fifth power of % .'. '2 is the fifth r(.ot ui •(KKKS2. Tho square root, cube root, fourth root, fifth root, of a given number is tliereforo tho base whoso S(juare, cube, fourth power, fifth power, ... is the given muuber. 104. Bvolutioll is tho operation of finding any root (»f a given number. It is therefore tho operation of finding fho base of which a given nund)er is tho power of given degree. 105. In Invohition, the base and the exponent (tho index of the degree of the power) are given and the power is to 1)0 determined therefrom. In Evt)lution, on tho other hand, the base is to bo determined, the power itself being given and also the exponent or index of its degree. Evolution is therefore an inverse of Involution. 106. There are two ways of denoting Evolution. In tho first or older notation, tho S(juare root of a given number is denoted by prefixing tho symbol i^J to tho given number ; the cube root is denoted by prefixing %/ , the fourth root by prefixing \J , the fiftJi root by prefixing y , and all other roots are similarly denoted, viz., by i)refixing to tho given number tho root-symbol ^/ combined with an index number indicating which root is to be taken. Examples. ^^ 04 denotes the square root of 04; •,i/64 denotes the cube root of 64 ; ijSl denotes the fourth root of 81 ; and ^ .^.j denotes the fifth rot>t of .^^. O 98 ARITHMETIC. denotes the square root of 49 ; . 1 •125 ; ij^^j) denotes the tenth root of 125^ denotes the TS^i ; and The second or modern notation for evolution employs fractional exponents to denote the roots of numbers. The exponent of the s(|uare root is |, that of the cube root is J, that of the fourth root is I, and, generally, the exponent of any root is the reciprocal of the exponent of the corresponding power. Examples. 49^ cube root of 81 * denotes the reciprocal of the fourth root of 81. [107. The root-symbol ^ is merely a variant form of the letter r. The employment of an index number with ^ is of -comparatively recent date , the old notation was ^q for the square root, ,^c for the cube root, Vqq for the fourth root, V^qfor the fifth root, ^/cc for the sixth root and so on for other roots. After this came the notation x/[6] for the sixth root, /J [7] for the seventh root,, and a similar notation for other roots. Later still came the notation ,>/", V ^ , 3-6. 3. 16* =2. ». 1-728^ = 1-2. 15. 11* < 2-224. 4. 81^ = 3. 10. •008^= -2. 16. l-16'«>l-05 5. 1000^ = 19. 11. 0001 •« =01. IT. •41^ > -8. 6. 100000 '=10. 13. 0000015=0-1 1§. 2«< 11487. r( EVOLUTION. 99 108. Evolution being an inverse of Involution a calculation in the former will be merely the reversal or undoing of a calculation in the latter. We require therefore a reversible process of involution {ind such a reversible process we have in Horner's Method. In it the required power is built up by successive increments as additions are made to the base or as it is enlarged figure by figure. To reverse this process we must withdraw the successive increments of the direct process, and since the increments may be added in any order (compare Example 2, p. 92 and Example If, p. 95), they may also be withdrawn wt any m'der. At the beginning of the calculation, the only digit of the root, the unknown base, of which we can be sure, ia the first digit on the left, therefore we commence by raising this digit to the degree of the power which the given number is to be of the required root, and subtracting this power from the given number. In determining this first digit of the root, it must be remembered that each figure subsequently added to the root or base gives two additional figures in the square of that root, three in the cube, four in the fourth power, five in the fifth power, and that for each figure to the right of the decimal point in the root there will be two to the right in the square of that root, three to the right in the cube, four to the right in the fourth power, five to the right in the fifth power, Hence, in preparing to extract any root of a number, we begin at the decimxil point and mark off the figures left and right in pairs in case of the square root^ in sets of three in the case of the cube root, in sets of four in the case of the fourth root, in sets of five in the case of the fifth root, This done, the set or period on the left will determine the first figure on the left of the root. 109. The following Table will assist in determining the first root-digit in cases of square root and cube root: — i I ^5. Root. 1, 2, 3, Square. 1, 4, 9, Cube. 1, 8, 27, 4, 5, 6, 7, 8, 9. 16, 25, 36, 49, 64, 81. 64, 125, 216, 343, 512, 729. Root. 1, -2, -3, -4, -5, -6, 7, 8, -9. 8(nmre. 01, 04, 09, 16, 25, 36, -49, 64, 81. (Me. 001, 008, -027, 064, '125, '216, '343, 512, '729. 100 ARITHMETIC. Ill Example 1. Find the square root of 5476. The root ia to bo squared, hence two working columns will be rec^uired. As the root is found its square is to be withdrawn or subtracted from 5476, therefore wo begin the second working column with — 5476, the prefixed - indicating the subtraction of the s72 but <82, . •. the square root of 54 > 7 but < 8, . '. the square root of 5476 > 70 but < 80, . •. the first figure of the root is 7. Write 7 in the place set apart for the root and then proceed with the calculation exactly as if the problem were to subtract 5476 from the square of a given base whose first digit is 7. This gives as the second line of calculation 1, 140, —576. To obtain a "trial digit " for the second figure of the root, divide the 576 in the last column by the 140 in the next preceding column. The ' quotient ' is 4. Write 4 as second figure in the root and proceed as in involution to find the value of 74^ - 5476. There is no 'remainder ' therefore 74 is the square root of 5476. Example 2. Find the square root of 12054784. (Compare the calculation with that of Example Jf, page 91, noting that there the square is built up, but that here the process is virtually the opposite.) EVOLUTION. 101 (0) 3 (60) 64 - 12054784(3472, sq. H. 305 256 (680) 687 (6940) 6942 -4947 4809 "13884 13884 The first period, 12, determines 3 as the first digit of the root. From 12 substr}vct3- and to the remainder, 3, 'bringing down' 05, the next period of the given number, and complete the formation of the second line of calculation 1, 60, -305. Divide 305 by 60. Tho quotient 5 is found on trial to be too large but 4 on trial proves to be tho right digit. Proceed as in involution to form the third line of calculation which will be found to be 1, 680, - 4947. Dividing 4947 by 680 gives 7 f \ 484471 442715 41756 37950 3806 3792 14 EVOLUTION. 103 Having found six figures of the root by the ordinary uncontractod process wo may find four or five figures more by contracting the process in exactly the same way as we contract in involution. In this example we divide 484471 by 63245 by contracted division, knowing that the figures rejected from the divisor will not affect the quf)tient figures, here root-figures, till the divisor is reduced to one or at most to two figures. The root thus found is correct to eleven figures. The general rule is that when the number of figures obtained by the unct)ntracted process is one more than half the number of figures rcijuired in the square root, than a third of the number recjuired in the cube root, than a quarter of the number required in the fourth root, than a fifth of the number recjuired in the fifth root, the rest oi the figures may be obtained by contracted operations. Example 5. Find the cube root of 1*25 correct to ten figures. 1 1 1-250 1077217345 1 2 307 3 32149 •250000 225043 314 3217 34347 3457219 24957000 24200533 3224 34T9787 3480433 348108' 756467 696087 60380 34811 25569 24368 1201 1044 157 139 18 Example /> page 95 is virtually an example of the extraction of the cube root of 116 correct to six figures and if each line in the fifth working-column oi Example o page 94 be subtracted from 10, the example will exhibit the operation of extracting the fifth root of JO to ten figures, 104 ARITHMETIC. EXERCISE VIII. Find the s(][uarti root of : — 1. a. 676. 1849. 3. 103041. 4. 10-3041. ft. 3321-3124. 6. -0050367409. Find tho cube root of : — r. 8. 380017. 814780504. 9. 70u227072. 10. 700227 072. 11. «-1990«3253. 12. -oooi6oio;;(H>7 Find, correct to six significant figures, the value of : — 13. 2\ 19. 40^. 2ft. 123456'1 14. 20^. SM>. 40002. 26. 123-456-*. 15. 200^, 91. -42. 27. 2^. 16. 2000^. *ia. 7449^. 28. 20l 17. •2l 23. 10002. 29. 200-1 1§. •02^. 24. 609800 1922. 30. -2401^. ll 31. Find ^^ correct to six significant figures and hence prove that 2- ^yS is the reciprocal of 2+ ,,/3 to six figures. 32. Prove to six significant figures that „J3x ^5=/iJ15 and that the product of y/5+ y/3 and ^5- ,^3 is 2. 110. The square of a given fraction has for numerator the square of the numerator of the given fraction and for denominator the square of the denominator of the given fraction. Hence inversely the square root of a given fraction has for numerator the square root of the numerator of the given fraction and for denominator the square root of the denominator of the given fraction. The cube, fourth power, fifth power of a given fraction has for numerator the cube, fourth power, fifth power of the numerator of the given fraction and for denominator the cube, fourth power, fifth pov/er, of the denominator of the given power. Hence inversely the cube root, fourth root, fifth root, of a given fraction has for numerator the cube root, fourth root, fifth root, of the numerator of the given fraction, and for denominator the cube root, fourth root, fifth root, of the denominator of the given fraction. EV0H3TI0N. 105 2=* Examples. f '^ 1 '■" 111 J u--' l9 J h 4^ 9^ 2 3 i 121 r 49^ ^^ 1 49'^ I 8 J 8« 512 1 121 r_27_i 1 512 J 121^ 612-^» 11 8 111. If in extracting any root of a given fraction, it is found that tlio root of the denominator cannot be obtjiined exactly, the fraction may be reduced to decimal form aiid the root extracted to any rec^uired degree of accuracy. Another method is to nmltiply both terms of the given fraction by any factor that will make the denominator an exact power of degree the reciprocal of the degree of the root ; the root of the resulting fraction is then extracted. This process is called rationalizing the denominator. It can often bo used with advantage to obtain a rapid approximation to a required root of a small number. Excmiple 1. Extract the square root of ^ correct to four figures. 111 J 11. ^=•36363636^ =-6030 + r 4x11^ 11x11 J 4£2 6-6333- 11 6030+. Example 2. Find the value of 1" r5 12 J (.12 = •41666667 =-645497 + . correct to six figures. li2J - f 5 1 2_ li2J ~ r 5 x3 15-i 3-872983 + 1 ^_15-_ J -"6 1-12x3 UeJ ~ I 62 J •645497 + . 1 12 J and < 6 6x4x2 4^x2-1 31 r < error 6 4'^x2-l' (i> 6x4x2 6x8 and 106 AKITMMETIC. fi X 8 X ai X 2 31 < error < x 6x8 31- X 2-1 ; (ii). .-. r 5 ) •' 31- X 2-1 1921 , < — = , and U2J 6x8x31x2 0x8x62 6 X 8 X 62 X 1921 x 2 1921 < error < - — - — —. x 1921- X 2-1 12 J '^ 6x8 X 62 X 1921 x 2 "" 6x 1 6x8x62 1921-5X2-1 7:^{8()481 ; (iii)- 8 X 62 X 3842 •645497224368, and ,. 6 X 8 X (52 X 3842 x 7380481 x 2 7380481 < error < , (iv). 6x8x62x3842 7380481'-' x 2 - 1 Tn this 3° method we first rationalize the denominator and then we try successively 15 xl^, 15x22, 15x3^, 15x4-, till we find a product that differs but little from a square number. Such a produc'^ is 15 x 1- which differs from 4- by 1. We therefore ., fl5^ write l36J in the form 4--1 62 J from which we obt.iin at once - as a first approximation to the required root with an 6 error in excess somewhat greater than , as may be ])roved 6x4x2 , .4 1 ,v ^ . 1 4 1 42x2-1 by sfiuarnig We next take = ^ ^ ''6-6x4x2 6 6x4x2 6x4x2 31 6x8 as a second approximation in excess with an error somewhat greater than 6 X 8 X 31 X 2 as may be i)roved by squanng - 31 6x8 6x8x31x2 mi,- 31 1 Ihis gives ^ -^ 6x8 6x8x31x2 312x2 — 1 1921 _' -J = -' — _- as a third approximation in excess 6 X 8 X 31 X 2 6 X 8 X (52 with an error somewhat greater than as may be proved by equaring -^ 1921 6x8 X 62 X 1921 X 2' 1 6x8x62 6x8x62x 1921 x 2 EVOLUTION. 107 by ccess x2' 2" This* givos 11)21 1921- X 2-1 fi y 8 X «2 « X 8 X «2 X 1921 x 2 « x 8 x 62 x 1921 x 2 7'i8()48I = ,. ^' ^- H8 jv fourth Hi)nroximHti<)ii to tho ruiiuired root. () X 8 X «2 X 3842 ^ Tho first i4)proxinmtion is corroct to ono (lecinml place, tho second is correct to three decimal places ; tho third, to six decimal places ; and tho fourth, to fourteen decimal places. It should bo noticed that IK 42 _ 1 (1.) comes from = , 86 6- /•• X , 15 16x8^ 31=^-1 (u.) comes from =^ - T»' (iii.) comes from 36 36x8^5 G'-^xS'^ 15 15x8i5x62'-i 192155-1 36 36 X 8- X 62- 6'-^ x 8- x 62- - and 7380481- - 1 .. V ,15 15 X 8- X 62'-^ X 3842-* (iv. ) comes from — = • = „ ..,..^.. ^ ^ 36 36 X 8- X 62- X 3842'-i 6- x 8- x 62^ x 3842-* r 2 v^ Example ->'. Find an approximate value of ! — } . 2 22 5- -3 11 11x11 21 ^ 5 11- nJ "ii'^^Sr^rs^ < error < 11 5*^x2-3 . f 2 V- 5-'x2-3 • LllJ ^ 47 , and 11 X 5 X 2 11 X 10 9 < error < 11 X 10 X 47 X 2 X 9 11 J '"iixioxir 11x10 47-X2-9 4409 , ., and 81 11x10x94x4409x2 2^^ 4409- X 2 -81 X 2 11 X 10 X 94 4409 8 1 . < error < ^^ 10^94^ 44092 ^ 2-81 ' .v-^ 38878481 11 J 11x10x94x4409x2 11x10x94x8818 6561 The next correction would ho 11 X 10 X 94 X 8818 x 38878481 x 2 tho lumierator 6561 being 81-. From this example we may^see that if possible a numerator should be found that difiers from a square number by but 1 or 2. This might easily have been done in 108 AllITHMETIC. UuH cuH« by Holoctiii^ 22x.'i- um luiiiiuriitor i«> W(»rk fiDiu. The culculutittii w«)ul(l ihuii liuvo uppoiinul hh followH : — 2^22^ JJ2 n'J IJW 14- +2 11 ~ If* "" iP ^ 3'^ ""33""" 33''' i Ui 14 33' and 2 14 2 14 3;Jx 14x2 ;Wxl4^ r> jj3>< 14.J ^^2 + 2 '33^ 14-J + l 111 J 33x14 -XixU 1 11)7 1 ... < error < x ; 3:Jx14x1»7x2 33x14 ll)7-x2-l .-. I'2yi^ ll)7';;_x2 1 _ 77«17 1 11 J " 33 X 14 X 1«7 X 2 " 33 x 14 x 3{)4' * ^' Horo tho tirst upproxiinatiou ia in ihoroforo iwUl the tii'Ht corroctioii. This corroction is in oxcohh, huucu thu uucond {4)[>r«)\iiimtion is in dufuct, furthur sincu thu numuruti)!' (»f thu tii'Ht corructiou was rudiicud to 1, thu munorators of all aubsotpiunt corroetions will also bu 1. In fad iho second approximation is obtainable from thu eijuality 2 _1JW^1J)8>7 , . . 4i>- < — --- =^ - . , iMid 7x47 7x47 2 2207 7 x"i7ir2207 "''■'■'"' "^^"^^ 2 7x47 22t)7--2' 4870847 ■ I'x l7""x 2207 " 7 X 47 X 2207 ' '"" , _ 2 _ _ 4870S^7_ 2 7">r4~7x 22^7 X 4870847 " "'*" "^ 7 x 47 x 22T)7 ^ 4870847- -2 ' Tho teru-f! .tf th*. ern)rs a.vd tho corrections aro reduced each time by division by the conniion factor 2. I EVOLUTION. 109 The 1 - + 1 iwlil tho L) socoiul tho iii'Ht L)so»jUunt uvtiuii is ;ed ouch In any ciwo in which a.'W33n.'i:j^ = -83555- { ^ !'^'= { -'^_ i"'= i 7x2x3;^ ) •'_12fP»^5 01330 (12) ~ \ 2^-1 X ;n "" 1 2'» X 3-» ( ~~ii <; = -83555 - 7_ 7 _7x2x3"^12fl_5-^4-l 2^x3^ 12 2- X 3 12) () ()••> ()••» > ---- and . > error. il2) 7 _12fix75' _ ^ 12 " (5"' X 75^ "" «^x 75-» 6^ X 75- (i). G X 5; X 3 1 5'»x34-l 6x6'-5x3 f}x5-x3 6x75 53156250 376» - -?^^i- = -835550 -• (ii). 376 h < VI 376 and 1126 () X 75 X 376- X 3 < error. («)• 1126 376^x3-1126 ()x75 (5 X 75 X 376'J X 3 6x76x376-ix3 150471002 «x 75x424 128 = •83564965 + . (iii). g ' ;;B g:e:rT3^aca ^-iV.l-i- fq \-gws;m 110 ARITHMETIC. (i) is correct to two figures ; (ii) is correct to five figures, the final fi being rejected without augmenting the preceding 5, on account of the sign < and the correction (a). 112. The process of forming a series of c 6, — is M'ritten in a line beside .r higher than the The next convergent is - — - = ^ ; ) -^ ( < », 3 is written in the lower line, the line of The next convergent 12 IS 7 + 5 12 ( 12 ) - „ — 7; = -^ : ) ^ c <6, .'.^ is written in the lower line. 3 + 2 5 ' ( 5 ) 5 The process thus far followed is continued until there is obtained a sufficiently close approximation to the required root. 3 5^ 27 49 267 485 T' ¥ 11' 20' 109' \m 2 7 12 17 22 71 120 169 218 T' 3"' 5"' 7"' 9' 29' 49' 69' 89' The principal convergents, as far as tlie series has been formed are tlierefore y? 5 22 9' 49 218 20' 89 ' 485 198' these being alternately less and greater than 6'-^. It is worthy of 3 notice that beginning with the superior initial — » there are thi'oughout the whole series two superior convergents followed by four inferior c(^nvergents, followed in their turn by two superior C(mvergents. This enables us to form with great ease and rapidity any recjuired number of principal convergents, after the first two are knt)wn. Thus, keeping to numerators alone, 5x44-2=- 22, 22x2+5 = 49, 49x4+22 = 218, 218x2+49 = 485. The denomina- tors may be similarly computed, thus the denominator following next after 198 is 198x4+89 = 881. The error committed in taking EVOLUTION. Ill 7 3 led [ipidity kt two 2 = 22, jomina- llowiug I taking 485 £0^ = 2-449495 for 6- is loss than — i < _ A„ < -OOOOO?, 198 X 881 IGOOOO henco (J- = 2*44949 — , correct to six figurofc;. Example ii. Form a series of convergents to the cube root of 6. 13<6<2'^, .'. we take \ and ^ as initial convergents, and form from them a series in the usual way, cubing each term to test whether it is a superior or an inferior convergent. 1 T' 2 1» 3 4 7 a Tr» A> 4» 5» V. !?, ih #, n, II. W, IJ 14 IGP •J. '5 > 3^8 ] r 5' 407 •i 5 7 • 1^^ =: 1 -8171206 -f , which is the cube root of 6 to eight figures. The next two principal convergents to 6-^ are 467x508 + 149 ^^^ 467x509 + 149 257x508+ 82 ''^'^ 257x509+ 8?' 113. This is the oldest and perhaps the simplest systematic process for obtaining fi series of approximations converging to the value of any required root of a given number. It is subject however to the distidvantage of being extremely tedious and laborious except where the law of immediate formation of the successive principal convergents is known, in which case it becomes an easy and rapid method of evolution. The following examples exhibit one method of directly computing the successive principal convergents to the square root of ,a given number. Example. 1. Find approximately the square root of 31. ,1 31^=5 + 1 5 5 6 1 14 5 5 3 2 13 5 5 3 3 4 15 5 6 1 1 1 10 5 6 1 1 4 &c. 5 3 &c. 1 3 &c. Quotients. Convergei its. 5 1 1 r. T> 0» 1> T> 1 3 5 T » a' » 1 3 rT8» 1 1 8 ~Z7S> 10 &(i. IfiOCS a S 8 5 • The first column always consists of 0, 1 and the greatest integer whose S(piare is less than the given number. In this example the first column will therefore consist of 0, 1 and 5. Let o, h and c denote the r umbers in any column ; a denoting the number in the 1st row ; 6, the number in the 2nd row ; and c, the number in the 3rd row. Let A, B and (7 denote the corresinrnding numbers in the next following column. The successive columns are formed each from the column next before it, thus : — 112 ARITHMETIC. Ill Mi! ill ^;! A = be - a ; B = . — - — , C = integral part f)f _ — , in which N denotes the number whose square root is required, in this examjUe SI, and / denotes the integral part of the square root of N, in this example 5. Thus in the first column a = 0, b = 1, c = 5 ; (A.- 1x5-0 = 5. 31-5" the second column is B = C=Int. f'll 1 5 + 5 = 6. = 1. Tu the second column a = 5, b = 6 and c = 1 ; rA= 6x1-5=1. the third column is B = C= Int. = 5. 31-1 " 6 " 6 This process of forming each column from the preceding column is continued until the second column occurs again, after which the several c«>lumn8 are repeated in the same order. The principal convergents to 31 -^ are obtained from the initials ? and I, by employing as ' (^uotien^s ' the numbers in the third row, viz., 5, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, &c. Example 2. Find a series of principal convergents to 6'-. The greatest integer whose square is less than 6 is 2, . *. the first column is 0, 1, 2. The succeeding columns are formed each from the immediately preceding column, thus : — A = bc — a. B - ^IrJ^, C - integral part of ^-±^. b B 6--i=2 + 1 2 2 2 2 1 2 4 2 2 2 4 2 4 Sec. Quotients. Convergent s. 1 2, 1 2 ' 0' 1' 2, 4, 5 22 2' 9' 2, 49 20' 4, 218 89' 2, 485 198' 4, 2168 881 &c. Compare with Example 1, § 112, p. 110. f 't .^i EVOLUTION. 113 &c. Find, 1. EXERCISE IX. correct to six signiticant Hgures, the value of :— 3. 3. 6. 7. §. 9. 2§. 29. 30. 31. 32. 33. f ^^ l9 r 250 ^ 1 2401 J L9J • ri7^ ^ 125 J • I32J • ^]\ 27 J r3 lid 10. 11 r3 l8 ] r 121 1 175 13. .45 J 13. 3^. 14. 5^. 15. 15^. 16. 17^. 17. 2A 1§. 26^. 19. 35^ 20. 372. 21. 7^. 22 11^. 23. 53^. 24. 77'-^". 25. 972. 26. 16012. 27. 24002. (7 111 r 9 1 2 „ _ I -~ , given — . =_J1 = '^20 J 20 1(X) 102x242" 2402" ' given ^-1!- 77x 402 3512-1 n 121 112x402 440^ 9 45 45x242 1612- 1 ri3) I24J given 13 78 _ 78x62 532-1 112, given 11 24 144 ~ 122^62 ~ ^22" 11x32 10^-1 32" - "3^^ • (J2, given 6 ^'>x 202^492^-1 52, given 5^ 202 202 5 X 42 92 - 1 42 42 » 114 ARITHMETIC. »• o- o 2x5- i-' + l 34. 2', given 2 = .■=' — — , 5- 5^ also 2 = 2 X 1^-' 17- - 1 35. [A^V 37. |i^^^ 2x29- 41 '- + 1 36. (' 729 I ■■■' U913J 3§. ■ 841 ) 87mX).' 29- 39 r2v^ 5 -^ ••' 40. 1.7^ J ■ 114. Two given (quantities are conimensurable if there be an integral inultii)le of one of them which is also an Integral multiple of the other. For exami)le, let there be two lines A. A and B of lengths such that a third J5. line which is five times the length of the line .1 is twelve tinies the length of the line B. Divide this third line into ox 12 = (50 eijual parts, then the length of any one of these parts will be ^ of five times the length of the line .1, i. e., the sixtieth part of the- third line will be -^.i of the line A or be contained twelve times in the lino -I. But the length of the same part will be ^ of twelve times the length of the line B, i. e., the sixtieth part of the third line will be 1 bf the line B or be contained five times in the line B. Hence a sixtieth part of the third line will measure both the line A and t]ie line B, i. e., the lines A and B have a common measure or arc commensurable. Expressed in symbols the preceding example is : — If 0A-I2B 6 A 12 B oxl2~5xl2' A B 12 A 12 fB and B = 6 f I 5 J . •. 'J and B are cmnmensuivable, 1 of B being a common measure or common unit. EVOLUTION. 115 Measure 116. If either of two commensurablo quantities be expressed in terms of the other as unit, the number expressing tlieir ratio or relative magnitude will be an integer, a fraction with integral terms or with terms reducible to integers, or a mixed nundier consisting in part of an integer and in part of an integral-termed fraction. For this reason integers, integral-termed fractions and integral-termed mixed numbers, whether decimally expressed or otherwise, are called commensurable or rational numbers. 116. Two given quantities are incommensurable if no integral multiple of one of them ia an integral multiple of the other. If either of two incommensurable quantities of the same kind be expressed in terms of the other as unit, the number expressing their ratio or relative magnitude will not be expressible exactly by any integer, integral-termed fraction or integral-termed mixed number whatever. For this reason a number which cannot bo expressed exactly by any integer or any fraction or mixed number with integral terms ia called an incommensurable or irrational number. If the length of the diagonal of a square be expressed in terms of the length of a side of the square as unit, the immber expressing their ratio or relative magnitude will be the S(piare root of 2. Now, in extracting the square root of 2, whe<-her as a decimal number or aa a fraction, there is always a remainder i. e., it is impossible to find a rational or commensurable number of which the square is exactly 2, Hence 2- is an incommensurable number, and the lengths of the diagonal and the side of the same square are relatively inconuuensurable quantities. Other examples of incommensurable numbers are 3-, 5-, 10-, 2'^ 5*, 9\ 100^, 2S 4S lOOS sK 117. Every number formed by ccjmbining a definite number of ones (or of integers) by means of the operations of addition, subtraction, multiplication and division, and of these only, is reducible to an integer or to a fraction, proper or iuq)roper, Avith integral terms, i. e., every number- so formed is a commensurable number, hence no ineommerusnrahle tLiimher can be ex2)re.s)ied bij rombining a definite number of commensiirable nnmbers by additions, subtractions, mvUiplications and divisions and. these only. li 116 ARITHMETIC. 118. IiiconiinunsurHblo numbers which can be formed from a deiinito number of commensurable numbers combined by means of the operations of addition, subtraction, multiplication, division, involution and evolution, are sometimes called surd numbers or surds to distinguish them from incommensurable numbers which cannot be so formed. The latter are called transcendental numbers. Examples. 2 h 3^, 1 + 2^,3 + 2^ 5=i X 6*, 8^-r4^, are surds. **• .... The ratio of the circumference of a circle to its diameter is a transcendental number as aiso is the exponent which expresses the degree of the power which 20 is of 10. (See Loijdrithmts.) 119. Involution is tha operation of raising a given base to a power of given degree. In the examples of this operation hitherto considered, the exponent or index of degree of the power has been either an integer or, in the case of roots, the reciprocal of an integer. But no such; restriction need be laid on the values of exponents ; these may be integral or fractional, commensurable or incommensurable, positive or negative, provided that the terms degree and power be interpreted in accordance with this extension and provided that the laws laid down for operating upon and with these generalized powers are consistent with each other and include as particular or special cases, the laws governing operations upon and with powers of integral degrees and their corresponding roots.. These laws which thus constitute the Fundamental Theorems of Involution and Evolution are ; — XXIV. If equals he raised to eqmd degrees, (have equal exponents), the powers are equal. (Equal-degreed roots of equals are equal. ) XXV. Equal powers of equals are of equnl degree, (Ixavc equal exj>on€nts. ) (Equal roots of equals are of equal degree.) XXVI. Raising the base to any degree raises the power to the power of itself of that degree. (Extracting any root of the base extracts the equal-degreed root of the power. ) XXVII. Multiplying the exp~ V or 6"^^ = {6"y. Examples of TJieorems XXVIII mul XXIX. 2 3 2+a 6 7^_x7_=-7 =7 , 7"x7'' = (7x7)x(7x7x7)=7''. 042 X 64* = 64'^^* = G4^. :64«. 642=8 and 643 = 4. 8 X 4=32-2 °=(64V =< 7 J ^ 7i*=7J+f =7iil^7 X 71^. S 2 5^2 3 3 ^3 =3 =3 , 3''-:-3^=(3x3x3x3x3)h-(3x3) = 3x3x3 = 3" 6*4-65 =6« 5=61''. C -3 -5 3 6-3 5 x5 =5 4-5 =5 =52. 6 -3 5 x5 =(5x5x5x5x5) X (l-i-5^5-7-5) 5 3 = (5x5x5x5 x5)-f(5x5x5) = 5 -=-5 . 1 -.-I ..I -.1 -.1-1 X 64-i X 64 :642-^64-»=642 •'=64«. 64'^ X 64"-^ = 8 X 1 = 8 ^ 4 = 64' -V- 64«. 4 _2 i 2 4_2 2_ 115x11 « = 115 4-ll'i = 115 ■=> = 11T5. = 2 2-.I 2-525 3 -3 2 x2 =2 -=-2 =l4-2 =2 , -3 —2 3 2 3 2 3+2 -5 3 x3 =14-3 ^3 =l-f-(3 x3 ) = l-^3 =3 a -4: 3 4 3+4 7 3 -=-3 =3 x3 =3 =3 . -3 -4 -3 4 4 3 4-3 3 ^3 =3 x3 =3 4-3 =3 =3. ■■\ /: ■"•'■■- ■'-.^■'■■^it^/i'Ti/, y:. ^Tf-^r? ' "'•':'?(■" EVOLUTION. Hi) Examplea of Tlieorem XXX. 1. 3^5''= (3x5)^=15'' for 3 % 5^" - (3 X 3) X (5 X 5) = (3 X 5) X (3 X 5) = (3 X 6) *. 3. 4-^x9^= (4 X 9) ^ = 30^ for 4'^ = 2, 9'^ - 3 and 6 = 36'^ ,-. 4^x9^=2x3=6=36^ and 4. 5. 2\3*=(2x 3)^=62; 2^ = 1-414214 X, , 3^ = 1-73205 + , 6^ = 2*44949 -, 1 -414214 X 1 -73205 = 2 44949 - . •3 -3 7 xll •3 •» :(7xll) =77 . 8 V 27*^ ==(8 +27)^ ' -2 -3 22 7. 7 +11 =11 +7 = rill a l7"J {1^-^ - luJ §. ifsJ re J =■- r«x-^.r^ r2^ s ^ Lr2J ="- ii5"r2J -"- 19J [121. The fundamental theorems of addition and subtraction set forth in §42, those of multiplication and division set forth in §^62 and 63 and those of involution and evolution set forth in ^§118 and 119, may by mere counting be proved to bo true in every instance in which the numbers to be combined are all commen- surable, but they cannot be thus proved if the numbers to bo combined or operated upon are incommensurable. In the latter case we practically assume or postulate the truth of these theorems which thus contain i^pplicitly, or rather actually become the definitions of, the generalized operations of addition, subtraction, multiplication, division, involution and evolution. For instance, we may prove by mere counting that twice three is equal to thrice two, that one-half of one-third is equal to one-third of one-half, that the square root of four multiplied by the square root of nine is equal to the square root of nine multiplied by the square root of four, but we canaot by such method prove absolutely and completely that the square root of two multiplied by the square root of three 120 ARtTHMETlO. I: I I' is equal to the square root of three multiplied by the square root o£ two or even that twice the B -r64^ ~(}4 ^=rfi4a-i-fi4^~ru~i 90. -1 ]^-[ igJ 10 J -I [Ar=[ IV. EJ -Ux5; -20 . 7 7/ \7 7 3 x6 =(3x6) -18 . _ (5 5 / \r> 6 _ 10 10 / \10 93. 3 x4 =(3x4) =12 . 9T. 7 x3 =(7x3) =21 91. 3 x4 =(3x4) =12 93. 3 x4 3x4; =12 95. 4 x5 90. 10 2 2/ \2 2 94. 5 x3 =(5x3) =15 . »». (jrx(ir-(ixjr=(i) 98. (0'x6' = (j-xo)'-2' 100. 2:x(|y=(2x|r=(|)\ 109. a)va)^=(H^)=(i)' 101. 3■-^4' = (3-^4)" = (|)^ 110. 2V(l)' = (2-fl)'=8'. 111. 3 x4 112. 3 x4 102. 3%4'' = (3^4r = (|) 103. 3 -j-4 =(3-f-4) =(^) U 3 / \2 104. 5 H-3 =(r;) . •J 2 /" \2 105. 4 -r5 =(i) . 3V6^ = (3^6)' = (.0.' 115. 4" _10_10/\10 — •> — '1> -a -3 (3x4)" 12 12 -a -f> -5 113. 3 x4 =12 114. 5 — ••> o 106. 10 10 / \ lor. 7 ^3 =(5) 116. 3 ' x(5 '=(3x0) ' = / \8 »^ /, \s / \8 _ 10 -10 10 10 108. (i) ^6 =(HC/ -dV !!''• 7 ^3 =7 x3 = 18 1 21 -7 10 EVOLUTION. 123 -8 10^. 130. 3'5x6^ = 16'^. 131. 5^x7- = 35-^. 133. 5^x10^=50^. 133. 2^x3^x6^ 30 V 134. 2^x2^=4^ = 2'^. 135. 2^x3«=6^'. >h iQo.2-%(i)^=rx(ir=(r=(.i)^=2\ Find, correct tt) six figures, the squaro roots of 2, 3, 5, 6, 7, 8, 10, 12, 15, 18, 20, 24, 27, 30, 35, 50 and the c\ibo roots of 2, 3, 4, 5, 6, 10, 12, 15, 16, 24, 25, 135, 25fi, and employing thcso roots and actually perf\E3l lae. (4Ar=(4^K=4l 19T. (5ir = {5^K=5.l ia§. 2^x3^ = 6^ 136. 2^x4-^ =8*, i. e., 2-^ x2'5 = 2'^ = 2 13T. 2^x5^=10*. 141. 12 138. 3^x4^ = 12*. 149. 18 139. 3^x5'^ = 15i 143. 20'^ = 5^ x 45 = 6=^x2,= 2^5 140. 8^ = 2^x4^ = 22x2,= 2/2. 144. 502=2^x25^=2^x5, =5^2. 145. 24^=6^x4^ = 6^x2,=2V614r. 16^ = (2*)-' = 2^ x2, = 2V2 146. 27^=3^ = 3^x3,= 3 V3. 14§. 24^=3^ x 8* =3^x2,= 2-jy3 149. 135* = 5* X 27* = 5* X 3, =3%/ 5. 150. 256^ = (2**)* = 2%2M4V4. 122. If a number which is correct to but a few significant figures, be either very large or very small, it may in general bo most conveniently written as the product of two factors, one factor being the number expressed by the significant figures with the decimal point between the first and second of them, the other factor ?^=32x42=3Jx2,= 2V2 i = 2'^x9'^=2^x3,= 3V2 124 ARITHMETIC. li ■1! 1 I M 1 1 I! being the power of 10 required to yieltl the propoacil number as the product of tlio two factors. Tho oxp(»nent of 10 in tho second factor ifj called tho characteristic of tho numhur to 10 as base. K.nunpie 1. Tho sun's mass is 3.'>0,000 times that of tho earth and its distance from tho earth is about 91,400,000 miles ; these numbers might bo written S.TxlO" and 9*14 x 10' rc^spectively. Tho characteristic of tho first number is 5, that of tho second is 7. JCx.ampU ii. Tho velocity of light is about 186,800 miles per scc(»nd and tho wavo-length of green light is about '0000208 of an inch. These (juantities may bo written 1 •803 x 10"^ miles ])er second and 2*08 x 10"" iiu^h, respectively. Tho characteristic of the wave, length number is )n'(jatiw,. Example S. Find the cube of 15876 con^ct to five significant figures. 1! 2 4 5 ~() 7 8 9 10 15876 ~31752 47628 ():i504 79.'J80 95256 111132 127-;J3()-x 10' =5-91311). X 101 «. Wt. of ojirb. iicidiriis iu this {ur = 5.91311). x lO^o x 3-5 x 10-' x 1-52 Wt. of carbon in this jias = 3151b. xlO'. = 3 1511). X 10" X3-M1. = 8-Hll). xlO". = 3,800 T. InqjerUd. EXERCISE XI. jting on a What is tho characteristic factor of 1. 33240. 2. 7WKKKK). 3. 2098()00(KKK) Writo in ordin 4. •(MM)0.'j;J5. 5. -OOOOiMJOSi. 6. 12750-78 iry notation 7. l-(XK374xlO §. 1-27418x10^ -0 10. 6x10 11. 10832 X 10' 21 9. 2-26x10 -\ 12. 3-04763 X 10 Find to live signilicant ligui-oa tho valuo of : — 13. 9-14x10 X 1-60933x10 14. 4-73x10 X 1-0089x10 -8 15. 10 ^1-2759. 16. 3-98x10 ao ^( 4.374x10 ifi 17. 1-863x10 X 6-336x10 -^( 208x10 18. 10"-j-(981x8-837xl0 "'). 19. 4\3 20. (1-27418x10) x3•1416-^6 y •33092 X ( 6-37x10 21. 1-96x10 X 283x11-22x10 h -5\2 22. U-6xl0 X 6-3709x10 x2K 2)' -.s\t 23. 1,2-37x10 ;^x 1-4707x10 ( 24. \6- 25x10 -11 25. (3003x10 -10\1 )^^(3-1416x 3-956x10 26. (4xl0')-^^(4xl0 "), 27. (l- 275678 X 10^ X 1-275584 x 10^ x 1271278 x 10^)-\ -t-^-w i I I 1 [ I I ! M i LOGARITHMATION. 123. The Logarithm of a given number to a given base is the exponent of the power which the given number is of the given base. T/ie terms logarithm mid exponent are therefore tnerely different names for the same thing. Thus, instead of saying "the exponent of 100 to base 10 is 2" we say "the logarithm of 100 to base 10 is 2 ;" instead of saying " the exponent of 32 to base 2 is 6 " wo say ' * the logarithm of 32 to base 2 is 5 ; " and instead of writing 100 = 10^ and 32 = 2^ we may write log 100=2 and log 32 = 5. If the base is , , 10 . . . 2 . 10 it is nsually omitted both in loriting and in reading logarithms. Examples. 81 = 3 , 125 = 5^ 10 1024 = 2 , 2401 = 7 , 1331 = 11^ log 81=4. 3 10=10 , loglO=l log 125 = 3. 5 100=10 , logl00=2. log 1024=10. 2 1000=10^, logl000=3. log 2401=4. 2= 8*, log 2 = i 8 log 1331=3. 11 f 1r.« 1 ••71 -^ i 4= 8^ log 4 = 5. 8 27= 9^'\ log 27 = 1-5 for I ! log l-7J 5 5 1-7 <5i but 1-71>5?5 i. e., l-7^<5 but 1-71>5 for 1-7 ^=4-913 and 1-71 = 5 000211. EXERCISE XII. Prove the truth of the f(tllowing statements, and express them in logarithmic mutation : — 5. 3,125 = 5". 9. 2 = 16'^'. «. 7,776 = 6"''. 10. 4 = 16 ^ 7. 14,641 = 11*. 11. 8 = 16*^". S. 1,000,000=10". la. 32 = 16^"^° 1. 128 = 2 . 2. 256=1*. 3. 729=3^ 4. 729=9'. LOGARITHMATION. 127 3S8 them in 1.5 13. 04-16 14. 1024=16""^ 15. 125-25'" 16. 279936 = .% i^V = 2 -fl -3 ir i§. ^4=4^ 19. ^ = 8\ 20. <^\= '16' iJ)j=3 -5 21. 22. ^1^ = 9 23. 0.1 = 10 24. 0.0001 = 10 -1 -4 Prove the truth of the following statements and express them in exponential notation : — ■ 25. log 8 = 3. 26. log"G4 = 3. 27. log 512 = 3. 28. log**343 = 3. 29. log '2187 =7. 33. log 1024 = 3?5. 34. log** 5= -5. 35. bg 9 = §. 37. log^(;5V)=-5. 3. 41. log 10 = 1. 42. log 1000: 43. log 100000 = 5. 4i. log 1 = 0. 45. log 0-1= -1. 30. log 10077696 = 9. 38. log (Jj)= -4. 46. log 0.01 = 31. log 20736 = 4. 12 39. log (o-4V)=-4. 47. log 0.001= -3. 32. log" 16.777216 = 6.40. log (70^^)= -3^.48. Iog0.00001= -5. 49. Prove that log 2-154< ,:^ hut that log 2155 > J. 50. Prove that log 2 is somewhat greater than '3. [124. The word logarithm means ratio-immbcr, and logarithms were so named because they record the number of successive multiplications (or successive divisions) by a fixed base, a conunon ratio or rate of progrossicm as it was at first called, the initial multiplicand (or initial dividend) being in every case 1. Thus, 2 is the fixed base, the common rate of progression by multiplication, of the series of numbers 1, 2, 4, 8, 16, 32, 64, 128, 2.56, 512, 1024 aud 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 aio the corresponding logarithms recf)rding the nimiber of successive multiplications by the ratio 2. The fixed base or common ratio of progression by multiplication is 3 in the series <.f numliors. and 1 1 1 27' 9' 3 3, -2, 1, 9, 27, 81, 24.3, 729 0, 1 , 2, 4, are the corres[)onding logarithms. The sign - preceding the first ''< V. "tt^-- ! ) 1 i i! 1 I I i i 1 ■ 128 ARITHMETIC. three of these logarithms denotes tliat successive divisions, not multiplications, are recorded. The common rate of progression by nuiltiplieation is 10 in the series 00001, 0001, 001, 01, 1, 10, 1(H), 1 = .og 256 = 8-3 + 5=log 8 + log 32. i.e. 8 = 2', 32-2^ s 3+5 a 5 8x32-256=2 =2 =2x2. Example 2. log 4 = 2; t.iU 5 log 4= 10 = log 1024 = l«»g '2 2 2 2 4 = 2 ; 10 2x5 s 2 =1024 = 2 =4 . (4'). LOGARITHMATION. 129 EXERCISE XIII. Prove the truth of the following statements : — 1. log (16xl28) = log 16+log 128. 2. log^(16 X 128) = log^ 16 + log 128. 3. log\l6 X 128) = log^lfi + log'^ 128. 4. log^(512-^64)=log^512-log 64. 5. log^(512^64)-log^512-log^64. 6. logV4-^256) = log'^64-log 256. 'ST. log^(2x32^4) = log 2+log'^32-log 4. 8. log ^7 X 243) = log 27 + log '243. 9. log^(27x243)=log^27 + log'243. 10. log® (27 -f243) = log** 27 - log 243. 9 9 9 11. log 8^=3 log 8. 16. 12. log 8^ =3 log 8. 17. 13. log 8 =3 log 8. 1§. « 3 14. log 8 =3 log 8. 19. 16 16 15. log 82=1 log 8. 20. 4 4 loc: 8'-^"^ ilo^S. log 3 = -25 log 3. 9 . 9 •2 log 3. 9 = -4 log 27. 9 log 3 9 log 27" log -25 = -7 log '25. 8 8 126. Logarithmation is the operation of finding the logarithm of a given number to a given base. It is, therefore, an inverse both of involution and of evolution ; for in involution a base and an exponent are given and the power of the base denoted by the exponent is required , and in evolution a power of an unknown base and the exponent of t'lat power are given and the unknown base is to be found, but in logarithmation there are given a base and a number conb.idered as a power of that base and the exponent which denotes that power is to be determined. For example, involution and evolution would furnish answers to the questions * What is the fourth power of 3 ? ', * What is the cube of the tenth root of iO ? ' ; but logarithmation is required to answer the questions ' What power of 3 is 81 r, ' What power of 10 is 2 ? '. \r .^^ I h I i ! I ! i i!^ I !i ! ; .1'. r 130 ARITHMETIC. 127. Thus of the seven fundamental operations of Arithmetic, addition and subtraction are each the inverse of the other ; so also are inultiplication and division inverse to each other, but the three remaining operations, viz. , involution, evolution and logarithmation, are so related to one another that each has the other two operations as its inverses. 128. There are several methods of computing logarithms, but we shall give examples of only two of them. Of these, the first was one of the methods proposed by Napier the inventor of logarithms, and was the method by which the first publisned tables of logarithms to base 10 were calculated. 129. First or Napier's Method. Extract the square-root of the base correct to three figures more than the number of decimal places to which the logarithms are to be correct. Next extract the square-root of the root just found, then extract the square-root of this last-found root, and so continue until there has been formed a table similar to Table I which follows. In forming this table, 10 having been selected as the base, tho roots were extracted to ten decimal places and eight decimal places retained. TABLE A. 10^ • 5 = 10 =3-16227766. 3 •16227766" • 1 5 = 10 = 1-77827941. 1 •77827941'^' • 12 5 = 10 = 1-33352143. 1-33352143^ ■ 0025 = 10 = 1-15478198. 1-15478198^ .03125 = 10 = 1-07460783. 1-07460783^ • 015025 = 10 = 1-03663293. 1-03663293^^ .0078125 = 10 = 101815172. 1-018151722 .00390025 = 10 = 1-00903505. 100903505^ .0019S3125 = 10 = 1-00450736. 1-00450736^ • 000970503 = 10 = 100225115. 1-00225115^ • 000488281 -10 = 1-00112494. 1-001124942 •000244141 = 10 = 1-0CD5623L 1-000562312 •000122070 = 10 = 1-00028112. 1-00028112^ .000001035 = 10 -100014066. LOGARITHMATION, 131 130. The exponont8-A\-hich would follow -OOOOGIOSS in order m the preceding Table, are obtained by taking ^, }, ^, ^^, &c. of •000061035, and the decimal parts of the corresponding powers, correct to eight decimal places, by taking i, J, J, ^^y &c., of •00014055, the decimal part of 1-00014055, the power of which •000061035 is the exponent. Hence the logarithm to base 10 of any number greater than 1 but less than 1 00014055 is, to nine decimal •000061035 , , . ,. -000061035 . ,, P^^'"'' •0Wi405V^"'^^'^^-'''"'''^PP'^^""^^^^^"' -06(lU()^ ^^ ^^'^ decimal or fractional part of the number. 131. The fraction -—r:r7T7r,.-T=- which is equal to -434273+ is •000140545 an approximation, correct t() four decimal places, to a number called the Modulus of logarithms to base 10. If any number other than 10 had btei. nade the base in Table A, a different number would have been obtained as the modulus , e. r/., had the base been 2 718281828459, the modules would have been 1, i. e., the loga- rithm to this base of any number greater than 1 but less than 1 •OOOl is simply the decimal part of the number, correct to eight or more decimal places. Had the roots in Table A been calculated to 32 decimal places, it would have been necessary to extend the columns to fifty-five terms before the decimal parts of the roots would be proportional to the exponents,* but in such case, the modulus would have been obta.ined correct to some eighteen decimal places. It has been computed to 136 decixaal places t ; to twelve places it is •434294481903. 132. If the powers in the third column of Table A be considered as given numbers, the exponents in the second column of the Table will be their logarithms to base 10. In Table B which follows, ti.e * Such a table was actually computed by Henry Briggs, Savilian Professor of -64 Geometry at Oxford. The fifty-fifth exponent or 2 he found to be 0-000,000,000,000,000, )S5, 511, 151, 231, 257, 827 and the corresponding root, the result of fifty-four successive extractions of the square root, to be I "000,000, 000,000, 000, 127, 819, 149, 320,032, 35. Briggs was the first to compute and pubLsh logarithms lO the base 10. t By means of the series the earlier terms of which are given in problem 65, page 90, The modulus is the reciprocal of that series. ^1' !r Ml . ,11: ,1 I ! ' I I ill 'I'li 132 ARITHMETIC. ])ower8 are tabulated as numbers and the exponents as the loga- rithms of these numbers. TABLE B. Number. 1 Lor.ARITHM. Number. Logarithm. 10 1 100903505 •00390625 3 16227766 •6 100450736 •001953125 1-77827941 -25 • 1-00226115 •000976563 1-33352143 -125 100112494 •000488281 1-15478198 •0625 100056231 •000244141 107460783 •03125 100028112 •000122070 1-03663293 •015625 100014055 •000061035 101815172 •0078125 10001 -000043427 TAB] LiE C. Multiples of the Modulus ^^^t^ig 1. -43427. 4. 1-73709. 2. -86855. 5. 2-17137. 3. 1-30282. 6. 2-60564. 7. 3-03992. 8. 3-47419. 9. 3-90846. Example 1. Find the logarithm of 2 correct to eight decimal places ; t. e. , find the exponent of the power to which 10 must be raised so that the result may be 2. From the columns of Numbers in Table B, (Col. Ill, the column of powers in Table A,) select the largest number less than the given number 2, and divide 2 by the number thus selected. From the columns of Numbers in Table B select the largest number less than the quotient just obtained, and divide that quotient by this second selected number. From the columns of Numbers in Table B select the largest number less than the last obtained quotient and divide lOGARITHMATION. 133 that quotient by this tji^a selected number. Continue thus to select and divide until there is obtained a quotient less than 1 00014055. These operations resolve 2 into a series of factors all of which, except the last, are numbers in Table B. Consequently the logarithms of these factors, except that of the last factor, are given in Table B and the logarithm of the last factor can be obtained by multiplying the decimal part of the factor by the modilus '43427. The logarithms of the factors being known, the logarithm of 2, their product, may be found, being the sum of the logarithms of the factors. 2 -^ 1 -77827941 - 1 12468265 1 12468265 -f 1 '07460783 = 1 04659823 1 04659823 -r- 1 '03663293 = 1 00961314 1 00961314 -7 1 '00903505 = 1 '00057292 1 00057292 -f 1 '00056231 = 1 '00001060 . '.2=1 '77827941 x 1 '07460783 x 1 03663293 x 1 '00903505 X 1 00056231 X 1 00001060 • 25 .0U125 .015025 •00390625 = 10 xlO xlO xlO • 000244141 .000010fiX.4;U27 X 10 X 10 .2 6 + .0;il2 3 + ^0150 2 5+^OOa90«2 5+^000244141+.000004604 = 10 = 10- •301029995 or log 2= '30103000, correct to eight decimal places. Written in logarithmic instead of in exponential notation, the latter part of the preceding calculation would be log 2= log 1-77827941 + log 1 '07460783 + log 1 '03663293 + log 1 -00903505 + log 1 '00056231 + log 1-00001060 = -25+ -03125 + -015625+ 00390625+ '000244141 + -0000106 X -43427 = -301029995, .-. log 2=3*0103000, correct to eight decimal places. [log 2= -301029995663981, correct to fifteen decimal places.] Example 2. Find log 48847, correct to eight decima^ places. Write 48847 in the form 4-8847x10* and resolve 4 '8847 into factors selected from the columns of Numbers in Table B. 4-8847 +3 16227766 = 1-54467777, 1 -54467777 + 1 -33352143 = 1 -15834491 r III." 1 1 1 1 i ! 'I ■PI 134 ARITHMETIC. 1 -15834491 -r 1 15478198 -= 1 00308637 1 •(KJ308637 -r 1 00225115 = 1 00083235 1 00083235 -H 1 -00056231 = 1 00026988 1 00026988 -r 1 00014055 = 1 00012931 ; . '. 48847 - 10* X 316227766 x 1 33352143 x 1 15478198 x 1 00225115 X 1()005()231 X 1-00014055 x 1 00012931 ; .-. log 48847 = log 10 • +l()g 3 -16227766 + log 1-3.'«52143 + l()g 1-1 5478198 + log 1-002251 15 + log 1-00056231 + log 1 -00014055 -I- log 1 00012931. =4+ -5+ -125+ -0(525+ 000976563+ 000244141 + 000061035+ -434273 x 00012931 = 4 688837895, correct t(j within 1 in tho last figure. .*. log 48847 = 4-68883790, correct to eight places of decimals. 133. The logarithm of any number may be found by this method independently of finding tho logarithm of any other number, but in foi-ming a table of logarithms, the logarithms of prime numbers alone need be computed, the logarithm of any composite number being the sum of the logarithms of the factors t)f such composite number and the loga "ithm of a power being the product ci the logarithm of the base of the power and the exponent of the power. Thus knowing log 2= -3010300, we obtain log 4 = log 2^=2 log 2= -6020600, log 8 = log 2'' =3 log 2= -9030900, &c. 134. The knowledge of the logarithm of one number will (jften greatly aid in computing the logarithm of another number which diifers but little from tho number wht>8o logarithm is known. Example .'/. Find log 81 correct to eight decimal places, given log 80=1-903089987. 81=80 + 1=80 x(l + ^o) = 80x 1-0125. Resolve 1 '0125 into factors selected from the columns of numbers in Table B. 10125 +1-00903505 = 100343393 1 00343393 + 1 00225115 = 1 00118012 1 00118012 + 1 -001 12494 = 1 00005512 . -. 81 = 80 X 1 •00903505 x 1 ■002i^^ 115 x 1 -00112494 x 1 -00005512 . -. 1( .g 81 = log 80 + log 1 •00{M»35( H) + log 1 -00225115 + h .g 1 • X)112494 + log 1-(HM)05512 = 1 •1H)3089987 + -(M>35K>625 + -000976563 + -0004S8281 + -43427 X ■(KKK)5512 = 1 -908485018, correct to within 1 in tho last figure. LOOARITHMATION. 136 4 -2x 4 = :8 3+1 4 3 -1 = =8x 10 -10 = 80 -1 .*. log 81 = 1 •JK)848502, correct to eight pliicoa of decinmls. From log 81 wo luiiy obtain log 3 for 81 = 3*, . •. log 81 = log 3* = 4 log 3. .-. 4 log 3 =1-90848502, . •. log 3 = '47712125, correct to eight decimal i)lace8. [logs = -477 1212647 1^K><)2, correct to 15 decinml places.] This problem is virtually, — Find log 3, given log 2. Wo proceed thus ; — 3-1 = 2 and log 2 is given, and log 4 = 2 log 2, and log 8 = log 2 + log 4 ; and log 10 = 1, and log 80 = 1( )g 8 + log 10 ; 3 =80 + l = 80x(l + Jo) = 80x 1-0126. Tlid remainder of the calculation is that already given. Example 4- Find log 7 given log 2 and log 3. 7-l = G andlog6 = log2 + log3 , 7 + 1 = 8 and log 8 = 3 log 2. .-. 7" -1 = 48 andlog48 = log6 + log8 7%1 = 50 andlog60 = logl00-log2 . •. 7* - 1 = 2400 and log 2400=log 484-log 50 . •. 7* = 2 :00 + 1 = 2400 X (1 + o 4^(7) = 2400 x 1 -00041667. 1 00041667 + 1 -00028112 = 1 00013551 .-. 7* =2400x100028112x1-00013551 .-. log 7^= log 2400+ log 1-00028112 + log 1 00013551. . •. 4 log 7 = 3-380211242 + 000122070+ -43427 x 00013551 = 3-380392160 .-. log 7= -845098040. [log 7 = -846098040014267, correct to 15 decinuil places.] Example 5. Find log 11, given log 2, log 3 and log 7- 99=11x3^ .-. log 99 = log 11 +2 log 3 = log 11 + -954242509. 99 - 1 =98, log 98 = log 2 + log 49 - log 2 + 2 log 7, 99 + 1 = 100, log 100 = 2; I M 'I ',1 ! I 136 AUITHMETIC. ! .-. m -1 = 9800, I()g9800=2 + l()g2 + 2 1()g7; . % 99" = 9800 + 1 = 9800 x (1 + <, ^o) = ^800 x 1 -00010204 ; . •. log 99^ - l(ig 9800 + log 1 •(K)010204 . •. 2 log 99 = 2 + log 2 + 2 h)g 7 + -4.34273 x -00010204 -=3-991270389. .'. log 99 = 1-995(535195 . •. log 1 1 + -954242509 = 1 -996«35 195 . •, log 11 = 1 -995035195 - -954242509 — 1-04139208(5, correct to oiglit plucoa of ducinmls. [log 11 = 1-041392085158225, correct to the 15th docinuU.] 135. If the number to bo resolved into factors selected from the columns of Numbers in ThI)1o B or any (quotient arising in the course of its resolution be but very little less than one of the tabular factors, it will in general be better to use such number or such quotient as next divisor and the tabular factor next greater than it as dividend. The tabular factor then becomes a divisor, not a multiplier, in the resolved form of the given number. Ejcample 6. Find log 3-14159265, correct to eight decimal places. 3-16227766 -f 3 -14159265 = 1 -00658424, 1 00658424 ^ 1 00450736 - 1 00206756 1 -00225115 -v- 1 -00206756 = 1 00018321 1 00018321 -r 1 00014055 = 1 -00004266 .•.314169266 = 3-16227766-^1 -00450736 4-100225116 x 100014055 X 1-00004266 .-.log 3 -14159265 = log 3-162Cr/66-log 1 -00450736 - log 1-00225115 + log 1-00014055 + log 1-00004266 = -5 - -001953125 - -0(X)976563 + -000061035 + -43427 X -000042(56 = -497149873, correct to the last figure. [log li- 14159265= -497149872694134- .] Had 3-14159265 been resolved into -a pi-oduct of factors, as 2 was resolved in Example 1 and 48847 in Exawple i2, no less than nine divisions would have been re(|uired to effect the resolution instead of the four divisions required in the resolution just given. LOGARITHM ATION. 137 EXERCISE XIV. Find, correct t<» 7 dociuial places : — 1. loglOOOOi. 4. log l'(XX)07. 9. log 1-mm. «. log 1'0T.j- -f ' 138 ARITHMETIC. 1 <10 . A 2 > 1 B 1 x2 =2 =2 <10 AxB 2 x2 =2 =4 <10 AxB^ 2* x2 =2^* = 8 <10 C = AxB^ 2* x2 =2* = 16 >10 BxC 4 3 7 2 _ 2 2 x2 =2 = 128 >10 BxC 7 3 10 .3 3 2 x2 -2 = 1024 >10 D = BxC 10 3 13 4 2 x2 =2 = 8192 <10 CxD 13 10 23 7 2 2 x2 =2 = 8388608 <10 CxD 23 10 33 10 3 2 x2 =2 = 85898346.. <10 CxD 33 10 43 13 _.4 2 x2 =2 = 87960930 <10 €xD 431063 16 5 2 x2 =2 = 90071992 <10 CxD 53 10 03 _ 19 6 2 x2 =2 = 91209720 <10 CxD 6 3 1.0 > 7 3 2 2 7 2 x2 -2 = 93398754 <10 CxD 73 10 S3 _ 25 8 2 x2 =2 = 96714066 <10 CxD 83 10 93 28 9 2 x2 =2 = 99035203 <10 E=CxD 9 3 10 103 31 2 x2 =2 = 101412048 >10 DxE 103 9 3 198 5 9 2 2 x2 =2 = 100433628 >10 F = DxE 190 9 3 28 9 87 2 x2 =2 = 99464647 <10 ExF 289 190 483 146 2 2 x2 =2 = 99895964 <10 G = ExF 485 196 681 205 2 x2 =2 = 100329130 >10 FxG 1051 485 2130 643 ' 4 2 x2 =2 = 100016289 >10 H=FxG 11165 2136 13301 4004 2 x2 =2 = 99993628 <10 J = GxH 16437 13301 28738 „ 8661 2 2 x2 =2 = 100003544 >10 K=HxJ 28738 13301 42039 12666 2 x2 =2 = 99997172 <10 L=JxK 42029 28738 70777 21806 2 x2 =2 =100000716 >10 M=KxL 183593 70777 254370 76573 2 2 x2 =2 =99999320 <10 N = LxM 264370 70777 3 2 5147 97879 2 x2 =2 =100000036 >10 P=MxN LOGARITHMATION. 139 A B AxB AxB C = = AxB BxC BxC D= =BxC CxD CxD Cxd"^ €xl>^ Cxd' :L = Lx m' = Mx N 1 10 Writing A in the form 2 < 10 and B in the form 2 > 10 , we have A =2 <10 1 B =2 >10 C=AxB'' = 2^ <10^ 3 10 3 D = BxC =2 >10 93 28 E = CxD =2 <10 2 196 50 F-DxE =2 >10 2 485 146 a=ExF =2 <10 4 2136 643 H=FxG =2 >10 6 13301 4004 J=GxH =2 <10 3 28738 8661 K=HxJ =2 >10 42039 12655 L=JxK =2 <10 70777 21300 M=KxL=2 >10 3 354370 76673 N=LxM =2 <10 325147 97879 P=MxN=2 >10 say 2<10o .-. 2>10^ .-. 2<10^ .-. 2>10i^ .;. 2<10^^ .-. 2 > 10^9^ .-. 2 < 10^5 .-. 2>10i^^ .-. 2<10T^§8t . •. 2 > 10^^'^''^ ... 2 < 10^2^.11 . •. 2 > 10^^^^ ... 2<102^^*^^^^ or log2^ .. log2^ .. log2TVk .. log2^^^ .. iog2fJf?f log2 through < to > again or vice versa ; thus they record without repetitions the number of such multiplications. Quotients, 3, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1 ; 1 3 28 59 146 643 4004 Convergents,!-. -^, -, _, _, _, — , 2136' 13301* 4 8651 12655 21306 28738' 42039' 76573 97879 70777' 254370' 325147' 11 III Ml i V' e•v!»^.5?^"'^'■^-^^i^''?^^*^?^jai^ 'vv^'^WiKS^'*'' ;■ "''''^^jJ-^- ^'''^^'^;t;"■/^l^ 140 ARITHMETIC. The next quotient the 13th. cannot be less than 1, and for 1 as 13th. quotient, the upper limit of error of the 12th. convergent is 1 _.,... 1- • 1 325147 X (325147 + 264370) which is 300000x600000 -12 <6xio ; hence 97878 does not differ from log 2 by so much as 6 in the 325147 twelfth decimal place. But the 11th. and 12th. convergents being close approximations to log 2, the required number, it is not necessary, in order to determine the 13th. quotient, to actually perform the multiplications which that quotient records. Consider for example how the fourth convergent quotient may be determined by the powers of 2 denoted by D and ^, page 138. The fourth convergent-quotient is simply the number of successive multiplications .of 1024, the D-power of 2, by 99035 , the E-power of 2, which are required to produce 3 the F-power of 2, and 1024 is approximately 10 , 99035 28 approximately 10 and the F-power of 2 approximately an integral power of 10 ; the number of these multiplications will therefore be less than the quotient of 1024 -f- lo' - 1 divided by 1 - 99035. . . -r 10^ ^ i. e., than '024 -^ "00965, but will be approximately equal to this quotient. We may therefore use the integral part of '024-7- "00965 as a convergent quotient to form the fourth or F-convergent ; and in point of fact the integral part of -024 -r "00965 is 2, the fourth convergent quotient. The correctness of the foregoing argument may be seen at once, if the proper method of multiplying by 99035 .... be adopted, viz., that described in § 69, xii, page 36. It should however be noticed that if the terms of the division, here "024-i- "00965, are not both very small the convergent-quotient sought may be greater than the quotient arising from the division. For example had we sought to determine the third convergent from (1- •8)-T-(l "024-1) we would hav^e obtained 8 as the third c< »nvergent-quotient instead of 9 the correct value. In like manner from the powers of 2 and 10 yielding any two consecutive convergents after the fourth, the quotient determining the third consecutive convergent may be obtained, and consequently the 13th. convergent may be computed from the powers of 2 and 10 yielding the 11th. and 12th. convergents. Thus ■'•fX' ,'■!■ ^".y ' l ■■" '.T^^.-'.^t, '■■>''",;;;■'■-''':■- ->!,, i'T:«-j7.->^^c;-.^-!,.-.:'.;.>fi.-,i.;?-,_-;. -V.'V -lit ,'.'|-,-,'jAr LOGARITHMATION. 141 as 6 in the the dividend obtained from N is 1 - 99999320 . . . = -00000680 the divisor obtained from P is 1-00000036 . . . .- 1 = -00000036 . '.the quotient is 680 -^ 36 = 18 + .,, iQ.r, .. T o- 97879x18+76573 1838335 .'.the 13th. convergent to log 2 is — —l -^ — - - = 325147x18 + 264370 6107016 An upper limit of error for this convergent is 1 6107016 X (6107016 + 325147) which is •.log 2: 6000000x6000000 1838335 <3xl0 -14 6107016 Example 2, Find log 3. Powers of 3. - 1 3 • 3 9 27 243 2187 19683 177147 1594323 1434891. 1291402.. 1162261... 1046035... 941432... 984771... 1030105... 1014418... 998969... 1013... •3010299956634, correct to 13 decimal places. Multipliers producing the next power. 3 3 9 9 9 9 9 9 9 9 9 9 1046035,.. 1046035... 984771... 984771... 1014418... The multiplier 3 occurs twice, 9 occurs ten times, the others twice, twice and once respectively; hence the first five convergent quotients to log 3 are 2, 10, 2, 2 and 1, and the sixth quotient will be the integral part of (1 '014418 - 1) + (1 - -998969) = 14418 + 1031 = 13 '9 + , which is 13. The characteristic of 3 to base 10 is 0, therefore th© initial convergents are -^r and -r- ; hence we have for log 3 142 ARITHMETIC. Quotients ; Converr- *■■ 1 d 2 1 10 10 2 21 2 62 1 73 13 1001 1' 2' 21' 14' 109' 153* 2098' or log 3 = )1 j98 of this convergent is = '477121, correct to six decimal plac6s, for the error <3xlO -7 209n X (2098 + 153) 4000000 Had 13*9 been used instead of 13 as sixth convergent quotient, the resulting convergent would have been 73.13-9+ 52 10667 .^^^^^^^_ 153x13-9 + 109 22357 which is a closer approximation than even 1001 2098' Example 3. Find log 48847. Powers of 48847. 1 48847 48847 2386029 . . 1165504 . . 569314 . . 663537 . . 773355 . . 901348. . 1050525 . . 946889 . . 994730 . . 104 The first five convergent quotients are 1, 2, 4, 1, 2, and the sixth is 9, the integral part of (1 '060525 - 1) + (1 - -994730) = 50626 + 5270. The characteristic of 48847 to base 10 is ^, and therefore the initial convergents are — and- • Multipliers producing the next power. 48847 48847 48847 1166504 • • • 901348 • • • 1050525 . . . Quotients ; Convergents ; 12 4 12 1 4 5 14 61 76 211 1 1 3 13 16 46 421 9 1974 The error of the l^^zt of these convergents is < 6x10 -0 421 X (421 + 45) 180000 1974 log 48847 =^t2:j- =4*68884- , correct to five decimal places. a;ii?i«B..,jiB,£^^li.iVJ^ir.i„i..iv ■■.■■>:f>iZV''---:^n LOGARITHMATION. 143 tor the error EXERCISE XV. Obtain, correct to 4 decimal places : — 1. log 7. 3. log 31. 2. log 6. 4. log 6-6. 5. log 2-72. 6. log 1-371. 137. Many other methods of calculating logarithms have been proposed, the greater number of them being merely t ariations of one or other of the two processes already described, but all of these methods are so tedious and involve so much labor in thei^ applica- tion that were it necessary to calculate a logarithm anew every time it was required, computation by the aid of logarithms would be a useless curiosity. To overcome this objection to their employment, the logarithms of all integral numbers from 1 to 200,000 have been calculated and recorded to seven places of decimals, once for all. A small part of this record, being a Table of Logarithms correct to six decimal places, is given at the end of this volume. In Table I are entered the logarithms to base 10 of all numbers from 1 to 100, in Table II are given the logarithms to base 10 of all numbers from I'OOO to 9-999 by increments of '001, and Table III contains the logarithms to base 10 of all numbers from 1 to 1-0999 by increments of -0001. The logarithms entered in Tables II and III are all decimals, but in printing the tables the decimal point has been omitted as unnecessary. The deciiaal pari of a logarithm is termed the mantissa of the logarithm, and the integral part, the characteristic of the logarithm. (See § 122.) 138. The following examples will show how to use Tables II and III either to find the logarithm of a given number cr to find the number corresponding to a given logarithm. Example 1. Find log 4-884, log 48840 and log '04884. We glance along the columns marked N**. until we find 488, the first three digits oi the given number ; we then pass horizontally along the line of 488 to the column headed 4, the fourth digit of the given number ; in that column we find 8776, these are the last four figures of the mantissa of the required logarithm. The first two or leading figures are 68, they will be found standing over the blank space which appears in the line of 488 in the column headed 0, 'W '.;■■■"«• ""■:'^' '=:- I I ilMf i i! I II' ji ! []^ 144 ARITHMETIC. Hence log 4-884= -688776. ^ (A.) 48840 = 4-884x10* log 48840= log 4-884 + log 10* -•668776+4=4-688776. (B.) •048vf =4-884x10-2 .-. log-04a. -log4-884+logl0-2 = -688776 - 2 =2 688776. (C.) . It will be noticed that when the characteristic is negative, as it is in (C), the minus sign is written above the characteristic, not in front of it. The mantissa in (C) is positive, being the logarithm of the factor 4-884. Example 2. Find log 4 076, log 407 '6, log 40760, and log -0004076. We firpt find 407 in the columns marked N°. and then run horizontally across to the column headed 6 in which we find "''■0234, the last four figures of the logarithm sought. The * in front of these figures indicates that the two leading figures of the logarithm are at the foot o£ the blank space in the column headed O. Looking there we find the leading figures to be 61, hence log 4-076= -610234. 407 6=4-076x102, log 407 -6 = -610234 + 2 = 2-610234. 40760=4-076x10*, log 40760 = -610234 + 4 = 4 -610234. •0004076 = 4-076 x 10-*,_ . *. log-0004076 = -610234 -4=4 -610234. 139. It may be seen from these examples that changing the position of the decimal point in a number changes the characteristic but does not change the mantissa of the logarithm of the number. The characteristic of the logarithm of a given number may and should be wi-itten down before the mantissa is found in the Table of Logs. , for the characteristic to base 10 is simply the number of places which the first significant figure of the given number is from the ones' figure of the number, the ones' figure itself not being counted, i. e. , it is considered as standing in the zeroth place. If the first significant figure of the given number stands to the left of the decimal point, the characteristic will be positive or zero ; if it stands to the right of the decimal point, the characteristic will be negative* "« If^v"-";*-^-'. !!'•'■:••■■ '■' •'■['^■■;.^,: ■■^>-;''\'r '■■■'" ■.■.::?V':?,-*:'^vv^ LOGARITHMATION. 145 (A.) (B.) (0.) . ve, as it is iic, not in jarithm of 5-0004076. then run bid *0234, n front of logarithm Looking iging the acteristic mber. may and he Table imber of r is from ot being lace. If e left of ro ; if it ! will be 11. 23. 16. 4-321. 21. 12. 230. ir. 4321. 22. 13. 2300. i§. •04321. 23. 14. 2-3. 19. 6789. 24. 15. •023. 20. 7810. 23. 26. 1178x101*. 27. 1-63 xl0*2. 2§. 6-48 xlO-8. 29. 4-496 xlO-«. 30. 7-604 xlO-«. EXERCISE XVI. Find from Table I the logarithm of : — 1. 7. 2. 37. ». 59. 4. 79. 5. 97. Write dow .i the characteristic of : — 6.723. y. -07)23. §.200000. 9.0*00002. 10.372-68. Find the loa;arithm of : — 7246. -007246. 676700. 67-67. 200-2. 140. Example 3. Find log 4*8847. Log 4-8847 is not given in the Tables but evidently it lies in value between log 4-884 and log 4-885 both of which are given and it may be computed from these tabular logarithms as follows : log 4-885= -688865 log -4 -884 =-688776 log 4 -885 - log 4 -884 = 000089 " -7 of -000089= -000062 log 4 -8847 = -688776 + -000062 - -688838 We need not have actually subtracted log 4 "884 from log 4-885 to obtain the ditference -000089, for this diflference is recorded in the right-hand column of the Table, the column headed D. Thus if we glance along the horizontal line in which we find log 4-884, we shall find the * difference ' 89 in the column D, and the computation of log 4-8847 will then appear as follows : — • log 4-884 =-688776 D=89- 7, 62, 3 log 4-8847 = 688838 141. This method of computing the logarithm of a number intermediate in value to two tabular numbers is merely a variation of the method of calculating logarithms exhibited in Examples 3, 4} and 5 of § 134, for 4 -885 = 4 -884 + -001 = 4 -884 X (1 + j .|tti-) log 4-885 = log4-884 + log (I + jbVt) = -688776-|-j8«54of -4343 = -688776+ -000089 = -688865, as given in Table IT. (^.) II n^! i 1 ! 146 ARITHMETIC. 4-8847 -4'8840+ •0007=4-884 x (1 + j.S/^Y) log 4-8847=log 4-884+log (l+j^li) = •688776+4^1 J of -4343 = -688776+ -7 of j^j of -4343 = -688776 + -7 of -000089. (B. ) = •688776+ -000062 = -688838, as found in Example 3. Now the * diflference ' -000089 obtained above in {A) is given in the Table of Logarithms, and knowing this difference and log 4-884, we may at once write down the line marked (JB). Example J^. Find log 2-718282. log 2718 =-434249 D=160 2 32,0 8 12,80 ,320 fSee § ij.J log 2-718282= -434294. EXERCISE XVII. Find the logarithm of : — 1. 7-3254. 6. 676767. 11. 6-37839x10 3. 59512. T. •186825. 13. 6-35639x10 3. 47763. §. 80008. 13. 1-0832 X lo' 4. •0049056. 0. •00457009. 14. 4-30725x10 5. 295-947. 10. 30033000. 15. 304763x10 142. To find the numb r corresponding to a given logarithm, we simply reverse the process of finding the logarithm of a given number. Example 1. Find the number of which -656769 is the logarithm. We look in Table II along the columns headed O till we find 65, the two leading figures of "656769, the mantissa of the given logarithm, and in the columns between the line led by 65 and that led by 66 we lock for 6769, the remaining figures of the given mantissa. We find these four figures in the line of the number 453 and in the column headed 7, hence ■•■ml^a^,. ».07\ M4 / 1 • LOGARITHMATION. . 147 •656769= log 4-537. >,/■( Had the given logarithm been 3*650769, we should have found the number 4" 537 by means of the mantissa and then have moved US, ) is given in nd log 4-884, B9xl0^ 9 X 10^ . 27 xlO . 5xl0~\ 3 X 10~^ . pgarithm, we of a given logarithm. we find 65, the given |65 and that the given lumber 453 the decimal point three places farther to the right as indicated by the characteristic 3, thus obtaining 3- 656769= log 4537. In like manner may be found 5- 656769= log 463700. 4-656769=log0004537. Example 2. Of what number is -497150 the logarithm ? On looking for "497150 among the logarithms of Table II we cannot find this mantissa, we therefore take out the logarithm next smaller than '497150 the given mantissa, and also the number corresponding to the logarithm taken out. This gives us •497068= log 3-141. Then subtracting -497068, the tabular logarithm from "497150, the given mantissa, we obtain '000082 as difference. From the column of differences we find that log 3^142 -log 3*141 =-000138, t. e., a difference of ^-000138 in the logarithms makes a difference of •001 in the corresponding numbers, hence a difference of '000082 •000082 of "001 = 82 of in the logarithms will make a difference of '000138 138 '001 = -00059 in the corresponding numbers, hence -497068 + -000082 = log (3*141+ -00059), i.e., -497150= log 3-14159. The actual calculation will appear as follows : "497150 068 138)820 1300 58 =log 3-141 5 9 •497150 = log 3 14159 The division is performed by the method exhibited in Example i, §, 67, page 33. It is not carried farther than the quotient 9 because the ' remainder' 58 is practically within the limit of error of the tabular logarithm '497068, which is correct to but 6 figures and which represents all logarithms from -49706750 to •49706849 : consequently the ' remainder ' 58 may be too r.mall by 50 or too ■J'--.^'T, 148 ARITHMETIC. large by 49. In the former case, the figure next following 9 In 3*i4i59, would be 6, in the latter case it would, to nearest approximation, be i ; it is therefore indeterminate with the tables at our command and consequently we omit it, ending our computation with 9. 143. The process of computing the logrtrithiu of a number inter- mediate in value to two tabular numbers or inversely of computing the number corresponding to a logarithm intermediate in value to two tabular logarithms is termed Interpolation of logarithms or of numbers as the case may be. 144. In the early part of Table II, the differences between consecutive logarithms are comparatively large and they change rapidly ; as a consequence interpolation will not in this part of the Table yield accurate results. This difficulty may however be avoided by employing Table III for all numbers and logarithms within its range. The method of using this Table is the same as that of using Table II. EXERCISE XVIII. Find the numbers corresponding to the following logarithms : — 1. •480007. 6. •817342. 11. 2-830083. 16. -000000. 3. •734960. 7. 1-817342. 13. 4-830457*. 17. 2-000204. 3. •740047. §. 5-817342. 13. 3-900000. 1§. -3010.^0. 4. 2-477121. 9. 1-817342, 14. 3^301000. 19. 3 010300. ft. •937700. 10. 5-817342. 15. 1-500005. 30. •030103. Find the characteristic-factor and cofactor of the numbers cor- responding to the following logarithms :— 31. 11716671. 33. ■5-534626. 33. iO-817037. 34. 14 660000. 3ft. 100-000123. COMPUTATION BY HELP OF LOGARITHMS. 145. The use of logarithms in lessening the labor of computation depends on the theorems numbered xxvii (a) and xxviii (a) of § 126. These may be restated as follows : — A. Tlie logarithm of a product is the aggregate of the logarithms of the factors of the prodiict. iW4«.Sij-( J'^-v ;:•;.; ■■•^6974 3-366974 2 -889078= log 774-6 .% 3-728 X -4837 -r( 02383 X 09769) =774 6. •571476+ 1-684576= -571476+ 684576 - 1=1-256052 - 1= -256052. a-377124 + 2 989850= I -366974 - 4 = -366974 -3=3 -366974. -256052-3-366974= -256052+3 - -366974=3-256052 - -366974=2-889078. Notice that at/din^ 1 is changed into subtracting i, and that subtracting'j, is changed into adding 3. Uxample 4. Find (a) the 77tli. power of 74-13 and (b) the 110th. power of 41^ §• («) log 74-13 = 1-869994 (Multiply log 74-13 by 77.) 77 13 '089958 log 74 -137 7 = 143 -989638 = 143 + h )g 9 762, 77 14a 74-13 =9-762x10 . (6) log 4205 = 3-623766 log 7413 = 3-869994 i -753772 110 28 -914920 = log 8 2209 - 28 fUji ' ':r">" "■{•■■■., LOOARITHMATIOX. 151 31)6Ji4 ttiKl 45021 .74J3J iiu =8*2201) X 10 -88 8-2200-1-10 (log 4205 - log 7413) X I io=( 753773 - 1) X 1 10 =83*914930- 110 = 28 914930. Example 5. Find («) tho 11th. root of 35480, Qi) tlie 7th. root of -00075367, uml (<;) the 7th. i)owor of tho 11th. root of •757«. (a) log 35480^" I = T^r of log 35480 = ^»j of 4 *549984 = -413636 -log 2*55)2, 35480''^ = 2 -592. {h) 1( )g •00075367 ' = } of log •00O753({7 = \ oi 4 *877181 = J of (7 +3*877181)= 1 + •553883=1-563883- iog*358, 00075367^ =-368. ((•) log -7676^'^ = i\ of log -7576 = /j of 1 *879440 = /^ of (11 -1-10*879440) = 7 +6*923280 =1 923280-1. .;*83807. •7576t'"i = -83807. In {b) we do not at once divide the cluiruoteristic 4 by 7 the root-index, for this would introduce a negative frauion into tlie quotient, in addition to the (positive) fraction arising from the division of tlie mantissa *877i8i, and a negative fraction in the quotient must be avoided if the required root is to be expressed decimally. To overcome the difficulty of a negative fraction we add 3 to the characteristic 4, thus making the negative part of the dividend an exact multiple of the divisor 7 and to counterbalance the addition of 3 we add 3 to the mantissa -877181. The corresponding operation on the number -00075367 - 7 '5367 . 75367 of which 4-877181 is the logarithm, is the change of — ^^ mto ^-^^« Had we divided 4 and •877181 separately by 7, th'-> c • iired root would have been obtained in the form of a fraction, the numerati// being the 7th root of 7*5367 and the denominator the 7lh rootjof loooo. In [c) we add 10 to the characteristic i, to mr.ke the negative part of the logurithm exactly divisible by the root-index ti, and we counterbalance this addition of lo to the characteristic by addiiig 10 to the mantissa ; i.e., we change ^7'570+io / mto \7 -576x10 +10 / =\7-S76xio / +to 6 7 -1 =8*3807x10 +10 =8-3807+10 —•83807. Example 6. What power of 1 'OS is 2 1 log 1*05 X exponent = log 2. exponent = log 2+-log 1 *05 = -301030-=- *021189 = 14 *207 14 .207 1*06 =2. ir)2 AIM'l'IIMMri(\ EXERCISE XIX. I i Apply lo^iiritliniH (o oMaiii approxiniiiU) viiliittH of (lio fi)l]owin^ indimttnl pru(liu;t.n, (niotJonlH, powum imd roitlH : I . :{ •74Hf» X 4H •:«>(> X :m 4 1 m. si. 2 •!m;:i74 x 4 himlT) x •2H4(i:{1). 3. •;{72Hr>(5 X •IU!>74r» X •:iH«;42l> x •47«».*W. 4. 4:j-h«;2!>x •(M)4Hr>7!>x •27H4(;x i'4<.)r.Mn. «. 7Hr>4!> X -(MLMMinH x i'A-^HOX; x •()()Hf.247. «. 4!»;i7Hr»7:«-r-4 -7(1845. W. •.'{7lMI48-5-f.7 048;{. 0. J •(I2!m4 :- 047285. 10. •02!)<;8.'{-f- •(M)2.*W(}7. 1 1 . :{»-«;452 X •()847<».'{ :- •42785!>. iil. •27<5:M X •()()284(;.'{ :- •05840«. IJI. 4-.S785 :-4!)8(}-4:;x •2!)7:{i>. 14. 8-97(Jxio'^ x2H(;48x ]()'" : (7-2!):{x 10") 15. I'478:ixl0 ' x2'<)(;5;{x 10 {(.•{•4!)()5x 10 '"') 10 16. 48-7:4:p'5i. 30. •008(;4:p^ 31. -l'. 3a. •0240(>*^ 33. •(M)478 30. (1)^ «''. (.0 .') (I rt 34. 1:M)54x10 -H\2a 35. 14 •058x10 -i(>\.;i; 3N. tm. 40. (,',) 1 1 V;niii/ ■ 01 \ ;{ It 2 7 / » ;» 41. 248-7 x3 1415J> 4a. 248-72\.'M4l5!)'f 44. •0:5702 ^f-2 ->785" 45. •174581' x0:{005 :i II (I a 43. •r>702^x^02785'i^. From log 2, log.'{, log 7, log 1 1, luul log 1:5 takmi from Tiil.lo I obtji Ml, 40. log:{2. 4r. log 48. 48. log 40. 40. loLc-(525. 50. log 1024. 51. log 2401. 5a. loL' 1-701. 53. log 07(5-070. i '* v ^ ' ■ l-(MJAIMTIIMATlON. 1 5;^ an. Show Ihtil 57. KxprnHM H) From lug .M47 1 = log lOxlng 7 7 lu jiHa powor of 'J, » l>HHo ,10 given 1. /. in «'., from 1( Tal.lo 11 )g 2 obtain log ohtuiii log JO to 10. Mid bllMO, " - «a. 'U'M7. 05. log 12 X log -0478 log -0478. 5«. n-47. 00. n470. «9. :J4 7 01. •:{47. l*rovt) tliiit, 03. log l2xIog.'{ -log.'t. 04. log 1*2 X log i -.'J? . -'log I -.'J?. 00. lionco hIiow how, from Ji tahhi of log/irithmn to Ihiho 10 a taMo of logarithiiiH to any othui* Idiho may bo (lompiitod. 07. Provo that lug 2.'{xlog J4xl«.g » - log 0. 10 1 -i It) 10 ■j:i 11 10 ON. What powor of 2 ih 7 V 00. What powor of 7 in 2? 70. What p.>wor of 7 ''Mi\ \h 94 -Hf).'} ? 71. What powor of UiHM jh 7-.'W(J ? 7a. Wlmfc ix.wor of 20 84 jh 47.' W ? 7J«. What powor of 4 "708 in •047wt)r of •02H:{7 jh J -05 '/ 7ft. What powor of '0470 in •0art of tho dovolupod valuo of, — 12 3 I an 117 iH'tt'i NO. 4-7 . 87.1-1 . NN, 2.'J78 " . 89. .'{•570 - How many /oroH ar») thtirt. botwison tho decimal ])oint and the hift-hand digit in the devolopod value of,— 90. •l04^ oa. •047^'^'*. O'l. 'iHrnm'"^^. 100 :« 2 1.7 91. -2 9:1. •(M)070 9«. -0477 What is the decinud onler of tlm iirHt digit on tho hift in tho doveloped value of, — ili illij'il !■ I If! 'ill !■ ! ; ill ! ■ill ill iiiii M ■I lip' I:,l!ii' ! t 154 -3 7 96. 2-843 . . 9§. 97. 4768-3~'-^^. 99. ARITHMETIC. •47083 \ •048()2~"^;j' 100. -000074 -■0473 147. It is now necessary to examine the degree of precision attainable in calculations made by means of Tables T, II and III. The mantissiB entered in these tables are not absolutely correct, they are merely the nearest representations of the correct values attainable with six decimal places, and they are in some cases in excess, in other cases in defect, but the excess or the defect is never greater than 5 in the sevcntio decimal place, i. c, the error in a tabular logarithm never exceeds "OOOOOOS. Now by means of Table B p. 132 we find that "0000005 is the logarithm of 1 "00000115, hence any logarithm actually entered in Table I, Table II or Table III may be the logarithm of its corresponding tabular number divided by 1 "00000115, or of the tabular number nmltiplied by 1 "00000115, or of any number between these limits. Hence the error in a tabular number never exceeds the •OOCXX)! 15 of the tabular number itself. If a logarithm be obtained by interpolation, the operation of interpolation may itself introduce an errcjr not greater than '0000005, and this error may be on the same side as the tabular error and consequently added to it, so that a, logarithm obtained by interpolation may be in error by •000001. If then we perform any calculation by help of logarithms, the result is liable to an err^r of the '00000115 part (say the one nine- hundred thousandth part) of itself for every logarithm employed and for every interpolation made in the process of calculation. If a logarithm be multiplied by any number, we must multiply the possible error from that logarithm by the multiplier of the logarithm. This is assuming that the errors lie all on one side, i. e. are all in excess or all in defect, and that each error is nearly at its limit. The cases in which this will occur will be comparatively rare, yet rare as they may be, we nmst take them into account in estimating the limit beyond which our result cannot err. In ordinary com])utations by the help of 6-figure logarithms, we may count on the result as almost certainly correct to 5 significant figures and as probably correct to 6 figures. We exclude cases of involution to high powers. LOGARITHMS. 155 EXERCISE XX. 1 . The squares of the times of revolution of the planets round the sun are as the cubes of their mean distances from the sun, i.e., A and B being two planets, if a fraction be formed having A'a time of revolution round the sun as numerator and ^'s time of revolution as denominator and a second fraction be formed having A 's mean distance from the sun as numerator and B'a mean distance as denominator, the square of the former fraction will be e(iual to the cube of the latter. Mercury performs a revolution about the sun in 87 '969 days ; Venus performs a revolution in 224*701 days ; the Earth, in 365 '256 days ; Mars, in 686' 98 days ; Jupiter, in 4332"586 days, and Saturn, in 10759"22 days ; determine the mean distance from the sun of Mercury, Venus, Mars, Jupiter and Saturn resi)ectively, taking the mean distance of the earth from the sun as the unit of length. Express these distances in miles, assuming the mean distance of the earth from the sun to be (a) 91,430,000, (h) 92,780,000. 2. A pupil who was "strong at figures" undertook to multiply 15 by itself on the first day of his holidays, to multiply the product by itself on the second day, to multiply the second product by itself on the third day, to multiply the third product by itself on the fourth day, and so to continue to do each day (Sundays and Saturdays excepted) to the end of his holid«ys which were to last four weeks. How many figures would there be in the twentieth product thus formed ? Determine the first five and the last ten figures of this product. How long would it take the boy to write down this product at the rate of three figures per second? How many figures would there be in the partial product formed in computing the twentieth product from the nineteenth, assuming that in the nmeiyeenth product the several figures 0, 1, 2, .... 9, occur each an equal number of times except that 5 occurs once oftener than any of the others ? Find to the nearest number of days how long it would take 100 men to compute these partial products, working at the rate of two figuics per second for six hours per day for 313 days per year. [Obtain log 16 to fifteen places of decimals from log 2 and log 3 J ' ':»' ' !' Ill'- is I i I ' i III' iii'i'' I «' ! if; ill!:: iiii 15G ARITHMETIC\ which uro given corroct to fifteen places, the fcirmer on p. 133, the latter on p, 135.] 3. In J^ 131, it is asserted that *' had the base [in Table A] been 2-718281828459, the modif ;s would have been 1." Test the truth of this assertion by forn in^ a table with exponents the same as those in Table A but w't' 2 < 1828 as base instead of 10, and with the calculations carried to six decimal places instead of to eight. (Extract the roots with the aid of Tables of Logarithms II and III. ) Show how to employ the table thus formed to calculate logarithms to base 2-71828. 4. Form a six-^ecimal-place table similar to Table A but with 12 as base instead of 10 and show therefrom and from the Tables of Logarithms that the modulus of logarithms to base 12 is equal to log 2-71828-7- log 12 which is equal to log 2-71828. 12 Show how to employ this table to calculate logarithms to base 12. 5. A sevfcii-tigure table similar to Table A but with 2-71828 as base instead of 10, having been formed, show that if the exponents in the second colunm be all multiplied by "4342945, the modulus of logarithms to base 10, the numbers in the first and third columns remaining meanwhile unchanged, the common base of the second colunin will be changed from 2' 71828 to 10. 6. A seven-figure table similar to Table A but with 2-71828 as base instead of 10, having been formed, show that if the numbers in the first and third columns Ue retained unchanged but the exponents in the second column be all multiplied by the modulus of logarithms to base 12, the common base of the second column will be changed from 2-71828 to 12. 7. State and prove the general theorem of which the theorems of Probs. 4 and 5 are particular cases, and thence show that the modulus of the logarithms to any given base may be used as a constant multiplier to convert logarithms to 2*71828 as base into the corresponding logarithms to the given base. 8. Hence show that the modulus of the logarithms to a given base is the logarithm of 2 71828 to the given base. (See Exercise XIX, Prd). 66.) Example, '434294 = log 2 71828. 0. If the difference between the logarithms of any two numbers be divided by the difference between the numbers and the quotient LOGARITHMS. 157 be multiplied by each of the two numbers, the [jroducts will bo one greater the other less than the modulus of the logarithms. Test the accuracy of this theorem in the Ciiso of logarithms to base 10 by applying it to numbers and their logarithms selected from the Tables of Logarithms I, II and III. This theorem seems to fail in application to many pairs of numbei'S selected from Table III and from the latter part of Table II, show that these may bo cases of seeming and not of real failure of the theorem. Example ; — Table II gives log 7 "001 - log 7 = •000062 which is correct to six decimal places, but to ten decimal places the difference is '0000620376 ; the theorem fails if '000062 is taken as the difference between log 7 001 and lbg7, but it does not fail if -0000620376 is taken as the fi-flference. 10. Show that the theorem of Prob. 8 may be deduced from the last theorem of § 130 in all cases in which the difference between the numbers is less than the <^n-thousandth part of the smaller number. (In the case of six-figure logarithms, it will be sufficient if the difference between the numbers is not greater than the thousandth part of the smaller number as will at once apj)ear if the numbers in the third coiumn of Table A be reduced to six decimals.) 1 1 . Show that the theorem of Problem 8 enables us to calculate the differences of the logarithms in Tables II and III directly from the modulus '4342945, without any previous calculation of the logarithms themselves and tiictt oonsequently a table of differences having been thus computed. Tables II and III may be formed by mere additions. [This "method of differences" is the method which is now employed whenever it is found desirable to extend a table of lo- arithms or, for the puri>oses of verification, to recalculate any part of sucli a table. In actual practice, the differences of the logarithms are not obtained directly by division of the modulus as here proix)sed, but are themselves computed from second differences. The number of divisions which must be made, is thus greatly reduced.] 12. The modulus of logarithms to base 10 is '43429448 and log 49 is 1*69019608 each correct to eight decimals, determine tlierefrom the logarithms of 4901, 4902, 4903, 4904, 4905, correct in each case to six decimal places. ;«:cA~;ti^; Ms.;ir,Aiwm w'a I i ! Ill I |l I" I ill' III i! 1'':' ! I i ililiil!! W liii 158 AlllTHMETIC. 13. Show that the exponenta of the powers to which the bases 1-01, 1-02, 103, 1-04, 1-05, 1-06, 107 must Koveiall> bo raised to produce 2 are approximately equal to 7<-' di^!(i;d by 1, 2, 3, 4, 5, 6 and 7 respectively, and to produce 3 the several exponents are approximately the (juotienu of 110 di\i.'.ed )>j, tl:o same seven nuuiberru In the following problen)/?, thi i-alues of he logarithms which are stated to be ' given ' are to be iaken from the Tables of LogHi'ithms, and the values of the logarithms to be computed aro to be found con i.v't" to six places of uecinuils. 14. C.Lven l,i/2 and log 3 and 2" x 3 " x 7^-1000188, find log 7. 16. Given iog;> ,v\., 3^'- 177147, ll" =1771561, find h)gll. 16. Given log 2, log 3, log 7 and h)gll and 2'^ x7 x 11 = 1232, 3 X 13 -- 12320 3 , find log 13. « 17. Given log 2 and log 7 and 7 " x 17 = 2000033, tmd log 17. 18. Given log 2 and h)g3 and 2 x 3% 19^ = 10(?00422, find log 19. 19. Given log 2, log3 and log 11 and 2x3^x11^=71874, c " x 23 = 71875, find log 23. .■> 2 2 20. Given log 2, log 3, log 7, log 11 and log 13 and 2x5x7 = 9800, 3* X 11^ =9801, 2 x 13^ x 29 = 9802, find h)g 29. 21. Given log 2, log 3, log 7, log 11 and log 13 and 2x3x 7 x 11 X 13-6006, 5* x31^ =600625, find log 31. 22. Given log 3, log 7, log 11 and log 13 and 3" x 7 x 11 x 13 x 37 =999999, find log 37. 23. Given log 17, log 19 and log 23 and 17'^ x 19 ''x 23^ = 41000681 4589, find log 41. 24. Given log 2, lo^ 3, log 7, log 11 and log 13 and 2*' x 3* x 43 = 24768, 7xlf xl3 =2476803329, find log 43. 25. Given h.g3 and log 17 and 17 x 47'^ = 30004847, find log 47. CHAPTER V. MENSURATION OR METRICAL GEOMETRY. 148. To Measure any magnitude is to determine what multiple or part or multiple of a part the magnitude is of a specified magnitude of the same kind selected as a standard or unit of measurement. The number which expresses what multiple or part or multiple of a part the measured niagnitude is of the unit, ' is termed the Measure of the magnitude. The relation which is determined or smujht to he deteiinined by such measurement is called the Ratio of the magnitude measured to the unit of measurement, 149. If the first of two quantities of the same kind be divided by the second, the quotient will be the measure of the first quantity in terms of the second quantity as unit. Thus 4 is the measure of 12 ft. in terms of 3 ft. as unit, for 12 ft. T-3 ft. =4 or, as it may otherwise be expressed, 12 ft. =4 (3 ft.) The measure of 3 oz. in terms of 8 oz. as unit is f or "375 for 3oz. -f8 oz. =§='375 or, otherwise expressed, 3 oz. =| (8 oz.) = -375 (8 oz.) 150. Four mascnitudes are said to be proportional, to be in proportion or to form a proportion^ if the ratio of the first magnitude to the second is the same as the ratio of the third magnitude to the fourth. 161. Hence if four magnitttdes he in proportio^i and if the first maynitnde he a multiple of the second, the third magnitude imll he the same midtiple of the f mirth ; if the first magnitude he a paH of the second, the third magnitude unll he the same part of the fourth ; if the first magnitude he a multiple of a part of the second the third magniivA?' will he the same multiple of the same part of the fourth. 152. A, B, (7 and D denoting four magnitudes of which A and J5 are of the same kind and C and /> also of the same kind, but not necessarily of the same kind as A and B, the expression A:B::C:D, read " ^ is to £ as is to D," denotes that the magnitudes A, B, C and D are in proportion in the order named, i.e., that if ^ is a I ^1 33BS am 100 ARITHMETIC. JMllte I : \i I I I i M ■ » I ill' ii I i 1 1 ii! : |s •'! lii! "!:'(, multiplu of /^, C is tlio samo luultiplo of L>; if A is h part, of 7>, ^' is tho same part of L> ; if A is a multiplo of a part of Ji, (■ is tho sanio imiltii)le of tlxe same part of JJ ; and generally that tho ratio of vl to B is the same as tho ratio of (j to iK Thus 12 in. = 4 (3 in. ) and 20 lb. - 4 (5 lb. ) .-. 12 in. :3in. :: 2011). :41b., read "12 in. is to 3 in. as 20 lb. is to 4 lb." So also, 15 gal. =ii (35 gal.) and 1^ min. = 2 (3J|! niin.) 15 gal. : 35 gal. :: Ih min. : 3^ niin., read '* 15 gal. is to 35 gal. as 1| niin. is to 3| min. 153. Tho measure of tho length of a lino is the number which expresses the ratio which tho measured line bears to a lino selected as tho unit of length. The unit of length or linear unit is usually either («»)> f^ fundamental unit, or {h), a multiple or a fraction of some fundamental linear unit. The yard and the metre which are both defined by physical standards (see pp. 12 and 17,) are example's of fundamental linear units. The mile and the kilometre are examples of units which are multiples of these fundamental units ; the inch, the foot and the centimetre are examples of units which ar6 definite parts or determinate fractions of fundamental units. Example 1. A certain rope is stated to be 37 yd. long. Hero the unit of measurement is the linear unit, a yard, and tiio measure of the declared length of rope is the number 37. Example 2. The length of the circumference of a certain circle is found to be 47 "85 in. Here the number 47 "85 is the measure of the length of the circumference, and the linear unit, an inch, is the unit of measurement. 164. The measure of the area of a surface-figure is the number which expresses the ratio which the measured figure bears to some determinate surface-figure chosen as the unit of area. The unit of area generally selected is either ((x), a square whose side is some specified unit of length, or (fc), a multiple «>f such a square. Example 1. The area of the floor of a certain hall is 240 sq. yd. Here the measure of tho area of the floor is 240 and the unit oi MENSURATION. 161 measurement is the areal unit, a square yard, t. e. , a square whose sides are each a yard in length. Example 2. The area of a certain field is found to be 7i^ ac. Here the measure of the area of the field is 7 J and the unit of measurement is the areal unit an acre which is equal to 10 square chains or 4840 squarf-i yards. 155. The measi.e of the volume of any solid or space-figure is the number which expresses the ratio which the nieasured figure bears to some determinate space-figure chosen as the unit of volume. The unit of volume is either (a), a cube whose edge is some specified unit of length, or (6), the volume of a given mass of some specified substance under stated conditions, or (c), a multiple or a fraction of this volume. Example 1. Tbe volume of air in a certain school-room is 560 cu. yd. Here the measure of the volume of air is 560 and the unit of measurement is the volume-unit a cubic yard, i. e. a cube whose edges are each a yard long. Example 2. A certain pitcher will hold f of a gallon of water. Here the measure of the capacity of the pitcher is f and the unit of measurement is a gallon, i.e., the volume often Dominion standard pounds of distilled water weighed in air against brass weights with the water and the air at the temperature of sixty-two degrees of Fahrenheit's thermometer and with the barometer at thirty inches. (>' 156. Two plane rectilineal figures are similar if to every angle in one of the figures there is a corresponding equal angle in the other, and if also the sides about the angles in one figure are proportional to the sides about the corresponding angles in the ther. The sides extending between corresponding angular points are termed cm'vesponding or homoloywis sides. 157. Hence if the lengths of two sides of a triangle b? given and also the length of one of the corresponding sides of a triangle similar to the former, the length of the second corresponding side of the latter triangle can be determined. 158. If two tria)igles have two angles of the f the sides K L and K M. K B C L M By § 158 the triangles are similar and the corresponding sides are A B and K L, B C and L M, C A and M K ; KL : LM : : AB :BC. The length of AB is 6 and that of B C is 8 sixteenths of an inch, • • AB = flofBC • • • KL=OofLM = 1 of 12 sixteenths of ai = 9 II II II Similarly, " • KM :ML : : AC :CB and AC = |iof CB • • KM = |of ML = 1 of 12 sixteenths of an inch 13* II II EXERCISE XXX. 1. ABC and KLM are similar triangles, the angles A and K being equal to one another and the angles B and L also equal to one another ; the side A B is 9" long, the side B C is 10" long and the side KL is 22 -5" long, find the length of the side L M. SJ. ABC and GHK are similar triangles, A and G being corresponding angles and B and H also corresponding angles ; the lengths of the sides AB, AC, GH and HK being 7", 15", 5-25" and 15" respectively, find the lengths of BC and H K. 3. A B C and GHK are similar triangles, the angles A and G being equal to one aiK )ther and the angles B and K also equal to one another; the measures of the sides are AC = 25, GH=44. HK=35andKG — 75. Find the measures of the sides ABandBC. MENSURATION. 163 an^le B , also lot 3onths of ich long. ing sides f an inch, A and K. jual to one ig and the G being igles ; the 15", 5-25" A and G |o equal to GH=44, iBandBC. 4. In AB, a side of the triangle ABC, a point D is taken, and the straiglit lino DE is drawn parallel to the side BC ; find the length of D E, the length of A B being 35', that of B 24' and that of A D 11' 2'. 5. The construction being the same as in problem 4, find the measure of D E, given that the measures of A D, D B and B are 7, 23 and 18 respectively. 6. The construction being the same as in problem 4, find the measure of A D, given A B = 45, B C = 20 and D E = 8. 7. The construction being the same as in problem 4, determine the length of BC, given that AD is 24yd. long, DB, 30yd. long, and DE, 18 yd. hmg. §. The construction being the same as in problem 4, determine the length of B C, A B being 104 ft. long, B D being 44 ft. long, and D E being 80 ft. long. 9. The construction being the same as in problem 4, what will 1)0 the length of A D if B D be 36 chains long, B C, 36 chains long, and D E, 15 chains long. 10. A stick 3' in length placed upright on the ground is found to cast a shadow 2' 6" long, what must be the height of a flagpole which casts a shadow 28' in length ? ' » 11. A gas-jet is 12 ft. above the pavement, how far from the ground-point directly beneath the jet must a man 6 ft. 8 in. in height stand that his shadow may be 6 ft. long. 12. The vertical line through a gas-jet 9' 4' above the sidewalk is 10' 6" from a man 5' 10" in height, find the length oi: his f^hadow. 13. An electric light is 15 ft. above the pavement, what will be the length of the shadow of a man 5 ft. 10 in. in height if he stand 30 ft. from the vertical line through the light ? 14. The parallel sides of a trapezoid are respectively 27 ft. and 35 ft. in length and the non-parallel sides are respectively 18 ft. 7 in. and 23 ft. 11 in. long. The latter sides are produced to meet ; find the respective lengths of the produced sides between the point of meeting and the shorter of the parallel sides of the trapezoid. 15. The lengths of the parallel sides of a trapezoid are 10*75 and 12*35 chains respectively ; four straight lines are run across i f 1G4 ARITHMKTir. thu tmpeznid pArnllel to tlumu Hiint intorvulH ; tind thu lungths of thusefour linoa. 10. The lengths of the imu-jvUoI Hi«lo8 of a tnipozoid nro 15 and 28 inches veHpectivoly, and of the non-parallel Bides 12 and 20 inches respectively ; through the intersection of the diagonals of the trapezoid a straight line is drawn parallel to the parallel sides. Find the lengths of the sections into uhich this lino divides the non-parallel sides. I T. Taking the diameter of the sun to bo 880,000 miles and the sun's distance from the earth to be 02 400,000 miles, what must be the diameter of a circular disk that it may just hide the sun when held between the eye and the sun and 21 inches in front of the eye? 1§. Three men A, B and stand in a row on a level pavement, ^'s height is 5' 3^", B'b is 5' r and Cs is 6' 1^"; if .-1 stand 10' to the right of B, how far to the left of 7> nuist G stand that the tops of the heads of the three men may range in a straight line? 19. The lengths of the sides of a triangle are 7 yd., 11 yd. and 12 yd. respectively and the perimeter of a similar triangle is 25 ft. ; find the lengths of the sides of the latter. 20. The jierimeters of two similar triangles are 25 ft. 6 in. and 56 yd. 2 ft. reaf actively." A side of the smaller triangle is 7 ft. long and a non-corre8i)onding side of the larger triangle is 17 yd. Find the lengths of the other sides of Iho triangles. 1 ft. in length. 169. Any one of the sides of a parallelogram having been selected as the base of the figure, the altitude of the parallelogram is the perpendicular distance between the base and the side parallel to the base. One of the sides of a triangle having been selected as the base of the figure, the opposite angle becomes the vertex, and the altitude of the triangle is the Uiujth of the perpeiidicular from the vertex on the base, or the base produced. 160. A polyhedron is a solid-figure enclosed by plane polygons. A polyhedron enclosed by four polygons, (in this case, triangles) is called a tetrahedron ; by six, a hexahedron ; by eight, an octahedron ; by twelve, a dodecahedron ; by twenty, an ioosahedron. -..I**' MENSUIIATION. 105 The faces of a polyhedron aro the enclosing prjygonn. If tho faces are all ecjual and regular, the polyhedron \h regular. The edges of a polyhedron are the lines in which its faces meet. The summits of a polyhedron are tiio points in which its edges meet. A Prismatoid is a polyhedron two of whoso faces are polygons situated in pjirallel planes and whose other faces are triangles having tho sides of tho polygons as bases and having their vertices at tho angular points of tho polygons. Tho polygcms situated in parallel j)lane8 aro called tho ends of *\ the prismatoid, and if one of them bo taken as tho buxe of tho solid, tho other becomes tho opponHc pamUel fmo. Tho other faces aro called tho htteral /hcch and their common edges ai'o named tho lafuml ed(jex. ( A B C D E F G is a prismatoid on tho quadrilateral base A B C D, tho opposite parallel face is tho triangle EFG.) The midcross-section of a i)rismatoid is its section by a plane parallel to tho planes in which aro situated tho end polygons and midway between these planes. Tlxe midcrosti-Hection. therefore bisects all the lateral edges of the prismato'ul. (HKLMNPQ is tho midcross-secticm of the prismatoid ABCDEFdr. Tho angular points H, K, L, M, N, P, Q are the mid-points of A F, BF, B G, CG, D G, D E, and A E respectively.) If the bases of two adjacent lateral faces of a prismatoid are parallel the two faces lie in one plane and together form a trapezoid. A Prismoid is a prismatoid whoso lateral faces aro all trapezoids. Tho end polygons must therefore have tho same number of sides and each corresponding pair must be co-parallel. (ABCDEFG H K L is a j)rismoid with pentagonal ^(^ cuds A B C DE and F G H K L.) ■ii-'"-,' 11 S= ssa i'itiil li'iir' III 1 i ' ' i iiiiii! l|! III ililli' I..;. 11 m 166 ARITHMETIC. A Wedge is a solid enclosed by five plane Fgures, the base is a trapezoid, two of the lateral faces are trapezoids and the other two lateral faces are triangles. A wedge is therefore ^^ a prism atoid on a trapezoidal base, in which the face opposite the base has become reduced to a e' straight line parallel to the two co-parallel sides of the base. (ABCDEF is a wedge ; the base ABCD is a trapezoid, the sides BC and AD being parallel to each other ; EF is parallel to both BC and AD, hence BCEF and ADEF are both trapezoids.)' A Prism is a polyhedron two of whose faces are parallel polygons, and the other faces, parallek)gram8. The bases or ends of a prism are the parallel polygons. The altitude of a prism is the pcrpendictdar distance between the planes of its bases. A right "prism is one whose lateral edges are perpendicular to its bases. A parallelepiped is a prism whose bases are parallelograms. A parallelepiped is therefore a solid contained by six parallelograms of which every opposite pair are parallel. A quad or quadrate solid is a right parallelepiped with rectangular bases. It is therefore contained by six rectangles. A cube is a quad whose faces are all squares. 161. A cylindric surface is a surface generated by a straight line so moving that it is always parallel to a fixed straight line. A cylinder is a solid enclosed by a cylindric surface and two parallel planes. The bases of a cylinder are the parallel plane faces. The altitude of a cylinder is the perpe'udicidar distance between the planes of its bases. A right cylinder is one in which the generating lines of the cylindric surfaces are perpendicular to the bases of the cylinder. A right circular cylinder is a right cylinder whose bases are circles. MENSURATION. 167 A Cylindroid is a solid bounded by t^vo parallel planes and the surface described by a straight lina which simultaneously describes two closed curves, one in each of the parallel planes. The plane figures enclosed by the curves in the parallel planes are called the ends of the cylindroid. A Sphenoid is a prismatoid or a cylindroid, one of whose ends has become reduced to a line. 162. A pyramid is a polyhedron one of whose faces, called the base, is a polygon and whose other faces are triangles whose bases form the sides of the polygon and whose vertices meet in a point called the vertex of the pjrramid. A pyramid is therefore a prismatoid one of whose parallel ends has become reduced to a point. A regular pyramid is one whose base is a regular p(jlygon and whose other faces are equal isosceles triangles The altitude of a pyramid in the length of the perpendicular let fall from the vertex on the plane of the base, 163. A conical surface is a surface generated by a. straight line which so m()^es tltu- it always passes through a fixed point called the vertex of the surface. A cone is a solid enclosed by a conical surface and a piano. It is therefore a cylindroid one of whose parallel ends has become reduced to a point. The base of a cone is the plane face opposite the vertex. The altitude of a cone is the lemjth of the perpendmdar let fall from the vertex on the plane of the base. A right circular cone has a circle for its base, and the *raight line joining the vertex of the cone and the centre of the base is perpendicular to the plane of the base. The frustum of a pyramid or of a cone is tho portion inclu(lr the cunc parallel to the base. 111 li :ii 168 ARITHMETIC. iiiiiii m (, ■£ 164. Two polyhedra are similar if to every solid angle in one of them there is a corresi)onding equal solid angle in the other, and to every face of one of them there is a corresponding similar face in the other. The corresponding edges of similar polyhedra are those which are corresponding sides of corresponding faces. 166. Similar surface-figures need not be rectilinea., they need not even be plane surfaces. Thus all circles are similar to one another, parallel piano sections of a cone are similar figures, all spherical surfaces are similar to one another, aitd generally the complete surfaces of similar solids are themselves similar. If two plane surface-figures are similar, they are, or they may be so placed as to be, parallel plane sections of a pyramid or else of a cone. Similar solid-figures are not necessarily bounded by plane surfaces ; e. - yd. by the length of the lot. The sum of the areas of th-^se two rectangles is 441 sq. yd. ; . •. ^5^- of the measure of the length of the lot = 441 ; the measure of the length of the lot=441-r^5^, = 46; . •. the length of the lot is 45 yd. Example 2. A rectangular park is 400 yd. by 660 yd. It is surrounded by a road of uniform width the whole area of whicli is one-sixth vi the area of the park. Determine the width of the road. The area of the park is (400 x 660) sq. yd. - 264000 sq. yd. The area of the voud is J of 264000 sq. yd. =44000 sq. yd. Therefore the area of the rectangle composed of both road and park = 264000 sq. yd. + 44000 sq. yd. = 308000 sq . yd. The park is a rectangle 260 yd. longer than it is wide. 3IENSU RATION. 171 WDTH V 2fi) YD •■ * WIDTH » ;:;;■ no yd o «9 130 YO. When the road is inchided with the park, both the length and the width of the rectangle is increased by double tbu width of the road ; the resulting block of land is therefore still a rectangle 260 yd. longer than it is wide. Hence if the length of the block be reduced by 130 yd. and the width of the block when thus shortened be increased by 130 yd. , the resultinst rectangle will be a SQUARE whose sides will be each 130 yd, longer than the vndth of the original block. Reducing the length of the block by 130yd takes from the block a rectangle 130 yd. by the width of the block. Increasing the width of the shortened block adds to this block a rectangle 130 yd. by lo^Oyd. more than the width of the oiiginal block, i. e.y it adds a rectangle 130 yr". by the width of the original block and a square 130 yd. square. Hence the two operations of reducing the lengjjjb of the original block and increasing the width of this shortened block increase the area of the resulting figure as compared with the area of tlie original block, by the area of the * completing square ' of 130 yd. square, i. e., by an area of 130^ sq. yd. =16900 sq. yd. The area of the original block was found to be 308000 sq. yd. The area of the completing square has been found to be 16900 sq. yd. Therefore the area of the completed square or square block 'vill be 308000 sq. yd. +16900 s.}. yd. = 324900 sq. yd. Therefore the length of the side of the square block = (324900)^yd. = 570 yd. Therefore the length of the rectangular })lock — 570 yd. + 130 yd. = 700 yd. The length of the park -660 yd. Therefore double the width of the road - 700 yd, - 6^0 y,l. ^ 40 yd. Therefore the width of the road -- 20 yd. M m h'^i W M 172 ARITHMETIC. EXERCISE XXII. '!ii V- i!j 1. Find to the nearest inch the length of the side of a square whose area is an acre ? 2. A square field contains exactly 8 acres. Determine the length of a side of the field, correct to the nearest link. 3. The area of a chess-board marked in 8 rows of 8 squares each, iii iOO sq. in. Find the length of a side of a square. 4. On a certain map it is found tbat an area o? 16000 acres is ' ^presented by an area of 6 '25 sq, in. Give the scale of the map in miles to the inch and also in the form of a ratio. 5. A rectangle measures 18' by 30' ; find the difierence between : ' - irea and that of a square of equal perimeter. ^. Six sheets of paper measuring 8 in. by 10 in. weigh an ounce ; find the weight of 120 sheets of the same kind of paper, each sheet measuring 11 in. by 17 in. T. Two rectangular fields are of equal area, one field measures 15 chains by 20 chains, the other is square. Find the length of a side of the latte? field, correct to the nearest link. 8. How many stalks of wheat could grow on an acre of ground, allowing each stalk a rectangular space of 2" by .3 " 'i 9. How many pieces of turf 3' 6" by 1' 3" will be required to sod a rectangular lawn 28' by 60' ? 16. Sidewalks 12 ft. wide are laid on both sides of a street 440 yd. long. Find the cost of the sidewalks at $1'35 i»er square yard for the pavement ■ ud V^ cents per lineal yard tor curbing ; deducting thi ie crossings of 54 ft. each ov both sides of ijhe street. 11. The Pj-ea of a rectangular field is 15 acres ; the length of the field is double the width, find the length of the field. 12. How maiiy yards of fencing- wire will be required to enclose a rectangular field thrice as long as it is wide, if the field contain 10 acres and the fence be made 5 wires high ? 13. The lengths of the sides of a rectangular piece of land are as 3 to 8, and its area is 60 acres. Find the lengths of the sides. 14. The perimeter of a rectangle is 154 in. , and the difierence in MENSURATION. 173 length of two adjacent sides is 11 in. Find the area of the rectangle. 1 5. The length of a rectangle is 88 ft. ; if the width were increased by 8 ft. , the area of the rectangle would in such case be 616 sq. yd. Find the width of the original rectangle. 16. The area of a certain rectangle is 1980 sq. yd. If the length of the rectangle were increased by 12 ft. , the area would be 2100 sq. yd. Determine the lengths of the sides of the rectangle. IT. Find the difference between the perimeter of a square field containing 22*5 acres and the perimeter of a rectangular field of equal area, the length of the latter field being to its width as 5 to 2. 18. A rectangular block of building-lots is 660 ft. long by 108 ft. wide. Find the area covered by an eight-foot sidewalk around the block just outside of it. 19. A six-foot sidewalk of Sin. planks is to be laid around a rectangle 266 ft. 8 in. by 480 ft. , the inner edge of the sidewalk to be twelve inches out from the sides of the rectangle. Find the value at $14 the M, board-measure, of the planking for the sidewalk. 20. Find the areas of the outer and the inner surface of a hollow iron cube measuring Sin. on the outside edge, the iron being -^ in. thick. 31. Find the area of the inside surface of a hollow quad measuring 3' 2" by 2' 8" by 2' 1" externally, the enclosing walls being 1^" thick. 32. The length of the base of a parallelogram is 46 ft. ; the length of the perpendicular on the baB(> from the opposite side is 28 ft. ; the length of a side adjacent to the base is 36 ft. ; find the length of the perpendicular on this side from the side opposite to it. 83. The adjacent sides of a parallelogram moasure 132 ft. and 84 ft. respectively and the area of the parallelogram is two-thirds of that of a square of equal perimeter. Find the perpendicular distance between each pair of parallel sides. 24. Find the cost of painting the gable-end of a h.nise @ 22 ct. per sq. yd., the width of the house being 32 ft. ; the hei-^ht of the eaves above the ground, 36 ft. ; and the peri)endicular height of the ridge of the roof above the eaves, 15 ft. 25. Find the area of a field in the form of an isosceles right-angled y m Ul I!- 174 ARITHMETIC. triangle, tho length of the perpendicular on the hypothenuso being 7 '50 chains. 36. The length of one of the diagonals of a quadrilateral is 27*7 ft. and the lengths of the jjerpendiculars on this diagonal from opposite angles of tho quadrilateral are 18*5 ft. and 113 ft. respectively. Find the area of the quadrilateral, 1°, if the diagonal lies wholly within the quadrilateral ; 2", if the diagonal lies wholly without the (luadrilateral. 97. The lengths of the diagonals of a courtyard in the form of a rhombus are 40 ft. and 26 ft. How many bricks 9" by 4h" w-ll be reipiired to pave the c<.>urtyard ; adding 6 % t«> the area to allow for broken bricks and for waste at the sides of tho courtyard ? 2§. One of the diagonals of a parallelogram measures 819 ft. aad the perpendicular on it from an opposite angle of the parallelogram measures 287 ft. Find the area of the parallelogram. 29. Tho area of a (piadrilatoral is 7956 sq. yd. , the length of one of the diagonals is 416 ft, and the length of tho perpendicular on this diagonal from an opposite angle of the quadrilateral is 192 ft. Find the length «>f the perpendicular froni the other opposite angle, 1°, if the diagonal is internal ; 2°, if it is external. 30. The area of a (quadrilateral is 12 '48 acres and the length of one of the internal diagonals is 19 '50 chains. Find the sum of tho lengths of the perpendiculars on this diagonal from the two opposite angles. 31. The area of a quadrilateral is 906 "5 sq. yd. ; theiength of one of the internal diagonals is 147 ft. ; and the difference between the lengths oi:' the perpendiculars on Ihis diagonal from the opposite angles of the quadrilateral is 33 ft. Find the lengths of these perpendiculars. 32. A B J D is a quadrilateral, A B = 400 ft., B C = 203 ft., CD = 396 ft., and DA = 196ft. ; the angles at A and C are right angles. Find the area of the quadrilateral. 33. Find the area of a trapezoid whose parallel sides measure 12' 7" and 19' 3" respectively, the perpendicular distf.nce between them being 8' 5". 34. ABCD is a quadrilateral; A B =37 '48 chains, B = 21-85 chains and CD = 29 '64 chains. AB is jwirallel to DC and the li;i MENSURATION. 175 ith of iposite angle at C is n right angle. Determine the areas of the triangles ABD and ACD and <»f the quadrilateral. 35. Find the area of a (^uadii lateral one of whose sides measures 23*29 chains and the perpendiculars on this side from the opposite angles of the quadrilateral 17'75 chains and 13 "45 chains respectively, the distiinces of the feet of these perjiendiculars from the adjacent angles being 3 '64 chains and 2 "40 chains respectively . 36. The area of a trapezoidal field is 3} acres and the sum of the lengths of the parallel sides is 440 yd. Find the porpendicuhir distance between these sides. The lengths of the sides being in the ratio of 5 to 6, find these lengths. 37. The area t)f a trapezoid is 9750 sq. yd. and the perpendicular distance between the parallel sides is 234 ft. If the length of one t)f the parallel sides be 410 ft., what will be the length of the other parallel side ? 3§. The area of a trapezoid is 47 '142 acres. One of the parallel sides is 6*12 chains longer than the other and the ])erpendicular distance between the parallel sides is 11 "64 chains. Determine the lengths of the two parallel sides. 39. The lengths of the parallel sides of a trapezoid are 12 ft. and 17 ft. and the perpendicular distance between these sides is 8 ft. A straight line is drawn across the trapezoid parallel to the parallel sides and midway between them. Find the areas of the two parts into which the trapezoid is thus divided. 40. The lengths of the parallel sides of a trapezoidal field are 15 '80 chains and 18*70 chains respectively and the perpendicular distance between these parallel sides is 14*40 chains. Four straight lines are drawn across the field parallel to the two parallel sides and dividing the distance between these sides into five equal parts. Find the areas of these five parts of the field. 41. The area of a triangle is 551 sq. yd. and the length of its l)ase is 95 ft. Tw«j straight lines a,re drawn across the triangle parallel to the base and dividing into three equal parts the perpendicular fi'oia the vertex on the base. Find the areas of the parts into which the tiiangle is divided by these lines. 42. A trapezoid with parallel sides whoso lengths are to be as 4 to 3 is to be cut from a rectangular board 14 ft. long. Find the !i ' i1 176 ARrrHMKTiC. ! I i.ji, m loiigths of tho iMirivllel sidos that tho trapezoid may be oi third of the board, tho tra])ezoid to lie of tho same width us tho nDard. 43. Tlio length of a rectangle is to its width im 7 to 4, and if its length be din.inihhed by 3 ft. while its width is increased by Jift., its area will bo increased by 198 8(j. ft. Find the length of the rectftngle. 44. Tho length of a rectangular piece of land is to its breadth as 9 to 5 ; if its length bo increased by 4 ft. and its breadth be diminished l)y ti ft. , its area will bo diminished by 355 sq. ft. Find the length and tho breadth of the piece of land. 45. The length of a rectangle is to its width as 16 to 9 ; if its length be diminished by 2 ft. and its width diminished by 3 ft. , its area will be diminished by 720 sq.ft. Find the area of the recfamgle. 46. A rectangular field 200 yd. long is surrounded by a road of the uniform width of 60 ft. The total area of both field and road is 9 A. 1240 sq. yd. Find the width of the field. 47. A rectantfiiiar field 780ft. in length is surrounded by a road of the uniforu! v' iisUi of 50ft., the area of the whole road being 15000 sq. j.'d. Fiiid the area of the field. 4§. A rectangular field 180 yd. by 150 yd. is surrounded by a walk of unifoim width, the whole area of the walk being 10000 sq. ft. Find the width of the walk. 49, Around a rectangular park runs a path of uniform width ; paths of the same width cross the park dividing it into four equal rectangles. The total length (f the park, including paths is 330 yd. ; its area, including paths, is 15 A. ; exclusive of paths the area is 13775 A. Find the width of the paths. ftO. The areas of twt) squares differ by 64 sq. yd. and the leng 'is of their sides difler by 2 yd. Find their areas. 51. The sum of the perimeters of two squares is 200 ft- and the difierence of their areas is 400 sq. ft. Find their areas. 53. The area of a rectangle is 945 sq. ft. and that of a square of equal perimeter is 961 sq. ft. Find the lengths of the sides of the rectangle. Nil \ MENSITHAT[()N. 177 jno;' hs 53. Tho areu of a luctanglo in .'i7245>8(j. ft. unci its lengtli exccocls its breadth by 40 ft. Find its length. 54. The area of a triangle is 2 A. 2152 sq. yd. and the length of tho base oxcoeds the altitude of tho triangle by 38 yd. Find tho length of tho base. 55. A certain rectangular tield of are A. is surrounded by a road of the uniform width of 56 ft.. t' voa of the road being 2^ A. Find the length and tho wiu ill 1. 170. In tho formuho which follow / .md M denote tho measures of the volume, the altitude, the area (»f tho base and the area of tho midcross-soction respectivoly, and 7, />, c, 1/, A-, /', d and m subscribed to V are to be read severally (juad, prism, cylinder, pyramid, cone, frustum uf pyramid or of cone, prismatoid (or prismoid) and wedge. I. The measure of the volume of a 2 = "'i>, hi which &i and />a denote the measures of adjacent edges of the base and consequently 6, J>2 = ^• (Special case, — cube. ) II. The meas'ure of the volume of a piifnn, i.s the. jjfoidict if the measures of the cdtltwle ami the cwcrt of the Ixise, or (Special cases. — quad, and cylinder.) i\ This theorem is true of the obli(|uo jjarallelepiped fur every <)l)li'J c,/, g and h respectively. Transfer the solid efghahcd froniiend to end of ABCDabcd thus transforming this parallelepiped into the parallelepiped EFGHc/grfe on the rectangular base EH^e which is equal to the base AD da. Through A;, a point in the edge eh, pass a plane at right angles to the edges ehy/g, EH, FG, and cutting these edges in the points A;, Z, K, and L respectively, < Transfer the solid EFLKc/ifc from side p*. to side of "EFGHefgh, thus transform- \ ing this parallelepiped into the rectangu- lar parallelepiped K.liM.lSklmn. The measure of the volume of AChd is the same as the measure of the volume of KM In, the two parallelepipeds being made up of the same parts differently arranged. The measure of the volume of KM. In is, by Theorem I, the product of the measures of its altitude and its base-area. Hence the measure of the volume of AOhd is the product of the measui*eB of ihe altitude and base-aiea of K M Z n, which is the same as the product of the measures of the altitude and base-area of ACbd itself, for the altitude of the parallelepipeds remains unchanged during the transfers and the base K N ti A; is merely the base AD da with its parts transposed. The theorem is therefore true of parallelepipeds. -. 2°. The theorem is true of a prism on a triangular base, for two similar and equal prisms on triangular bases may be so joined together as to form a parallelepiped with both volume and base doub.d the volume and base of either prism. Hence double the measure of the volume of a prism on a triangular base is the product of the measure of the altitude and the measure uf double the area of the base, and therefore the measure of the volume is the product of the measures of the altitude and the area of the base. 3". The theorem is true of prisms whose bases have five or more sides for by passing planes through any one lateral edge and through all the other lateral edges except the two immediately adjacent to the first edge, any such prism will be resolved into an ■ ';-^;Si\ .■^V ,£' d from vend 3piped into >ase EH/ie „_- -."» 1 1 the measure r made up of he volume of >f its altitude ae of A06rf base-a'i.ea of Lsures of the tude of the ifers and the ransposed. »e, for two „ ^na /.< /! se. Ifive or more stl edge and J immediately lived into an 179 f ■ same \ \ » / / / t » • \ \~ / / / \ \ / \l ^ t Jl MENSURATION. a/ggregskte of trian^lar based prisms which have all the altitude as the resolved prism and whose triangular bases together make up the base of the resolved prism. All prisms are included under one or other of 1°, 2° or 3°, therefore the theorem is true generally. [The student should make models of the solid- figures here considered and also of those considered under theorems III and FV which follow. Solid- figures can very easily be cut out of potatoes or turnips.] in. The meastire of the volume of a pyramid is ONE-THIRD of the product of the m^eaafurea of the altitude and the wrea of the hase^ or Yj=\aB. (Sx)ecial cases, — cmie and sectw of a aphercj including sphere itsdf. This theorem is true of tetrahedra (triangular pyramids) for any triangular prism, e.gr., ABCDe/, can be divided into three tetrahedra of which two, ABOD and De/C, will be of the same altitude as the prism and will have the triangular faces of the prism as their respective bases ; the third tetrahedron BCDe, may be seen to have an altitude and a base equal to each of the other two by resting the prism first on the face Ae and next on the face B/. . The theorem is true of pyramids with bases which have four or more sides ; for, by passing planes through any one l,ateral edge and all the other lateral edges except the two adjacent to the first edge any such pyramid will be resolved into an aggregate of tetrahedra which have all the same altitude as the pyramid and whose bases together make the base of the pyramid. Hence the theorem is true of all pjrramids. '' [The preceding proof assumes that two tetrahedra on equal and similar bases and of the same altitude are of equal volume, a proposition which is a particular case of Euclid xn, 5. The proposition may also be proved as follows : — 7t .% :r%; ii - ..'m '*f^ lik ''•f'ii .♦;t. -. '^^■.'.■r^'i^S^^fl^Ti'. -/■■'■ > 180 ARITHMETIC. Divide one of the lateral edges of each tetrahedron into any^ number of equal parts, the same number in both tetrahedra, and through the points of division pass planes pai j'llel to the bases. All * the sections of the first tetrahedron are triangles equal and similar to the corresponding sections of the second tetrahedron. Beginning with the base of the first tetrahedron, construct on the base and on each section as base a prism with lateral edges parallel to one of the edges of the tetrahedron and with altitude equal to the perpendicular distance between the sections. Beginning with the first section above the base of the second tetrahedron, construct on each section as anti-base or upper triangular surface, a prism with lateral edges parallel to one of the edges of the tetrahedron and with altitude equal to perpendicular distance between the sections. The aggregate of the first-constructed series of prisms is greater than the first tetrahedron and the aggregate of the second series is less than the second tetrahedron, therefore the dilierence in v« e between the tetrahedra is less than the difference in vo a between the prism-aggregates. But, by II p. 178, each prism in the second tetrahedr; -u is equal in volume to the prism in the first tetrahedron next above it in order numbering from the prisms on the bases of the tetrahedra. Therefore the diflference between the prism-aggio^ptes is the basal prism in the first aggregate. Now the volume of this basal prism may be made less than any assignable volume, for the measure of its volume is the product of the measures of the altitude and the area of the base. The base is constant being the liase of the tetrahedron, but the altitude being the perpendicular distance between the sections may be increased or diminished by changing the number of the sections. By doubling the number of the sections the altitude and with it the volume of the prism will be diminished by one-half of itself. If we ' K-»^' ■ K MENSURATION. 181 3 any' if and . All imilar on the to the greater Iseries is v< e vo <3 [equal in order l-ahedra. le basal |ian any iduct of base is le being greased . By it the I£w© again double the number of sections, we shall again diminish the volume of the prism by one-half of itself. Repeating the doubling we repeat the subdividing, and the process may be continued till the volume of the'basal prism is less than that of any assigned solid however small. Hence the tetrahedra can have no assigTiable difference of volume, and, both being constants, they cannot have a variable difference ; therefore they are of equal volume.] IV. The measwre of the volume of a prismatoid is ONE-SIXTH of the product obtained by midtiplying the measure of the altitude by the sum fanned by additig the measuren of the areas of the parallel faces to four times the measure of the area of the midcross-sectiou, or Vi=ia(Bi+^M-\-B,,). (Special cases, — prism^ld and cylind/roid, wedge and sphenoid^ prism and cylinder, pyramid and cone and frusta of pyramid and cone, ellipsoid and frustum of dlipsoid by planes perpendicular to an axis.) Let ABCDEFG be a prismatoid and denote the measure of its altitude by a, the measure of the area of the base ABCD by ^i, and the measure of the area of the face EFG, the face parallel to the base, hy B2. Bisect the lateral edge BG in the point H and through H pass a plane parallel to the base AB C D, cutting, and therefore bisecting, the other lateral edges in K, L, M, N, P and Q respectively. Tho polygon HKLMNPQ ' is the midcross- section of the prismatoid and itd perpendicular distance both from the base ABCD and from the parallel face EFG is one-half of the altitude of the prismatoid. The measure of that distance is therefore ^a. Let M denote the measure of the area of the midcross-section. ■ .^/.<"fc1w35v!ilP .V"^/'..>;^- 182 ARITHMETIC. ^■■ I',. I; In tho piano of tho midcross-sectiun take any point R and pass; !^ / ^ planes through R and each edg^ of tho prismatoid thus resolving that solid into the nine pyramids RABCD, REFG, RBCG, RGCE, RCDE, RDAE, REFA, RABF, RFGB, a pyramid '' on each face of the prismat id. ^ ' The measuro of the volume of RABCD is one-third "of the ^jr product of ^a and B^i.e.^ ^aB^. * U Tho measure of the volume of REFG is one-third of the product of \a and JBg, *.e., \0'B^. To determine the measure of the volume of the other pyramids join RH, RK, RL, RM, RN, RP and RQ, also join BK. Let the measures of the areas of the triangles RHK, RKL, RLM, RMN, RNP, RPQ, RQH be denoted respectively by mj, W2, m,, m4, m^, m,, and ini7, therefore w-i -hma +W3 -l-m4 +W5 +mg -I-W7 = Jf. Because CG is bisected in K, the triangle BCG is double the ^ triangle BKG. Because BG is bisected r'n H, the triangle BKG is double of the triangle HKG. Therefore the triangle BCG is four times the triangle HKG. Therefore the pyramid RBCG is four times the pyramid RHKG. But ttvking G as the apex and RHK as the base of RHK^, the altitude of this pyramid is one-half that of the prismatoid therefore the measure of the volume of RHKG is one-third of the product of \ a and Wj., i, e., \amx' Therefore . the measure of the volume of RBCG is %am-^, l^: In like manner it may be shown that ' the measure of the volume of RGCE is ^ama, ti II It II II II RCDE is iarriQ, I. II II II It It RDAE is ^w.4, It II II II II II REFA is lams, II II It II 11 It RABF is ^m^, II It II It It It RFGB is ^aruj. ' . ^ Hence the sum of the measures of the volumes of the pyramids on the lateral faces of the prismatoid is ;^ ^{m^+m2-\-m^+m^-\-m^+m>fi+m,j)=^ aM. >, Adding to this sum the measure of the volume of the basal <.'^.. )* ?"V- .;J,U, .-j .' '■4" ^v'" ,■•■' ■.'V -i'-t>'-" ■'•■ ^Z:>;:r':kl^:J--\ l-^ -X : '.< '^) ;;*-ai,.^-r^; i;Ai :$ A'i^ A^iff'i. .v»ii.;y'»"n ; MENSURATION. 183 pyramids RABCD and REFO, the measure r»f the volume of the whole prismatoid is ix^iud to be . or Vi^^B^+^M+B^). This is known as the Prisinoidal Formula. It is of the very highest importance, nearly all the elementary formulee in stereometry being but special cases of it. IV, a. The meaamre of the volume of a frustttm of a pyramid is ONE-THIBD of the prodiuit formed by midtiplying the measure of tlie altitude by the sum obtained by adding the measures of the areas of the twoparalld /aces to the square root of the prodrict of these two measwres; or This theorem may easily be deduced from the Prismoidal Formula, but it may also be proved independently as follows : — All cases of frusta on bases having four or more sides can be reduced to the case of frusta on triangular bases ; for, by passing planes through any one lateral edge and all the other lateral edges except the two adjacent to the first edge, any frustum whoso base has more than three sides will be resolved by these planes into an aggregate of triangular-based f rusta which have all the same altitude as the given frustum and whose triangular bases together make up the base of the given frustum. Hence it will be sufficient to consider only frusta on triangular bases. Let AB C D E F be the frustum of atetrahedron Let a denote the measure of its altitude ; B^y the measure of the area of the base ABC ; and B^i the measure of the area of the face D £F. Pass a plane through AG and F and another plane through EF and C, thus resolving the frustum into the three tetrahedra ABCF, DEFC, ACEF. Let F^, V^ and V^ denote C the measures of the volumes of these several tetrahedra, therefore if .J ..■^ ''\'i.: ■ ■'.!. i M ':v"fifi ■M->- '••«<■: ■?*«■" 'V* J'li"':^'^ .■^:V I ► i?'- 184 ARITHMETIC. Complete the pyramid of which ABCDEP is a frustum h^ producing the lacoral faces, — and thereby producing the lateral edges, — to meet in a common point G. Taking the triangle ABO as the base of ABCF, the tetrahedron and the frustum have the same altitude ; therefore the measure of the volume of ABCF is }inB^, or Taking the triangle DEF as the base of DEFC, the tetrahedron and the frustum have the same altitude ; therefore the measure of the volume of DEFC is JftBa, or Taking C as the common summit of the tetrahedra ABFC and AEFC, tliese two pyramids will have the same altitude, and therefore in determining the ratio of their volumes their common altitude may be omitted as being merely a common factor. The volumes of the tetrahedra will therefore have the same ratio as the areas of their bases have, or ABFC ABF AEFC AEF (1) Taking F as the common summit of the tetrahedra CAEF and CDEF, these two pyramids will have the same altitude, and therefore in determining the ratio of their volumes, their common altitude may be omitted. The volumes of the tetrahedra will therefore have the same ratio as the areas of their bases have, or CAEF CAE CDEF CDE (2) AB and EF being parallel, the triangles ABF and AEF have the same altitude, viz., the perpendicular distance of EF from AB ; the areas of these triangles will therefore have the same ratio as the lengths of their bases AB and EF have, or ABF AB .gv Also AEF EF AB AG EF~EG (4) « • \ ► ,,l.- MENSURATION. 185 CA and DE "boing parallel, tho triungloH CAE and ODE havo the same altitude ; the areas of these triangles will therefore have the same ratio as tho lengths of their bases A and DE have, or CAE CA Also CDE DE CAAG DE~EG* (5) Collecting v,he equalities numbered (1), (3), (4), (6), (5) and (2) and arranging them in the order here indicated, we obtain ABFC ABF AB AG CA CAE CAEF therefore AEFC AEF EF ABFC AEFC EG DE CDE CDEF (7) AEFC DEFC For the volumes of the three tetrahedra ABFC, DEFC, AEFC substitute the measures of these volumes in terms of a common unit and (7) becomes V V. r,=^}iaB,+iaB^ + la(B^B,) h IV, 6. TJie measure of the vohcme of a ireehfe is ONE-SIXTH of the continued prodtict of the me.os^ire of the altitiide of tlie wedge, the msasftire of the width of the base and the smn of the meamires of tlie lengths of the three parallel edges, or in which a denotes the measure of the altitude of the wedge, h denotes the measure of the width of the base and ftj, 6^ and h.^ denote the measures of the respective lengths of the three co-parallel edges. ;^. -.:% , * ,,' ■'■'"'■t 186 ARITHMETIC. Lot ABCDEF )iu a wudgo on the base ABCD. It may be troatod as a prisniatoid whose base i»a trapezoid and whose face parallel to the base has become reduced to the straight line £F. Let NP Q be a plane section of the wedge at right angles to the edge EF and therefore Q C also at right angles to the edges AB and CD which are parallel to £F. The length of the line PQ is the width of the base and the length of the perpendicular from N on PQ is the altitude of the wedge. Hence the measure of the length of PQ is 6, and the measure of the length of the perpendicular from N on PQ is a. Let G H K L be the midcross-section of the prismatoid and let it cut the triangle NPQ ii^ the straight line RS which will therefore be parallel to PQ. R is the mid-point of NP and S is the mid-point of NQ, therefore BS=^PQ, and therefore the measure of BS is ^h. The measure of the length of AB is &i, that of the length of EF is &3, and G and H are the respective mid-points of AE and BF, therefore the measure of the length of G H is ^(&| +h^). The measare of the length of CD is &a, that of the length of F E is &s> Aii^ ^ ^^^ ^ ^^^ the respective mid-points of C F and D E, therefore the measure of the length of K L is|(&a 4-63). Applying the Prismoidal Formula, the measure of the volume of the wedge is Ja(^i+*^+^2) (1) Bi is the measure of the area of the trapezoid ABCD. Tlie measures of the lengths of the parallel sides of this trapezoid are bi and b^ respectively and the measure of its width is b, M is the measure of the area of the trapezoid GHKL. The measures of the lengths of the parallel sides of this trapezoid are ^&i+&3) and ^^2+^3) I'espectively and the measure of its width is ^ &, .-. AM=b{^b^+^b,, + b.^). (3) '■: .,'•• : »";.., .''*!fsi^f. ;<\^r li^iri^iltilfejf, MENSURATION. 187 ume of (1) (2) The !oid are of its 2?a is tho measure of the orca of the lino EF, /i3 = 0. (4) Substitute in (1) the values of ^j, 4i»f and B^ given in (2), (3) md (4), = J^rt6(bi+&„+&3). In the case of the common wedge or wedge on a rectangular base, b and b^ are the measures of the lengths of adjacent basal edges and 62=^1 ^ rV, c. The meamre of the vdume of a tetrahedron is TWO- THIRDS of the prodiuit of the measure of the perpeiulictdar distaiice between any tvx> opposite edges ar I the meamre of tlie area of the 2^rallelogram whose angular points are the mid-points of the other four edges of the tetrahedron, or r,= laM. The tetrahedron is a prismatoid whose parallel faces are reduced to two straight lines, and the mid-parallelogram is its mid- cross-section, therefore by the Prismoidal Formula, Ft=^aitf. Each side of the mid-parallelogram is equal to half of the edge of the tetrahedron parallel to the side, therefore, if the midcross-section of the tetrahedron be a rectangle and V|=§aCiC2, in which c^ and Cj denote half the measures of the lengths of the two edges parallel to the midcross-section. In this case the midcross-section divides the tetrahedron into two wedges (hemitetrahedra) whose altitudes are equal as also ar6 their volumes. It is worthy of notice that if a prism, a hemitetrahedron and a pyramid are on equal bases and are of the same altitude, the volume of the prism is thrice and the volume of the hemitetrahedron is twice that of the pyramid, or ■J- J' *,;'■ . 4 ■:.*>■■ ■■.■4V' tWi^- ;fe«i^ser&iui ^K'-i-feSy: itrs/a^^i^iri^iKL-l'Si . ■ ' ■■'i^lMM^ >tl ;^*ii:*fei*ife r 188 ahithmetic. ■«,■ . Kxamph'. 1. An iron limk in tho form of a hollow ciiho whoso RiduB, b«)ttoni tviul top uro nil tuid evorywheru of tho samo thicknesB, hiiH i\ oipncity of 381 gallons. Thu lungth of an outside edge of the tank is 4 ft. Find the thickness of the sides. The onpHcity of tho tank is 381 gal. = 277 118CU. in. X381. =(277'll«x881)cu. in. The length of tho edge of a cube of this capacity is (277-118x381)?»in'! The cube root of 277*118x381 may be obtained directly by multiplication and evolution, or it may be computed by the aid of logarithms thus : — , log (277 118 X 381)^ = Jf the floor is the sum of the areas of thi' throe trapezoids ABCM, MCDN, NDEF. The area of ABCM is ^ of 6(28 + 32) s«i. ft. - 180 8(i. ft. II II II MCDN is I of 16(32 + 34) sq. ft. = 496 wi. ft. II II II NDEF is i of 8(34+30)8(1. ft. =266 sq.ft. .-. the area of the floor is (180+495 + 256) sq. ft. =931sq. ft. (1) The area of the ceiling is ^ of 12(22 + 23) sq . ft. = 270 sq. ft. (2) The area of the midcross-section is the sum of the areas of the fimr trapezoids PQBW, WRSX, XSTZ, ZTUV. To determine the ureas of these trapezoids, the lengths of their parallel sides and of the normal distances between these sides must first be found. PQ=i(AB+GH)=^(28+22)ft.=25ft. WR = \ (MC + GH) = 1(32 + 22) ft. = 27 ft. XS =^(ND+GH) = i(34 + 22)ft. =28 ft. ZT=-J(ND+LK)=i(34 + 23)ft.=28ift. VU=J(FE + LK)=l(30+23)ft.=26ift. PW = lAM = iof 6ft.=3ft. WX = J MN«=i of 15 ft. =7i ft. XZ=|GL=lofl2ft.=6ft. ZV=lNF = Jof 8ft.=4ft. The area of PQRW is ^ of 3(25+ 27) sq. ft. =78 sq. ft. M M .. WRSXis^of 7^27 + 28) sq.ft. =206i sq.ft. ., ., M XSTZ is I of 6(28 + 28i)8q. ft. - 169^ sq. ft. ZTUV is I of 4(28i + 26i)8q. ft. = 110 sq.ft. The area of the midcross-section is (78 + 206|+169^ + 110)sq. ft. = 563| sq.ft. " (3) .-. the capacity of the attic is ^ of lOi-^ 931 + 4(563]) + 270 J-cu. ft. = 6048 cu. ft. EXERCISE XXIII. ■■%. ^ I. Find the number of cubic inches in the volume of measuring a foot by a yard by a metre. a ( ^uad "^ J^Jjf', JSAi t'H •■*•■ w ".»<*,. V,r '^•■•f' 190 ARITHMETIC. d. Find txi the nearest gallon the vuluniu of a (juad measuring 76 in. by 87 '5 in. by 12(i-875 in. fl. Find, correct to four significant figures, the length t)f the inside edge of a cubical vessel which will just hold 10 gallons. 4. Find, correct to four significant figures, the length of the inside edge of a cubical vessel which will just hold 100 gallons. 5. One aero of a certain wheat-field yielded 21001b. of wheat weighing 7 lb. 10^ oz. per measured gallon. At this rate what was the yield in cubic inches per scjuaro yard oi the field, and what would be the length of the edge of a cube equal to the yield of a square inch of the acre i 6. A quadrate reservoir is 147 ft. 8 in. long, 103 ft. 6 in. wide and 11 ft. 9 in. deep. When the reservoir is nearly full of water, how many ci;oic feet of water must be drawn off that the water-surface may sink 4 ft. 4 in. '{ 7. Find to the nearest gaUon the capacity of an open quadrate tank measuring 7' C" by 6' 4" by 5' 8" externally ; tho material of which the tjuik is made being 1\ inches in thickness. §. Three cubes of lead measuring respectively A, f^ and ^ of an inch on the edge were melted together and cast into a single cube. Find the length of the edge of the cube thus formed, neglecting loss of lead in melting and casting. 9. Four cubes of lead measuring respectively 6, 7> 8 and 9 inches on the edge were melted together and cast into a single cube. Find the length of the edge of the cube thus formed, if 4 per cent, of the lead was lost in the melting and casting. 10. Three cubes of lead measuring respectively 3*1, 3*6 and 3 7 inches on the edge were melted and cast into a uiiigle A„' ■•V,4,, ^y* vlJ*v.A "ff? .,'■■>> »-*,::^' liU^.;*; ""■,,■•" ■.,^-' :-i-- "':.'_■. J- "••■ -'.f ' \ ;, .-s- . ; -- ,'-sr" , . i4l'' -^ ■;,■;,»■'; '."■.T'^ l.^y"^--- i'; MENSURATION. 191 -I 13. The three adjacent edges of a quad are to one another as 2:3:5 and its volume is a cubic ntutre. Find the length of the edges and the areas of the faces of the (£uad. 14. Find the volume of a cube tjie area of whose surface is 100'86sq. in. 1ft. The surface of a cube measures 30s(i[. in. Find the area t)f the surfAce of a cube of tiv) times the volume of the fornior. 16, The volume of a cube is 30 cu. in. Find the volume of a cube whose^ surface has an area tivo times the area of the surface of the former cube. 17. A cube measures 5 in. on the edge. A second cube is of thrice the volume of the first. By how much does the length of an edge of the second cube exceed that of an edge of the first cube ? 1§. A cube measures 5 in. on the edge. Find the volume of a cube whose surface-area is thrice that of the former cube. 19. A quadrate cistern i%5ft. wide by 6 ft. long by 4 ft. 2 in. deep. Its width and its length are each increased by 6 inches. How much deeper must it be made that the total increase of its capacity may be 250 gallons s 20. If a quad has its length, its breadth and its height respectively a twelfth, a thirteenth and a fourteenth as long again as the corresponding dimensions ri another quad ; show that the volume of the first quad will be a quarter as large again as the volum,^ of the second quad. 31. By raising the temperature of a cube of iron, the length of each of its edges was increased by "5 per cent. Find correct tt) four decimals the ratio of increase in the vt>lume of the cube. 33. Each edge of a cube is diminished by a tenth of its length. By what fraction of itself is the volume diminished ? By what fraction of itself is the area of the surface diminished 'i 33. By taking the decimeter as equal to 4 in. what percentage of error is introduced into («), linear measurements ; (6), are^d measurements ; (c), volume measurements ? 31. The height of a solid six-inch cube of India-rubber is diminished by pressure to 5 85 in. If the volume of the solid remain the same and the lateral expansion be uniform throughout, what will be the dimensions of the new base 1 192 ARITHMETIC. 35. The length of a quad is 7 in., its height is 3 in., and the total area of its surface is a square foot. Find the volume of the (j[uad. J16. The length of a quad is 13 'Sin., its width is 8 4 in., and the total area of its surface is 466 "5 sq. in. Find its volume. 2T. The width of a quad is 7 05 ft., its height ia 3 13 ft. and the total area of its surface is 30 sc^. yd. Find its volume. S8§. The width of a quad is 371 nmi., its height is 284 mm., its volume is a cubic metre. Find the area of its base. 29. Find the area of the surface of a quad 31 "62 cm. wide by 38 "73 cm. long and of "03 cubic metre volume. 30. The area of the base of a (juad is 71*288 sq. in., that of a side of the quad is 56*868 sq. in. and that of an end is 52 "65 sq. in. Find the volume of the quad. ttl. The length of the perimeter of the base of a quad is 20 in. ; , the area of the base is 20 "16 sq. in. ; and the total area of the surface of the quad is 90*32 sq. in. Find the volume of the quad. 32. The perimeter of the base of a quad measures 25 '4 in., the area of one end of the (juad is 23*1 sq. in., and the volume of the quad is 166*32 cu. in. Find the lengths of the edges of the quad. 33. Find the measure of the length of the edge of a cube the measure of whose volume is equal to the measure of the area of its surface. 34. The measure of the volume of a quad two of whose edges measure 3 in. and 4 in. respectively, is the same as the measure of the area of the whole surface of the quad. Find the length of the third edge. 35. The area of the surface f>f a quad on a square base is 192 sq. in. The area of the base is ecjual to the sum of the areas of the two sides and two ends. Find the volume of the quad. 36. The volume of a quad on a square base is 6572 cu. in., the height of the quad is 11*8 in. Find the length of an edge of the base. 37. Find the weight <>f the air in a rectangular room measuring 27' 8" by 23' 5" by 12' 4", the weight of the air being the 001295 of the weight of an equal volume of water. :;\ >ti\:.;:i^--lif:af:'mP-^ /:.■.-*•«' A,;.-; ..-*>•■ d the 3f the \d the lid the m., its dde by )£ a side 1 sq. in. s 20 in. ; e surface ill., the ,e of the quad. ube the ea of its . Ise edges lasure of of the base is areas of in., the \q of the lieasunng )1295 of MENSURATION. 193 3S. If a cubic foot of gold weigh 12001b., find the thickness of gold-leaf of which 1200 leaves 3J inches sijuare weigh an ounce troy. 89. A quadrate block of stone measuring 5 •297" by 7 472" by 9*57" weighs 38-14 lb. Compare the weight of any volume of the stone with the weight of an equal volume of water at 62' F. 40. What will be the weight of 36 iron rods each 14 ft. long and of cross-section f of an inch square, if the specific gravity of the iron be 7 "7 ? 41. What length of a bar of iron will weigh 10 lb. , the cross-section of the bar being a rectangle measuring f in. by 1;^ in. ? 43. What weight will justkeepunderwatera stick of square-timber measuring 36 ft. by 10 inches square, the specific gravity of the wood being '725 ? 43. What sized cube of iron placed on a quad of dry i)iiie measuring 8 ft, by 6*6 ft. by 4 in. will just sink the quad in water, the specific gravity of the pine being '472 and that of the iron 7 '7 '^ 44. An open quadrate tank is 5 ft. 6 in. long, 4 ft. 3 in. wide and 3 ft. 8 in. high, the sides and bottom of the tank are % in. thick. Find the number of cubic inches of material in the vessel. 45. Find the weight of a hollow iron cube measuring 2'73P inches on the outer edge, the thickness of the iron being "167 of an inch and its specific gravity 7 "7. 46. Find the thickness of the sides of an iron box in the form of a hollow cube, which weighs 266 lb. when empty and 566 lb. when filled with water ; the sides, bottom and top being all of the same thickness and the specific gravity of the iron 7 '7. 4T. The sides, bottom and lid of a quadrate box have a uniform thickness of § in. The outside measurements of the box are 8 in. by 12 '6 in. by 16*25 in. How many cubes each f of an inch on the edge, will the box hold. 4§. Find the thickness of the material of which a closed hollow iron cube is constructed, if the cube weigh 33 lb. 4 oz. and measure 10*5 in. on an outside edge, the specific gravity of iron being 7 "7. 49. An iron cube is coated with a uniform thickness of gold. Find the thickness of the gold if the coated-cube is 3 inches long M .:s| .*;aJ^ 194 ARITHMETIC. and weighs 7 '525 lb., the specific gravity of the gold being 19 '25 and that of the iron 7 '7. 50. Find the volume of a right triangular prism 8 in. long, the terminal triangles being right-angled and the lengths of the sides containing the right angle being 1*2 in. and 2*1 in. respectively. 51. The normal length of a triangular prism is 79 mm., the length of an edge of one of the terminal triangles is 43 mm., and the length of the perpendicular on that edge from the opposite superficial angle is 29 mm. Find the volume of the prism in cubic centimetres. (The normal length is the length measured at right angles to the parallel ends. If one of these ends be taken as the base of the prism, the normal length will be the altitude of the solid.) 53. The altitude of a prism is 17 '3 in. and its base is a parallelogram of length 25 "76 in. and normal width 9 "7 in. Find the volume of the prism. 53. The normal length of a trapezoidal prism is 97 ft. 6 in., the lengths of the parallel edges of the trapezoidal ends are 37 ft. 5 in. and 23 ft. 4 in. respectively and the perpendicular distance between these edges is 9 ft. 6 in. Find the volume of the prism. 54. The length of a prism is 9 ft. 4 in. A right cross-section of the prism is a quadrilateral, one of whose diagonals measures 3 ft. 7 in. and the perpendiculars on that diagonal from the opposite angles of the section are respectively 1 ft. 5 in. and 1 ft. 7 in. long. Find the volume of the prism. 55. Tho fabled wall of China was said to be 25 ft. wide at the bottom, 15 ft. wide at the top, 20 ft. high and 1600 miles long. How many cubic yards of material would such a wall contain. 56. How many cubic yards of earth must be removed in the digging of a ditch 147 ft. long, 8 ft. wide at the top, 5 ft. wide at the bottom and 4 ft. 6 in. deep, the ends of the ditch being vertical/ 57. How many gallons of water will fill a horse trough 7 ft. 6 in. long, 10 in. deep, 14 in. wide at the top and 11 in. wide at the bottom ; the ends of the trough being at right angles to the bottom and sides ? 5§. How many prismatic bars of laad each 10*5 in. long must be melted down to make a cube 6*25 in. on the edge, a right cross- -^->A. -"■^' : V-- >■'' J -.-,. MENSURATION. 195 section of each bar being a trapezoid measunng 1*3 in. and 'Gin. respectively on the parallel sides and '5 in. in perpendicular distance between these sides ; '5 per cent, of the lead being lost in the melting ? 59. 7843 cu. yd. of earth were removed in digging a ditch 2 ft. 9 in. deep, 4 ft. 6 in. wide at the top and 3 ft. wide at the bottom. Find the length of the ditch assuming that the ends were vertical. 60. The cross-section of a canal is 36 ft. wide at the surface of the water and 20 ft. wide at the bottom ; what must be the depth of the water if 100 yd. in length of the canal contain 589315 gallons of water ? 61. A ditch 125 yd. long is filled to a depth of 1ft. 9 in. by 10571 gal. of water. What must be the width of the ditch at the bottom if the width at the surface of the water be 3 ft. and the ends of the ditch be vertical ? 62. A stream flows at the rate of 3 miles per hour through a trough whose cross-section is a trapezoid. The width of the bottom of the trough is 21 inches, the depth of the water is 4 5 inches and the width of the surface of the water in the trough is 25 inches. How many gallons flows through the trough per minute ? 63. A prism of 21 inches altitude weighs one ton. Find the area of the base, the ifiaterial of the prism weighing 5241b. per cubic foot. 64. The volume of a prism is 6cu. ft. ; its height is 9 in., and its base is an isosceles right-angled triangle. Find the lengths of the edges of the base. 65. A shed with a single sloping roof is 22 ft. long by 12 ft. wide ; the height of the roof above the floor is 12ft. at the front and 8 ft. at the back. Find the total capacity of the shed. 66. A school-room with attic ceiling is 32 ft. long by 28 ft. wide. The ceiling at the side walls is 10 ft. above the floor and slopes upward until it attains a height of 14 ft. 6 in. and then becomes level, the width of the level part being 12 ft. The ceiling meets the end walls at right angles. Find the air-capacity of the school-room. 67. A parallelepiped is cut by two planes which neither meet the ends nor intersect. The area of a right cross-section is 96 sq. in. and the lengths between the cutting planes of the four parallel r^'S. ■.^ 196 ARITHMETIC. ! 1 1 i edges are respectively 6in. 7 '5in. lOin. and 8*5 in. Find the volume of the portion of the parallelepiped between the cutting planes. 6§. The base of a pyramid is a triangle one side of which measures 15 '3 in. ; the length of the perpendicular on that side from \he opposite angle of tlie base is 9 Gin. and the altitude of tho pyramid is 125 in. Find the volume of the pyramid. 60. Find the volume of a tetrahedron whose base is a right- angled triangle, the sides of the base containing the right angle measuring 17 in. and 19 in. respectively and the altitude of the tetrahedron being 18 in. ' 70, Find the volume of a tetrahedron whose base is a right-angled isosceles triangle, the altitude of the tetrahedron being 7 ft. 5 in. and the length of the hypothenuse of the base being 5 ft. 7 in. TI, Find the weight of the pyramid formed by cuttinc; off a corner of a cube of lead by a plane passing through three adjacent corners, the length of an edge of the cube being 2 '5 in. and the specific gravity of lead being 11*4. 73, One of the corners of a quad of gold is cut off by a plane which meets the three conterminous edges, 2*7 inches, 4*3 inches and 3 6 inches respectively from their common point. Find tho value of the piece cut off, the specific gravity of the gold being 17*66 and its value $18*95 per ounce troy. » 73. The base of a pyramid is a square whose side is 3*45 ft. long. The altitude of the pyramid is 4 75 ft. Find the volume of the pyramid. • 74. Find the volume of a pyramid whose altitude is 4 ft. 5 in. and whose base is a rectangle measuring 3 ft. 4 in. by 3 ft. 9 in. 75. The altitude of a pyramid is 2 ft. 3 in., its base is a trapezoid whose parallel sides measure 1ft. 9 in. and 1ft. 3 in. respectively, the perpendicular distance between these sides being 1ft. 4 in. Find the volume of the pyramid. 76. The base of a pyramid is a square 2 ft. 7 in. long and its volume in 3'2cu. ft. Find the altitude of the pyramid. 77. The volume of a pyramid on a rectangular base is half a cubic yard. The length of the base is 3 ft. 9 in. and the altitude of the pyramid is 32 ft. Find the width of the base. 78. The volume of a pyramid on a square base is a cubic yard and its altitude is a yard. Find the length of au edge of the base. \:k. '^/..i^it;.. --...-i^-'^y .^V-fi'-.-W^?. 1. .■■.-■■,■ ^ - T" "J ' ;;r:^-7*j";'-vr'.tr\'..*:^« V(PPifi'-";''"^'.:7'.V*''r^ ■*''»*'' ■^^"•.'•^t- '•^'H^ ■'?"■ yy*^. "'T'-f.'""-'/^' '*''"' - . ' - 'J •^- MENSURATION. 197 volume les. leasures rom ^ilie pyramid a right- ht angle e of the it-angled 7 ft. 5 in. 7 in. tinfli off a I adjacent . and the y a plane 4-3 inches Find the told being .ft Jong. ne of the • 4 ft. 5 in. 9 in. : trapezoid ^pectively, 1ft. 4 in. ig and its is half a iltitude of pubic yard the base. 79. The volume of a pyramid on a square base is 30*87 cu. in. and the altitude of the pyramid is equal to the length of an edge of the base. Find the altitude. 80. , The volume of a pyramid is 77 cu. in. The base of the pyramid is a tiuadrilateral ; the length of one of the diagonals of the base is 15 in. and the lengths of the perpendiculars on this diagonal froni the opposite angles of the base are 10*6 in. and 9 in. Find the altitude of th o pyramid. St. The base of a pyramid is a square 15 in. long and the altitude of the pyramid is 16 in. The base of another pyramid is a rectangle 16 in. long by 12 '5 in. wide. Find the altitude of the second pyramid, the volumes of the two pyramids being equal. 82. The Great Pyramid of Egypt when complete was 480 ft. 9 in. in height, and its base was a square 764 ft. in length ; in its present condition the pyramid is 450 ft. 9 in. high and its base is a square 746 ft. long and wide. Find to the nearest cubic yard the volume of the pyramid in its complete and also in its present state. 83. The representative gold pyramid in the International Exhibition of 1862 was 10 ft, square at the base and 44 ft. 9^ in. high. Find the volume of the pyranud, and the weight and the value of the gold represented by it, taking the specific gravity of the gold at 19 "25 and its value at $20 "67 per ounce troy. 84. Since the construction of the pyramid mentioned in problem 83, about 25,000,000 ounces troy of gold have been mined ; how luucli higher would the pyramid require to be made to include this (juantity ? 85. The base of a pyramid is a square whose sides are 25 in. long. The altitude is 16 in. A plane parallel to the base divides the pyramid into parts of equal volume. Find the perpendicular height of the plane above the base. 86. The base of a pyramid is a trapezoid whose parallel sides measure 19 5 in. and 13 '7 in., the perpendicular distance between them being 12 6 in. The altitude of the pyramid is 14 in. At what height above the base must a plane parallel t the base be drawn, that it may bisect the pyramid ? 87. The base of a pyramid is a square whose area is 7 sq. ft. The altitude of the pyramid is one yard. A plane parallel to the base so divides the pyramid that the volume of the frustum between Ids ARITHMETIC. tlie base and the piano is double the volume of the pyramid abov the plane. Find the height of the frustum. §§. The altitude of a pyramid is 15 in. A plane parallel to the base divides the pyramid into two parts whose volumes ase such that thrice the volume of the frus' un between the plane and the base is equal to five times the volume of the pyramid above the plane. Find the height of the frustum. 89. Find the volume of a prismoid whose top and bottom are rectangles the corresponding dimensions of which are 3 ft. by 2 ft. and 5ft. by 3 "5 ft., the altitude of the prismoid being 3 "5 ft. •O, Find the volume of a prismoid whoso top and bottom are rectangles the corresponding dimensions of which are 3 ft. by 2 ft. and 3 "6 ft. by 5 ft., the altitude of the prismoid being 3 5 ft. 91. Find the <;apacity of a cart the top of which measures 4' 3" by 3' 8" ; the bottom, 3' 9" by 3' 2" ; and the depth, 2' 3". 92. How many gallons of water will fill a ditch 2 ft. deep, the top and bottom of the ditch being rectangles whose corresponding dimensions are 148 ft. by 3 ft. 4 in. and 146 ft. 6 in. by 2 ft. 3 in. ? 93. Find the weight of an iron shaft whose ends are rectangles, one end measuring 10*5 in, by 17 in., the other end measuring 7 in. by 12 in. , the length of the shaft being 13 ft. 6 in. and the specific gravity of the iron, 7 "7. 94. What weight will just sink a scow in the form of a hollow prismoid with rectangular base, the length of the scow over all being 14 ft. 1 in. ; its width, 3 ft. 8 in. ; its full depth, 2 ft. 11 in. ; the length of the bottom outside, 12 ft. ; the width of the bottom 3 ft. and its weight 920 lb. 95. Find the volume of a pile of broken stones, the base of the pile being a rectangle measuring 13 ft. 6 in. by 7 ft. 5 in. ; the top of the pile a rectangle measuring 12 ft. 2 in. by 6 ft. ; and the height of the pile being 2 ft. 10 in. 96. It is usual to take as the measure of the volume of a pile of broken stones the product of the measure of the altitude of the pile and the measure of the area of its midcross-section. By how much would the volume thus calculated be in defect of the actual volume in the case of the pile described in the problem immediately preceding. A. ,-''?'.' .;^'^;•„:.;^,!;i=^^i^:. \'n:A .irSsi.:, .:vtia.v;; 'a,- f fi>- 'p'it^- MENSURATION. 199 97*. Find the number of cubic yards in a railway cutting in the form of a prismoid with trapezoidal ends ; the lengths of the parallel '^ides at one end being 124 ft. and 33 ft., and the distance between them 28 ft. ; the corresponding dimensions of the other end being 104ft., 33ft. and 21ft. respectively; and the distance between the ends being 235 '5 yd. 9§. A straight ditch with a fall of 1 ft. in 300 yd. is to be dug in level ground. The sides are to slope 1 in 1, the bottom is to be 4 ft. wide, and the depth at the upper end is to be 3 ft. 6 in. Find the number of cubic yards of earth that will require to be removed in digging the first 1000 yards of the ditch. 99. How many cubic yards of earth will be excavated in making a railway cutting through ground whose surface is an inclined plane rising in the sa,me direction as the rails, the length of the cutting being 123 yd. ; the width at the bottom 33 ft. ; the width at the top at one end, 66 ft. ; at the other end, 100 ft. ; and the depths of these ends, 22 ft. and 48 ft. respectively ? 100. A railway-embankment is made on ground which falls at 20 ft. per mile in the same direction of the rails, which themselves fall 1 in 800. The length of the embankment is 2100 yd. ; its widtli at the top is 33 ft. , the slope of the sides is 1 in 1 and the height at the upper end is 1ft. 8 in. Find the number of cubic yards of earth in the embankment. 101. The ends of a prismoid are rectangles whose corresponding dimensions are 17 '3 in. by 11 "4 in. and 9 5 in. by 6*6 in. ; the altitude of the prismoid is 21 "6 in. The prismoid is divided in two i^arts by a plane parallel to the ends and midway between them. Find the volume of each part. 102. The ends of a prismoid are rectangles whose corresponding dimensions are 7 ft. by 5 ft. and 3 ft. by 2 ft. The prismoid is divided by planes parallel to the ends, into three prismoids each 1ft. 8 in. in altitude. Find the volume of each of these three prismoids. 103. A prismoid, One of whose ends is a rectangle measuring 15 in. by 12*5 in., the opposite end measuring 9*6 in. by 8*4 in,, and whose altitude is 2 ft. is cut into two wedges by a plane which passes through the longer edge of one end and the opposite longer edge of the other end. Find the volumes of the wedges. .'■fr 200 ARITHMETIC. Hi .' 104. Tlio heiglit ui n wodgu is 18 in., tho longth of the odgo is 16 in. , and tho diniunsions of tho bngo which is h rectangle, are 12 in. by 8 in. The wedge is divided into two paits by a plane parallel to the baBe and midway between the base and the edge. Find the volume of each part. 105. Had the wedge described in problem 104 been bisected by tho plane parallel to the base, what would have been the height of the piano above the base ? 106. Tho length of the edge of a wedge is 8*5 in., the length of the base which is a parallelogram is 6*3 in. and its normal width is 4 5 in., tho height of the wedge is 15 in. The wedge is divided into three parts of equal height by planes parallel to the base. Find tho volume of each part. lOT. The ends of a prismoid are rectangles whose corresponding dimensions are 18 in. by 15 in., and 10 in. by 18 in. ; the height of the prismoid is 7 ft. 4 in. The prismoid is cut by a plane parallel to the ends and at a distance of 2 ft. from the larger end. Show that the section is a square, and find the volumes of the parts into which the plane divides the prismoid. 10§. Find the number of cubic yards of earth in an embankment from the accompanying plan and following data : — The base ABCD is a (|uadrilateral and the top EFGHK is a pentagon. Tne edges AB, EF, KH and DO are all parallel to each other, AD and £K. are in a plane at right-angles to AB, and LG is paraUel to EF. AB=96yd., DC = 124yd. ; EF=84yd., LG = 98yd., KH = 90yd. ; AE = 18ft., EL = 16ft., LK=18ft., KD = 12 ft., the last four measurements being * in plan ', i.e. being tho horizontal distajices between verticals through the points A, E, L, K and D. The height of the embankment is 18 ft. 109. The lengths of two opposite edges of a tetrahedron are 7 "2 in. and 5 6 in. respectively and the perpendicular distance between them is 6 4 i^. The midcross-section is a rectangle. Find the volume of the tetrahedron. \ MENSURATION. 201 1 10. A rcctHiigulHi' tank 3 ft. long by 2 ft. 4 in. wido by 2 ft. « in. ileoj), rested on props 3 in. high, a proj) at each corner. By accident one of the props was knocked out of its jdaco and the cistoni was tilted on the adjacent two until the unsupported corner touched the ground. How much less water would the tank liold in that position than it would hold when level ? 111. The base of a wedge is a rectangle measuring 3 '6 in. by 2*4 in., the length of tlie opposite edge is 3in., the height of the wedge is 8 in. Find the volume 1° if the throe-inch edge is parallel to the longer side of the base ; »2'' if it is parallel to the shorter side of the base. 119. The base of a sphenoid is a square whoso sides are 10 in. long ; the opposite edge is parallel to the diagonal of the base, and of the same length as the diagonal ; the altitude of the sphenoid is 16 in. Find the volume of the sphenoid. 113. The lengths of the three parallel edges of a wedge are 7*5 in., 5*7 in. and 6*9 in. respectively and the area of a section at right angles to these edges 76 sq. in. Find the volume of the wedge. 114. The base of a wedge is a rectangle measuring 13*5 in. by 11*2 in., the length of the opposite edge is 5 '4 in., this edge being parallel to the longer side of the base ; the perpendicular distance of this edge from the plane of the base is 18 in. The wedge is divided into two pieces by a plane which intersects the edge opposite the base at a point distant 7 "5 in. from one end and which cuts the two edges parallel to this edge at points distant 5 "25 in. and 7 '5 in. from the ends corresponding to that from which the 7 "5 inches was measured. Find the volume of each part. 115. The base of a wedge is a rectangle measuring 18 in. by 15 in. ; the opposite edge is parallel to the hmger side of the base and is 10 in . long ; the length of the perp.endicular from this edge on the base is 21 in. Find the volume of the parts into which the wedge is cut by a plane passing through one end of the edge opposite the base and which is parallel to the triangular face at the other end. 116. The base of a wedge is a trapezoid whose parallel edges are 3 ft. and 1 ft. 9 in. long respectively and whose width at right angles to these sides is 15 in., the length of the edge opposite the ■^A ■--<^l i —.'Ms;" J<^.-.. .*,■■ .!^*■.-4i-■■.^■.■,l«i.^ n I 1 202 AIUTFIMETIC bune Ih IHin., aiul the volumu of tho wedge in 2cu. ft. Find the nltitudo of tho wodge. IIT. The lengtli rtf a side of the huso r>f h frustum of a R<{uarc pyramid is 3' 0", that of a side of the top is 1' 8", the altitude of tho frustum is 2' 6". Find the volume of the frustum. 118. Find the number of cubic feet in a stick of square timber 18" square at one end, 14" scpiare at tho other ond and 36' long. 119. Find the weight of a frustum of a scjuare pyramid of marble, tho height of tho frustum being i\ ft. 6 in. ; the length of an edge of the base, 4ft. 4 in., and of an edge of the top; 2 ft. 8 in., the weight of a cubic foot of marble 172 lb. 190. In the frustum of a square pyramid w^hose base-area is 2 8(1. y^' ^^*^ whose altitude is 4ft. 6 in., the lengths of the basal edges are to those of the top edges as 3 to 2. Find the volume of the frustum. 131. The areas of the base and top of a frustum of an iron pyiitmid are 1 8([. ft. 488(j. in. and 1 sq. ft. 3 sq. in. respectively and the weight of the pyramid is 8881b. Find the height of the pyramid, the specific gravity of the iron bciiig 7 11. 193. The altitude of a frustum of a scpiare pyramid is equal to the length of a side of the base and is double of the length of a side of the top. Find the altitude, the volume of the frustum being 4 cu. ft. 193. A frustum of a pyramid has the area of its base nine times the area of its top. Compare its volume wit!i that of a prism whose altitude and base-area are respectively the same as the altitude and the base-area of the frustum. 194. A frustum of a pyramid has the area of its base four times the area of its top. Compare its volume with that of a pyrar.:id whose altitude and base-area are respect)^ ely the same a** tJie altitude and the base-area of the frustum. 19$. Find tho volume of the frustum of a pyramid on a rectangular base i:^iea8uring 4 ft. by 2 ft. 8 in., the height 6i the frustum being 5 ft. 3 in. and the length of the top, 3 ft. 6 in. 196. The voluir-'^ -^f the frustum of a pyramid on a rectangular base is 3*6 cu. ft. l^h& ]jng i of one side of the base is 1 ft. 6 in., the length of the oorre.'ipcnding sid; , of the top is 10 in. , the height of the frustum is 1 '■'}. 4 u. Find the lengths of the other sides of the base and top. >i'X,' :i:.,;'/;\ ■'^,■'"% ;:-U-':^-,^'4fc.^u^^; "■ ■> ,«%' ■ ;"' "^n'T'T '' v —^ 'y," ■• MENSUKATION. 203 lilT. The base <>f tho frustiuu of a pyniinid in (i loctanglo whoso length iH double its width ; the nrvn oi the top is half t)io area of tho base ; tho height of the frustum is 3ft. 8 in. and its volume is a cubic yard. Find the length of the base. 138. The base of a frustum ()f a pyramid is a trapezoid the lengths of vvlioso parallel sides are 275 cm. and 225 cm. respectively, the diutuii between them being 192 cm. The height of the frustum is ?S75 mm. and the width of its top is 148 cm. Find tho volume ot Mie frustum in cubic metres. 129. Find the area of tho surfaco of a s([uaro pyramid whose basal ot!.{ed are each 3' 4" long, the slant height*^f each side being 3' H". 130. Find the area of the surface of a frustum of a square pyramid, the length of a side of tho base being 18 in. ; that of a side of the top, 6 in. ; and the slant height of each of tho lateral faces being 27 in. 131. Find the height and the width of a quad whose length is 3 ft. whose volume is 9 cu. ft. and the area of whose surface is 288q. ft. KWsq.in. 133. A horse-trough 9 ft. long, 15 in. wide at the top and 10 in. wide at the bottom, and 12 in. deep, is full of water. If 30 'allons of water be drawn off by how many inches will the surface of the water in the trough sink ? (The ends of the trough are vertical ; the calculation is to be made to the nearest tenth of an inch.) 133. The cross-section of a canal is 33 ft. wide at the bottom and 58 ft. wide at a height of 10 ft. from the bottom. At what depth nmst the water in the canal stand that 1000 yd. in length of the canal may contain 4,545,725 gallons ? i34. The volume of the frustum of a square pyramid is 172 cu. ft., the height uf the frustum is 36ft. and the length of a side of the base is 2 ft. 8 in. Find the length of each side of the top. 1 35. A covered rectangular tank whose dimensions are 3' 6" by 2' 11" by 1' 9 " w ill hold j ust 81 gallons. What nmst be the thickness of the material "i; which the tank is made, the bottom, sides and top being h,L ,1 the sant*» t^^hickness ? {This problem is of a type which is the inverse of tht «yj*e to which piohlems 9 and 10 of Exercise VI helwig. The calculatiom for these three problems toill tJierefoi-e follow parallel lines.) #■■ 's » ■ ■■■'-.- "-1 i :-^.>l.V^? ill i 1 204 ARITHMETIC. 171. In any right-angled triangle, the S(]uare8 on the sides , containing the right angle are together equal to the square on the hypothenuse or side opposite the right angle. (Euclid I, 47.) Let ABC l)e a triangle right-angled at C, and let a, b and c be the MEASURES of the lengths of the sides oppoaite the angles A, B and C respectively ; . '. a", ft* and c- are the measures of the areas of the squares on these sides and it follows from this and the preceding proposition that a3+6- = c2, (1) and .-. c = (a- + b^)^, (A) ® that is ; — In (tuij riylit-awjled triaiujU', the mcaswe of the letujth of the hypotheionsc is the square ,oot of the sum of the squares of the ineasurcK of the lengths pf the sides containi)i ^ mm . = (2025x49)'^ mm. = 315 mm. . •. AD = 1590 mm . - 315 mm . = 1276 mm. .-. AC = (12752+9882)^mm. = (1625625 + 976144)^ mm, = 2601769^ mm. = 1613 mm. 172. If m and n be any two whole numbers then shall m'^—n^f 2mn and m^+n^ • be the measures of the lengths of the sides of a right-angled triangle . For (w" —n^)" —m*' — 2m^tt,^ +n^, (2m'>t)^ = Am^ti^ and (m^ +n"y--=m^+2n^n^ + }i'^. EXERCISE XXIV. 1. Find the length of the hypothenuse of a right-angled triangle whose other sides are 65 in. and 72 in. long respectively. 2. Find the length of the hypothenuse of a right-angled triangle, the lengths of the other sides being 77' mm. and 464 mm. 3. Find the length of the diagonal of a scjuare whose side is one foot long. 4. Find the length of the diagonal of a cube whose edge is one foot long. 1 Hi . -4 ,-..•«, '^^^ >,'4'»»?""' 20(> ARITHMETIC, I I I i I ; ff. What \H the length of the Hide of ii H(|UHre whoso dingonal is one foot long? a. VVlmt is the length of the edge of a cube wlutse diagonal is one fot»t long ? 7. What is the length of the edge of the largest cube that can })e cut out of a sphere (J inches in diameter ? 8. What is the length of the diagonal of a cube if the length of a diagonal t»f one of the faces of the cube is 3 ft. ? 9. What will be the length of the diagonals of the faces of the largest cube that can be cut t)ut of a sphere 3 inches in diameter ? 10. A quad measures 24 in. by 11 '7 in. by 4 4 in. Find the lengtlia of the diagonals of its faces. 11. A quad is 14 ft. hmg by 5ft. wide by 2 ft. thick. Find the length of its diagonal. 19. A(piadmea8ure8()325ni. by5'79fim. by •528 m. Determine the length of its diagtmal and the lengths of the diagonals of its faces. 13. The lengths of the diagonals of the faces of a quad are 22 ft., 8 ft. and 3 ft. respectively. Find the length of the diagonal of the tjuad. 14. The diagonals of the faces of a (i[uad are respectively 25 in., 23'79in. and 9'7'Hn. long. Determine the lengths of the edges of the quad. • 15. The lengths of the sides of the base of a triangular pyramid are 38 '8^3 in. , 30*92 in. and 26*95 in. respectively. The lateral edges meet at right angles at the vertex. Find the volume of the pyramid. 16. The lateral edges of a pyramid are all equal to one another. The base is a rectivngle 4 ft. 8 in. long by 4 ft. wide. The height of the pyramid is 3 ft. 9 in. Find the area of the surface. 17. The hypothenuse of a right-angled triangle is 16*13 in. in length and one of the other sides is 12*76 in. long. Determine the length of the third side. 1§. Two sides of a triangle are 218 ft. and 241 ft. in length and the perpendicular from the included angle on the third side ii 120 ft. long. Find the length of the third side. 19. A ladder 25 ft. long stinds vertically against a wall. How far nuut the foot of the ladder be drawn out horizontally from the wall that the top of the ladder may be tlrawxi down one foot 'i '^■r,-.^.-aj;:<^-^-^^if^;^^: ■jrr^i.-^^^eif^-^ j^/fiK ■J MENSUUATION. 207 90. A rope hanging loose from a hook 20 ft. above level ground, just reaches the ground. How high above the ground will the lower end of the rope be when it is drawn 10 ft. aside from the vertical ? 31. Iwo cf the sides of a triangle are 1450 ft. and 1021ft. long respectively. From the contained angle a perjjendicular is let fall on the third side, and the segment of that third side between the foot of the perpendicular and the shorter of the first mentioned two sides is 779 ft. in length. Find the area of the triangle. 33. The base of a pyramid in a rectangle 12 in. long by 10 in. wide. The lateral edges are each 31 in. long. Find the volume of the pyramid and the area of its surface. 33. A flagpole 53 ft. 4 in. in height is broken by the wind and the top falling over strikes the ground 14 ft. 8 in. from the foot of the pole before the pieces part at the place of breaking. Find the length of the piece broken off, the ground being level. 34. A B and C are three houses standing at the angles of a right-angled triangle. A is 80 ch. east of C, and B is north of C and 51'20ch. nearer to it than to A. Find the distance from A to B. 35. The lengths of the four sides of a trapezoid taken in order are 608 ft. , 554 ft. , 250 ft. and 520 ft. Find its area and the lengths of its diagonals. 173. In any obtuse-angled triangle, the squares on the sides containing the obtuse angle are together let>s than the square on the third side or side opposite the obtuse angle by twice the rectangle under either of the two sides containing the obtuse angle and the projection on it of the other of these two sides. (Euclid II, 12. ) Let ABV be a triangle obtuse- B angled at C, and let a, b and e be | the MEASURES of the lengths of • the sides opposite the angles A, B | and C respectively, and let <^» be ' the meositre of the length of CD, 35 ~""^^" C b the projection of CB on AC produced, tnen will «2, b^ and c^ be the measures of the areas of the squtlres on the sides and 6/_» will ; ^:i, it «t:;r 208 ARITHMETIC. i ! bo the meoinire of the area of the rectangle contained by CA and CD. It follows from this and the preceding proposition that a2+6'-i-+26/_.=c3 (3) and .•. (aa+62+2?><(..)-=(;, (C) that is ; — In, any obtuse-angled triangle, if to the sum of th^ squares of the measures of the lengths of the sides coitainiag the obtme angle there he added twice the prodm't of the measures of the lengths of either of these sides awl the projection on it of the other of these sides, the squ<=c2 (4) and . •. (a- + 6» - 26/j^ = c, (D) that is ; — In an,\i triangle, if from the sum of the squares of the measures of the lengths of the sides containing an acute angle there he subtracted tivice the prodxcct of the m^ea^SMres of the lengths of either of these sides and the projection on it of the other of these sides, the square root of the remainder ivill be the measure of the length of tlie side opposite the acute angle. 175. If the angle BCD he^)ie-ihird of two right angles, i.e., if it be equal, to the angle of an equilateral triangle, the line CD will i ! ~*ii «^^:^-':J^^i;1v^f MENSURATION. 209 CA and lat « (3) (C) 'use aiujlc s 0/ either sides, the f the side ainiiig an liird side, der either jtion on it the sides 1 contained [reposition (4) (D) Ires of the \ile there he 'either of \the sqiiare hf tJie side les, i.e., if eCDvill be equal to half of the side CB ; therefore twice the rectangle under CD and CA will be equal to the rectangle under CB and CA tuid consequently (C) of § 173 will become c = (a2 + 69+a6) h (Cc) and (D) of § 174 will become c=(a2 + 62-a6)^ (Dd) It should be noticed that in the casa of (Cc) the angle BCD is an external angle of the triangle ABC and the internal obtuse angle is two-thirds of two right angles. EXERCISE XXV. 1. The lengths of the sides of a triangle are 125 ft., 244 ft. and 267 ft. respectively. Find the area of the rectangle under each side and the projection on it of either of the other sides. Find also the lengths of the sides of three squares equal in area to the three rectangles thus obtained. 3. The lengths of the sides of a triangle are 84 ft. 1 in., 158 ft. 2 in. and 188 ft. 3 in. respectively. Find the length of the projection of the shortest side on the longest. 3. The lengths of the sides of a triangle are 595 mm. , 769 mm. and 965 mm. respectively. Find the length of the projection of the shortest side on each of the others. 4. The lengths of the sides of a triangle are 26 in., 39 in., and 40 in. respectively. Find the lengths of the projection of the shortest side on each of the others and the lengths of the perpen- diculars on these sides from the opposite angles. 5. In a right-angled triangle, the lengths of the sides containing the right angle are 30 ft. 4 in. and 52 ft. 3 in. Find the lengths of the segments into which the hypothenuse is divided by the perpendicular on it from the right-angle, and also the length of that perpendicular, and prove that the product of the measures of the lengths of the segments of the hypothenuse is equal to the square of the measure of the length of the perpendicular. 6. The lengths of the sides of a triangle are 13 yd., 14 yd. and 15 yd. respectively. Find the lengths of the perpendiculars on the sides from the opposite angles. N .^■v/'^ ;; ■..»J~■'2•'^>'ii'iiCJiiv^!.te^>l s>iS^ 4,' V'^K , ; •Ai ■• ^\: 210 ARITHMETIC. 7. Show that tho triangle whose sides are respectively 25ft.t 39 ft. and 66 ft. long, is obtuse angled and find the lengths of the projections of each side on the other two. 8. From a point 0, three lines OA, OB and 00 whose lengths are respectively 195 ft., 264 ft. and 325 ft. are drawn making equal angles with one another in the same plane. Find the lengths of the linos AB, BO and CA. 9. From a point O, three lines OD, OB and 00 whose lengths are respectively 440 ft., 264 ft. and 325 ft. are drawn making equal angles with one another in the same plane. Find the lengths of the sides of the triangle BCD. 10. The lengths of the sides of a triangle are 21 ft. 2 in., 21 ft. 10 in. and 26 ft. 4 in. respectively. Find the lengths of the medians of th3 triangle. (See Mackay's Euclid, Ap. II, Prop. 1.) 176. If the lengths of the sides of a triangle are known, the propositions of §§ 173 and 174 will enable us to determine the length of the perpendicular on any side from the opposite angle and consequently to find the area of the triangle. It is not however necessaiy to compute the length of the perj^endicular on a side in oi-der to find the area of the triangle, this may be determined directly from the lengths of the sides as follows : — iii, a. Frora the measure of the leiigtJh of the semiperimeter of the trimigle subtract the measfiire of the le^igth of each side separately, multiply together tJie three remahiders and the common^ mivAieud, the sqxiare root of the product will he the measure of the area of the tria^igle ; ' S,= '{s(s-a)(3-h){s-c)y^ in which 8^ is the measure of the area of the triangle, a, h and c are the measures of the lengths of the sides and s is the measure of the semiperimeter; i.e., 2s = a + 6 + c. Let X, y and h denote the measures of the lengths of ADy CD and BD respectively, see Figs, of §§ 173 and 174, then will Fig. of § 173. Fig. of § 174. a — i/ = b x + y=^b =c^+h^ = ^2+7^8 .ji^ '^'^-n: "^■^'*,r v'**i ily 26ft.» bha of the ,e lengths king equal lengths of iBB lengths king equal lengths of 2 in., 21 ft. ihe mediaxis .) known, the le the length ) angle and lot however on a side in determined imeter of the e separately, niwn^i(d, the the triatigle ; h and c arc basure of the of AB, CD will and MENSURATION. X* — y*=c' — rt* a;*'»-i/a=c2-a» {X- -y)(x + y) = c^-a^ (x + y)(x-y)=c^-a^ b{x + y) = c^-a^ h(x-y)=c^ -a^ ^x-y)=b^ b(x + y)=b^ 211 2hx = 62 +c3-a2. 26a; ^h^+c^-a^. h^=zc^-x^=(c+x)(c-x) 4.b^h^=(2bc + 2bx){2hc-2bx) =26c + 62 + c2-a2)(26c-62-c2+a2) = <|(6 + c)2-a» y ^a2-(6-c)2 )- = (b + c + a)(b + c-a)(a + b-c)(a-b + c) = 2s(2s - 2a) (2s - 2c) (2s - 26) = 16s(s - a) (s — 6) (s — c). .N-. ib^h^=s(s-a)(s-b)(s-c)', . - ^bh — •{ s(s - a) (s - a)(s - c) y^. But 8^=\hh S,= is(s-a)(s-b)(s-c)y'^. An important advantage of this method of computing the area of a triangle is that it can be adapted to calculation by logarithms for it yields at once log>Sf, = ^-{ logs + log(s-a) + log(s-6) + log(s-c) y. Example. Find the area of a triangle the lengths of whose sides are 13*14 m., 14*15 m. and 16*13 m. respectively. 2s = 42 -42 ^ s=21*21 .-. log s =1*326541 s-a= 8*07 log(s-a)= •906874 s- 6= 7-06 log(s-6)= *848805 s-c= 6*08 log (s-c)= -783904 2)3*866121 . '. log S = 1-933062 = log 85 '716 .". the area of the triangle is 85*716 square metres. 177. If the measures of the lengths of the sides of a triangle bo kl (m2 4- w2), m)i (fc2 + ^2)^ (fc,^ + ^^) (fc^ _ ^rt), the measure of the area of the triangle will be khnn (kn + Im) (km - In) k,l,m and n denoting any numbers whatsoever. T '.■ ■ - jt, ■ ^= i m ^i^ifi^^ii :'^"i?;'^!^'->f, :^*V;:-^;/r ■''' v;-.-;^;^i" n.-y/if ■ '^a'' x::^ ''"? ' ' • ' "'j:f 212 ARITHMETIC. Tho tiiiiugle can bo roaolvod into two right-nnglcd trinngles, fclio intMVBuros of tho lengths of tlio sides of tho tirst buing and the measures of the lengths of the sides of the second being and U (m'' ~n^) + mn (fc» - 1^)^ (kn + Im) (km - la). u BXBRCISB XXVI. Find the area.) of the triangles the lengths of whose sides are respectively 1. 13 yd., 10 yd. and 13 yd. a. 13 yd., 24 yd. and 13 yd. 3. 13 ft., 4 ft. and 15 ft. 4. 13 ft., 14 ft. and 15 ft. 6. 13 in., 11 in. and 20 in. lili 6. 13 in., 21 in. and 20 in. 7. 13 m. , 37 m. and 40 m. 8. 13 m. , 45 m. and 40 m. 9. 1.23 ch., 5-95 ch. and 6 76 ch. 10. 73-2 ch., 45-5 ch. and 87 -6 ch. 11. What will be the value at $73 per acre of a triangular piece of land the lengths of whose sides are 478*5 chains, 329*6 chains and 237 *4 chains respectively ? 13. A triangular piece of land the lengths of whose sides were •1234 miles, '2345 miles and *2086 miles respectively was sold for $975. What was the price per acre ? 1 3. The lengths of the sides of a triangle are respectively 212 ft. . 225 ft. and 247 ft. A straight line is drawn across the triangle joining the mid-points of two of the sides. Find the area of the trapezoid thus formed. 94. The lengths of the sides of a triangle are 126 m., 269 m. and 325 m. respectively. Straight line^ are drawn across the triangle parallel to one of the sides and joining points of trisection of the other two sides. Find the areas of the parts into which the triangle is thus divided. 15. The length of the side of a square is 44 ft. A point is taken within the square distant 12 '9 ft. and 37*7 ft. respectively from the ends of one side. Find the areas of the triangles formed by joining iihe point to the four corners of the square. \>. ',„«V !fvl.=^^L, >. Jl 'S^*-V ^V*/ * Ai/^.^ '"'"i,"Svj«;t '^- "V i^i,. •.■vn* •P' •>"T'.-''V:''.* ♦■; MENSURATION. 213 glo8, the md being e Bides are 20 in. 40 m. 40 m. and 6-76 ch. and 87 -6 ch. mgular piece 6 chains and ie sides were was sold for tively212ft.. the triangle le area of the _., 269 m. and la the triangle iBection of the Ich the triangle point is taken lively from the led by joining 19. The lengths uf two adjacent sides of a rectangle are 349 ft. and 2'A7 ft. A point is taken within the rectangle distant 225 ft. tod 164 ft. respectively from the ends of the longer side of the rectangle. Find the areas of the triangles into which the rectangle is divided by lines joining its angular points to the given point. 17. The lengths of two oi the sides of a triangle are 55 ch. and 39 ch. respectively and the angle contained between these sides is two-thirds of a right-angle. Find the area of the triangle. 18. Find the area of the gable end of a bam 66 '2 ft. wide, the height of the eaves being 19 ft. at the front of the bam and 8 ft. at the back, and the lengths of the rafters being 29 ft. on the front and 56 '2 on the back, the bam standing on level ground. 19. The lengths of the sides of the triangle ABC are 6983 mm., 17079 mm. and 18574 mm. Find the area of a triangle whose sides are equal to the medians of the triangle ABC. 90. The lengths of the medians of a triangle are' 16*45 ch., 47 '77 ch. and 60*52 chains respectively; find the area of the triangle. 178. Given tlie leiufth of the radius of a circle and the length of the chord of any arc of the circle, to find the le^tgih of the chord of half the arc. Let r, k and ^2 denote the measures of the lengths of the radius, the chord of the arc and the chord of half the arc respectively, then ^^^^ fc3 = ^2ra-»<4r2-fc?)H*. Let ABK be the circle, C its centre, ADB the arc and D the mid-point of the arc. Join AD, AB, AC and CD ; the radius CD will bisect the chord AB at right angles, say in E. The measures of the lengths of AC, AB and AD are respectively r, ki and k^. Let q denote the measure of the length of CE. CE2=:CA2-AE*- 'If a :fii\( •■■ 'Jf '-_'^;' C(£^ Vh V >.■. tv" . .■-_)t*ftsv^V.- t'~^':C-''t^'T'^if':'] '■' ^' .'■!;■■■ (1) 214 ARITHMETIC. AD» = AC* + CD* - 2 CD CE -2CD2-2CDCE fca=2r2-2rg (2) by(l) =2r2-r(4r2-jt;)i ifc2 = -{2r»-r(4r«-fci)H^. (3) Example 1. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle, find the length of a side of the inscribed regular convex dodecagon. In this case we are given %i =r, fca = ^ 2ra-r(4r2-r2)i J-i --=(2r«-3^r3)* = (2-l-73205081)^r = •26794919^r = -siressoor. Therefore the length of a side of a regtdar convex dodecagon inscribed in a circle is '51763809 of the length of the radiiis of the circle. The length of the semiperimeter of the dodecagon is six times the length of a side and '51763809 r x 6 = 3-1058285 r, therefore The length of the semiperimeter of a regidar convex dodecagmt inscribed in a circle is 3*1058285 times the length of the radius of the circle. Example 2. Find the length of a side of a regular convex polygon of 24 sides, inscribed in a circle. In this case h^ is the measure of the length of a side of regular convex polygon of twelve sides, inscribed in the circle, .". by Example J, fci = '51763809 r, h^=-{ 2 r2-r(4r2- -51763809^2)^ y^ = (2 r2 -3-7320508^2)^ = -26105238 r. LilM:'.MlSil i aia iB ' t aftMii M 6i«i'Lrffa>P niU i !**■*■ MENSURATION. 215 Therefore the length of a aide of a reg^dnr convex ^SJ^-gon inscribed in a circle is '26105238 of the length of the radius of the circle. The length of the semiperimeter of the 24-gon is 12 times the length of a side and -26106238 r x 12 = 3 1326286 r, therefore Ttie length of the semiperimeter of a regular convex S/^-gon inscribed in a circle is 3*1326286 times the length of the radius of the circle, 179. Given the lengths of tlie radius of a circle, of the chord of any arc of the circle and of the chord of h(df the arc, to find the sxim of the lengths of the tangents from the ends of the half-a/rc to their point of intersection. Let r,ki and fcj denote the measures of the lengths of the radius, the chord of the arc and the chord of half the arc respectively, and let t denote the sum of the measures of the lengths of the tangents from the ends of the half-arc to their point of intersection, then will Let ABK be the circle, ADB the arc and D the mid-point of this arc. Join AD and AB and draw tangents to the circle at A and D and let them meet in G. Draw the diameter D K bisecting the chord AB in E. Join KA and produce KA and DG to meet in F. Then because GA and, GD are equal, being tangents from the point G, and DAF is a right angle, therefore the angle GAF is equal to the angle GFA, therefore GA is equal to GF and consequently the tangents AG and DG are together equal to DF. The measures of the lengths of AB and AD are ki and fcj* and the sum of the measures of the lengths of AG and DG which is equal to the measure of the length of DF, is t. EA is parallel to DF, both being at right angles to DK, therefore the angle EAD is equal to the angle ADF, also the angle AED is T) 'i' i IM ■- '4' .'■ii .1 ^ \ I 'if I 1,8 I iliO ! i ■; li ■f'T.:p^. ,*?» ^^"■^ "' ' ' '■/■"".-J ' 216 ARITHMETIC. equal to the angle DAF, both being right angles, therefore the triangle AED is similar to the triangle DAF FD : DA : : DA : AE • • V 9 K^ • • /Cq • 2 1 t 2fea 2k: *= Example. Find the length of a side of a regular convex dodecagon circumscribed about a circle. Let r be the measure of the length of the radius of the circle, then will A.^ = r and k^ ^-. -61763809 r. (See Example 1, p. 214.) * = 2( •51763309 r)a-rr = -63589838 r. Therefore the length of^ side of a regtdar convex dodeca{ion, circumscribed ahmit a circle is -63689838 of the length of the radius o/ the circle. The length of the semiperimeter of the dodecagon is six times the length of a side, and -53589838 r x 6=3-2153903 r ; therefore !Z7i6 length of tJie semiperimeter of a regidar co^ivex dodecagon, circumscribed about a circle is 3-2163903 tim^es Die length of the radius of the circle. EXERCISE XXVII. Find the length of a side and also the length of the semiperimeter of a regular convex polygon inscribed in a circle, the unit of measurement being the radius of the circle and the number of the sides of the polygon being I. 48. 3. 96. Find the length of a side and also the length of the semiperimeter of a regulfl convex })olygon circumscribed about a circle, the unit of measurement being the radius of the circle and the number oi the sides of the polygon being 3. 24. 4. 48. $. 96. '^irTSU' IMJ'irfS&!;*A:i^fc^*'i^v' i,'-:S:i.- ■■'< ,;"Vi. \x '^»:;:i,;ii ,vi-gon will be n (HM) which is equal to 2n (HD). The perimeter of the circumscribed 2H-gon will be 2n (FD). Now AED being a right angle, AD is greater than AE, 2n(AD)>2n(A]p), i.e., the perimeter of the inscribed regular 2n-gon is greater than the perimeter of the inscribed regular ?t-gon. Because FD is less than HD 2n(FD)<27i(HD), (.t! , the perimeter of the circumscribed regular 2/i-gon is less than the perimeter of the circumscribed regular n-gon. 1 "iiAii ■V 4. . i .*sl ili ... 1 . 218 ARITHMETIC. The angle DAF being a right angle, FD is greater than AD 2n(FT))>2n(AD) i.e., the perimeter of the circumscribed regular 2/i-gon is greater than the perimeter of the inscribed regular 2»i--gon. If then a regular hexagon be inscribed in a circle, and a similar hexagon be circumscribed about the circle, the perimeter of the circumscribed hexagon will be greater than the perimeter of the inscribed hexagon. If next a regular convex dodecagon be inscribed in the circle in which the hexagon was inscribed and a similar dodecagon be circumscribed about the same circle, the perimeter of the inscribed dodecagon will be greater than the perimeter of the inscribed hexagon, and the perimeter of the circumscribed dodecagon will be less than the perimeter of the circumscribed hexagon but will be greater than the perimeter of the inscribed dodecagon. Hence the difference in length between the circumscribed and inscribed dodecagons is less than the difference in length between the circumscribed and inscribed hexagons. If next a regular 24-gon he inscribed in the circle and a similar 24-gon be circumscribed about the circle, the perimeter of the inscribed 24-gon will be greater than the perimeter of the inscribed 12-gon, and the perimeter of the circumscribed 24-gon will be less than the perimeter of the circumscribed 12-gon but greater than the perimeter of the inscribed 24-gon. Hence the difference in length between the perimeters of the circumscribed and inscribed 24-gons is less than the difference in length between the perimeters of the circumscribed and inscribed 12-gons. If next a regular 48-gon be inscribed in the circle and a similar 48-gon be circumscribed about the circle, the difference in length between their perimeters will be less than the difference in length between the perimeters of the circumscribed and inscribed 24-gons. By continuing this process we shall obtain a series of pairs of polygons whose perimeters become more and more nearly equal at each doubling of the number of their sides. Now as the circumference of a circle is greater than the perimeter of any regular convex polygon inscribed in the circle but is less than the perimeter of any similar polygon circumscribed about the circle, the lengths of the perimeters of these polygons may be taken •Miiiii rtiiiiiiLkiaiiiiMMi'**'* similar similar length length 4-gons. jairs of qual at rimeter iss than ut the taken MENSURATION. ,:-x¥f«:i^TP':i^r!^f^'m>^-^:'^ 219 as limits between which the length of the circumference must lie. But it has been shown that, beginning with a regular hexagon, as the number of sides of the inscribed and circumscribed polygons is successively doubled the diflference between the lengths of their perimeters becomes less and less. In other words, by repeatedly doubling the number of the sides of similar inscribed and circumscribed polygons, the limits between which the length of the circumference lies, are made continually to approach each other * and therefore a nearer and nearer approach may be made to the exact length of the circumference. As an example let us take the measures of the lengths of the semiperimeters of the inscribed and circumscribed regular convex polygons of 12, 24, 48, and 96 sides respectively, which are given in the Exampl,es of §§ 178 and 179 and in the answers to the problems in Exercise xxvii, and, tc denoting the measure of the length of the semicircumference when the radius is the unit of measurement, we shall obtain from ^he 12-gons 3*10 < from the 24-gon8 3 13 < from the 48-gons 3 139 < from the 96-gons 3*141 < Since 3^ < 3 141 and 3 1428 < 3^^ 3f?<^<3}8. These are known as Archimedes' Limits of the ratio of the semicircumference of a circle to its radius, or of the circumference to its diameter. [181. Had we carried our calculations beyond the 96-gons to the 12,288-gons we should have obtained 3 1415926 <7t< 3-1415927. Vieta, doubling the number of sides 16 times successively computed to 10 places of decimals the lengths of the perimeters of the inscribed and circumscribed regular 393,216-gon8 and found that 31415926535 <7C< 31415926537. * We do not here enquire whether the limits thus found may be made approach each other indefinitely, nor is it necessary to ascertain whether they do so, for we seek not an exact but only an approximate rectification of the circle. It < 3-22 It < 3 16 It < 3 1461 Tt < 3 1428. 7\ r'l iiii ■ m ii I . t > .1^ I •riiiV^'";,-.' tfc'.. -Sri.a' 'W"**«S ■'T.?* '*'. "•i"'^' 220 ARITHMETIC. Ludolph van Ceulen starting from squares and successively doubling the number of sides 60 times, determined 7t to 35 decimal places. 182. The method which has been described of approximating to the value of re depends on the proposition that the arc AD (see Fig. on p. 215) is greater than the chord AD but is less than the sum of the tangents AG and GD, i. e. , less than FD. This method is extremely tedious if Jt is to be computed to more than three or four decimal places, but the following theorems aflford a means of greatly reducing the labor of calculation. r. The arc AD > chord AD + J (AD - AE) ; 2°. The arc AD < chord AD + J (DF - AD). If for these magnitudes we substitute the measures of their lengths in terms of the radius aS unit of measurement, and multiply throughout by 2 n, we shall obtain 2 nr > 2 »<^2 4- |(2nfc£ — nfcj), 2 TT < 2 wfca + J {2nt - 2 nk^). If Pn and P^n denote the measures of the lengths of the perimeters of the inscribed regular n-gon and 2 n-gon respectively and Q^a denote the measure of the length of the perimeter of the circumscribed regular 2ri-gorf, in terms of the radius as unit of measurement, then will ^ Pu = nk,^, P2n = 2nfc2 and Q2n=2nt and the preceding limits may be written 2;r>P2„+x(P2„-P.), As an example of the closeness of these limits let us take the case in which Pgn is the measure of the length of the perimeter of the inscribed regular 96-gon, then will iP.=3'1393502, iP2„= 3 -1410319, and ^(22„ = 3 1427146 and .-. ;t>3-1410319 + ^(31410319-31393502) but It < 3 -1410319 + J (3 1427146- 3 1410319) i.e., *> 3-1415925 but ?r< 3-1415928. K'iS.':-^ :'^t ''v^>■^^^J-S imSiM 'mm !(ii4,*4(^-^.V^^i'*^fh^W(*' ^ 'm^?,— r •''' ^^ ■•'''?^"^'''*''^>s>>'-'^^'' ' j^s ■ ■■■?;"5?'^->*^5*fiW'i-".":;p*'<^T,y^;.;:v^^ UTl^^'i^'^^fl 'W MENSURATION. 221 1 83. Numerous other geometrical constructions of the approximate length of an arc have been proposed for the evaluation of tt, the best being one which yields 30nr<8g4n+8P2n-P« This was published in 1670 by James Gregory who at the same time laid the foundation of the modem methods of computing it by proving that the series to be continued endlessly. Twenty -nine years later, Machin announced that ' ' ' •••) i«=4(l- 5 3x6» 6x55 1 + 7x67 1 ) • • • t ■ ' 239 3 X ?39a 5 x 239^ 7 x 239^ and computed re thereby to 100 places of decimals. Recently W. Shanks, employing this series, has calculated it to 707 places of decimals . The rapidity of convergence of Machin's series gives it a great advantage over Gregory's for purposes of calculation, but it and the many other series which have been proposed and used for the evoluation of it, can be er^sily deduced from Gregory's series. It may here be mentioned that it has been pro.ed that » is a transcendental number, i.e., it camwt be exactly expressed by a definite number of integers combined by the operations of addition, subtraction, midtipUcation, division, involution and evolution.] 184. Let Pe be the length of the circumference of a circle and R be the length of the radius, therefore, since 2it r.^ the measure of Po in terms of R as unit of measurement, Pj will be equal to 27r radii, which we shall denote by Po-27r(R). If we now adopt any other unit than R, say U, and if p, be the measure of Pe and r be the measure of R both iii terms of U, then will P.=P=(U)andR=r(U), and.-. p,(U) = 2it{r(U)]- =(2;rr)(U), jp,=2jrr. ■* !:, ■^-f ^ ->i;'!l m K**'^Nw. -rstflSii. rt,'j. ■' V .• • 222 ARITHMETIC. that is, the measure of the length of the eireumfereivce of a circle is the proihtct of It and txoice the measure of the length of the radius, / ^ correct to 3 significant figures, TC being < 3 '1416 n n 6 r n M II 7 n II m 185. If 2 rt. 2 6 and p, denote the measures of the lengths of the major and minor axes and of the perimeter of an ellipse and if a* -6* be small compared to a^, then will p,>ita\l + a2+3 63 3aa+6a J but j9,<7r^ 2(a2 4.6a) [5 [186. If a conical spiral beginning with a radius of r, units, advance in n revolutions through a distance of h units measured on the axis of the cone, and have then a radius of r^ units, the measure of the length of the spiral will be roughly approximate to •{n^nHr^ + r^Y^h'^).^. If ro==rn, the curv^e is a cylindric spiral or helix, (the edge of the thread of a screw) and the rectification is exact. Jih—Qy the curve is the common spiral or spiral of Archimedes.] EXERCISE XXVIII. 1. The inner diameter of a circular drive is 210 ft. in length and the width of the drive is 28 ft. Find the length of the inner and of the outer odge of ^he drive. 9. "VN'hat will be the cost of the wire at $1.25 per 100 yd. for a barbed- wire fence five wires high around a circular fish-pond 60 ft. in diameter ? 3. The minute hand of a clock measures 1ft. 3|in. from the centre of its arbor to the tip of the hand. Find the distance travelled by the tip of the hand during the course of 365 days. 4. Find the length of the radius- of a wheel which made 1600 revolutions in rolling 3 25 miles. 3. A circular path is 400 yd. in length on its inner edge. What will be its length 5 ft out from that edge all around ? ./'■ ^:.5»|*/^,. . V^^ •'■?j; JitllWilMW" ^'^'■:M''^"'W^>i^ "isH*' ^t :;'■•*%:' rde is the IS, bha of the se and if : r, units, measured units, the umate to ge of the imedes.] Ingth and Inner and |yd. for a id 60 ft. trom the travelled Ide 1600 What -"•V jv MENSURATION. 223 6. The length of the hypothonuse of a right-angled triangle is 2*9 in. and that of one of the other sides is 2'lin. Find the length of the radius of a cii'cle whose circumference is equal to the sum of the lengths of the circumferences of circlps described on the three sides of the triangle as diameter. 7. The diflforence in length between the diameter and the circumference of a circle is 2 ft. 6 in. ; find the length of the diameter. 8. If Mercury describe round the sun in 87 '97 days a circle whose radius is 35,700,000 miles in length and Saturn describe in 10759*22 days a circle whose radius is 882,000,000 miles long, what n'^U be the orbital speed in miles per minute of each of these planets i 9. Find the length of the arc which subtends an angle of 60° at the centre of a circle of 10 in. radius. 10. Find the length of the arc which subtends an angle of 36° at the centre of a circle of 25 in. radius. 11. Of how many degrees will the angle be which an arc whose length is 1ft., subtends at the centre of a circle of 2 ft. radius. 13. How many degrees will there be in the angle subtended at the centre of a circle of 1 ft. radius, by an arc whose length is 2 ft. ? 13. How many degrees will there be in the angle subtended at the centre of a circle by an arc whose length is equal to the length of the radius, if the length of the radius be (a) 1 ft., (6) 2 ft., (c) 3ft., (d) 7 ft., (e) 27-3 in. 14. The length of the vadius of a circle is 17 '5 in. ; find the length of the perimoter of a sector of which the angle is (a) 90% (b) 270°. 15. WhatVill be the length oi the perimeter of the segment of a circle of 18 in. radius, if the arc of the segment subtend an angle of 45° at the centre of the ' "; -^le ? 16. The length of the perimeter of a semicircle is 5 ft. ; find the length of the diameter. 17. The length ot the perimeter of a sector of a circle if 7 '2 ft. ; find the length of the radius the angle of t'.ie sector being 30°. 18. The length of the perimeter of the segment of a circle is 7 -2 ft. ; find the length of the radius if the arc of the segment subtend tn angle of 30° at the centre of the circle. I I II ;; «t' ■='"■■ I #1 kj^- |:i *'? •»"«>, , I; llli < ! pi ■! M ' li iiril n 224 ARITHMETIC. 19. Find the length of the perimeter of an ellipse the lengths of whose axes are 12 in. and 10 in. respectively. ilO. Find the length of the quadrantal arc of an ellipse whose semiaxes measure 11*9 in. and 7 "9 in. respectively. til. Find the length of the quadrantal arc of an ellipse whose semiaxes are 10*199 m. and 9'799 m. respectively in length. 33. Find the length of ^he radius of a circle whose circumference is of the same length as the perimeter of an ellipse whose semiaxes are 40*399 yd. and 39 599 yd. long respectively. 33. Find the length of the equator ; 1°, assuming it to be an ellipse tho lengths of whose semiaxes are 20,926,629 feet and 20,925,106 feet respectively ; 2°, assuming it to be a circle of 20,926, 202 feet radius. 94. The French metre was originally defined to be the 10,000,000th part of the length of a meridian quadrant taken from the equator to the pole. Had this definition been retained what would be the length of a metre in inches, if the length of the polar axis of the earth be 41,709,790 ft. and the length of the equatorial diameter be 41,852,404 ft. 35. Mars revolves around the sun in an ellipse, the centre of the suii being one of the foci of the ellipse. Find the lengths of the %emiaxes and of the perimeter of Mars' orbit if the greatest and the least distance of the planet from the sun be respectively 154,000,000 miles and 128,000,000 miles. 36. Find the average speed in miles per minute of Mars in his orbit, given the data in problem 25 and that his periodic time is 687 days. 3T. Assuming the earth to be a sphere of 7913 miles diameter, find the length of a degree of longitude in latitude 60°. 3§. Assuming the earth to be a sphere 7913 miles in diameter, find the length i. a degree of longitude in 45° north latitude. 39. The leigth of the perimeter of an ellipse is 383 in. and the length of tho axes are as 10 to 7 ; find the lengths of the axes. 30. The difference between the lengths of the radii o' a front and a hind wheel of a carriage is 7 in. What must be the lengths of thase radii if the front wheel make 70 revolutions more than tho hind wheel makes in rolling a mile. j3Ji.^j-f'ii4--V ,jgS'';?i?^v. g^'Vcw.'. ^»)>tfej ■g^jr''.Sfeliij^«^-«»'t*v\ MENSITUATION. 225 igths of ) whose e whose i. oference semiaxes to be an feet and circle of be the ken from ned what ' the polar equatorial itre of the bha of the Istandthe ^,000,000 tars in his lie time is 1 diameter, 1 diameter, ide. and the ixes. front e lengths le than the 10^ a 187. Let ABK be a circle ; C, its centre ; AF, half oi n side of a regular n-goii cir- cumscribed about the circle ; B F, half of an adjoining side oi' the n-gon. JohiABand CF. CF wiU bisect the chord AB at right angles, say at E, and will bisect the arc A B, say at D. Draw GDH, tangent to the circle at D and meeting AF in G and BF in H. Then GH is equal tc side of a regular 2i(.-gon circiunscribed about the circle ABK. Also AG =GD-DH = HB. Bisect AF in M and draw GL and MN parallel to FC jmd njeeting AE in L and N respectively. Because the angle GDF is a right angle GD LN 4AG in the Hgnrein the preceding section. AG GF, . •. the triiingle A(!D ; the triangle (JFD, . •. double the triangie A(iD < the triangle AFD, . •. 2(triangle AGD) - 2(segnient AD) < triangle AFD - segment AD ; . •. triangle AGD -segment AD ' i (triangle AFD - segment AD); . •. Hgure A(jiDC - sector ADC - A (triangle AFC - sector ADC ;) . •. figure AGHBC - sector ADBC <- i (figure AFBC - sector ADBC) : i.e., the excess of a sector of the circumscribed regular 2/(--gon (»\t'V the corresponding sector of the circle is less than half of the excess of the corresponding sector of tin circumscribed reguliir u-gon ovit the sector of the circle. Applying this theorem to all the sectors of the circtimscribed polygons and taking the aggregates of the excesses, we tind that TJie excess of the area of a rciiidur .^n-ijon riroitnscriheil ahoul , and 'Jafa = ^ rq, . '. Sa—i^rpt can, by sufficiently increasing u, ho made less than any proposed numher however small. 101. By a train of reasoning similar to that in the preceding section, but applied to areas instead of to lengths of perimeters, it may be proved that by doubling the number of the sides of a regular n-gon circumscribed about a circle the excess of the area of the n-gon over the area of the circle is reduced to less than half of what it was before the doubling took place, and that by repeating the doubling a sufficient nnmber of times, the excess of the area of the circumscribed n-gon over the area of the circle can be mtlde less than any explicitly assigned area however small. This result expressed in terms of the measures of the areas instead of in terms of the areas themselves is, — /Sb - Sg can, by sufficiently increasing n, be made less than any proposed number however small. But it was sht wn in the preceding section that 8a -^ rpt can, by sufficiently increasing n, be made less than any proposed number however small, .•. {Sa-\rp^-{S„ — S^ can, by sufficiently increasing u, be made less than any proposed number however small. But(^„-^rp,)-(^„-;8,)=^.-^r?>. Now 8„ r and p^ are constants, and increasing it, can have no effect on them, .'. /S, --^77?o must be less than any proposed number howevtr jmall ; and it cannot be variable, .-. S,-^rp, = 0, .'. Se = ^rp„ that is ; — The measure of the area of a circle is one-half of fJic wodtict of the measiiresofthe leiujthtt of the raJivs and the circumferenre f the circle. 192. Hence, by Euclid, VI, 33, . iii, h. The measure of the area of a sector of a circle is o'i,e-hiilf of th^e prodvct of the meuMires of the length s of the radius mtd the arc jf the sect&r. MENSIIIIATIOW. 229 s than anj preceding imeters, it sides of ii ihe area of lan half of repeating ;he area of 1 be nijlde rhis result i)f in terms 3 than any )s than any »(, be made ,ve no effect r howevor ■half of ih' rriinifetrm'i' 5 in oie-hii[j and the arc 193. Substitute Jrcr for p, in the eiiuation and it becomes V.' Th^e meAUiurr of the. (ireu of a OIRC'LK w the proiluct of tc ami the mpmrc of tke measure of tlie leiujth of the radhis of the eireie. 194. Let 2a. denote the measure of the length of the major axis and 2/- denote the measure of the length of the minor axis of an ellipse and let tai, denote the area of the ellipse. The ratio of the area of an ellipse to the area of the circle described on the major axis as diameter is the same as the ratio of the length of the minor axis to the length of the major axis. But the length of the minor axis is hia of the length of the major axis, tlierefore the area of the elli])8e is hja of the area of the circle described on the major axis as diameter. (S*e= of Tta'-, a H, = itab. vi. The meantire of the area of an ellipse is the rontinned product if It atul the. measures of tike leuijths of the semiaxes of the ellipse. EXERCISE XXIX. [In the following problems it may be taken equal to 3 •1416 and log flr= -497150.] 1, Find the area of a circle the length of whose radius is 3*75 in. 3. Find the area of a circle (^f 7 ft. diameter. 3. Find the area of a circle whose circumference is 13 '09 cm. in length. 4. Find the length of the radius of a circle whose area is an acre. 5. Find the length of the diameter of a circle whose area is a square mile. 6. Find the length of the circumference of a circle whose area is 18 "7 acres. 7. How much will it cost to gravel a circular piece of ground 51 ft. in diameter, at 7 cents per square yard 1 ■< >.'rf"1 : vr M 230 AIJITIIMr.TIC. S. Find tin* IriiL,; It •>! iiu< rinliuM "f t. cii'i ItM\ Ims*- ini a in •.•i|nnl to Mio Hiiiii of tlu> mviis (if fout'cii'cluH of 10 in., loin., JHin. iindlit in. rmliuH ivH|u'i'tivi'ly. O. Kind Mut totnl pn^Hnnro on n \Aniv )itt inclum in diuiuntur, tlio ])n>HHiM'o |)or H(|Maro inch lioin^ (Sn lit. 10. Tim I'ircnmfcn'ncd of Iho fifcnliir hiiHin of h fountuin nu>)iHUi'i that tlio ^V(M^dlt «,»f the platt) \h roducod hy 40 \}v.r cont. Find Mio length of tho nuliiiH of tho hole. lil. .V roctnnmdin" room, 27' 0" hy I.M'd", has a Hinnicironlar bow-window H' 1" in «liainf\n', thrown out, at tho Hide. Fitul the area of the tloor of the whole room. l«I. The area of a Hoiuicirclo i.s l.'M nq. in. Find the length of itH periuu^ter. 14. The lengths of the sides of a triangle are IJJft. 14 ft. and J 5 ft. res )ectivoly. Find the ditleronce hetwoun the area of the triangle and that of a circle of equal perimeter. 15. Tho periujetera «>f a circle, a Htpiare ami an ecpiilateral triangle are each () ft. in length. Find hy how nuich the area of the circle exceeds the area of each of the other tigures. 10. Find the ditVerenco l)etween the area of a circle of 5m. radius and that of a regular hexagon of etpial perimeter. 17. Find the length of the diam«>^er of a circle whoso area i: equal to that of a square whose sides are each lli ft. long. IN. The length of the difuueter <»f a circle is lH7yd. Find the length of the side of a stpiare whose area is e. A circle is inscribed in a sipiare whose sides are each 17 in. long. Find tlu! area between the sides of the scpiare and tho circumference of the circle. ilO. A s(piare is inscribed in a ciicle of 11 ft. radius. Find the area between the circumference of the circle and the sides of the square. '■Jit. Find the dilleren*^^ between the area of a circle of 7'Vni- radius and that of a regular inscribed hexagon. of a ri of tlui Hcmici Sinn o as dial ill. ii:j-4ci Kind ilO. is 2f» ft of the '17. ;iiilillll MKNSritATloN. 2:)1 T)!)). •V 111. t|^. Kiiitl t |i<< .'lit'.. Mriiiit it( li< «l«<| mi I he liy|M)HuuiiiHi> of 11 ri^lil uii^IimI Iriuiii^li) uh i', tlio loii^thii (if (ho otluirHidoH uf tlio triim^hi Imiiij^ 7 ft. i»m(1 17 fl. r»!H|MMJivt»ly. Sl!l. Show lliut. ill Hiiy ri^lil iin^lcil Iriuii^lr, llit^ ai'cii of tlid sctiiii'ii'clo (IcNcrilx'it nii tlio liypnt Ikmiiiho iih (liHiimttM' Ih ( radii of mi hiiiiiiIiih or |)|tiiMi riii|^' ufo 'J.'l'4ciii. Htid .'l(i'<»('iii i'( within ono foot of Mil! odj^o of tho pond. tH>. Around a oironlar lawn containing' U'.'M) acroH nuiH a walk of iiiiiforni width containing a (piartcr of an aoru. Find tho width of lliu walk. !I0. A circular lawn OH yardH in dianiotor Iihh a drivo of Miiif«»r!ii width around it. Kind that width, if tho aroa of tho drivo iH jiwt. Ii.ilf that of tho lawn. ;tl. What will it cost to iiavo a circular courtyard of) ft. in diaiiiotor, at (>()(!. jior Hcpiaro foot, loavinj^ in tho contro unpavod a licxa iiiiii. ;W. Tho area of a sector is 1 1 ''Hi\. ft. and the angle of tho sector 1.^ IS 30 . Find tho length of tho radius. Ilii, 232 ARITHMETIC. I' 111 36. The ureiv of a sector is eipiHl to tlie area of the square on the radius of the sector. Find the number of degrees in the angle of the sector. S7. A sector of an annuhis is 12 inches broad and the lotigths of its bounding arcs are .% in. and 28 in. resi»ectively. Find the area of the sector, its angle and the lengths of its radii. The length of the radius of a circle being one foot find the area of a segment which subtends at the centre of its circle an angle of 3§. iiO\ 39. 120° 40. 90°. 41. The length of the radius of a circle is 24 in. Two parallel chords are drawn both on the same side of the centre, one subtending an angle of GO" at the centre, the other subtending there an angle of }K)°. Find the area of the zone between the chords. 42. Show tliat the ch>. eeniiolliptic arch is 72 ft. and the clear height is 24 ft. Tae thickness «)f the arch at the crown is H ft. and at the sprintjing it is 7 ft. (5 in, Find the area of the face. 195. The mantel of a cylinder or of a cone is the lateral or curved surface of the cylinder or the cone. (iif denoting the measure of the area (jf a surface, <•(/, meij, rk, )n,ik\ » and ;:; subscribed to ^', are to bo read of a cylinder, of the mantel of a cylinder, of a right circular cone, of the mantel of a right circular cone, of a sphere and of a /.ouo of a 8j)here, respectively. 106. vii. The mvaHura of the avca of the mantd of n (;vlini>er i» the 'pi'o(bu:t of the inciDiiire of the leaijth of the mnntd oud the measure of the length of the perimeter of a r'ujht cross-seetion of the cijlinder, or The truth of this theorem will a])pear at once on developing or unwrapping the mantel by rolling the cylinder on a i)lane surface. The develoi)ed mantel can by a single transposition of parts ])e transformed into a i)arallelograni v/hose base is a generating line of the mantel and whose width at right angles to the base is the length of the perimeter of a right cross-section of the cylinder. In the case of the right cylinder the mantel develops into a rectangle. vii, a. In the case of the right qircular cylinder 2> = 27tr, (Sn„.y = 27r/vr in which r is the measure of the length of the radius of the base and a is the measure of the altitude of the cylinder. vii, h. Adding the sreas of the ends to the area of the mantel, gives for the area of the whole surface of a right circular cylinder S,.y-=^2irr{a + r). 197. viii. TJie WAiasnre of the area of the tn^tdel of a right CIRCULAR CONE Ls the votd'inned prodvct of it the measure of the dmd height of the cone and the measure of the length of the radius of the hase, or S„,^=^7Crl, If the cone be rolled on a plane surface, the mantel will develop into the sector of a circle whose radius is the slant height of the I , ■ is- 1? Vm ' '..f.i-'r...'; , ',K.:' 234 AIUTHMETIC. cono nnd wlioao arc is nqwil in lenytli to Mio circinnforenco of the base of the cone ; and the measure of the length of thii t circumference is 2 7Cr. viii, a. In the case of a frustum of a right circular cone, the mantel d-^velopa into the sector of an annulus, therefore AS'„f=7r/(ri+r.j) in which Cj, r.^ and r„ !irc tlie measures of the lengths of tlie radii of the ends and of the midcross-section of the frustum. viii, b. Adding the area of the base to the area of the mantel 198. ix. The ineamm'. of the area of the surfa<.;k of a sphehk Isfourtimeti the proilnet of rr anl, the measure of the area of the surface will be ^ ,.. f k^+2l-^ "-^""'[ik^w] but <2nj/i (]Ji+fi). EXERCISE XXX. 1. Find the area of the mantel of a right cylinder of 3 ft. altitude and 15 in. perimeter of base. 2. The slant height of a cylinder is 39 in. and the length of the perimeter of aright cross-section is 40 in. Find the area of the mantel. 3. The length of a cylinder is 22 ft. and its least girth is 22 in. Find the area of the mantel. 4. Find the area of a right circular cylinder of 25 in. altitude and 12 in. radius (>f base. 2:ls AlllTHMETir', 5. 'Yhii ftxes cf tlio bnso of a r'ujht elliptic oylinTler nro 15 in. and 12 in. long rospoctively and tho lengtli of tho cylind-or is 7 ft. <)in. Find tlie area of the mantel. 6. Find the area r. o whole surface of a right circular cylinder of 15 in. radius and ^ altitude. 7. P^ind tho area nl uie irfmh' surface of a cylindric jnpo 8 ft. (> in. long and an inch and a (piarter thick, the length of tho internal diameter being lOi in. §. Find tho area of the whole surface of a right ellijjtic cylinder 6 ft. long, the lengths of the axes of the base being 12 in. and 10 in. respectively. 9. The area of the mantel of a cylinder is Ssq. ft. and the length of the ])e:'imetei v.f a right cross-section is .'Ht. Find the length of the cylinder. 10. Tho area of the mantel of a right circular cylinder is 2 sc|. ft. 117 sc}. in. and the length of the radius of Jie base is (>*75in. Find tho length of tho cylinder. 11. The area of the whole surface of a right circular cylinder is 21 stj. ft. and the height of the cylinder is e(|ual to the length of the diameter of the base. Find the length of the diameter. 1*2. Tlie area of tho whole surface of a right circidar cylinder is 27 sc]. ft. and tho length of the cylinder is thrice the lengtli of the radius. Find *^he lengtli of the radius. Find the area of tho mantel of a right circubir cone whose dimensif b".se 10 in. the er is the I'hose 4 ft. 11. MKNsrUATlON. 239 ''ft. 20. Altitude 5 ft., length of circuinferonet) of hsvHe Oft. 11 in. 21. Tiio firea <»f the nmntel of ii riglit. circular cone in 5 h(\. ft. and the leniith of the circuuifere'.ico of the haso is 46 in. Find the slant height of the conj. aft. The area of the niivntel of a right circular cone is 7.s(|. ft. 72s(i. in. and the length of the circumference of the base is 5 ft. Find the altitude of the cone. 211. Find the slant height of a right circular cone whose mantel has an area of 15 Hi\. in. and whose base-radius has a length of 1 •5in. 24. Find the altitude of a right circular cone, given that the area of its mantel is 5s((. ft. and the length of the radius of its base is ()in. 2«S. The area of the mantel of a right circular cone is 2*5 scj. ft. and its slant height is 25 in. Find the length of the circumfereiuje of the base. 2tt. The ai'ea of the mantel of a right circular cone is ]5a(j. ft. and the slant height is 2 ft. Find the length of the radius of the base. 27. The area of the wlude surface of a right circdar cone is 2 S(|. yd. and the slant height is twice the length of the diameter of the base. Find the length of the diameter of the base. 25. How many yards of car.vas 45 in. wide will be re((uired to make a conical tent 10 ft. wide and Oft. high ? 29. How many yards of canvas 32 in. wide will be re(|uired to make a conical tent 15 ft. wide and 10 ft. high, if 10 ;' of the canvass is cut away or turned in, in the making of the tent. 30. The area of the mantel of a I'ight circular cono is twice the area of the base. Find the vertical angle. 31. A right circidar cylinder and a right circular cone stand on ef 21 inches radius, the height of the segment being 10 inches, and :he distance of its base from the centre of the sphere, 11 inches. 46. Find the area of the whole surface of a zonal segment of a sphere of 12 in. radius, the distances from the centre of the sphere vt-.u\\ MENSURATION. 241 of the tenninal circles of the i,oue being 5 in. ^nd Oin. both on the sanje side of the centre. 47. The length of the diameter of a sphere is .'>0 in. and the length of the radius of the base of a cap-segment of the sphere is 5 in. Find the height of the cap at right angles to it& base. 4§. A 8])hero is 30 inches in diameter. What fraction of the whole surface will })e visible to an eye placed at a distance of 10 ft. from the centre of the sphere ? 49. At what distance from the centre of a sphere of 9 in. radius must a luminous point be placed to light up (me-third of the surface of the sphere ? 50. Find in 8({uare miles the area <»f the surface <»f the earth assuming it to be practically an oblate spheroid the lengths of whose semiaxes are 20,926,202 feet and 20,854,896 feet respectively. 202. If tit'o soliiiii oil equal liases and of equal altitiidt^ are sncli that all plane sections of the solids pandlel to and at eipial distances from their bases are equal to one another, the section of one solid at each and every distance from its base equal to the section of the other solid at the same distance from, its base, then will the solids be equal in volume. This proposition may be shown to follow from Theorem II p. 177, by applying a method of demonstration similar to that employed on pu. 180 and 181 to prove that tetrahedra on equal and similar bases and of the same altitude are of equai volumes, and on pp. 225 to 228 to obtain the quadrature of the circle. 203. II, a. The measure of the volume of a cijlinder is the product of the measures of the altitiide of the cylinder and the area of its base, or V,y = aB. This proposition follows iuunediately from Theorem II, p. 177, and § 202. In the case of the right circular cylinder, by 5^ 193 B=7rr'^ and .'. Frcy = ^a''". 204. Ill, a. The measure of the volmne of a cone is ONE-THIBD of the product of the ineasnres of the altitude of the cone and the area of its base, or V, = -^aB. T> ■'M II ;:liHl I tr' I' Si- f ■ j 242 AIUTHMETIC. t Tliis i»i(>p(j8ition follows iimauiliaceiy from Theorom III, p. 17", and ^ 202. In the CH80 t»f the right circulur c(»no, by i^ 193 B=irr-^ and .-. V;^ = )^7r(tr^. For tho measure of the volume of a frustum <>f a right circular cono, IV o, p. 183, gives Kkt= h^a (r J + rji'a + r'i). 206. V. The measure of the volimw of an dlipsoid is FOUE- THIRDS of the continvctl prod net of it and the measnren of the IciKjths of the ttemiaxes of the ellipsoid, 244 AHITHMKTir. i!r, Bui tli'.t iiiuHHiiri! of this voiiiUK! of tito .tplicru in e<|uul to thu 1UUH8U1-U of tliu vmIiiiiiu of thu hollowed cylinder, Foi- tho hoilowod cylitjdor in tho inticedinfj; pivtof, tlu'iu nuiy Ito Huhstituted a totnihodron whoHo nltitudo (distnnco l)etwt!uu ii puir of oppoaitu odgos) in o*[ni\.\ to a roceding proof of Y,»(, oni}>loying, liowover, an oUiitsoid instoad of a H[)horo, a right elliptic hollowed cylinder instead of a right circular hollowed cylinder and ellipses instead of circles. CA and CD <>f the tigure should bo seniiaxes of the ellii)Soid. 207. It should 1)0 noticed that if the sphere l)o inscribed in the hollowed cylinder and two planes parallel to the ends of the cylinder be drawn cutthig tho figures, nt)t only will the volumes of the sphere-segment and hollowed cylinder between the cutting planes be e(iual, the areas of the /one of the si)here and tho mantel of the cylind(^r between the cutting planes will also be eipial. 208. If a right circular cylinder, a hemisphere and a right circular cone be on e(jual bases and of tho same altitude, the volume of the cylinder will be thrice and the volume of the hemisphere will be twice the volume of the cone, or Vr,.y^7er\ r,.==|7rr', V,,^l7trK Compare these relations in volume with th(jse of the prism, the hemitetrahedron and the pyramid, given on page 187. 209. X. The measure of the area of a tore ar iiiNd /« the pnxlncl of the measures of the length of the jxirim^ter of a right crofts-sectlon and the length of the axis of the tore. VI. The measure of the volume of a tore is the product of thr measure of the area of a right cross-sectimi and the measure of the length of the axis of the tore. 210. T//e areas o/ SIMILAR plane figures or of similar surfaces are to one another as the squares of the measures of the lengths of their corresponding linear dimensions. The volumes of similar solids are to one another as the cuhes cf the measures of the lengths of their corresponding linear dimensions. \ MENSURATION. 245 EXERCISE XXXI. I. Thu lungth of tliu nuliiia of tliu hntm of a right circulur cyliiulur in 5 in. and thu altitude of thu cylineen estimated to be only the millicmth of an inch in diameter. How 246 ARITHMETIC. many miles of such a fibre would a grain of sand make, the grain being a right circular cylinder one-hundredth of an inch long by one-hundredth of an inch in diameter ? 13. Find the volume of a hollow right circular cylinder, the length of the radius of the inner surface being 3o in. ; of the radius of the outer surface, 4*125 in. ; and of the cylinder, 7 ft. 6 in. 14. Find the thickness of the lead in a pipe of three-quarter inch bore, if 10 ft. of the pipe weigh 21 lb. and a cubic foot of lead weigh 712 lb. 15. A hollow right circular cylinder of rast iron 15 feet in length and 4 feet in diameter of outer surface, is set u]/iight and bears on the top a weight of 250 tons. Determine the thickness of the metal so that the };ressnre on the biise may l)e 15(X)lb. per square inch, the weight of a cubic foot of cast iron being 444 lb. 16. Find the volume of a hollow-elliptic cylinder 75 ft. in length, the lengths of the axes of the inner surface l)eing 5 ft. and 3 ft. respectively and the thickness of the walls being 8 in. 17. Find the volume of a cone whose altitude is 15 in. and whose base is a circle 10 in. in diameter. 1§. The volume of a cime is 3 "5 cubit feet and its altitude is 5 feet. Find the length of the radius of the base which is a circle. 19. Find the volume of a cone whose slant height is 05 in. and whose base is a circle 32 in. in diameter. HO. Find the volume of a cone whose altitude is 35 in. and whose slant height all round is 37 in. 21. Find the volume of a cone on a circular base of 5 in. radius, the area of the mantel of the cone being a square foot. 32. Find the volume of a cone on a circular base, the altitude of the cone being 10 in. and the area of the mantel being a square foot. 33. Find the volume of the frustum of a cone on a circular base, the height of the frustum being 10 "6 in. and the lengths of the radii of the ends being 5 in. and 2 in. 34. The slant height of a frustum of a right circular cone is 10 in. and the lengths of the radii of the ends are IHin. and 10 in. respectively. Find the volume of the frustum. 25. Find the volume of the cone from which the frustum iu problem 24 was cut. MENSURATICK 247 36. The lengths of the radii of the ends of a frustum of a right circular cone are 6 ft. and 9 ft. res[)ectively and the altitude of the frustum is 4 ft. Find the volumes of the two frusta formed by cutting the frustum by a plane parallel to the ends and midway between them. ft7. The lengths of the radii of the ends of a frustum of a right circular cone are 4 ft and 6 ft. respectively and the altitude of the frustum is 3 ft. Find the volumes of the three pieces produced by cutting the frustum by two planes parallel to the ends and trisecting the height of the frustum. 2§. A pyramid 15 inches in altitude is divided into three parts of e(|ual volumes by planes parallel to the base. Find the altitudes of the three parts. 29. The lower portion of a haystack is in the form of a frustum of a right circular cone with the end of shorter diameter below, the upper part of the stack is in the form ot a cone. The total height of the stack is 25 ft., the length of its greatest circumference is 54 ft., the height of the frustum is 15 ft. and the length of the diameter of the base is 15 ft. How many cu])ic yaj-ds are there in the stack ? 30. The area of the whole surface of a right circular cone is 25 sq. ft. Find the volume of the cone, the slant height being live times the length of the radius of the base. 31. The volume of a right circular cone is 7854 cubic inches. Find the area of the whole surface of the cone, the altitude being thrice the length of the radius of the base. 99. A vessel in the form of a right circular cone whose slant depth is equal to the length of the diameter of its mouth, just holds a gallon. Find the slant depth. 3JI. Find the volume of a sphere 12 inches in diametei . 34. Find the volume of a sphere a great circle of which is 33 in. in circumference. 35. The area of the surface of a sphere is a square yard. Find the volume of the sphere. 36. How many gallons will a hemispherical bowl 18 Inches in diameter hold? if ' H \w 248 ARITHMETIC. :l ii 97. What will be the weight cf n sphoricnl shot of cast iron 5'5 inches in diameter if a cubic foot of iron weigh 4441b. 'i 3§. Find the weight of a sj)here of lead 3"75 inches in diameter, tlio lead weighing 7121b. per cubic foot. 39. What weight of gunpowder will till a spherical shell of 7 in. internal diameter, if 30 cubic inches of the gunpowder weigh a pound ? 40. Find the volume 1° of the greatest sphere, 2° of the greatest hemisphere, that am be cut out of a cube of wood measuring 7*5 inchus on the edge. 41. The largest possible cube is cut out of a sphere one foot in diameter. Find the length of an edge of the cube and the volume of material cut away in making the cube. 4^. Find the weight of a s])herical shell 1*75 in. thick and of 8 inches external radius, the material composing the shell weighing 4901b. per cubic foot. 43. The length of the greatest circumference of a spherical shell is 25 in. and the length of the internal diameter is 575 in. Find the weight of the shell, the substance of which it is composed weighing 5001b. per cubic foot. 44. A sj)horical shell weighs 131b. and the lengths of the external and internal diameters are ()in. and 4 in. respectively. Find the weight of a shell of the same substance but of 8 in. external and 5 in. internal diameter. 45. Find the volume of a solid in the form of a right circular cylinder with hentisj)herical ends, the length of the diameter of the cylinder being 3 ft. ii in. and the extreme length of the solid being 25 feet. 46. A cylindrical pontoon with hemispherical ends is constructed of sheet-iron "125 in, thick, the extreme length of the pontoon is 22ft. and the length of its outside diameter is 2ft. Gin. Find the weight which the pontoon will support when half innnersed and also the greatest load it will bear assuming the sjiecific gravity of sheet-iron to be 7 '75 and taking the weight of water at 62 '5 lb. per cubic foot. 47. Find the thickness of an 8-inch shell if it weigh half af- much as a solid ball of the same diameter and of like material. HBBP KENSURATION. 249 4§. A spherictil alioll 10 in, in dijiineter weighs '9 as much as a solid ball t)f tho sanio diameter and substiUice. Find the length of the internal diameter. 49. A cast iron shell Sin. in diameter, is tilled with gunpowder and plugge«l with iron; the whole then weighs To'oll). Find the thickness of the shell, supi)osing the iron to weigh 4441b. per cubic foot and tlu! guni)owder to weigh r)7*<) per cubic foot. 50. If the nature of the eartli's crust be known to an average dej)th of 5 miles, what proportion of the whole volume of the earth is known, assuming the earth to be a sj'here 7^12 miles in diameter? 51. If the ocean coVer 7''i"t> per cent, of the earth's sin-face and its average depth be 2 miles, what projMirtion will its volume bear to the volume of the whole earth considered as a sj)here 75*12 miles in diameter 1 52. If tho atmosphere extend to a height of 45 miles above the earth's surface what proportion will its volume bear to that of the earth assuu»ed to be a sjjhere <»f 7912 miles diameter '( 551. The radius of the base of a right circular cone is 2 inches and the volume of the cone is ecpial to that of a s])herical shell of 4 in. external and 2 in. internal diameter. Find the altitude of the cone. 54. A Stilton cheese is in the forni of a cylinder, a Dutch cheese is in the form of a sphere. Find the length of the diameter of a Dutch cheese weighhig 91b., a Stilttm cheese 8 inches in diameter and 7 inches high weighing (5 lb. 55. The length of the radius of the base of a segment of a si)here is 2 in. and the length of the radius of the sphere is (>in. Find the volume of the segment. 56. The height of a segment of a sphere is (Jin. and the length of the radius of the base is 8 in. Find the volume of the segment. 57. The lengths of the radii of the ends of a zonal segment of a sjjhere are Sin. and Sin. respectively, and the height of the segment is 3 in. Fhid the volume oi the segment. 5§. Find the volume of a zonal segment of a sphere, the ends of the segment being on opposite sides of the centre of the sphere and distant from it 10 in. and 15 in. respectively, tlie length of the radius of the sphere behig 20 inches. 260 ARITHMETIC. ft 9. A section pamllei to the liase of a homisphere bisects its altitude. Find tlie ratio of the volumes of the segiuents. 60. A sphere whose volume m a cuhic yard is divliled l)y a plane into segments whose altitudes are as 2 to 3. Find the volumes of the segments. 61. How much water will run over if a heavy globe of 2 in. diameter be dropped into a conical glass full of water, the diameter of the mouth of the glass being 2"6 in. and its depth Ji in. ? 6'2. Find the volume of the i)rolate spheroid genei'ated by an ellipse of 12 in. major and 10 in. minor axis. 63. Find tlie volume <»f the earth assuming it to be an oblate splieroid of 41,709,790 ft. polar axis and 41,852,404ft. e(|uatoriaI diameter. 61. Find the volume of the earth assuming it to ])e an ellipsoid the lengths of ./hose semiaxes are 20,5)2<),()29ft., 20,925, 105 ft. and 20,854,477 ft. res[)ectively. Find also the length of the mean-radius or radius of a sphere of the same volume as the earth. 65. Find the length of 100 complete coils of a wire (me-tenth of an inch in diameter coiled closely iipon a cylinder of 5 in. radius. 66. On examining and taking the dimensions of a steej) cistern, which was supposed to be ])erfectly cylindrical, I found the bottom cr«tss diameters to be 70 inches each, but the toj) diameters were 08 and 72 inches respectively. The depth of tlie vessel was 05 inches. What is the diflerence in the capacities of the true cylinder at 70 inches diameter and the one exandned ? 67. A steep cistern in the form of a frustum of an elliptic cone, the cross diameters at the bottom being 84 and 04 inches, and the diameters at the top 72 and 57 inches, is 50 inches deep, and is filled to the depth of 25 inches with dry barley. How many cubic inches does it contain ? 6§. A cylindrical iron tank, 20 feet kmg and 4 feet H inches in diameter, was placed horizontally on a Hat car and tilled with oil at Petrolia. When it arrived at Tonmto, it was found upon l)eini; dij)ped frcmi the top, to be 10 inches to the surface of the oil. What was the wantage in gallons 1 CHAPTER VI. PROPORTIONAL AND IRREGULAR DISTRIBUTION AND PARTNERSHIP. 211. If four magnitudes be in proportion and if the first magnitude be a multiple of the second, the third magnitude will be the same nmlti])le <»f the fourth ; if the first magnitude be a part of Lhe second, the third magnitude will be the same part of the fourth ; if the first magnitude be a multiple of a part of the second, the third magnitude will be the same uiulti[)le of the same i)art of the fourth ; and, generally, according as the first magnitude is greater than, ecpial to <»r less than auy umltiple or i)art or uudtiple of a part of the secoml, the third magnitude is also greater th.an, e(pial to or less than the sauie luultiple or the sauie part or the same uiultiple of the sauie part of the fourth ; and, conversely ; (/ these roitditliytis are Hati{fied the four mmiititiules are In propoiiioti,. (See .S§ 148 to 162, p'^ 159 and KJO.) 212. Hence if four quantities be in proportion the first and second (quantities will also be proportionjd to any equiuiultiples of the third and fourth cpiantities or to any ecpiifractional parts of these (quantities, I.e. the third and jourth (piantities un^y both be multiplied or both divided by the same nuui^cr without affecting the proportion. Example. $12 =^ | of $18 and 64 lb. = § ( )f 96 lb. , $12: $18: :641b :961b. Dividing both 641b. and 961b. by 7 will not aft'ect the fj in the statement 64 lb. == § of 96 lb., nor will nmltiplying the two (quotients l)y 4 affect the H, $12:$18: :911b. :13ilb. and $12 : $18 : : 364 lb. : 54 ii lb. So also if four (quantities be in ])r()[)ortion, the first and second (quantities may both be nudtiplied or both divided by the same nvnnber without affecting the proportion. Thus in the preceding example, multiplying both $12 and $18 251 ...i ^ ■]i '\ If' li '•^^ ^'^' 252 ARlT>rMETIC, !; by 2 and dividing the products l»y 5 will not allect tho !^ in the Htjitoinont $i2--i of .si 8, r of *12: ? of $18; :fi411). :m5ll)., i.e. *4-80:.'*7-20 • :()411). iJHilb., and . •. $4 -80 : ^7 20 : ; Ml lb : 541! \)>. Honco, generally, if four (luantitiea br in j^oportioi any eciuiniultijdes oreciuifractional parts wf the tirs* itud second ^juantities will also bo ])roi)ortional to any e<{i inuiltii»it.!H or eijuifractioual ])arts ot the third and fourth (juaiitities. 213 If four quantities be in proportion and if any e<]!iinniltiph!H or efpurractional pa- i« of tlu tirat jmkI third (luanhiies bo tal ^n and also any e(pjii ;Jti\>:(S or e in jvjoportion. Example. 15 vi'.~ v of 25 hi. and 57 gal. ~ ? of 95 gal. !(■> in, -. 25 in. : : 57 gal. : H5 gal. Multiplying l)oth 15 in. and 57 gal. ))y <> will niultij)ly t (io ;i by in hoth the statements, 15 ;'u. = 'i; of 25 in. and 57 gal =--= '} of 1)5 gal. vvliich thus becoiae 15 in. X <) = ? X () of 25 in. and 57 gal. x fi = ? x of 95 gal. Mf^iiplying both 25 in. and 95 gal. by 7 will divide the ;.J x S by 7 in botii these statements which thus r)ec(»me 15 in. X ()= ? X 5 of (25 in. x 7) and 57 gal. x 6 = -^ x of (95 gal. x 7) (15 in. X ()) : (25 in. x 7) : : (57 gal. x 0) : (95 gal. x 7), /. ( . 90 in. : 175 in. ': : 342 gal. : dm gal. 214. If four quantities be in proportion and if tho first and second quantities bo expressed in terms of one and the same unit and tho third and fourth cpiantities bo also expressed in terms of one and tho same unit, the unit of the first and second (quantities not l)eing necessarily tho same as the unit of the third and fourth quantities, it follows from the ju'eceding section that tho i)roduct of the mearures of the first and fourth (piantities is e(jual to the product of the measures of tho soc«)nd and third (piantities. For, if the first and third (piantities both bo multiplied by tho measure of the fourth (piantity, and the second and fourth (juantities both be multiplied by the measure of the third (piantity, in the proportion PROI'OUTIONAL AND IRUEGULAU DTSTIUllUTION. 253 the. ill. by 7 x7) fonntxl by these multiiilos tlie third uud fourth (HuuititieH will ])o e(]U!il to one another unci therefore the first and second roi)ortion formed by the multiples, is the product of tl:*.' measures of the first and fourth ([uantities of the original proportion, and the measure of the sec«)nd cpiantity in the new proportion is tho product of the measures of the second and third (juantities of tho original proportion. Hence the product of the measures of tht first and fourth (piantities of tho original proportion is e(iual to tho product of the measures of tho second and third (luaiititios of the original proportion. Example, .f .'35 = § of $6ii aiul 55 yd. = g « .f H8 yd. $35 :$b(i : :55 yd. :8«yd. Multiply $35 and 55 yd. both by 88; the measure of 88 yd., the fourth quantity or term of the proportion. Also nmltiply .f5f) and 88 yd. ])oth by 55, the measure ()f 55yd., the third (piantity or term of the proportion. Then by 5^ 2i;'» $.'35 X 88 : 156 x 55 : : 55 yd. x 88 : 88 yd. x 55 But 55 yd. x 88 =88 yd. x 55 $35x88 = $56x65. 215. If four (piantities form a proportion, the (juantities are called the terms of the proporticm ; the first and fourth (quantities jire called the extreme tenns or the extremes of the proportion and the second and third quantities are called the mean terms or the means of the proportion. Enq)loying this phraseology and with the inqjlication of the conditions regarding the units of the terms, the theorem of §214 may be briefly stated under the form T}ie prudnct of the mea.svres of the extremes of a proportUm, is equal to the product of the measures of the means of the pnqyartio'H. tho moan terms of a proportion be e"s as 3 to 4, and B's money to Cs as 4 to 5, How much money has A compared to G '/ 266 ARITHMETIC. ill. A grocer lias 8411). (if a iiiixturo of greoii find bluck teas, the weight of green tea in tlie mixture being to the weight of black tea in it as 5 to 1 ; how many pounds of y)lack tea nnist bo added to make the weight of green to that of black as 4 to 1 ^ 39. Milk is worth 20 cents a gallon, but by watering it the value is reduced to 15 cents a gallon. Find the jn-oporticni of water to milk in the mixture. 33. Divide $4500 between two persons in proportion to their ages which are 21 and 24 years. 34. Two men receive $15 for doing a certain jtiece of work. Now one n>an had worked but li days while the other had worked 5 days on the job. If the money is to be divided in proi)ortion to the lengths of time the men worked, how much should each receive ? 35. A farm is divided into two ])arts whose areas are as to 13, and the area of the larger i)art exceeds that of the smaller by 18 A. 880 sq. yd. Find the area of the farm. 218. Let there be r.ny number of (quantities, say A, B, C, D, , all of one kind and an eijual nuud)er of ([uantities, say a, b, C, d , also all of ; . •, if 1 bo divided into jMirtH proportional to 4. 5 and 1 ->» .'. if ^720 1)0 dividod into ])art,s proi)ortional to 4, 5 and (5, those partH will 1)0 ,K ,,f $720, /\ of $720, and /V (.f $720. /. of !8?7:iO = $lU2. ,^i, of . $720 = $240 ,", of *720 = Ji;288 Proof. $W2 + $240 + $288 = $720, Ala Also and $192-4 of $240 $240=;; of $288 !.>'., $192: $240: :4 :5 l.r., $240 :$288 : : 5 : 01b. : : .\ 1 . I 5 • S' •i ■ VI iisr, M •'-I,"'#''"^e-- ' ' 258 AmTMMKTK*. EXBRCIfcJl!] XXXIII. Divide — I. 1.'{.'U into parts proportional to 2, 4, 5. a. li) T. 112011). into partH jnoportional to },, }., ]. Jl. $57 into i)artH proportional to ], r, '. 4. $10J)'()r) into parts i)roportional to 1, 2, 3, 3, 4. .'5. $UMi4 into parts proportional t(» 2, 2], 2;^. . 6. $1720 into parts proportional to 10, 2\, 1, \, .'j. ■ 7. 1 HO 11). into parts proportional to .'{•.'{, "7, '^k 5. $253 in tlio jjroportion of it, 7, and 10. ». $(5:{:«) in tho proportion of i, ;.J, and -7. 10. 15223 in tho proportion of f, I, ^% -{\^, ,^,. 11. Hu|^ar is composed of 4{)*85() jiarts oxygen, 43 '2(15 carl )on, and ()*87t> hydrogen ; how many ])ound8 of each is there in 1300 11 >. of sngar ? 13. Gunpowder is composed of nitre, charcoal and sulphur in thc^ proportion of 33, 7 and 5. (1.) How many Ih. of sulphur are there in 180, lb. of jjowder? (2,) How many lb, of powder can bo made with 30 lb. of sulphur '. (3.) How much nitre and sidphur nnist bo mixed with 1121b. ot charcoal to form gunpowder 1 13, A man divides $3300 amongst his three sons, whose ages aro Ifi, 19, and 25 years, in sinus proportional to their ages : t\v(»yoHvs afterwards he similarly divides an ecpial svnn, and again after three years more ; how irmch does each receive in all ? 14, Two sums of money are to be divided among three jiersons, one sum ecjually and the other in the i)roportion of 3, 5, and 8, The shares of the first two amount to $r)4"o(> and $81'36 resi)ectively. Determine the sums, 15, I want an alloy consisting of 19 parts by weight of nickel, 17 of lead, and 41 of tin. The only nickel 1 can obtain is 101b. nf an alloy containing 11 j)arts of nickel to 7 parts of tin and 5 of lead. How nmch lead and tin nuist I add to make up the alloy I want? lor ? phuv '. 11). of nickel, H , , Oil), nf ■ 1(1 T) of ■ and iilloy I 1 and 1M{(UH)RTI<)NAI, AND IUUE(JTT|,AU DISTHIHrTION. 25}) Hi. Two ptTHonH trHvelliiiL5 toguthur j'-j/rvj t(> ;»iiy oxpensoH in tlio vatio 'if !^'t m !ii. A i)ays tho first day's bill which amounts to •iJWJ'lO; ]i the second, which amounts to ^H'dd ; and C tlio third, which amounts to $1)'24. How must they settle accounts '( IJ>, A founder is recpiirod to supply a ton (22401b.) of fusible metal consisting of 8 parts by weight of bismuth, 5 of lead, and .'{ parts bismuth, 4 lead and li tin. How much of tho alloy must ho take, and how much lead and tin must he add to make u]) tlio order? Example S. Divide 53 "5 A. among three men so that the first man may receive 7 A. as (jfton as the second receives 8A., and tho second may receive 5A. as often as the third receives 4A. Share of 1st : share of 2nd : : 7A. : 8A. share of lst= J of share of 2nd Share of 2nd : share of 3rd : : 5 A. • 4 A. share of 2nd = J of share of 3rd, share of lst = | of | of share of 3rd. share of 1st + share of 2nd + i)lns share of 3rd = {1; of 2 + 4 + 1) share of 3rd, = (ijO+^t + ii^) share of 3rd, = J.j^/ of share of 3rd. 53-5A = -»:[lf of share of 3rd of 53 '5 A = share of 3rd share of 3rd = IG A. share of 2nd = 2 <*f share of 3rd = 20 A. share of lst=§ of f of share of 3rd = 17'5A. and lor > II jf 260 ARTTHMKTTC. EXERCISE XXXZV. 1. Divide fl050 among A, B, C and D so that A's share may he to B's as 2 to 3, B's share to C's as 4 to 5, and C's to D's as to 7. 3. Divide £28. 13s. 8d. among A, B and C, so that for every shilling given to A, B gets 10s,, and C a half-guinea. (21s, = 1 guinea.) 3. Divide 32 gal. 3(|t. lipt. into four measures so that the first shall he to the second as 9 to 14, the second to the third as 21 to 25, the third to the fourth as 20 to 23. 4. An assemhlage of 700 persons consists of 5 men for every 2 children, and 3 children for every 7 women. How many of each ? 5. The j<»int capital of four partners, A, B, C, D, is $12()00 ; A's investment is $10 for every $17 of B's, C's is $34 for every $(55 of D's, and B's is half as much again as C's. Re(iuired the amount <»f the investment of each 6. Divide $3274*70 among A, B and C, giving A five per cent more than B, and six per cent, less than C, 7. A's rate of working is to B's as 7 to 5, B's to C's as 4 to 3, C's to D's as 5 to 6 ; time A works per day is to time B works per day as 1) to 10, time B works to that C works at 10 to' 11, that of C to that of D as 10 to 7 ; number of days A works to number B works as 15 to 7, number B works to number C works as 11 to 20, and number C works to immber D works as 7 to 5. How should $1220, the sum paid for the work, be divided among them ? Example. 4- Divide the number 429 into three parts such that five times the first part may be equal to seven times the second and t • nine times the third. First X 5 = second x 7 = third x 9 and first + sec< »nd + th ird 429 first X 7 X 9 + second x 7 x 9 + third x7x9=::429x7x9= 27027 first X 7 X 9 + first x 5 x 9 + first x 7 x 5 - 27027 first X 143 = 27027 fi.rst - 27027 -r 143 = 189, and second = first x 6 -=- 7 = 135, and third = first x 6 ^ 9 = 106. PROPORTIONAL AND IRREGULAR DISTRIBUTION. 261 EXERCISE XXXV. 1. Divide ^9*fi() between A and B so that 'A times A's share may be equal to 5 times. B's. a. A, B, and C have together $1740 ; if .%\7 of A\s-^»o of B's = ^j}(f of C's, find tlie share of each. 3. A pound of tea, a pound of coft'ee, and a pound of sugar together cost $1 "B? ; find the i>rice of each having given tliat 71b. of tea cost as much as 16 lb. of coflee, and 31b. of coffee as uiuch as 111b. of sugar. 4. Divide $1650 into two jjarts, such that the simple interest «ai (me t>f thenv at 4| % for 3 years would be ecjual to the simple inteiest on the other at 6 % for 2j years. 5. Divide $1560"50 into three such parts that the amount of the first for 2.' years at 5 % may be eipud to the amount of tlic second for 2h years at 3| % and also to the amount of the third for 4 years at 4 %, simple interest. 6. A father leaves $15000 to be d' vided among his three sons, aged respectively 16, 18, and 20 years so that if their res])ective shares be put to simple interest at 6 /', they may have equal shares on coming of age. How io tln> money to be divided ? 7. Divide 365 into thruc parts, such that twice the first, 5 times the second, and 24 % of the third, may oe equal to one antjther. §. Three coal M'^agtms contain 195 cwt. of coal in such ])roportions thfit 10 times the load in the first, 12 times that in the second, and 15 times that in the third, are e(][ual (quantities. What weight does each wagon carry ? 9. A man, a woman, and a boy finish in a day a piece of work for which $4*65 is paid. Find the share of each on the sui>position that 2 men do as nuich as 3 women or 5 boys, and that the pay is l)roportional to the v*rork dtme by each. 10. Divide the number 80 into four Huch j)arts that the first increased by 3 the second diuiinished by 3. the third multiplied by 3 and the fourth divided by 3, may give eipial results. 262 ARITHMETIC. Example 5. The daily wages of men, 11 women and 12 boys is $53 '40. Find the daily wages of each man, on the supposition that 3 men do as much work as 5 women, and 4 women as much as 5 boys. Assume the work done by one woman in one day as the unit of work. Then the 11 women do 11 units of work the 9 men do ^| of 5 = 15 units of work the 12 boys do ^?' of 4 = 9| units of work Hence the money must be divided in the proportion of 15, 11, and 9|, which is in the jn-oporticm of 75, 55 and 48. the 9 men's daily wages = j"/»j of .f53-4() each man's daily wages = ^ of /y^g of ii?53'40 = f2"5(). EXERCISE XXXVI. 1 . Divide $490 among 2 men, 8 women, and 10 children for work done, on the supposition that 1 man does as much as 3 women or 5 children. 2. A, B, C, rent a ])asture for $92 ; A ])uts in horses for 8 weeks, B, 12 oxen for 10 weeks, C, 50 cows for 12 weeks. If 5 cows are reckoned equivalent to 3 oxen, and 4 oxen to 3 horses, what shall each pay ? 3. Three workmen, A, B, C, did a certain piece of work and were paid daily wages according to their several degrees of skill. A's efficiency was to B's as 4 to 3^ and B's to C's as 6 to 5 ; A worked 5 days, B, 6 days, and C, 8 days. Tlie whole amount paid for the work was $36 '26. Find each man's daily wages. 4. Three men, working respectively 8, 9, 10 hours a day, receive the same daily wages. After working thus for 3 days, each works one hour a day longer, and the work is finished in 3 days more. Tl $114 '05 is i^aid for the work, how much should each man receive' 5. Three mechanics. A, B, C, are to divide among them the proceeds of a job valued at $125*50, and finished in 9 weeks, the share of each being proportional to the work dcme by him. B car do half as much again in the same time as C, an'^ \ twice as much. PROPORTIONAL AND IRREGULAR DISTRIBUTION. 263 C works steadily 8 iiours a day ; B works 7 hours a day for the first 2 weeks, 5 f(jr the next 2, 3 for the next 4, and 11 for the last. During the first 7 weeks, A works only 2 hours a day for 4 days of the week, and during the last 2 he works 14 hours a day, but finds that in the last 4 hours of each day he can get through no more work than C could. How much should each receive ? Example (I. A drover bought oxen at .^40, cows at $30, and sheep at $10 a head, paying for all $1440. There were 2h times as many cows as oxen, and 5 times as many sheep as cows, how many did he buy of each ? No. oxen : No. cows : : 1 : 2^ No. cows : No. sheep : : 1 : 5 No. oxen : No. cows : No. sheep .•. as (»f ten as he expends $80 in purchasing oxen he will expend $150 in i)iuchasing cows, and $250 in sheep ; Hence tlie money must be divided in the proportion of 80, 150, 250, vvhich is in the proportion of 8, 15, 25 ; cost of oxen = -^^ of $144<) = $240 No. oxen = $240 -f- $40 = 6. 2 5 25, 5 : 25 ; M II i m 1 EXERCISE XXXVII. 1. A person bought wheat at 80c, barley at 75c, and oats at 40c a bushel, exi)ending for barley half as much again as for wheat, and for oats twice as much ;is for wheat. He so-1 the wheat at a gain of 5 %, the barley at a gain of 8 %, ''nd the oats at a gain ()f 10 7» and received altogether $9740. How many bushels of each did lie )uy Si. Suppose that $95*10 is to be divided among a certain number of men, women and l)oys ; that there are 10 boys for every 3 men, and If) men f( >r every 39 women, that each boy receives 5 cents, each woman 10 cents, and each man 25 cents ; find the number of men, of women, and of boys. tj. A debt of $170 is paid in $5 })ills, $2 bills, and $1 bills, the mnnber of each denomination being i)roportional to 4, 7 and 10 ; how many were there of each '. 264 AUITHMETIC. 4. A debt of $350 is paid in $10 bills, ^ bills, and $2 bills, there are | as many ten's as five's and 2;^ times as many two's as five's. How many wei'e there of each denomination ? 5. A merchant paid $84 for 100 yd. of clo»^h of three different kinds. For every 4 yd. of the first kind he had Sh of the second and for every l.Jt'yd. of the second he had 1] yd. f>f the third ; if 2 yd. of the first cost as much as 3 yd. of tlie second, and oyd. of the second as much as 4 yd. of the third ; find the i)rice per yard of each kind of cloth . Example 7. Divide $784070 among A,|B, C and D, giving ^ $77 '74 more than 40 % of what B and D receive ; B $88 less than ? of what C and D receive ; and C $99 more than 33 J % of what D receives. Assume D's share as the "init, that is, exi)ress the shares of the others in terms of D's share and known (iiiantit'cs. Then, since D's share =; D's share, C's „ - 1 D's , +$'.)9, / B's share ^ ^ ( C's + D's) - $88 = f D's ,. - $28 "HO, A's „ -|(B's + D'«) + $77.74=.V^D"s . +$H(i-.30. sum of shares ^ -'^-^^-D's share 4-$l."'*>"70. 3A-t D's share + $13(>-70 = $7840-70 ; D's share - ($7840 70 - $136 70) x .,' , -^ -$2700. EXERCISE XXXVIII. I. Divide $3000 among A, B, C and D so that A may receive $40 more than 33;^ y of what B, C and D receive ; B $50 less than 60 7 of the united shares of C and D ; and C r of D's share and $3<- besides. 24. Two men A and B, make a bet on the result of a walking match, the total sum staked being $105. A's stake is to B's as B's original money is to A's. If A win he will have 2\ times as mucli money as B will liave left, ])ut if he lose he will have left \'i; of tlu' sum B will then have : how much had each at first? PROPORTIONAL AND IRREGULAR DISTRIBUTION. 265 3. Four men own a tiuil)cr limit, which they sell for $7200 ; the first receives $1)00 more than | of what the other thn.c ^et ; the second $<5(X) less than 70% of the joint shares of the third and fourth ; and the third $400 more than 'i of a sum which exceeds the share of the fourth by $2300. How mucli do eajh receive,' after paying their proportionate share of the expenses of the sale which amount to $360 ? 4. Divide $52" 50 among A, B and C so that IVs share may be half as much again as A's, and C's one-third as much again as A's and B's together. 5. Divide $252'50 among A, B, C and D so tliat the sum of the shares of A and B may l)e § of the sum of the sliares of C and D, and that B's share may be j'^ of A's, and C's j'jj of B's. 6. In a certain factory the nund)er of men is j"',, the number of l)oys, and the nuud)er of wouien 3H 7 of the whole number of |)eisons employed. If to g' ve each boy (5d., each woman Js., and each man 2s. (id. refpiires £47. lis., find the number of men, women, and boys. 7. A, B and C engage to hoe an acre of corn for $4'()8. A alone could hoe it iv 48 hours ; B, in 3() hours ; and C, in 24 hours. A begijis first and ■.»()rks alone 10 hours ; then B connnences and A fiiul B work together hours, when C begins and all work togetlar till the job is finished. H<)w much should each receive ? S. Two men, A and B, hired a si)an of horses and a carriage for $7 to go from M to 11, a distance of 42 miles. At N, 12 miles from M. they took in C, agreeing to carry him to R and back to N for nis |)ro])ortionate share of the expenses. At P, 24 miles from M, tliey took in D, agreeing to take him to Tl and ])ack to P for his proportionate share of the expenses. What should each person j)ay ? ((Jive briefly the arguments for and those against each of the two commcmly presented solutions of tliis problem.) 9. $1200 is to be distributed among A, B and C. From part of it tliey are to receive e(pial amounts, and of the rest B's shares is to l)c 10 : more than A's, and C's 10% mor> than P's. Altf)gether 1V8 share is S/j'ii ' i»«>i'o than A's and 7U % l^^^s thax> C's. Find the part of the $1200 that was e(|ually divided. r i '1 j i 1 \ 1 1 lit 266 AlllTHxMKTIC. PARTNERSHIP. 219. A Partnership is a voluntary association of two or more . persons wlio combine their money, goods or other property, tlieir hvhor or their skill, any or all of thijse, for the transaction (jf business or the jcnnt prosecution of any occtipation or calling, such as the carrying on of any manufacture or trade or the j)ractice of any nvofpssion, i;p;!U an agrctnnent that all gains and losses shall bo slini ad in certain specified proportions among the persoiisconstituting the partnership. .■:>uch an association is styled a Firm, a Compa)iii, or a Horse and (lie persons iniiting to constitute the association are caliud the Partners of the Firm. ? Investment of a partner in a lirm is the money or, property contributed by him to the tirm. The Capital of a firm is the total of the investments of the partneis. The Net Gain within a certain period is the excess of the total gains of a firm over its total losses within the period. The Net Loss within a certain period is the excess of the total losses of a firui over its total gains within the period. A Dividend is the share of the net gain or of any sum divided among the meml)ers of a firm or a company^ which belongs to any partner. The dividends to the several partners are gen'^rally in proportion to their investments. 220. In a partnership n whiuii the gains and losses are to be divided among the partners in proportion ' their investments, tn find each ])artnor's share of any net gain or net loss : — i. If the iiircsfments are contrihvted for equal times, divide the ml gain or the ntt loss in proportion to the investments. ii. If the investments are co)drihuted for nneqnal times, imdtiplu each inrestment hii the 7m', Th. Sinclair, C. Harvey and H. Stevens enter into partnership, Sinclair investing $37,500, Harvey $28,600, and Stevens $24,000, ;ind they agree to share all gains and all losses in proportion to their investments. At the end of the year the resources of the firm lire $124,368-50 and the liabilities are $37,429-50. Stevens now wishes to withdraw from the firm and sells to his partners his uiterest in the business in shares proportional to their interests in it. How much should he receive from each ? <». T. Allan and E. Jamieson engage in business with a joint capital of $19,200 and agree to share gains and losses in proportion f It-'. I I 268 ARITHMETIC. to tlieir investmouts. At the end of a yenr Allan receives n dividend of ^1100 and Jnuiieson a dividend of ^1300. What was the amount of the investment of each ? T. D. Rowan, F. Galbraith and J. Muiiro enter into partnershi]) and agree to share all gains and all losses in proportion to their several investments. They gain ^ToOO of which Rowan receives $2100, Galbraith $'M0O, and Munro the balance. How nmch did Rowan and Galbraith respectively invest if the amount of Munro's investment was $18,000? *. Three r^^rchants enter into partnership, the first iwvestH ^1855 for 7 months, the second invests .f 887 "50 for 10 months and the third invests ^770 for 11 months ; and they gain .$434. What should be each partner's share of the gain ? 9. L, M and N entered into partnership and invested respectively $19,200, $22,500 and $28,300. At the end of 5 months L invested $3800 fidditional ; M, $2500 ; and N, $3700. At the end of ;i year the net gain of the firm was found to l)o $7850. What was each partner's share of this, if all gains and all losses v/ere shared among the partners in proportion to their average investments ? 10. (J raves and Barr form a partnership. Graves investing $7000 and Barr $8000. At the end of 3 months Graves increases his investiuent to $9000 but at the end of T) months nun-e he withdraws $4000 from the business. Barr, 4 months after the formation of the partnershi[), withdraws $2000 of liis investment but 5 months later increases it by $4000. At the end of the year the resources of the firm are $27,850 and ikx liabilities are $8460. What is tli.' amount of each ])artner's interest iii the business Jiow, the net gain being div'ded between the imxtners in proportion to their average investment!.^ ? 11. Stuart and Moss enter into partnership, Stuart contributing $5000 more capital than Moss. At the end of 5 months Stu.ut withdraws $2500 of his capital and 2 months later Moss increases his investment bj"^ $2500. At the end of their first year of ])artnershi|>, their assets exceed their liabilities by $24, 800 and on ^iividing their net gain in the ratio of their average investments, Stuart's interest in the business is found to exceed that of Moss by $461 "54. Find the amount of the original investment of cn-Ji. 13 7 CHAPTER VII. I. PERCENTAGE. 221. The phrase per cent, which is a shortened form of the Latin p!/' centum, is equivalent to the English word hundredths. Hence a rate per cent, is a rate or ratio per hundred and a number expressing a rate per cent of any ([uantity expresses simply so many hundredths of the quantity. Thus 5 per cent, of any sum of money is 5 hundredths of tlie sum ; 7i percent, of a given length is 7i hundredths of the length ; and 225 per cent, is 225 hundredths. 222. The symbol % is frequently employed to denote the words per cent., and may therefore be read either percent, or hnmlredths. Thus 5% = -05, 25% =25, i%=005, 133,^ / - l-3.\;t, 7 J % of 84Q = 075 of 840 = (>3, 145 % of $(540 = 1 -45 of $(>40 - 1928. 11 J t J EXERCISE XL. 1. A lawyer collected $287 '50 and charged 5"/ for his services ; how much did he retain, and how much did he pay over? What per cent, is the amount paid over of the amount collected ? 3. On Jan. 10, a merchant buys goods, invoiced at $<870'40 on the following terms : 4 nios. , or less 6 % if paid in 10 days. What sum will pay the debt on Jan. 15 ? 3. A house is sold for $10,400, and 25% of the purchase money is paid down, the balance to remain on mortgage. How nuich remains t)n mortgage ? 4. A man invests 42 % of h'.s capital in real estate and has $53,070 left ; what is his capital ? 5. A horse was sold for $f)58 which was 10^ % more than its cost ; how much did it cost ? {}. A bankrupt's assets are $23,625, and he pays 40% of his liabilities ; what are his liabilities ? 269 270 AlUTHMETIC. r.r. 11 7. A ])ivyi>msti'r recoivus i85ir>0,(K)0 from Mio tro.iaury Imf f, ils to account for }ti5225() ; wlmt 18 thu porceiitngo of loss to t lu) gov ■ riiaioht i fti. ^(540 iiicrciist'd ])y n curtain jier cunt, of itsolf ociualti ^7iiO ; ro(iiiirecl tlio rato \wv cei i. 9. A tea merchant mixes 40 11). of tea at 45ct. ]»er 11>. witli 50 lb. at 27ct. j)er 11). and sells the mixture at 42ct. i)er Ih. What per cent, profit does ho make ? 10. A merchant buys a hill of dry goods, Aj)!. 10, am(^unting to $(>377'84, on the following terms : 4mos,, or less 5/ if paid within JJOdays. How nnich would settle the account on May 1(5? The amount paid May H't is what % of the full amount of the hill ? 11. On Aug. 1(), a merchant buys a hill of goods amounting to $2475 oil the following terms : 4mos., or less 5% if paid in 30 days. Sept. 15, he makes a ])aymentof $1000, with the iniderstanding tliat lie is to have the benefit of the discount of 5 %. With what amount should he be credited on the books of the seller? How much would be due at the expiriition of the 4mos. ? lij. Paid $()fi4"2o f..>v transportation on an invoice of goods amounting i.> .'?8S(M', V\ hat per cent, must be added to the invoice price to male a protit of 20% on the full cost ? 13. A business firms resources consist of notes, merchandise, j)ersonai accounts, Arc, to the amount of $5)117 "(Jl, and a ])alance, which is 44% of their entire capital, on deposit in bank. How much is on deposit? I'l. At a forced sale a bankrupt's house was sold for $8000, which was 20% less than its real value. If the house had been sold for $12,000 what per cent, of its real value would it have brought? -lo**. The jiopulation of a town of (>4,000 inhabitants increases at the rate of 2i % in each year, find its jjopulation (i) 1, (ii) 2, (iii) .'I years hence. 16. The population of a city increases at the rate <»f 2% yearly. Tt now has 182,051 inhabitants ; how many had it (i) 1, (ii) 2, and (iii) 3 years ago? 1 7. A ship depreciates its value at the beginnin in value each year at the rate of 10 % of g of the year, and its value at the end of 3 years is $14,580 ; v.hat was its original value ? l)W of of I'LK* KNTA^iK 271 U i I L V in tlie stuto of ico .15-9% of leud. t" lujul oro is not 1§. A man in business Iohu.s in his liist your 5 / of liis cii])itiil, l)ut ill liis second year lu' i^uins (i ', of what he liatl at the end of the liist year, and liis eai»ital is now )ii^'\/ . How nin< ' was removed from 504 gal. of the on i> 20. Tho stulf out of a lead mino con After washing, by which j rocess the a i5;', % greater than in 1885, while the amount deposited in 1887 ceeded the average of the three ])revious years by 20"/. The aggregate of the four years was $1 50,0,'{7 '50. Find the amount deposited in i ich year. 23. In 1871 the ixjjmlations of Toronto, Hamilton and St, Thomas were severally 5<)01)l, 2(5710 and 2107. In the next ten years tliey increased 54%, o4'0%, and 280 -8 / resjiectively. Determine the increase per cent, of their united population. 23. The cattle on a stock-fann increase at the rate of 18;J % per annum. In 1880 there wei o 0850 head of cattle on the farm ^how many were there in 188(t ? 2-1. In a certain election A polled 88% of the votes jn-omised him, and B polled 00 % of those promised him, and 15 was elected by a majority of 3 votes. Had each candidate received the full nundier of votes j)romised him, A wouhl have been elected by a majority of 25. How many votes did each candidate receive ? 25. The delivery of letters in a certain town is carried on by four postmen, two of whom deliver on 14 streets and twM) on 17 streets, l)ut the work of the latter two is 20 ^ less ])er street than that of the former tAVo. A fifth man is })ut on to help them. In what ratio should he help the two pairs of men so that all five shall have equal work ? '''""^h. ij li ..>s^'^. ^>. IMAGE EVALUATION TEST TARGET (MT-3) % id. 1.0 I.I 40 12.0 1.8 ^ ||M 1.6 .< 6" — ► <^ /^ Photographic Science^ Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 o t/j «^ -,' fV^'^^e*;'!;;?^ M'^?^;;^3iir'-^:;iiv(y l . »»v;-,jj^-: ^' -^r 1^ 272 ARITHMETIC. ,•«■■ 4:. OS II. PROFIT AND LOSS. a 223. The Prime Cost of merchandise or other property is the net sum paid by the purchaser thereof to the seller thereof. The Gross Cost of merchandise or other jjroperty is the sum of the prime cost, all charges for purchasing, and Jtll expenses for freight, storage, handling, and such like. 224. Profit is the amount by which tlie selling price exceeds the cost price. Net Profit or Gain, is the anK)unt by which the selling price excee-■• <'.«''*'^< S /ti^-'i^*^' i.r.'1fi^<^iri^^'^^^'r.>irf%y■■^^i^>^'''• ■ 'f .'-■ 274 AlUTHMETIC. ■% r -y life- 17. 4 liorses and 7 cows cost ^(M) ; l)ut, if the price t)f the horses were to rise 26 % and that of the cows 16 % they would cost $466 •50 ; find the cost of a horse and of a cow. 18. The cost of freight and insurance on a certain (juantity of goods was 15% and that of duty 10% on the original outlay. The goods were sold at a loss of 5%, but had they brought $3 more there would have been a gain of 1 %. How much did they cost ? 19. A bookseller sold a book at *7% below cost, but had he charged 50 cents more for it, he would have gained V %. Find tlie cost of the book to the bookseller, and the price at which he sold it. 30. A man buys pears at 35ct. a score, and after selling 7 dozen at 46ct. a dozen (giving 13 to the dozen) he finds he has cleared his original outlay. If he then sell the i-emainder at the rate of 2 for a cent, what will he gain % on the whole transaction ? 31. I buy two cows for $66 ; if I sell the first at a loss of 6% and the second at a gain of 6%, I should gain y^ % ; what was the price of eUch cow ? 33. I bought a lot of coffee at 12ct. per lb. Alhjwing that tho coffee will fall short about 5 % in roasting and weighing it out, and that 10% of the sales will be bad debts, for how much per pound must I sell it so as to gain 14 % on the cost ? 33. A grocer mixed together two kinds of t rid sold tlio mixture, 144 lb., at an advance of 20% on cost, .living for it $62*10. Had he sold each kind of tea at the same price per pound as he sold the mixture he would have gaiiJ^d 15% on the one and 25% on the other. How many pounds of e.'ich were there in the mixture, and what was the coat of eac\. per pound? 34. The manufacturer of an article charged 20% profit, the wholesale dealer charged 25 % of an advance on the manufacturers ])rice and the retail dealer charged 30% of an advance on tlio wholesale price. Find the cost to the manufacturer of an article for which the retail dealer charged $23 "40. 35. I sold for $296, two horses which had cost me $280. The gain per $100 on one of them was e(jual to the loss per $100 on the other and also equal to the difference in cost of the two horses. Find the cost of each. .'S3'r?i.*ii.,. \,i1K-','iV •ij'*"?-:'''!^^';,''",'^*?;-:,'- ;iS^~'';;''--'';-t~->''i/iY''>-v*'"iij'''''^''"*^^^ f^^'-^: INSURANCE. 275 III. INSURANCE. 226. Insurance is a contract by which one party, the insurer, in consideration of a sum of money received from another party, the insured, engages to pay a stipulated sum on the happening of a particular event or undertakes to indemnify the insured or his representatives for loss or damage arising from certain specified causes, if sustained within a stated time. The instrument or document setting forth the contract is termed an insurance Policy. The sum paid by the insured to the insurer is styled the Premium. It is generally a tixed percentage of the amount insured. The Term of an insurance is the period for which the contract is made and the risk assumed. 227. The ordinary kinds of insurance are Fire Insurance, Marine Insurance and Life Insurance. 228. In Fire Insurance, the insurer undertakes to indemnify the insured up to a specified sum, for loss or damage that may occur to certain property described in the policy, if caused by fire, within a stated time, generally one, two or three years. 220. In Marine Insurance, the insurers contract to indemnify the insured up to a stipulated sum for any loss or damage that may occur to a certain ship, cargo or freight, any or all of them, by storms or other perils of navigation during a particular voyage or within a specified period not usually exceeding twelve months. 230. In Life Insurance, the insurer engages to pay on the death of the insured, i sum specified in the policy. In an Endotmnent Policy, the stipulated sum is payable to the insured if he should survive a specified number of years, but should he die before the expiration of the period named, the sum assured is to be paid to the representatives of the insured or to a pei^^on named in the policy. 231. Fire and life insurances are usually undertaken by companies or corporations organized to carry on such business ; ,^l ''.■^Hfrw/v*';':'^,'^ ^.'iS' •,>^'.\^:.-*p:>^.^'^'X\i/' %^^\-,<\^^-j ^■'A:iji-r . w^y*. ■>■•■ -r-T 276 ARITHMETIC. '%^ '% marine insurance is undertaken both by companies and by private persons. A marine insurance by private individuals is generally undertaken by several parties and each of them writes his name under or at the foot of the policy, and engages on his own account to indemnify the insured to the amount set opposite his name : on this account inuividual marine insurers are cnlled iimlenoriters. 232. In an ordinary fire i)olicy, if the loss is only partial, the insurer undertakes to pay the full value of the property destroyed or the full amount of the depreci^ion of the property damaged, provided it does not exceed the sum covered by the insurance. In marine policies there is commonly an average clause which declares that the indemnity for a partial loss of property not insured to its full value will be the same part of the loss as the sum covered by the insurance is of the full value of the property. 233. If a property is insured in two or more companies or by two or more underwriters, the insurers are liable for the indemnity for a partial loss, in sums proportionate to the amounts of the risks severally assumed by them. EXERCISE XLII. 'Mr 1. A factory valued at $35,000 was insured for f of its value, the rate of insurance being f % for one year. What was the amount of the premium ? 3. A warehouse valued at $62,500 was insured for f of its value, the rate of insurance was 1| % for three years, and the cost of the policy and the agent's expenses were $2*50. What was the amount paid for the insurance ? 3. What will be the cost of insuring a cargo of 24,000 bushels of wheat valued at $106 per bushel, the insurance covering f of the value of the cargo, the premium rate being 1|% and the other expenses of the insurance being 2| ^ of the premium 1 4. A merchant's stock was insured for $42,000, h of this amount being at f % , § of the remainder at | % and the remainder at ^ %. Find the total amount of premium jjaid. :■■■■- ': .;,^'fif'-';fr'-'-i4fii INSURANCE. 277 5. A building and contents are insured as follows : — $12,000 in the Imperial, ^C^HK) in the National and $5000 in the Livnoashire Insurance Company. Were a loss to the extent of $3500 to occur through fire, what portion of the loss should each company bear ? 6. Merchandise valued at $63,000 was insured in the Phoenix Insurance Co. for $15,000, in the North British and Mercantile Insurance Co. for $12,000 and in the Norwich Union Fire Insurance Society for $8000 ; if the merchandise is damaged by fire to the extent of $10,500, how much of the damage should each company pay ? 7. A merchant insured his stock for $33,000 for one year at |%. Six months thereafter the policy was cancelled at the request of the insured. Find the amount of premium returned, the short, rate for six numths being §%. §. A factory and the machinery therein is insured for $65,000 ; 2" of this sum is at |% premium and the remainder is at §%. What is the average rate per cent, of premium paid on the whole ? 0. A fire insurance company insured a building for $60,000 at | % premium and reinsured one-half of the risk in another company at §% and one-third of the risk in a third company at |%. What amount and what rate of premium did the company net on the remainder of their risk ? 10. A steamboat worth $60,000 is insured in three companies, in two to the amount of $15,000 each and in the third to the amount of $20,000. For what sum would each company be liable if the vessel were to sustain damage to the extent of $6600 ? 11. A ship worth $56,000 was insured for $15,000 in one insurance company at J% premium and for $32,000 in another company at I %. The vessel received damage in a storm to the extent of $7500. What amount had each company to pay to the owners of the vessel and by how much did each amount exceed the premium received by the company paying that amount ? 12. A fire insurance company charged $196*88 for insuring a house for $17,500. What was the rate per cent, of insurance ? 13. A merchant's stock was worth $120,000 ; he insured it at § its value paying $700 premium. Wluxt was the rate per cent, of insurance ? What was the rate in cents per $100 ? ^■'■■.•^•^v^.fki^'''yr,'filiv^^ . }' 1 ,;*!-v X-- ■ ■'&■ "■,4 ■.■■^h-- ;-if> ■ B '■'''t--' fl -#'"v ;;*•■■■■ ■^'S * :j^: , '.^-^i ' 'J*' - ^t" % ■ '-t . '.'^-T-i ■ V?^> '■V ■. n -;■*:'■ M ■■i-;- i ■■%; .;">>,; i!', ! 278 ARITHMETIC. .,V ^■- '*?' ^4*;- *- hi^: 1 J. A shipment of goods is insured for $7500 and $18*75 is paid as premium. At that rate, what would be the amount of the premium on $18,760 1 1ft. The sum of $286 was paid for the insurance at | of its value of a ship worth $60,000. What was the rate per cent, of premium, if $3*75 was charged for the policy and the preliminary survey ? 16. For what sum was a house insured if the premium paid was $17 '60 and the rate of insurance ^ % ? 17. For what sum was a shop insured if the rate of insurance was 66 cents per $100 and the premium paid was $81 '26 ? IS. A fire insuranoe comiMvny received $350 for insuring a factory at 1|% premium, and charged 1% for insuring a less hazardous property of the same valuation as the factory. What w.as the amount of the premium paid on the second property ? 19. A merchant owns ^ of a steamship and insures f of his interest at f %, paying $337 *50 premium. What was the value of his interest in the steamer ? If during the con^^inuance of the policy, the vessel be damaged in a collision to the extent of $36,000, what sum will the merchant be entitled to receive from the insurance company ? 30. The invoice price of a shipment of goods is $1845. The shipper wishes to insure the goods for such a sum as will, in case of loss, cover both invoice price and amount of premium. For what sum should the shipment be insured if the rate of insurance is §.% ? 31. The value of a consignment is $4250. For what sum should it be insured that the owner may receive both the value of the consignment and the amount of the premium in case of total loss, the rate of insurance being 55 cents per $100 ? 39. For what sum should a cargo worth $18,750 be insured to cover the value of the cargo, the cost of insurance at 1% and $2 50 for the policy and broker's charges ? 23. A cargo of wheat invoiced at $9930 is insured for $10,000 which sum covers not only the invoice value of the wheat but also the premium paid and $5 for expenses. What was the rate per- cent, of the insurance ? 34. A shipment of goods is insured for $6000, which sum covers the value of the goods, the premium at 1| % and $2 '50 for expenses. What was the value of the gooda ? S. i'j"' ".•,■"*'■■'■"' -"'■"'■»' ■,-,1'?"- .«.rj COMMISSION AND HUOKERAGE. 279 IV. COMMISSION AND BROKERAGE. . 234. An Agent is a person au^^horized to transact business for another. The person for whom the agent transacts business is called his Principal. 236. A Coramission Merchant is one who buys or sells goods for other jjersons by their authority. Conmiission merchants are usually i)luced in possessicn of the goods bought «tr sold. 236. A Broker is a person who, in tlie name of his principul, C'tt'ects cimtracts to buy or to sell. The broker is not in general placed in possession of the goods bought or sold. The title Broker is also applied to i)er8( n?? who deal in stocks, bonds, bills of exjhange, pr' aissory notes, <&c., and to mercantile agents "ivho transact the business for a ship when in port. 237. Commission is the charge made by an agent for transacting business. 238. The Gross Proceeds of a sale or of a collection is the total anu>unt received by an agent f(U' his principal. 230. The Net Proceeds of a sale or of a collection is the sum due the principal from the agent, after deducting his commission and all other charges. These charges include freight , liandling, storage, advertising, and such like. 240. The Prime Cost of a purchase is the net sum paid by an agent for merchandise or other property and does not include his commission or other charges. 241. Commlnsion is tisually reckoned at a rate per cent, mi, the ijrons proceeds of sales and collections, on the prime cost of purchases, and on the net amovid of inveHtments. EXERCISE XLIII. I. A commission merchant sold 270 l>arrels of flour at $6 a barrel, and received 6 % commission. What was his commission ? How much did he remit to his employer ••"f ^ ■ I*-;-' ■affife--vK"'.<4'-<^^ !!i>;-^.>jM&% JjC^;^ii\^Affii»i; J >- ij^ifeS&iSif^a&'^fitS^ afej> .U::- --s^UisSiir; ■ _- . --1,-Vj* 280 AIUTHMETIC. 'V ■i'. ■ 9. A commission of 1|24258 waa clmrgetl for Holling $.'i77*2 wt)rth of goods. What wuh tho into of commission V 3. A grain-deiiler charged lU % f(»r selling a quantity of wheat, and received for his conunission $218*40 ; for how much did he sell the wheat ? 4. A real-estate broker sold a house on 6| % commission, and sent to the owner $3000. What was the broker's commisnion, and what sum did ho receive for tho house ? ft. A mei'chant sent $32.')8ti0 to New Orleans to be expended in cotton. Tho broker in New Orleans charged (J% commission. What sum was paid for tho c«ttton ? 6. If $512 50 include tho ])rice T>aid for certain goods and 2J/' conunission to the agent, how nmch money does tho agent expend in purchasing the go<»ds ? T. An agent sold 210 bush, of oats at 60ct. a bush, and charged $.'i'78 for doing so. Find his rate of commission. 8. How many yards of cloth at 90ct. a yd. can an agent buy witli tho connnissiim received from the sale of 300 bush, of potatoes at oOct. a bush., his rate of conunission being li % ? 9. A man bought a horse and carriage for $450, which sum was his commissicm at 2^ % on the sale of a farm. For how much was the farm sold ? 10. A broker is offered a commission of 5i % for selling wool and guaranteeing j)ayment, or a C(»nnuission of 3| % without guaranteeing payment. He acctipts the 6| % and guarantees ])ayment. Tlu' sales amount to $17,000, and the bad debts to $295.50. How much did he gain by choosing the 5^ % ? 1 1 . Sent to a commission merchant in Guelph $2080*80 to invest in flour, his ci further dividend uf 25%, on which he allows his attorney 6%. What is the net amount that he receives ? lA. An agent sold a quantity of cotton amounting to $7317*83, and charged a commission uf 2^ %. He was instructed to invest the proceeds in dry goods, after deducting a commission of IJ % on the amount so expended. What was his tobvl connuission ? 16. An agent sold 300 bales of cotton, averaging 4(52 Ih. to the bale, at 15'7ct. per lb., his commission lK!ing25ct. jujrbiilc, imd the charges ])eiiig $1()1. He purchased for the consignor dry goods amounting to $2570 '37) charging a com mission t>f 1^ %. How much was still due the consignor ? 17. A comrission merchant sold a consignment of bacon at 11^ ct. i)er pound and invested the proceeds, less his connnission, in tea at 38 ct. per pound. His commission on the two transactions at the rate of 5 % on the sale of the bactm and 2% on the purchase of the tea amounted altogetlier to $52'50. How many iiounds of bacon did he sell and how many pounds of tea did he buy ? 1§. A iniller sends 4000 bbl. of flour to a commission merchant with instructions to sell the flour and remit the net proceeds by draft. The consignee pays $40240 for freight and other expenses, sells the flour at $(i"75 per barrel, charges 3 % commissitm and pays \ % premium for draft. Find the amount of the draft. 19. The owner of certain property pays his agent 2^ % for collecting his rents, insurance and repairs cost him ()!| % of his ne.t income but on this sum he pays no income tax, his income tax at 17^ mills on the dollar amounts to $153 "73. Find the gross rents from his property. 90. An agent sold a consignment of boots and shoes for $3825 and invested the proceeds, less his commission, in leather. His total commission on the two transactions amounted to $150. What rate did he charge, the rates on both sale and purchase being the same ? 91 . An agent sold a consignment of fish for $2460 and invested the proceeds, less his commission, in flour. The commission on the sale exceeded the commission on the purchase by $3. What rate did he charge, the rates being the same on the two transactions ? '^A-4 ^1. .v'*-j,i;,*i*.'»^-wfi.;u.ji.;->ii'i,"i.i»' M* 282 ahithmktk;. -A- V. DISCOUNT. ' 242. Discount In an ahntetru'iit or infiiiiioa from the n(tmiiuil pri4:*i or Vidue of aiufthiiiif ; hh, for eXHinplo, from tho cutaloguo or lint price of an urticlu, from tliu nmouiit of u bill or invoice of goodH or of a debt, or from tho face yalue of a promisBory note. 243. The Bate of Discount is usually stated as a rate pet- cout. of the aiHount from which the (lisv;>\int in made. 244. Trade Discounts are rettiictvNiH made from the catcdoijHe or lint ftrires of (jniKts. In some branches of business the manufacturers and the wholosalu dealers catalogue their goods at fixed prices, usually the retail selling price, and then all(»w retail dealers reductions or discounts from these catalogue jmces. These discounts generally depend on the amount of the purchase and the terms of payment, whether cash or credit. By varying the rate of discount, the niainifacturer ciiu raise or lower the price of his goods without issuing a new catiilogue. 245. Very often two or even more successive trade disc;ounts are to be deducted. In such cases the^^ns^ rate denotes a percentage of the cataloyiie price; the second rate denotes a percentage of the reniaiiuler after the first di^'fCoHnt has been, made ; the third rate, a percentjvge of the remainder after the secmul discount lias been nuule ; and so on. Thus, discounts of 20% and 5% in succession oflf any amount, or, as it is generally expressed in business, i^^O and 5 off, means that "20 of the amount is to be deducted from it, and then from the remainder "05 of that remainder is to ])e taken. EXERCISE XLIV. 1. What is the difference between discounting a bill of $30f0 at 40%, and then taking a discount off the remainder of 5 % for cash, and discounting the whole at 45 % ? 2. An invoice of crockery, amounting to $147"»'20, was sold Jan. 3, at 90 days, subject to 40% and 10% discount, with an •Vi _ fVi. ' ■ *A> V""! " • * niSCOUNT. 283 additional (liHcoiiiit of u thu marking priuu ho that a murchant, in closing out a fuilu, may Holl broadcloth coHting ^{'HOayard at 10 /, below cost, and yet bo ablo to allow 40% off thu marking j)rico ? 4. A cabinet dealer directed his BiiloHmau to mark a set* of furniture so that, l)y allowing 20% otF the marked ])rice he may realize a gain of 25 X. The salesman marked the set by mistake at iB200, or at a loss to thu dealer of 20% »>£ the wile. How much less than thu re<{uired marking price was the set markerice and sells them at !iO and 10 off the list price. What is his gain i)er cent.V 7. A manufacturer sells certain goods at 30 and 10 ofii, and gains thereby 12i %. What is the list jjrice, if the goods cost ^28? 8. I purchase books at $2 each, less 33.\%, and 6% for cash. What is the net cost? What % discount may be given off the list price so that I may sell them at a net pn^fit of 10 % ? O. Show that successive discounts of specified rates may be taken off a list price in any order without affecting the net price. Thtis 20 and 10 off is equivalent to 10 and 20 off, so also 30 and 10 and o off, JO and 30 and 5 off, and 5 and 30 and 10 off are all equivalent. 10. 20 and what rate off are eijuivalent to 40% off? 25 and what rate off are equivalent to 40 / off? 30 and what rate off are equivalent to 40 % off? 20 and what rate off are e(|uivalent to 33_^ % off? What rate taken off twice in succession is e«iuivalent to 36% 11. 19. 13. 14. off? 15. What rate taken off twice in 8uccessiuivalent to 44% off? 16. What rate taken oft' thrice in succession is ecpiivalent to 48 "8 % off? 17. What rate tasen off thrice in succession is e^;,*M .R >- 284 ARITHMETIC. 246. A Promissory Note (often called briefly a Note) is a written promise to pay, unconditionally, on demand or at a lixedor a determinal)le' future time, a si)ecitied sum of money, to a particular person named in the note, or to a person named or his order, or to bearer. 'A note which is, or on the face of it purports to be, both made and payable within Canada, is an inland note : any other note is a foreign note. 247. The Maker of a uote is the person who signs the promise. The Payee is the person to .whom or to whose order the note is made payable. The Holder or Bearer of a note is the person who lawfully possesses it. The Pace Value (or simply the Face) of a note is the sum of money (exclusive of interest) which the maker promises to pay. 248. A jiromissory note may be made by two or more makers, and they may be liable thereon jointly, or jointly and severally according to its tenoi*. If a note runs " I promise to pay," and is signed by two or more persons, it is deemed to be their joint and several note. 249. An Indorser of a note is a person who writes his name on the back of the note. By so doing he guarantees its payment and l)ecomes responsible therefor, uidess when indorsing he writes above his signature the words "without recourse." A note payable to itriler must be indorsed by the payee when transferred to anyone else, but a note payable to bearer need not be indorsed. A special indorsement specifies the perscm, called the indorsee^ to whom, or to whose order, the note is to be payable. An indorsement in blank specifies no ind )rsee, and a note so indorsed becomes payable to beai'er. Whtfi a note has been indorsed in blank, any holder may convert t'.ie blank indorsement into a special indorsement I y writing above the indorser's signature a direction to pay the note to or to the order t)f himself or some other i)erson. An indorsement is restrictive which prohibits the further negotiati«m of the note or which expresses that it is a mere autliority to deal with the note as thereby directed, and not a DISCOUNT. 285 transfer of the ownership thereof, as, for example, if a note be indorsed " Pay D only," or ''Pay D for the account of X," or "Pay D or order for collection." A restrictive indorsement gives the indorsee the right to receive payment of the note and to sue any party thereto that his indorser could have sued, but gives him no power to transfer his rights as indorsee unless it expressly authorise him to do so. Where a restrictive indorsement authorises further transfer, all subsequent indorsees take the note with the same rights and subject to the same liabilities as the first indorsee under the restrictive iudx^iocment. 250. A Negotiable Note is one which may be sold or transferred by the payee to anyone else ; and a note is negotiated when it is transferred from one person to another in such a manner as to constitute the transferee the holder of the note. A negotiable note may be payable either to order or to bearer. A note is payable to bearer which is expressed to be so payable, or on which the only or last indorsement is an indorsement in blank. A note is ])ayable to order which is expressed to be so payable, or which is expres ;ed to be payable to a particular person, and does not contain words prohibiting transfer or indicating an intention that it should not be transferable. Where a note either originally or by indorsement, is expressed to be payable to the order of a specified jierson, and not to him or his order, it is nevertheless payable to him or his order, at his option. A note payable to bearer is negotiated by delivery. A note payable to order is negotiated by the endorsement of the holder completed by delivery. Where the hcjlder of a note payable to bearer negotiates it by delivery without indorsing it, he is called a transferor h\j delivery. A transfercjr by delivery is not liable on the instrument. A transferor by delivery who negotiates a note thereby warrants to his immediate transferee, being a holder for value, that the note isniiat it purports to be, that he has a right to transfer it, and that at the time of transfer he is not aware of any fact which renders it valueless. When a note contains words i)rohibiting transfer, or indicating an intention that it should not be transferable, it is valid as between the parties thereto, but it is nob negotiable. 286 AUITUMETIC. 251. Where a jiromissory note is in tlio body of it mado i)ayable at a particular place, it must l»e presented for payment at that placy in order to render the maker liable : in any other cjise, ])resentment for payment is not necessary in order to render the maker liable. Presentment for payment is necessary in order to render the indorser of a note liable. Where a nt)te is in the body of it made jjayable at a particular place, prefentment at that place is necessary in order to render an indorser liable ; but when a ])lace of payment is indicated by way of memorandum only, presentment at that place is sufficient to render the indorser liable, l>ut a i>resentment to the maker elsewhere, if sufficient in other respects, will also suffice. 252. Maturity (properly Date of Maturity) is the day on which the note becomes legally due. Where a note is not payable on demand, the day on which it falls due is determined as follows : — Three days called days of grace, are, in every case where the note itself does not otherwise provide, added to the time of payment as fixed by the note, and the note is due and payable on the last day of grace. Whenever the last day of grace falls on a legal holiday or nim-juridical day in the Province where any such note is payable, then the day next following, not being a legal holiday or non-juridical day in such Pro\'ince, is the last day of grace. A note is payable on demand, which is expressed to be payable on demand, or on presentation, or in which no time for payment is expressed. 253. Where a bill is payable at a fixed period after date, after sight, or after the happening of a specified event, the time of l)ayment is deteiuiined by exqjuding the day from which the time is to begin t) run and by including the day of payment. The term " M5 = 2 -758003 + -845098 -2 + 2 -045323 - 2 -562293 = 1 •086131 = log 12-194.] •' if 11 288 ARITHMETIC. EXERCISE XLV. Find the date . $586 •(>7. 28th Dec. 1891. 4 mo. 15th J",n. 8 %. 6. Find the proceeds of the following note discounted in Toronto (m 1st May 1890, at 7%, exchange ^ %. {fn Ottawa, 1st May, 1800. Three months after date I promise to pay to the order of Thomas A Stuart, Three Hundred and Ninety j^/^ Dollars, at the Bank of Connnerce here. Value received. James Henderson. 7. A note for $250 was discounted 40 days before maturity and the proceeds were $247 "80. What was the rate of discount, there being no exchange ? §. A note for $742*76 was discounted 93 days befoi'e maturity and the proceeds were $730 '47. What was the rate of discount, the rate of exchange being ^ % ? 9. For what sum nuist a note be drawn in order that if discounted .SO days before maturity, the proceeds may be $425 ; the rate of discount being 7% and there being no exchange ? J O. For what sum must a draft payable thirty days after sight be '^rawn in order that if discounted on day of drawing the proceeds ;:iay be $745 '25 ; the rate of discount being 7| % and that of exchange | % ? II. A i)romissory note for $385 '20 was discounted on 1st March, 1890, at 7 % discoinit and l^ % exchange and tlie proceeds were $377 "70. Determine the date of maturity of the note. S ^%, i^-;t' ,.,:.■:.>,, INTEREST. 289 VI. INTEREST. 261. Interest is the sum which the lender of money charges the borrower f(»r the use <>f the sum borrowed, or which f the money he pays as interest at the end of each period and ihv lender gains the use of it, and the value of this use is assumed to be interest at the rate paid on the principal. Hence, in calculations concerning periodic paytnents, the methods^ not of simple hnt oj compownd iaterest, shoidd be emploijcd. For example it is usual with savings banks which pay annual interest to credit each depositor ut the end of every interest year with all interest on his deposit accrued but undrawn, treating such interest as a new deposit, the net result being that the banks pay compound interest. 271. If interest which is by agreement to be paid at specified intervals, is not so paid, and the lender has to collect it by process of law, the courts have authority to grant at their discretion simple interest on the accrued periodic interest. The interest upon interest, if thus granted, is styled damages and its maximum rate is the legal rate of six per cent, per annum. Simple Interest. 272. Problems in simple interest involve the consideration of principal, rate, time, interest and amount ; and any three of these being known the other two may be determined, for by definition ; — 1°. The interest is the continved prodiu-t of the principal, the rate per unit and the tneasnre of the time. 2°. The amount is the sum, of the principal and tlie interest. 273. Expressed in general symbols these statements are r. I=Prt, 2°. A = P + I, the letters J, P, t and A denoting severally the measures of the interest, princijjal, time and amount, and ■/• denoting the rate per v/nit. INTEREST. 291 EXERCISE XL VI. Find the simple interest on and the amount of I. $473-28 for 3 years at 6 "/. a. $385-35 for 1.^ years at 5%. 3. $628-25 for 185 days at 4|%. 4. $935-68 for 66 days at 6 J %. 5. $147 -50 for 3 years 93 days at 7%. «. $250 from 9th July to 18th Aug. at 8%. 7. What principal will yield $43-25 interest in 2J years at 6^% ? §, What principal will in 95 days yield $9 20 interest at 7 % ? 9. What princii)al will yield $10 as interest at 6 % from 1st. May to Slat. Oct. of the same year ? 10. What principal will amount to $1000 in 4^ years at 4^ % ? II. What principal will amount to $73*56 in 66 days at 8% ? 12. A debt due on 3rd March was not paid and interest at 6| % was charged on it from that date. On 6th June following, the debt amounted to $100. Wliat was the sum due on 3rd March ? 1 3. At what rate will $375 '50 amount at simple ^interest to $441-21 in 2^ years? 14. At what rate will $222-66 yield $21 simple interest in 1 year and 94 days ? 15. At what rate will $438*88 borrowed on 17th Ap. amount at simple interest to $446*93 on 29th July next following ? 16. At what rate will a sum of money at simple interest double itself in 20 years ? 1 7. At what rate will a sum of money at simple interest quadruple itself in 50 years 1 l§. In what time will $273-85 yield $28-86 simple interest at 6%? 19. In how many days will $733-65 amount to $743 70 at 5% simple interest ? 30. A debt of $175 became due on 13th June after which date interest was charged at the rate of 7 %. When the debt was paid the interest accrued on it was $4*10. When was the debt paid ? I ': ! m l\ :i:| 292 ARITHMKTIC. 91. In what time will a sum of money double itself at 5% simple interest? il3. In what time will a sum of money triple itself at 8 % simple interest 1 33. The proceeds of a r^te for $137 "50 discounted 40 days before maturity, were $136 'SO. vVhat was the rate of discount charged on the face of the note and what was the rate of interest paid on the proceeds? 34. Find the discount off $385*77 due 86 days hence, (i) at 8 % discount, (ii) at 8 % interest. Show that tlie difference between amounts (i) and (ii) is the interest at 8 % on (i) or the discount at 8%off(ii). 3ff. What rate of interest is- equivalent to .10% discount, tlio term of discount being one year ? •26. What rate of interest is equivalent to 10% discount, the term of discount being 96 days ? [274. The Present "Worth at a specified rate of interest of h bill or a promissory note is the sum of money which put out at interest at the specified rate will when the bill is due or the note matures amount to the sum due on bill or note. The difference between the present worth at a specified rate of interest of a bill or a promissory note and the amount of the Dill or the note when due, is by some writers termed the True Discount, and the specified rate of interest is called the Bate of Disamnt. Considered as an abatement or deduction made fronv the amount of the bill or the note, the so-called True Discount is certainly a discount, hut so would be any other abatement, but to call the rate at which the present worth increases by interest, a rate of discount, i.e., a rate of counting offy is a perversion of the term which is not sanctioned by commercial usage and which leads to 'needless confusion when pupils go from the class-room to the counting-house. The problems which are commonly given under the head of True Discount are properly problems on Interest and were they correctly worded and proposed as problems on Interest they would be perfectly legitimate and unexceptionable. Thus Prob. 10, Ex. xlvi, j), 291, may be put under the form : — What is the present worth at 4*5% interest of $1000 due 4*5 years hence ?] ^ff^: .-ft-/ .:,<: INTEREST. 293 Averaging Accounts. 275. When one i)er8()n owes another several amounts due at different times, the date on wliich all these debts may be discharged by payment of thjir sum, without loss of interest to either the debt(jr or the creditor is called the Average Date or Equated Time. Example. A bought goods of B as follows : — May 17, $200 at 30 days' credit ; June 3, $260 at 60 days' credit ; June 12, $210 at 90 days' credit. On July 5, A paid B $300 on account. Find the equated time for paying the balance. Had A paid B $200 + $250 + $210 = $660 on May 17, B would have gained the interest on $200 ft)r 30 days, the interest on $250 for 77 days and the hiterest on $210 for 116 days. But if A delay from May 17 to July 5 to pay $300 of the $660, B'& gains will be reduced by the interest on the $300 for 49 days, the number of days of delay. And if A defer the payment of the $360, balance of the $660, until the equated date, B will lose the balance of the interest ho would have gained had all the payments been made on May 17. Interest on $200 for 30da. =Int. on ($200 x 30-$ 6000) for 1 da. „ „ 250 M 77 I. = .1 t. ( 250 X 77= 19250) „ „ „ „ 210 M 116 „ = „ M ( 210x117= 24.360) „ „ $49610 for Ida. Interest on $300 for 49da. = Int. on ($300 X 49= 14700) m n $360 $360 )$34910( 97 Interest on $34910 for Ida. =Int. on $360 for (34910^360) days - Int. on $360 for 97 days. Equated time = 97 days after May 17 = Aug. 22. 276. Should any of the items include cents, omit the cents in the calculation, and take the nearest number of dollars to the amounts of the items. 277. The method of determining the equated time of an account, which is exhibited in the preceding solution, is bas(Kl on the assumption that what the debtor gains by retaining certain sums i 294 ARITHMETIC. after they become duo he loses by paying other sums before these become due, but as })oth gains and losses are cunii>uted on the full amounts of the items, while the actual [^ain is the interest on the amounts of the deferred payments and tlie actual loss is the interest on the i)resent worth of the anticipated payments, it is evident that the solution is not absfJutely exact. However, in ordinary business transactions, the error is too snrUl to materially affect the result. EXERCISE XLVII. Find the equated date of payment of 1. Sep. 3, $350 @ 60 da. 2. Aug. 27, 8^25 @ 60 da. M 1520 @ 90 da. Sept. 20, $280 @ 30 da. M $175 @ 30 da. Oct. 31, $786 @ 90 da. 3. On May 2, goods amounting to $1250 were purchased on the following terms ; $400 payable in 30 days, $500 })ayable in 60 days and the bfl lance payable in 90 days. Find the equated date for the payment of the whole bill. 4. On Sep. 19, a commission merchant received a consignment of 600 barrels of apples. He sold 120 barrels at $2*25 on Sept. 24 ; 75 barrels at $2-30 on Sep. 27 ; 150 barrels at $2-40 on Oct. 7th ; 150 barrels at$2-35onOct. 22 ; and the balance at $2' 20 on Nov. 18. Find the equated date of the total sales. 5. Henry Simpson sold A. Thomson & Co. merchandise as follows : Sep. 1, 225 bbl. flour @ $6, on 30 days' credit ; Sep. 9, 180 bbl. of pork averaging 2081b. ® ll^^ct., on 60 days' credit ; Sep. 17, 150 doz. eggs @ 16 ct. per dozen on 2 months' credit ; Oct. 7, 572 lb. bacon @ 13.] ct. on 3 months' credit ; Nov. 10, 4601b. butter @ 21^ ct. on 90 days' credit. Find the equated date for the payment of the sum-total of the several bills. 6. A holds three promissory notes made by B, one is for $245*60 payable in 3 months from Feb. 13, 1889 ; another is for $425 payable 60 days after date of Mar. 5, 1889 ; and the third, is for $186 25 and is dated Ap. 3, 1889, and payable 90 days after date. On Ap. 17, 1889, B offers to pay $500 on the notes, and give in exchange for them a single note for the balance'on them unpaid. When should the single note be payable ? PARTIAL PAYMENTS. 295 Partial Payments. 278. A Partial Payment Ih a paynient <»f only a part of a debt and its accrued intereHt. 279. A Receipt Indorsement is hi» Hcknowledginent of the receipt of a partial payment written on the ])aek of a note, mortgage or other documentary evidence of debt, stating the amount and the date of the payment. 280. When partial payments have been made on an interest- bearing note or other obligation, the balance unpaid and due at any given date may be found as follcjws : — Find the interest on the jyrincipid from the date of the note or other obli(j(dion to the date of the fir d partial payment. (ff) If the first partial payment is equal to or exceeds the interest fhii.H foiiitdy sid>tra<:t the first payment from the sum of the principal and its accraed interest, and consider the remainder as a new principal. (h) If the first partial payment is less than the interest thus found, find the interest on, the pi'incipal to the date of the next or of the earliest sxibsequent partial payment at which the sum of the payments eqiuds or exceeds the hderest due at such date, and subtract the sum, of the payments to that date from the sum of the principal and its accnied interest to that date, and consider the remainder as a new principal. Similarly find the interest on the n«w principal to the date of the next partial payment. If that payment be equal to the interest thus found or if it be greater than, the itderest, proceed as in (a); but, if the paxjment be less than the interest, proceed ^ully rliiii'KO ilittM'imt ii|i(in i\\\ «IoI)In I'roiii tlu) liiiio (liit' liuliuii^iuH mtooiiiilM uiiil hIIhw iiilmmt (o llio huiiio (iiiio iipMii till |MU'liii) * piiynioiilN frotii llio liino lli<y llirti ili> tlio Niiin of tlio piii'tiiil |wiyiii«iiitN iiixl lli(>it' in'cnuMi iitl«ir« fVniii tlio niiiii *>f tlio (loltlH uimI llioii' HccniiMl iiitoitml, llio loiiiaiiiiKtr lloill^ llio Ixkluiioo duo. I 'poll tlii'* I)uIhiio(% if llio (uhmiiiiiI. lut iml, moiiiiwliilo |Miiil or rloNo«l hy iinti<, iiiloroMt. in cliiu'^od tit tlio liiiio wlioii tlio lUHMltllltM HI'O IV^uin WullUIOOtl, IUI«l Ih hIIoWOiI to llio NlilllO tilllO llpitll nil porliiil puyinoiilH from llii« liiiio tlii.'l ; Koh. It, IHHU, #21(118 ; •lunolJ. IHH'.I, i|i;OI(l; Sop. 'J, IHHi>, #|H;{*Jf.. Mow iiiuoli wandiio on tho nolo on Nov. 11, IHH'.r/ 51. On a niort^aj^o for li|»:i7r»0 dat.»..'. iVlay U\, ISM7, and lioarin^ intoro.s| ut ir , Ihoro woio paid May HI, 1888, j||C{r»(» j Sopl.. 18, IHHH. #'JHO; .Ian. liiJ, I88t>, HjirMk ; May III, I88U, #Ul>f. ; Oot. 'M, 188',>, JJ^MH). What muiu was tlu«» on llio iiiort)j;ago »»J? Jan. li, .I81KI ; •I. How luuoli waa duo on tho ftdlowin^^ nolo, on Ool. Jil, I88U1 ijlHW). Toronto, Oil. Ill, 1HH7 For vahui root*ivod, 1 pr«)uiiNo t.o pay .Mox. 'riioiupHon or ordur, on doinnul, Miirht. hundrod and lifty l)ollar«, with inloroHt. from ilato at six por ooulum. John Stuart. On this noto tho f«>llowinj( payinonts woro ind«»rHod. April L'O. IHHH, #125. Nov. 20, 1888, |(lijr>. ,lan. 21, 1HH5>, r' • July 20, 188J), #420. roMi'miNU iN'ii:m:.s'i'. 21>7 Oomt>'>vuid Iiitereat. 283. Ooiupound Int.nroMt in iiil«tr«ml, wliirli In rri(M)u nol, drawn il. will l»u placud to Mm crudifc of Mm duportitor, making' lii.s dopoHit !i{(|.'l(M)-50 at. Ilmhuginning of Miu Miird period of mx niontJiH. Tim intifoA for Miu third half yujir will hu (ioiiiput.ud on Mm !||«i;{(MI-5() «top..Hil, and will Mmruf(.ru ]>u 02 of .iio not,.lrawn, will he added to Mm SiHi;K|J)-50 niakiii}; .^ i.of,ul of 111. '?■')* 5 1 at Mm cnxlit, of tlm duixmitor at tho ond of 18 inontliH. 298 ARITHMETIC. Thus $1250 at 4 / interest compounded semi-annually will in a year and a half amount tt) $lIi2H'51 ; and the compound interest at the specified rate and for the stated time will be #132H 51-^1250 Gompntathm. $1250 =original amount or jtrincipal. \.'{)2 — ri(tr, of increase in amount. ~^ 25 (K) 121)0 $1275 = amount at end of 1st [leriod. 102 25-50 1275 $1300 '50 = amount at end of 2nd })eriod. 102 2«0106 1300-50 $1326 •51 = amount at end of 3rd period. EXERCISE XLIX. Find the amount and tlie compovnid interest of : — 1. $800 for 3 years at 5 % compounded annually. 2. $425 for 4 years at 4 % compounded annually. 3. $250 for 2 years at 6 % compounded semi-annually. 4. $366 •67 for 2.V years at 4 % compounded semi-annually. 5. $722 '50 for 1^ years at 4 % compounded quarterly. Find correct to six significant figures the amount of $1 at compound interest at 6 % for one year, interest compounded. 6. annually. 7. semi-annually. §. (quarterly, Find correct to six significant figures the rate of increase in the amount of $1 at 5 % interest compounded annually for 9. three years. 10. five years. 11. seven yeai's. Find correct to six significant figures the rate of increase in the amount of $1 at 4 % interest com})ounded (quarterly for one year. 13. two years. 14. three years. COMPOUND INTEREST. 299 286. Problems in Compound Interest involve the consideration of origvMil umou'nt or principal, rate, number of compmindings, final amount and interest, and any three of these being known the other two may be determined. Let r denote the nominal rate of interest per unit ; t the measure in years of the length of time between two successive compoundings ; n the number of compoundings ; A^ the measure of the original amount, the principal ; A^ the measure of the amount after n compoundings ; and I^ the measure of the interest after n compoundings ; then will A^ = Aq{1 + H), A,^ = A^ {l + rt) = A^{l^rty^ A^ = A.{l + rt) = A^,{l + rt?r' A^ = A,,{l + rt) = A^{l + rtY • ^„ = ^._i(l+r*) = ^o(l + '0", {A.) and . '. log ^n = log AQ+n log (1 + rt) ; (.4a. ) and J„ = ^„-^o- {B.) 287. If there should occur a broken period whose measure in years is ti, t^ being ns. Debentures frecjuently charge certain specified property with the repayment of the money borrowed on them ; in such cases the debentures are practically mortgages on the property. 298. An Interest Coupon is an interest certificate payable to bearer, printed at the bottom of bonds and debentures giveix for a term of years. There are as many coupons attached to each bond as there are instalments of interest to be paid on it, a coupon for each instalment. Each coupcm is cut off and presented for payment when the interest for the period mentioned in it becomes due. 299. Consols, i.e., Consolidated Annuities are British Govern- ment securities bearing 3 % interest. These with the other British Gf»vei'nment securities for which permanent provision has been made, the most important of which are the Reduced Annuities and New three per cent. Annuities, are in England termed the Fublic Funds. 300. Rentes (i.e. Annuities) are French Government securities bearing various rates of interest. 301. Stock Brokers are persons who deal in stocks, bonds and similar securities. When a st(jck broker buys or sells for a principal he chai'ges a commission, technically termed brokerage, which ranges, according to circumstances and previous agreement, from ^ of 1 % to ^ of 1 % of the par value of the securities bought or sold, the most common rate being I of 1 %. Occasionally special rates are agreed upon and paid. 304 ARITHMETIC. In England, stock brokers do not deal directly with each other but sell to or buy from stock-jobbers who act for themselves and make their profits out of the turn of the market. 302. In Canada and the United States, stock quotations usually state only the rates per cent, which the luai-ket values of the stocks and bonds quoted bear to their par values ; but in England (quotations of other than government securities generally give the price per share or per bond. The following is an illustrative example of a stock report and quotations : — The closing prices on the Toronto Stock Exchange to-day (6 Dec, 1889), were as follows : Stocks. Banks. Montreal . , Ontario . . . Molsons . . , Toronto . . Merchants' Commerce Imperial . . Dominion . Standard . , Hamilton . 3J 3 rj > ?3 ° 200 100 50 200 100 50 100 50 50 100 w^ ^ 5 3i 6 5 1 P. M. 02 1— I Si ^ 2254 132 168 219 142 121i 152f 222.1- 138" St* 2241 i3ii 213 139 121 150 221 1371 146 4 P. M. 225^ 131^ 156 220 140 121i 153 223 138 224| 131 212* 139 121 150f 222 137^ 147 It will be seen from this report that in Toronto, on 6 Dec. , 1889, sellei's of Bank of Montreal stock were offering it at the rate of $225*25 crts/i/ per $100 sfocfc and -as each share represents f 200 of stock, sellers were really asking $450*50 per share. The report also shows that buyers of Bank of Montreal stock were offering for ic 224^ % to 224| % of its par value, i.e., were offering $449 to $449*50 per share for it. STOCKS AND BONDS. 305 Other forms of rei)<)rt may be seen in the ' Financial Columns ' of the Tonmto and Montreal daily newspapers and in the ' Share Lists ' (jf any Stock Exchange. Example, 1. Find the price at 140| of 40 shares ($40 each) of Western Assurance Co. stock, brokerage J %. Par value ( )f stock = $40 x 40 = $1H0^. Rate paid = 140| % + J % - 1 •40|. Cost of stock = $1600 X 1-407- =$2254. Example 2. I sold 500 shares of Bank of Montreal stock at 224| and invested tlie proceeds in Bank of Commerce stock at 124|, paying \ % brokerage on each transaction. Find the increase in my annual inccjme, the Bank of Montreal paying a half-yearly dividend of 5 %, the Bank of Commerce a half-yearly dividend of Par value of B. of M. 8tock = $200x500 = $100,(XK). Rate received = 224| % - \ % =2-24|. Amount to be invested = $100,000 x 2'24g = $224,625. Rate paid f or B. of C. stock = 124^ % -f J % = 1 "24^ Price of 1 share of B. of C. stock =$50 x 1 -241 -$62-1875. Number of shares bought is the integral part of $224625^$62-1875 which is 3612 Mid there is $3-75 of cash over. Par value of 3612 shares of B. of C. stock =$180,600. 2 dividends at 5 % each on $100,(M)0 of B. of M. stock = $100000 X -10 =$10000. 2 dividends at 3J % each on $180600 of B. of C. stock =$180600 X 07 = $12642. Increase of annual income = $12642 - $10000 = $2642. "' :i \ 30G AIUTHMETK!. EXERCISE LI. 1. 25 s a. 18 3. V5 4. 2)0 5. 910 6. ?.>0 Find the chsIi vnluo of 25 sliures Onturio Bank at l.'ii. Htjindiird Bank at 11^7^. Bank of Toronto i\t 218. (^50) Dominion Telegraph Co. at83|. ($100) Canadian Pacific R.R. at 72^. ($24-35j) North West Land Co. at 79|. 7. S )ld through a broker 1500 shares ($100) of Jersey Central R.R. stick at 121^, brokerage J %. What were the net proceeds of the sale '{ 8. Bought through a broker 1600 shares ($100) St. Paul R.R. stock at 69|, br<>kerage J^ % . What was the gross cost of the stock ? 9. A si)eculator bought 36500 sliares ($100) Reading R.R. stock at 39| and sold them at 40^. What was his gain ')n the transaction ? 10. A man bought through a broker 1900 shares ($100) Canada Southern R.R. stock at 54;^ and sold them at 55§. What was his net profit on the transaction, brokerage each way J % ? 11. A man bt)Ught through a broker 7600 shares ($100) of Lake Shore R.R. stock at 107t and sold 2400 shares at 107| and the remainder at 107|. What was the amount of his losses on the transactions, brokerage being I- % each way ? lis. A bank declared a dividend of 3| %. How much should a stockholder owning 120 shares ($50) receive ? 13. An insurance company declared a dividend of 6 %. What rate is that on the market value of tlie shares which are at 185 14. Ct)mpare the rates on the cash values of 6 % on stock at 216 and 3| % on stock at 125. 15. Sold 37 shares ($25) B. and L. Association stock, receiving therefor $1019*81. At what rate was the stock sold ? 16. Bought through a broker 750 shares ($50) in the Farmers' Loan and Savings Society paying therefor $43968 "75. At what quotation were they bought, brokerage ^ % ? 19*. Sold through a broker 215 shares ($50) in the Dominion Savings and Loan Society receiving from him for them $9728 "75. At what quotation did the broker sell them, brokerage ^ % ? STOCKS AND BONDS. 307 lid H Hiafc 216 iving inion •76. 1§. Bcmght stock at 197^ and sold it at 194|, having meanwhilo received a dividend of 6 % on it. My net gain by the transaction after paying J % brokerage each way, is $336. How many shares ($40) did I buy ? 19. A man received $495 as dividend at 4^ % on his bank stock. He sold 40 shares ($100) at 143| and the remainder at 144^, paying I- % brokerage. What were the net proceeds of the sale ? 20. A capitalist had $20000 to invest. He purchased $8700, par value, of Canadian 4 % bcmds at 103 and $7300, par value, of Canadian 3| % bonds at 93i and invested the balance as far as he could in bank stock (shares $100) at 149J, paying half-yearly dividends of 4 % each. What was the gross amount of his investment he paying | % brokerage for buying each class of securities ? What was his annual income from these investments ? What average rate per cent. i)er annum did he receive on these investments ? 21. The difference between the annual income derived from a certain sum invested in 7 % stock at 150 and that from an equal sum invested in 9 % stock at 202^, is $40. What is the amount invested in the 7 % stock and what is th6 annual income therefrom 'i 23. A shareholder receives a dividend of 6 % on his stock and pays thereon an income-tax of 16_| mills on the dollar. Next year he receives a dividend of 6^ % and pays an income-tax of 12h mills on the dollar. He finds that his income is $830 more in the latter year than it was in the former. How much stock does he hold ? 23. A man invests a certain sum in 3 % stock at 90 and an equal sum in 4 % at 95. Each stock rises 5 % in price ; the investor then sells out and invests the proceeds of each stock in the other. The stocks fall to their former value and he again sells out at a total loss of $1943.90. Find the sum he originally invested. 24. What sum invested in the three per cents at 95 will in 17^ years amount to £10000, the price of the funds having risen meanwhile to 100^ ; interest to be payable and compounded half yearly ? 25. If money be worth 5 %, what should be the price of 6 % bonds which are to be paid oflf at par 3 years after the date of purchase, the interest on the bonds being payable half-yearly. Ill CHAPTER VIII. EXCHANGE. 303. Elzchange is the system by vihich accounts between persons in distant ])laces are settled witliout the necessity of sending large sums of money or largo quantities of gold or silver from one plarie to the other, thus avoiding the risk and expense of transportation. For example, suppose that A of Halifax owes B of Toronto f7<"»00 for wheat and that A' of Toronto owes Y of Halifax $76lX) for h'l'nX fish. In such case, B in Toronto can draw on A in Halifax for $7500 and sell the draft to X who transmits it to Y who in turn presents it to A who thereupon i)ays Y. Thus instead of A sending $7500 from Halifax to Toronto to pay B, and X sending $7500 from Toronto to Halifax to pay Y, X of Toronto pays B in Toronto and A of Halifax pays Y in Halifax, the itbts being as it were excluinged. Domestic or Inland Exchange is exchange carried on between two cities in the same country. Foreign Exchange is exchange carried on between two cities in different countries. » 304. A Draft or Bill of Exchange is a written order by one person, called the drawer, directing a second person, called the drawee, to pay a specified sum of money to a third person, called the payee, or to the payee's order. A Domestic or Inland Bill of Excha'age, usually called a Draft, is one of which drawer and drawee reside in the same country. A Foreign Bill of Exchange is one of which drawer and drawee reside in different countries. Foreign bills of exchange are usually drawn in sets of three, called respectively the First, the Second and the Third of Exchange, and are of the same tenor and date and so worded that when one of the set is paid, the others become void. The object of thus drawing the bills in sets of three is to provide against loss in transmission. The bilL or two of them are sent either by different routes or by the same route at different dates. EXCHANGE. 309 305. An Acceptance is an agreement by the drawee to pay the sum specified in the draft or bill of exchange. The usual mode 3f accepting a bill of exchange is for the drawee to sign his name under the word *' accepted" written across the face of the bill. If bhe bill be payable a specified number of days after sight, the date of acceptance should be inserted. 306. If the drawee of a bill refuses acceptance or if, having accepted, he fails to make i)ayment when it is due, the bill is immediately protested, i.e., a written declaration is made by a public officer called a Notary Public, at the request of the holder or person in legal possession of the bill, notifying the drawer and the indorsers c»f its non-acce])tance or non-payment. 307. Bills of exchange are negotiable or non-negotiable upon the same conditions and are subject to the same indorsements as promissory notes. The date of maturity of bills of exchange is ascertained in the same manner as that of notes ; see § 252, p. 286. 308. The Pace or Par of a bill of exchange is the sum specified in the bill, exclusive of interest, premiums, discount, or commission. When bills of exchange on a given place sell for more than their par value, exchange on that place is said to be above par or at a premium ; when they sell for less than their face value, exchange on that place is said to be heloio par or at a discount. 309. Exchange is usually conducted through bankers or brokers who buy commercial bills on distant cities and mail them for collection to their correspondents or agents in those cities. Drafts or bills of exchange are then drawn on the correspondents for the whole or fi T any required part of the sums thus placed to the credit of the principals and sold to persons who wish to use money in those cities. Bankers and their correspondents also draw on each other for sums required by persons dealing with them and at stated periods strike a balance of the sums thus drawn. 310. The Par of Exchange between two countries is the value of the monetary unit of one of the countries expressed in terms of the currency of the other. The intrinsic par of exchange is the real or intrinsic value of coins estimated by the weight and purity of the metals of which they are composed. " \ 1 ( 310 AIUTHMETIC. The legal par of exchamje, m the par ostablishod under authority of statute. The dollar of Canada is defined by statute to be of such value that four dollars and eighty-six cents and two-thirds of a cent shall .be equal in value to one pound sterling; thus 04 8())| per £1 is the legal par of exchange between Canada and Great Britain. There being no Canadian gold coinage and the silver and bronze coins of Canada being only a token coinage, there is no intrinsic par of exchange between Canada and Great Britain. The intrinsic value of the sovereign, the coin which determines the value of the pound sterling of Great Britain, in terms of the gold dollar, the monetary unit of the United States of North America, is $4 •8^6564— ; for, 1869 sovereigns contain 211200 grains of pure gold and the United States gold eagle contiiins 232*2 grains of pure gold and 211200 -^ 1869 -^ 23 22 = 4 -866564-. The value determined at the United States Mint and proclaimed by the Secretary of the Treasury is $4 '8665, a sum which approaches the intrinsic value far within the * remedy ' allowed on the sovereign. The intrinsic value of the ten-franc gold pieces of France, Belgium and Switzerland is $1 93. The intrinsic value of the ten-mark gold piece of the German Empire is $2"8c. 311. The Bate of Foreign Exchange is the market or commercial value of the monetary unit of one country expressed in terms of the currency of another. The following quotations were given by the New York Agents of the Canadian Bank of Commerce as indicating the rates for actual business in sterling exchange on 7 Dec. , 1889. Prime Bankers, 60 days 4-80^ Demand 4*84^ Cables 4 -841 60 days 4-79|-^ do. 4 -781 @ 4-7^. Sterling Bills are those drawn by first-class bankirg houses in New York on first class banking-houses in London, England. Cummercial Bills are those drawn by merchants or commercial houses of good standing in America on their correspondents abroad. do. do. Commercial Documentary Prime Bankers' EXCHANGE. 311 A Dooinuntnnj Commercitd Hill is i\ bill drawn by a Hhippor upon his cted at 9j^ it is meant that tho rate of exchange is ^"441* x 1"095 per £1, i.e., ^4"8(}3 per £1 which is the ■)iew par. So also sterling exchange at 9 means $4-44* x 109 per £1, i.e., $4-84§ per £1. 313. The usage of Canadian bankers is to draw bills of exchange on London payable either at (iO days after sight or on demand, but as tho greater part (>f the business is done in tho former class of bills, quotations are assumed to be for sixty-day bills unless it is specifically stated to be otherwise at tho time of making them. 314. A Circular Letter of Credit is a letter issued by a banking-house to a person who purposes to travel abroad and addressed to bankers generally and to the agents and correspondents of the banking-house in particul.ar in the several countries which the traveller is about to visit, requesting them to supply tho traveller with m(mey as he requires it until a total amount has been paid him not exceeding the sum specified in the letter. The sums paid to tho traveller from time to time are indorsed on the letter. A letter of credit is not transferable from one person to another. Example 1. What will a bill of exchange on Lcmdfni for £6000 realise in Toronto exchange at 8i ? The %\ here means Si % premium on the old par of exchange of ^4*44^ which gives $*„ttxl'086 as tho rate of exchange for the transaction. . •• £6000 is equivalent to %S^ x 1 085 x 6000 = $28933 "33. i 312 ARITHMETIC. Example 2. Exchange at New York on London is 4 '842, and at London on Paris it is 25 25 francs per £1. What sum remitted from New York through London to Paris will pay a debt in Paris of 12500 francs ? 25-25fr.=£l = $4-84| 12500 fr. = $4 -841 X 12600-r-25-25 = $2.399-75. EXERCISE LII. What will a bill (m London for £'75 cost, exchange at 9^ ? What will a bill on Londcm for £225 cost at 9^ ? What will be the value of a bill for £()0 at 8 ? What must be paid for a bill on London for £15 7s. (kl, at 10 ? What sum bterling will be equal to $100 Canadian, exchange 1. 3. 3. 4. 5. 6. What sum sterling should I receive for $5500 Canadian, exchange 9| ? T. The Government of Canada purchased the following sterling exchanges: For transmission to Messrs. (lilyn, Mills & Co., £50,000 at 8| and £10,fKX) at 8| ; for transmission to Messrs. Baring Brothers &Co., £20,000 at 9 and £40,000' at $4-846 per £1 stg. ; and for transmission to the Bank of Montreal, Londker in Boston a bill of exchange on Liverpool for £300 ])aying the broker $1457 "04 for it. At what quotation was the bill purchased, allowing | % for brokerage ? 16. I paid a broker $1511*90 for a bill of exchange (m Bremen for 6400 marks. At what (juotation was the bill purchased allowiiig I % for brokerage ? 17. I sold through a broker a bill of exchange on Manchestt;r for £600 and received $2912*35 as the net proceeds. At what rate • 0.0 35(10-1) + 2 352-35 317 .30^- 005 g^ g^ ,^^, 27( 1000-1) +5 83 _ 27583- 27 _ 27556 99900 99900 ~ 99900" m •27583= •27§|§=- 316 APPENDIX. Example 1. Prove that 5= 55= 555= 5555= '55= 555= '5555. f= -55 = -555 = -5555 = - , X. e. -5 =55 =-555 =-5555 = Also, f= |f=-5Sf^-55§§ = , i. e. '5 ='55 ='555 ="5565 = Example 2. Prove thab 237 = '2372= '23723= 237237 = -237237 I. e., -237 =2372 =23723 =237237 =-2372372= Also 23T = 2M24I=.23.ia3ia=-23I23I23_ . . _13r2.a7.2.37=. . Aiau, j,jj,, 999959— '^ff9990» ■^'^99l»9^9 999999999 i. e. , -237 = -237237 = 2372372 = -23723723 = - - = 237237237 = - - These two examples exhibit a property of fractions whose denominators are expressed by 9's only or by one or more 9's f'« •;•. wlt^A-tiJi: . ,..-.^- .t..''-.t.,^tJ'j'*,.---'i.'-f"^, l..r.>^J - x"^' LOGj^RITHMS. — TABLE II. 321 262 261 259 258 256 254 253 252 250 249 248 246 245 243 242 D. Ko 1 2 3 4 6237 6 6 7 8 D. 380 255273 5514 5755 5996 6477 6718 6958 7198 7439 241 181 7679 7918 8158 8398 8637 8877 9116 9365 9594 9833 239 182 260071 0310 0.48 0787 1026 1263 1501 1739 1976 2214 238 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4682 237 184 4818 5054 5290 5625 5761 6996 6232 6467 6702 6937 235 185 7172 7406 7641 7875 8110 8844 8578 8812 9046 9279 234 186 9513 9746 9980 »0213 *0446 »0679 •0912 *1144 *1377 ♦1609 233 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 232 188 4158 4381) 4620 4850 5081 5311 5542 5772 6002 6232 230 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8526 229 190 8754 8982 9211 9439 9667 9895*0123 *0351 *0578 *0806 228 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 li>3 6557. 6782 6007 6232 6456 6681 6905 7130 7354 7578 225 194 7802 8026 8249 8473 8G96 8920 9143 9366 9589 9812 223 195 290035 0257 0480 C702 0925 1147 1309 1591 1813 2034 222 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 221 197 4466 4687 4907 5127 5347 5567 6787 6007 6226 6446 220 198 6665 6884 7104 7323 7.'^42 7761 7979 8198 8416 8635 219 199 8853 9071 9289 9:07 9725 9943 »0161 *0378 *0595 *0813 218 200 301010 1247 1464 1681 1898 2114 2331 2647 2764 2980 217 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 202 5351 55G6 6781 5996 6211 6425 6639 6854 7068 7282 215 203 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 213 204 9030 9843 *0066 »0268 *0481 »0693 »0906 *ill8 *1330 *1642 212 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 211 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 207 5970 6180 6390 6599 6809 7U18 7227 7436 7646 7854 209 208 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 4282 4488 4694 4899 5105 5310 5516 5721 6926 6131 205 212 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 213 8380 8583 8787 8991 9194 9398 9601 9805 *0008 *0211 203 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 2438 2640 2842 3044 3246 3447 3649 3850 4061 4253 202 216 4454 4H55 4856 5057 5257 5453 5658 58-9 6059 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 218 8456 8656 8855 9054 9253 9451 9650 9849 *0047 ♦0246 199 219 340444 0642 0841 1039 3 1237 1435 1632 1830 2028 2225 198 No O 1 2 4 5 6 7 8 D. ■-1 :^^,.,\-:,>^?-v;i. ,^4./:Jt? i'^t'r.yi^v^'^ 322 LOQARITHMS.— TABLP: II. No 1 2 2817 3 4 6 6 7 8 D. 220 342423 2(»20 3014 3212 3409 3606 3802 3999 4196 197 221 4392 4589 4785 4981 6178 5374 5570 57(i6 5%2 6157 196 222 6353 5549 6744 6939 7135 73;io 7525 7720 7915 8110 195 223 8305 8500 8694 8889 9083 9278 9472 9(')66 9860 *0054 194 224 350248 0442 0636 0829 1023 1216 1410 1603 1796 1989 193 225 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 226 4108 4301 4493 4685 4876 5068 52()0 5452 5643 5834 192 227 6026 6217 6408 6599 6790 6981 7172 7;i63 7554 7744 191 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 229 9835 *0025 »0215 »0404 »0593 •0783 •0972 *1161 *1350 *1539 189 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 188 231 3612 3800 3988 4176 4363 4551 4739 4926 5113 530: 188 262 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 234 9216 9401 9587 9772 9958 *0143 *0328 *0513 *0698 *0883 185 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 184 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 237 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 *0030 181 240 380211 0392 0573 0754 0934 1115 1296 1476 1056 1837 181 241 2017 2197 2377 2557 2737 2917 3097 3277 345f> 3636 180 242 3815 3995 4174 4353 4533 4712 4891 5070 6249 5428 179 243 6606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 8989 178 245 9166 9343 9520 9698 9875 *0051 *0228 *0405 *0582 *0759 177 246 390935 m2 1288 1464 1641 1817 1993 2169 2345 2521 176 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 248 4452 4627 4802 4977 5152 5326 6501 5676 6850 6025 175 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 7940 8114 8287 8461 8634 8808 8981 9154 9328 9501 173 251 9674 9847 *0020 *0192 *036o *0538 *0711 *0883 *1056 *1228 173 252 401401 1573 1745 1917 2089 2261 2433 2605 2777 2949 172 253 3121 3292 8464 3635 3807 3978 4149 4320 4492 4663 171 254 4834 5005 6176 5346 5517 5688 5858 6029 6199 6370 171 255 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 256 824C 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 257 9933 *0102 *0271 *U440 *060« ^3777*0946 *1114 *1283 VAf>l 169 258 411620 1788 1956 2124 229;, 2461 2629 2796 2964 •{132 168 259 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 167 No 1 2 3 4 5 6 7 8 9 D. ■^^rV' . ^;' - ■■". ■ •.^.■.,-., :t'-''^- .'idA .v-i.:,-''-.v -*.-■?:' ARITHMETIC. 323 Ho 1 2 3 6474 4 5641 6 5808 e 7 8 D. 2f)0 414973 5140 5307 5974 6141 6308 6474 167 261 6641 681 »7 6973 7139 7306 7472 76;i8 7804 7970 8135 166 262 8301 8467 8633 8798 8964 6129 9295 9460 9626 9791 165 263 9956 *0121 ' »0286^ »04fil ' »0616 »078l^ »0945 *1110 ♦1276 ♦1439 166 264 421604 1768 1933 2097 2261 2426 2590 2754 2018 3082 164 265 3246 :«io 3574 3737 3901 4065 4228 4392 4655 4718 164 266 4882 5045 5208 5371 65;J4 5697 5860 6023 6186 6349 163 267 6511 6074 6836 6999 7161 7324 7486 7648 7811 7973 162 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 162 269 9752 9914 *0076 »0236 »0398 ''oseo »0720 •0881 •1042 ♦1203 161 270 431364 1625 1686 1846 2007 21C7 2328 2488 2649 2809 161 271 2969 3130 3290 3450 3610 3770 3930 4090 4249 4409 160 272 4569 4729 4888 6048 6207 5367 6626 5685 5844 6004 159 273 6163 6322 6481 6640 6799 6957 7116 7-275 7433 7692 159 274 7751 7909 8067 8226 8:^ 8542 8701 8859 9017 9175 168 275 9333 9491 9648 9806 9964 »0122 •^0279 •0437 •0594 ♦0762 158 276 140909 1066 1224 1381 1638 1695 1852 2009 2166 2323 157 277 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 278 4045 4201 4357 4613 4669 4825 4981 5137 5293 5449 lf6 279 5604 6760 6915 6071 6226 6382 6637 6692 6848 7003 165 280 7158 7313 7468 7623 7778 7933 8088 8242 8397 8652 155 281 8706 8861 9J16 9170 9324 9478 9633 9787 9941 ♦0095 164 282 450249 0403 0657 0711 0866 1018 1172 1326 1479 1633 164 283 1786 1940 2093 2247 2400 2553 2706 2859 3012 3166 153 284 3318 3471 3624 3777 3930 4082 4236 4387 4540 4692 153 285 4845 4997 6160 6302 5464 5606 6758 6910 6062 6214 152 286 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 287 7HJ2 8033 8184 8336 8487 8638 8789 8940 9091 9242 161 288 9bj^2 9543 9G94 9846 9996 *0146 •0296 •0447 •0697 ♦0743 151 289 46C898 1048 1198 1348 1499 1649 1799 1948 2098 2248 150 290 2398 2548 2697 2847 2997 3146 3296 3446. 3594 3744 150 291 3893 4042 4191 4340 4490 4639 4788 •1936 5086 6234 149 292 5383 5532 5680 5829 5977 6126 6*?74 6423 6571 6719 149 293 6868 7016 7164 7312 7460 7608 7756 7904 8062 8200 148 294 8347 8495 8643 8790 8938 9085 9233 9380 9527 9676 148 293 9822 9969 *01 16*0263 *0410 *0657 *0704 •0861 ♦0998 ♦1145 147 296 471292 1438 1585 1732 1878 2025 2171 2318 2464 2610 146 297 2756 2903 3049 3195 3341 3487 3633 3779 3926 4071 146 298 4216 4362 4608 4653 4799 4944 6090 6236 6381 6526 146 299 5671 6816 1 5962 2 6107 6252 6397 6642 6687 6832 6976 146 No 3 4 6 6 7 8 D. ■v: %vweu. 324 LOGARITHMS. — TABLE IL No 1 7266 2 7411 a 4 6 7844 e 7 8 D. 300 477121 7666 7700 7989 8133 8278 8122 146 301 8506 8711 8865 8999 9143 9287 94H1 9676 9719 9863 144 'M2 480007 0161 0294 0438 0582 0726 0869 1012 1166 1299 144 303 1443 1686 1729 1872 2016 2169 2.302 2446 2588 2731 143 304 2874 3016 3169 3302 .3446 3687 3730 3872 4015 4167 143 306 4300 4442 4586 4727 4869 6011 6153 6295 5437 6679 142 306 6721 5863 6005 6147 6289 6430 6572 6714 68.55 6997 142 307 7138 7280 7421 7663 7704 7845 7986 8127 8269 8410 141 308 8651 8692 8833 8974 9114 9255 9:<96 9537 666 6789 7898 8021 9126 9249 0351 0473 1572 1694 2790 2911 4004 4126 5215 5336 1862 1990 2117 2245 3136 3204 3;i91 3518 4407 4034 4601 4787 6674 5800 5927 605;i 6937 7063 7189 7316 8197 8322 8448 8574 9462 9578 9703 9829 0705 0830 0955 1080 1953 2078 2203 2327 3199 K32S 3447 3671 4440 4564 4688 4812 5078 5802 6925 6049 6913 70;i6 7159 7282 8144 8267 S3ii9 8512 9371 9494 9616 9739 0595 0717 0840 0962 1816 Wm 2060 2181 3033 3155 3270 3398 4247 4368 4489 4610 5457 5578 5699 5820 2372 2800 2627 3646 3772 3899 4914 6041 6167 6180 6306 6432 7441 7667 7693 8699 8826 8951 9954 *0079 *0204 1205 13;i0 1454 2452 2576 2701 3696 3820 3944 6423 6544 6664 6785 6905 7026 7627 7748 7868 7988 8108 8228 8829 8948 9068 9188 9;108 0428 *0026 *0140 •0265 *0385 *0604 *0624 1221 1340 1459 1578 1698 1817 2412 2531 ZZZQ 2700-2887 3000 3600 3718 3837 3950 4074 4192 4784 4903 5021 5130 5257 5376 5966 6084 6202 0320 6437 6555 7144 7262 7379 7497 7014 7732 8319 8436 8554 8G71 8788 8905 9491 9608 9725 9842 9959*0076 0660 0776 0893 1010 1126 1243 1825 1942 2058 2174 2291 2407 2988 3104 3220 3336 3402 3568 4147 4263 4379 4494 4610 4726 5303 5419 5534 5000 5765 5880 6457 0572 6687 6-*02 6917 7032 7607 7722 7836 7951 8066 8181 8754 8868 8983 9097 9212 9326 4936 6172 7405 8635 9861 1084 2303 3519 4731 5940 7146 8349 9548 *0740 1936 3126 4311 5494 6673 7849 9023 *0193 1309 2523 3684 4841 5996 7147 8295 9441 6060 6183 6296 6419 7529 7652 8758 8><81 9984 0106 1206 1328 122 2426 2547 122 3640 3702 121 4852 4973 121 6061 6182 121 72tJ7 7387 120 841)9 8589 120 9'J67 9787 120 *0863 *0982 119 2055 2174 119 3244 3362 119 4429 4548 119 5612 5730 118 0791 0909 118 7967 S'-'U 118 9140 9257 117 *0309 *0420 117 1476 1592 117 2639 2755 116 3800 3915 116 4957 5072 116 6111 6226 115 7202 7377 115 8410 8525 115 9555 9669 114 128 127 127 126 128 126 126 126 125 124 124 124 123 123 123 326 LOGARITHMS.— TABLE 11. No 1 2 **12 3 *0126 4 *0241 6 6 7 8 9 D. 380 579784 9898 *0355 *0469 *0583 *0697 *0811 114 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 114 382 5063 2177 2291 24Q4 2518 2631 2745 2868 2972 3085 114 383 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 113 394 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 385 6461 5574 5686 6799 5912 6024 6137 6250 6362 6475 113 286 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 112 387 7711 7823 7935 8047 8160 8272 8384 8496 8008 8720 112 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 389 9950 **61 *0173 *0284 *0396 *0507 *0619 *0730 *0842 *0953 112 390 591065 1176 1287 1399 1510 1621 1782 1843 1955 2066 111 391 2177 2288 2399 2610 2621 2732 2843 2964 3064 3175 111 392 3286 339V 3508 3618 3729 3840 3950 4061 4171 4282 111 393 4393 4503 4614 4724 4834 4945 5055 6165 5276 5386 110 394 5496 6606 5717 5827 5937 6047 6167 6267 6377 6487 110 395 6597 6707 6817 6927 7037 7146 7266 7366 7476 7586 110 396 7695 7805 7914 8024 8134 8243 8363 8462 8572 8681 110 397 8791 8900 9009 9119 9228 9337 9446 9556 9666 9774 109 398 9883 9992 »0101 »0210 *0319 *0428 •0537 *0646 *0755 *0864 109 399 600973 1082 1191 1299 1408 1617 1625 1734 1843 1951 109 400 2060 2169 2277 2386 2494 2603 2711 2819 2928 3036 108 401 3144 3253 3361 3469 3577 3686 3794 3902 4010 4118 108 402 4226 4334 4442 4650 4658 4766 4874 4982 5089 5197 108 403 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 108 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 107 405 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 406 8526 8633 8740 8847 8964 9061 9167 9274 9381 9488 107 407 9594 9701 9808 9914 **21 »0128 *0234 *0341 *0447 *0554 107 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 iei7 106 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 106 411 3842 3947 4053 4169 4264 4370 4475 4681 4686 4792 106 412 4897 5003 6108 6213 5319 5424 6529 5634 5740 5845 105 4i;^ 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 105 414 7000 7105 7210 7315 7420 7526 7629 7734 7839 7943 105 415 8048 8163 8257 8362 8466 8571 8C76 8780 8884 8989 105 416 9093 9198 9302 9406 9511 9615 9719 9824 9928 0032 104 417 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 104 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 104 419 2214 2318 1 2421 2 2525 3 2628 4 2732 2835 2939 3042 3146 104 No 6 6 7 8 9 D. 51 ARITHMETIC. • 527 No O 1 3353 2 3 4 3663 5 3766 6 7 3973 8 9 D. 420 623249 3456 3559 3869 4076 4179 103 421 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 103 422 5312 5415 5518 5621 5724 5827 5929 6032 6135 6288 103 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 103 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 102 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 102 426 9410 9512 9613 9715 9817 9919 **21 *0123 *0224 *0326 102 427 630428 0530 0631 0733 0835 0936 1038 1139 1241 1342 102 428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2366 101 429 2457 2559 2660 2761 2862 2963 3004 3165 3266 3367 101 430 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 100 431 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 100 432 5484 5584 5685 5785 5886 5986 6087 6187 6287 6388 100 433 6488 6588 C688 6789 6889 6989 7089 7189 7290 7390 100 434 '7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 99 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 99 436 9486 9586 9686 9785 9885 9984 **84 *0183 *0283 *0382 99 437 640481 05S1 0680 0779 0879 0978 1077 1177 1276 1375 99 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 439 2465 2563 2662 2761 2860. 2959 3058 3156 3255 3354 99 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 98 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 442 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 98 443 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 444 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 98 445 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 446 9335 9432 9530 9627 9724 9821 9919 **16 *0113 *0210 97 447 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 97 448 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 97 449 2246 2343 2440 2536 2623 2730 2826 2923 3019 3116 97 450 3213 3309 3405 3502 3598 3695 3791 3888 3984 4080 96 451 4177 4273 43C9 4465 4562 4658 4754 4850 4946 5042 96 452 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 9o 453 0098 6194 6290 6386 6482 6577 6673 6769 6864 6S60 96 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 455 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 96 456 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 95 457 9916 **11 *0106 *0201 *0296 *0391 *0486 *0581 *0676 *0771 95 458 660865 0900 1055 1150 1245 1339 1434 1529 1623 1718 96 45S 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 95 No 1 2 3 4 6 6 7 . L 8 9 D. 328 LOGARITHMS.— TABLE 11. 1 No 12 3 4 6 6 7 8 94' 1 460 66^2758 2852 2947 3041 3135 3230 3324 3418 3512 3607 ■ 461 37.11 3795 3889 3983 4078 4172 4206 4360 4454 4548 94 K 462 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 H 46:5 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 94 H 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 94 I' 465 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 93 ^K 466 8386 8479 8572 8665 8759 8852 8945 9038 9131 9224 93 ■B 467 9317 9410 9503 9596 9689 9782 9875 9967 **60 *0153 93 HA 468 670246 0339 0431 0524 0617 0710 0802 0895 0988 1080 93 ^K 469 1173 1266 1358 1451 1543 1636 1728 1821 1913 2005 93 F 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 92 |i 471 3021 3113 3205 3297 339U 3482 3.i74 3666 3758 3850 92 472 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 ^M 473 4861 4953 50-15 5137 5228 5320 5412 5503 5595 5687 92 P 474 5778 5870 6962 6033 3145 6236 6328 6419 6511 6602 9J p. 475 6694 6785 6876 8968 V059 7151 7242 7333 7424 7516 91 476 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 91 477 8518 86C9 8700 8791 8882 8973 9064 9155 9246 9337 91 «5{r 478 9428 9519 9610 9700 9791 9882 9973 **(}3 *0154 *0245 91 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 91 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 90 481 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 B 481 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 90 485 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 89 486 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 487 7529 7618 770 r 7796 7886 7975 8064 8153 8242 8331 89 K 488 8420 8509 8598 8(^87 8776 8865 8953 9042 9131 9220 89 489 9309 9398 9486 9575 9664 9753 9841 9930 **19 *oior 89 490 690196 0285 0373 0462 0550 0639 0728 0816 0905 0993 89 491 1081 1170 1258 1347 1435 1524 1612 1706 1789 1877 88 ^^v. 492 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 88 493 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 88 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 Ej-""- 495 4605 4093 4781 4868 4956 5044 513j 6219 5307 5394 88 49b 5182 5569 5ti57 5744 5832 5919 6007 6094 6182 6269 87 497 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 87 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 . 499 8101 8188 8275 8362 8449 8535 8622 8709 8796 8883 87 D. No ° 12 3 4 5 6 7 8 9 : 5 ARITHMETIC. 829 1 No 1 9057 2 3 4 6 6 7 8 9 D. 50; 698970 9144 9231 9317 9404 9491 9578 9664 9751 87 601 Ji838 9924 **11 **98 *0184 *0271 *0358 *0444 *0531 *0617 87 60;^ 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 86 503 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 86 604 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 .86 505 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 86 506 4151 423t> 4322 4408 4494 4579 4665 4751 4837 4922 b07 5008 5094 5179 5265 5350 5430 5522 5607 5693 5778 86 508 5864 5949 60a5 6120 6206 6291 6376 6462 6547 6632 85 509 6718 6803 6888 6974 7050 7144 7229 7315 7400 7485 85 510 7570 7655 7740 7826 7911 7996 8081 8166 8251 8336 85 511 8421 8506 8591 8676 .8761 8846 8931 9015 9100 9185 85 512 9270 9355 9440 9524 9609 .^694 9779 9.«63 9948 (033 85 513 710117 0202 0287 0371 0456 0540 0625 0710 0794 0879 85 514 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 84 1807 1892 1976 2060 2144 2229 2313 -2397 2481 2565 84 ■ N : 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 84 017 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 84 618 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 519 • 6167 5251 5335 5418 5502 5586 5669 5753 5836 5920 84 520 6003 6087 6170 6254 '6337 6421 6504 6588 6671 6754 83 1 521 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 1 522 7671 7?54 7837 7920 8003 8086 8169 8253 8336 8419 S3 I 523 8502 8585 86ij8 8751 fc834 8917 iJOOO 9083 9165 9248 83 1 524 9331 9414 9497 9580 9663 9745 9828 9911 9994 **77 -83 i 625 720159 0242 0326 0407 0490 0573 0655 0738 0821 0903 83 1 520 C986 1068 1151 1233 1316 1398 1481 1563 1646 1728 82 1 527 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 528 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 . ! 529 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 1 530 4276 4358 4440 4522 4604 4685 4767 4849 4931 5013 82 I 531 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 82 532 5912 5993 6075 6156 6238 6320 6401 6483 6564 0646 82 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 534 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 1 635 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 1 536 9165 9246 9327 9408 9489 9)70 9651 9732 9813 9893 81 j ' 537 9974 **55 »0136 »0217 *0298 *0378 *0459 *0540 *0621 *0702 81 1 538 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 81 j 539 1589 1669 1750 1830 3 1911 1991 2072 2152 2233 2313 81 D. ^ 1 No 1 2 4 5 6 7 8 9 S30 LOGARITHMS. — TABLE IL Nr 1 2 3 4 6 6 7 8 D. 510 73239'' :474 2555 263d 2715 2796 2876 2956 3037 3117 80 541 31!^ 3278 3358 3438 3518 3598 3679 3759 3839 3919 80 542 39t/^ 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 643 4800 4880 49C0 5x.i^ 5120 5200 5279 5359 5439 5519 80 544 5599 5679 5769 5838 5918 5998 6078 6157 6237 6317 80 545 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 546 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 547 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 79 548 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 549 9572 9651 9731 9810 9889 9968 **47 *0126 *0205 *0284 79 550 7403fi3 0442 0521 0600 0678 0757 0836 0915 0994 1073 79 551 1152 1230 1309 1388 1467 1546 1624 1733 1782 1860 79 552 19.i9 2018 2096 2175 2254 2332 2411 2489 > , 2568 2647 79 553 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78 554 3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 557 5855 6933 6011 6089 6167 6245 6323 6401 6479 6556 78 558 6634 6712 6790 6868 6945 7023 7101 7179 72r>6 7334 78 559 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 77 561 8963 9040 9118 9195 927« 9350 9427 9504 9582 9659 77 562 9736 9814 9891 9968 **45 »0123 *0200 *0277 *0;^54 *0431 77 563 750508 0586 0663 0740 0817 0894 0971 1048 1125 1202 77 564 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 77 565 2048 2125 2202 2279 2356 2433 2509 *2586 2663 2740 77 566 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 567 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 568 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 569 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 5875 5951 6027 6103 6180 6256 6">2 6408 6484 6560 76 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 76 573 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 574 8912 8988 9063 9139 9214 9290 9366 9441 9517 9592 76 575 9668 9743 9819 9894 9970 ##45 *0121 *0196 *0272 *0347 75 576 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 75 577 1176 1251 1326 1402 1477 1552 1627 1702 1778 18)3 75 578 1928 2003 2C7S 2153 2228 2303 2378 2453 2529 2604 75 579 2679 2754 2829 2904 2978 3053 3128 3203 3278 3353 75 No 1 2 3 4 6 6 7 8 9 V. D. 80 80 80 80 80 80 79 79 79 79 79 79 79 78 78 78 78 78 78 78 77 77 77 77 77 77 77 77 70 76 76 76 76 76 76 75 75 75 75 75 v. -» i: h-' ARITHMETIC. 1 < 331 No 1 3503 2 3578 3 3653 4 5 6 7 8 9 D. 580 763428 3727 3802 3877 3952 4027 4101 75 581 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 75 582 4923 4998 5072 5147 5221 5296 6370 6445 5520 5594 76 583 5669 5743 5818 5892 5966 6041 6115 6190 62(54 6338 74 584 6413 6487 0562 6636 6710 6785 6859 6933 7007 7082 74 585 7156 7230 7304, 7379 7453 7527 7601 7675 7749 7823 74 586 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 74 587 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 588 9377 9451 9525 9599 9673 9H6 9820 9894 9968 0042 74 589 770115 0189 0263 0336 0410 0484 0557 0631 0706 0778 74 590 0852 0926 099: 1073 1146 1220 1293 1367 1440 1614 74 591 1587 1661 1734 1808 1881 1955 "2028 2102 2175 2248 73 592 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 594 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 4517 4590 4663 4736 4809 4882 4955 6028 6100 6173 73 596 5246 5319 5392 5465 5538 5610 5683 6756 5829 5902 73 597 597^1 6047 6120 6193 6265 6338 6''11 6483 6556 6629 73 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 7427 7499 7572 7644 7717 7789 7862 7934 SOOii 8079 72 600 8151 8224 8296 8368 8441 8513 8585 8653 8730 8802 72 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 602 9696 9669 9741 9813 9885 9957 •C029 *0101 *0173 *0245 72 603 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 72 604 1037 1109 1181 1253 1324 1396 2114 1468 1540 1612 1684 72 605 1765 1827 1899 1971 2042 2186 2258 2329 2401 72 606 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 607 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 71 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 6259 71 610 5330 6401 5472 5543 5615 5686 5757 6828 5899 5970 71 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 613 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 71 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9610 71 616 9581 9651 9722 9792 9863 9933 »»»4 **74 *0144 *0215 70 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 619 1691 1761 1 1831 1901 1971 2041 2111 2181 2252 2322 70 No 2 3 4 5 6 7 8 D. 8d2 LOGARITHMS. — TABLE IL No 12 3 4 6 6 7 8 9 D. 620 792392 2402 2532 2602 2072 2742 2812 2882 2952 8022 70 621 3092 3' • 3231 3301 3371 3441 3511 3581 3051 3721 70 622 3790 38 39;«) 4000 4070 4139 4209 4279 4319 4418 70 623 4488 4* 4627 4097 4707 4«36 4906 4976 6045 5115 70 624 5185 SHo-i 5324 5393 5163 6532 6602 6672 5741 6811 70 625 5880 5949 6019 6088 6158 6227 6297. 6306 6436 6505 69 026 6574 6044 6713 6782 6852 6921 6990 7060 7129 7198 09 627 7268 7337 7406 7475 7545 7614 768;i 7752 7821 7890 09 628 7960 8029 8098 8167 82.«i 8305 8374 8443 8513 8582 09 629 8651 8720 8789 8858 81)27 8996 9065 9134 9203 9272 69 630 9341 9409 9478 9547 9616 9685 9754 9823 9892 9961 69 631 800029 0098 0107 0236 0305 0373 0442 0611 0580 0648 09 632 0717 0780 085 1 0923 0992 T061 1129 1198 1260 1335 09 633 1404 1472 1541 1009 1678 1747 1815 1884 1952 2021 69 634 2089 2168 2226 2295 2303 2432 2500 2668 2037 2705 09 635 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 636 8457 3525 3594 3602 3730 S798 3867 3935 4003 4071 68 637 4139 4208 4276 4344 4412 4480 4548 4(>16 4685 4753 08 638 4821 4889 4957 5025 5093 51(51 6229 52^)7 6365 5433 68 639 5501 6569 5637 57o5 5773 5841 5908 5976 6044 6112 08 640 6180 6248 6316 6384 6451 6519 6587 6665 6723 6790 89 641 6858 6926 6994 7061 7129 7197 7264 7332 7400 7407 89 642 7535 7003 7070 7738 7800 7873 7941 8008 8076 8143 08 643 8211 8279 8340 8414 8481 8549 8616 8684 8751 8818 07 644 8886 8953 9021 9088 9150 9223 9290 9358 9425 9492 C7 645 9560 9027 9694 9702 ^829 9896 9964 **31 **9n *0165 67 (346 810233 0300 0367 0434 OoUl 0509 0636 0703 0770 0837 67 ' (i9 1 69 69 69 69 69 69 68 68 68 68 68 89 89 68 67 67 67 67 67 67 67 67 67 67 66 66 66 66 66 66 66 D. Sb^ No 660 661 6<)2 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 67i> 680 681 662 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 No 6 819544 82020 L 0858 1514 2168 2822 3474 4126 4776 6426 6076 6723 73i)9 8015 8660 9304 9947 830589 1230 1870 2609 3147 3784 4421 6056 6691 6324 6957 7588 8219 8849 9478 840106 0733 ia59 1985 2609 3233 38:^5 4477 9010 9076 9741 0307 9873 9939 02()7 0;*33 0399 0464 0530 0595 0924 0989 1055 1120 1186 1251 1579 1615 1710 1775 1841 11>06 2233 2299 23i)4 2430 2495 2560 2887 2952 3018 ;1083 3148 3213 3539 3605 3670 3735 3800 ;^65 4191 4256 4321 4386 4451 4516 4841 4906 4971 5036 5101 6166 6491 6556 5621 6686 6751 5815 6140 6204 6269 6334 6399 6404 6787 685'2 6917 6981 7046 7111 7434 7499 75G3 7628 7(592 7757 8080 8144 8209 8273 8333 8402 8724 8789 8853 8918 8982 9046 9368 9432 9497 9561 9626 9090 ♦*11 **75 *0139 *0204 *02(i8 *0332 0653 0717 0781 0815 0909 0973 1294 1358 1422 1486 1550 1614 1934 1998 2062 2126 2189 2253 2573 2637 2700 2764 2828 2892 3211 3275 3338 3402 3466 353) 3848 3912 3975 4039 4103 4106 4484 4548 4611 4675 4739 4802 5120 5183 5?17 5310 6373 6437 5754 5817 6881 5944 6007 6071 6387 6451 65.4 6577 6641 6704 7020 7083 7146 7210 7273 7336 7652 7715 7778 7841 7904 7967 8282 8345 8408 8471 8534 8597 8912 8975 9038 9101 9164 9227 9541 9604 %67 9729 9792 9855 0169 0232 0294 0357 0420 0482 0796 08^9 0921 0984 1046 1109 1422 1485 1547 1610 1672 1735 2047 2110 2172 2235 2297 2360 2672 2734 2796 2859 2921 2983 3295 3357 3420 3482 3544 3606 3918 3980 4042 4104 4166 4229 4539 4601 4664 4726 4783 4^50 6 0661 1317 1972 2626 3279 3930 4581 5231 6880 6528 7175 7821 8467 9111 9754 *0396 1037 1678 2317 2956 3593 4230 4866 6500 6134 67()7 7399 8030 8()60 9918 0545 1172 1797 2422 3046 3669 4291 4912 8 **70 *0136 0727 0792 l:W2 1448 2037 2103 2691 2756 3344 3409 39% 40(>1 4646 4711 6296 5361 6945 6010 6593 6658 7240 7365 7886 7951 8531 8595 9175 9239 9818 9882 *0460 *0525 1102 1166 1742 1806 2381 2445 3020 3(}57 4294 4929 6564 6197 6830 7462 8093 8723 9362 9981 0608 1234 1860 2484 3108 3731 4353 4974 3083 3721 4357 4993 6627 6261 6894 7525 8156 8786 9415 **43 0671 1297 1922 2547 3170 3793 4416 6036 D. 66 66 m 65 66 65 65 65 65 66 66 65 65 64 64 64 64 64 64 64 64 64 64 64 63 63 63 63 63 68 63 63 63 63 63 62 62 62 62 62 8 9 D ■'T>.':T»>'f'' lt*v" 'i>"a> . A ;-.■ -V 334 I LOGARITHMS. — TABLE IL > No 12 3 4 6 6 7 8 9 D. B I 700 846098 5160 52J2 6284 5346 5408 5470 5532 6594 6656 62 ■ A ■ ■ 701 6718 6780 5842 6904 5966 6028 6090 6151 6213 6276 62 ■ '■ ■■> 702 6337 6399 6461 6523 65&6 6646 6708 6770 6832 6894 62 ■ • if," ;o3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7MI 62 ■ 704 7673 7634 7696 7758 7819 7881 7943 8004 8066 £128 62 m •■ 1 706 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 1 1 ;.. * "* •' 706 8805 8866 8928 89^9 9051 9112 9174 9235 9297 9358 61 M Iv. ■ •707 9419 9481 9542 9604 9()05 9726 9788 9849 9911 9972 61 W ^4,;- I 708 850033 0095 0166 0217 0279 0340 0401 0462 0524 0585 61 B 1 ¥"-■ 709 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 ■ 1 1 i-'i; , 710 1258 1?20 1381 1442 1503 1564 1626 1686 1747 1809 61 » t 1 -'A ' 711 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 ■ ■ -4, 712 2480 2641 2602 2663 2724 2785 2846 2907 2968 3029 61 ■ f ». ■' .^- 713 3090 3160 3211 3272 3;J33 3394 3455 3616 3577 3637 61 ■ f 714 S698 3769 3820 3881 3941 4002 4063 4124 41&r 4245 61 ■ r 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 48r2 61 I 9 ™ . 716 4913 4974 5034 5095 5156 5216 5277 5337 6398 54139 61 ■ 9 ,".'■ / 717 5519 6580 5640 6701 5761 5822 5882 5943 6003 6064 61 ■ 718 6124 6185 6246 6306 6366 6427 6487 6548 6608 6668 60 ' T-.:":, ■ 719 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 ■*■'(' - . .. i'. . 720 7332 7393 7453 .7513 7574 7634 7694 7755 7815 7875 60 ^f .- ' 721 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 . ■^ ," 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 :r^'"'"- 723 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 724 9739 9799 9859 9918 9978 **38 **98 *0158 *0218 *0278 60 "■."r" - ' 725 860338 0398 0458 0618 0578 0637 0697 0757 0817 0877 60 '. '■ 726 0937 0996 1056 1116 1176 1236 1295 1355. 1415 1475 60 ■\-.v' 727 1634 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 ..■'V ■.',- 728 2131 2191 2251 2310 2 370 2430 2489 2549 2608 2668 60 ■ >r' ■ ' ■ 729 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 ;>'. ■ 730 5323 3382 3442 8501 3561 3620 3680 3739 3799 3868 59 1 tV 731 3917 397V 4036 4096 4155 4214 4274 4333 4392 4452 59 1 ■■\v 732 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 ■ .^V- '733 5104 5163 5222 5282 5341 5400 5459 6519 5578 66;i7 69 ■ ] 734 5696 6755 5814 6874 59a3 5992 6051 6110 6169 6228 69 ■ ; "'': •' 735 6287 6340 6405 6465 6524 6583 6642 6701 6760 6819 59 m ^ :v- 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 K 'If:. 737 7467 752(i 7585 7644 7703 7762 7821 7880 7939 7998 69 ■ f !'". .' ^^ 738 8056 8115 8174 8233 8292 8350 8409 8^68 8527 8586 59 m 739 No 8644 8703 8762 8821 8879 8938 8997 8066 9114 9173 59 M D. 1 12 3 4 5 6 7 8 ] ! Yi^iky ]'-^^ ^'liV'^^-'^iW ' v^;'^. ^■^'- :J -^.'^^ :>K . ,. I .kfJit'xx^ -ii't Jlif .'^^'■V . '^V';' i^^- t^B ARITHMETIC. 335 62 62 62 62 62 61 61 61 61 61 61 61 61 60 60 60 60 60 60 60 60 60 60 60 60 69 69 69 59 59 69 59 59 59 69 No 1 2 3 9408 4 9466 6 6 7 8 9 D. 740 869232 9290 9349 9525 9584 9642 »701 9760 69 741 9818 9877 9935 9994 **53^ *0111 ' »0170 ♦0228 ♦0287 ♦0345 69 742 870404 046^ 0521 0579 0638 0696 0755 0813 0872 0930 58 743 0989 104/ 1106 1164 1223 1281 1339 1398 1456 1516 68 744 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 68 745 2156 2215 2273 2331 23? 9 2448 2506 2664 2622 2681 68 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 58 747 3321 3379 3437 3495 3553 3611 3669 3727 3785 8844 58 748 d902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 749 4482 4540 4698 4656 4714 4772 4830 4888 4946 6003 68 750 . 5061 5119 5177 5235 5293 5351 5409 6466 5524 6582 68 751 6640 5698 5756 5813 5871 6929 5987 6045 6102 6160 58 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 68 753 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 58 764 7071 7429 7487 7644 7602 7659 7717 7774 7832 7889 58 755 7947 8004 8062 8119 8177 8234 8292 c349 8407 8464 67 756 8622 8579 8637 8694 8752 8809 8866 8924 8981 9039 67 757 9096 9153 9211 92(i8 9325 9383 9440 9497 9555 9612 67 758 9669 9726 9784 9841 9898 9956 **13 **70 ♦0127 ♦0185 57 759 880242 0299 03ft6 0413 0471 0528 0585 0642 0699 0756 67 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 57 761 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 57 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 763 2525 2581 2638 2695 2752 28o9 2866 2923 2980 3037 57 764 3093 3150 3207 3264 3321 3377 3iM 3491 3548 3605 57 765 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 67 76fe 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 67 767 4796 4852 4909 4965 5022 5078 ol35 6192 5248 6305 57 768 5361 6418 5474 5531 5587 5644 5700 6757 6813 5870 67 769 5926 5983 6039 6096 6152 6209 6266 6321 6378 6434 66 770 6491 6547 6604 6660 6716 6773 6829 6885 6942 6998 66 771 7054 7111 7167 7223 7280 7336 7392 7449 7505 7661 56 772 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 66 773 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 66 774 8741 8797 8a53 8909 8965 9021 9077 9134 9190 9246 56 775 9302 9368 9414 9470 9526 9582 %38 9694 9750 9806 56 776 9862 9918 9974 **30 **86 *0141 *0197 *0253 ♦9309 ♦0365 66 777 890421 0477 0533 0589 0645 0700 0756 0812 0868 0924 66 778 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 56 779 1537 1593 1649 1705 1760 1816 i8za 1928 1983 2039 66 No 1 2 3 4 5 6 7 8 D. ''I ■•■■«!;.. - ■•[■ ■:■& '.^r rifi 'M ~>i [ 336 LOGARITHMS. — TABLE IL No 1 2 3 4 6 6 7 8 D. '^ 780 892095 2150 2206 2262 2317 2373 2129 2484 2540 2595 66 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 56 782 3207 3262 3318 3373 3429 M8i 3540 3595 ;W51 3706 66 783 3702 ;«i7 3873 3928 3984 4039 4094 4150 4205 4261 55 ;. 784 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 65 ';; 7a5 4870 4925 498J 5036 6091 5146 5201 6257 5312 5367 65 r 786 5423 5178 6533 6588 5644 5699 5754 5809 5864 6920 65 787 5975 6030 60H5 6140 6195 Ii25l 6306 6361 6416 6471 65 \ 788 6526 6581 6636 6692 G747 6802 6857 6912 6967 7022 65 789 70/7 7132 7187 7242 7297 7352 7407 7462 7517 7672 66 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 66 791 8176 8231 8286 8:i41 8396 8451 8506 8561 8615 8670 65 792 8725 8780 8835 8890 8914 8999 9054 9109 9164 9218 56 793 9273 9328 9383 9437 9192 9547 9602 9656 9711 9766 65 794 9821 9875 9930 9985 **39 **94^ »0149 *0203 *0258 *0312 65 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 55 796 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 65 797 1458 1513 1567 1622 1076 1781 1785 1840 1894 1948 64 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 64 . 79y 2547 2601 2656 2710 2764 2818 2873 2927 2981 3036 64 800 3090 3144 3199 3253 3307 3361 3416 3470 3524 3678 54 801 3633 3687 3741 3795 3819 3904 3958 4012 4066 4120 54 1 «02 4174 4229 4283 4337 4S91 4415 449i^ 4553 4607 4601 54 803 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 1 5256 5310 RJ64 5418 5472 5526 5580 5034 5688 5742 64 i 805 5796 5850 5904 5958 6012 60G6 6119 6173 6227 6281 64 80G ti335 6389 6443 6497 655 L 00l>4 6658 6712 6766 6820 54 807 6874 6927 6981 7035 7089 7143 7196 72iO 7304 735S 54 } 808 7411 74U5 7519 7573 7626 7080 7731 7787 7841 7895 54 1 809 1 7949 8002 8056 8110 8103 8217 8270 8324 8378 8431 64 !. 810 8485 8539 8592 8646 8699 8753 8807 8860 8914 8967 64 ^ «ll 9U21 9J74 9 J 28 91«1 9235 9289 9342 9396 9149 9503 54 81. 9556 9610 9003 9716 9770 9823 9877 99 iO 9984 **37 53 8L3 910091 0144 0197 0251 Om 0358 0111 0464 0518 0571 53 814 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 63 816 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 63 816 Ids^ 1743 1797 1850 1903 19a0 2O09 2063 2116 2169 53 817 . 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 53 818 2753 2S06 2859 2913 2966 8019 3072 3125 3178 3231 63 819 3284 8337 3390 3443 3496 3549 3602 3655 3708 3761 63 1 "■ No 1 2 3 4 5 6 7 8 D. N 8: « -ia»'^- •j;:<'J?L i:^;^.;**. ■f%'^:j.s:y::. <■■ ■■■...-:■_ •■;'V(^-.,-, ■/■■■---i7.,f/iS- ARITHMETIC. 337 66 66 66 65 65 65 65 65 65 66 65 55 55 55 55 55 55 54 64 64 54 54 54 54 54 54 54 54 54 54 54 54 53 53 53 No 913814 1 2 3 3973 4 4026 6 6 7 8 D. 820 3867 3920 4079 4132 4184 4237 4290 63 821 4343 4396 4449 4502 4555 4608 4(;60 4713 4766 4819 63 822 4872 4926 4977 5030 5083 5136 6189 6241 6294 6347 63 823 6-^00 64 ■>3 5505 6558 5611 6»>64 5716 6769 6822 6876 63 824 6927 6980 6033 6085 61 .« 6191 6243 6296 6349 6401 63 826 6454 6607 6659 6612 66H4 6717 6770 6822 6875 6927 63 826 6980 7033 7086 7138 7190 7243 7296 7348 7400 7453 63 827 7506 7558 7611 7663 7716 7768 7820 7873 7926 7978 62 828 80S0 8083 8136 8188 8240 8293 8345 8397 84rt0 85r)2 62 829 8565 8607 8659 8712 8764 8816 8869 8921 8973 9026 62 830 9078 9130 9183 9236 9287 9340 9392 9444 9496 9649 62 831 9601 9653 9706 9758 9810 9862 9914 9967 ♦*19 **71 62 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 62 833 0615 0697 0749 0801 0853 0(K)6 0958 1010 1062 1114 62 834 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 62 835 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 62 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 62 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 62 838 3244 3296 3343 3399 3451 3503 3656 3607 3658 3710 62 839 3762 3814 38f*6 3917 3969 4021 4072 4124 4176 4228 62 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 62 841 4796 4848 4899 4951 50O3 5054 5106 6157 6209 5261 62 842 5312 5364 5415 5467 5518 5570 6621 5673 6725 5776 62 843 6828 5879 5931 5982 6034 6085 6137 6188 6240 6291 61 814 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 61 845 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 61 846 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 61 847 7883 793) 7986 8037 8088 8140 8191 8242 8293 8345 61 848 8390 8447 8498 8519 8601 8652 8703 8754 8805 8857 61 849 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 51 850 9419 9470 9521 9572 9623 %74 9725 9776 9827 9879 51 851 9930 9981 ♦*32 **83 »0134 -0185 »0236 *0287 *0338 *0389 61 852 900440 049 i 0542 0592 0643 0694 0745 0796 0847 0898 51 853 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 61 854 1458 1609 1560 1610 1661 1712 1763 1814 1865 1916 51 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 61 856 2474 2524 2576 2626 26''7 2727 2778 2829 2879 2930 61 857 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 51 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 61 859 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 61 No 1 2 3 4 5 3 7 8 p. :''<»■•- 38$ LOOAUITHMS— TAIJLE II. No 1 4549 2 3 4 6 e 7 8 D. 860 934408 4699 46-)0 4700 4751 4801 4852 4902 4953 60 801 6003 5054 6104 6154 6205 5255 6306 53)6 6106 6457 50 84i2 5607 ft5r>8 5ii08 f.();-)8 5709 6759 6809 5800 6910 6960 60 963 6011 6061 Olll 0162 6212 6262 6313 6363 6413 6463 50 864 6514 6564 6614 6605 6715 6706 6816 6865 6916 6966 50 8fi5 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 60 mi 7518 75()8 7618 76()8 7718 7769 7819 7809 7919 7909 50 867 8019 8069 8119 8109 8219 82«;9 8;J20 8370 8420 8470 60 868 8520 8570 8rt20 8670 8720 8770 8820 8870 8920 8970 50 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 r>o 870 9513 9569 9619 9669 9719 9769 9819 9869 9918 9908 50 871 940018 (i0()8 0118 0168 0218 02(57 0317 0307 0117 04(17 50 872 0516 05GG 0616 0666 0716 0705 0815 0805 0915 0964 50 873 1014 1064 1114 1163 1213 12()3 1313 1362 1412 1402 50 874 1511 1561 1 1611 1660 1710 1760 1809 1859 1909 1958 50 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 50 876 2501 2554 2603 2(i53 2702 2752 2801 2851 2U01 2950 50 877 :muo 3049 3099 3148 3198 3247 3297 3346 3396 344) 49 878 3495 3544 3593 3043 36P2 3742 3791 3841 3890 3939 49 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 49 880 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 49 881 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 882 54G9 5518 5507 5616 5605 5715 5764 5813 5802 5912 49 88;i 5961 (iOlO 6059 6108 6157 6207 6256 6305 6354 6403 49 884 6452 6501 0551 6600 6649 6098 6747 6796 6845 6894 49 88.') 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 88G 7434 7403 7532 7581 7030 7679 7728 7777 7b26 7875 49 887 7924 7973 8022 8070 8119 8108 8217 8206 8315 8364 49 888 8413 8462 8511 8500 8609 8657 8706 8755 8804 8853 49 889 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 49 891 9878 99:^6 9976 **24 **73^ *0121 ' *0170 *0219 *0267 *0316 49 892 9503OO 0414 0402 0511 0560 0008 0657 0706 0754 0803 49 893 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 49 894 1338 1386 1435 1483 1532 1580 1029 1677 1726 1775 49 896 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 48 896 2308 2356 2405 2453 2502 2550 2599 2647 2096 2744 48 897 2792 2841 2889 2938 2986 3034 3083 3131 31 SO 3228 48 898 3276 3325 3373 3421 3470 3518 3566 3015 36(J3 3711 48 899 3760 i8L8 1 3856 3905 3953 4001 4049 4098 4146 4194 48 No O 2 3 4 6 6 7 8 D. ARITHMETIC. 339 No 1 4291 2 4339 3 4387 4 6 6 4532 7 8 D. 900 954243 4433 4484 4580 4628 4677 48 901 4725 4773 4821 4869 4918 49()6 6014 6062 6110 6158 48 902 6207 6253 6303 6361 6399 5447 6495 6543 6592 6640 48 903 6688 blM 6784 6832 5880 6928 6976 6024 6072 6120 48 904 6168 6216 6266 6313 6361 6409 6467 6606 6653 6601 48 906 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 908 .)375 9'?3 9471 48 911 9518 9u(i6 9614 9661 9709 9757 9864 v>852 9;>U0 994 48 912 9995 **42 **90 ♦0138 *0185 *0233 *0280 *0b28 ♦0376 ♦042 48 913 960471 0518 0666 0613 0661 0709 (.756 0804 085] {■'J.' 48 914 0946 0994 1041 1089 1136 1184 1231 1279 1326 IJli 47 915 1421 1469 1616 1563 1611 1658 1706 1753 180 1848 47 916 1895 1943 1990 2038 2085 2132 2180 2227 2276 2322 47 917 2309 2417 24<)4 2511 2559 2606 2653 2701 2748 27J»5 47 918 2843 2890 2937 2986 3032 3079 3126 3174 3221 3268 47 919 3316 3363 3410 3457 3604 3552 3599 3646 3693 3741 47 920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 47 921 4260 4307 4354 4401 4448 4495 4542 4590 1637 4684 47 922 4731 4778 4825 4872 4919 4966 6013 5061 5108 5156 47 923 6202 5249 5296 5343 6390 6437 5484 5531 5578 6625 47 924 5672 5719 5766 6813 5860 5907 5954 6001 6048 6096 47 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 47 926 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 927 7080 7127 7173 7220 72(i7 7314 7361 7408 7454 7501 47 928 7548 7595 7642 7688 7735 7782 ;■& 7875 7922 7969 47 929 8016 8062 8109 8156 8203 8249 ^:'--G 8343 8390 8436 47 930 8483 8530 8576 8623 8670 8716 8763 8810 8866 8903 47 931 8950 8996 9043 9090 9136 91*!3 9229 9276 9323 9369 47 932 9416 9463 9509 9556 9i;02 9b49 9695 9742 9789 9835 47 933 9882 9928 9975 **21 *%f. ^ *0114 »0161 ♦0207 ♦0254 ♦0300 47 934 970347 0393 0440 0486 0533 0h79 0626 0672 0719 0765 46 935 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 46 936 1276 1322 1369 1415 1481 1508 1554 1601 1647 1693 46 937 1740 1786 1832 1879 1975 1971 2018 2064 2110 2167 46 938 2203 2249 2295 2342 2338 2434 2481 2527 2573 2619 46 939 2666 2712 2758 2804 2851 2897 5 2943 2989 3035 3082 46 No 1 2 3 4 6 7 8 9 D. ■ ■i4 340 LOGARITHMS — TABLE IL No 2 3 6 940 973128 941 3590 942 4051 94ti 4612 944 4972 945 6432 946 6891 947 6350 948 6808 949 7266 960 7724 951 8181 952 8637 953 9093 954 9548 955 980003 957 0458 957 0912 958 1366 959 1819 960 2271 961 2723 962 3175 963 3626 964 4077 965 4527 966 4977 967 6426 968 5875 969 6324 970 6772 971 7219 972 7666 973 8113 974 8559 975 9005 976 9450 977 9895 978 990339 979 0783 No 3174 3220 3266 3313 3359 3405 3451 3636 3682 3728 3774 3820 3866 3913 4097 4143 4189 4235 4281 4327 4374 4558 4604 465(* 4696 4742 4788 4834 5018 5064 PliO 5156 6202 6248 5294 6478 6524 6570 661 B 5662 6707 5753 5937 5983 6029 6075 6121 6167 6212 6396 6442 6488 6533 6579 6625 6671 6864 6900 6946 6992 7037 7083 7129 7312 7368 7403 7449 7495 7641 7686 7769 7815 7861 7906 7962 7998 8043 8226 8272 8317 8363 8409 8454 8500 8683 8728 8774 8819 8865 8911 8956 9138 91«4 9230 9275 9321 9366 9412 9594 9639 9685 9730 9776 9821 9867 0049 0094 0140 0185 0231 0276 0322 0503 0549 0594 0640 0685 0730 0776 0957 1003 10*8 1093 1139 1184 1229 1411 1456 loOl 1547 1592 1637 1683 1864 1909 1954 2000 2045 2090 2135 2316 2362 2407 2452 2497 2543 2588 2769 2814 2859 2904 2949 2994 3040 3220 3265 :3310 3356 3401 3446 3491 3671 3716 3762 3807 3852 3897 3942 4122 4167 4212 4257 4302 4347 4392 4572 4617 4662 4707 4752 4797 4842 6022 5067 5112 6167 6202 5247 5292 5471 5516 5561 5606 5651 6696 5741 6920 5965 6010 6055 6100 6144 6189 6369 6413 6458 6503 6548 6593 6637 6817 6861 6906 6951 6996 7040 7085 7264 7a09 7353 7398 7443 7488 7532 7711 7756 7800 7845 7890 7934 7979 8157 8202 8247 8291 8336 8381 8426 8604 8648 8693 8737 8782 8826 8871 9049 9094 9138 9183 9227 9272 9316 9494 9539 9583 9628 9672 9717 9761 9931) 9983 *'28 **72 *0117 *0161 *0206 0383 0428 0472 0516 0561 0606 0650 0827 0871 0916 0960 1004 1049 1093 2 3 5 6 8 9 D. 3497 3543 46 3959 4005 46 4420 4466 46 4880 49:^6 46 6340 6386 46 6799 5845 46 6258 6304 46 6717 6763 46 7175 7220 46 7632 7678 46 8089 8136 46 8546 8591 46 9002 9047 46 9457 9603 46 9912 9958 46 0367 0412 45 0821 0867 45 1276 1320 45 1728 1773 46 2181 2226 45 2633 2678 45 3086 3130 45 3636 3681 45 3987 4032 45 4437 4482 45 4887 4932 46 5337 5382 45 5786 5830 46 6234 6279 46 6682 6727 46 7130 7176 45 7577 7622 45 8024 8168 45 8470 8514 46 8916 8960 46 9361 9405 45 9806 9850 44 *02f.O *0294 44 0694 0738 44 1137 1182 44 8 > D. Tli AlUTHMETIC. 341 D. J 46 > 46 ) 46 > 46 i 46 46 46 46 46 46 46 46 46 46 46 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 44 44 44 44 D. No O 1 2 3 4 5 € 7 8 9 D. 980 991226 1270 1315 1359 1403 1448 1492 1536 1580 1625 44 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 983 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 988 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 44 989 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 44 990 5635 5679 5723 5767 5811 5854 5898 5942 5986 6030 44 991 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 994 7385 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 44 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 997 8695 8739 8782 8826 8869 8al3 8956 9000 9043 9087 44 998 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 • 44 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 1000 OOOOOO 0043 0087 0130 0174 0217 0260 0304 0347 0391 43 1001 0434 0477 0521 0564 0608 0651 0694 0738 0781 0824 43 1002 0868 0911 0954 0998 1041 1084 1128 1171 1214 1258 43 1003 1301 1344 1388 1431 1474 1517 1561 1604 1647 1690 43 1004 1734 1777 1820 1863 1907 1950 1993 2036 2080 2123 43 1005 2166 2209 2252 2296 2339 2382 2425 2468 2512 2555 43 1006 2598 2641 2684 2727 2771 2814 2857 2900 2943 2980 43 1007 8029 3073 3116 3159 3202 3245 3288 3^il 3374 3417 43 1008 3461 3504 3547 3'00 3633 3676 3719 3762 3805 3848 43 1009 3891 3934 3977 4020 4063 4106 4149 4192 4235 4278 43 1010 4321 4364 4407 4450 4493 4536 4579 4622 4665 4708 43 1011 4751 4794 4837 4880 4923 4966 5009 5052 5095 5138 43 1012 5181 5223 5266 5309 5352 5395 5438 5481 5524 5567 43 1013 5609 5652 5695 6738 5781 5824 5867 5909 5952 5995 43 1014 6038 6081 6124 6166 6209 6252 6295 6338 6380 6423 43 1015 6466 6509 6552 6594 6637 6680 6723 6765 6808 6851 43 1016 6894 6936 6979 7022 7065 7107 7150 7193 7236 7278 43 1017 7321 7364 7406 7449 7492 7534 7577 7620 7662 7705 43 1018 7748 7790 7833 7876 7918 7961 8004 8046 8089 8132 43 1019 8174 8217 8259 8302 8345 8387 5 8430 8472 8515 8658 43 No 1 2 3 4 e 7 8 9 D. 342 LOGARITHMS. — TABLE III No 1 8643 2 3 4 8770 6 6 7 8 8941 9 D. 1020 008600 8085 8728 8813 8850 8898 8983 43 1021 9026 90118 9111 9153 9196 9238 9281 9323 P366 9408 42 1022 9451 9493 9536 9578 9621 96(53 9706 9748 9791 9833 42 10'.>3 9876 9918 9961 *0003 *0045 *0088 *0130 *0173 *0215 *0258 42 1024 010300 0342 0..85 0427 0470 0512 0554 0597 0639 0681 42 1025 0721 07i)6 0809 0851 0893 0936 0978 1020 1063 1105 42 1026 1147 1190 1232 1274 1317 1359 1401 1444 14S6 1528 42 1027 1570 1613 1655 1()97 1740 1782 '1824 1866 1909 1951 12 1028 1993 2035 2078 21.0 2162 2204 2247 2289 2331 2373 42 1029 2415 2458 2500 2542 2584 2626 2669 2711 2753 2795 42 1030 2837 2879 2922 2964 3006 3048 3090 3132 3174 3217 42 1031 3259 3301 3343 3385 3427 3469 3511 3553 3596 3638 42 1032 3680 3722 3764 3806 3818 3890 3932 3974 4016 4058 42 1033 4100 4112 4184 4226 4268 4310 4353 4395 4437 4479 42 1034 4521 45(J3 4605 4647 4689 4730 4772 4814 4856 4898 42 1035 4940 4982 5024 5066 5108 5150 5192 5234 5276 5318 42 1036 5360 6402 5444 5485 5527 5569 5611 6653 6695 6737 42 1037 5779 582 L fi863 5904 5946 5988 6030 6072 6114 6156 42 1038 .6197 6239 6'?81 6323 6365 6407 6448 6490 6532 6574 42 1039 6616 6657 6699 6741 6783 6824 686t) 6908 C950 6992 42 1040 7033 7075 ni7 7159 7200 7242 7284 7326 7367 7409 42 1041 7451 7492 7534 7576 7618 7659 7701 7743 7784 2826 42 1042 7868 7909 7951 7993 8u34 8076 8118 8159 8201 8243 42 1043 8284 8326 8368 8409 8451 8492 8534 8576 8617 8659 42 1044 8700 8742 8784 8825 8867 8908 8950 8992 9033 9075 42 1045 9116 9158 9199 9241 9282 9324 9366 9407 9449 9490 42 104(1 9532 9573 9615 9656 9698 9739 9781 9822 9864 9905 42 1047 9947 9988 ' *0030 ' ^0071 ^ »0113 ' »0154 ' ^0195 *0237 *0278 *0320 41 1048 020361 0403 0444 0486 0527 0568 0610 0651 0693 0734 41 1049 0775 0817 0858 090O 0941 0982 1024 1065 1107 1148 41 1050 1189 1231 1272 1313 1355 1396 1437 1479 1520 1561 41 lOSI 1603 1644 ^(585 1727 1768 1809 1851 1892 1933 1974 41 1(52 2016 2057 2098 2140 2181 2ii22 2263 2305 2346 2387 41 1053 2428 2470 2511 2552 2593 2635 2(576 2717 2758 2799 41 1039 2841 2882 2923 2964 3005 3047 3088 3129 3170 3211 41 1055 3252 3294 3335 3376 3417 3458 3499 3541 3582 3623 41 1056 3(564 3705 3746 3787 3828 3870 3911 3952 3993 4034 41 1057 4075 4116 4157 4198 4239 4280 4321 43(13 4404 4445 41 1058 4486 4527 4568 4609 4650 4691 4732 4773 4S14 4855 41 1059 4896 4937 4978 5019 5060 4 5101 5 5142 6183 5224 5265 41 No O 1 2 3 6 7 8 9 D. AIUTHMETIC. S48 D. 43 42 42 42 42 42 42 12 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 41 41 41 41 41 No 6 8 9 1060 025306 lOGl 5715 10'i2 6125 1003 6533 1064 6942 1065 7350 1066 7757 1067 8164 1068 8571 1069 8978 1070 9384 1071 9789 1072 030195 1073 0600 1074 1004 1075 1408 1076 1812 1077 2216 1078 2619 io7y 3021 1080 3424 1081 3826 1082 4227 1083 4628 1U84 5029 1085 5430 1086 5830 1087 6230 1088 6629 1089 7028 1090 7426 1091 7825 1092 8223 1093 8620 1094 9017 1095 9414 1096 9811 1097 040207 1098 0602 1092 0998 D. No' O 5347 5388 5129 5470 5511 5552 5756 5797 5838 5879 5920 5961 6165 6206 6247 6288 6329 6370 6574 6615 6656 6697 6737 6778 6982 7023 7064 7105 7146 7186 7390 7431 7472 7513 7553 7594 7798 7839 7879 7920 7961 8002 8205 8246 8287 8327 8368 8409 8612 8653 8693 8734 8775 8815 9018 9059 9100 9140 9181 9221 9424 9465 9506 9546 9587 9027 9830 9871 9911 9952 9992*0033 0235 0276 0316 0357 0397 0438 0040 0081 0721 0762 0802 0843 x0i5 1085 1126 1166 1206 1247 1449 1489 1530 1570 1610 1651 1853 1893 1933 1974 2014 2054 2256 2296 2337 2377 2417 2458 2659 2699 2740 2780 2820 2860 3002 3102 3142 3182 3223 3263 3464 3504 3544 3585 3625 3665 3866 3906 3.146 3986 4027 4067 4267 4308 4318 4388 4428 4468 4669 4709 4749 4789 4829 4869 5069 5109 5149 6190 5230 5270 5470 5510 5550 5590 5630 5670 5870 5910 5960 5990 6030 6070 6269 6309 6349 6389 6429 6469 0669 6709 6749 6789 6828 6808 7008 7108 7148 7187 7227 7207 7466 7506 7546 7586 7626 7665 7865 7904 7944 7984 8024 8064 8262 8302 8342 8382 8421 8461 8060 8700 8739 8779 8819 8859 9057 9097 9J36 9176 9216 9255 9454 9493 9533 9573 ,9612 9652 9850 9890 9929 9909*0009*0048 0246 0286 0325 0365 0405 0444 0042 0681 0721 0761 0800 0840 1037 1077 1116 1156 1195 1235 6 6 6593 6002 6411 6819 7227 7635 8042 8449 8856 9262 5634 5674 6043 6084 6452 6492 6800 6901 7268 7309 7676 7716 fc.083 8124 8490 {^531 8896 8937 9303 9343 9668 9708 *0073 *0114 0478 0519 0883 0923 1287 1328 1691 1732 2095 2135 2498 2538 2901 2941 3303 3343 9749 *0154 0559 0964 D. 41 41 41 41 41 41 41 41 41 41 41 41 40 40 1308 t 40 3705 4107 4508 4909 5310 5710 6110 6509 6908 7307 7705 8103 8501 8898 9295 9092 *0088 0484 0879 1274 3745 4147 4548 4949 5350 5750 6150 6549 6948 7;w 7745 8143 8541 8938 9335 1772 2175 2578 2981 3384 3786 4187 4588 4989 5390 5790 6190 6589 6988 7387 7785 8183 8580 8978 9374 9731 9771 *0127 *0167 (523 0563 0919 0958 1314 1353 8 9 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 39 D. ^:N"SA7srE:E2;S. (The atisivers of ExerciaeH I, II and /K are due to Mr. Thomat McJanet of Ottawa ; those of Exercises XXXII to XLVI, to Mr. Thomas Kirkcomiell, Mathematical Master of Fort Hope High School ; the latter (jentleman also tested the answers of Exercises I and III.) Exercise I. 1. 222. 2. 3625. 3. 2222 aq. yd. 2 sq. ft. 4. (i), 9' 11 " ; (ii), (a) 8' 2if' , (/>) 8' «.Vr". 6. 7V\ mi. per lir. ; 8| mill, per mile. 6. (i), } ; (ii), th- 7. 21^^ gal. 8. 67f ^ da. 9. 36 ct. 10. ^20-46. 12. 1 A. 361 sq. yd. 7 sq. ft. 13. !$388-23. 14. 5'1". 16. 2I35V0. nil 16. (i\ 170h gal. ; (ii), 46| gal. 17. 229-6 lb. ; 33-26 c. ft. 18. 25 da. 19. $1080. 20. $6 -58. 21. $70; $30800. 22. 968-7627 aq. in. 23. 6' 4". 24. (i), l:28;jV p.m , (ii), UgV"? min. 25. 16 years. 26. ^Vo ; $1974, 2T 0-4. 28. B in 37^ da., G in 26 da. 29. 36%. 30. $89 20. 31. (i), 635-90 Km. ; (ii), 62-31 Km. ; (iii), 142-17 Km. ; (iv), 122-16 Km. ; (v), 52-6'c;Km. ; M), 72036 Km. ; (vii), 730-66 Km. 32. 684* sq. ft. 33. lOj^ nun. 34. ';i), 2-2645 sq. ft. ; (ii), 2-004 s(i. ft. ; (iii), l-«<«08 sq. ft. ; (iv), 1-4428 sq. ft ; (v), 1-2625 sq. ft. ; (vi), 0-8906 sq. ft. ; (vii), 0-7462 sq. ft. 35. 34^11 yd. 36. $112-29. 37. $9-60. 38. 19ii|yd. 39. (a), 36yr. ; (/>), 60yr. ; (c), 36 yr. 40. 4* %. 41. 60417 and 425-425. 42. ^i), $20-67 ; (ii), $18-84. 43. 10 aq. ft. 92^ sq. in. 44. 22-86 gal. 46. 6 times per 2 sec. 46. $2700. 47. 66 yd. per min. 48. $114. 49. 64yr. 60. $742-38. 51. r^^. 52. 3y\A. 53. 1 hr. 16 min. 64. 37^^ mi. pex- hr. 55. ^ ; $92 40. 56. (i), 16 sec. ; (ii), 9| sec. ; (iii), 37^sec. 57. $600. 58. 9,'^%- 59. 6^/. 60. (i), 4if % ; (ii), $525. 61. A. $21, B, $1680. 62. 705301^ cubic miles. 63. 000645 in. 64. 29ff mi. per hour. 65. 120 subscribersj g263.5a. 66. 20 mi, per hour. 67. $34-05. 68. ^4's $6800; 346 ARITHMETIC. Ba 84600 ; 0-7931. 69. 2C0 da. 70. ^100. 73, F; ])'y 4tl\, paid $20-88. 74. Betweei oo and 37 miles per hour. Bocvveen t-i .-vivi 36 miles per hour. 75. (a) If. (6) {J. 76. i. 19. 'i, 8 ; .ui, 3. 168 marbles. 77. 19 mi. per hour. 78. $3:;r»-()5 ; iim %. 79. 255 days. 80. $8 00. 8i?. 40 times. 83. 6227 ?jg c. ft. ; 502 1^. 4427§t gr. 84 323 yd. 85. N. v. 28t'., paid $16*24. 86. 120 acres. 87. (• ) If ('>) 2^. 83. A, by T^yd. 89. $432-34. 90. $7750-51. 91. 2880 revolutions, 04 and 15 ^-evclat.io .s. 92. (a)24-27 o, in., (/.)38-92c -a., (c)100-465c. in., ((/)154-280c. iii., ^.*;)320c, ill , C/)38«)-2c. in. 93. 1649851b. 13|^.^-oz. 94. ,; vr2-55. 95. (i), f'. by 24f yd., (ii), B, by 47§^ yd. 96. 4 hr. 97. (a) 5|. {b)2l t:-3. 45''J yd. 99. $115 -68. 100. 28f 't Exercise <) 2 Q 1 J ^_ s 14 J].I a 1 ? ^ i; "J 4 7 •> ' 1 5 .1 101) TaTjy II ! 1 i- *• 'i 7 :i.- <-'• •^ ff lit 2TI4 ii.i.'i r4() • '-'•8 ^5 108 il 7 1 3 7^ A iS| ?*S 2(!.5 770 {)• *• ^ .-J lU i;) S.'i B">8 Jsi 1111(7 48 %(S^ _7 7 9, lr,i4 _ni3_l^ ].;(.-i i;jT2H' 8. 1 !JL ..ajt _2_8_0 1 TiT T3U 155 44!) I*lt7^- 1 3 yi 37 71 2&0 i.?21 1814 4 200 • 2 T .!• 8 TiT .5 58T >>'I70 43561 lf,t 11 1873 4..T nT7:r5 5- ••■'-'• 3 7 10 377 387 llol «T4^ ;75813' -'••'•• at 17 41 00 14_Q 230 7011 10 O (; I. 19 2fi XI 97 2 n V yfiS fl89 234fl T^ 2 9 M 9 9 T6 9 5000- ^^' ^ S A 11 To 4T 55 Ti>.> l09 5 71 1351 3.3^29 JM^.Q 3iQ.i.l 13 5 2-2 49 2JLa 485. 2Ji_58 204.1 1144 17^>31 .*,_ "539 20000- J-'-*' 2 9 20 89 198 881 T079 3039 Tl 57 t2.5.Q8 .nj4Sl 214949 14 l73o3 2.549 10(T(JOO- ^^' 249Xi T0T9? -9 01JL _4JL5_8 a<(T(T9i I 3 X 10 J 7 14W9 15Q6 : 5 7 Iff 23 39 34Tt 3455 1 4^589 -IK 1 1 1 5 29 3 4 131 296 723 114 2 1Q 1 TUdOim- ■•••-'• -■■ S 7 9 5^ ?5l •.'35 531 1^97 3125* ■■■"• ^(5 'A 401 _4_?1. -4JJ 8441 " J_2jl (i8 9 25000rt0- 19. 3 I 884 2 1 17. I 11 Jj'. S«3 90H1 fi.92 3_ 83 18157^ T989 1.8984 .2^_9D7 4c89i I LI Tilt n »* ■ ^i\ K o TT (1 8T ITOSffOgStS tOO(TO(.5oa- 26. t H m m rm nm mm ; mim uhnn^ mum- 26. 1 i H M- ti iei m m \m mn Mm nmrtmu ANSWERS. 347 mim mnn nnm- 27. m^ mh mn mm mm mim mmi 28. 1 m mi um mu mm mm mm mmi umu mmi mnn mum- 29. 2 h ft w w in- 30. i q^ ^ xVs fW ^g ij\^ ifi¥3 *fM HJ2§ tvwr Mj^ii'v h^mi Amv5. Exercise III. 1. 31017414. 2. 67 0509. 3. 1503-543. 4.3-20424. 5.31. 6.1301030. 7.2-477121. 8.3. 9.21556589. 10. 2. 11. 1-0019656. 12. 99999970. 13. 4856. 14. 1161. 15.6770. 16.67-7. 17.70-7107. 18.795 775. 19.0 0264675. 20. 0013964. 21. 0'301030. 22. 0477121. 23. 0-318310. 24.2-30269. 25.2-22398. 26.16 5304. 27.1-39642. 28.2 7183. 29.0-36788. 30.14107. 31.2. 32 1-5. 33.1-25. 36.8000 m. 36. 1926 yd. 37. 4047 centiares. 38. 6789 sq. yd. 39. 40 47 hectares. 40. 247 1 A. 41. 132 1 gal. 42. 599*4 litres. 43. 384,300 Km. 44. 147,100,000 Km. 45. \, Y, V> W» W- 46. \\ Y, ¥, V, W-. W, ¥.¥• 18-03 yr. 47. tV ^\. ^, ^, #5, iV^. 48. 1, f ^, \h M, H. fit. 49. \, h ^^, ij, if Exercise IV. 1. 30197 in. 2. 1000-796875 lb. 16051 c. ft. 3. 3 min. 26^ sec. ; 3min. 26 ^^^^\ sec. 4. 1100 bu. 5. ^in. 6. 78 % of copper, 22 % of zinc. 7. 0-2532 %. 8. $9000-49, $123-09. 9. 2nd Nov., 1889. 10. 8 %. 11. (i), 9^ min. ; (ii), 8§ min. 12. 2-89 times. 13. 120-4264 lb. 14. 209 da. §g| ; §§|. 15. Friday at 3,15 a.m. ; 3,20 a.m. 9,05 ^^ a.m. 16. $5-40. 17. Iqt.; f 18. ^r\Wr%' $514-93. 19. 2nd Ap., 1889. 20. 7%. 21. 23f ft. per sec. ; 471f yd. per min. ; 16,1^ mi. per hr. 22. 3762 revolutions ; 7 mi. 160f yd. : 21if g mi. per hr. 23. 632|| lb. 24. us 5%o5 ; 23f ^ grammes per millier. 25. Oct. 24 at 2 a.m. 26. 31igal. 27. 27ct. 28. 12«gtf %. 29. $477-34. 30. 16,ct. ; 16 ct. 31. 480^ia^yd. per min. ; 16/^,^ mi. per hr. ; 439-467 m. per min. ; 26-3()8 Km. per hr. 32. 1 min. 9-6 sec. 33. 180041 lb. 34. ^^^ ; $91 64. 36. 256791 lb. 36. 271296 ; 247643/^^. 37. 46 lb. @ 26 ct. 38. 61 -35 %. 613-5 per 1000. 39. 8^g %. 40. $836-53 ; $78309 ; 1365 %. 41. 4 hr. 5 min. ; m I SI I 348 ARITHMETIC. 4 hr. 4 min. 42. 61^ yd. 16 roUs 28 yd. 43. $7 04. 44. 8 -878 lb. 46. 13-75ct. per hr. 46. (a) 276-812 c. in. ; (6) 173 lb. ; (c) 26 -363 lb. ; (d) 316-357 lb. 47. Gains $160-75 ; 30-63 %. 48. $2-34. 49.$340-86. 60.(a)7i}%;(6)7-834%; (c)7-91%. 61. 5-911 T., 6-2115 T., per sq. in. 831-2 kilog. ; 873 4 kilog., per sq. cm. 62. $4769. 63. ^. 64. 949J lb. 66. 54 times ; l^g qt. ; f . 66. $28-66. 67. 5 hr. 68. 775 marks ; 79J« % ; 125§^ %. It Virould reduce the 775 marks to 620 marks but would have no effect on the percentages. 69. $206 56. 60. $192 95. 61. $16 77. 62. $661-12. 63. $35-94. 64. 210 071b. e6.7imi.per.hr. 25f mi. 66. ^. 67. 20 boys. 68. 5789-658 T. 69. $78304. 70. $863-65. 71. 2210 tiles ; 58' 0" ; 13'. 72. 1531-46 lb. 73. 1:596 mi. 74. $1-71. 76. i. 76. 12 men. 77. 7ff hr. 78; 3 hr. 79. 66 doz. 48Jf %. 80. Net proceeds $3061-71. 81. $1-21; $1-48. 82. $153-92. 83. (a) 12 c, ft. 1534 c. in. ; (h) 12 c. ft. 192-4 c. in. 84. 667 strokes ; 25 strokes. 86. y^. 86. '^. 87. $7-56; $14-40; $18-75. 88. $1-08; 27tV %• 89. 21tV % ; $23-84. 90. $1095-78 ; $1097*97. 91. A, 48 men ; jB, 72 men ; 0, 60 men ; A 80 men. 92. $25-59 ; 40 536 c. ft. 93. 32 sq. ft. 102 sq. in. ; 10 c. ft. 1620 c. in. ; |if § ; £^j. 94. 1 hr. 11 min. 41*84 sec. ; 3 hr. 5 min. 41*4 sec. '96. ■^. 96. $511-25. 97. $840; $504; $3-60. 08. 35-3 %. 99. 19-264 %. 100. $1164-14. Exercise V. 1. 28. 2. S^. 3. 5*. 4. 10*. 6. 01*. 6. 2-36. 7. 08. 8. (iy. 9. 3x3x3x3. 10. 12x12x12. 11.16x15. 12.25x25x26x25x25. 13. 2-5 x 2*6 x 2-5 x 2-6 x 25 14. 0-25x0 •25x0-26x0 •26x0- 26. 16. §xfx§x§. 16. Hx Hx^. 17.2x2x2x3x3x5x7x7. 18.64. 19.36. 20.626. 21. 1024. 22. 5640626. 23. 202572273617. 24. 6554671841. 26.14348907. 26.1-331. 27.000001. 28.0000000000064. 29. 1-126162419264. 30. 2401. 31. 24 01. 32. 0-2401. 33. 13144-256. .34. 13 144256. 36. 0000013144256. 36. f. 37. a. 38. ^. 39. Ul 40. iff 41. 648. 42. 54000. 43. 5292. 44. 6298292000. .46. 117649. 46. 626. 47. 72. ANSWERS. 349 48.23x32x5x7. 49.2^x33x53x13. 60.2io. 51.23x72x29. 52. 2' x3« x7 X 13. 53. 0'25. 64. 02. 55. 01. 56. 009706. 57 to 61. 0-7854. 62. 2014. 63. 04966. 64. 0'6931. 65. 2-303. . . Exercise VI. 1. 4,529,55225,5555449,566922084,56592679961.- 2. 64, 79507, 8.*^ '.53453, 83740234375, 83791924694479. 3. 144,1728: 16129, 2048383 ; 1633284, 2087336952 ; 163481796, 2090278243656. 4. 2646-999(i01, 136185 482471849, 7006606 887694149201. 5. 0018496, 00251^A56, 000342102016, 0000465125874176, 000006327518887936. 6. 0-01. 7. 006139. 8. 002. 9. 00686. 10. 90-744. Exercise VIII. 1. 24. .2. 43. . 3. 321. 4. 3-21. 5. 4818. 6. 07097. 7. 73. 8. 934. 9. 888. 10. 88-8. 11. 1837. 12.0-0543. 13.1-41421. 14. 4-47214. 15.14-1421. 16.44 7214.^ 17. 0-447214. 18. 0141421. 19. 6 -32456. 20. 63 2456., 21. 0-632456. 22. 86 3076. 23. 31-6228. 24. 780 897. 25. 49-7933. 26. 497933. 27. 1-25992. 28. 2 71442. 29. 5-84804. 30. -621455. Exercise IX. 1. 6. 0-530330. 7. 10. 0-612372. 11. 14.2-23607. 15. 18. 5 09902. 19. 22. 3-31662. 23. 26. 40-0125. 27. 30. 0-73598. 31. 34. 1-41421. 35. 39. 0-87358. 40. i 2. 11. a 0-769800. 0-261861. 3-87298. 5-91608. 7-28011. 48-9898. 3:31662. f 36. ^. 0-893904. \l 4.0 745356. 5. 8. 547723. 9. 12.0-788811. 13. 16. 4-12311. 17. 20. 6-08276, 21. 24. 8 -7 7496. 25. 28. 0-797724. 29. 32. 2-44949. 33. 37. 0-609884. 38. 0-824621. 0-846164. 173205. 4-89898. 2-64f75, : 9-84886. 0-670820.( .2.23607. ; 0-471957. Exercise XI, 1. 10*. 2. 10«. 3. lO^o. 4. 10-^ .5. 10-^ 6, 10*.; 7.10,007,400. 8.12741-8. 9.0 000226. 10.0-000,000,006. 11. 1,083,200,000,000,(300,000,000. 12. 0^000,030,476,3. 13. 1-4709x1011. 14. 4-8721x10-3. 15-. 78376x108. 16. 90992x103. 17. 5-G75xlOi*. 18. 11635x107. 850 ARITHMETIC. 19. 8 5534x10^0. 20. 10832 x 10^'. 21. 3-7207x10*. 22. 4-5162 xlO«, 23.2-2641x10*. 24. 6-3611 xlO-^o. 26. 6-6966x10-*. 20. 8-5499 xlfr-*. 27- 1-27418x10-. Exercise XIV. 1. 0000043. 2. 0000087. 3. 0000130. 4. 00000304. 6. 0-0000686. 6. 00001084. 10. 0-4771213. 14. 1-1139434. 18. 1 11 3943. 22. 1-462398. 2. 0-7782. 3. 0-4914. 4. 08195. 8. 0005208. 9. 0012576. 12. 0-4342495. 13. 1 4913617. 16. 0-845098. 17. 1-230449. 20. 1-278754. 21. 1-301728. 24 . 1 -361728. 26. -301030. Exercise XV. 1. 0-8451. 5. 0-4346. 6. 0-1370. Exercise Xyi. 6. 10". 7. lO"". 8. 10«. 9. 10-« 11. 1-361728. 12. 2-361728. 13. 3361728. 14. 16. 0-635584. 20. 3-892651. 24. 1-830396. 28. 8-811575. 7. 0-0004341. 11. 0-8450980. 15. 1-2304489, 19. 1-278754. 23. 1-612784. 15. 2-361728. 19. 3-831806. 23. 5-830396. 27. 42-212188. 17. 3-635584. 21. 3-860098. 25. 2-301464; 29. 6-652826. 10. 102. 0-301728. 18. 2-635584. 22. 3 -860098. 26. 14-071145. 30. 6-881042. Exercise XVII. 1. 0-864831. 2. 2774604. 3. 4679092. 4. 3-690692. 5. 2471214. 6. 5830439. 7. 1-271435. 8.4-903133. 9. 3-659925. 10.7-477599. 11.8 804711. 12. 8-803211. 13. 27 -034709. 14. 7 634200. 15. 5483962. Exercise XVIII. 1. 3-02. 2. 5-432. 3. 6-496. 4. 300. 5. 8-6636. 0. 6-5666. 7. 65-666. 8. 656660. 9. 0-65666. 10. 0000065666. 11. 67621. 12. 67680. 13. 00079433. 14. 1999-9. 15. 0-31623. 16. 1. 17. 100 047. 18. 2. 19. 1024. 20. 10718. 21. .5-208 xlO^. 22. 3-4247 xl0-«. 23.6-562x10-10. 24. 4-5709xlOi*. 25. 1 -00028 Xl">-i°«. Exercise XIX. 1. 499-27. 2. 4 0798. 3. 0089064. 4.0-088785. 5.86-898. 6.561-33. 7.0 62614. 8.0 0066856. 9. 34 464. 10. 12-4368. 11. 7-8541. 12. 0013446. 13.0-00026113. 14.0 0035259. 15. 12637 x 10- '5. 16.115779. ANSWERS, 361 17. 17300. 18. 0024513. 19. 4 06735. 20. 9'8019xl0-". 21. 1-2589. 22.108791. 23.0-29587. 24.0-44402. 26.0-61439. 26. 0-46986. 27. 74989. 28. 1-24732. 20. 0- 96691. 30. 0-1J8921. 31. 0-79433: 32. 076522. 33. 1423. 34. 6-4143 X 10-2^ 36. 5-4184 x 10"^ 36. 080274. 37. 0-5848. 38. 0-491516. 39. 99718. 40. 10000026. 41. 1917791. 42. 23 097. 43. 0-168792. 44. 034278. 46. 2 -2339. 46. 6 log 2. 47. 4 log 2+ log 3. 48. 2 log 7. 49. (2-41og2)- 1. 60. 10 log 2. 61. 4 log 7. 62. 6 log 3+ log 7 -3. 63. 2 log 2 + log7 + logll + 31ogl3-3. 64. 41og3+21ogll-log2-21og7-2. 66.(41og2 + log7 + logll + 3-61og3-21ogl3)-l. 67. 2»-3^2 = 10; l4-log2. 68.1-8507. 69.0-64921. 60.0-28246. 61.-2-1765. 62. -0-68512. 68. 2 -80736. 69. 0-35621. 70. 2-2766. 71. 0-43924. 72. 2-4923. 73. -194843. 74. -0-0136958. 76.2-5124. 76.20-149. 77.17.673. 78.118956. 79.10-2448. 80. 9-58435. 81. 27267. 82. 302. 83. 48. 84. 206. 86. 44. 86.83. 87. 7. 88. 46. 89. 27. 90. 6. 91. 69. 92. 164. 93. None. 94- None. 96. 32. 96. -17. 97. -1. 98. 1. 99. 10. 100. 0. Exercise XX. 1 . Mar. , 0-3871 ; Ven. , 0-72333 ; Mars, 1 52369 ; Jup., 6-2012 ; Sat., 9-538. (a), 35,916,000 ; 66,134,000 ; 139,310,000 ; 475,540,000; 872,060,000. (6), 35,916,000; 67,111,000; 141,370,000 ; 482,560,000 ; 884,930,000. 2. 1233222 figures ; 1,169,649 18,212,890,625; 114 hr. llmin. Usee. 342,188,706,078 figures; 263 yr. 21 da. Exercise XXI. 1. 25". 2. 20^' ; 11-25". 3. 19-886" ; 42-614". 4. 7' 7-886". 6. 4-2. 6. 18. 7. 40-5 yd. 8. 138' 8". 9. 25714 ch. 10. 33' 7-2". 11. 6' 8-5". 12. 17' 6", 13. 19' 1091". 14. 62' 8-625", -.0' 8-625". 15. 11 07011., 11-39 ch., ll-71ch., 1203 ch. 16. 4-186", 7 814'; 6-977", 13 023". 17. 2". 18. 8' 2-182". 19. 5' 10", 9' 2", 10'. 20. 7 ft. 9-6 in., 10 ft. 2-4 in. ; 15yd. 1ft. 8 in., 23yd. 2ft. 4 in. Exercise XXII. 1 . 69 yd. 1 ft. 9 in. 2. 8 -944 ch. 3. 1 -26 in. 4. 2 mi. to 1 in. 1 : 3520. 6. 36 sq. ft. . 6. 46-76 oz, 7. 17-32 ch. 352 ARITHMETIC. 8. 1045440 sbvlks. 9. 384 pieces. 10. $4747 '80. 11. 17-32 ch. 12. 5081 yd. 13. ir>cl)., 40ch. 14. 1452 sq. in. 16. 55 ft. 16. 30yd. by ee yd. 17. «-41ch. 18. 13984 sq. ft. 19.^384-38. 20. 316 -3758(1. in. 21. 378(i.ft. »l-5Hq.in. 22. 3«ft. 23. 68-9ft.; 92-57 ft. 24. 834-(«. 26. 6-fi25 A. 26. 1 ' 402 73 sq. ft. ; 2°, 99 -72 sq.ft. 27. 18«7 bricks. 28. 4 A. 2207 sq. yd. 29.1% 152-25 ft. ; 2°, 53«-25ft. 30. 12*80 ch. 31. 127 5 ft., 94 -5 ft. 32. 79194 sq.ft. 33. 13.*{8(i. ft. 139 8(i. in. 34. 40-9409 A, ; .32-3817 A. 36. 31 7495 A. 36. 200 yd, 240yd. 37. 340ft. 38. 43-50 ch., 37-44ch. 39. 53 sq. ft., 03 sq. ft. 40. 4-63392 A., 4-80096 A., 4-968 A., 5 13504 A., 5 30208 A. 41. 551 sq. ft., 1653 sq. ft, 2755 sq. ft. 42. 5' 4", 4'. 43. 161ft. 44. 441 ft., 245 ft. 46. 17424 sq.ft. 46. 146 yd. 2 ft. 47. 413600 sci. ft. 48. 5 ft. 49. 11 ft. 60. 289 sq. yd. , 225 s(i. yd. 61 . 841 h<\ ft. , 441 sq.ft.. 62. 35 ft., 27 ft. 63. 194 0335 ft. 64. 174 yd. 66. 484 ft., 330 ft. Exercise XXIIT. 1. 17008 c. in. '2. 3005 gal. 3. 14 05 in. 4. 30-26in. 6. 15 -704c. in. ; 0-2297 in. 6. 66228- 5 eft. 7. 1549gal. 8. lin. 9. 12 in. 10. 2 014 in. 11. 133 -3334 mm. by 44 -4445 mm. by 44 -4445 mm. 12.2 2894 yd. 1 3. 643 -66 mm. by 965 •49mm. by 1609 -15 mm. ; 0*6214, 1 0357 and l-5536centiare8. 14. 68-921 c. in. 16. 87-72sq.in. 16. 335 '410. in. 17. 2-211 in. 18.649-52c.in. 19. 5-4 in. 20. 00151 to 1. 21. (a^ 0-271; (fc) 0-19. 22. In reductions from metric expressions a * calculated length ' will be in excess by 1-599% of the actual length and should therefore be decreased by l-576% <>f itself; a 'calculated area' will be in excess by 3-224% of the actual area and should be decreased by ."-123% itself; and a 'calculated' volume will be in excess by 4-874 % of the actual volume and should be decreased by 4-648 % of itself. In reductions to metric expressions a * calculated length ' will be in defect by 1 -574 % of the actual length and should be increased by 1599% of itself; Ac. 23. 6-8457 in. square. 24. 1071c. in. 25. 625 -683c. in. 26. 244 798 eft. 27. 3 -521 lea. 28. 0-3447 ca. 29. 461 -4680. in. 30. 50-4c.in. 31. 7-2in. ; ANSWERS, {153 5Bin. ; 4-2in. 32. 6. 33. 12 in. 34. 128c. in. 36. 230 in. 36. «45-2251b. 37. 0000()0(;7. 38. 27JKW to 10000. 39. 230 •321b. 40. 7«'777in. 41. 584 59 lb. 42. 12()%in. 43. COGflc. in. 44. 1 •841b. 46. iVin. 46. 24149cubo8. 47./„in. 48. 00102 in. 40. 10-08 c. in. 60. 492565 millilitres. 61. 43211c. in. 62. 28134 c. ft, 1458c. in. 63. 50c. ft., 288 c. in. 64.117,333,- 33.'JJ c. yd. 66. 159-26 c. yd. 66. 406 gal. 67. 49 2 bars. 68. 6844 -8 yd. 69. lift. Sin. 60. 2ft. 2 in. 61. 394 4 gal. 62. 218l88c. in. 67. 306 c. in. 68. 969 c. in. 69. 3.S293 4 c. in. 70. ] •07 lb. 71. .1i^l235-31. 72. 18-8456 c. ft. 73. 18 c. ft., 696 c. in. 74. 15 eft. 76. 16846 in. 76.3 ft. 4-5 in. 77. 5-196 ft. 78. 45243 in. 79. 157 in. 80. 18 in. 81. 3464344 c. yd. ; 30!M)908 c. yd. 82. 1492 36 c. ft. ; 895 T. 13631b. 1722 gi". ; $539984058 47. 83. 42ft. lOJ in. 84. 33 in. 86. 2-888 in. 36. 11 04 in. 87 2175 in. 88. 39375 c. ft. 89. 40-25c.ft. 90. 30^484c.ft. 91. 5128" 27 gal. 92. 5776 •711b. 93.70081b. 94. 244 -369 eft. 96. 771c. in. 96. 47272 2640. yd. 97. 5365-226C. yd. 98. 28246 722 c. yd. 99. 101138 343 c. yd. 100. 1699-38 c. in. ; 972-972c.in. 101. 47 6850. ft. ; 29352c. ft. ; 15 eft. 102. 165c. in. ; 95 76c. in. 103. 696 c* in. ; 264 c. in. 104. 5-654 in. 105. 124 542 c. in. ; 82791c. in. ; 30042 c. in. 106. 3618 c. ft. ; 8 045 c. ft. 107. 10138 c. yd. 6 c. ft. 108. 43 008 c. in. 109. 11 gal. 110. 32 64 c. in. ; 37-44 c. in. 111. 1000 c. in. 112. 509 -2 c. in. 113. 647 234 c. in. ; 441 -406 c. in. 114. 1575 c. in. ; 840 c. in. 115. 18 432 in. 116. 19c. ft. 418c. in. 117.64^c.ft. 118. 13954 -3 lb. 119.57c.ft. 120. 2048 in. 121. 18998 ft. 122. 13 to 9. 123. 7 to *. 124. 53c. ft. 362c. in. 125. 2 -9 ft. 126. 4 -474 ft. 127. 8987 Htres. 128. 34 sq.ft. 64sq.in. 129. 11 5 sq.ft. 130. 2' 8"; 1' 1^". 131. 6-6 in. 132. 6 ft. 133. 1'8". 134. l|in. Exercise XXIV. 1. 97 in. 2. 905 mm. 3. 1ft. 5 in. 4. 1ft. 8| in. 5. 8Mn. 6. 6 -928 in. 7. 3 -464 in. 8. 3-674 ft. 9. 2-45 in. 10. 267 in., 24 -4 in., 12-5in. 11 . 15ft. 12. 8-595m. ; 354 ARITHMETIC. 8-579m., 6-347m., 5-82m. 13. 14. 4-29in., 8-8in., 23-4 in. 15. 962-676 c. in. 16. o6sq. ft. 408q. in. 17. 9-88 in. 18.391ft. 19.7ft. 20. 2.ft. 21. 6831288q. ft. or 168988 sq.ft. 22. 1200 c. in. ; 790*9 sq. in. 23. 28ft. 8*2 in. 24. 36-9 ch. 25. 231668q.ft, ; 696 ft., 630 ft. Exercise XXy. 1. 1936 sq.ft., 57600 sq.ft., 13689 sq. ft. ; 44ft., 240ft., 117ft. 2. 46ft. 5-494in. 3. 9-2 mm., 359-5 mm. 4. 7 in., 8-8 in. ; 24in., 23-4in. 5. 15 ft. 2*753 in., 45ft. 2*247 in. ; 26ft. 2-797 in. 6. 12*923 yd., 12yd., ll*2yd. 7. 19*8 ft., 12 -692 ft., 20ft., 44*8 ft., 36 ft., 51*692 ft. 8. 399ft., 455 ft., 511ft. 9. 616ft., 665 ft., 511ft. 10. 17 ft., 21ft. 3. in., 21ft. 9 in. Exercise XXVI. 1. 60sq.yd. 2. 60sq.yd. 3. 24 sq.ft. 4. 84sq. ft. 6. 66sq. ft. 6. 126sq. in. 7. 240sq. in. 8. 252sq. in. 9. 2-9274 sq.ch. 10. 166 417 A. 11. 8260653. 12. $118-68. 13. 16672 *5sq. ft. 14. 18*2 Ares, 54*6 Ares, 91 Ares. 15. 227 -04 sq. ft. , 804* 32 sq. ft. , 740 * 96 sq. ft. , 163 * 68 sq. ft. 16. 14760 sq.ft., 17352 -28 sq.ft., 28341 '5 sq. ft., 25749 -22 sq. ft. 17. 92-8812 A. 18. 1698*8 sq.ft. 19. 44*7154 sq. metres. 20. 37-08^3 A;* Exercise XXVII. 1. 0*130806; 3-13935. 2. 0*065438; 3*14103. 3. 0-26330R; 3*15966. 4. 0*131087; 3*146086. 6. 0*0654732 ; 3*142715. Exercise XXVIII. 1. 659 734 ft. ; 835*664 ft. 2. $3*93. 3. 2mi. 1710yd. 2ft. 3in. 4. 1ft. 8Hn. 5. 410yd. l|ft. 5in. 6. 3|in. 7. 14 in. 8. 1770*7 mi. per min. ; 357.7 mi. per rain. 9. 10-472 in. 10. 15*708 in. 11. 28° 38' 52-4". 12. 114° 35' 29*6". 13. 57U7'44*8". 14. 62-489 in. ; 82*167 in. 15. 27 '914 in. 16. 1-945 ft. 17. 2-853 ft. 18. 6-915 ft. 19. 34*6 in. 20. 15*8 in. 21. 15-7 m. 22. 40yd. 23. 43,827,033yd.; 43,827,735yd. 24. 1-093827 yd. 26. 141,000,000 mi. and 140, 400, 000 mi. 26. 899mi. per min. 27. 34 -527 mi. 28. 48 -83 mi. 29. 141*244in. and 98-87 in. 30. 28 45 in. and 35-45 in. ANSWERS. 355 Exercise XXIX. 1. 44 18 sq. in. 2. 153 -94 sq.ft. 3. 13-636 80. cm. 4. 117 -76 ft. 5. 6957 84 ft. 6. 3199 -41 ft. 7. $15-89. 8. 35in. 9. 319071b. 10. 829-58sq.ft. 11. 4-427 in. 12. 386-1468q. ft. 13. 14848 in. 14. 56 37 sq. ft. 15. 0-615 sq. ft. ; 1-133 sq.ft. 16. 7-31 sq.m. 17. 13 54 ft. 18. 16 572 yd. 10.62-O2sq.in. 20. 138 13 sq.ft. 21. 32 •225sq.m. 22. 93-46 sq. ft. 24. 2488-14sq.cm. 26. 15-708sq.ft. 26. 189-69ft. 27. 28-274sq.ft. 28. $9610. 29. 3-1 yd. 30.11yd. 31. $141147. 32. 44 1 in. 33. 40-15sq. in. 34. 136-358q.cm. 35. 6742 ft. 36. 57° 17' 44-8". 37. 33°25'21". 38. 0-0906sq.ft. 39. 0-6142 sq.ft. 40. 0-2854sq.ft. 41.112-2sq.in. 42. 158 57 sq. in. 44. 92-88 sq. in. 45. 2905sq.in. 46. 62-832 sq. ft. 47. 9*487 sq. ft. 48. $206-91. 49. 1269-21 sq. ft. 60. 692 -72 sq.ft. Exercise XXX. 1. 3sq. ft. 108 sq. in. 2. 10 sq. ft. 120 sq. in. 3. 40 sq. ft. 48 sq. in. 4. 19-37 sq. in. 6. 384 sq in. 6. 49 sq. ft. 120-6 sq. in. 7. 52 sq. ft. 134-7 sq. in. 8. 18 sq. ft. 94-89 sq. in. 9. 3ft. Sin. 10. 9 -55 in. 11. 25 -33 in. 12. 12 44 in. 13. 8sq.ft. 45sq. in. 14. 17 sq.ft. '>6-69sq. in. 15. 64sq. ft. 104-66sq. in. 16. 21sq. ft, 111-2 sq. in. 17. 92775 sq.ft. 18. 3026 sq.ft. 19. 7sq. ft. 122-97 eq. in. 20. 33 sq. ft. 118-55 sq, in. 21. 2ft. 8 in. 22. 2 ft. 10-7 in. 2ci. 3-183 in. 24. 3 ft. 1 72 in. 25. 2 ft. 4 8 in. 26. 2 -387 ft. 27. 25 -C9 in. 28. 14§yd. 29.41yd. 30.60°. 31. 2 to 1. 32. 136-5 sq. in. 33. 753-98 sq. in. 34. 1021 -02 sq. in. 35. 130 -627 sq. in. 36. 78 -63 sq.ft. 37. 880 -78 sq. in. 38 8-343 in. and 12-457 in. 39. 45 783 sq. m. 40. 113 1 sq. in. 41 . 45 837 sq. in. 42. 3-39 in. 43. 82 467 sq. ft. 44. 47 124 sq. in. 45. 16-144sq. ft. 46. 6 -065 sq.ft. 47. 0-858 in. or 29 -142 in. 48. t^. 49. 27 in. 50. 196,940,000 sq. miles. Exercise XXXI. 1. 628 32 c. in. 2. 226-194 c. in. 3. 6 77 in. 4. 2 ft. 6. 2Jin. 6. 7 -927 ft. 7. 7 -0663 in. ; 196 0844 sq. in. 8. 185-336 mm. 9. 503 08 mm. 10. 383 92 mm. 11. 13 mi. 1566-2yd. 12. 15783 mi. 13. 1347 45 c. in. 14. 0-1502 in. 15. 2-4 in. 16. 733 037 c. ft. 17. 3927 c. in. 18. 9 81 in. 19. 16889-24 c. in. 20. 5579 47 c. in. 21. 201 •16 c. in. S56 ARITHMETIC. 22. 186-7c.in. 23. 428'83c.in. 24. 4322-84c.in. 25. 571&-108c.m 26. 428 -828 eft. 27. 101 c. ft. ; 78-66c. ft. ; 59-1 eft. 28. 1 -8963111.; 2 -7033 111.; 10*4004 In. 29. 141-87 c. yd. 30.7-836c.ft. 31.2408-66sq.lii. 32. 10 -8573111. 33. 904 -78c. In. 34. 606 -863c. In. 35. 4387 -14c. In. 36. •5096gal. 37. 22 -283 lb. 38. 11 -377 lb. 39. 5-98651b. 40. 110446 c. In. 41. 6 928 In. ; 574-226 cm. 42. 101-274 lb. 43. 47-546 lb. 44. 33-1 lb. 45. 229-303 eft. 46. 2378-9 lb. ; 6626-9 lb. 47. 0-825 hi. 48. 4 -64 in. 49. 1-42 in. 50.0 003787. 51. 1 to 900. 52. 103 to 1000. 53. 7 in. 54. 10-01 in. 55. 20 -123c. in. 56. :a-662 c. in. 57. 433-541 c. in. 58. 25525 -4 c. in. 59. 13 to 6. 60. 0-36 c. yd. ; 0-64c.yd. 61. 3-28392c.in. 62. 928-32c.in. 63. 2-598817c.mi. x IQi? 64. 2-598682c.ini. x lO^i ; 20902046ft. 66. 3173in. 66. 68-G68c.in. 67. 99041 0. in. 68. 253 gal. Exercise XXXII. 7. 25 gal. 8. 6 da. Q. 2|^ wk. 10. 4 A. 3740 sq. yd. 11. §f<)z. 12. 3^ 13. JI7-8125. 14.3:6.5:3. 15. 9:4. 4:9. 16. 17:24. 17. 5:4. 18. 17:7. 19. 9:8. 20. 5:8. 21. 3|lb, 22. 1 to 3. 23. $2100, $2400. 24. $5-62|, $9-37^. 25. lOOA. ' * Exercise XXXIII. 1.242,484,605. 2. 18055 -4 lb., 12036 -9 lb., 9027-7 lb. 3. $17-45, $27-92, $11-43. 4. $13-05, $26-10, $39-15, $39 15, $62-20. 5. $320, $360, $384. 6. $1200, $300, $120, $60, $40. 7. 132 lb., 281b., 201b. 8. $66, $77, $110. 9. $1860. $2112, $2464. 10. 2925, 3640, 4212, 1950, 2496. 11. 648 -1281b. of oxygen, 562 -445 lb. of carbon, 89*427 of hydrogen. 12. 201b. 2701b. 5281b. of nitre and 801b. of sulphur. 13. $2704, $3151, $4046. 14. $134-40, $118-08. 15. 24^3'vlb. of lead, 17Ji^lb. of tin. 16. $6-02. 17. $5000, $8750, $11250. 18. A to C, 30ct. ; B to C, 36 ct. 19. 1991 If lb. ; 202|lb., 46.|lb. Exercise XXXIV. 1 . $160 , $240, $300, $360. 2 . £1 68. 8|d. , £13 6s. 9|d., £14 Os. 2d. 3. 40-3088pt., 62 -7026 pt., 74-6459 pt., 85 -8428 pt. 4. 300 1?)., 120c/i., 280^'. 5. $2100, $3570, $2380, $4550. 6. A, $1085 70; B, $1034; C, $1155. 7. A, $640; B, $200 ; C, $300 ; D, $180. ANSWERS 357 Exercise XXXV. 1. $6, $3 60. 2. $600, $840, $300. 3. 88 ct., 38^ ct., 10^ ct. 4. $900, $750. 6. $522, $536, $502 '50. 6. $4507 -06 ; $4965 -41 ; $5527 '53. 7. 37 5, 15, 312- 5. 8. 78001b. , 65001b., 52001b. 9. $225, $150, 90 ct. 10. 12, 18, 5, 46. Exercise XXXVI. 1. $147, $196, $147. 2. $10-82, $2029, $60-89. 3. $2-50, $l-87i U'^H- 4. $38-25, $38, $37-80. 5. $40, $42-30, $43-20. 1. 2500 bu., 4000 bu., 10000 bu. 16, 28, 40. 4. 15, 20, 50. 6. 99-8ct., Exercise XXXVII. 2. 144 7n. 351m, 4806. 3, 66-53 ct., 8317 ct. Exercise XXXVIII. 2. A, $240 ; B, $180. 3. $13-50, $30. 5. $79-98, ^ 1. $780, $801-25, $426-79, $991-96. $2565, $1425, $1710, $1140. 4. $9, $17 14, $11-99, $143-38. 6. 144 m., 480 6., 351 -u;. 7. $2-10, $1-50, $1-08. 8. $2-23, $2-23, $1-59, 95 ct. or $2-42, $2-42, $1-41, 75 ct. 9. $369, $399, $432. Exercise XXXIX. 1. $656-25, $109375. 2. $107030, $789, $1296-70. 3. $18004-83, $13495-17. 4. $20419 64, $34486-50. 5. $13138-05, $10019-95. 6. $8800, $10400. 7. $16434-78, $24260-87. 8. $185 '81, $126 99, $121-40. 9. $2216-98, $2480-08, $3152-94. 10. $9368*20, $10021-80. 11. $13000, $8000. Exercise XL. 1. $14-38. 2. $823-82. 3. $12300. 4. $91500. 5. $564. 6. $59062-50. 7. 1|- %. 8. 12* %. 9. 20 %. 10. $6058-95, 95%. 11. $105263, $1422-37. 12. 21-5%. 13. $7163-84. 14. 120 %. 15. 65600, 67240, 68921. 16. 130050, 127500, 125000. 17. $20000. 18. $2000. 19. 72 gallons. 20.3538641b. 21. $31250, $32812 50, $43750, $43125. 22.53-7%. 23. 4096. 24. 1122:1125. 25. 37 .32 in work. Exercise XLI. 1. $2425. 2. 150%. 3. 10%. 4. 4% loss. 6. 38f%. 6. 12% gain. 7. $2400. 8. $210. 9. 24«. 10. 14/^%. 11. 18ct. 12. 9d. ; 512. 13. $140. 14. 71-':5T. 15. 2ct. 16. 12|%. 17. $45, $30. 18. $40. 19. $2 08. 20.14 13%. 21. $25, $30. 22. 16 ct. 23. 751b. at37|ct., 691b. at 34Ut 24. $12. 25. $160, $120. 358 ARITHMETIC. Exercise XLII. 1. «131-25. 2. $471-25. 3. 1232-47. 4. ^32-60. 6. $l68b, $1120, $700. 6. $4500, $3600, $2400- 7. $82-50. 8. 0-8%. 9. $187 '50; If %. 10. $1980, $2640. 11. $2393-62, $5106-38. 12. 1^ %. 13. 1% ; 87ict. per $100. 14. $46-87^ 15. i %. 16. $2000. 17. $12500. 18. $245. 19. $88333-33, $14000. 20. $1851-94. 21. $4273-50. 22. $18918 23. 65 ct. per $100. 24. $5930. Exercise XLIII. 1. $81, $1539. 2. $6-43. 3. $6240. 4. $3264. 6. $3055. 6. $500. 7. 3%. 8. 3 yd. 9. $18000. 10. $2. 11. $480. 12. 6%. 13. $61200. 14. $1388-62. 15. $288-39. 16. $18909 18. 17. 66521b., 19011b. 18. $25663 44. 19. $9653-38. 20. 2 %. 21. 2^ %. Exercise XLIV. 1. 2. $747-80. 3. $5-40. 4. $175. 5. 28%. 6. 31^%. 7. $50. 8. $1-27: 3^%. 10. 25%. 11.20%. 12. 14^%. 13. 16|%. 14. 20%. 1^ 25 17% 16, 20%. 17. 13%. 18. 20%. Exercise XLV. 1. $308 02. 2. $960-12. 3. $445-17. 4. $78-54. 5. $573-04. 6. $382-90. 7. 8%. 8. 6%. 9. $432-38. 10. $751-28. 11. June 4. Exercise XL VI. 1. $85-19. 2. $28-90. 3. $14-33. 4. $11. 6. $33-61. 6. $2-19. 7. $34949. 8. $504-96. 9. $333-33. 10. $831-60. 11. $72-51. 12. $98-40. 13.7%. 14.7*% 15. 6|%. 16. 5%. 17. 6%. 18. lyr. 276 da. 19. lOOda. 20. Oct. 13. 21. 20 yr. 22. 25 yr. 23. 8%. 8-034%. 24. $7-27; $7-14. 25. 11^%. 26. 10-267%. Exercise XLVII. 1. 12 Nov. 2. 17 Dec. 3. 30 June. 4. 10 Oct. 5. 2 Nov. 6. 4 July or 8 July, 1889. Exercise XL VIII. 1. $22-58. 2. $2364-33. 3. $71-41. Exercise XLIX. 1. $92610; $12610. 2. $497*19; $72-19. 3. $281-38; $31-38. 4. $404-83; $38-16. 5. $766-95; $44-45. 6. $106. 7. $1-0609. 8. $1061364. 9. 1 -J 67625. 10.1276281. 11. 1-4071. 12. 1040604. 13. 1082857, 14. 1126825. ANSWERS. 359 Exercise L. 1. $1559-20; |«09-20. 2. ^9-23; $294-23. a $2770-89; $1520-89. 4. $48 90; $12-65. 5. $650-17 ; $222-67. 6. $4792 -20 ; $4667 20. 7. $1470268 ; $1470268 ; $1 796076 x 10^ * ; $1-829594x102 2. 8. $456 39. 9. $257 20. 10. $256-61. 11. $62-50. 12. $41-67. 13. $4826. 14. $48-72. 15. $48-95. 16. 8>r. 17. 6yr. 314 da. 19. 29 yr. 325 da. 20. 29 yr. 129 da. 21. 29 yr. 25 da. 22. 2i %. 23. 9%. 24. Nearly 5%. 25. 53yr. 29 da. 26. 114 yr. 34 da. 27. 6-167%; 6-183%; 6-184%. 28. 3-6^%. 38-94. Exercise LI. 1. $3275. 2. ^1237-50. 3. $32700. 4. $10468-75. 5. $69231-25. 6. $6792-04. 7. $182062-50. 8. $111 JOO. 9. $22812-50. 10. $712-50. 11. $4200. 12. $210. 13.324%. 14. 125:126. 15. IIOJ. 16. 117i. 17. 90§. 18. 280. 19. $15807-63. 20. $19989; $827*50; 4-14 %. 21. $18000; 840. 22. $160000. 23. $35938-44. 24. £5625. 25. 102|. Exercise LII. 1. $36500. 2. $109750. 3. $288. 4. $7517. 5. £20 lOs. ll|d. 6. £112^< lis. 3d. 7. $241111-11, $4838889; $96888-89, $193840; $96833-33, $96722-22, $96750, $96805-56, $?02000, $292000, $389333 '33. 8. $464-89. 9. $464-02. 10. $570-75. 1 1 . 95f . 12. The drafts at 60 days' sight. $648 51. 13. $2098 -05. 14. $298-89. 15. $4-84§. 16. 94§. 17. 9^ 18. 5 18. 19. 95i. 20. £986 13s. 4d. 21. 19552 fr. 22. $28-86. 23. $38-48. 24. $30538 93; 358 664 gr. ^ ' !!■ ir 360 ARITHMETIC. • . ,1: tk CORRECTIONS. Page 12, line 5 up ; after length iusert is the yard which. Page 77, Prob. 20 ; after 1889 insert and payable 9 July, 1889. Page 82, Prob. 70 ; after discounted insert at 8 % . Page 83, Prob. 81 ; for $6*60 read $29*40. The amwers ivill then be $5-39 and $6 '60. Page 84, Prob. 90 ; for payable in read drawn at. Page 98, line 7 up ; omit of e quality. Page 123, Prob. 141 ; /or ^2 read ^S. Page 137, Prob. 24; insert x3^3 x78xll x 13^2' ^5^-f-17. Page 151, line 13up ; m second deiiominator, fv (> readO. Page 151, line 5 up ; /or -^10-l read x 10"^. Page 155, line 9 up ; for partial product read partial products. Page 162, line 5 up ; for HK read GK. Page 187, line 10 ; for ah(2hi H-ftg) read lab(2bi+bs). Page 192, Prob. 34 ; /or 3 in. read 6 in. Page 200, Prob. 108, figure ; join FB and HC. Page 206, Prob. 13 ; for 22 ft., 6ft. and 3 ft. read 53 ft., 48 ft. and 43 ft. Page 213, line 15 up ; for k read /vj. Page 223, Prob. 6 ; for diameter read diameters. Page 249, Prob. 49 ; /or 75 -Sib. read 45 •51b. awl insert Ih. after b7'Q. Page 307, Prob. 23 ; for at a total loss of $1943-90 read at a loss of $3514*75 on the amount realized by his former sales. The a>iswer will then he $64980. Page 316, line 12 up ; for '574 recti* "574. ■.-:;a^' ■,jrj*.- ih. ly, 1889. .;■■,/ /,ot> //:: .y . .'■