B 
 
 m 
 

 ^ 
 
SYSTEM 
 
 OF 
 
 PRACTICAL ARITHMETIC, 
 
 APPLICABLE TO THE 
 
 PRESENT STATE OF TRADE, 
 
 * 
 
 \ AND 
 
 MONEY TRANSACTIONS: 
 
 ILLUSTRATED 
 
 tp 
 
 BY NUMEROUS EXAMPLES UNDER EACH RULE ; 
 
 FOR 
 
 THE USE OF SCHOOLS. 
 
 BY THE REV. J. j'&YCE. 
 
 THE FOURTH EDITION, REVISED AND CORRECTED. 
 
 LONDON: 
 
 PRINTED FOR RICHARD PHILLIPS, 
 
 NEW BRIDGE-STREET. 
 
 ^nd sold by SHERWOOD and Co..^ ^Paternoster-row ; WILSON and O,, York ; 
 M oz LEY, Gainsborough ; STODDART and Co., Hull; DoicandCo., 
 Edinburgh; KEENE, and WOGAN and Co., Dublin; 
 and by all other Booksellers ; with a full 
 Allowance to Schools. 
 
 1812. 
 I Price Three ShUUngs and Slxp&ice, Bound.] 
 
James GjJtet. Printer, mvn-gourt, Fleet-stfeet, London, 
 
T6 
 PREFACE, /e/2. 
 
 IN presenting a new System of Arithmetic to the Public, 
 some account of its plan and execution will be expected. 
 It is hoped, that the title of the present Work will briefly 
 explain the views of the Author, who, from his own ex- 
 perience in the business of education, has long been con- 
 vinced, that, among the excellent introductory books to 
 this science, no one is sufficiently adapted to the occasions 
 of common life. Some are too abstruse for novices, while 
 others are defective in such examples as point out the ap- 
 plication of the several rules to transactions of real business. 
 
 If the Author of this System of Arithmetic has not de- 
 ceived himself, he has supplied these efficiencies; raid he 
 appeals, without apprehension, to that Public, whose can- 
 dour and liberality he has already experienced, to deckle 
 upon this attempt to render the elementary rules of Arith- 
 metic at once practical and popular. 
 
 There are few children who <Io not experience some dis- 
 gust in passing through the first four rules; occasioned, with- 
 out doubt, by the paucity of examples, and by the want of 
 interest in those that are given. The Author has therefore 
 devoted a large portion of this work to the earJy rules, aad 
 has illustrated them by miscellaneous questions, in which 
 will be found much useful information, applicable in the 
 progressive stages of life. 
 
 The modes of treating the' Rule of Three, of illustrating 
 
 Vulgar and Decimal Fractions, Practice, &c. &c., will best 
 
 speak for themselves. But a reason may be demanded for 
 
 the introduction of Logarithms, and for the particular 
 
 ^method adopted in those parts in which the doctrine -of 
 
IV PREFACE. 
 
 Annuities, Reversions, Leases, &c. i& illustrated. The 
 working of Logarithms is extremely simple, and it appears 
 to be high time to introduce that great discovery into the 
 ordinary processes of Arithmetic, and no longer to leave its 
 advantages to those only who have leisure to cultivate the 
 higher branches of the Mathematics. The uses of Loga- 
 rithms are eminently great in facilitating many calculations 
 and being, at the same time, of such easy attainment, it 
 is matter of wonder that they have not been taught long 
 ago in the operation of Raising Powers and Extracting Roots, 
 and in their consequent connection with the other parts of 
 Arithmetic. At any rate it appeared to the Author to be 
 highly proper, if not indispensable, .to give, in one view, 
 the different modes of calculating Compound Interest, and 
 other similar rules, by COMMON and DECIMAL ARITH- 
 METIC, and also by LOGARITHMS. 
 
 A Book of Arithmetic for Schools should contain eveiy 
 thing necessary to be known previously to the study of 
 Algebra; and for the sake of those who wish to proceed to 
 that science, it should be introductory to it. This, it is 
 believed, will be found to be one of the characteristics of 
 the Work now offered to the Public. There are, however, 
 thousands who never trouble themselves to learn beyond 
 the elements of the arithmetic which they acquire at school, 
 who, looking to trade and commerce as the objects of their 
 future lives, seek only for that knowledge which, in some 
 way or other is applicable to those objects. These, in al- 
 most every rank and situation, have frequent occasion to 
 calculate the INTEREST of money and the DISCOUNT of 
 bills; and to ascertain the value of ANNUITIES on single and 
 joint lives; of SURVIVORSHIPS; of LEASES; of REVER- 
 
PREFACE. V 
 
 SIONARY INTERESTS; of FUNDED PROPERTY; and of 
 
 FREEHOLD ESTATES. 
 
 On the doctrine and usage of Exchanges, so much of the 
 theory is given as will, it is presumed, render the subject 
 familiar and interesting ; and so much of the practice as 
 shall enable a youth, on his entrance into a counting-house, 
 to apply his knowledge to any particular country. In this 
 article, as in that which relates to the buying and selling of 
 Stocks, are given illustrations and explanations of the va- 
 rious items in the Stock and Exchange Tables, which are 
 to be found in the public papers. 
 
 For the assistance of Masters, the Author has prepared 
 a Key to the entire Book, containing solutions to many, 
 and answers to all the questions ; to which lie has subjoined, 
 aPractical System of MentalArithrnetic, that may be taught 
 to boys before they leave school : and in the new edition 
 will be found a new and peculiar method of setting exam- 
 pies in the first rules, now originally published, wholly 
 calculated to relieve the labour of the teacher, as by this 
 means the number of the examples may be increased indefi- 
 nitely, to which the answers will be instantly known by the 
 preceptor, though to the pupil they will be inscrutable., 
 
 It is the intention of the Author of this work speedily to 
 publish a treatise on Algebra, of the size of the present vo- 
 lume, and on a similar plan of practical utility. In this 
 will be given demonstrations of the rules of common Arith- 
 metic, which necessarily depend on Algebra for solution, 
 and which, on that account could not, with propriety, be 
 introduced into a System of PRACTICAL ARITHMETIC. 
 
The Attention of the Conductors of Schools, and of Bo&k- 
 sellers in general, is invited to the following List 
 
 OF NEW AND VALUABLE ELEMENTARY BOOKS. 
 
 . s. d. 
 
 Blair's Practical English Grammar, with copious Exercises, 
 containing every thing, essential, and nothing superfluous, 
 complete in this single Book . . . . . 026 
 
 Class Book, or 365 English Lessons . . . 056 
 
 v Ileading Exercises, for Junior Classes . . 0-26 
 
 Gram mar of Natural Philosophy, by which the various 
 
 branches of Science may, for the first time, be practically 
 
 taught in Schools . . . . . . . 036 
 
 -' ' Grammar of Chemistry, new edition . . 4 O 
 
 ' First Catechism, containing common things necessary 
 to be known by all Children . . . . . 009 
 
 School Dictionary, for- learning by rote, containing 
 
 none but important words, and omitting derivative, vulgar, 
 obscene, and trivial words . . . . O 2 C 
 
 '^Smith's Grammar of Geometry . . . . . 036 
 
 Mavor's Circle of the Arts and Sciences, containing the^first 
 
 Principles of every Branch of Knowledge . . . 050 
 
 Goldsmith's Grammar of the Laws and Constitution of Eng- 
 land, for the Use of Schools and Junior Students of Law 4 X) 
 
 Grammarof Geography, including the Use of the 
 
 Globes, numerous Exercises, Questions, &c., &c., with 
 
 maps, &c. . . . . . . . 036 
 
 Geographical Copy Book, or Skeleton Maps, to 
 
 be filled in by the Student. Parti 3 o 
 
 Do. Do. Do. Part II. . . . . o 3 O 
 
 ' " . Popular Geography, containing all the interesting 
 
 Features of that Science, with 60 plates . . . 14 o 
 School Atlas, or Key to the Geographical Copy 
 
 Books 050 
 
 Robinson's Grammar of History, by which that Subject may 
 
 now be practically taught in Schools . . . . 036 
 
 Mortimer's Grammar of Trade and Commerce . . 036 
 
 Irving' s Elements of English Composition . . . O 7 6 
 
 Gregory's Letters on Literature and Composition, 2 vols. o 13 O 
 
 Thornton's Grammar of Botany, with numerous Plates 070 
 The Book of Trades, with sixty Engravings, descriptive of all 
 
 the useful Arts, 3 vols., each . .. . . . -036 
 
 Morrison's Elements of Book-keeping, by Single and Double 
 
 Entry, with Copper -plate Forms, &c. - . . 070 
 Crocker's Land Surveying in all its Branches and Varieties, 
 
 with coloured Plates, &c. . . -, . . o .'7 '-0 
 
ARITHMETIC. 
 
 ARITHMETIC is the science which explains the various 
 methods of computing by numbers* 
 
 All its operations are performed by Addition* Subtraction, 
 Multiplication and Division. 
 
 NUMERATION OR NOTATION. 
 
 JVhen two or more figures are placed together 9 the first, 
 or right-hand figure is taken for its simple value; the se- 
 cond to the left signifies so many tens; the third so many 
 hundreds ; and the fourth so many thousands ; and so on, 
 according to the following Table : * 
 
 *o ri *o 43 .2 of 
 
 s i 4 1" * * ! ' s .* 
 
 ^ Jf V * S: P -a ^ 
 
 gS C o rs JS c -^ Js *5 
 
 & *g ~ ^ X" H ~ H ffi H > 
 
 5 431 26978 
 
 Tims figures, besides their common value, have one which depends 
 upon the place in which they stand when joined to others; 6 and 5 arc 
 read six and five; but if they stand together, 65, they are read sixty-five. 
 The figure 5 on the right hand denotes its simple value only, but the 6, 
 from its situation, becomes ten times greater than its simple value, or 
 sixty, therefore the two together are called sixty-five. 
 
 If there be three figures, as 973, the first denotes its simple value, as 
 eight ; the second a value ten times greater than its simple value, as 
 seventy ; and the third is a hundred times greater than its simple value, 
 as nine hundred : the figures together are read nine hundred and seven* 
 ty-eight. 
 
 In this manner, the value of each figure to the left is always ten 
 times greater than it would be if it stood in the next place on the righ*$ 
 
 * The Tutor is recommended to direct the Pupil to commit to me- 
 mory y.11 the passages which are printed in Italic characters, and like- 
 wise all the tables. 
 
 B 
 
2 NUMERATION* 
 
 thus 6C66, the first figure 6 is simply six, the next is sixt) r , the third 
 six hundred, and the fourth six thousand ; the whole number is readj, 
 Six thousand six hundred and sixty -six. 
 
 The frst six figures in the table above are read, One hundred twenty- 
 six thousand nine hundred and seventy-eight. The whole period of 
 nine .figures is thus read, Five hundred and forty -three millions, one 
 hundred and t\\enty-six thousand, nine hundred and seventy-eight. 
 
 The enumeration of figures may be carried much further 
 according to the following Table : 
 
 rt 
 to 
 
 o 
 
 IS 
 
 IS 
 
 . 
 
 1^1 
 
 '6 
 
 
 8 
 
 CJ tn 
 
 c 
 
 
 la 
 
 e 
 
 o 
 
 IM 
 
 O 
 
 1 
 
 Jj c* J 
 
 o 
 
 
 .2 
 
 IL . 
 
 
 *Cl ^5 
 
 -o 15 
 
 
 
 c 15 
 
 8 
 
 ll 
 
 c .2 
 
 3 
 
 15 
 
 8 | 
 
 ill If, 
 
 5 e S 3 5 
 
 undreds 
 
 | 
 
 1 w . 
 0.1 
 
 C 7^ 
 
 M -c w 
 
 "0 C 3 "O 
 
 C rt 2 o G 
 3 M *r ^ 3 
 
 g'S 
 
 a 
 
 
 
 H 
 
 
 
 3 
 
 E H H 
 
 
 
 
 H S 
 
 ffi t-> ^ S 
 
 ht) 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 6 
 
 4 6 7 
 
 9 
 
 
 
 5 1 
 
 4 627 
 
 5 S 
 
 V 
 
 
 
 
 
 "*- ^ 
 
 *^<: 
 
 ae 
 
 -^ 
 
 S_ - ^i 
 
 ,, > 
 
 vr *m* ^~ 
 
 In large numbers it is common to divide them into periods of six 
 figures each, and half period? of three figures. The foregoing three 
 periods are read One hundred twenty -three thousand, four hundred 
 and fifty-six billions, four hundred eighty-seven thousand, nine hun- 
 dred and fifty one millions, tour hundred sixty-two thousand, seven 
 hundred and fifty- three.* 
 
 Hence the following- general 
 
 RULE. To the simple value of each figure, join the name 
 qfits place according to the situation in the series, as 
 jrethy thousands, millions, bijlion$ 9 trillions, fyc. 
 
 * The names of the higher period* after Billions, are Trillions, Quad- 
 rillions, Quint-, tit ous, Scxtillions, Septiliiovs, OctiHions, ond Nonil- 
 limiSy each period consisting n/ six places of figures. The first three 
 of every period are so marry Units of it, and the Latter, or left hand party 
 so many Thousands. The following Table contains the whole series: 
 
 TABLE. 
 Nonillions Oc illions. Septillions. Sextillions. Quintillions. 
 
 1-23,456 456,789 567,345 321,234 458,764 
 
 Quadrillions. Trillions. Billions. Millions. Units. 
 
 874*321 374,532 459,876 538,764 459,579 
 
NUMERATION. 
 
 S 
 
 EXAMPLES IN NUMERATION AND NOTATION* 
 
 Head, or write down in "words, the value of the following 
 Numbers : 
 
 Ex. 
 
 l. 
 
 2, 
 3. 
 4. 
 5. 
 6. 
 7. 
 8. 
 9. 
 
 19 
 244 
 3045 
 45060 
 69305 
 93614 
 564875 
 4500342 
 5687041 
 6843700 i 
 
 Ex. 21. 340 
 
 22. 436901 
 
 23. 36Q45 
 
 24. 9874000 
 
 25. 654328 
 4328764 
 856540 
 43760000 
 37004 
 
 26. 
 27. 
 
 28. 
 29 
 
 30. 85600341 
 
 Ex. 11. 40005 
 
 12. 324060 
 
 13. 400369 
 
 14. 76*5 
 
 15. 564001 
 
 16. 43976* 
 
 17. 9300042 
 
 18. 70000021 
 19- 35000 
 20. 50000000 
 
 Ex. 31. 456074328 
 5900007643 
 68670/49004 
 
 34. 876430786453 
 
 35. 1000000843213 
 
 36. 34876543218764 
 
 37. 594632171834765 
 87643285176487589 
 1234507S90001259 
 987654321123456789 
 
 "vVrile down 'the figures answering to the following Examples, 
 
 : Ex, 1. Thirty-nine. 
 
 2. Four hundred and sixty-nine. 
 
 3. Two thousand and one. 
 
 4. Thirty -five thousand and twenty-eight. 
 
 5. Three hundred and seventy-six thousand. 
 
 6. One million and' fifty-nine. 
 
 7. Eighty- seven millions, five hundred and eighty thousand, one 
 
 hundred and nine. 
 3. Five huadred seventy six millions, three hundred twenty-five 
 
 thousand, three hundred and ninety-one. 
 9. Eight hundred millions and eighty. 
 10. Three hundred and three millions and thirty-one.* 
 
 32. 
 33. 
 
 38. 
 39. 
 40. 
 
 NOTE. 
 
 '* Besides these ten examples, it will be desirable that the pupil should, 
 after having written the preceding forty examples into word-;, wrire 
 them back again into figures, without the assistance of the book; He 
 should likewise be desired to mention the value of each line in the sub- 
 sequent examples in Addition, as well as the sum total ; by these means 
 Numeration will, in both its parts, become perfectly familiar to him, 
 B 2 
 
4 NUMERATION. 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. 1. By the late enumeration of the people, the number of in- 
 habitants in England is put down at nine millions, three hundred forty- 
 three thousand, five hundred and seventy-eight ; and the number found 
 to be in London was eight hundred eighty-five thousand, five hundred 
 and eighty-seven ; How are these numbers expressed in figures? 
 
 Ex.2. The world was created two thousand three hundred and forty- 
 eight years before the Deluge; three thousand two hundred and fifty-one 
 years before the building of Rome ; four thousand and four years before 
 the birth of Christ, and five thousand and fourteen years before the pre- 
 sent time [l 81 1] : Let each of these numbers be expressed in figures. 
 
 Ex. 3. Express in words the distances of the primary planets from 
 the Sun, which are as follow : 
 
 Mercury .... 37,000,000 Venus .... 66,000,000 
 
 The Earth . . . 95,000,000 Mars .... 145,000,000 
 
 Jupiter .... 493,000,000 Saturn . . . 903,000,000 
 
 TheHerschel . . i,8!3,ooo,ooo miles.* 
 
 FRACTIONS, or broken numbers, are expressed in the following 
 manner : A halfpenny is denoted by J ; a farthing, by %, being the 
 one-fourth of a penny; and three farthings by f, being three-fourths 
 of a penny. Thus it appears that a fraction is any part or parts of 
 an unit, and is expressed by two numbers separated from each other 
 by a short line. The lower number shows how many parts the unit 
 is divided into, and the upper figure points out what number of 
 these parts are contained in the fraction : thus ^, when standing for 
 
 * The ancient "Romans, in their Notation of Numbers, made use of 
 the following five letters : I, V, X, L, and C, which, singly, stood for 
 one, five, ten, fifty, and a hundred. By repeating and combining 
 these, any other numbers were expressed: thus II, signified two', III, 
 i\rec\ XX. twenty \ CC, (wo hundred, and so on. The rules for 
 Eornan Notation are as follow : 
 
 1. The annexing a letter of a lower value to one of a higher, in- 
 creases its value, or denotes the sum of both , as VI, signifies six ; XII, 
 denotes twelve; LV, fifty-five; LXX VI, seventy- six; CLII, one hun- 
 dred and fifty-two. 
 
 2. The prefixing a letter of a lower value, to one of a higher, sub- 
 tracts their values, or shows their difference thus, I prefixed to V, or 
 IV, is /our ; IX, nine ; XL, forty ; XC, ninety , &c. 
 
 For the sake of abbreviation, the Romans introduced these marks:- 
 JQ , five hundred ; ClQ, a thousand : these, in process of time, were 
 written DM, so that now the D signifies five hundred, and the M. a 
 thousand ; but in the titles of many old books we find the ether mode 
 cf Notation. The following table will exhibit every thing necessary to 
 be known on this subject : 
 
 Table. 
 
NUMERATION. 5 
 
 three farthings, shows that a penny is divided into four parts, the 3 
 determines the number of the parts, and we call it three-fourths of a 
 penny. 
 
 Inches are usually divided in eighths, or eight parts in each inch *, 
 and the fractional parts are thus expressed: 
 
 I means three-eighths. | means five-eighths. 
 
 I means seven-eighths. | means four-eighths, equal to one half. 
 
 Sixteenths are likewise in common use, and we say, 
 
 T ~ live sixteenths. ~ eleven sixteenths, 
 -g- three sixteenths. 44 fifteen sixteenths. 
 
 I 
 
 
 TABLE. 
 LX 
 
 , . . 60 
 
 11 
 
 ... 2 
 
 LXX 
 
 . . . 70 
 
 Ill 
 
 
 I XXX . .... 
 
 
 IV, or IIII , 
 
 ... 4 
 
 xc 
 
 . . 9O 
 
 
 
 c 
 
 190 
 
 VI . . 
 
 ... 6 
 
 ci 
 
 101 
 
 VII 
 
 
 ccc 
 
 . . . SOO 
 
 VIII 
 
 
 
 . . . 500 
 
 IX 
 
 
 I3C, or DC . . 
 
 . . . 600 
 
 X 
 
 
 IQCCC or DCCC 
 
 
 XI 
 
 ... 1 1 
 
 IQCCCC, orDCCCC or CM . . 
 
 . . . 900 
 
 XII 
 
 12 
 
 
 . . . 100O 
 
 XIII . . . , 
 
 . .13 
 
 CI^C or MC 
 
 . , . 1100 
 
 XIV 
 
 
 MM, orTT* 
 
 
 XV 
 
 XVI 
 
 ... 15 
 
 1 fi 
 
 100 t> or V 
 
 . . . 500O 
 
 XVII . . . 
 
 ... 17 
 
 
 . 600O 
 
 XVIII . . 
 
 . ... 18 
 
 IOOMMM or VIII 
 
 8000 
 
 XIX . . . 
 
 XX 
 
 ... 19 
 
 o o 
 
 
 . . . 10000 
 
 XXI . . . 
 
 . ... 21 
 
 CCI / '~\ r i]yi or XI 
 
 1 1 00O 
 
 XXX . . . 
 
 , ... 80 
 
 
 5000O 
 
 XL .... 
 
 . ... 40 
 
 T^^^ IV1 IVf 
 
 5^000 
 
 XLI . . . 
 
 ... 41 
 
 
 . . 101000 
 
 L . 
 
 . 50 
 
 r 1 !^ T^rrr 1 VT nr TM "nrrr 1 . YT 
 
 1 O 1 T 
 
 * The word thousand is often expressed by a line drawn over the top 
 
 of a number; thus X signifies ten thousand, and M a thousand thou- 
 sands, 
 
 f The annexing 3 to the number j^, increases its value tea times: 
 thus 33 is 5000, and 1333 is fifty thousand. 
 
 J The prefixing C, and at the same time annexing a 3 to the num- . 
 berCIC, makes its value ten times greater; CCI33 is 10,000, an 
 i is 100,000. 
 
a 
 
 ADDITION. 
 
 ADDITION teaches the method of finding- the sum o* 
 total of several numbers. 
 
 RULE. (1.) Place the numbers under one another, $&>- 
 thai units may stand under units, tesi-s under tent,, tVc. 
 
 (2.) Add up the figures in the row of twits: set down 
 what remains above the even tens, or if' nothing remains, a 
 cypher, and for the tens carry as many ones to the next 
 column** 
 
 (3.) Add up the other rows in the same manner, and in 
 the last column put down the whole sum contained m it. f 
 
 Ex. 1. What is the sum of 3684, 4863, 365, 29, 56874, 
 and 600 ? 
 
 3684 
 
 4803 
 
 365 
 
 29 
 
 56874 
 609 
 
 Answer .... 66424 is the sum total. 
 
 PR^ OF. Add the numbers together in a contrary order, beginning at 
 the top instead of the bottom. 
 
 * Ten on the right-hand line is equal only to one, or unit, in the 
 next line on the left of it, as ws have seen in Numeration : when 
 therefore the. sum of any column amounts to, 01 exceed- ten, or any 
 number of tens, we carry unit for every ten to the next column ; for Q 
 being the highest digit, any number above it icquires more thon one 
 place to express ir, which is done by removing the tens as so many 
 units to the next place. 
 
 f* The following Table is thought by some persons to be proper to 
 be committed to memory. The use of it maybe easily explained to 
 children of five years old, and when once learnt completely, no difficulty 
 will be found in Addition; for if the pupil knows, at first thought, the 
 sum of any two of the digits, the rest is easy : for instance, if he knows 
 that 6 and ; are thirteen, he will know that 36 and 7 are 43, because c 
 
345 
 489 
 204 
 695 
 731 
 27 
 
 2491 
 
 ADDITION. 
 
 EXAMPLES. 
 
 8776 
 6734 
 5709 
 9564 
 3218 
 4507 
 
 38508 
 
 293068 
 
 and 7 being 13, he knows there must be a three in the answer to ths 
 Question of how many are 36 and 7, or 46 and 7, and so on. 
 
 ADDITION TABLE. 
 
 1 
 
 z 
 
 3 
 4 
 
 2 
 4 
 
 5 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 10 
 
 9 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 11 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 32 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 5 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 6 
 
 8 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 7 
 
 11 
 
 12 
 13 
 
 13 
 
 14 
 
 15 
 
 16 
 
 8 
 
 10 
 
 11 
 
 12 
 
 14 
 
 15 
 
 36 
 
 17 
 
 9 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 17 
 
 18 
 
 To use this table : Take the greater of the two digits, whose sum is 
 sought, in the upper line, and the lesser on the left-hand column ; in 
 the same iine with this, and underneath theother,stands the sum sought, 
 If I want to know the sum of 8 and 5, 1 look for 8 on the head line, and 
 on the same row of figures with s on the left hand side stands 13, the sum. 
 
 Thistaole may be concerted into a SUBTRACTION TABU, (seep. 12) ? 
 and the use of it, in this way, is " To find the difference of any two 
 numbers." Look for the largest number in the same line in which the 
 least stands on the left hand column, and the difference will be found in . 
 the head line over the largesi number. Thus if I want the difference 
 between 7 and 16, I look for 16 in the same line in which 7 stands, in 
 the left fiand column, and in the. head line, above the i& I find 9, ths 
 
ft ADDITION, 
 
 Ex. J. 2 Ex. 2. 4 Ex.3. 7 Ex.4. 3 Ex.5, 4 Ex. 6. 4 
 458406 
 004553 
 236694 
 9 4 2 7 81 
 223865 
 5 4 7 7 * 
 375527 
 214945 
 
 Ex, 7. 8 Ex.8. 3 Ex.9. 5 Ex.io. 1 Ex. 11. 9 Ex.12. 81 
 
 13 7 S6 42 8 32 
 
 4 28 7 3 7 3 
 56 6 44 4 1 4 
 
 553525 
 
 940 66 3 00 
 
 5 38 85 207 
 16 21 2 7 5 8 
 
 4 5 1 29 4 3d 
 
 Ex, 13. 2* Ex. 14. 3 Ex.15. 4 Ex. 16. 5 Ex. 17. 6 Ex, 18. 7 
 2 3 4 5 a 7 
 234567 
 234 567 
 234 567 
 234567 
 234567 
 234567 
 S 3 4 5 6 7 
 
 NOTE. 
 
 * This and the seven following sums may be rendered very useful 
 in Chewing the pupil the foundation of the Multiplication Table ; thus 
 he may be desired tu take two or th-ee rows of each of the eight sums 
 < n bis ^ate, and add them up, he will then see how three twos, or d 
 times 1 make 6 ; how three fours, or three times 4 make 1 2, and so of 
 the rest. When he hh- dom* the eight sums consisting of two figures, 
 he may be required to comrui the results to memory, which will be 
 rendered a very easy business, when he sees the results hefrue him on 
 the slate. Let hir> then proceed to the eight sums consisting of three 
 figures each ; then wi ii tour figures, and 60 un till the whole nine figures 
 are finished, and the table learnt* 
 
ADDITION. 
 
 Ex. 19. 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 
 Ex. 20. 9 Ex. 21. 24 Ex. 22. 56 Ex. 23, 
 
 35 
 64 
 28 
 7* 
 19 
 45 
 35 
 64 
 
 65 
 
 74 
 48 
 82 
 98 
 33 
 66 
 59 
 
 8? Ex. 24. 
 
 33 
 
 29 
 
 86 
 
 39 
 
 45 
 
 20 
 
 31 
 
 99 
 
 25 
 
 64 
 88 
 64 
 77 
 25 
 66 
 33 
 
 Ex. 25. 5162 
 4876 
 4008 
 3079 
 1234 
 2341 
 3168 
 
 Ex.26. 7640 
 39 
 
 5784 
 4304 
 9865 
 6543 
 2871 
 
 Ex. 27. 49325 
 24609 
 
 37485 
 16004 
 23348 
 32946 
 329 
 
 Ex. 28. 5432 
 5789 
 1234 
 5678 
 9123 
 4009 
 5746 
 
 Ex. 30.*49603 
 
 50792 
 
 4652 
 
 49S5J 
 
 654 
 
 * 78432 
 
 * The teacher may, from the three examples in p. 10, form for his 
 pupil an indefinite number, by desiring him to copy on his slate the 
 first three, or four, or five, or any other number of lines : or he may 
 desire him to take only a. single column, or half a column, or the half 
 of two or of three columns, according to the progress 'he has already 
 made. 
 
 To make young persons ready and accurate in Addition, which is of 
 vast importance in almost every situation of life, the master may call a 
 class round him , who have the same sum on their slates, and desire them 
 to add each a figure till the sum is done ; a place in the class to be lost 
 whenever there is a mistake or a pause. The sum neatest set down to 
 take precedence in the first instance. 
 
 B5 
 
10 
 
 ADDITION. 
 
 Ex. 31. 
 
 67543 
 
 896/3 
 
 56789 
 
 22345 
 
 67890^ 
 
 12932' 
 
 45764 
 
 S5365 
 
 J2345 
 
 54321 
 
 67854 
 
 58108 
 
 4Q328 
 
 08/65 
 
 43200 
 
 87219 
 
 Ex. 32. 
 
 93217 
 
 Ex.33. 8542 
 
 76213 
 
 39764 
 
 34567 
 
 78912 
 
 89002 
 
 34567 
 
 45678 
 
 91874 
 
 345 
 
 43604 
 
 67890 
 
 51871 
 
 45632 
 
 20302 
 
 12349 
 
 99SS7 
 
 56789 
 
 44556 
 
 48672 
 
 17280 
 
 24 
 
 50776 
 
 514O3 
 
 43509 
 
 46/95 
 
 49312 
 
 31274 
 
 64 18 
 
 4567O 
 
 43004 
 
 MISCELLANEOUS EXAMPLES IN ADDITION. 
 
 Ex.1. Add together the following sums: 98764, 897652, 8/6,, 
 459321, 21, 80 ; and 76942. 
 
 Ex.2. Add39764, 47652, 34291, 225, 48,. 764871, and 10000 
 together. 
 
 Ex. 3. What is the sum of thirty-five thousand and four; five hun* 
 4red and forty thousand, thrte hundred and nine,; four hundred and 
 .twenty-seven ; fifty thousar.d mne hundred and eighty '$ two millions 
 and five; and seven hundred and seventy-seven ? 
 
 Ex. 4. When will a child, born in 1806, be forty-nine years old ? 
 
 Ex. 5. How many days are there in the first eight months of the year, 
 When it is not leap-year ? 
 
 PvX.G. How old is the world this year, 180S, supposing it was created 
 4004 yeais before the birth of Christ ? 
 
 Ex. 7. A personal his death left 3237 /. to his widow; to his eldest 
 son he bequeathed 5250/- ; and to each of five other children he left a 
 thousand pounds less than to the eldest son : Vie left also to a nephew 
 \Q5l. r and the same sum to be divided among four distant relations; 
 How much money did he leave behind.htm ? 
 
 Ex. 8.- The lease of my house was granted me in the .year 1793, fcj 
 ninety-nine .years.; .v/hen.w-iJl itexpjrc? 
 
 Ex. 9. Hew. many days ..wilJithefe be.betvreen J-amia/jC'ths first aarl 
 
 ITov^iTib.cr thaSfith., 3 6 08. -being leap .year:, bcth.days iaclvLsh'e ? : 
 
ADDITION, 11 
 
 Ex. 30. What do the following sums amount to, 1268 + 8612 -f 10013 
 4-27^4-9194- 84-550099?* 
 
 Ex. 1 1 . How many chapters are there in the several books of the New 
 Testament ?f 
 
 Ex. 12. How many chapters are there in the several books of the Old 
 Testament ? 
 
 Ex. 13. How many chapters are there in the Bible, which consists 
 of the Old and New Testaments ?- 
 
 Ex. 14. In travelling from London to Bath in a post chaise, for how 
 many iniies shall I have to pay? The distance from London to Houn- 
 slow is 10 miles, from Hounslow to Maidenhead is 16. miles, from Mai- 
 denhead to Reading 13 miles, from Reading to Speenhamland 16 miles, 
 from Speenhamland to Marlborough is 19 miles, from Marlborough t 
 Chippenham is 10 miles, and from Chippenham to Bath is 13 miles. 
 
 Ex. 15. How far is it from London to Harwich? To Romford are 11 
 miles, from thence to Ingatestone 12 miles, from Ingatestone toChclms- 
 ford 6 miles, from Chelmsford to Colchester aie 21 miles, and from Col- 
 chester to Harwich 20 miles. 
 
 Ex.16. In travelling post to Margate I pay a shilling a mile: How 
 many shillings.shall I have paid at the end of rhe journey ? The distance 
 from London to Dartford is 15 miles; from thence to Rochester is 14 
 miles; from Rochester to Sittingb^urne is 11 miles; from Sitting- 
 bourne to Canterbury is 15 miles, and from Canterbury to Margate is 
 17 miles. 
 
 * The pupil may new be taught that -the character .-\- 9 which is 
 called plus, is used to denote Addition, and shews that the numbers 
 between which it stands are to be- added together: thus 94-3 shews 
 that nine is to be added to three. Two lines placed thus :, signify 
 equal to, therefore when we write 9-f-3iri2, it is the same thing as 
 saying in woids, nine added to three are equal to twelve. Again 54-J2 
 4-4 4- 9i=:3o 5 that is , 5 and 12 and 4 and g being added together, are :. 
 equal to <3O, . 
 
 f- The learner must refer to the table of contents of his Bible, te ena- 
 ble .him to. answer this- and-the two .following examples* 
 
SUBTRACTION, 
 
 BY SUBTRACTION we find the difference between two 
 numbers.* 
 
 Rule (1). Place the tetter number under the greater, so 
 that units may stand under units, tens under tens, fyc. ; be~ 
 gin at the right hand, and take each figure in the lower 
 line from the figure above it, and set down the remainder. 
 
 (2). If the figure in the lower line be the greater, add ten 
 to the upper one, and then take the lower one from the sum, 
 set down the remainder and carry one to the next lower 
 figure, with which proceed as before, f 
 
 (3). When the figure in the lower live is equal to that 
 above it, the difference is nothing, for which a cypher must 
 le set down. % 
 
 NOTES. 
 
 * This character , called minus, when placed between any two 
 numbers, denotes that the smaller number is to be subtracted, or taken 
 from the larger : thus 9 5 shews that 5 is to be taken from the 9 , and 
 we say, 9 5m4 ; that is, 5 subtracted from 9 is equal to 4. Again, 
 21504 13695ZT7809. 
 
 ^ This operation is commonly called borrowing, and as ten in the 
 light-hand line is equal to only one in that which precedes it on the 
 left, one is only carried to that line. The pupil may ask why the one 
 is added to the lower line, instead of diminishing the upper line by the 
 ne borrowed ? The question is very proper : and he will see. if he try 
 it, that either mode of operation produces the same result; but the usual- 
 method is thought to be the best in practice : thus, if I have to take 28 
 from 45, 1 say S from 5 I cannoc, but 9 from 15 and there remain 7, I 
 carry 1 to the 2, and say, 3 from 4 and there remains l ; the answer is 
 17. It will be the same if I sav 8 from 15 and there remains 7, and 
 then making the 4 into 3, by taking from it the one I borrowed, I say 
 9 from 3 and there remains i ; the answer is still 17. 
 
 J See Table, p. 7> with explanation. 
 
SUBTRACTION. 
 
 13 
 
 From^. . 87469S 
 Take . . 561436 
 
 Remainder 31326-2 
 
 EXAMPLES. 
 
 765087 
 425436 
 
 339651 
 
 762134 
 5Q7082 
 
 165052 
 
 PROOF. Add the remainder to the last line, and if the sum be equal 
 to the first, the work is right. 
 
 From . . 658742 390076 431267 
 
 Take . . 346l'2l 184193 280795 
 
 Remainder 3i26'2i 
 
 Proof . . 65S742 
 
 205883 
 
 390076 
 
 EXAMPLES FOR PRACTICE. 
 
 Ex. 1. 4867434 2. 67HQ491 3. 58/6486 4. 3390761 
 25:;4213 5458354 3564214 1478490 
 
 Ex. 5. 7052673 
 
 3860749 
 
 d. 9276807 7-7231607 8. 9104008 
 4859434 5987465 9031648 
 
 Ex. Q. 6734078 10, 520133-2 11. 60O0342 12. 1COOOOO 
 5^43769 4876543 5999343 9*9999 
 
 Ex.13. 4002103 14. 3874205 15. 9000123 16. 5301864 
 3987654 1796432 8123456 99 
 
 Ex, 17. 7962038 18.91111118 19.46810^6 20. 8302007 
 6498100 80000009 930C6 3912934 
 
 Ex. 91. 60001234 32. 71216003 23. 30061217 24. 26013032 
 49993490 39876543 19996642 19125340 
 
1'^ SUBTRACTION 
 
 Ex.25. 98/43205 26. 50237480 2/. 49764321 28. 93816030^ 
 >V 9999999 41926321 15875492 Q27Q08- 
 
 ^p- *** , 
 
 Ex.29- 942S6730 30. 923708OO 31. 4260130* 82. 27000019 
 32199739 4812719 22500894 4102094 
 
 Ex33. 76253922 34. 33861400 35. -94681039 36. 6901090 
 3.44939 23713509 3041316 v 1860018 
 
 Ex. 37. 591040029 33,271216904 39-97348098 40. 97468901P 
 490300019 28391767 $290412 31689247 
 
 Ex.41. 543902742 42.913062138 43. 7972C0833 44^1/0909009 
 312003/17 44823165 62310079 2-47J0905 
 
 Ex.45. 99326104 46". I93909u9 47,30921090 48. 1 1 l66/783o 
 21281299 2109109 1937099 38103475 
 
 MISCELLANEOUS EXAMPLES IN SUBTRACTION. 
 
 Ex i. The invention of gunpowder was discovered in the year 1302 : 
 How long is is if ince to the present year, i 8 \ l ? 
 
 2. What is ihe difference betneen thirty-five thousand three hundred ' 
 and nine, and nine thousand and ninety-nine. 
 
 3. How rmich does .-even hundred six thousand and four exceed 
 fourteen thousand nine !rind;ed and thirty seven ? - 
 
 4. How much does fifteen thousand and five want of twenty-three 
 thousand? 
 
 5. The art of printing was discovered in the year one thousand four, 
 hundred forty-nine : How long is it since ? 
 
 6. Coaches weienrut used in England in.the year 158O: How many 
 years is i; since ? 
 
 7. Needle making was introduced into England from India in the 
 year i ."> ;5 : How many years was that before tne present king.came to 
 his throne, which \vr.s in ] 700 ? -.S'e tne Book of Trades. 
 
 9. Required the answers- of the three following sums ; 13345 999 5 ', 
 2059 928; and Q58764 498/6, 
 
 10, How many, more chapters are .there in the Old Testament ?*IFU \ 
 ia:the N 
 
15 
 
 MULTIPLICATION: 
 
 MULTIPLICATION is a short method of Addition, and il 
 teaches us to iiud what a number will amoiflit to, when it is 
 repeated a certain number of times. 
 
 RULE. The number to be -multiplied is called the Mul- 
 tiplicand-, mid the number multiplied is called the Multi- 
 plier. The number found is called the Product. 
 
 MULTIPLICATION TABLE..* 
 
 2 times, 
 
 3 times 
 
 4 times 
 
 5 times 
 
 6 times 
 
 7 times 
 
 or twice 
 
 1 are 3 ; 
 
 1 are 4 
 
 1 are 5 
 
 1 are c 
 
 1 are / 
 
 1 are 2 
 
 2 . . Ci 
 
 
 o . jo 
 
 O ~[ *~* 
 
 o . . 14' 
 
 2 . . 4 
 
 3 . . 
 
 3 . . 12 
 
 3 . , 15 
 
 3 . . 1^ 
 
 3 . . 21 
 
 3 . 6 
 
 
 
 
 
 4 . . 8 
 
 5 . i-15fl 5 . . 20 
 
 5 . . 25 
 
 5 . . 3C 
 
 5 . . 35 
 
 5 . . 10 
 
 6 . . u. 
 
 6 . . 24i 
 
 6 . . 30 
 
 . . 36 
 
 6 . . 42 
 
 6 . . 12 
 
 7 2JJ 
 
 7 . 2b 
 
 7 . . 3.i 
 
 7 > 49 
 
 7 49 
 
 7 . . 14 
 
 8 . . IA\ 
 
 8 . , -J 
 
 8 . 1 ' ) 
 
 
 8 . *> 6 i 
 
 S . . 16 
 
 9 . . 27J 
 
 9 . 36 
 
 9 . 45 
 
 9 5-.1 
 
 9 . . 63 
 
 9 . . 18 
 
 10 . . 30 ! 
 
 10 . . 40 
 
 10 . . 50 
 
 10 . . 60 
 
 10 . . 70 
 
 10 . . 20 
 
 11 . .33 
 
 11 .. 4-1 
 
 .1 . . 55 
 
 11 .. arJ 
 
 il . . 77 
 
 11 . .22 
 
 12 . . 36 
 
 12 . . 4 
 
 12 . . 60 
 
 12 . . 7^; 
 
 12 . . 84 
 
 12 . .24 
 
 j 
 
 
 
 
 
 t 
 
 8 times 
 
 9 times I 
 
 10 times 
 
 11 times | 12 times 
 
 1 are 8 
 
 l are 9} 
 
 1 are 10 
 
 l are n !| l are 12 ' 
 
 2 . . 16 
 
 2 . . IS 
 
 2 . . 20 
 
 2 . . 2- 
 
 2 . . 24 
 
 3 . . 24! 
 
 3 . . 27 
 
 3 . . 30; 
 
 3 . . 33 
 
 3 . . *C> 
 
 4 . . 32 
 
 4 . . 36 
 
 4 . . 40; 
 
 4..4d 
 
 4 . . 48 
 
 5 . . 40 
 
 5 . . 4-'. 
 
 5 . . 50 
 
 5 .. 53 
 
 5 . . 60 
 
 6 . . 
 
 6 . . 5-i 
 
 6 . . so ! G . . efei 
 
 6 . . 72 
 
 7 . 56 
 
 7 . . 63 
 
 7 . . 7^! 7 . . 77! 
 
 7 . . fil 
 
 8 . 
 
 | 8 . . 72 
 
 8 . . 80 ! 8 . . 88| 
 
 8 . . 06 
 
 9 . . ; 2 
 
 ; 9 . . 81 
 
 9 . . 90 
 
 9 . . <?9{ 
 
 9 . .JOS 
 
 10 . . 80 
 
 10 . . iJO 
 
 10 . .100. 
 
 10 . .110J 
 
 10 . .120 
 
 11 . . 88 
 
 11 . . 99 
 
 11 . ,110 
 
 11 . .121J 
 
 11 . .132 
 
 12 . . 96 
 
 12 . JOS' 
 
 12' . .120: 
 
 12 . .132J 
 
 12 . .144 
 
 NOTE, 
 
 * Trie above Table ii.iist be iirst learnt in Gisch a manner that the 
 pupil may be able to. answer, in an instant, any possible question that 
 c.an be formed from it ; and if he has attended 10 v/hat was said in Addi- 
 ^tODjp; e^thcgieuteipartei -.': overcome. 
 
16 
 
 MULTIPLICATION. 
 
 I. When the Multiplier does not exced 12. 
 
 RULE. Multiply every figure in the multiplicand from 
 right to left, consider kow many tens there are in each pro- 
 duct, the remaining units set d , im under the figure multi- 
 plied, and carry the tens as so mam/ ones to the next pro- 
 duct. The last product is to be wholly set down. 
 
 Ex. 
 
 420847 
 8 
 
 3366776 
 
 EXAMPLES. 
 
 Ex. 2. 94564875 
 
 472824375 
 
 Ex. 3. 3476819 
 12 
 
 41/21828 
 
 Thus in the first example, I s-r, s times 7 are 56, in wb:ch there arc 
 five: tjn nndsix over, 1 put down the six, and say a tiiiif's 4 ure 32, and 
 
 As soo -. ;.">e pupil has learnt chc table in columns, let him learn it 
 as it stands below. 
 
 MUJ'IPIJ dTIONTA^LE. 
 
 1 
 2 
 
 3 
 6 
 
 2 
 
 4 
 
 5 
 
 i j 
 
 4 
 
 b 
 
 7 
 14 
 
 6 
 
 ;2 
 
 18 
 
 8 
 16 
 
 11 
 
 2 
 
 10 
 20 
 
 12 
 24 
 
 3 
 
 9 
 
 6 
 
 15 
 
 j J 
 
 21 
 
 18 
 
 27 
 
 2 -i 
 
 33 
 
 30 
 
 36 
 
 
 12 
 
 8 
 
 20 
 
 16 
 
 28 
 
 21 
 
 36 
 
 3-J 
 
 14 
 
 40 
 
 48 
 
 5 
 
 15 
 
 10 
 
 25 
 
 20 
 
 j5 
 
 30 
 
 45 
 
 40 
 
 55 
 
 50 
 
 60 
 
 6 
 
 7 
 
 18 
 21 
 
 12 
 14 
 
 30 
 35 
 
 2 
 28 
 
 & 
 49 
 
 36 
 42~ 
 
 54 
 6, 
 
 56 
 
 66 
 
 77 
 
 60 
 
 -70 
 
 72 
 
 84 
 
 8 
 
 24 
 
 16 
 
 40 
 
 32 
 
 06 
 
 4 a 
 
 72 
 
 64 
 
 88 
 
 
 96 
 
 9 
 
 27 
 
 18 
 
 45 
 
 36 
 
 63 
 
 >4 
 
 81 
 
 72 
 
 CO 
 
 90 
 
 10S 
 
 10 
 11 
 
 30 
 
 33 
 
 20 
 22 
 
 50 
 55 
 
 40 
 
 44 
 
 70 
 
 77 
 
 60 
 66 
 
 3E 
 
 99 
 
 SO 
 
 88 
 
 110 
 
 " 
 
 100 
 110 
 
 120 
 
 pr 
 
 12 
 
 36 
 
 24 
 
 60 
 
 48 
 
 84 
 
 72 
 
 108 
 
 96 
 
 132 
 
 120 
 
 Ul 
 
 To enable the Teacher to exercise his pupil in all the combinations* 
 I shall add the three following series, which will be found veiy useful in 
 examining boys and girls in clas es. Thus he combines all the 12 num- 
 bers with the figures in rows, as 8 times 3 ; 8 times 2, and soon : he may 
 then do the same in columns, as 5 times 3 ; 5 times 9 ; 5 times 5, &c, 
 
 Series 1. 3, 2, 5, 7> 4, 8, 6, 12 9, 11, 10 
 Series 2. 9, 8,10, 6,11, 4, 2, 3,10, 5,14 
 Series 3, 5, 11, 0, 2, 7, 9* 12, 6> 8, 4,10 
 
MULTIPLICATION. 17 
 
 elding the 5 from the last product, I have 37 ; I put down the 7, and 
 carry the 3 for the three tens : I then say 8 times 8 are 64, and 3 are 67, 
 7 and carry 6 : 8 limes is o, but put down the 6 brought from the last 
 product : 8 times 2 are 16, put down the 6, and then 8 times 4 are 32, 
 and the i brought forward are 33, which, as being the last prod uct, must 
 be set down. 
 
 EXAMPLES FOR PRACTICE. 
 
 x. 1. 4653245 Ex. 2. 8756894 Ex. 3. 4986587 
 234 
 
 Ex. 4. 3390/63 Ex. 5. 7052673 Ex. 6. 0276807 
 5 l> 7 
 
 Ex. 7. 7231607 Ex. 8, 9)34908 Ex. 9. 6734078 
 8 9 1O 
 
 Ex., 10. saoisaa Ex, n, to pa 173 Ea.ia, 
 
 11 12 u 
 
 Ex.13. 8302-2CQ7 Ex. 14. 53918(51 Ex. 15. 4681953 
 
 12 11 1*5 
 
 Ex.15, 95743205 Ex, 17. 509474<?fi Ex. 18, 49764329 
 
 3 8 7 
 
 x. 19, 5972804- Ex.20. 509764S Ex. 21. 587*496 
 
 564 
 
 Ex.22. 5*39027 Ex.23. 9999999 Ex. 24. 88SH838 
 
 7 8 r 
 
 Ex.25. 9734895 Ex..26. 920/085 Ex. 27. 594^367 
 
 989 
 
18 MULTIPLICATION. 
 
 This character x > which is called St. Andrew's cross; is used to tig- 
 note Multiplication, and when it stands between two numbers, it signi- 
 fies that those numbers are to be multiplied into one another: thus 9X6 
 ~54, is read, nine multiplied by-six is equal to fifty-four. Again 12X 
 11ZZ1U2, that is 12 multiplied by 11 is equal to 132. 
 
 EXAMPLES.* 
 
 Ex, 1. 528318769 X 5 Ex. 2. 956728314 X 3 
 
 Ex. 3. S259 ; 34685 X. 7 Ex. 4. 4868/5294 X Q. 
 
 Ex. 5. 4Q6745832 X 9 Ex. 6. 6536-37544 X 8 
 
 Ex. 7. 578940245 X 2 Ex. 8. 759654318 X 11 
 
 Ex. 9. 987234617 X 6 Ex. 1O. 867122456 X 12 
 
 Ex. 11. 716-132978 X 9 Ex. J2. 687649321 X 7 
 
 Ex. 13. 795483206 X 11 Ex. 14. 779368245 X 9 
 
 "Ex. 15. 91872648 X 12. Ex. 16. 980049005 X 5 
 
 Ex. 17. 856/8654 X 4 Ex. 18. 39005/864 X 6 
 
 Ex. 19. 894367542 X 8 Ex. 2O. 765438958 X 4f 
 
 II. To multiply by 10, add an to the multiplicand : 
 
 tmis 5G7 x 10 is 5670; and 567 x 100 is 50700; and. 
 0489 X 100QO = 64890000. Therefore, to multiply a 
 given number of one denomination, by a number whose sig- 
 nificant figures do not exceed 12, having a cypher or cyphers 
 joined to it : 
 
 RULF.. Write down the cypher or cyphers for the first 
 part of the product tow irds the right hand, and then mid- 
 t'tpl.!/ every figure in the multiplicand by the significant 
 figures of the multiplier, as in the preceding case. Thus 
 
 * If the pupil be not sufficiently ready in multiplying, after he has 
 wo; V;:;l rhe following twenty sums, the preceptor may, by changing the 
 multipliers only, increase the number of examples to any extent. 
 
 f- When 13 is the multiplier, the sum may be done in a single line, 
 by multiplying each figure in the multiplicand by 1C-, and to each pro- 
 duct add the number to be carried, if any, and also the figure which is 
 multiplied. 
 
 EXAMPLES. 
 
 Ex.1, 400642 Ex.2. 526287 Ex. 3. 345682 
 
 13 13 13 
 
 6417346 7491731 4493866 
 
 Here, in the first example, I say 12 times are 24, and 2 are 26 ; 6 
 and carry two: 12 times 4 are 48 and 2 are 50, and 4 are 54 ; 
 carry 5 : 12 times <S are 72, and 5 are 77, and 6 are 83, &c c 
 
MULTIPLICATION. 19 
 
 rj4C9456 x 50 = 173472800, and 98765432 X 8000 =: 
 TD0123456000, for 
 
 3469456 98765432 
 
 50 8000 
 
 173472800 790123450000 
 
 Ex. 
 
 Ex. 
 Ex. 
 Ex. 
 
 i. 
 
 3. 
 5. 
 
 7. 
 
 6754328 
 8329674 
 6470078 
 7856423 
 
 X 
 X 
 X 
 X 
 
 EXAMPLES 
 
 70 Ex. 
 
 1 1 Ex. 
 900O Ex. 
 1000 Ex, 
 
 f\ 
 
 4. 
 
 6. 
 
 3. 
 
 QS7654329 
 567SC943 
 i) tin 7 654 
 
 7490134 
 
 X 
 X 
 X 
 X 
 
 800 
 120 
 110O 
 600 
 
 III. When the multiplier consists of several figures. 
 
 RULE. The multiplicand must be multiplied by each 
 .figure of the multiplier Separately, beginning with the right- 
 hand figure, and the first figure oj every product must stand 
 exactly under the figure multiplied by. Add these products 
 together for the whole product. * 
 
 * To multiply by any nuhibcr betv.'crn 13 and 19. 
 
 RULE. , Multiply theunits figure of the multiplicand, by theright-hand 
 
 digit of the multiplier ; set down the unit's figure cf the picduct, and re- 
 member what is to be carried. Multiply the second figure of the multi- 
 plicand ; to the product, add what was to be carried, and also the first 
 figure of the multiplicand. Then set down the unit's figure, and retain 
 in your mind the number to be carried, as before. Multiply the third 
 figure of the mul'iplictiad ; add the number to be carried, and also the 
 second figure of the multiplicand, and so on: thus 
 
 743(55487595 
 17 
 
 12 642 132 89 132 
 
 Here I say 7 times 6 are 42 ; I put down the 2 and carry 4, and say ? 
 r times 9 are 63, and l are 67, then add the 6, which makes 73 ; pat down 
 the 3, and say 7 times 5 arc 35, arid 7 are 42, to which add the 9, which 
 make 51, put down 1 and carry 5, and so on, till the last figure, when 
 I $ay 7 times 7 are 49, and 3 to be cunied are 52, take in the -1 . 
 make 56, put down 6, and add 7 to the 5., and set down 12. 
 
20 MULTIPLICATION. 
 
 EXAMPLES. 
 
 37864320 35964837 
 
 579 846 
 
 520778Q61 215788962 
 
 405050303 143859308 
 
 289-321645 287718616 
 
 33503446491 30426243642 
 
 PROOF. The readiest woy of proving the truth of sums in 
 Multiplication is, by casting out the nines. 
 
 RULE. Make a cross like that which is used to denote 
 Multiplication : add tog-ether thi3 fjniu-s in the multiplicand, 
 casting out all the nines in the sum as often as they amount to 
 9, and put the remainder down on one side of the cross ; do the 
 same with the multiplier, and put down the remainder on the 
 other side of ', he cross. Multiply the two remainders toge- 
 ther, am! casting- out the nines of their product, will leave 
 the same remainder as the nines cast out of the answer, when 
 the work is right.* 
 
 EXAMPLES.. 
 
 4303267 O 
 
 5f 8 X I 
 
 U6746136 
 27550602 
 32960305 
 
 2608075656 6538013578 
 
 To prove the second example, I say 7 and 6 arc 13 ; 4 above ninf, 
 (emit the 9) : 4 and 2 are 6 and 8 are 14 ; 5 above nine, (omit the 9) ; 
 5 and 5 are 10,1 above 9, l and 4 are 5 : I place the 5 on the left-hand 
 of the cross, and say 8 and 5 are 13,4 above 9 ; 4 and 7 arc 1 1 , 2 above 
 9; the 2 I put on the right-hand of the cross. Now 5X2 gives 10, 
 which is 1 above 9, I put the 1 it the top of the cross, and then cast 
 
 NOTE. 
 
 * The method of proof depends on thejproperty of the number 9, 
 which belongs also to the number 3, but to none of the other digits; 
 viz. that any number divided by 9, will leave the same remainder as the 
 sum of digits divided by 9: thus 8769 divided by 9, leaves 1 as are* 
 mainder; and so will 8 -f- 7+6 -f- 7, or 28, divided by 9. 
 
MULTIPLICATION. 21 
 
 out the 9*s of the whole product, and I find the remainder is i, which 
 answering to the l at the top of the cross, leads me to conclude that the 
 operation is right. 
 
 IV. When cyphers are intermixed with the figures in the 
 multiplier, 
 
 RULE. Omit the cyphers, and let the first figure of each 
 product be placed under its multiplier. 
 
 EXAMPLES. 
 
 Ex.1. 7650329 'Ex.2. 446534S 
 
 600509 1 7000603 -3 
 
 . 5X2 7X3 
 
 68S52961 1 35722784 3 
 
 38251645 26792088 
 
 45901974 3125743(3 
 
 4594091417461 31260150931584 
 
 Ex 0. 84S2/5 X 706 Ex. 4. 973648 X 8005 
 Ex. 5. 597384 X 830004 Ex. 6. 364759 X 2709 
 Ex.7. 245918 X 703006 Ex. 8. 609483 X 95007 
 
 V. When the multiplier is the product of two or more 
 numbers in the table. 
 
 RULE. Multiply the multiplicand by one of the compo- 
 nent parts, and that product by the other, and so on: thus 
 if I have to multiply a given sum by 64, 1 find S X 8 ~ 64 ; 
 instead, therefore, of multiplying by 6 and 4 in the usual 
 %utiy> I multiply first by 8, and then that product by S again,. 
 
 EXAMPLES. 
 
 39746285 X 168* 
 7 
 
 3 
 
 8X6 
 
 3 
 
 35321088 166Q34397O 
 
 4 
 
 6077375880 
 
 NOTE. 
 
 * Here 1C8 7X6X-*, 
 
22' M U LTI P LI C ATIOK. 
 
 EXAMPLES IN ALL THE CASES. 
 
 Ex.1. 93365497 X 13 Ex.2. 54962S7* X '2fi 
 
 3. 35729876 X 56 4. 47893062 X 48 
 
 5. 73167482 X 77 6. 8274386 X 96 
 
 7. 39745371 X 86 8. 5487962 X 357 
 
 g. 72983456 X 99* 30. 3891307 X 464 
 
 11. 737394 X 4^67 12. 3oS46 X 46S-2 
 
 13. 329357 X 2839 14. 5S427 X 3957 
 
 15. 462875 X 6874 16. 47683 X 3456 
 
 17. 594326 X 5936 18. 87493 X 7892 
 
 19. 486752 X 4608 20. 29687 X 357Q 
 
 21. 8739690279 X 39/829 20. 7936820O56 X 500634 
 
 23. 2576432874 X 613487 24. 9~lt>7403258 X 65300O 
 
 25. 872694325 X 2900008 26. 715976032 X 350706 
 
 >7. 526730169 X 590*34 28. 37945687 X 999999 
 
 29. 74714328 X 345627 30. 46382719 X 50000092 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. 1. Multiply three millions thirty-nine thousand and three, by 
 thirty-five thousand and twenty-eight. 
 
 2. Multiply six biHions, six hundred thousand and sixty-five, by eight 
 thousand and th.rty-nine. 
 
 0. There are eleven hundred hackney coaches in London ; suppose, 
 on the average, each coach earns thirteen shillings a day, how many- 
 shillings will be expended in the hire of these carriages in a year of 3C5 
 days, Sundays being exempted ? 
 
 4. In Jamaica only there were imported, annually, not le?s than ten 
 thousand eight hundred negroes from the coast of Africa: How many- 
 slaves have free bon Englishmen made in that island, since the year 
 1799 to the year 18O7, in which the infamous traffic was abolished. 
 
 NOTE. 
 
 '* To multiply by any series of 9*s. Add as many cyphers to the 
 multiplicand as there are 9*5 in the multiplier, and then subtract the 
 original multiplicand. 
 
 EXAMPlES. 
 Ex. 1. 843628 X 9Q9 Ex. 2. 3475962 X 90999 
 
 843628000 347596200000 
 
 813628 3475962 
 
 842784372 347592724038 
 
 In the first example I add three cyphers, and subtract from the mul- 
 tiplicand, thus increased, the original multiplicand 5 and the difference 
 is the true arvswer. 
 
MULTIPLICATION. 23 
 
 5. A boy can point sixteen thousand pins in an hour: How many 
 "will he do in six days, supposing he works eleven clear hours in a dav ? 
 See Blair's Universal i receptor. 
 
 6. What is the continual product of 25, 19> 705, and 999?* 
 
 7. How many changes can be rung on twelve bells? -f- 
 
 3. Multiply the difference between 50487 and <3 0056, by the sums of 
 850, OO67, and 800 ? 
 
 9. The sum of two numbers is 30355, and the greater number is 
 25251 : What is their pioduct ? J 
 
 10. The sum of two numbers is 4584, and the less is 1876: What is 
 the.ii product? 
 
 11. What is the difference between twelve times fifty-seven, and 
 twelve rimes seven and fifty ? 
 
 12. How many miles will a person walk in sixty-six years, supposing 
 he travels, one day with another, six miles, and there are 365 days in a 
 year ? 
 
 13. How many cubic feet does this room contain, which is fifteen 
 feet long, fourteen feet wide, and thirteen feet high? 
 
 NOTES. 
 
 * The continual product of any given numbers is found by multiply- 
 ing them into one another: thus 8 X 5 X 9H: 360, is the continual 
 product of 8, 5, and 9. 
 
 -f* The continual product of i, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, will 
 be the number required : thus the number of changes thai can be rung 
 on six bells is 1 X 2 X 3 X 4 X 5 X 6 72t>. By ihe same method we find, 
 that the number of changt-s that can be made of the twenty-four letters 
 
 621654 561827 891919 360000 
 in the alphabet is equal to v - v"-^ v ~ v * v ~- -v - ; v --~ v-*-' The 
 
 43-21 
 
 letters in the English alphabet may be reckoned 24, as formerly the i 
 and j were expressed by the same character, and so were the u and v. 
 
 % From the sum subtract the greater number, and the remainder will 
 be the lesser number: the answer will be found by multiplying the 
 greater and lesser numbers together. 
 
 To find the number of cubic feet in any r^om, &c., the continual 
 product of the height, length, and width must be fourvi, 
 
DIVISION. 
 
 BY DIVISION, we find how often one number is contained 
 in another of the same denomination ; this is a short method 
 of performing Subtraction. 
 
 The sum to be divided is called the dividend; the figure, 
 or figures, by which \Te divide, is called tho divisor; and 
 the result is called the quotient. 
 
 In this Rule, as in Multiplication, there are several dis- 
 tinct cases. 
 
 I. When the divisor does not exceed 12. 
 
 Write the divisor on the left-hand side of the di~ 
 nd* make a curve, and consider how often the divisor is 
 in the first figure, or in the first two or three 
 figures, and set the quotient under it; and for every unit 
 remaining after subtraction, curry TEN to the next figure 
 of the dividend. 
 
 EXAMPLES. 
 Ex. 1. 4)786543-28 Ex. 2. 0)85674327 
 
 19663582 
 
 95193690 
 
 Ex.3. 11)10976541 
 
 988770 i 
 
 Ex. 4. 12)11272489 
 
 939374 1 
 
 In the second example, I say there are 9 nines in 85 and 4 over; 
 I put down the nine and carry the 4, as 4O to the 6, and the y's in 46, 5 
 times and 1 over; put down the 5 and carry l, as 10, and say the Q'S 
 in 17, once and 8 over; put down the i an<t carry 8 as 80; 9'sin 80, 9 
 time*- and two over, and ?o on : at the last figure there are 6 remaining, 
 this put down beyond a small line. 
 
 It is usual, in giving the answer, to mate a short line under the 
 remainder, and place under it the divisor: thus the answer to the 
 
DIVISION. 
 
 2o 
 
 second sum is 9519369|: that of the third sum is 988776^., and that 
 of the fourth 9393/4; and the three remainders are fractions, 
 which we read six-ninths, five-elevenths, and one -twelfth. See p. 4 
 and 5. 
 
 EXAMPLES. 
 
 5)7639487 
 1527897f 
 
 7)440295 
 62899f 
 
 This character ~, when placed between two numbers, signifies that 
 'the one is divided by the other : thus Q5 -= 8 ~ 11| ; and we read 95 
 'divided bu 8, gives 11 and s even- eighths over ; that is t there are 11- 
 eights in 95, and seven remaining. 
 
 Ex. l. 
 
 EXAMPLES. 
 
 5087 -r 7 Ex. 2. 
 
 Here 5687 ~ 7 = 81 2f 
 For 7)5687 
 
 49876 ~r 3 
 
 49375 .i. 3 = 16625|. 
 3)49876 
 
 16625 1 
 
 Ex. 3. 87240322 - 
 
 - 3* 
 
 Ex. 4. 62304678 - 
 
 - 4 
 
 5. 74009654 - 
 
 - 5 
 
 0. 26730217 - 
 
 - 6 
 
 7. 5^9234600 - 
 
 - 7 
 
 . tf. 37026541 - 
 
 - 9 
 
 9. "468/2135 - 
 
 - 8 
 
 30. 56438/52 - 
 
 - 7 
 
 31. 45900361 - 
 
 - 9 
 
 1-2. 3256487 - 
 
 - 8 ' 
 
 J3. 59764218 - 
 
 - 10f 
 
 14. 3532640 - 
 
 - 11 
 
 15. 32753742 - 
 
 - 12 
 
 16. 3333333 - 
 
 - 12 
 
 17. 44444444 - 
 
 - 11 
 
 18. 5598764 - 
 
 - 12 
 
 19. 98897603 - 
 
 - 9 
 
 20. 9330048 
 
 8 
 
 PROOF. The method of proving the 
 to multiply the answer by the divisor, 
 
 truth of sums in Division, is 
 and take in the remainder, the 
 
 result will be equal to the dividend. 
 
 
 * When any number of the dividend is less than the divisor, put 
 down a cypher, and carry u as a remainder to the next figure, unless 
 it be the last figure in the dividend, then it is put down as a remain- 
 der ; thus 7226 -7- 4 = 1806|; for I say 4 in the 7, once and 3 
 over ; 4 in the 32 goes 8 times ; 4 in 2, I cannot, but 4 in 26, 6 times 
 and 2 over. 
 
 f To divide by 10. Cut off the last figure in the dividend as a re- 
 mainder, and the thing is done; thus, 764S9 ~- 10 = 7648 9. 
 And 694312 ~ 10 = 69431 2. 
 
 c 
 
26 DIVISION. 
 
 Ex. 7959467834 -* 7. 
 
 7)7959467834 
 
 Quotient - 1137066803 3 7X2 
 
 7 8 
 
 Proof - - 7959467834 
 
 Another method is by casting out the nines, as in Multiplication,- 
 RULE. Cast away the nines in the divisor, and put the remainder on 
 one side of the cross ; do (he same with the quotient, for the other 
 side of the cross : then for the top figure multiply these two numbers 
 together, cast away the nines, and add the excess of nines in the re- 
 mainder after division, and the excess of nines in this sum will be equal 
 to the e,xcess of nines in the dividend, if the work is right. See the 
 preceding example, where I put down the 7 on one side of the cross ; 
 the excess of nines in the quotient is 2, which I put on the other side 
 of the cross, then I say 7 times 2 are 14, and the remainder 3 make 17, 
 which is 8 above nine, this I put at the top of the cross, and I find that 
 8 is the excess above the nines in the dividend, therefore I conclude 
 the operation is right. 
 
 II. To divide a number of one denomination, by another 
 number whose significant figures do not exceed 12, having 
 a cypher or cyphers joined to the right hand. 
 
 RULE. Cut off the cyphers from the divisor, and the 
 same number of figures from the right-hand of the dividend; 
 then divide the remaining figures of the dividend by the 
 remaining part of the divisor, and the result is the answer. 
 
 To the remainder, if any, join those figures of the divi- 
 dend, which were first cut off, and the whole will be the 
 true remainder.* 
 
 Divide 4685321 by 800; and 326441 by 1200. 
 
 8,00)46853,21 13,00)326)4,41 
 
 5856 521 272 41 
 
 Of course the true answers to these sums are 5856^i-i, and 2 7 a...!.!-.. 
 
 * When nothing remains after the division, the figures that were 
 cut off will be the true remainder; thus 76543-246 -~ 1200 = 
 
DIVISION. 
 
 EXAMPLES. 
 
 Ex. 1. 3476521 
 
 60 
 
 Ex. 2. 
 
 8543009 - 
 
 - 700 
 
 3. 2937648 - 
 
 - 800 
 
 4. 
 
 9003456 - 
 
 - 9000 
 
 5. 5620042 - 
 
 - 1100 
 
 6. 
 
 7641121 - 
 
 - 500 
 
 7. 4052079 - 
 
 - 1200 
 
 8. 
 
 8496551 - 
 
 - 12000 
 
 9. 7921164 - 
 
 - 90 
 
 10. 
 
 9939216 
 
 - 8000 
 
 11. 46201132 - 
 
 700 
 
 12. 
 
 1234567 - 
 
 - 120 
 
 III. To divide a given number of one denomination, by 
 si divisor which is compounded of two or more numbers in 
 the Multiplication Table. 
 
 RULE. Divide the given number by one of thvse parts, 
 and the quotient by the other component part, and so on till 
 each of the component parts has been used as a divisor; 
 thus 46875815777 -r 105 is performed as follows: the di- 
 visor 105 is equal to 7 X 5 X 3 ; I therefore divide the di- 
 vidend first by 7, and the quotient by 5, and this second 
 quotient by 3. 
 
 7)46875815777 
 5)6690545111 
 
 3)1339309022 
 
 Answer - - 446436340 
 
 EXAMPLES. 
 
 Ex.K 
 
 84596543 -r 36 
 
 Ex.2. 
 
 345069549 - 
 
 - 42 
 
 3. 
 
 45897642 - 
 
 - 56 
 
 4. 
 
 945960542 - 
 
 - 99 
 
 5. 
 
 39200761 - 
 
 - 66 
 
 6. 
 
 8/932874 - 
 
 - 768 
 
 7. 
 
 38426587 - 
 
 - 550 
 
 8. 
 
 44444444 - 
 
 - 121 
 
 NOTE. 
 
 * Here are two remainders, l and 3", to be accounted for, and the 
 general rule in finding the value of remainders, in this mode of con- 
 tracted division, is : 
 
 RULE. Multiply the last remainder by the preceding divisor, and 
 to the product add the preceding remainder, and place the sum thus 
 found over the two divisors multiplied into one another. Here 2, the 
 last remainder, is to be multiplied by. 5, which is 10 ; then add the 
 first remainder i, and set the n over 5 X 3, or 15 ; "the true answer 
 s 44G436340.1.L = 446436340 7 7 ^, because I multiply the 11 and 
 the 15 by the first divisor 7. 
 
 If there be many remainders, instead of two, multiply the sum 
 first found by the next preceding divisor, and to the product add 
 C2 
 
28 
 
 DIVISION. 
 
 9. 
 
 28476974 - 
 
 - 720 
 
 10. 
 
 55555555 - 
 
 - 37 g 
 
 11. 
 
 56342872 - 
 
 - 132 
 
 12. 
 
 33992288 - 
 
 - 288 
 
 13. 
 
 34/65982 - 
 
 - 144 
 
 14. 
 
 98453392 - 
 
 - 432 
 
 15. 
 
 24853274 - 
 
 - 512 
 
 16. 
 
 83547552 - 
 
 - 99 
 
 17. 
 
 43333999 - 
 
 - 343 
 
 18. 
 
 54954335 - 
 
 - 720 
 
 19. 
 
 5555556 - 
 
 - 729 
 
 20. 
 
 25574538 - 
 
 - 343 
 
 IV. To divide by a number consisting of two or more 
 digits, which number is not compounded of those in the 
 table. 
 
 RULE. (1). Draw a curved line on the right and left 
 of the dividend, and write the divisor on the left. 
 
 (2). Find how many times the divisor is contained in as 
 many figures of the dividend as are just necessary, and 
 place the number on the right for a quotient. 
 
 (3). Multiply the divisor by the quotient figure, and 
 place the product under the above-mentioned figures of the 
 dividend, subtract this product from that part of the divi- 
 dend under which it stands, and bring down the next figure 
 in the dividend, or more if necessary, to the right-hand of 
 the remainder, and proceed as before, till the ivhole is 
 finished. This is called LONG DIVISION. 
 
 the next preceding remainder, and so on till all the divisors and re- 
 mainders are gone through: suppose the sum be 8476521 to be divided 
 by 5^25 = 5 X 5 X 7 X 3. 
 
 5)8476521 
 
 5)1695304 1 , 
 7)339060 4 
 
 3)48437 1 
 
 16145 2 
 
 Multiply 2 by 7, and take in l = 15. Now 15 X 5 and take Jn 
 4 ^=: 79. And 79X5 and take in 1 = 396. The true answer, there- 
 fore, is I6145?|f ; for the Arithmetician will easily find that the frac- 
 
 tion -jf-f is equal to the sum of the several fractions y + TT 
 
 yVr + T^T 
 
 yr ~- Ttf 
 
jJIVISION. 
 
 Ex. 5537049 ~ 954, 
 
 i>54) 5537049(5804 Quotient 
 4770 - . 
 
 Remainder. 
 
 Here, the divisor not being contained in the first three figures, I 
 consider how often it is contained in the first four,-f* and find it to be 5 
 times, the 5 I put in the quotient, and multiply the divisor by it, setting 
 the product under the dividend. I now subtract this product, and to 
 the remainder 767,1 bring down the-o, and find that the divisor is con- 
 tained 8 times in 7670, the 8 I place in the quotient, and proceed to 
 multiply the divisor by it*; the product subtracted leaves only 38 ; I 
 now \>ring down the 4, but the divisor not being- contained in 384, I 
 put down o in the quotient, and bring down the 9> the remaining figure 
 in the dividend, and proceed as before. + 
 
 * "When more figures than one are brought down to the. remainder, 
 to make it larger than the divisor, a cypher must be written in the 
 quotient for every figure so brought down. 
 
 f The only difficulty in Long Division, is finding readily how often 
 the divisor is contained in each of the parts of the dividend ; in long 
 sums it will be found very convenient to beginners to construct a table 
 of the products of the divisor from 1 to 9, whence the several quotient 
 figures, and their corresponding products will be seen at first sight : 
 thus, if the divisor be 954, the table is as follows: 
 
 954 
 1908 
 2862 
 3816 
 4770 
 57^4 
 6678 
 7032 
 8586 
 
 "l. The pupil will always remember, that as many figures of the 
 dividend must be taken as will contain the divisor ; under the last 
 of these let him place a dot, and likewise under every subsequent 
 figure, as he makes use of it, to the end of the dividend : of course- 
 
30 
 
 DIVISION. 
 
 EXAMPLES. 
 
 Ex. 1. 78654321 76 
 
 3. 68742164 87 
 
 5. 77755562 654 
 
 7. 53430-132 - 7654 
 
 9. 57678443 8439 
 11. 564320376 3976 
 13. 677744032 5186 
 35. 627432871 4967 
 17. 44444444 5555 
 J9. 33333333 999 
 Ex. 21. 48/264325S76 
 
 22. 876842987621 
 
 23. 918318296542 
 
 24. 507843276549 
 
 25. 877896543210 
 
 26. 444444444444 
 
 27. 2220003330046 
 
 28. 540965328762 
 
 29. 32899438654 
 
 30. 784303254871 
 
 After the pupil has gone through all the foregoing examples in 
 Long Division, he should be taught the ITALIAN METHOD, as it is 
 usually called, of which the following is an example, worked at length ; 
 and as the Jtalitn method is so much neater, and with practice full 
 as easy> and taking up only half the space, it is recommended that 
 the learner should repeat the former examples by this mode of opera- 
 tion. 
 
 Ex. Divide 6452800 by 765. 
 
 765)d452600(8435 
 3328 ' ' 
 2680 
 3850 
 
 25 Remainder. 
 
 The answer is 
 
 Ex. 2. 569-1327 S - 
 
 - 97 
 
 4. 84365487 - 
 
 - 69 
 
 6. 45687403 - 
 
 - 187 
 
 8. 56943286 - 
 
 - 429 
 
 10. 58456942 - 
 
 - 32/9 
 
 12. 92876487 - 
 
 - 7392 
 
 14. 46859210 - 
 
 - 1437 
 
 16. 55555555 - 
 
 - 7777. 
 
 18. 888000999 - 
 
 - 999 
 
 20. 111111111 - 
 
 - 7777 
 
 56780909 
 
 
 90956843 
 
 
 56400032 
 
 
 64785321 
 
 
 92836058 
 
 
 - 750000564 
 
 
 708385032 
 
 
 - 5406057 
 
 
 - 10010432 
 
 
 - 90834360 
 
 
 the number of digits in the quotient will be equ.il to the 'number of 
 dots in the dividend. 
 
 2. Every remainder must be less than the divisor ; for, if it be equal 
 to or greater than the divisor, the quotient figure which has produced 
 the result is too small. 
 
 3. When the product of the divisor by the quotient figure is greater 
 than the particular part of the dividend used, the quotient figure is too 
 large. 
 
DIVISION. 31 
 
 ILLUSTRATION. I find the divisor is contained 8 times in the first 
 four figures, I accordingly put 8 in the quotient, and multiply in this 
 manner, 8 times 5 are 40, and put down as a remainder 2, the differ- 
 ence between o the units place of the number gained by multiplica- 
 tion, and the figure in dividend, from which it was to be subtracted, 
 carry 4 : 8 times 6 are 48 and 4 are 52, the difference now between 
 the 5 in the dividend and the 2 in 52 is 3, which I put down and carry 
 5 : 8 times 7 are 56 and 5 are 61, between which and 64 is 3, which 
 I put down. Bring down the next figure 8, and proceed: the divisor 
 will now go 4 times, I put 4 in the quotient, and say 4 times 5 are 20$ 
 put down 8 and carry 2 : 4 times 6 are 24 and 2 are 26, the difference 
 between 26 and 82 are 6, which I put down and carry 3 : 4 times 7 
 are 28 and 3 are 31, which taken from 33 leaves 2. Bring down o, 
 the divisor now is contained three times in the dividend, the 3 I put 
 in the quotient, and say 3 times 5 are 15, this taken from 20 leaves 5, 
 which put down and carry 2 : 3 times 6 are 18 and 2 are 20, which 
 taken irom 28 leaves 8, carry 2 : 3 times 7 are 21 and 2 are 23, which 
 taken from 26 leaves 3. Bring down the other 0, the quotient is now 
 5, with which proceed as before. 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. 1. Divide fifty millions by four thousand and seventy-nine. 
 
 2. The planet Mercury goes round the sun in 88 days, which is 
 the length of her year, how many years of Mercury would make 50 
 of our years, supposing each year contained exactly 365 days? 
 
 3. It is estimated that there are a thousand millions of inhabitants 
 in the known world : if one thirty-third of this number die annually, 
 how many deaths are there in a year ? 
 
 4. The national debt, at present, cannot be less than five hundred 
 millions sterling : how long would that be in paying off, at the rate of 
 two millions and twenty-five pounds per annum? 
 
 5. The taxes annually collected amount to full thirty- three mil- 
 lions of pounds : how many poor families or six persons each would that 
 sum support, supposing the annual expenses of the father and mother 
 to be 20 J., and of each child 7 1. ? ^ 
 
 6. My friend is to set sail to Jamaica on the first of March, 1812; 
 the distance is reckoned to be 3984 miles: at what rate will he go, 
 supposing he reaches the Island on the 10th day of April, that is, in 
 41 days? 
 
 7. What is the difference between the 12th part of 20,100, and 
 the 5th part of 9110? 
 
32 DIVISION. 
 
 8. The prize of 30,OOOZ. of the last Lottery became the property of 
 15 persons : how much was each person's share, after they had allowed 
 750/. to the office-keeper for prompt payment ? 
 
 9. The sum of two numbers is 1440, the lesser is 48: what is their 
 difference, product, and quotient? 
 
 10. The crew of a ship, amounting to 124 men, have to receive, 
 as prize-money, 1890/. ; but as they are to be paid off, they determine 
 to make their commander and boatswains present, the one of a piece 
 of plate, value ibl. ; the other of a whistle, which is to cost 5/. : how 
 much will each receive after these deductions are made ? 
 
 11. In all parts of the world a cubical foot of water weighs ] ooo 
 ounces : how many pounds are there, supposing 16 ounces make a 
 pound ? 
 
 12. A cubical foot of air weighs one ounce and a quarter, how 
 many pounds avoirdupois of air does a room contain, which' is 10 
 feet high, 14 feet wide, and 16 feet long?* 
 
 13. Hydrogen gas, or, as it was formerly called, inflammable air, 
 that is, the gas with which balloons are filled, is full nine times lighter 
 than the common air which we breathe : how much less would a bal- 
 loon, containing 27,000 cubical feet, weigh if filled with hydrogen gas, 
 dhan if filled with common air ?^ 
 
 14. At what rate per hour and per minute does a place on the equa- 
 tor move, supposing the great circle of the earth to be 25,000 miles, 
 and the earth to turn on its axis exactly in 24 hours ? 
 
 * The number of cubical feet is obtained by multiplying the height, 
 width, and length together, or ]0 X 14 X 16, and the product mul- 
 tiplied by 1 J gives the number of.ounces, which, divided by 16, is the 
 answer, viz. 175 pounds avoirdupois. This circumstance cannot fail 
 of exciting surprize, to conceive that in a moderate sized room, the 
 air, which is invisible and scarcely observed to exist, should be known 
 to weigh more than three half-hundred weights. 
 
 7" The answer will be found to be 1875 lb.; of course the balloon 
 would ascend, with several persons in its boat; because it will ascend, 
 when the balloon and persons are together, lighter than an equal bulk. 
 :J common air. 
 
REDUCTION. 
 
 REDUCTION is the method of converting- numbers froi 
 one name, or denomination, to another of the same value ;.- 
 and it is divided into Reduction descending, and Reduc- 
 tion ascending* 
 
 When numbers of a higher denomination are to be 
 brought to a lower, it is called Reduction descending, and 
 it is performed by Multiplication. 
 
 When numbers of a lower denomination are to be brought 
 to a higher denomination^ it is called Reduction ascending^ 
 and is perform&lby Division. 
 
 DEDUCTION 
 
 OR CONCERTING GREAT .INTO SMALL. 
 
 RULE. Multiply the given number by as many of the 
 lower denomination as malce one of the higher. 
 
 Thus, in reducing 55/. into shillings, I multiply the 55 
 by 20, and the answer is 1100 shilling's; in both cases the 
 value is the same, that is, 55/ a is equal to 1100 shillings. 
 
 REDUCTION ASCENDING, 
 
 OR CONVERTING SMALL INTO GREAT.' 
 
 RULE. Divide by as many of the lower denomination as 
 make one of the next higher. 
 
 Thus, in bringing 890 pence into shillings, I divide the 
 number by 12, and the answer is 74 shillings and two pence 
 over.* 
 
 NOTE. 
 
 * The remainders, if any occur, are always of the same denomine* 
 lion as the respective dividends. 
 
 tifr 
 
34 REDUCTION. 
 
 TABLE. 
 
 4 farthings (9) - - make i penny d. 
 
 12 pence - make l shilling .. 
 
 20 shillings - make l pound . 
 
 21 shillings - make l guinea.* 
 
 -J denotes a farthing : ^ two farthings, or a halfpenny: and f three 
 farthings. 
 
 EXAMPLES. 
 . S. d. 
 Ex.l. Reduce 29 6 sf into farthings. 
 
 20 I multiply the 29 by 20, and take 
 
 in the 6, thus I find 566 shillings are 
 
 586 shillings equal to 29^. 6s. To reduce the shil- 
 
 12 lings to pence, I multiply the 586 
 
 by 12, and take in the 8, which give 
 
 7040 pence 7040, the number of pence equal to 
 
 4 29^.6*. 8d. I next multiply 7010 by 
 
 4, and take in the f , and find the 
 
 Answer 28163 farthings, answer is 28163 farthings, equal to 
 the given sum 29/. 65. sf</. 
 
 Ex. 2. In 28163 farthings how many pounds sterling? 
 
 4)28163 I divide the 28163 farthings by 4, because 
 
 4 farthings make a penny ; the answer is 
 
 12)7040 J 7040, and 3 over, which are farthings, be- 
 
 cause the remainder is always of the same 
 
 2 4 0)5 8,6 8d. denomination as the dividend. I next divide 
 
 the 7040 by 12, and the answer is 586 and 8 
 
 Answer .29 6 8-| over, which are pence; and now 586 divided 
 by 20, gives 29 and 6 over, which are shil- 
 lings ; the true answer is therefore 29Z. 6s. S^c/.-f- 
 Ex. 3. Reduce 28 shillings to pence. 
 
 4. Bring 56 pounds into shillings. 
 
 5. Reduce 672 pence into farthings. 
 
 6. How many pence are there in 105/. ? 
 
 7. In 1000 guineas how many shillings? 
 
 8. In 47041. how many pence ? 
 
 NOTES. 
 
 * . s. d. ard q. arc the initials of the Latin words Librae, Solidi, 
 Denarii, and Quadrantesj signifying pounds, shillings, pence, and 
 farthings. 
 
 f Hence it is evident, that the best method of prc^v^pg the truth of 
 examples in Reduction is, to reverse the operation, ^t^will also be the 
 best method of teaching the pupil to depend ofi Tiis own exertions, by 
 making him prove each example, and not permitting him to begin the 
 second till hdhas made the first come right. 
 
TROY WEIGHT. 35 
 
 Ex. 
 
 9. In 3995Z. how many farthings ? 
 
 10. In 7968 guineas, how many farthings? 
 
 11. Hovy many farthings are there in 75 guineas? 
 l2.*Reduce 576/. into farthings. 
 
 13. In 99^. how many shillings, pence, and farthings ? 
 
 14. Reduce 567^. gs. g^d. into farthings. 
 
 35. How many halfpence are there in 157Z. 7$. 7jd. 
 
 16. In 1084890 pence, how many pounds? 
 
 17. In 8410896 pence, how many guineas? 
 
 15. In 4808764 farthings, how many pounds? 
 
 19. How many seven-shilling pieces are there in a thousand guineas ? 
 QO. How many groats are there in a hundred guineas ? 
 
 21. Bring 3110456 pence into groats, shillings, and pounds, 
 
 22. How many crown-pieces are there in jgl. 15s ? 
 
 20. How many half-crowns are there in 85/. I2s. 6d. ? 
 24. In 769 guineas, how many sixpences ? 
 
 TROY, 
 
 OR GOLDSMITHS' WEIGHT, f 
 
 TABLE. 
 24 grains (gr.) . . make . . 1 pennyweight, dwt. 
 
 20 pennyweights 1 ounce, . , . oz. 
 
 12 ounces 1 pound, . . . tt>. 
 
 Ex. 1. How many grains of gold are there in a cup weigh- 
 ing 3 Ib. 9 oz. 6 dwts. 18 grs. ? J 
 
 NOTES, 
 
 * These ten examples are all in descending Reduction ; bat as they 
 furnish the pupil with an equal number in ascending Reduction, it will 
 be advisable that he should make himself perfect in these, befor* he 
 proceed to others rather less simple. 
 
 f* By this weight, gold, silver, jewels, and precious stones are 
 weighed. It is also used in ascertaining the strength of liquors. 
 
 J If the pupil has made himself master of the foregoing examples 
 in money, he will meet with ud serious difficulty in this and the fol- 
 lowing articles in Reduction. Let him Ic.arinrnind^ that, in all cases, 
 to reduce a greater name, to lesser he must multiply ; and to bring a 
 lesser denomination to a greater he must divide ; and the talic at the 
 top of each article shoius at once ivhat numbers are to le used in the 
 multiplication and division. If I want to bring pounds troy into grains, 
 I multiply by 12, by 20, and by 24. If I wart to bring grains into 
 pounds troy, I reverse the operation, and divide by 24, 20, und 12. 
 For the same reason, as will be seen hereafter, if I want to kno\v how 
 
36 TROY WEIGHT*. 
 
 lb. oz. uwts. gr. 
 
 3 Q 6 18 
 12 
 
 45 Here I multiply the 3 t>y 12, and take in 9 for tire 
 
 20 number of ounces : I then multiply 45 by 20, and 
 
 take in the 6 for the pennyweights : and afterwards 
 
 906 the 906 by 24, and take in l 8 i\.r grains. 1 take the 
 
 24 8 in when I multiply by 4, and the 1 when I nml- 
 
 .' tiply by 2. 
 
 3632 
 1813 
 
 21762 
 
 Ex, 2. How many pounds Troy are there in a million ef* 
 grains ? 
 
 4)1,000000 
 
 6)250000 
 
 2,0)4166,6 4*~ 16 grains. 
 12)2083 6 
 
 173 7 Answer 173 Ibs. 7 oz. 6 dwts. 16 grs. 
 
 Ex. 
 
 3. In 36lb. 10 oz, 12 dwts. 16'grs. how many grains ? 
 
 4. How many pounds troy are there in 5987 pennyweights? 
 
 5. In 1434 lb. oz. o dwts. 19 grs. how many grains? 
 
 6. How many pounds are there in 45065 grains? 
 
 7. Reduce 105 Ibs. troy into grains. 
 
 8. In 495 spoons, weighing 103 Ibs. 1 oz. 10 dwts.,. how many 
 grains ? f 
 
 NOTES. 
 
 many minutes there are in 36 days, I multiply that number by 24, 
 because 24 houis make a day ; and then by 60, because 60 minutes 
 make an hour. 
 
 * Ingtead of dividing by 24 in long Division, I .have . divided by 
 the component parts 6 and 4, (see Division, p. 27.) In the second 
 -division the^e is a remainder of 4; to find the value of which I mul- 
 tiply it by the first divisor, of course the true remainder is 16 grains. 
 The resuk. would have been the same, though differently expressed, 
 had I divided by 6 and by 4, instead of 4 and 6.: in the former case 
 there would have been two remainders, .viz. 4 and 2, , to find the 
 value of which I should multiply 2, the last remainder by 6 the 
 first divisor, and add 4^ the first remainder, making .together, 16, as 
 before.- 
 
 f* It will be very desirable, that the pupil should prove the truth t 
 af .all the examples in Reduction j and if at the end o'f ''each articJe", 
 
3T 
 AVOIRDUPOIS; 
 
 OR GROCERS' WEIGHT.* 
 
 TABLE. 
 
 16 drams (dr.) - make l ounce, oz. 
 1 6 ounces - 1 pound, Ife. 
 
 8 pounds - 1 stone of meat.-f* 
 
 14 pounds - l stone, horseman's weight. 
 
 28 pounds - l quarter, qr. 
 
 4 quarters, or I12lb. - l hundredweight, cwt. 
 
 20 hundred weight - l ton. 
 
 NOTES. 
 
 he be not quite expert in working them, he will find an advantage in 
 repeating the operations before he proceed to the next. 
 
 * By this weight almost all coarse and b-eavy goods are weighed ; 
 such as butcher's meat, grocery, cheese, butter, &c. ; wax, pitch, 
 tallow, and all metals, excepting gold and silver. Avoirdupois w r e;ght 
 was first used in Henry VHIth's reign ; and was introduced expressly 
 for weighing butcher's meat, and other coarse and heavy articles. 
 
 The Avoirdupois ounce is less than the Troy ounce ; but the Avoir- 
 dupois pound is greater than the pound Troy. 175 Troy ounces are 
 equal to 192 Avoirdupois ounces; but 144 Ib. Avoirdupois are equal to 
 175 pounds Troy. Therefore. one pound Avoirdupois is equal l Ib. 
 2 oz. ll dwts. 16 gr. Troy. 
 
 Hence the following Table : 
 
 144 Ib. Avoirdupois m I75lb. Troy. 
 192 oz. - - iz 175, oz. 
 
 Ib. oz. dwts. gr. 
 
 lib. - - ~ l 2 11 16 m 7000 grs, troy, 
 i oz. - - ~ o o is 5 437 
 l dr. - - zz o l '31 zz 27.35 
 
 Ib. oz., dr. 
 
 l Ib. Troy - zz O 13 2j nearly Avoirdupois, 
 l oz. - - o l i 
 
 Hence the .difference between the Ib. Avoirdupois, and the Ibs. 
 Troy and Apothecaries' weight, is, that the first contains 7060 grs. ; 
 both the others only 5760 grs. 
 
 f- The stone of meat in some counties is 12 Ib., in others 14, and 
 even 16 Ib. 
 
 $ Thiistons-is the-standard at Newmarket; .that is> the persons 
 Tiding races are weighed by this stone, and a jockey that is said to 
 weigh 7j stone, weighs 105 Ib. avoirdupois. By the avoirdupois 
 ounce andlb.) hay and bread are weighed, according to the following;, 
 
3S 
 
 AVOIRDUPOIS WEIGHT. 
 
 Ex. 1. How many drams are there in 225 tons, 17 cwt. 
 
 3 qr. 24 Ib. 12 oz. 8 dr. ? 
 
 tons, cwt. qr. Ib. oz. dr. 
 
 225 1J 3 24 12 8 
 20 
 
 4517 I multiply by 20 and take in the 17 cwt., 
 
 4 because 20 cwt. make one ton; then by 4 
 
 '" and take in the 3, because 4 quarters make a 
 
 180/1 cwt,; then by 28 and take in the 24, because 
 
 28 28 Ib. make a quarter ; then by 16 and take 
 
 in the 12, because 16 ounces make a pound : 
 144572 and again by 16 and take in the 8, because 
 36144 16 drams make an ounce. 
 
 Answer, 129539272 drams. 
 
 129539272 
 
 NOTE. 
 
 HAY WEIGHT. 
 56 Ibs. of old hay .... 
 
 60 Ibs. of new hay - 
 
 36 trusses - - - - - ' 
 
 Of straw, 36 Ibs. make the truss, and 36 trusses the load, 
 
 1 truss. 
 1 load. 
 
 BREAD. 
 
 A peck loaf weighs 
 
 A half ditto 
 
 A quartern ditto - 
 
 A peck of flour weighs 
 
 A bushel 
 
 A sack, or 5 bushels 
 
 Ib. oz. 
 
 17 6 
 
 8 11 
 
 4 5 
 
 14 O 
 
 56 
 
 280 
 
APOTHECARIES' WEIGHT. 39 
 
 Ex. 2. How many tons are there in 259078544 drams ? 
 
 4)259078544 I divide by the same numbers with 
 
 . which I multiplied in the last example, 
 
 4)64/69636 only in the reverse order: and instead 
 
 of dividing by 16, 16, and 28, by long 
 
 4)16192409 division, I divide by their component 
 
 parts, 4X4;4X4;7X 4. In bring- 
 
 4)4048102 1 ~) 07,. ing the ounces into pounds, I have two 
 remainders, viz. l and 2, to find the 
 
 7)1012025 2j value of which, I multiply the last re- 
 
 mainder 2, by the first divisor 4, and 
 
 4)144575 take in the 1, which make 9 ounces. 
 
 - Ib. For the same reason the remainder 3 is 
 
 4)36143 3 zz 21 equal to 21 Ib. See note to pages 27 and 
 
 28. 
 
 2 4 0)903 t 5 3 
 
 tons, cwt. qrs. Ib. oz. 
 
 451 15 3 21 9 Answer 451 15 3 21 9 
 
 Ex. 3. In 179 cwt. how many pounds ? 
 Ex. 4. Reduce 8345 tons into quarters. 
 
 Ex. 5. How many ounces are there in 4 tons, 15 cwt. 2 qrs. 12 Ib.? 
 Ex. G. In 233076 ounces of sugar, how many cwt. 
 Ex. 7. How many drams are there in 53 tons, 14 cwt. l qr. 12 Ib. 
 14 oz. 8 dr. ? 
 
 Ex. 8. In 32384818 drams, how many tons weight? 
 
 APOTHECARIES' WEIGHT.* 
 
 TABLE. 
 
 20 grains (gr.) - make l scruple 9 = 20 gr. troy. 
 
 3 scruples - - l dram 3 = 60 
 
 8 drams - - l ounce J = 48O 
 
 12 ounces - - 1 pound ft = 5760 
 
 NOTE. 
 
 * By this weight Apothecaries mix their medicines, but they buy 
 their drugs by Avoirdupois weight. The pound and ounce made use 
 of by Apothecaries, and the pound and ounce Troy weight, are the 
 same, but the smaller divisions are different. See note to Troy weight. 
 Physicians write their prescriptions according to the following table 
 and characters : 
 
 20 grs. troy - zz l scruple 9j 
 
 60 - zz 1 dram jj- zz Biijj 
 
 480 - zz l ounce Jj zz jviij 
 
 5760 - zz 1 pound Ibj zz Jxij 
 
40 APOTHECARIES WEIGHT; 
 
 Ex. 1. How many grains are there in 2 Ib. 5 oz. 4 dr. 
 1 scr, 17 gr. ? 
 
 Ib. oz. dr. scr. gr. 
 
 2 5 4 1 17 
 
 12 
 
 2Q T multiply the pounds by 12, and take in 
 
 8 the 5, bejause 12 ounces make a pound : 
 
 afterwards by 8, 3, and 20, taking in the 
 
 236 several drams, scruples, and grains, as in the 
 
 3 former articles. 
 
 709 
 
 20 
 
 14197' Answer - 14197 grains. 
 
 Ex. 2. In 42591 grains, how many pounds ? 
 
 2,0)4259,1 
 
 The multipliers in the last example 
 arf made divisors in this, in the reverse 
 
 order. 
 
 22)88 5 
 
 7.4 5 2 11 
 
 Answer - 7 Ib. 4 oz. 5 dr. 2 scr. 1 1 gr. 
 
 Ex. 3.. In 51 Ib. 2 oz. of rhubarb, how many scruples? 
 Ex. 4. In 234876 grains, how many pounds ? 
 Ex. 5. How many pounds are there in 1000 ounces_of opium ? 
 Ex. 6. In 239. Ib. 9 oz. 2 dr. 2 scr. 14 gr., how many grains ? 
 Ex. 7. How m any scruples are there in one hundred ami three 
 ounces of Peruvian bank ? 
 
 Ex. 8. In 126794 grains, how many pounds? 
 
 Apothecaries make use of the following characters also: 
 R recipe, take. 
 
 a, aa, or ana, of each the same quantity. 
 f5> or ss, signifies the half of any thing, 
 cons;, congius, a gallon, 
 coch. cochleare, a spoon fuL 
 M manipulus, a handful. 
 P. pugiU as much as can be taken between the trlumb 
 
 two fore fingers, 
 ^s.^a sufficient quantity,- 
 
44 
 WOOL WEIGHT. 
 
 l-i pounds* . make . l stone, st. 
 2 stone, or 2 8 pounds, l tod . . t. 
 
 TABLE. 
 
 6j tod, or 13 stone, or i wey, or weigh, w. 
 2 weys i sack, s. 
 
 12 sacks . L . ..... i last, . 1. 
 
 6 Ex. i, How many stone are there in 6 weys of wool ? 
 
 . 6 i 
 36 In multiplying by 6 J, I first take the 6, as usual, and then to 
 
 multiply by ^, I divide the multiplicand by 2, place the quotient 
 ~ 3 under the former product, and add them together for the 
 
 answer. 
 
 2 
 
 78 Ex. 2. In 786 stone of wool, how many weys ? 
 
 The readiest way of working this example is, to divide the stone by 13)786 
 13 ; because 13 stone make l wey. The answer is, 60 weys and GO^-G 
 
 6 stone, or 60 weys and 3 tods. 
 
 Ex. 3. In a pack of wool, weighing 3 cwt. 2 qrs. how many tods are there ? 
 
 Ex. 4. How many ounces are there, in a.tod of wool? 
 
 LONG MEASURE. f 
 
 TABLE. 
 
 3 barley corns (b.c.) . l inch, 
 
 2 inches % 1 foot, ft. 
 
 3 feet, or 36 inches . . i yard, 
 
 5 yards, *f[ or 1 6 J ft. l pole, or rod, p. 
 40 poles,or 220 yds. l furlong, fur.** 
 8 furlongs, or 1760yds. l mile,m.-f f- 
 
 2 yds. or 6 ft. or 72 inch, l fathom, fth.| 3 miles, or 528oyds. 3 league, lea. 
 
 NOTES. 
 
 * The table usually begins with 7 pounds make l clove ; but the clove differs 
 n different counties, and, besides, is not often used in the wool trade. The stone 
 s also different in different counties : in Gloucestershire it is l 5lb., but in Here- 
 irdshire it is only I2lb. By a statute of Hen. VII. it was made 1 4lb. as above. 
 
 f The origin of Long Measure is taken from a grain of barley, of which 3, se- 
 ated out of the middle of the ear and well dried, make, l inch; accordingl}, l 
 arley -corn is the least measure. But in this, as in other weights and measures, 
 ic standard to which all are referred, is preserved at Guildhall, London ; and 
 icrefore we have no need to look after grains of wheat, or of barley, to get accu- 
 te weights and measures, 
 
 J Four inches make a hand*, which is used in estimating the heightcf horses, 
 
 || The English yard is said to have been taken from the arm of king Hen. I., 
 
 the year 1101. 
 
 The French toise answers to the English fathom, with this difference, that 72 
 rench inches are equal to 76.736, or nearly 7 6j English inches. 
 
 ^[ The pole is different in different counties : in Lancashire, 7 yards rnak a 
 ole, and in Cheshire, an adjoining county, they reckon 8 yards to a pole ; and 
 i some counties, 6 yards go to a pole. 
 
 ** A furlong being 220 yards, or 660 feet, or 7920 inches; a chain, used in 
 ind-measuring, is the tenth part of a furlong, or 792 inches ; and there being 
 00 links in a chain, the link is 7.92 inches. 
 
 The mile is of different lengths in different countries. The ancient 
 
42 LONG MEASURE. 
 
 Ex. 1. How many yards are there between London and 
 Bath, the distance of which is 108 miles? 
 
 309 In multiplying by 5j, I multiply first by the 
 
 8 5 as usual; then to multiply by J, I divide the 
 
 multiplicand by 2, add the quotient so found, 
 
 864 to the product already obtained : a shorter way 
 
 40 would be to multiply 108 by 1760, the number 
 
 1 of yards in a mile: thus . . 1760 
 
 108 
 
 100080 Answer . . . 1Q0080 yards. 
 
 190080 
 
 NOTES. 
 
 Roman, and modern Italian mile, contained 1000 paces, miUe passus t 
 whence the term mile U derived. The following table will shew the 
 length of the mile, or league, in the principal nations of Europe, ex- 
 pressed in yards. 
 
 yards 
 Mile of Russia ..... is . . . noo, or 5 furlongs 
 
 - Italy ..... . . . 1467 nearly, or 5-6thsof an Eng. mile 
 
 ..... England ... ... 1760 
 
 Scotland and Ireland . . 2200, or i English mils and a quarter 
 
 Small league, of France,* . . 2933, or i English mile,and 2-3dsof 
 
 a mile 
 Mean league of ditto ...... 3666, or 2 English miles, and i-i2th 
 
 of a mile 
 
 dito 
 
 Spain ......... 5028, or 3 English miles nearly 
 
 Germany ....... 5866, or 3 English miles and a half 
 
 ~ Denmark * " I ' * ' ?' 233 or 4 mi " es and l " 6th ^ a m ' lle 
 - Hungary ........ 8800, or 5 English miles. 
 
 * The French, in all these measures, now make use of the Metre, 
 which is equal to 391 inches nearly, or to one yard, three inches, and 
 one-third of an inch. 
 
CLOTH MEASURE. 
 
 Ex. 2. In 7G0329 feet, how many leagues ? 
 
 3; 7 C03 2 9 
 
 I first bring the feet into yards, by di- 
 viding by 3 ; then, as I cannot divide by 5^, 
 I multiply the last qu * lent by 2, to bring it 
 into half-yards, and divide by 11, because 
 there are 11 half-yards in a pole, I find a 
 remainder of 6, which are halt-yards, equal 
 to 3 yards. 
 
 11)506886 
 
 4,0)4608,0 6 3 
 
 8) 11 52 
 8)144 
 
 43 Ans. 48 lea. m. fur. op. 3yds. 
 
 Ex, 3. How many inches are there in 1009 miles ? 
 
 Ex. 4. Reduce 57 rn. 4 fur. ss p. 3 yds. 2 ft. 3 in. 1 b. c, into 
 barley-corns. 
 
 Ex. 5. In 100004 poles, how many inches? 
 
 Ex. 6. In 4og683 feet, how many furlongs ? 
 
 Ex, 7. How often will the wheel of a coach turn round in going 
 from London to Sheffield, or in 160 miles, supposing the circumference 
 of the wheel to be 16 feet? 
 
 Ex. 8. Suppose on an average I step two feet and a half ; how many 
 steps shall 1 take in walking from London to Richmond, a distance of 
 10 miles? 
 
 CLOTH MEASURE. 
 
 TABLE. 
 
 2j inches * - - ~ - make l nail, n, 
 
 4 nails, or 9 inches ' - - l quarter, qr. 
 
 4 quarters, or 36 inches - 1 yard, yd. 
 
 5 quarters, or 45 inches 1 English ell, E.e. 
 
 Ex, 1. How many inches in length are there in 156 ells 
 
 of cambric ? 
 
 156 
 5 
 
 780 
 
 1560 
 195 
 
 In multiplying by Qj, I first multiply by the 
 2, then divide the multiplicand by 4, which is 
 the same as multiplying by % ; add the sums 
 thus found for the true answer. 
 
 1755 
 
 Answer - 1755 inches, 
 
44 SQUARE, OR LAND MEASURE. 
 
 Ex. 2. In 1000 inches of cotton, how many yards are there? 
 9)1000 Here I divide by 9, because 9 inches make 
 
 - a quarter of a yard, and it is easier to divide 
 
 "4) 1 1 1 1 by 9, than by 2 j, and then by 4. 
 
 27 3 1 Ans. 27 yds. 3 qr. o n. l in. 
 
 Ex. 3. How many English ells are there in three thousand and fifty- 
 ftve nails? 
 
 Ex. 4. In 15yds. 2 qr. 3 n. l in., how many; -half inches ? 
 Ex. 5. How many inches are there in 10056 yards ? 
 Ex. 6. Reduce 546 English ells to nails. 
 
 SQUARE, OR LAND MEASURE.* 
 
 
 TABLE. 
 
 
 144 square inches 
 
 make l square foot 
 
 
 9 square feet 
 
 - 1 square yard 
 
 
 1 00 square feet 
 
 1 sq. of flooring, 
 
 roofing, &c. 
 
 1 00 acres 
 
 l hide 
 
 
 301 square yards 
 
 l perch, p. 
 
 
 40 perches 
 
 l rood, rd. 
 
 
 4 square roods, 
 
 or 4840 square yds. 1 acre-f- 
 
 
 640 acres 
 
 l square mile. 
 
 
 NOTES. 
 
 * Square-measure is used to estimate all kinds of superficies, such 
 as land, paving, plastering, roofing, tiling, and every thing that has 
 length and breadth ; thus, if I want to measure a room 25 feet long 
 and 18 broad, I multiply 25 by is, and find the square measure equal 
 to 450 feet. 
 
 Land is measured by a chainv called Gunter's chain, which is 4 poles, 
 or 22 yards, or 66 feet long ; and it consists of 100 equal links. Ten 
 chains in length, and one in breadth, make an acre, that is, 100 X 
 100 X 10 = 100000, equal the number of links in an acre. If there- 
 fore I have a field to measure sixty-three chains and fifty-five links 
 long, and twenty- five chains twelve links wide, I put them do w n thus : 
 63.55 X 25.12 = 159.63760, the answer is 159 acres, and 63760 links, 
 over. The chains and links are separated by a dot, and then they are 
 multiplied as in common Multiplication, taking care to cut off, by a 
 dot, five figures of the product towards the right-hand, which is the 
 sarne thing as to divide by 1 00000. 
 
 If I wish for greater accuracy, I multiply the remainder 63760 by 4, 
 the number of roods in an acre, and cut off five figures again as a re- 
 mainder, which must then be multiplied by 40, the number of perches 
 ina rood: thus6376oX4 = 2.55040: now 55040X 40 = 22. 01600. The 
 true answer, therefore, is l 59 ac. 2 r'd. 22 p., and a remainder of 1600. 
 
 f A piece of ground, as a field, garden, &c., that measures rather 
 
SQUARE, OR LAND MEASURE. * 4J 
 Ex. I. How many yards are there in 5604 acres ? 
 
 5604 
 4 
 
 22416 
 
 40 Here, as in a former case, to multiply by aoj, 
 
 I first multiply by the 30, and then divide the mul- 
 
 896640 tiplicand by 4, which is the same as to multiply by 
 
 30|; a ; and adding together the two sums thus found 
 
 for the answer. 
 
 o 7123 36o Answer ' * 27123360 yards. 
 
 Ex. 2. In 6534 square feet, how many perches ? 
 
 9)6534 
 
 726 
 4 
 
 As we cannot divide by 30^, I multiply the 
 
 121)2904(24 726 yards by.4, to bring them into quarters, and 
 
 242 then divide by 1.21, because there are 121 quar- 
 
 ters in 30^ yards. 
 
 484 
 
 484 
 
 * * Answer - 24 perches. 
 
 Ex. 3. How many roods are there in 382 perches? 
 Ex. 4. In 561 acres of ground, how many perches and yards ? 
 Ex. 5. In 2967400 inches, how many acres ? 
 Ex. 6. How many perches are there in 997 acr. 2 rd. 10 p.? 
 
 When length, breadth, and thickness, are to be taken into con^ide- 
 ration, it is called cubic, or solid measure, which is used to estimate 
 the quantity of stone or marble in blocks, or of timber in trees. Hence 
 the following Measure : 
 
 more than 69| yards in length, and as much in breadth, contains just 
 an acre. A garden of half an acre is comprised in a square, whose 
 sides are 49 yards 5 in. long : and the sides of one of a quarter of ail 
 acre will be nearly 35 yards long. 
 
 A square is a geometrical figure of foiiF equal sides and angles : and 
 a square number is produced by multiplying any number into itself, 
 thus; 49 and 144 are square numbers, being produced by multiplying 
 7 and 12 into themselves, as 7 X 7 n: 43, and 12 X 12 zz 144. 
 
CUBIC, OR SOLID MEASURE.* 
 
 TABLE. 
 
 1728 cubic inches - - make 1 cubic foot 
 27 cubic feet - - - - l yard 
 40 feet of rough timber - -V bad Of 
 50 feet of hewn timber J 
 
 42 feet ----- l ton of shipping. 
 
 Ex. 1. In 36 solid yards, how many inches ? 
 
 '36 
 27 
 
 252 
 72 
 
 972 
 1728 
 
 Answer - 1679616 inches. 
 
 Ex. 2. How many solid inches are there in 2 tons 12 feet of hewn 
 timber ? 
 
 Ex. 3. In 1259712 solid inches, how many yards ? 
 
 NOTES. 
 
 * A cube is a solid body, that has length, breadth, and thickness, 
 of equal dimensions : it contains six equal sides, A cubic number is 
 produced by multiplying any number twice into itself; 27, 125, and 
 512, are cubic numbers, being produced by multiplying 3, 5, and 8, 
 twice into themselves, as 3 X3X3 = 27;5X5X5r=i 125 ; 
 8X8X8 = 512. 
 
 -f* A cubic yard of earth is called a load : 128 cubic feet, that is, a 
 pile of wood 8 feet long, 4 broad, and 4 deep, make a cord of wood j 
 but los cubic feet make a stack. 
 
47 
 
 WINE MEASURE.* 
 
 TABLE. 
 
 4 gills - make 1 pint, pt. 
 
 2 pints .... i quart, qt. 
 
 4 quarts - - - . j gallon, gal.-f* 
 
 63 gallons i hogshead, hhd. 
 
 84 gallons i puncheon 
 
 2 hogsheads, or 126 gallons - l pipe, or butt, p. 
 
 2 pipes, or 252 gallons i tun, t. 
 
 Ex. 1. How many gallons are there in 5 pipes of wine ? 
 
 5 
 
 2 
 10 
 
 63 
 Answer - 63 o gallons. 
 
 NOTES. 
 
 * Wines, spirits, cider, perry, oil, vinegar, and milk, are bought 
 and sold by this measure, which extends only to the gallons, for in 
 different kinds of wine the measures are very different, as follow: 
 Claret, 63 gallons 1 hhd. 
 
 Madeira, 110 ditto 1 pipe 
 
 Vidonia, 1-20 ditto 1 ditto 
 
 Sherry, 130 ditto l ditto 
 
 Port, 138 ditto 1 ditto 
 
 Buna's,} '"' > di 
 
 The pipe of Port is seldom accurately 138 gallons, and it is cus- 
 tomary to charge what the vessel actually contains. 
 
 f- By an act of parliament passed in the reign of Queen Anne, 
 the wine gallon is fixed at 231 cubic inches. 
 
 Hence a pint is - 28.875 cubic inches 
 
 a quart is - 57.75 do. 
 
 It is ascertained that 12 wine gallons of distilled water weigh exactly 
 100 pounds avoirdupois. 
 
 The origin of liquid measure, was from Troy-weight : eight pounds 
 Troy of wheat gathered from the middle of the ear, and well dried, 
 were,, by a statute made in the reign of Henry 111., ordained to be a 
 gallon of wine measure. No other liquor measure but this was used 
 for ages ; and it would, perhaps, be difficult to ascertain how the 
 several changes have obtained in the country. 
 
48 ALE AND BEER MEASURE. 
 
 Ex. 2. In 7006 pints, how many gallons ? 
 2)7006 
 
 4)3503 
 
 875 3 Answer - 875 gal. 3 qts. 
 
 Ex. 3. In 31490 pints, how many gallons? 
 
 Ex. 4. In 3 tuns, l hhd. 49 gallons of claret, how many quarts ? 
 Ex. 5. How many tuns of Port wine are there in 46088 gallons?* 
 Ex. 6. In ten thousand gills of Sherry, how many hogsheads ? 
 
 ALE AND BEER MEASURE. 
 
 TABLE. 
 
 
 2 pints - make 
 
 l quart, qt. 
 
 4 quarts - 
 
 1 gallon, gal.f 
 
 9 gallons - 
 
 1 firkin, fir.J 
 
 2 firkins, or 1 8 gallons - 
 
 l kilderkin, kil. 
 
 2 kilderkins, or 36 gallons 
 
 1 barrel, bar. 
 
 54 gallons - 
 
 l hogshead, hogs. 
 
 2 hogsheads, or 108 gallons - 
 
 l butt, bt. 
 
 * If the pupil should divide by 9 and 7> instead of 63 in Long 
 Division, he will find two remainders of 8 and 3, the value of which 
 183X9 + 8 35. Seep. 27. 
 
 J- One gallon, beer measure, contains 282 solid or cubic inches : 
 
 Hence a pint is - - - 3.5.25 cubic inches. 
 
 a quart is -. 70.5 do. 
 
 a barrel, or 36 gallons - .10152. o do. 
 
 a hogshead, or 54 gallons 15228.0 do. 
 
 Ten yards of inch pipe, (that is, of pipe whose diameter is one inch,) 
 contain exactly an ale gallon. 
 
 A cubic foot of water weighs 1000 ounces; of course 32 cubic 
 feet weigh 2000 Ib. which was formerly a ton. 
 
 J In the year 1689, a statute of excise was passed, which made a 
 firkin of ale or beer, without distinction, to consist of 8 gallons: 
 this has, however, been long in disuse ; and it was customary, till 
 within a few years, to make the firkjn of beer to consist of 9 gallons* 
 but that of ale only of 8 ; custom has now abolished the distinction, 
 and at present for beer and ale the firkin contains 9 gallons, and, of 
 course, w,e do not, in this work, retain any other measures in the 
 tables than are used in the existing business of life. 
 
CORN MEASURE. 
 
 Ex. 3 . In 500 barrels of ale, how many pints ? 
 Ex. 2. In 9065 butts of strong beer, how many gallons? 
 Ex. 3. How many quarts are there in 79 hogsheads of beer ? 
 Ex. 4. In 76459 quarts, ho\v many kilderkins ? 
 Ex. 5. In thirty thousand eight hundred pints of porter, 
 hogsheads ? 
 
 Ex.. 6. Mow many pints are there in 3 butts of beer ? 
 
 CORN MEASURE * 
 
 TABLE. 
 
 2 pints ------- make 
 
 4 quarts ---- --_ 
 
 2 gallons --------- 
 
 4 pecks -- 
 
 2 bushels --- - --- - - 
 
 5 bushels --- - ----- 
 
 8 bushels -.-- ----- 
 
 5 quarters, or 40 bushels - - - 
 2 weys, or so bushels - - - - 
 
 EXAMPLES. 
 
 Ex. i. In 57 quarters of corn, how many pecks ? 
 Ex. Q. In 24S456 pecks of oats, how many lasts ? 
 Ex. 3. How many pints are there in 19 bush. 2 p. of canary seed? 
 Ex. 4. In 2 weys and 4 quarters of barley, how many bushels ? 
 Ex. 5. How many quarters of corn are there in 50,000 gallons ? 
 
 quart 
 
 gallon 
 
 peck 
 
 bushel 
 
 strike 
 
 load of corn 
 
 quarter-f' 
 
 wey, or load of wheatj 
 
 last 
 
 * One gallon corn measure, contains 268. 6 solid inches: Hence 
 A pint is-------- 33.6 cubic inches nearly 
 
 A quart is - - - - - - - - a 67/2 do. 
 
 A peck, or 8 quarts - - - - - 537.6 do. 
 
 A bushel, or 4 pecks - - - - 2150.4 do. 
 
 A heaped bushel is one-third more. 
 A quarter, or eight bushels - 17203.2 do. 
 The standard bushel is a cylindric vessel ibf inches in diameter, 
 and 8 inches deep. 
 
 f A quarter of wheat was so called, upon the supposition that it 
 weighed 500 Ib. or a quarter of a ton. See note to Ale and Beer 
 measure. The bushel, or the one-eighth of a quarter, is equal to 
 looo ounces, or a cubic foot of water. 
 
 By this measure, corn, seeds, fruits, sand, salt, Newcastle coals, 
 oysters, &c. are measuied and sold. A bushel of wheat on the average 
 weighs 60 pounds; of barley 50 pounds ; of oat* 3 8 pounds. 
 
 J It will be observed, that 6 bushels and 40 bushels arc both called 
 loads ; the one is reckoned a man's load ; the other to be removwi 
 bv a cart. 
 
 D 
 
50 
 COAL MEASURE. 
 
 TABLE. 
 
 4 pfecks - make 1 bushel' 
 
 3 bushels - - - - - i sack 
 
 12 sacks, or 3 6 bushels - 1 chaldron 
 
 21 chaldrons - - 1 score.* 
 
 EXAMPLES. 
 
 Ex. 1. Mow many sacks are there in five score of coals ? 
 
 Ex. 2. How many bushels of coals are there in a vessel containing 
 1 5 score ? 
 
 Ex.3. In ten thousand and 12 pecks, how many chaldron are 
 there ? 
 
 Ex. 4. How many chaldron of coals are there in ten thousand and 
 five pecks ? 
 
 Ex. 5. In three score and ten bushels of coals, how many sacks ? 
 
 COMMERCIAL NUMBERS, 
 
 OR ARTICLES SOLD BY TALE. 
 
 12 articles of any kind 
 
 - 1 dozen 
 
 33 ditto - 
 
 - 1 long dozen 
 
 12 dozen - 
 
 - 1 gross 
 
 20 articles of any kind 
 
 - 1 score 
 
 5 score - , - 
 
 - l hundred 
 
 6 score ----- 
 
 - l great hundred 
 
 12 score - - 
 
 - 1 pack of wool 
 
 5 dozen skins of parchment 
 
 - 1 roll 
 
 ?2 words in Common law 
 
 - l sheet 
 
 so in the Exchequer 
 
 - l ditto 
 
 go in Chancery 
 
 ditto 
 
 84 sheets of paper - 
 
 quire 
 
 20 quires - - - 
 
 ream 
 
 21^ quires, or 5 16 sheets - 
 
 printer's ream 
 
 2 reams . - - - 
 
 bundle 
 
 NOTE. 
 
 * In the purchase of coals, to a single chaldron there are 12 sacks 
 only ; but if 5 chaldrons be ordered at one time, the seller must send 
 S3 sacks. 
 
ARTICLES BY TALE. 
 
 Folio is the largest size of books, of which, 
 
 2 leaves, or 4 pages, make a sheet. 
 
 Quarto, 4to. - - 4 leaves, or 8 pages, make a sheet, 
 Octavo, 8vo. - - 8 leaves, or 16 pages, ditto 
 Duodecimo, I2mo. - 12 leaves, or 24 pages, ditto 
 Octodecimo, 18 mo.. - 18 leaves, or 36 pages, ditto 
 
 EXAMPLES* 
 
 Ex. i. How many long dozen are there in ten thousand oranges ? 
 
 Ex.2. How many gross are there in one hundred and fifty thou- 
 sand corks ? 
 
 Ex. 3. In seventy thousand quills,, how many great hundreds are 
 there ? 
 
 Ex. 4. I have a deed containing 4 skins of parchment, and each 
 skin contains 650 words; for how many sheets shall I have to pay 
 the person who copies it, reckoning according to the common law 
 charge ? 
 
 Ex. 5. The writing of an Exchequer cause occupies 315 sheets: 
 for how many words shall I have tj pay the clerk who copies it for 
 me? 
 
 Ex. 6. A suit has been four years in chancery, and I. wish to have 
 a copy of all the proceedings ; for how many sheets shall I pny, 
 supposing it occupies I20i skins of parchment, and each skin 690 
 words ? 
 
 Ex. 7. How many sheets are there in 40 reams of paper ? 
 
 Ex. 8. How many common reams of paper are there in ten thou- 
 sand printer's reams ? 
 
 Ex. 9. What number of sheets less are there in 50O common reams 
 of paper, than there are in the same number of printer's reams ? 
 
 Ex. 10. What number of pages arc there in a folio coBtaining 211 
 sheets ? 
 
 Ex. 11. What will be the difference in the number of 
 whether I print in 12mo. or is mo., supposing my work \vill m 
 fourteen sheets ? 
 
 Ex. 12. What number of words are there in Dr. Gregory's." Dic- 
 tionary of Arts and Sciences, which contains 240 sheets 4to., and 
 each page contains 14784 words ? 
 
 Ex. 13. How many reams of paper were used in printing that 
 Dictionary, six thousand copies having been taken off? 
 
 Ex. 14. How many peas w r ere used in wrifring the said Dictionary,, 
 supposing each pen to write 840 words? 
 
 D2 
 
T I M E. 
 
 TABLE, 
 
 60 seconds (sec.) ..... make 1 minute, m. 
 60 minutes, or 3600 seconds . . . ] hour, h. 
 24 hours, or 1440 minutes .... 1 clay, d. 
 
 7 days, or 168 hours ...... 1 week, w. 
 
 4 weeks, or 28 days ...... 1 month, m. 
 
 12 calendar months, or 52 weeks, or 
 
 365 days, or 8766 hours . . 1 year.* 
 
 Ex. 1. In 4109 days, how many months and years? 
 
 Ex. 2, Reduce 150 days to hours and minutes? 
 
 Ex. a. In 70 years how many days, supposing each year to consist 
 
 Ex. 4. How many minutes, hours, arid days, are there in 5960034 
 seconds ? 
 
 Ex. 5. How many minutes are there in 1808 years, allowing 365^ 
 days make one year ? 
 
 Ex.6. How many seconds has a boy lived, who is 12 years, g 
 months, and 13 days old, reckoning 13 lunar months of 28 days each 
 to a year ? 
 
 * Thirteen months, each containing 4 weeks, arid each week con- 
 taining 7 days, make only 364 days ; but the common year is di- 
 vided into 12 calendar months, and it consists of 3654- days, some 
 Of the months having 30, and some J3l days, and February having 
 only 23 days, excepting on leap year, which is every fourth year, 
 when February has 2<^days: this adoption of one day in four years, 
 makes the reckoning 365^ days for each year: the following lines 
 will assist the memory in recollecting the length of each particular 
 month ; 
 
 Thirty days hath September, 
 April, June, and November ; 
 February has twenty-eight aloRC, 
 And all the rest have thirty one. 
 
 Though the year is usually reckoned at 365f days, yet that is not 
 perfectly accurate, it being fully ascertained, that the year consists of 
 365 days, 5 hours, 48 minutes, 48 seconds. 
 
 Leap-year maybe found by dividing the year by 4; if there be no 
 remainder it is leap-year; thus 1808 is divisible by 4, without a 
 remainder, and is leap-year. The year 1 800 was an exception, and so 
 will 1900, a day dropt, in an hundred years, being necessary to keep 
 the calculations accurate. 
 
ASTRONOMY.* 
 
 TABLE. 
 
 60 seconds' (CO*) make 1 minute, i 1 
 
 CO minutes l degree, 1 
 
 J30 degrees 1 sign 
 
 12 signs, or 360 degrees . 1 great circle. 
 
 Ex. 1. In 185 degrees bow many minutes and seconds? 
 Ex. 2. How many degrees are there in five thousand and 
 fifty-five seconds ? 
 
 Ex! 0. Hew many seconds arc there in a great circle ? 
 Ex. 4, In 548056 seconds, how many signs ? 
 
 5* How many seconds are there in s. 4 55' 5G V ? 
 Ex. 6. In 700809 seconds, how many degrees ? 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex, 1. In 195 pounds, how many shillings, pence, and 
 
 Ex. 2. In 77 guineas, how many shillings, pence, and 
 farthings ? 
 
 Ex, 3. How many crowns, half-crowns, shillings, and 
 sixpences are there in 354 ? 
 
 Ex. 4. In 4432127 farthings, how many pence, shillings, 
 pounds, and guineas ? 
 
 * This table is used in astronomical and geographical calculations. 
 The astronomical day commences at 12 o'clock at noon ; but the com- 
 mon or civil day begins at 1 2 o'clock the preceding night : of course the 
 astronomical day begins 12 hours later than the common day. 
 
54 MISCELLANEOUS EXAMPLES. . 
 
 Ex. 5. In 14 ingots of silver, each weighing 27oz. 5dwts*, 
 how many grains ? 
 
 Ex. 6. In three dozen of table spoons, each weighing 
 2oz. 9dwts., how many pounds? 
 
 Ex. 7. In 78 bags of hops, each weighing 3cwt., how 
 many pounds ? 
 
 Ex. 8. How many pounds andcwts. of tobacco are there 
 ia 75 hogsheads, each containing 3cwt. Iqr. 141b. ? 
 
 Ex. 0. In 98465 inches of broad-cloth, how many yards 
 and ells ? 
 
 Ex. 10. In five thousand yards of cloth, how many nails ? 
 
 Ex. 11. How many inches are there between London and 
 Bristol, a distance of 120 miles ? 
 
 Ex. 12. How many barley-corns will reach round the 
 earth, which is a great circle of 360 degrees, and each de- 
 gree contains 69 miles ? and how many quarters of barley 
 would be necessary to perform this, supposing 9200 barley- 
 corns to fill a pint measure ? 
 
 Ex. 13. How often will a wheel turn in going from Lon- 
 don to York, a distance of 198 miles, if the wheel be 2-J 
 yards in circumference ? 
 
 Ex. 14. How many perches are there in a field contain- 
 ing 105 acres ? 
 
 Ex. 15. If a field of 5 acres be taken from one of 56 
 acres, how many square yards will remain ? 
 
 x Ex. 16. How many pints and gallons are there in 39 
 hogsheads of cyder ? 
 
 Ex. 17. How many minutes have elapsed since the crea- 
 tion of the world to the year 1808, supposing the 
 world to have been created 4004 years before the birth of 
 Christ, and each year to consist of 365 . days ? 
 
COMPOUND ADDITION, 
 
 ADDITION OF MONEY. 
 
 
 PENCE AND 
 
 SHILLING TABLES. 
 
 
 Pence 
 
 *. d. 
 
 Pence 
 
 s. d., 
 
 Shill. 
 
 . s. d. 
 
 20 - 
 
 - are i 8 
 
 -.12 - 
 
 - are i o 
 
 20 - - 
 
 100 
 
 25 - 
 
 - - 2 1 
 
 1.8 - 
 
 - - 1 6 
 
 25 - - 
 
 1 5 
 
 SO - 
 
 - - 2 6 
 
 24 - 
 
 - - 2 
 
 30 - - 
 
 1 10 O 
 
 S5 - 
 
 - - 2 11 
 
 30 - 
 
 - - 2 6 
 
 35 - - 
 
 1 15 
 
 40 - 
 
 - - 3 4 
 
 36 - 
 
 - - 3 j 
 
 40 - - 
 
 209 
 
 45 - 
 
 - - 3 9 
 
 42 - 
 
 --36 
 
 50 - - 
 
 2 10 
 
 50 - 
 
 --42 
 
 48 - 
 
 - - 4 
 
 60 - - 
 
 3 O 
 
 55 - 
 
 - , 4 7 
 
 54 - 
 
 --46 
 
 70 - - 
 
 3 10 O 
 
 CO - 
 
 - - 5 
 
 GO - 
 
 - - 5 
 
 80 - - 
 
 400 
 
 65 - 
 
 - 5 5 
 
 66 - 
 
 - - 5 6 
 
 90 - - 
 
 4 10 O 
 
 70 - 
 
 - - 5 10 
 
 72 - 
 
 - - 6 
 
 100 - - 
 
 5 O 
 
 75 - 
 
 - - 6 3 
 
 78 - 
 
 - - 6 6 
 
 110 - - 
 
 5 10 
 
 80 - 
 
 - - 6 8 
 
 84 - 
 
 --70 
 
 120 - - 
 
 600 
 
 85 - 
 
 - - 7 i 
 
 90 - 
 
 - - 7 6 | 
 
 13Q - - 
 
 6 10 
 
 90 - 
 
 - - 7 6 
 
 96 - 
 
 --80! 
 
 140 - - 
 
 70O 
 
 95 - 
 
 - - 7 11 
 
 102 - 
 
 --86! 
 
 150 - - 
 
 7 10 
 
 100 - 
 
 - - 8 4 
 
 108 - 
 
 - - 9 
 
 160 - - 
 
 8 O 
 
 K>5 - 
 
 - - 8 9 
 
 114 - 
 
 - - 9 6 ; 
 
 170 - - 
 
 8 10 
 
 130 - 
 
 - - 9 2 
 
 120 - 
 
 - - 10 0! 
 
 180 - 
 
 9 O 
 
 115 - 
 
 - - 9 7 
 
 ' 132 - 
 
 - - 11 ] 
 
 190 - - 
 
 9 10 
 
 120 - 
 
 - - 10 
 
 144 - 
 
 - - 12 
 
 200 - - 
 
 10 
 
 COMPOUND ADDITION is a method of collecting 
 ral numbers of different denominations into one sum. 
 
 RULE. (1) Arrange the numbers so that those of the 
 same denomination may stand directly under each other y 
 and draiv a line under them. 
 
 (2). Add the numbers in the lowest denomination toge- 
 ther, and find how many units of the next higher denomi- 
 nation are contained in their sum. 
 
 (3). Write down the remainder > and carry the units to 
 the next higher denomination) and proceed so to the .end. 
 
66 COMPOUND ADDITION. 
 
 . s. 
 
 d. I first add together the farthings, which I 
 
 Ex. 468 19 
 
 4i find to be 14, but 14 farthings make Bid , I 
 
 123 16 
 
 ll-j put down the arid carry the 3 to the column 
 
 937 12 
 
 9 of pence, which I then add together, and find 
 
 654 13 
 
 7jp the sum to be 58, but by the table, 55 pence 
 
 123 17 
 
 4-| are 4,s. 7^-> therefore 58 pence are 4s. iorf., 
 
 456 13 
 
 loj I put down the 10 and carry the 4 to the 
 
 439 4 
 
 6-| column of shillings ; I now add the shillings 
 
 592 12 
 
 4$ together, and find the sum to be 115, but 115 
 
 3847 15 
 
 sruilings make 5Z. J 5s., I put down the 15, 
 10\ and carry the 5 to the pounds, and proceed as 
 
 
 in simple addition. 
 EXAMPLES OF MONT3Y. 
 
 . s. 
 
 d. . s. d. . s. d. . s. d. 
 
 Ex. 1. 55 3 
 
 8 2. 67 2 8 3. 95 2 9 4. 49 9 11 
 
 62 6 
 
 3 24 9 9 89 7 8 33 8 7 
 
 96 2 
 
 1 38 2 5 7-243 96 12 9 
 
 31 8 
 
 4 4259 67 92 7 5 34 
 
 43 7 
 
 5 78 6 6 51 8 9 51 8 9 
 
 10 9 
 
 8 , 64 6 45 6 4 12 1Q 7 
 
 
 . *. d. 
 
 ..*. d. . s. d. . jr. ,J. 
 
 5. 58 15 9 
 
 6. 42 16 g 7. 02 13 4| 8. 50 }g fc$ 
 
 7 W 5 5 
 
 37 15 11 84 14 9 97 ; ' 
 
 61 7 10 
 
 73 9 9 7 ; 3 18 4| 35 !4 2 
 
 64 36 3 
 
 62 10 6 69 17 10 4fl 1 
 
 $2 15 10 
 
 29 4 4 43 l.> 7 67 J9 2 
 
 19 12 8 
 
 1Q 17 11 35 14 1 if. 24 15 o.| 
 
 , 
 
 . s. d. 
 
 . 5. J. . -. d. . s, d. 
 
 ?. 54 17 0| 
 
 10. 67 16 8j 11. 18 14 8j 12. -41 15 ?-J 
 
 
 /I 13 9 93 15 1C.J 5f) 10 9" 
 
 
 84 11 8| 37 611 C 2 1 6 3| 
 
 
 32 19 3 78 16 5| 78 4 11 
 
 
 4>B I'O 4^ 69 1-2 74- 8/13 '/7>- 
 
 
 55 18 7 4-5 SI) 9-2 .19 c| 
 
 
 ? .24 I 2 1 ; 3 J 13167 
 
 
 
 ; on of th'.5 rule may be thus illustrated: if I have to 
 
 j any s im of mo cy, p;-nce are more c.^iVL-nient than far- 
 
 1^, and shtlUugs than pence j ai:d i:i iar^:e sums bank notes or 
 
COMPOUND ADDITION. 57 
 
 . 
 
 S. 
 
 d. 
 
 . 
 
 S. 
 
 d. 
 
 
 
 . S. 
 
 d. 
 
 . 
 
 S. 
 
 d. 
 
 13. 46 
 
 2 
 
 34 
 
 14. 45 
 
 19 
 
 94: 
 
 15.43 
 
 17 
 
 10* 
 
 36. 52 
 
 18 
 
 10 
 
 65 
 
 10 
 
 4j 
 
 63 
 
 17 
 
 ll:f: 
 
 50 
 
 14 
 
 0| 
 
 67 
 
 12 
 
 2^ 
 
 74 
 
 O 
 
 10 
 
 79 
 
 13 
 
 5^ 
 
 72 
 
 6 
 
 *I 
 
 77 
 
 14 
 
 9 
 
 81 
 
 17 
 
 8f 
 
 46 
 
 10 
 
 9 
 
 65 
 
 19 
 
 
 82 
 
 13 
 
 10* 
 
 39 
 
 15 
 
 10 
 
 35 
 
 8 
 
 7 
 
 91 
 
 5 
 
 3^ 
 
 98 
 
 32 
 
 11 4 
 
 23 
 
 10 
 
 *4 
 
 47 
 
 19 
 
 ioj 
 
 38 
 
 19 
 
 10 
 
 21 
 
 17 
 
 7? 
 
 39 
 
 14 
 
 7* 
 
 19 
 
 14 
 
 6 
 
 29 
 
 12 
 
 H 
 
 45 
 
 12 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 A'. 
 
 d. 
 
 . 
 
 S. 
 
 d. 
 
 . 
 
 A\ 
 
 d. 
 
 . 
 
 A'. 
 
 d. 
 
 17.77 
 
 15 
 
 4 
 
 38. 57 
 
 15 
 
 01 
 
 19- 446 
 
 19 
 
 9j 
 
 20- 48 
 
 14 
 
 '0 
 
 69 
 
 10 
 
 T 
 
 , 64 
 
 9 
 
 2 
 
 152 
 
 15 
 
 io4- 
 
 36 
 
 13 
 
 10 
 
 41 
 
 
 
 io| 
 
 76 
 
 17 
 
 104 
 
 695 
 
 12 
 
 oi 
 
 74 
 
 15 
 
 71 
 
 57 
 
 13 
 
 8 
 
 97 
 
 16 
 
 9 
 
 758 
 
 3 
 
 5 
 
 23 
 
 18 
 
 2j 
 
 87 
 
 9 
 
 io| 
 
 39 
 
 1 8 
 
 4 
 
 338 
 
 14 
 
 3^ 
 
 48 
 
 9 
 
 6 
 
 91 
 
 16 
 
 I 1-2- 
 
 45 
 
 jo 
 
 10 
 
 166 
 
 39 
 
 11 
 
 81 
 
 16 
 
 4^ 
 
 75 
 
 14 
 
 8 
 
 59 
 
 17 
 
 9 
 
 279 
 
 12 
 
 of 
 
 77 
 
 1 i 
 
 4 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 $ 
 
 rf 
 
 
 
 <; 
 
 r/ 
 
 . 
 
 S. 
 
 rf 
 
 . 
 
 V, 
 
 r/ 
 
 '21 . a 1 ; 
 
 IP 
 
 of 
 
 22. 12 
 
 14 
 
 
 23. 54 
 
 I 1 
 
 10 
 
 24. 414 
 
 19 
 
 9 
 
 
 10 
 
 9 
 
 93 
 
 16 
 
 10 i 
 
 22 
 
 19 
 
 6| 
 
 627 
 
 17 
 
 1 1?: 
 
 64 
 
 18 
 
 7* 
 
 17 
 
 12 
 
 11 
 
 61 
 
 16 
 
 y i 
 
 741 
 
 6 
 
 45 
 
 38 
 
 10 
 
 3 
 
 56 
 
 33 
 
 7* 
 
 34 
 
 17 
 
 o-J 
 
 865 
 
 14 
 
 8 
 
 49 
 
 15 
 
 i'li 
 
 91 
 
 19 
 
 11 
 
 53 
 
 12 
 
 1 1^ 
 
 9*7 
 
 6 
 
 io| 
 
 64 
 
 19 
 
 
 76 
 
 14 
 
 5; 
 
 72 
 
 >0 
 
 6 
 
 347 
 
 34 
 
 10| 
 
 9 1 - 
 
 17 
 
 8 5 
 
 14 
 
 i 1 
 
 3 
 
 76 
 
 14 
 
 11 
 
 449 
 
 13 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 A'. 
 
 A 
 
 . 
 
 S. 
 
 C/. 
 
 . 
 
 A'. 
 
 C/. 
 
 . 
 
 ft. 
 
 ^. 
 
 25. 427 
 
 IS 
 
 101, 
 
 26. 54 S 
 
 1 1 
 
 6 
 
 27.493 
 
 2 
 
 81 
 
 28. 412 
 
 9 
 
 ll{ 
 
 941 
 
 17 
 
 9 
 
 932 
 
 18 
 
 4-i 
 
 347 
 
 14 
 
 3 4 
 
 924 
 
 19 
 
 6 i 
 
 712 
 
 19 
 
 6 
 
 379 
 
 
 
 6i 
 
 729 
 
 '9 
 
 5 
 
 75,. 
 
 1 1 
 
 3 
 
 C25 
 
 12 
 
 ?4 
 
 414 
 
 17 
 
 2 
 
 672 
 
 5 
 
 8 3 
 
 627 
 
 19 
 
 of 
 
 51 1 
 
 Jl 
 
 30 
 
 573 
 
 4 
 
 5-1 
 
 548 
 
 10 
 
 3 
 
 438 
 
 10 
 
 4 i' 
 
 462 
 
 10 
 
 64 
 
 697 
 
 13 
 
 9\ 
 
 217 
 
 12 
 
 81 
 
 363 
 
 2 
 
 10^ 
 
 363 
 
 11 
 
 9! 
 
 551 
 
 6 
 
 11 
 
 974 
 
 1.. 
 
 7 1 
 
 221 
 
 15 
 
 8 
 
 
 
 
 
 
 
 146 
 
 5 
 
 ! I 
 
 147 
 
 1 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 guineas than shillings. If therefore a person sell 7 yards of tape -at 
 ^ farthings per yard, it is ra, re convenient to receive 5 pennv -pieces 
 and one farthing, than to bavc 21 farthings 5 and so of the higher de- 
 nominations, 
 
 J)5 
 
58 COMPOUND ADDITION. 
 
 . 
 
 S. 
 
 d. 
 
 . 
 
 *. d. 
 
 . 
 
 $. 
 
 d. 
 
 . 
 
 S. 
 
 d. 
 
 29. 152 
 
 15 
 
 4 
 
 30. 504 
 
 3 9j 
 
 31. 576 
 
 14 
 
 9 
 
 32. 8'27 
 
 18 
 
 lif 
 
 255 
 
 18 
 
 63: 
 
 636 
 
 19 5 
 
 613 
 
 12 
 
 1 14 
 
 550 
 
 11 
 
 8J 
 
 348 
 
 1-2 
 
 9* 
 
 421 
 
 2 7f 
 
 7!Q 
 
 13 
 
 4 
 
 938 
 
 9 
 
 4 
 
 410 
 
 
 
 10 
 
 347 
 
 12 10 
 
 914 
 
 14 
 
 6* 
 
 344 
 
 
 
 3 
 
 566 
 
 13 
 
 IT 
 
 383 
 
 
 271 
 
 10 
 
 9 
 
 615 
 
 16 
 
 1^: 
 
 6-31 
 
 6 
 
 4 
 
 848 
 
 x^ ^ 
 
 759 
 
 8 
 
 5| 
 
 471 
 
 o 
 
 7 
 
 781 
 
 3 
 
 10 
 
 710 
 
 8j 
 
 43S 
 
 i:> 
 
 
 214 
 
 15 
 
 TOj 
 
 949 
 
 16 
 
 7 
 
 483 
 
 10 4j 
 
 918 
 
 1 L 
 
 4 
 
 745 
 
 19 
 
 2 
 
 123 
 
 15 
 
 11 
 
 426 
 
 19 7 
 
 564 
 
 7 
 
 2 
 
 90 
 
 9 
 
 9 
 
 
 33. 79 2 
 
 10 
 
 3| 
 
 34. 88 
 
 16 11 J 
 
 35. 28 
 
 9 
 
 ** 
 
 36. 60 
 
 15 
 
 i 
 
 437 
 
 14 
 
 9| 
 
 26 
 
 14 5j; 
 
 54 
 
 17 
 
 9 
 
 48 
 
 13 
 
 4 
 
 354 
 
 10 
 
 10-J 
 
 9 
 
 7 2^ 
 
 6 
 
 
 
 1 1 
 
 93 
 
 IS 
 
 G 
 
 516 
 
 IS 
 
 4 
 
 36 
 
 12 4-f 
 
 28 
 
 13 
 
 5| 
 
 7 
 
 7 
 
 10 7 T f 
 
 209 
 
 13 
 
 10- 
 
 41 
 
 18 3 
 
 65 
 
 18 
 
 7* 
 
 35 
 
 i<) 
 
 4 i 
 
 521 
 
 17 
 
 sj 
 
 27 
 
 3 8^ 
 
 9 '2 
 
 6 
 
 4| 
 
 73 
 
 6 
 
 9 g 
 
 739 
 
 6 
 
 10 
 
 54 
 
 15 11-J 
 
 7 
 
 16 
 
 i 
 
 31 
 
 17 
 
 3" 
 
 365 
 
 2 
 
 6 i 
 
 12 
 
 19 6 
 
 14 
 
 5 
 
 10 
 
 59 
 
 14 
 
 icff 
 
 147 
 
 17 
 
 9 
 
 20 
 
 10 
 
 40 
 
 
 
 9 
 
 60 
 
 
 
 10 
 
 
 37. 94 
 
 l 
 
 i 
 
 38. 53 
 
 11 4^ 
 
 39- 68 
 
 19 
 
 if 
 
 40. 75 
 
 12 
 
 B 
 
 68 
 
 2 
 
 6 
 
 6 
 
 2 8 
 
 84 
 
 7 
 
 3 2 
 
 40 
 
 
 
 4 
 
 46 
 
 5 
 
 ll| 
 
 18 
 
 5 3^: 
 
 8 
 
 
 
 5f 
 
 8 
 
 17 
 
 4 
 
 29 
 
 If) 
 
 3 2 
 
 26 
 
 10 7j 
 
 25 
 
 ll 
 
 9^ 
 
 24 
 
 39 
 
 5f 
 
 48 
 
 12 
 
 O 
 
 42 
 
 4| 
 
 9 
 
 13 
 
 7 
 
 59 
 
 15 
 
 2^ 
 
 5 
 
 17 
 
 7 
 
 64 
 
 2 2 
 
 4? 
 
 15 
 
 6 i 
 
 82 
 
 6 
 
 ^T 
 
 61 
 
 13 
 
 3^ 
 
 71 
 
 18 10| 
 
 32 
 
 1 
 
 3 
 
 7 
 
 18 
 
 4 
 
 7 
 
 14 
 
 10* 
 
 3 
 
 14 ll 
 
 1 
 
 18 
 
 2 
 
 33 
 
 2 
 
 9J 
 
 12 
 
 18 
 
 5% 
 
 90 
 
 O 6| 
 
 2 
 
 16 
 
 4 
 
 8 
 
 10 
 
 0^ 
 
 
 
 
 42. 78 
 
 
 
 
 
 
 
 
 41, 39 
 
 14 
 
 4 f 
 
 12 5 
 
 43. 127 
 
 10 
 
 10| 
 
 '44. 515 
 
 14 
 
 63 
 
 97 
 
 12 
 
 
 37 
 
 14 8j 
 
 356 
 
 14 
 
 yf 
 
 943 
 
 17 
 
 
 73 
 
 15 
 
 io| 
 
 35 
 
 6 
 
 483 
 
 9 
 
 4^ 
 
 623 
 
 15 
 
 11^: 
 
 6 
 
 10 
 
 n'i 
 
 28 
 
 16 10| 
 
 849 
 
 7 
 
 11 
 
 417 
 
 19 
 
 3J 
 
 30 
 
 2 
 
 9 
 
 11 
 
 8 85 
 
 680 
 
 18 
 
 1 !- 
 
 338 
 
 24 
 
 10 
 
 16 
 
 12 
 
 6 f 
 
 49 
 
 15 7 
 
 774 
 
 19 
 
 7^- 
 
 385 
 
 18 
 
 11-1 
 
 58 
 
 16 
 
 li 
 
 6 
 
 11 4j 
 
 114 
 
 6 
 
 2| 
 
 764 
 
 13 
 
 6 
 
 2 
 
 13 
 
 7 
 
 ' 62 
 
 15 3 
 
 Q51 
 
 18 
 
 9^ 
 
 453 
 
 19 
 
 2 
 
 S2 
 
 
 
 Si 
 
 5 
 
 18 4$ 
 
 428 
 
 15 
 
 6 
 
 562 
 
 18 
 
 5 I 
 
 10 
 
 1-0 
 
 10 
 
 00 
 
 10 
 
 i67 
 
 16 
 
 2 
 
 223 
 
 14 
 
 2 
 
COMPOUND ADDITION. 
 
 . s. d. 
 
 45. 6^7 16 10^ 46. 
 734 17 4l 
 879 14 3-J 
 919 12 1OJ 
 131 19 11 
 235 7 6| 
 496 18 ;3f 
 587 9 5 
 673 11 10 
 820 19 4 
 
 . 5. d. 
 
 491 16 9 
 272 15 6 
 889 17 10^ 
 647 19 2f 
 398 16 7 
 563 16 10| 
 770 5f 
 945 17 7 
 420 13 9$ 
 150 10 
 
 47. 
 
 . s. d. . 
 
 722 10 9j 48. 477 
 966 4 8j 395 
 899 13 6 736 
 248 16 10|: 692 
 532 14 9 565 
 476 19 7jj: 937 
 744 12 9 ' 441 
 669 15 /| 760 
 593 15 11;J 672 
 150 10 40 
 
 S. de 
 
 16 4% 
 lb 2^ 
 5 11 
 14 9 
 13 5^ 
 17 
 16 4| 
 18 Oj 
 11 11 
 10 
 
 
 49. 
 
 494274 12 9f 
 /C5502 6 4 
 300089 2 2j 
 402193 17 9 
 375451 3 10 
 269440 18 6f 
 323428 15 1O 
 567865 11 9^ 
 910649 1O 6 
 
 50. 901442 16 10 
 234971 5 9|- 
 567352 14 ?1 
 912261 19 2.J- 
 345512 17 9j 
 6/8830 12 6 
 912887 19 10 
 4^6713 10 3j 
 891391 17 81- 
 
 . 
 
 S. 
 
 
 
 A\ 
 
 
 
 1 
 
 d. 
 
 d. 
 
 . 
 
 d. 
 
 . 
 
 51. 4567 
 
 14 
 
 n|* 
 
 52. 3256 
 
 19 
 
 6j 
 
 53. 3567 
 
 12 
 
 9| 
 
 4934 
 
 15 
 
 9 
 
 4397 
 
 10 
 
 iii 
 
 ,7960 
 
 17 
 
 10 
 
 2/65 
 
 16 
 
 101, 
 
 1974 
 
 1.2 
 
 9L 
 
 1234 
 
 15 
 
 7| 
 
 9876 
 
 Ifl 
 
 llL 
 
 7M6 
 
 8 
 
 4 
 
 5678 
 
 12 
 
 8 
 
 3497 
 
 9 
 
 5 
 
 3942 
 
 15 
 
 10^ 
 
 9-23 
 
 14 
 
 10 
 
 1234 
 
 10 
 
 8| 
 
 -4567 
 
 8 
 
 9? 
 
 4567 
 
 13 
 
 11-i 
 
 678 
 
 16 
 
 10 
 
 4567 
 
 17 
 
 1 l|- 
 
 ,5912 
 
 17 
 
 9 
 
 ^4376 
 
 8 
 
 9 
 
 9376 
 
 12 
 
 8 i" 
 
 ,3450 
 
 9 
 
 6 
 
 5794 
 
 15 
 
 4 
 
 4623 
 
 2 
 
 J^' 
 
 7891 
 
 10 
 
 4^ 
 
 7921 
 
 12 
 
 101 
 
 5932 
 
 5 
 
 4 
 
 2845 
 
 6 
 
 3 
 
 1764 
 
 13 
 
 .9$ 
 
 2487 
 
 7 
 
 3 
 
 6;89 
 
 12 
 
 5J 
 
 1 805 
 
 17 
 
 ,4 
 
 5:6* 
 
 16 
 
 llj 
 
 2345 
 
 13 
 
 11 
 
 1764 
 
 12 
 
 ,7 
 
 .3234 
 
 IS 
 
 2 
 
 789 
 
 16 
 
 Pi 
 
 3459 
 
 is 
 
 11 
 
 678 
 
 9 
 
 9 
 
 4972 
 
 15 
 
 10 
 
 2946 
 
 16 
 
 loj 
 
 fi012 
 
 17 
 
 10 
 
 , 3456 
 
 19 
 
 5 
 
 1796 
 
 14 
 
 10 
 
 3456 
 
 2 
 
 2 
 
 7891 
 
 16 
 
 7f 
 
 .4325 
 
 36 
 
 8 
 
 780J 
 
 1.4 
 
 5 
 
 23^5 
 
 14 
 
 11 J 
 
 5678 
 
 12 
 
 ll| 
 
 1234 
 
 13 
 
 10 
 
 6/82 
 
 1-2 
 
 9 
 
 4932 
 
 U 
 
 6 
 
 5678 
 
 15 
 
 7 
 
 4315 
 
 1 1 
 
 /i 
 
 ') 05 
 
 9 
 
 54 
 
 9 2-3 
 
 13 
 
 4 
 
 *105 
 
 8 
 
 6 
 
 * From these three examples the preceptor may make an almost 
 
60 
 
 COMPOUND ADDITION. 
 
 36271 
 
 EXAMPLES OF WEIGHTS AND MEASURES. 
 TROY WEIGHT.* 
 
 In adding up the column of grains I find 
 ths sum to be 12-2, which I divide by 24 to 
 bring it into pennyweights ; and 122 grains 
 make 5 pennyweights and 2 grains over ; 
 the 2 I put down, and carry the 5 to the co- 
 lumn of pennyweights ; I then add these to- 
 gether, and tind the sum to be 101, which I 
 divide by 20 to bring to ounces, 1 put down 
 the l and carry 5 to the column of ounces ; 
 then adding the ounces, I findjhe sum 79, 
 which, by dividing by 12, give 6 lb. 7 oz. the 
 
 lb. 
 
 76S4 
 11234 
 9876 
 1493 
 3587 
 2345 
 6789 
 3257 
 
 OZ. 
 
 9 
 11 
 
 8 
 9 
 10 
 7 
 9 
 11 
 
 dwts. 
 16 
 5 
 11 
 19 
 10 
 6 
 14 
 15 
 
 gr. 
 
 22 
 19 
 22 
 12 
 3 
 15 
 21 
 8 
 
 
 
 
 / i pu 
 
 11 UVJ V 
 
 
 1 1 u i a iJ 
 
 ly nit u lu 
 
 LUC p 
 
 U UUU3, 
 
 and proceed as in simple Addition.-}* 
 
 lb. 
 
 OZ. 
 
 dwt. 
 
 lb. 
 
 CZ. 
 
 dwt 
 
 . gr. 
 
 lb. 
 
 OZ. 
 
 dwt. 
 
 1. 414 
 
 9 
 
 14 2. 
 
 410 
 
 9 
 
 12 
 
 19 
 
 3. 526 
 
 10 
 
 19 
 
 617 
 
 5 
 
 13 
 
 342 
 
 11 
 
 16 
 
 12 
 
 712 
 
 9 
 
 17 
 
 715 
 
 10 
 
 9 
 
 912 
 
 3 
 
 14 
 
 14 
 
 944 
 
 6 
 
 14 
 
 322 
 
 7 
 
 15 
 
 751 
 
 6 
 
 10 
 
 22 
 
 633 
 
 10 
 
 11 
 
 413 
 
 2 
 
 10 
 
 626 
 
 10 
 
 17 
 
 16 
 
 319 
 
 4 
 
 10 
 
 514 
 
 11 
 
 15 
 
 427 
 
 4 
 
 11 
 
 23 
 
 247 
 
 9 
 
 12 
 
 976 
 
 8 
 
 7 
 
 123 
 
 1 1 
 
 17 
 
 12 
 
 123 
 
 10 
 
 17 
 
 
 lb. 
 
 oz. 
 
 dwt. gr. 
 
 
 lb. 
 
 oz. 
 
 dwt. 
 
 OZ. 
 
 dwt, 
 
 gr. 
 
 4. 940 
 
 10 
 
 19 15 
 
 5. 
 
 174 
 
 11 
 
 19 
 
 6. 174 
 
 19 
 
 23 
 
 738 
 
 6 
 
 4 23 
 
 
 74 
 
 10 
 
 13 
 
 714 
 
 11 
 
 14 
 
 614 
 
 3 
 
 17 13 
 
 
 944 
 
 9 
 
 14 
 
 714 
 
 O 
 
 18 
 
 546 
 
 7 
 
 16 19 
 
 
 74 
 
 11 
 
 19 
 
 74 
 
 1 
 
 22 
 
 321 
 
 10 
 
 5 22 
 
 
 944 
 
 1O 
 
 13 
 
 948 
 
 o 
 
 21 
 
 230 
 
 9 
 
 15 15 
 
 
 74 
 
 11 
 
 3 
 
 ^ 74 
 
 2 
 
 12 
 
 046 
 
 11 
 
 19 23 
 
 
 12 
 
 4 
 
 6 
 
 301 
 
 14 
 
 4 
 
 
 
 
 
 
 
 
 
 
 indefinite number, if the pupil has not already attained to accuracy 
 in adding up the foregoing examples. He may be desired to take on 
 his slate 3 or 4 or more lines of either example, or he may be desired 
 to take 2 or 3 or more lines of each example, and range them under 
 one another for a new example, and so proceed till he has performed 
 the operation as often as necessary. 
 
 * The reader is referred to the tables in the preceding pages, which 
 it is hoped he has already committed to his memory. 
 
 f This illustration for an example in Troy Weight, will be sufficient 
 for the various examples in the other Weights and Measures, which 
 differ only in the vahie of the divisors. 
 
COMPOUND ADDITION. 61 
 
 lb. oz. dwt. oz. dwt. gr. 
 
 
 7. 
 
 71 
 
 64 
 
 77 
 14 
 64 
 
 74 
 77 
 
 105 
 
 11 
 8 
 
 3 
 2 
 6 
 2 
 9 
 
 19 
 14 
 
 1 I 
 . g 
 14 
 13 
 12 
 
 
 1 
 
 1. *4 
 64 
 
 74 
 66 
 74 
 14 
 19 
 13 
 
 19 23 
 14 17 
 19 11 
 13 9 
 J4 11 
 10 3 
 1 1 14 
 17 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 AVOIRDUPOIS WEIGHT. 
 
 lb. 
 
 OZ. 
 
 dr. 
 
 
 tons 
 
 , cwt, 
 
 , qr. 
 
 lb. 
 
 lb. 
 
 oz. dr. 
 
 1. 318 
 
 10 
 
 10 
 
 
 2. 416 
 
 19 
 
 2 
 
 26 
 
 3. 539 
 
 I 3 1 f 
 
 436 
 
 9 
 
 8 
 
 
 313 
 
 10 
 
 
 
 20 
 
 SJ6 
 
 14 13 
 
 624 
 
 14 
 
 6 
 
 
 271 
 
 11 
 
 3 
 
 16 
 
 223 
 
 12 7 
 
 419 
 
 6 
 
 15 
 
 
 725 
 
 19 
 
 2 
 
 18 
 
 81 1 
 
 9 6 
 
 245 
 
 9 
 
 7 
 
 
 357 
 
 14 
 
 2 
 
 25 
 
 700 
 
 14 
 
 853 - 
 
 11 
 
 10 
 
 
 4-29 
 
 17 
 
 3 
 
 22 
 
 414 
 
 12 1-2 
 
 145 
 
 9 
 
 8 
 
 
 235 
 
 15 
 
 2 
 
 19 
 
 
 
 
 
 
 tons, 
 
 cwt, 
 
 ,qr. 
 
 lb. 
 
 tons, cwt. 
 
 qr. 
 
 cwt. 
 
 qr. lb. 
 
 4. 305 
 
 14 
 
 2 
 
 11 
 
 5. 
 
 174 
 
 19 
 
 3 
 
 6. 174 
 
 3 27 
 
 418 
 
 18 
 
 
 
 
 
 
 74 
 
 14 
 
 2 
 
 724 
 
 2 24 
 
 336 
 
 2 
 
 1 
 
 14 
 
 
 714 
 
 13 
 
 1 
 
 149 
 
 1 14 
 
 119 
 
 13 
 
 3 
 
 27 
 
 
 718 
 
 16 
 
 2 
 
 719 
 
 2 16 
 
 767 
 
 36 
 
 
 
 8 
 
 
 734 
 
 15 
 
 2 
 
 407 
 
 1 23 
 
 782 
 
 9 
 
 1 
 
 16 
 
 
 714 
 
 14 
 
 1 
 
 149 
 
 a 17 
 
 421 
 
 15 
 
 3 
 
 19 
 
 
 155 
 
 
 
 3 
 
 76 
 
 3 15 
 
 
 
 
 qr. 
 
 Ib. 
 
 OZ. 
 
 
 
 lb. 
 
 02. drs. 
 
 
 
 7- 
 
 44 
 
 27 
 
 15 
 
 
 i 
 
 5. 17 
 
 15 15 
 
 
 
 
 74 
 
 26 
 
 14 
 
 
 
 27 
 
 14 11 
 
 
 
 
 19 
 
 14 
 
 13 
 
 
 
 16 
 
 13 9 
 
 
 
 
 74 
 
 12 
 
 14 
 
 
 
 74 
 
 14 14 
 
 
 
 
 66 
 
 27 
 
 13 
 
 
 
 70 
 
 
 
 
 
 
 74 
 
 19 
 
 10 
 
 
 
 64 
 
 13 10 
 
 
 
 
 13 
 
 17 
 
 5 
 
 
 
 13 
 
 4 5 
 
 
62 COMPOUND ADDITION. 
 
 APOTHECARIES' WEIGHT. 
 
 lb. oz. dr. 
 
 oz. dr. sc 
 
 . gr. lb. oz. 
 
 dr. sc. gr. 
 
 1. 314 8 4 
 
 2. 22 3 2 
 
 19 3. 646 11 
 
 4119 
 
 210 11 4 
 
 56 O 1 
 
 13 715 3 
 
 7 1 14 
 
 766 10 2 
 
 43 2 2 
 
 11 934 3 
 
 4 O 12 
 
 555 9 6 
 
 54 7 
 
 17 373 10 
 
 529 
 
 417 8 1 
 
 765 2 
 
 14 216 5 
 
 1 2 16 
 
 324 7 3 
 
 45 6 1 
 
 159 2 
 
 5 14 
 
 lb. oz. dr. 
 
 oz. dr. sc. 
 
 
 
 dr. sc. gr. 
 
 lb. oz. dr. 
 
 -4< 47 11 7 
 
 5. 149 7 2 
 
 6. 749 2 19 7. 
 
 84 11 7 
 
 94 10 G 
 
 714 3 O 
 
 607 1 IB 
 
 74 10 6 
 
 74 10 4 
 
 619 2 1 
 
 714 2 17 
 
 37 5 4 
 
 75 9 3 
 
 74 6 2 
 
 40O X) 
 
 19 4 13 
 
 69 2 
 
 1-62 5 2 
 
 74 1 13 
 
 74 12 
 
 57 1 2 
 
 74 1 2 
 
 715 2 14 
 
 79 2 6 
 
 1821 
 
 777 6 l 
 
 64 1 18 
 
 19 2 4 
 
 3935 
 
 146 4 
 
 16 O 10 
 
 13 4 8 
 
 
 
 
 
 
 CLOTH 
 
 MEASURE. 
 
 yd. qr.nl. 
 
 E.e. qr. nl. 
 
 E e. qr. nl. 
 
 yd. qr. hi.. 
 
 1. 434 3 2 
 
 2. 511 4 2 
 
 3. 565 404. 
 
 543 3 2 
 
 527 .1 2 
 
 660 2 o 
 
 626 2 1 
 
 83t> 2 2 
 
 613 2 3 
 
 439 4 2 
 
 724 1 
 
 754 2 3 
 
 758 3 1 
 
 .337 1 2 
 
 -8*2 2 3 
 
 217 1.3 
 
 846 1 3 
 
 854 2 3 
 
 003 3 
 
 7.2 5 3 2 
 
 925 2 2 
 
 766 O 2 
 
 j 227 1 1 
 
 438 2 2 
 
 
 E.e. qr. nl. 
 
 E.e. qr. nl. 
 
 yd. qr.nl. 
 
 E.e. qr. nl. 
 
 .ri. 120 2 2 
 
 6, 537 2 
 
 7. 74 3 3 6 
 
 !. 77 4 3 
 
 394 4 1 
 
 916 3 1 
 
 64 2 1 
 
 14 3 2 
 
 110 2 
 
 328 3 3 
 
 74 1 3 
 
 74 2 1 
 
 - 481 1 2 
 
 457 1 2 
 
 49 2 1 
 
 49 1 2 
 
 556 4 3 
 
 646 3 2 
 
 74 1 2 
 
 74 2 i 
 
 664 3 1 
 
 287 4 2 
 
 44 3 1 
 
 44 1 2 
 
 779 2 3 
 
 561 2 2 
 
 16 2 3 
 
 -S4 2 
 
COMPOUND ADDITION. 
 
 LONG MEASURE. 
 
 miles 
 
 1. 427 
 
 689 
 
 ,fur. p. 
 
 6 23 
 5 26 
 
 yds, 
 "3* 12 
 5 
 
 yds, ft. 
 
 .214 2 
 183 2 
 
 in. bcC. 
 90 3, 
 11 2 
 
 lea. 
 
 520 
 623 
 
 mi. fur. p. 
 1 6 13 
 
 1 .7 27 
 
 322 
 
 7 oo 
 
 2 
 
 597 
 
 8 1 
 
 721 
 
 
 
 4 16 
 
 
 510 
 
 2 38 
 
 4 
 
 61 
 
 9 2 
 
 
 7 2 
 
 826 
 
 1 
 
 3 32 
 
 
 777 
 
 4 
 
 
 
 3 
 
 72 
 
 5 1 
 
 
 6 1 
 
 932 
 
 2 
 
 6 1 
 
 
 883 
 
 3 10 
 
 4 
 
 930 1 
 
 3 
 
 -315 
 
 1 
 
 2 28 
 
 
 126 
 
 24 
 
 
 
 49 
 
 2 1 
 
 
 4 1 
 
 409 
 
 1 
 
 5 39 
 
 
 412 
 
 7 39 
 
 4 
 
 291 2 
 
 10 2 
 
 376 
 
 2 
 
 7 27 
 
 
 
 lea. 
 
 m. fur 
 
 
 fur. 
 
 P- 
 
 yds. 
 
 
 p. yds. ft. 
 
 
 feet 
 
 in.b.c. 
 
 4. 17 
 
 2 7 
 
 
 5. 147 
 
 39 
 
 5 
 
 6. 
 
 177 5 2 
 
 7. 
 
 174 
 
 ll 
 
 ( 2 
 
 14 
 
 1 6 
 
 
 614 
 
 37 
 
 4 
 
 
 714 4 1 
 
 
 49 
 
 10 
 
 i 
 
 74 
 
 1 7 
 
 
 714 
 
 19 
 
 3 
 
 
 7-1 41 2 
 
 
 7-i 
 
 .11 
 
 2 
 
 68 
 
 2 4 
 
 
 674 
 
 17 
 
 1 
 
 
 613 1 
 
 
 64 
 
 9 
 
 1 
 
 74 
 
 1 
 
 
 719 
 
 27 
 
 2 
 
 
 714 1 'Jl 
 
 
 74 
 
 10 
 
 1 
 
 69 
 
 2 1 
 
 
 197 
 
 19 
 
 1 
 
 
 719 1 1 
 
 
 64 
 
 11 
 
 9 
 
 74 
 
 1 2 
 
 
 ' 724 
 
 14 
 
 3 
 
 
 437 2 1 
 
 
 74 
 
 10 
 
 
 
 9(3 
 
 2 4 
 
 
 604 
 
 29 
 
 5 
 
 
 610 4 
 
 
 94 
 
 11 
 
 2 
 
 
 LAND MEASURE. 
 
 
 ac. 
 
 r. 
 
 P- 
 
 
 ac. 
 
 T, 
 
 J. 
 
 ac. 
 
 r. 
 
 p> 
 
 
 i. 
 
 452 
 
 2 
 
 38 
 
 2. 
 
 982 
 
 o 
 
 24 3. 
 
 921 
 
 i 
 
 29 
 
 
 
 314 
 
 1 
 
 3.5 
 
 
 613 
 
 3 
 
 14 
 
 604 
 
 3 
 
 32 
 
 
 
 715 
 
 2 
 
 16 
 
 
 100 
 
 1 
 
 27 
 
 736 
 
 2 
 
 29 
 
 
 
 430 
 
 2 
 
 35 
 
 
 474 
 
 
 
 19 
 
 559 
 
 3 
 
 28 
 
 
 
 529 
 
 3 
 
 7 
 
 
 363 
 
 1 
 
 31 
 
 265 
 
 1 
 
 17 
 
 
 
 346 
 
 1 
 
 23 
 
 
 755 
 
 3 
 
 38 
 
 427 
 
 
 
 30 
 
 
 
 61 
 
 3 
 
 11 
 
 
 647 
 
 
 
 6 
 
 883 
 
 1 
 
 39 
 
 
 
 214 
 
 2 
 
 35 
 
 
 234 
 
 2 
 
 29 
 
 291 
 
 3 
 
 25 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * The pupil will recollect, that to bring- yards into poles, he must 
 multiply the yards by >, and then divide by 1 1 ; and if there be~a 
 remainder it will be half yards. 
 
64 COMPOUND ADDITION. 
 
 ac. r. p. ac. r. p. ac. r. p. ac. r. p. 
 
 4. 77 3 39 5. 714 3 39 6. 14 3 39 7. 174 3 3g 
 
 64 2 37 619 1 30 74 1 39 / 14 1 27 
 
 74 1 
 64 2 
 74 1 
 64 2 
 14 1 
 94 3 
 
 24 
 19 
 18 
 37 
 13 
 14 
 
 714 2 27 
 619 1 34 
 719 2 37 
 719 1 , 24 
 610 2 14 
 174 3 38 
 
 64 2 14 
 74 1 18 
 47 2 24 
 18 1 14 
 74 2 19 
 74 2 24 . 
 
 618 
 719 
 
 734 
 7!5 
 639 
 714 
 
 2 
 
 1 
 
 1 
 o 
 
 j 
 
 1-2 
 14 
 11 
 il-4 
 24 
 34 
 
 
 
 
 
 WINE 
 
 MEASURE. 
 
 hhd 
 
 . gal. pt. 
 
 tuns, hhd 
 
 g- 
 
 qt. 
 
 tuns, hhd. g. 
 
 qt. 
 
 1. 62(5 
 
 44 
 
 7 
 
 2. 522 
 
 i 
 
 39 
 
 3 
 
 3. 148 
 
 2 
 
 '25 
 
 3 
 
 753 
 
 17 
 
 I 
 
 257 
 
 3 
 
 34 
 
 2 
 
 
 513 
 
 
 
 42 
 
 3 
 
 438 
 
 52 
 
 6 
 
 763 
 
 -2 
 
 58 
 
 3 
 
 
 614 
 
 1 
 
 36 
 
 1 
 
 217 
 
 13 
 
 7 
 
 611 
 
 3 
 
 43 
 
 1 
 
 
 340 
 
 3 
 
 43 
 
 '2 
 
 135 
 
 45 
 
 
 
 937 
 
 1 
 
 16 
 
 3 
 
 
 416 
 
 2 
 
 56 
 
 1 
 
 497 
 
 56 
 
 2 
 
 238 
 
 
 
 31 
 
 2 
 
 
 9 5 '2 
 
 3 
 
 26 
 
 
 
 32 
 
 11 
 
 3 
 
 749 
 
 3 
 
 7 
 
 O 
 
 
 567 
 
 1 
 
 19 
 
 3 
 
 236 
 
 
 
 
 
 319 
 
 '2 
 
 59 
 
 3 
 
 
 7 Q '2 
 
 3 
 
 46 
 
 2- 
 
 
 
 
 
 
 
 hhd. 
 
 gal 
 
 
 
 
 
 tuns, 
 
 hhd. g. 
 
 
 pun. gal. 
 
 qt. 
 
 
 .qt. 
 
 gal. 
 
 qt. 
 
 pt. 
 
 4. 714 
 
 3 62 
 
 5. 
 
 714 84 
 
 3 
 
 6. 
 
 74 
 
 41 
 
 3 
 
 7. 14 
 
 a 
 
 1 
 
 614 
 
 2 61 
 
 
 615 81 
 
 '2 
 
 
 64 
 
 40 
 
 2 
 
 7 4 
 
 2 
 
 1 
 
 174 
 
 1 39 
 
 
 7J4 74 
 
 1 
 
 
 74 
 
 19 
 
 1 
 
 39 
 
 2 
 
 1 
 
 164 
 
 2 47 
 
 
 614 18 
 
 2 
 
 
 64 
 
 39 
 
 2 
 
 37 
 
 1 
 
 
 
 274 
 
 1 49 
 
 
 713 75 
 
 
 
 
 74 
 
 4O 
 
 1 
 
 39 
 
 2 
 
 
 
 175 
 
 2 37 
 
 
 614 17 
 
 1 
 
 
 69 
 
 16 
 
 1 
 
 77 
 
 1 
 
 1 
 
 375 
 
 1 49 
 
 
 715 14 
 
 3 
 
 
 17 
 
 39 
 
 2 
 
 39 
 
 3 
 
 1 
 
 704 
 
 64 
 
 
 919 68 
 
 
 
 
 28 
 
 44 
 
 3 
 
 24 
 
 *'2 
 
 
 
 
 ALE AND BEER MEASURE. 
 
 bar. fir. gal. 
 
 hhd. gal. qt. 
 
 bar 
 
 .fir. 
 
 gal. 
 
 pt. 
 
 bnr. 
 
 fir. 
 
 ft 
 
 . 130 3 
 
 5 2 
 
 . 666 
 
 '29 2 
 
 3. 
 
 278 
 
 2 
 
 o 
 
 6 4 
 
 . 381 
 
 2 
 
 6 
 
 471 3 
 
 7 
 
 883 
 
 42 2 
 
 
 154 
 
 3 
 
 5 
 
 3 
 
 37 
 
 3 
 
 7 
 
 348 1 
 
 6 
 
 S77 
 
 53 
 
 
 561 
 
 
 
 8 
 
 7 
 
 43 
 
 2 
 
 5 
 
 726 2 
 
 7 
 
 .044 
 
 26 3 
 
 
 S23 
 
 1 
 
 6 
 
 4 
 
 16 
 
 1 
 
 4 
 
 619 1 
 
 4 
 
 27 
 
 37 1 
 
 
 386 
 
 2 
 
 3 
 
 5 
 
 4 
 
 3 
 
 8 
 
 455 
 
 2 
 
 455 
 
 18 3 
 
 
 238 
 
 1 
 
 5 
 
 6 
 
 73 
 
 3 
 
 O 
 
 .327 2 
 
 7 
 
 610 
 
 52 1 
 
 
 117 
 
 3 
 
 
 
 2 
 
 5 
 
 1 
 
 2 
 
 234 O 
 
 1 
 
 757 
 
 4 3 
 
 
 792 
 
 2 
 
 4 
 
 
 
 8 
 
 3 
 
 7 
 
COMPOUND -ADDITION, 
 
 bar. 
 
 S, 71 
 
 14 
 16 
 17 
 
 '-29 
 17 
 
 4 1 
 67 
 
 fir. 
 3 
 2 
 1 
 1 
 2 
 1 
 2 
 
 
 gal. 
 
 8 
 7 
 
 4 
 3 
 
 7 
 6 
 6 
 
 bar. fir. gal. hhd. gal. qt. hhd. gal. qt. 
 6. 73 3 7 7. 714 47 3 8. 714 53 , 3 
 69 2 6 614 44 1 415 47 2 
 14 1 7 374 43 2 714 10 1 
 39 22 157 41 1 614 27 1 
 1916 719421 /15512 
 49 2 6 374 41 2 714 37 2 
 37 14 174 12 1 615 19 1 
 49 2 3 419 4-3 '2 714 48 3 
 
 
 
 
 
 
 
 
 
 
 
 
 CORN MEASURE. ' 
 
 qr. 
 
 b. 
 
 pec. 
 
 b. 
 
 P- 
 
 ga. 
 
 ch. 
 
 b. pcc. 
 
 
 qr. 
 
 b pcc, 
 
 J. 571 
 
 6 
 
 2 
 
 2. 506 2 
 
 1 3. 161 
 
 28 2 
 
 4. 
 
 87 
 
 5 3 
 
 9'36 
 
 4 
 
 a 
 
 524 2 
 
 1 
 
 394 
 
 13 1 
 
 
 29 
 
 4 1 . 
 
 693 
 
 7 
 
 3 
 
 914 
 
 3 
 
 1 
 
 465 
 
 16 3 
 
 
 66 
 
 5 '2 
 
 438 
 
 <5 
 
 i 
 
 393 
 
 2 
 
 
 
 03y 
 
 37 1 
 
 
 18 
 
 4 1 
 
 343 
 
 4 
 
 
 
 7 46 
 
 2 
 
 1 
 
 791 
 
 34 2 
 
 
 44 
 
 6 ' 
 
 297 
 
 2 
 
 3 
 
 673 
 
 3 
 
 
 
 537 
 
 1 5 2 
 
 
 a *2 
 
 3 
 
 749 
 
 1 
 
 
 
 25-2 
 
 
 
 1 
 
 631 
 
 23 1 
 
 
 70 
 
 7 .2 
 
 244 
 
 5 
 
 2 
 
 438 
 
 1 
 
 
 
 443 -19 2 
 
 41 
 
 5 1 
 
 248 
 
 3 
 
 1 
 
 0^7 
 
 3 
 
 1 
 
 594 
 
 12 3 
 
 
 59 
 
 7 3 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 P- 
 
 
 
 ch. 
 
 b. pec. 
 
 Ch. b. 
 
 pec. 
 
 score, ch 
 
 . b. 
 
 5 40 
 
 1 
 
 3 
 
 6. 01 
 
 24 
 
 3 7. 
 
 14 31 
 
 3 8 
 
 . 74 
 
 20 
 
 35 
 
 17 
 
 1 
 
 <J 
 
 74 
 
 16 
 
 2 
 
 74 31 
 
 2 
 
 49 
 
 19 
 
 3 a 
 
 2 i 
 
 1 
 
 7 
 
 43 
 
 14 
 
 1 
 
 64 30 
 
 1 
 
 64 
 
 37 
 
 3i 
 
 S3 
 
 
 
 4 
 
 56 
 
 28 
 
 3 
 
 74 27 
 
 2 
 
 74 
 
 34 
 
 10 
 
 47 
 
 i 
 
 7 
 
 23 
 
 5 
 
 1 
 
 64 19 
 
 9 
 
 39 
 
 13 
 
 9 
 
 59 
 
 
 
 1 
 
 73 
 
 27 
 
 3 
 
 7i 31 
 
 1 
 
 47 
 
 16 
 
 a 
 
 56 
 
 1 
 
 5 
 
 99 
 
 32 
 
 2 
 
 64 n 
 
 1 
 
 39 
 
 17 
 
 4 
 
 68 
 
 1 
 
 6 
 
 10 
 
 28 
 
 1 
 
 91 26 
 
 2 
 
 37 
 
 12 
 
 34 
 
 23 
 
 1 
 
 7 
 
 9 
 
 35 
 
 2 
 
 4 26 
 
 2 
 
 56 
 
 13 
 
 4 
 
 
 
 
 
 
 
 
 TIME 
 
 
 
 
 
 
 jno. 
 
 w. 
 
 d. 
 
 h. 
 
 W. 
 
 d. h. 
 
 mi. 
 
 d. 
 
 h. 
 
 mi. 
 
 sec. 
 
 1. 19 
 
 2 
 
 6 
 
 19 2. 
 
 ^7 
 
 4 23 
 
 38 
 
 3. 6-2 
 
 7 
 
 47 
 
 38 
 
 46 
 
 1 
 
 4 
 
 21 
 
 64 
 
 6 13 
 
 47 
 
 18 
 
 12 
 
 54 
 
 56 
 
 22 
 
 3 
 
 5 
 
 9 
 
 15 
 
 3 21 
 
 H) 
 
 76 
 
 21 
 
 16 
 
 49 
 
 57 
 
 2 
 
 3 
 
 21 
 
 35 
 
 1 18 
 
 15 
 
 34 
 
 9 
 
 20 
 
 31 
 
 62 
 
 1 
 
 6 
 
 12 
 
 7S 
 
 6 9 
 
 59 
 
 99 
 
 23 
 
 81 
 
 46 
 
 17 
 
 3 
 
 2 
 
 14 
 
 49 
 
 2O 
 
 6 
 
 53 
 
 22 
 
 28 
 
 32 
 
 11 
 
 3 
 
 4 
 
 16 
 
 71 
 
 3 34 
 
 43 
 
 15 
 
 4 
 
 58 
 
 23 
 
 29 
 
 1 
 
 3 
 
 21 
 
 23 
 
 3 7 
 
 24 
 
 64 
 
 16 
 
 13 
 
 10 
 
66 COMPOUND ADDITION, 
 
 yrs. 
 
 mo. 
 
 W. 
 
 
 mo. 
 
 W. 
 
 d. 
 
 days, hrs. 
 
 min. 
 
 hrs. min. sec. 
 
 4.737 
 
 .12 
 
 3 
 
 5. 
 
 64 
 
 3 
 
 6 6 
 
 . 714 23 
 
 59 
 
 7. 647 
 
 59 
 
 59 
 
 347 
 
 31 
 
 2 
 
 
 74 
 
 1 
 
 5 
 
 74 14 
 
 54 
 
 137 
 
 54 
 
 54 
 
 618 
 
 10 
 
 1 
 
 
 34 
 
 2 
 
 8 
 
 94 21 
 
 55 
 
 375 
 
 56 
 
 56 
 
 374 
 
 9 
 
 2 
 
 
 74 
 
 1 
 
 4 
 
 74 33 
 
 53 
 
 714 
 
 17 
 
 19 
 
 175 
 
 1 
 
 1 
 
 
 63 
 
 2 
 
 1 
 
 69 32 
 
 14 
 
 615 
 
 54 
 
 54 
 
 714 
 
 12 
 
 3 
 
 
 74 
 
 1 
 
 2 
 
 74 12 
 
 19 
 
 714 
 
 17 
 
 13 
 
 615 
 
 10 
 
 1 
 
 
 64 
 
 2 
 
 1 
 
 37 11 
 
 17 
 
 613 
 
 34 
 
 56 
 
 33 4 
 
 9 
 
 3 
 
 
 94 
 
 2 
 
 6 
 
 46 22 
 
 49 
 
 626 47 
 
 49 
 
 
 ASTRONOMY. 
 
 s. 
 
 
 
 / 
 
 // 
 
 
 S. 
 
 o 
 
 / // 
 
 S. 
 
 e 
 
 / 
 
 it 
 
 11 
 
 24 
 
 37 
 
 41 
 
 
 5 
 
 3 
 
 26 25 
 
 6 
 
 9 
 
 54 
 
 36 
 
 7 
 
 12 
 
 57 
 
 21 
 
 
 9 
 
 5 
 
 37 56 
 
 3 
 
 29 
 
 59 
 
 7 
 
 3 
 
 25 
 
 13 
 
 17 
 
 
 8 
 
 24 
 
 42 59 
 
 11 
 
 26 
 
 21 
 
 19 
 
 4 
 
 29 
 
 18 
 
 29 
 
 
 3 
 
 9 
 
 12 15 
 
 9 
 
 24 
 
 50 
 
 40 
 
 5 
 
 16 
 
 52 
 
 43 
 
 
 4 
 
 8 
 
 17 41 
 
 -11 
 
 18 
 
 29 
 
 27 
 
 3 
 
 19 
 
 47 
 
 51 
 
 
 3 
 
 26 
 
 9 8 
 
 5 
 
 13 
 
 51 
 
 46 
 
 n 
 
 29 
 
 51 
 
 36 
 
 
 5 
 
 16 
 
 8 27 
 
 6 
 
 7 
 
 1 
 
 9 
 
 ~9 
 
 18 
 
 30 
 
 30 
 
 
 11 
 
 20 
 
 40 50 
 
 10 
 
 12 
 
 24 
 
 36 
 
 3 
 
 4 
 
 44 
 
 44 
 
 
 10 
 
 9 
 
 55 37 
 
 7 
 
 21 
 
 42 
 
 56 
 
 7 
 
 * .' 
 
 25 
 
 36 
 
 51 
 
 
 4 
 
 22 
 
 44 56 
 
 5 
 
 23 
 
 51 
 
 46 
 
 
 
 
 
 
 
 
 
 
 
 
 MISCELLANEOUS EXAMPLES IN ADDITION. 
 
 1. What is the sum total, in shillings, of 54 guineas, 29 pounds^ 
 36 guineas, and 48 pounds? 
 
 2. Add together 16/. 125. id. ; 156J. gs. 9 $d. ; 203952. 125.; 24Z. 
 195. llj<2. ; 37 /. 65. 7<i ; 3-27 /. 18s. ; and 100 guineas. 
 
 3. In collecting an account of debts owing to me, I find Mr. A. 
 owes me 97*. 165.5 Mr. B. 1-25Z. 135. 0fd. ; Mr. C. 6/. 6s. ; Mr. D. 
 5O guineas ; and Mr. E. 71^. os. gd. ; what is the whole sum due to me ? 
 
 4. A gentleman rdered a service of plate from his silversmith, 
 and on receiving his bill, he finds that he had dishes and covers 
 weighing 45 Ib. Q oz, 12 dwts. ; plates weighing 70 Ib. 7 oz. 16 dwts. ; 
 spoons of different sizes, and ladles, 24 Ib. 9 z. 12 dwts. ; waiters 
 15 ib. 10 oz. ; salts and castors, 4 Ib. 4 oz, 3 dwts. ; candlesticks, 
 39 Ib. 11 oz. 17 dwts. ; and sundry smaller, articles 5 Ib. 3 oz. ; what 
 is the weight of silver he will have to pay for? 
 
 5. A carrier brings goods to a shopkeeper, viz. 8 bags of hops 
 weighing 19 cwt. 3 qrs. 14 Ib.-; cheeses weighing 15 cwt. l qr. 21 Ib.; 
 butter weighing 12 cwt. 2 qrs. ; two chests of tea, weighing l cwt. 
 each; and a sack of salt weighing 8 cwt. 2 qr. 12 Ib. ; how much 
 .weight wiii the carrier have to charge ? 
 
COMPOUND ADDITION. 67 
 
 6. The rent of my house is soZ. per annum; the house tax is three 
 pounds fifteen shillings; land tax 5/. ; windows I5l. 125. Od. ; poor's 
 rates io/, ; lighting, watching, and street rates 3/. QS. 3^d. ; how 
 much therefore do my house and taxes stand me in per annum ? 
 
 7. The following is an estimate of the repairs wanting to my 
 house ; how much is the whole sum ? Carpenter's bill 27 J. 95. g^d. ; 
 bricklayer's and plasterer's ijL 75. Qd. ; mason's 5 1. 55. ; painter's, 
 glazier's, and plumber's, fourteen guineas j smith's, for new rails, 
 111. ; and the slater's gi. 185. 
 
 8. A man purchased some goods for the country ; the first parcel 
 contained 25 yds. 2 qr. 2 nl, of broad cloth ; the second 126 yds. 
 2 qrs. of serge; the third a thousand yards or green baise ; and the 
 fourth 19 yds. 3 qrs. 2 nl. of shalloon ; what was the whole quantity ? 
 
 9. A wine merchant, retiring from business, takes an account of 
 the stock of wines in his cellar, and finds 5 pipes and 50 gallons of 
 port wine, 3 p. 2 hhd. of sherry; 10 pipes of Lisbon; 2 pipes cff 
 claret ; of Madeira he had 36 gallons ; of brandy 50 gallons ; of mm, 
 two hogsheads; and of Hollands, i hhd. and 12 gallons ; what quan- 
 tity of liquor did his cellar contain ? 
 
 10. A friend in Essex desired me to measure his farm, which he 
 holds on a lease ; the three -fields at the back of the house measured 
 59 ac. 2 r. 20 p. ; the large piece of ground in the valley measures 
 74 acres, three others measure each on an average 11 ac. 1 r. j36 p. ; 
 the field laid down in clover contains 7 ac. 3 r. 2 p. ; one sown with 
 caraways, I find to be 3'J acres-, arid the ground belonging to the 
 garden, out-houses, &c. makes about lj acres; how many acres 
 ought he to pay for? 
 
 11. A merchant sends to his banker on the 2d. day of the month, 
 in money and bills, to the amount of two thousand guineas ; on 
 the fifth he sends him 900/. 19$. 4d. ; on the eleventh he sends 5QO/.; 
 and in the course of the remaining days of the month he sends 
 151:>Z. 125. li^d.' y how much therefore may he draw as occasion re- 
 quires? 
 
 12. A gentleman's steward received the following sums of money 
 for rents; what was the gentleman's income? Of farmer A he re- 
 ceived 394/. !:>$. 6^., of B 97^. 145. 9^-, of .C 175/. 105.., of JD 99/, 45, 
 and of E 139/. 12s. 4d. 
 
 13. A person borrows of several friends the following sums of 
 money ; of the first 5OO/. ; of the second 225/. 125. ; of the third fifty 
 guineas; of the fourth seventy guineas and 2-2 crowns; of the -fifth 
 he had 150/. 75. Qd. ; how much will he have to pay interest, for ? 
 
 14. A man borrowed a sum of money, and paid at different times 
 75 guineas, .but he still owed ssl.Qt. 9^.: what WAS the ordinal 
 xiebt? 
 

 COMPOUND SUBTRACTION 
 
 Is the method of finding the difference between two 
 given compound numbers. 
 
 RULE I. Having arranged the numbers so that the 
 smaller may stand under the greater, subtract eat 
 ber in the lower line from that which stands above it, and 
 
 ' down the remainders. 
 
 2. When any of the lower denominations are greater 
 than the upper, increase the upper number by as many as 
 m&ke cue of the next superior denomination, from which 
 take the figure, in the lower line, set down the difference, 
 and c<irry MIC to the next number in the lower line, and 
 subtract as lefore. 
 
 Ex. Subtract 505/. 17s. Q\d. from COO/. 105. 7 id. 
 
 * </. I Jure I gay 2 farthings from 1, 1 cannot, 
 
 000 10 7jp but I adc! 4 to the 1, because 4 farthing* 
 
 595 37 oj make a penny, and 2 from 5 and there re- 
 
 -...- ...... ~ - mains }; I carry one to the 9 j 30 from 7 I 
 
 412 9} cannot, but I add 12 to 7, because 12 pence 
 make a shilling;, and 10 from 19 an : 
 
 Proof- 600 10 7 remain 9 I cairy i to 17 ; and 
 I, i. I cannot, but I add 20 to the KK 
 
 shillings make a pound, and 16 from 
 
 there remain 13 j I now carry one to the five, and go on as ic; 
 Subtraction. 
 
 The method of proof is the same as in simple Subtraction, 
 
 EXAMPLES. 
 
 Ex. i. 
 
 Answer 
 
 Proof - 
 
 Ex.4. 
 
 . 
 
 1-15 
 136 
 
 $. 
 19 
 
 17 
 
 df. 
 
 "c| 
 
 . 
 
 Ex. 2. 370 
 
 309 
 
 s. d. 
 
 17 7| 
 
 12 4 
 
 . 
 Ex, 3. 450 
 
 371 
 
 s. d. 
 
 12 6| 
 10 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 594 
 
 374 
 
 10 
 1Q 
 
 ?f 
 
 Ex. 5. 465 
 349 
 
 12 7^ 
 
 17 9| 
 
 Ex. 6. 564 
 375 
 
 1 2 2- 
 18 4^ 
 
COMPOUND SUBTRACTION. 69 
 
 . .v. ct. . s. d. . s. d. 
 
 Ex. 7. 371 19 2 Ex.s. 700 o o Ex.o. 476 19 4 
 
 199 17 Hj W 16 6| 374 12 9 
 
 Ex. 10. 473 18 7f Ex. 11. 249 9 9| Ex. 10. 376 17 7 
 291 12 /f 159 19 llj 299 14 4-J 
 
 Ex. 13. 594 O Ex. 14. 796 12 llf Ex. 15. 4J6 17 7 
 593 19 9| 669 8 3 399 19 11 j 
 
 Ex. 06. '599 2 2 Ex. 17. 209 18 8 Ex. 18. 500 
 177 12 7 1 159 19 9-J 409 19 11^ 
 
 Ex. 19. 422 3 6j Ex.20. 224 2 6f Ex. 21. 794 15 
 371 15 7f 156 6 6| 3t>7 16 
 
 Ex, 22. Q09 Ex.20. 764 15 4j Ex. 21, 674 6 oj 
 800 19 U'f 98 12 11 249 19 9$ 
 
 Ex. 25. 372 10 6 Ex. 26. 649 12 0| Ex. 27. 341 5 11 J 
 
 149 6 4f 597 19 H 230 9 4- 
 
 Ex.28. 846 9 8j Ex. 29. 124 9 10| Ex. 30. 90441 5 9 
 375 9 9j 109 10 Si 6^7217 13 10 
 
70 COMPOUND SUBTRACTION. 
 
 . s. d. . s. d. . s. d. 
 
 Ex.31. 438 7 10 Ex.32. 12427 16 11^ Ex. 33. 1654 12 7 
 
 099 16 9 7618 14 9| 585 9 10j 
 
 Ex.34. 14476 5 6j Ex. 35.222 18 9-J Ex. 36.96481 16 9 
 714 13 8| 142 7 lOg 3/68 10 9| 
 
 Ex.37. 164 17 8 Ex.33. 18149 14 oj Ex. 39. 417 4 10| 
 29 2 9f 17216 4| 519 11 7g 
 
 Ex. 40. 20412 13 9f Ex. 41.425 18 9 Ex. 42. 22425 14 9^ 
 19011 14 2j 139 10 9$ 21018 8 ll| 
 
 Ex.43. 183 9 lj Ex.44. 24463 13 llj Ex. 45. 421 16 0-1 
 24 14 10^ 17732 16 9 326 19 
 
 Ex.46. 86473 6 gj Ex. 47. 433 17 2} Ex. 48. 28446 17 9 
 56117 13 10 311 19 4 19994 14 8| 
 
 Ex.49. 194 12 gj Ex.50. 80490 9 9 Ex. 51. 4/4 19 
 117 12 9 24689 15 1 362 13 
 
 Ex. 52. 26475 13 9 Ex. 53.4559 16 9| Ex. 54. 34487 15 11 j 
 24716 18 II 2 3228 9 5^ 31767 19 10 
 
COMPOUND SUBTRACTION. 71 
 
 . s. d. . s. d. . s. d. 
 
 X. 55.2139 7 10 Ex, 56.36492 7 5j Ex. 57-3471 19 9| 
 1914 13 10| 20082 O 6-J 203 19 9| 
 
 Ex.58. 38410 14 9 Ex.59.4557 1'8 Oj Ex.60. 601273 11 7 
 28019 1910^ 3945 17 llf 462104 15 S 
 
 Ex.61. 5534 11 3 Ex. 62.424136 11 6j Ex. 63.786O O 
 559 12 7 379126 10 9f 3271 4 7 
 
 Ex. 64.441391 fl Oj Ex. 65.6234 6 6 Ex. 66. 1414 9 9|: 
 389091 9 8 309 12 loj 7 ; ^9 12 ll| 
 
 Ex. 67. 1173 14 9f Ex. 68. 484760 10 9 Ex. 69- 791 5 
 437 18 l!| 329189 19 9J" 261 19 
 
 Ex.70. 14112 Ex.71. 1345 19 9| Ex. 72. 4621 15 9^ 
 4612 19 l" 345 17 9^ 394 19* o| 
 
 Ex. 73. 396 19 9^ Ex.74. 254 14 o.^ Ex.75. 1214 5 
 29 19 9| 244 19 10j 8.8O 8f 
 
 EX.76. 564121 10 10| Ex.77. 4465 J0 ^J Ex.78. 4532 13 9 
 379178 16 10-f 304 llj 4319 15 ll 
 
72 
 
 COMPOUND SUBTRACTION. 
 
 Ex. 79- 408 19 
 
 254 1 
 
 4|- 
 
 Ex. 80. 60985 14 
 1427 19 
 
 . 5. cl 
 
 Borrowed 300 o 
 
 . *. d. 
 
 Borrowed 1000 o o 
 
 f 15 15 
 
 Paid at \ 89 7 7j 
 
 C 177 16 7-J 
 Paid at \ 105 3 
 
 different <^ 76 8 1 
 
 different < 5'2 10 11 
 
 times j 46 15 10 
 V 105 
 
 times J 216 9 9j 
 V 300 9 
 
 Paid - 2S3 6 6| 
 
 Paid - 881 18 4 
 
 Remain unpaid 16 13 5j 
 
 Unpaid - 118 1 8 
 
 Suppose a person is debtor to 
 sundry persons, in the follow- 
 
 And is creditor, by book-debfcj 
 from different people, in the fol- 
 
 ing sums. 
 
 lowing sums. 
 
 . s. d. 
 
 . ,s. d. 
 
 J678 14 9J 
 
 764 14 9-J 
 
 29 17 4| 
 
 39 14 4 
 
 550 
 
 500 
 
 1054 12 9j 
 
 99 5 9 
 
 26 5 
 
 2500 
 
 770 
 
 5505 5 11 
 
 95 19 9 
 39 11 3 
 
 3000 
 
 
 Cr 
 
 Dr. 
 
 Dr. 
 
 Balance in favour of Cr. 
 
 Required the balance of this 
 account ? 
 
 Required the balance of this 
 account ? 
 
 Dr. 
 
 Cr. 
 
 Dr. 
 
 Cr. 
 
 . s. d. 
 
 . S. d. 
 
 . S. d. 
 
 . .s. rf. 
 
 764 14 9 
 
 397 14 ll 
 
 /69 19 10-J 
 
 49 32 11 
 
 397 10J 
 
 267 n 9 
 
 643 4 4^ 
 
 1000 17 g| 
 
 210 19 9^ 
 
 726 13 8j 
 
 248 1] 7 
 
 3/66 5 5 
 
 467 16 7| 
 
 464 16 O 
 
 591 8 4 
 
 4 4 O 
 
 871 14 @ 
 
 215 12 6 
 
 9 19 <) 
 
 250 12 8f 
 
 564 12 6j 
 
 345 9 10| 
 
 .OO O 
 
 1/50 17 
 
COMPOUND SUBTRACTION. 73 
 
 EXAMPLES OF WEIGHTS AND MEASUilES.* 
 
 TROY WEIGHT. 
 
 Ib. oz. dvvt.gr. Ib. oz. dwt.gr. Ib. oz.dwt.gr. 
 
 Ex. 1. 187 9 12 -20 2. 256 6 22 3. 567 4 O 
 
 169 6 14 17 199 9 3 20 379 11 9 9 
 
 Ib. 02. dwt. gr. Ib. oz. dwt. gf. Ib. oz. dwt. gr. 
 
 4. 254 0005. 675 3 9 6. 423 5 15 14 
 
 253 11 19 20 576 9 17 16 246 11 18 23 
 
 
 Ib. 02. dwt. 
 7. 14 ll 9 
 
 11 10 14 
 
 oz.dwt.gr. 
 8. 74 12 18 
 64 14 37 
 
 Ib. oz. dwt. 
 9. 175 3 10 
 159 11 14 
 
 oz.dwt.gr. 
 I'O. 17 10 20 
 14 11 23 
 
 
 AVOIRDUPOIS WEIGHT. 
 
 tons,cwt.qr. 4 ib.oz. dr. tcms,cwt.qr.lb.oz. dr. tons.cwt.qr.lb.oz.dr. 
 1. 72 10 3 14 10 12 2. 64 15 2 15 10 9 3. 25 O O 
 9161 2 514 6 46153 51 -2 14 24 2 O 15 
 
 tons, cwt. qr. Ib. oz. dr. tons } cWt.qf.lb.oz.dr, tons,cwt.qr.lb.oz.dr. 
 
 4. 67 2 1 4 14 2 5. 36 7 1 1 1 1 6-76 300 04 
 
 2914 3 2 14 30 32555 671220144 
 
 tons, cwt. qr. cwt. qr. Ib. qr. Ib. oz. Ib. oz. dr. 
 
 7-M 1-2 2 8. 17 1 25 9- M3 22 12 10. 174 11 1O 
 
 1 14 3 14 2 27 74 19 14 39 1-2 13 
 
 APOTHECARIES' WEIGHT. 
 
 Ib. oz. -dr. scr. Ib. oz. dr. scr. Ib. oz. dr. scr. 
 
 1.456 940 2.269 8 3,2 3.987 4 4 O 
 
 309 472 178 11 3 1 379 10 5 I 
 
 NOTE. 
 
 The general student need not work the whole of these examples. 
 E 
 
74? COMPOUND SUBTRACTION. 
 
 Ib. o*. dr. scr. Ib. oz. dr. scr. Ib. oz, dr. scr. 
 
 Ex.4. 564 5.375 771 6. 394 2 2 O 
 
 469 332 369 472 299 11 7 2 
 
 Ib. oz. dr. oz. dr. scr. dr. scr. gr. Jb. oz. dr. 
 
 7.144 10 5 8. 27 4 I 9. 27 1 14 1O. 74 10 5 
 
 64 11 7 14 7 2 14 O 19 65 11 
 
 CLOTH MEASURE. 
 
 yds. qr. n. E.e. qr. n. yds. qr. n. yds. qr. n. 
 
 Ex. 1.218 20 2. 46 3.567 1 1 4.459 1 2 
 
 176 13 23 2 2 469 2 399 3 3 
 
 yds. qr. n. E.e. qr. n. E.e..qr. n. E.e. qr. n. 
 
 5. 174 216. 174 31 7. 171 1 3 8. 12 1 1 
 
 39 3 2 49 4 2 74 4 2 10 4 3 
 
 LONG MEASURE. 
 
 yds. ft. in. b.c. yds. ft. in. be. yds. ft. in. b.c. 
 
 Fx. 1.456 2 10 1 2.679 O O 3.267 111 
 
 379 1 11 2 599 1 1 1 199 222 
 
 lea. m. fur. p. lea. m. fur. p. lea. m, fur. p. 
 
 4.470 1 4 19 5. 367 O 6.225 1 1 1 
 279 2 7 23 179 2 5 23 167 244 
 
 lea. m. fur. fur. p. yds. p. yd. ft. ft. in. b.c. 
 
 7*. 31 24 8. 14 34 5 9. 14 6 1 10.17 11 2 
 
 326 12 39 5 942 14 11 1 
 
 LAND MEASURE. 
 
 ac. r p. ac. r. p. ac. r. p. ac. r. p. 
 
 E.I. 456 2 t5 2.457 1 29 3.356 39 4.594 1 1 
 
 399 29 374 3 39 279 3 39 259 3 17 
 
COMPOUND SUBTRACTION. 
 
 ac. 
 
 r. 
 
 P- 
 
 ac. 
 
 r. 
 
 P* 
 
 ac. 
 
 r. 
 
 P- 
 
 ac. 
 
 r. 
 
 P- 
 
 :.12 
 
 
 
 32 
 
 6. 112 
 
 1 
 
 31 
 
 7.12 
 
 1 
 
 25 
 
 8. 19 
 
 1 
 
 20 
 
 i 
 
 3 
 
 14 
 
 74 
 
 2 
 
 37 
 
 10 
 
 3 
 
 39 
 
 14 
 
 2 
 
 21 
 
 
 WINE MEASURE. 
 
 tuns,hhd.gal. qt. pt. tuns,hhd.gal. qt. pt. tuns,hhd.g. qt. pt. 
 
 1.456 24 1 1 2.257 3 10 1 1 3.467 2 O 
 
 399 3 46 3 1 199 50 3 1 299 3 32 2 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 tuns 
 4. 27 
 19 
 
 ,hhd.gal. punch, gal. qt. hhd. 
 
 2 54 5. 147 14 2 6. 14 
 3 62 79 83 3 ]2 
 
 gal. qt. 
 1 2 
 41 3 
 
 gal. 
 7. 24 
 18 
 
 qt. 
 2 
 
 
 pt. 
 2 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 BEER MEASURE. 
 
 
 butts, 
 
 hhd. g. qt. pt. 
 
 butts, hhd. g. qt. 
 
 pt. bar. fir. 
 
 gal. 
 
 qt. 
 
 1 
 
 . 256 
 
 
 
 39 1 
 
 1 
 
 2. 
 
 467 
 
 000 
 
 3, 
 
 ,376 2 
 
 6 
 
 2 
 
 
 198 
 
 i 
 
 51 3 
 
 1 
 
 
 299 
 
 111 
 
 1 
 
 371 o 
 
 8 
 
 3 
 
 
 
 bar. 
 
 fir. 
 
 g. bar. 
 
 fir. 
 
 g. bar. fir. g. - 
 
 'bar. 
 
 fir. g. 
 
 
 4. 14 
 
 3 
 
 5 5. 
 
 147 
 
 1 
 
 3 
 
 6.271 
 
 1 2 
 
 7. 143 
 
 1 
 
 2 
 
 
 12 
 
 3 
 
 7 
 
 39 
 
 3 
 
 8 
 
 55 
 
 2 3 
 
 97 
 
 2 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CORN 
 
 AND 
 
 COAL MEASURE. 
 
 qr.bush. 
 
 P, 
 
 ch. 
 
 bush. p. 
 
 ch.bush. p. 
 
 qr. bush 
 
 .p. 
 
 1 
 
 . 124 
 
 3 
 
 2 2. 
 
 109 
 
 18 
 
 2 
 
 3. 529 
 
 17 1 
 
 4.376 
 
 O 
 
 
 
 
 90 
 
 6 
 
 3 
 
 7 
 
 29 
 
 3 
 
 297 
 
 31 2 
 
 246 
 
 7 
 
 3 
 
 
 
 
 P- 
 
 
 
 
 
 
 
 
 
 ch. 
 
 b. 
 
 w. 
 
 qr. 
 
 b. 
 
 qr. 
 
 b. p. 
 
 score 
 
 ch. 
 
 b. 
 
 3 
 
 .74 
 
 31 
 
 3 6 
 
 17 
 
 3 
 
 1 
 
 7.147 
 
 6 2 
 
 8.47 
 
 1 
 
 12 
 
 
 47 
 
 31 
 
 2 
 
 14 
 
 3 
 
 7 
 
 94 
 
 7 3 
 
 14 
 
 20 
 
 35 
 
 
 
 
 
 
 
 
 
 
 
 
 TIME. 
 
 d. hr. niin. d. hr. min.sec. mo. w. d.hr. w. d. hr. m. s. 
 
 1.37 2 39 2.74 3 12 14 3.46 1 1 4 4.36 O 
 
 29 21 49 47 21 54 36 29 3 6 21 35 6 23 50 59 
 
70 CO-IMPOUND SUBTRACTION. 
 
 yrs. m. w. m. w. d. d. hrs. m. hrs. min. sec, 
 
 Ex.. 5. 17 10 Q 6.147 2 3 7.167 21 50 8.174 50 51 
 
 14 12 3 19 2 4 19 23 54 94 59 57 
 
 MISCELLANEOUS EXAMPLES IN SUBTRACTION. - 
 
 1. I borrowed of a friend five hundred guineas, and have paid, at 
 different times, three hundred and ninety pounds six shillings and seven- 
 pence three farthings : what have I still to pay ? 
 
 2. A horse and his harness are worth 49^. 4s. 6d. ; but the harness 
 is worth eleven guineas : I demand the value of the horse ? 
 
 -3. What sum, added to 150 guineas, will makeup 199/. 9s. 9%d.? 
 
 4. At an eclipse of the sun, the moon is situated between the earth 
 and sun : how far distant is the mcon from the sun, supposing the dis- 
 tance between the earth and sun 95 millions of miles, and that between 
 the earth and moon 240 thousands ? 
 
 5. The great bell at Oxford weighs 7 tons, 1 1 cvvt. 3 qrs. 4 Ib. ; that 
 at St. Paul's 5 tons, 2 cvvt. 1 qr. 22 Ib. ; and the great Tom of Lincoln 
 weighs 4 tons, 16 cwt. 3 qr. 16 Ib.: how much heavier than these to- 
 gether is the great bell at Moscow, which is 198 tons? 
 
 0. The Royal Exchange cost- so thousand pounds in building ; the 
 Mansion-house 40 thousand ; Blaokfriars-bridge, l 53 thousand ; West- 
 minster-bridge, 389 thousand ; and the Monument, thirteen thousand 
 pounds ; but the Cathedral of St. Paul's cost 800 thousand : how much 
 did this cost more than all the rest? 
 
 7. If my income is 367 /. *.'4d.and'my expenditure be 34'0 gui- 
 neas : how much can I lay by ? 
 
 8. A person, by great losses, was obliged to call his creditors together : 
 he found his whole property amount to 527/. l?s. s%d. ; but he owed 
 to one man ] 50^. ; to another 300 guineas ; to a third 20 crowns ; to 
 a fourth 55 1. 8*. 9Jd, ; arid to a fifth 200 guineas : liow much will they 
 be losers? 
 
 9. A nobleman leaves, between his two children, 3/jOOoZ. ; to the 
 younger he leaves fifteen thousand guineas : what was the fortune of 
 the elder ? 
 
 10. An apprentice has served of his term of seven years, three years 
 two months, three weeks, four days, seventeen hours : how much 
 longer has he to serve ? 
 
 1 1 . From a field of 6f acres, I. take out two gardens, one measuring 
 4 J roods, and the other 2 J roods, and a piece of ground for coach-house 
 and stables, that measures l rood arid 12 perches : what will be the size 
 of the field after these pieces are taken away? 
 
 12. A plumber puts lead upon the different parts of my house that 
 weighs 5 cvvt. 3 qrs.; and he takes away, in return, old lead weighing 
 2 cwt. 24lb. : what is the difference in the weight between the new and 
 the old lead ? 
 
IT 
 COMPOUND MULTIPLICATION 
 
 Is the method of finding the amount of any given number 
 of different denominations, by repeating it any number of 
 times : 
 
 I. When the given multiplier does not exceed 12. 
 
 RULE. Write the multiplier under the lowest denomina- 
 tion of the multiplicand, multiply even/ number of the mul- 
 tiplicand by the multiplier -, and bring the several products, 
 as they occur, to the next higher denomination. Write down 
 the remainders, aud carry ibe integers to the next product, 
 
 Ex. Multiply .768 14*. D:jd. by 9. 
 
 . $. d. I multiply first the J-by 9, but 18 farthings make 
 768 14 9~ 4 2^> ' P ut down the-,and carry 4; 9 times 9 are 81, 
 9 and 4 are 85 ; 85 pence are 7s, id, I put down the i, 
 and carry 7 ; 9 times 14 are 126, and 7 are 133 shil- 
 
 OQ18 13 lj fr n s > or 6/. 13*., put down the 13, and cany 6; 
 .___ times 9 are 72, and 6 are 78 ; and so of the rest, as in 
 simple Multiplication. 
 
 . s. d. . s. d. 
 
 Ex. 1. 3987 4 6j X 2 Ex.- Q. 0564 1O 7j X 3 
 
 Ex. 3. 2987 3 9f X 3 Ex. 4. 2648 16 8-| X 5 
 
 Ex. 5. 3487 12 8 X 6 Ex. 6. 3498 2 ! C$ X 7 
 
 Ex. 7. 56Q4 16 11 4: X 8 Ex. 8. 2691 1 8 M^ X 9 
 
 Ex. 9. 3764 12 8j X 10 Ex.10. 34(55 15 loj X 11 
 
 EX. 11. 4610 15 4 X 12 ' Ex. 12. 3591 19 Q X 4 
 
 Ex. 13. 1456 16 10 X 12 Ex. 14. 2761 144X6 
 
 Ex. 15. 3420 13 5 X 10 Ex. 16. 4694 12 7 X 8 
 
 Ex. 17. 2675 19 3^ : X 9 Ex. 18. 3476 17 8j X 5 
 
 Ex. 19. 4675 17 8j X 11 Ex 20. 4900 Q 9j X 7 
 
 II. When the multiplier is a composite number, and can 
 be resolved into two or more component parts; See p. 21. 
 
 RULE. Multiply by its component parts successively , 
 and the last product will be the answer. 
 
 Ex. Multiply .374 10s. 11 id. by 63. 
 . s. d. 
 
 374 10 llf X 63 = 9 X 7 
 
 9 
 
 ^ Here 9 X 7 = 63: I therefore multi- 
 3370 18 9j ply by 9, and that product by seven, which 
 7 gives the true answer. 
 
 Answer . 23596 11 
 
7S 
 
 COMPOUND MULTIPLICATION. 
 
 EXAMPLES. 
 
 
 . 
 
 s. 
 
 d. 
 
 
 
 
 . 
 
 s. 
 
 d. 
 
 
 
 Ex.1. 
 
 456 
 
 12 
 
 ei 
 
 X 
 
 15 
 
 Ex.2. 
 
 436 
 
 14 
 
 * 
 
 X 
 
 16 
 
 3. 
 
 784 
 
 15 
 
 4 
 
 X 
 
 18 
 
 4. 
 
 397 
 
 16 
 
 10 
 
 X 
 
 21 
 
 5. 
 
 674 
 
 j 8 
 
 io| 
 
 X 
 
 22 
 
 6. 
 
 487 
 
 10 
 
 i 
 
 X 
 
 24 
 
 7- 
 
 245 
 
 10 
 
 3 
 
 X 
 
 30 
 
 8. 
 
 376 
 
 15 
 
 Jl 
 
 X 
 
 30 
 
 9. 
 
 246 
 
 19 
 
 94 
 
 X 
 
 35 
 
 10. 
 
 489 
 
 18 
 
 ** 
 
 X 
 
 42 
 
 11. 
 
 397 
 
 13 
 
 a 
 
 X 
 
 48 
 
 12. 
 
 369 
 
 10 
 
 2 
 
 X 
 
 54 
 
 13. 
 
 384 
 
 35 
 
 104 
 
 X 
 
 56 
 
 14. 
 
 965 
 
 13 
 
 of 
 
 X 
 
 63 
 
 15. 
 
 592 
 
 1-2 
 
 9 
 
 X 
 
 66 
 
 16. 
 
 800 
 
 
 
 8 
 
 X 
 
 72 
 
 27 
 
 931 
 
 13 
 
 2f 
 
 X 
 
 84 
 
 18. 
 
 914 
 
 M 
 
 4 
 
 X 
 
 77 
 
 19 
 
 397 
 
 4 
 
 *| 
 
 X 
 
 96 
 
 20. 
 
 374 
 
 3!2 
 
 ** 
 
 X 
 
 103 
 
 21. 
 
 459 
 
 9 
 
 of 
 
 X 
 
 100 
 
 22. 
 
 279 
 
 13 
 
 3 
 
 X 
 
 12O 
 
 23. 
 
 376 
 
 15 
 
 4 
 
 X 
 
 121 
 
 24. 
 
 347 
 
 3 
 
 9 
 
 X 
 
 132 
 
 25. 
 
 3/6 
 
 4 
 
 ** 
 
 X 
 
 144 
 
 26. 
 
 567 
 
 14 
 
 7 
 
 X 
 
 45 
 
 27. 
 
 8W7 
 
 16 
 
 6 
 
 X 
 
 108 
 
 28. 
 
 675 
 
 13 
 
 3j 
 
 X 
 
 88 
 
 29. 
 
 487 
 
 19 
 
 11| 
 
 X 
 
 121 
 
 SO. 
 
 856 
 
 la 
 
 2 
 
 X 
 
 132 
 
 III. When the multiplier is not a composite number. 
 
 RULT. Take the composite number which is nearest to 
 it, and multiply ly the component parts, as before : then 
 add or subtract as many times the first line, as the compo- 
 site member is less or greater than the given multiplier. 
 
 (1) Multiply .324 12.9. flic/, by 394. 
 
 . s. d. 
 
 324 12 >l 
 
 8 
 
 2597 
 
 4 
 
 
 7 
 
 18179 2 
 
 4 
 
 
 7 
 
 12/253 16 
 
 4 
 
 649 5 
 
 1 
 
 X 394 = 8 X 7 X 7 + 2. 
 
 The nearest composite number is 392 = 8 
 X 7 X 7 ; I accordingly multiply by these three 
 figures, and to the product I add twice the ori- 
 ginal sum, which gives the true answer. 
 
 1279^3 l 5 
 
 . s. 
 
 Ex, 1. 574 12 
 
 3. 325 8 
 5. 226 18 
 7- 300 
 
 EXAMPLES. 
 
 . s. 
 
 Ex.2. 387 18 
 4. 222 12 
 6. 136 14 
 
 d. 
 
 38 
 
 d. 
 
 6 X 
 4 X 58 
 9f X 78 
 3j X 273 
 
 N. B, Compound Multiplication for Icrge numbers, may be per- 
 formed by the rule of Practice, as will be shewn further on, 
 
 X 
 X 
 X 
 
 46 
 
 68 
 94 
 
 8. 249 12 Oj X 356 
 
COMPOUND MULTIPLICATION. 79 
 
 
 . 
 
 S. 
 
 d. 
 
 
 
 
 . 
 
 S. 
 
 d. 
 
 
 
 Ex. 9. 
 
 525 
 
 16 
 
 Of 
 
 X 
 
 412 
 
 Ex. 10, 
 
 326 
 
 18 
 
 3 
 
 X 
 
 687 
 
 11. 
 
 239 
 
 9 
 
 9 
 
 X 
 
 740 
 
 12. 
 
 560 
 
 
 
 2^ 
 
 X 
 
 388 
 
 13. 
 
 660 
 
 15 
 
 4* 
 
 X 
 
 1004 
 
 14. 
 
 407 
 
 13 
 
 1 
 
 X 
 
 1325 
 
 15. 
 
 700 
 
 
 
 0| 
 
 X 
 
 1450 
 
 16. 
 
 110 
 
 10 
 
 11 
 
 X 
 
 1208 
 
 EXAMPLES OF WEIGHTS AND MEASURES.* 
 TROY WEIGHT. 
 
 lb. oz. dwt. gr. Ib. oz. dwt. gr. 
 
 Ex.1. 187 9 12 20 X 4 Ex.2. 256 6 22 X 5 
 
 3. 169 6 14 17 X 6 4. 379 11 9 9X7 
 
 5. 254 333X9 6. 2*53 11 4 20 X 8 
 
 7. 675 4 15 10 X 11 8. 375 O 17 X 12 
 
 AVOIRDUPOIS WEIGHT. 
 
 ton.cwt.qr.tb. oz. dr. ton.cwt.qr. Ib. oz.dr. 
 
 Ex. 1. 12 10 3 14 10 12 X 2 Ex, 2. 64 13 2 15 68X4 
 
 3. 25 02 8 4 4X3 4. 46 15 3 12 44X6 
 5. 75 13 18 6 30 X 8 6. 39 12 2 16 10 8 X 9 
 
 APOTHECARIES' WEIGHT. 
 
 lb. oz. dr. scr. lb. oz. dr. scr. 
 
 Ex. 1. 456 8 4 1 X 5 Ex. 2. 748 522X8 
 
 3. 534 7 6 2 X 12 4. 3/8 10 1 X 11 
 
 5. 321 5 4 1 X 1O 6. 491 5 7 2 X 9 
 
 CLOTH MEASURE. 
 
 yds. qr. n. E.e. qr. n. yds. qr. n. 
 
 Ex. 1. 210 2 1X4 2. 378 4 3X7 3. *596 3 1 X 12 
 
 4. 357 1 3X6 5. 738 3 2X9 6. 876 O 3 X 10 
 
 LONG MEASURE. 
 
 yds. ft. in. b.c. lea. m.fur. p. 
 
 Ex, 1. 456 2 10 1 X 5 Ex, 2. 379 1 6 20 X 7 
 
 3. 369 192X8 4. 376 2 5 37 X 9 
 
 5. 241 2 11 1 X 10 6. 674 2 ^ 18X6 
 
 LAND MEASURE. 
 
 acr. r. p. acr. r. p. 
 
 Ex. 1. 456 O 25 X 11 Ex. 2. 597 3 12 X 12 
 
 3. 371 2 18 X 4 4. 271 2 25 X 1O 
 
 5. 189 3 32 X 8 6. 430 12 X 8 
 
 WINE MSA SURE. 
 
 tuns, hhd.gal.qts.p. tuns,hhd.gal.qts. 
 
 Ex. 1. 456 3 28 2 1X4 Ex. 2. 456 3 46 2 X 6 
 
 3. 374 2 60 3 1 X 8 4. 350 2 25 1 X 2 
 
 5. 221 1 410X5 6. 124 3 50 3 X 10 
 
 * The compound rules relative to money are those which are 
 chiefly useful. 
 
SO COMPOUND MULTIPLICATION. 
 
 BEER MEASURE, 
 
 butts.hhd.gal.qts.pt. bar. fir. gal. qt. 
 
 Ex. 1. 250 1 20 2 0X8 Ex. 2. 375 8 6 3 X 6 
 
 3. 374 2 730X7 4. 676 2 8 2 X 9 
 
 5. 487 1 50 1 X 8 6. 169 3 2 1 X 12 
 
 CORN AND COAL MEASURE. 
 
 qr.bush.p. chal.bush.p. qr.bush.p. 
 
 Ex. 1. 124 32X4 2. 124 17 3 X 6 3. 46 7 2 X 5 
 
 4. 91 63X8 5. 178 34 2 X 7 6. 87 4 OX 1O 
 7. 594 3 0X9 8. 476 10 1 X 11 9- 31 a 2 X 12 
 
 TIME. 
 
 w. d. hrs. m. sec. yrs. mo. w. d. 
 
 Ex.1. 73 6 10 40 30 X 5 Ex.2. 594 12 3 4 X 7 
 
 3. 3S 4 151 15 20 X 9 4. 364 8 2 C X 8 
 
 5. 98 5 17 13 55 X 12 6. 443 10 3 3 X 11 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. What cost 12 lb. of tea, at 7s. erf. per lb.? 
 
 2. What cost ]&- lb. of sugar, at is. ijd. per lb.? 
 
 3. What is the value of 24 yards of Irish, at 3s. 6%d. per yard ? 
 
 4. What will 79 bibles come to, at 4s. 7-Jrf. each ? 
 
 5. What is the value of 85 gallons of brandy, at 19s. 9-Jrf. pot 
 gallon ? 
 
 6. What is the weight of 23 ingots of geld, each weighing 6 lb. 
 7 oz. 15 dwts. 20 gr. 
 
 7. What will 157 oxen cost, at 15/. 5s. gd. each ? 
 
 .. What is the value of 576 sheep, at ll. 6s. Qd. each ? 
 
 9. How much must I pay for 759 chaldron of coals, at 585. 6d t 
 per chaldron ? 
 
 10. What is the value of 199 firkins of ale, at 12s. 6d. per firkin ? 
 
 11. What is the value of 245 yards of broadcloth, at 195. yd. per 
 yard ? 
 
 12. What is the worth of a stack of hay, containing 7 5 loads, at 
 Ql. igs. Qd. per load? 
 
 13. What is the worth of 12 j lb. of coffee, at 4s. 6d. per lb. ?* 
 
 NOTE. 
 
 * The pupil will recollect, that to multiply by J is to divide by 2, 
 and to multiply by % i-s to divide by 4 ; therefore this question is 
 worked thus : 4s. erf, and the answer is 2/. 16s. Qd. 
 12 
 
 2 14 
 2 3 
 
 S. 16 ~3~~ 
 
COMPOUND MULTIPLICATION. 
 
 81 
 
 14. How many pounds sterling are there in 28 purses, each con- 
 taining 15 guineas, 15 half-guineas, 15 seven-shilling pieces, and 3 
 crowns ? 
 
 15. What, is the weight of 1000 guineas, each guinea weighing 
 5 dwts. oj gr. ? 
 
 16. I bought at a sale 47 * dozen of port wine, at ll. 5s. 6d. per 
 dozen, how much money must I send to pay for it ? 
 
 17. What is the value of 85 tons of iron, at is/. 17*. 9|fZ. per ton? 
 28. What do 79 packages of goods weigh, supposing that each 
 
 package weighs 3 cwt. 3 qrs. 15 Ib. ? 
 
 19. If one ounce of gold cost 3/. 165. 8a\, what is the- value of 436 
 ounces ? 
 
 SO. What shall I pay annually for 459 acres of land, at aZ. 12s. 6d. 
 per acre ? 
 
 21. What is the price of 185 gallons of rum, at 135. ojrf. per gall. ? 
 
 22. If a man spend 2s. >^d. per day, how much does he expend in 
 a year ? 
 
 23. If a bankrupt pay his creditors 12.<?. gd. in the pound, -bow 
 much money is required to pay debts to the amount of 1250/. ? 
 
 24. How many pounds sterling are there in a million seven-shilling- 
 pieces? 
 
 BILLS OF PARCELS, 
 
 A MERCER'S BILL. 
 
 . s. d< . s. d. 
 
 12 yards of silk, at - 
 
 
 
 15 
 
 2 per yard 
 
 
 
 
 114 Do. of flowered silk, at 
 
 
 
 18 
 
 74 - - 
 
 
 
 
 16 Do. ofTelvet, at - 
 
 1 
 
 2 
 
 4 - - 
 
 
 
 
 12 Do. of satin, at - 
 
 
 
 13 
 
 9 - - 
 
 
 
 
 27. Do, of brocade, at 
 
 
 
 J5 
 
 7 - - 
 
 
 
 
 14 Do, of lustring, at 
 
 
 
 6 
 
 3 - - 
 
 
 
 
 A STATIONS 
 
 tt's 
 
 
 
 BILL. 
 
 
 . 
 
 S. 
 
 d. . s. d. 
 
 250 Reamsxrf paper, at 
 
 i 
 
 3 
 
 6 per ream 
 
 
 
 
 112 Do. do'. at - 
 
 2 
 
 4 
 
 6 - - 
 
 
 
 
 34 Do. of imperial brown, at 
 
 1 
 
 35 
 
 - - 
 
 
 
 
 500 Dutch quills, at 
 
 
 
 3 
 
 9 per had. 
 
 
 
 
 
 2500 Do, common, at - 
 
 
 
 2 
 
 3 - - 
 
 
 
 _ 
 
 __ 
 
 If I have to multiply by f, I first divide the original sum by , 
 and then that quotient by 2, and add them together: What is the 
 value of 8f yards, at 6s, gd. per yard ? 
 8-J 
 
 2. 14 
 3 
 1 
 
 Answer - 2 19 
 
 E5 
 
COMPOUND MULTIPLICATION. 
 
 A CARPENTER'S BILL. 
 
 s. d. 
 65 'cubic feet of oak, at - - 4 3 per foo t 
 125 Do. wrough and framed, at - 5 8 - - 
 176 Do. fir framed and moulded, at 3 6 
 15 square shed roofing, at - 5 6 per square 
 8 Do. hip and valley roofing, at 8 3 - - 
 70 feet water trunk, at - - o 10 per foot 
 364 feet ovolo wainscot sashes, at -09-- 
 124 Do do. mahogany, at - - l 4 - - 
 10 men's labour, lor 25 da^ s, at - 4 8 per day 
 
 . 
 
 s. 
 
 d, 
 
 i 
 
 
 
 
 A BRICKLAYER S BILL. 
 
 . s. d. 
 
 39 rod of grey-stock brick-work, 13 33 per rod 
 7 Do. in party-wall, at - 7150 - - 
 105 feet of 18 inch drain, at - o 3 per foot 
 loso Do. of pointing old work, at o 5j| - - 
 1500 giey stocks, at - -046 per hdd. 
 125 pan-tile?, at - - - o O l each 
 45 hods of mortar, at- -007 - - 
 13 Do. of tarras, at - - o 4 2 - - 
 15 bricklayers, 25 days, at - 4 6 per day 
 12 labourers, ditto, at - 3 - - 
 S6 load of rubbish carted away, at Q 6 per load 
 
 . 
 
 S. 
 
 d. 
 
 
 
 
 
 A SLATERS BILL. 
 . S. d. 
 
 9 square of Westmorland 
 slating, at - - - 2 19 6 per squ. 
 7 Do. of Welsh ladies, at - l 17 4 - - 
 6 Do. of Welsh coumess, at 1 18 3 - - 
 35 Do. ripped and rubbish 
 cleared, at - - - 2 7 - ~ 
 12 slaters 7 days, at - 4 5 per day 
 6 labourers, do. at - -029-- 
 5050 clout nails, at - - - 4 per hdd. 
 
 . 
 
 ! 
 
 S. 
 
 d. 
 
 
 
 
 
 PAINTER AND GLAZIER'S BTLL. 
 
 s. d. 
 
 3 035 yards of painting 3 times in oil, at o 7| per yard. 
 565 Do. do. and sand, at V; 3 - - 
 36 sash frames, at - --on each 
 432 sash squares, at - - - o* 8j per doz. 
 19.65 feet of best Newcastle glass, at i 7 J per foot 
 356 Do. large size, at - - -2J-J-- 
 1000 Do. in lead work, at .- l o.| 
 
 . 
 
 S. 
 
 d. 
 
83 
 
 COMPOUND DIVISION 
 
 Is the method of finding how often one given number is 
 contained in another of different denominations ; or, to di- 
 vide a given compound number into any proposed number 
 of equal parts. 
 
 I. When the given divisor does not exceed 12. 
 
 Ru LE. Place the divisor to the left-hand of the dividend, 
 Divide the highest denomination of the dividend by the di- 
 visor > and write down the quotient ; reduce the remainder 9 
 if any, into the next lower denomination, adding to it the 
 number ivhich stands in that place of the dividend^ and 
 divide as before, and so proceed to the end* 
 
 Ex. 1695/. 14*. 4fc*. -r 8. 
 
 , .s. 
 
 8)1695 14 
 
 211 19 
 
 d. I divide the pounds, as in simple Di 
 
 4% vision ; the remainder is 7, which I re- 
 
 duce to shillings, that is 140, to which I 
 3j 2 add the 14, and say, the 8's in 154 will 
 8 go 19 times and 2 over; I put down the 
 
 19, and bring the remainder, 2 shillings, 
 into pence, and add to it the 4 ; the 8's 
 in 28 will go 3 times, and 4 over; reduce 
 the 4 pence to farthings, and take in the 
 
 -g, and the 8's in the 18 will go twice, and 2 over : thus the answer is 
 $\ll. IQs. Qd. 2. 
 
 Proof. The method of proof is by Co<mpound Multiplication* 
 
 EXAMPLES, 
 
 Proof 1695 14 4% 
 
 
 . 
 
 J 
 
 d. 
 
 
 . 1. 
 
 457 
 
 8 
 
 9j - 
 
 - 3 
 
 3. 
 
 396 
 
 IS 
 
 ?J- ~ 
 
 - 4 
 
 $' 
 
 474 
 
 12 
 
 10 - 
 
 - 6 
 
 7. 
 
 897 
 
 16 
 
 4 - 
 
 - 8 
 
 9- 
 
 759 
 
 
 
 - 
 
 9 
 
 11. 
 
 101 
 
 15 
 
 9'j - 
 
 - 11 
 
 13. 
 
 900 
 
 
 
 O - 
 
 - 8 
 
 15. 
 
 800 
 
 10 
 
 2 - 
 
 - 7 
 
 17 
 
 464 
 
 d 
 
 94 - 
 
 - 
 
 
 
 f. 
 
 S. 
 
 d. 
 
 
 Ex 
 
 . 2. 
 
 579 
 
 18 
 
 4% - 
 
 2 
 
 
 4. 
 
 768 
 
 2 
 
 6 2 ~ 
 
 - 5 
 
 
 6. 
 
 934 
 
 14 
 
 5 - 
 
 - 7 
 
 
 8. 
 
 256 
 
 17 
 
 10| - 
 
 - 10 
 
 
 10. 
 
 694 
 
 19 
 
 6 - 
 
 - 12 
 
 
 12. 
 
 496 
 
 
 
 - 
 
 - 12 
 
 
 14. 
 
 500 
 
 5 
 
 5 - 
 
 - 4 
 
 
 16. 
 
 270 
 
 17 
 
 7* ~ 
 
 - 
 
 
 18. 
 
 901 
 
 \, 
 
 1 - 
 
 - 9 
 
COMPOUND DIVISION, 
 
 II. When the divisor is a composite number. 
 
 RULE. Divide by the component parts of the divisor 
 successively, and the last quotient will be the answer. 
 
 Ex. 148. 8s. 8.i</. -5- 27 = 3 x 9. 
 
 5. d. Here the division is first by 3, and 
 
 3)l48 8 8^ then by 9, as in the former example; 
 
 and I find two remainders; I there- 
 
 9)49 9 6f l"J fore, as in Simple Division, multiply 
 
 - >m - 2 j the last remainder 6, by the first di- 
 
 5 9 H;f 6 ) visor 3, and take in the first remainder 
 
 3, and then place under it the common 
 divisor 27. The answer is 5/. QS. 3 l%d.*g. See p. 27. 
 
 4 s. d. 
 Ex. 2. 769 
 
 4. 594 7 6 
 6. 333 10 10J 
 8. 498 9 
 10. 596 
 12. 465 
 14. 564 
 16. 678 
 18. 999 
 20. 564 
 22. 248 
 24. 505 
 
 23. 
 25. 
 27. 
 
 &)l350 10 11 
 
 The answer is 5/. QS. 
 
 
 d. 
 
 >7 12 
 
 6 2 ~ 
 
 - 14 
 
 9 15 
 
 8^ - 
 
 - 15 
 
 J6 9 
 
 9 - 
 
 - 16 
 
 J7 
 
 f " 
 
 - 18 
 
 9 5 
 
 
 - 24 
 
 9 18 
 
 7 
 
 - 27 
 
 7 9 
 
 9| - 
 
 - 30 
 
 >7 4 
 
 6 - 
 
 - 33 
 
 4 3 
 
 6} - 
 
 - 42 
 
 7 14 
 
 4 - 
 
 - 48 
 
 7 12 
 
 11 - 
 
 - 72 
 
 5 13 
 
 3j - 
 
 - 88 
 
 4 18 
 
 8^ 
 
 - 108 
 
 5 3 
 
 3 - 
 
 - 1,32 
 
 9 8i 
 
 26. 564 
 
 28. 8b8 
 
 11 
 
 3^ 
 8 
 6 
 
 
 20 
 25 
 28 
 32 
 36 
 44 
 49 
 54 
 56 
 63 
 84 
 99 
 120 
 144 
 
 jEx. 3. 167 
 3. 339 
 o. 486 
 7. 987 
 9. 439 
 11. 379 
 1-3. 487 
 15. 
 17- 
 
 19- 327 
 21. 3 
 5 
 6 
 4 
 
 When there are three component parts. 
 Ex. 1350. 10^. lid. -r- 240 =5 x 6 x S. 
 
 , s. d. The division in this example fol- 
 
 lows the same rule as before, but 
 there are three remainders, to find 
 6)270 22 4 C the true value of which, I multiply 
 
 the third l , by the second divisor 6, 
 
 8)45 4^ 2") arid take in the second remainder, 
 
 ( 8 ^ 2 4 4 4 o 2 > tilat is > once ^ is 6, and 2 are 8 ; 
 
 5 12 '6j 1 J L then this product, 8, I multiply by 
 
 5, the first divisor, and take in the 
 
 first remainder, that is, 8 times 5 are 40, and 4- are 44 r so that the true 
 answer Is 5/. 125. 6|t/. 4 5 , as will be proved in the next page by Long 
 Division. See also p. 27 and 28. 
 
 * The arithmetician-will easily perceive that the fractions | -f- -| -f- 
 5 are equal'^g -f- ^ 4- -^ ~ (because now they have each a com- 
 
 mon denominator) 
 
COMPOUND DIVISION. 35 
 
 Ex. 1. 5527. 10$. 6^.^243. 2. 18508. 12;. .lid..*. 1296. 
 
 With these two examples the preceptor may form almost any num- 
 ber, by varying the divisors. The several numbers which were before 
 made use of as multipliers, may now be used as divisors. 
 
 III. When the divisor is greater than 12, and not a com- 
 posite number ? 
 
 RULE. The several quotients must be found by f he method 
 of Long Division, (see pp. 28 and 29), reducing the remain- 
 ders to the next lower denomination, and taking in those 
 numbers of the dividend which are of the same denomination. 
 
 Ex. Divide 1350. 10*. lid. by 240. 
 . s. d. 
 
 240) 1350 10 ll(5l. 
 1200 
 
 Having divided th$ pounds by 240, I find a re- 
 
 
 150 
 
 
 
 Having dm 
 
 
 2O 
 
 
 
 mainder of 15 
 
 
 
 
 
 in the 10 anc 
 
 240) 3010(125. 
 
 is 130, which 
 
 
 2880 
 
 
 
 11, and then ( 
 
 
 13O 
 
 
 
 131, which I 
 
 
 12 
 
 
 
 before ; the la 
 
 240)1571(6'^. 
 
 place the divisc 
 5/. 125. 63^.53 
 
 
 1440 
 
 
 
 
 
 131 
 
 
 
 
 
 4 
 
 
 
 
 240) 524(J 
 
 
 480 
 
 
 
 
 
 44 
 
 
 
 
 
 
 . 
 
 S. 
 
 d. 
 
 
 Ex. l. 
 
 985 
 
 18 
 
 9 
 
 19 
 
 
 3. 
 
 465 
 
 16 
 
 4j - 
 
 29 
 
 
 5. 
 
 565 
 
 13 
 
 3 - 
 
 37 
 
 
 7. 
 
 800 
 
 8 
 
 8 - 
 
 41 
 
 
 9. 
 
 987 
 
 14 
 
 4 
 
 46 
 
 
 11. 
 
 598 
 
 12 
 
 6 - 
 
 - 67 
 
 
 13. 
 
 483 
 
 6 
 
 6 - 
 
 73 
 
 
 15. 
 
 9S6 
 
 5 
 
 Pf - 
 
 8 9 
 
 
 17- 
 
 1465 
 
 19 
 
 2 - 
 
 - 1O7 
 
 
 19. 
 
 2690 
 
 12 
 
 3 - 
 
 - 166 
 
 
 21. 
 
 6259 
 
 11 
 
 6 - 
 
 - 215 
 
 
 23. 
 
 9654 
 
 7 
 
 7| - 
 
 - 649 
 
 
 25. 
 
 5942 
 
 17 
 
 3* - 
 
 - 757 
 
 
 27- 
 
 46-28 
 
 5 
 
 9 - 
 
 - 1001 
 
 
 29. 
 
 145<3 
 
 16 
 
 7 - 
 
 - 3761 
 
 
 . 
 
 ,?. 
 
 (?. 
 
 Ex. 2 
 
 1001 
 
 12 
 
 Hi - 
 
 23 
 
 4. 
 
 2468 
 
 13 
 
 3-| - 
 
 39 
 
 6. 
 
 5/46 
 
 9 
 
 6 - 
 
 59 
 
 8. 
 
 6321 
 
 3 
 
 3J - 
 
 61 
 
 30. 
 
 4268 
 
 12 
 
 8 
 
 69 
 
 12. 
 
 "4821 
 
 9 
 
 7i - 
 
 87 
 
 14. 
 
 5 CM 3 
 
 16 
 
 6 .- 
 
 97 
 
 16, 
 
 3648 
 
 4 
 
 "6 - 
 
 97 
 
 18. 
 
 4683 
 
 15 
 
 4- 
 
 - 376 
 
 20. 
 
 5649 
 
 9 
 
 9 - 
 
 - 439 
 
 2!2. 
 
 3604 
 
 10 
 
 10 - 
 
 - 509 
 
 24. 
 
 6534 
 
 16 
 
 si- 
 
 - 606 
 
 26. 
 
 4593 
 
 1'2 
 
 4 - 
 
 - 1585 
 
 28. 
 
 5349 
 
 
 
 O - 
 
 - 4/S6 
 
 30, 
 
 9504 
 
 1 
 
 if - 
 
 - 8073 
 
86 
 
 COMPOUND DIVISION. 
 
 V. When the divisor consists of a number not exceeding 
 
 12, with one or more cyphers. 
 
 RULE. Cut off., by a line, as many places in the pounds 
 as there are cyphers in the divisor, and divide by short di- 
 vision; then reduce the remainder to the next lower deno- 
 mination, as in the last rule. 
 
 Ex. Divide 5645. 14s. 4%d. by 1200. 
 
 12,00)56,45 14 4 
 
 845 
 
 20 
 
 12 00)169 14 
 
 s. 14 114 
 
 12 4 00)13,72 
 
 688 
 
 have stood 4/. 145. 
 
 Having cut off two figures in the pounds 
 to answer to the cyphers in the divisor, I di- 
 vide by 1 2 ; the remainder is 845, which I 
 reduce to shillings, ;and take in the 14, and 
 divide as before: the second remainder is 
 114, this I multiply by 12, and take in the 
 4j an( j divide: the remainder is now 172, 
 which, reduced to farthings, gives 688 ; 
 this not being equal to the divisor, 1 set 
 down the answer 4 1. 14s. id. fgfe. But 
 as it was obvious, from inspection, that the 
 remainder, 172, would not, when reduced, 
 contain the divi or once, the answer might 
 : for the value of fjfod. is equal to /*&$.* 
 
 EXAMPLES OF WEIGHTS AND MEASURES.f 
 TROY WEIGHT. 
 
 lb. oz.dwt. gr. Ib. oz. dwt. gr. 
 
 Ex. 1. 287 9 12 20 4 Ex. 2. 356 6 22 -j- 5 
 
 3. 269 6 14 7 6 4. 379 119 ~ 7 
 
 5. 354 3 3 39 6. 356 11 4 20 ~- 8 
 
 7. 675 4 15 10 11 8. 775 17 ~ 12 
 
 NOTES. 
 
 * When the divisor is 1, with any number of cyphers, there is no 
 division care is only necessary in cuttingoffthe true number of figures 
 in each separate dividend. Ex. 869874^. J2s. gd. ~- 1000. 
 . s. d. 
 
 1,000)869,879 12 9 
 20 
 
 T7~llir Answer - 869*. 17** 
 
 In all questions of interest, commission, buying and selling of stock, 
 &c. &c. the divisor is 100 ; of course care must be taken, in cases of 
 those kinds, to cut off the two right-hand figures in each part of the 
 (dividend. 
 
 f The student need not dwell on these varieties, 
 
COMPOUND DIVISION. 
 AVOIRDUPOIS WEIGHT. 
 
 87 
 
 tons.cwt.qr. Ib. oz. dr. tons.cwt.qr. lb.oz.dr. 
 
 Ex. 1. 412 10 3 14 10 12 -r 2 Ex. 2. 664 13 1 12 6 84 
 
 3. 529 001866-7-3 4. 464 3 27 36 
 
 5. 678 22 2 8 2-7-8 6. 591 5 O 43122 
 
 APOTHECARIES' WEIGHT. 
 
 lb. oz.dr. sc. Ib. oz. dr. sc. 
 
 Ex. 1. 591 841 5 Ex. 2. 748 570-7-8 
 
 3. 639 1 1 2 12 4. 392 10 6 -7- 11 
 
 5. 487 200 10 6. 421 4 5 1 ~- 9 
 
 yds. qrs. n. 
 
 Ex. 1. 5210 214 
 3. 3976 I 2 6 
 5. 4721 00 8 
 
 CLOTH MEASURE. 
 
 E.e. qrs. n. 
 
 Ex. 2. 5964 31 11 
 4. 7645 4 2 12 
 6. 3492 03 9 
 
 LONG MEASURE. 
 
 yds. ft. in. b.c. lea. m. fur. p. 
 
 Ex. 1. 5946 2 101 5 Ex. 2. 3795 2 7 30-7-7 
 
 3. 4736 1828 4. 4965 1 3 18-7-9 
 
 5. 2005 11 2 10 6. 6743 26 4 -~ 6 
 
 LAND MEASURE. 
 
 acr. r. p. 
 
 Ex. 1. 654 2 24 11 
 
 3. 371 018 4 
 
 5. 891 3 32 8 
 
 WINE MEASURE. 
 
 aer, r. p. 
 
 Ex. 2. 958 3 12 ~r 12 
 4. 379 25 -f- 10 
 6. 496 1 1-^-8 
 
 tuns,hhd.gal.qts.pt. tuns,hhd.gal.qts. 
 Ex. 1. 456 3 27 2 14 Ex. 2. 656 3312-7 
 3. 594 30 3 08 
 5. 271 O 2 
 
 butts,hhd.gal.qts.pt. 
 Ex. 1. 294 1 12 3 1 
 3. 379 1 730 
 5. 469 1 50 1 
 
 6 
 
 4. 391 2 25 1 -7- 3 
 6 6. 421 3 50 3 -^- 10 
 
 BEER MEASURE. 
 
 bar. fit. gal.qts. ~* 
 -4 Ex. 2. 976 3 6 3 -r 6 
 - 7 4. 224 1 9 
 
 0. 796 2 1 -*- 12 
 
 CORN AND COAL MEASURE. 
 
 quar.bush.p. chal.bush.p. qr.bush. p. 
 
 Ex.1. 224 3 2 4 2. 124 17 3 6 3. 46 7 2 
 4. 991 63 8 5. 387 34 2 
 
 7 954 3 
 
 9 8, 47<3 20 1 - 
 
 7 6. 37 4 lo 
 
 - 11 9. 31 02 
 
 12 
 
88 COMPOUND 
 
 TIME, 
 
 w. days. h. min. sec- yftf, mo, w. d. 
 
 Ex. 1. 7/9 6 20 40 25 ~ 5 Ex. 2. 591 12 2 4 7 
 3. 391 4 12 16 12 ~- 9 4. 954 6356 
 5. 913 4 5 ~ 32 6. 348 1O 3 3 11 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. i. If 17 yards of cloth cost 19/. Qs. gd., what is it per yard? 
 
 2. What is the price of one pound of sugar,- if sib. cost nine shil- 
 lings ? 
 
 3. The expenses of a journey amounting to 97Z. os. fid. are to be 
 defrayed by six persons : how much will each have to pay ? 
 
 4. I have bought 12 gallons of wine for "jl. iGs. 6d. : how much is 
 that per gallon ? 
 
 5. Twelve boys are to have a guinea and a half divided among them : 
 what will be each boy's share ? 
 
 6. A- hundred and twenty-five sailors have taken 8465 /. prize-money : 
 how much will each man be entitled to ? 
 
 7. I have bought 144 pair of stockings for 27/. : at what rate can I 
 sell them so as to gain by each pair one shilling ? 
 
 8. What did I pay apiece for sheep, having bought 75 for 135J. ? 
 0. Cheese at 3/. 125. 6tZ. per cwt. : how much is that per Ib. ? 
 
 10. If 81 oxen cost 178 1/. 125. 6d. : what is the value of one? 
 
 11. If a pipe of wine cost 95/. : how much is that a dozen, which 
 contains three gallons? 
 
 1 2. Bought 50 dozen of wine for a hundred guineas : how much is 
 that per bottle ? 
 
 13. Divide a thousand guineas between 23 people.,. and. see how 
 much it is for each ? 
 
 14. If 12 pieces of linen cloth contain 250 yards, wh:it is the 
 length of a single piece ? 
 
 15. Mow much can I afford to spend a day, a week, and a month, 
 if my income be 500/. per annum ? 
 
 16. If 12 tea-spoons weigh 9 oz. 17 dwt. 12 gr, : what is the weight 
 of each spoon ? 
 
 Miscellaneous Questions. 
 
 Ex. 1. It is said that Syrius, or the Dog Star, is the nearest of all 
 the fixed stars, and that its distance is computed at 2,200,090,000,000 
 miles : how many years, (each containing 065 days, 6 hours exactly,) 
 would a cannon ball be in passing from the earth to Syrius,. supposing 
 it travelled at the rate of 480 mixes per hour ? 
 
MISCELLANEOUS QUESTIONS. S|> 
 
 Ex. 2. The planet Mercury is about thirty-seven millions, of miles 
 from the Sun ; Verms sixty-eight millions ; the Earth ninety-five 
 millions; Mars a hundred and forty-five millions ; Jupiter four hun- 
 dred and ninety-three millions ; Saturn nine hundred and eight, and 
 the Herschel one thousand eight hundred millions of miles from the 
 Sun : put these several distances down in figures, and add them toge- 
 ther as a sum in Addition. 
 
 Ex. 3. How much nearer the Sun is Mercury than Mars ; and how 
 much farther is the Herschel than the Earth ? See Ex. 2. 
 
 Ex.4. The beautiful planet Venus travels, in its annual journey 
 round the Sun, at the rate of 75,000 miles in an hour : how many 
 miles does she travel in one of her years, or in 228^ days ? 
 
 Ex. 5. The Earth travels, in her annual course, at the rate of 68,400 
 miles in an hour: how many miles therefore do we move in a second? 
 
 Ex. 6. There are in the Old Testament 39 books and 929 chapters, 
 and in the New there are 27 bcoks, and 260 chapters : how many 
 books and chapters are there in the Bible ? 
 
 Ex. 7. There are 23214 verses in the Old Testament, and 79-59 in 
 the New: how much therefore do the verses in the former exceed those 
 in the latter? 
 
 Ex. 8. There are 592439 words in the Old Testament, and 181253 
 in the New : how many words are there in the Bible ? 
 
 Ex. 9. In ihe Old Testament there a.e 2,728,100 letters, and in the 
 New tliere are 838,380 : what are the sum and difference of these tw 
 numbers ? 
 
 Ex. 10. There are in the Bible 3, 566,480 letters : how long would 
 a person be in counting them, supposing he could count 200 in a 
 minute ? 
 
 Ex. 1 1 . A printer charges 5|d for every 1 000 letters that he sets up : 
 how many thousand must. he set up to earn i/. 15.. per week. 
 
 Ex. 12. If a printer set up 8500 letters per day, how long would he 
 be in composing the Old Testament, and how long in composing the 
 whole Bible ? See Ex. 9 and 10. 
 
 Ex.13. If a printer be desired to set up the Bible in Lstin, how 
 much would he earn in the business, at the rate of b&d. per looo letters, 
 supposing there are as many letters in Latin as there are in English ? 
 
 Ex. 14. Ifthere be as many letters in the Greek Testament as there 
 are in the English, how much would a printer earn in setting it up at 
 8f d. per thousand ? 
 
 Ex. 15. The name of JEHOVAH occurs 6855 times in the Old Testa- 
 ment : what proportion therefore does this word bear to all the other 
 words in that book ? 
 
 Ex. 16. The word and occurs in the Bible 46227 times : what pro- 
 portion does that bear to the other words ? See Answer to Ex. 8. 
 
 Ex. 17. There are in the northern side of London 126 houses newly 
 built, and unlet, the average rent of which is 85/. ; and 75 houses at 
 50Z. each, and 68 at 30 guineas each : what is the total annual loss of 
 these empty houses to the proprietors ? 
 
 Ex. 18. There are lioo hackney coaches in London, each of which 
 earns on an average i 8s. per day : how much is expended weekly^ 
 monthly, and annually, on these vehicles ? 
 
90 MISCELLANEOUS QUESTIONS. 
 
 Ex. 19. What are 256 reams of paper worth, at 33s. 6d. per ream ? 
 
 Ex. 20. Fifty thousand larks have been sold in a single season in 
 London : what did they fetch, supposing they were bought at 1 fycl. 
 each ? 
 
 Ex. 21. The circumference of the Earth, in the latitude of London, 
 rs 15,120 miles, which is the space we pass over in 24 hours, by the 
 diurnal motion of the earth : how much space do we pass over in a 
 minute? 
 
 Ex. 22. Three thousand ounces o gold are imported into England 
 annually : how many pounds and grains are imported in 50 years, at 
 this rate, and what is the value of it at si. l 8s. per ounce ? 
 
 Ex. 23. To work the silver mines in South America, 40,000 negroes 
 are imported annually : how many of these poor creatures have perish- 
 ed in this work during the last century ? 
 
 Ex. 24. The duty on hops amounted, at l-J<7. per lb., in a certain 
 year, to 2 6,3 57 /. 9s. Qd. : how many hops were grown that season ? 
 
 Ex. 25. The battering ram employed by i'itus to demolish the 
 walls of Jerusalem, weighed 1 00,000 Ibs. : how many tons did it con- 
 tain ? 
 
 Ex, 26. The copper mines in the island of Anglesey produce 1500 
 tons annually, and those in Cornwall 4000 tons : what is the value of 
 the whole at g^d. per lb. ? 
 
 Ex. 27. Mr. Bolton coined 40,000,000 penny-pieces, each weighing 
 an ounce : how many pounds of copper were used for them : how 
 much was the value of these in pounds sterling; and what was gained 
 by this coinage, supposing the copper and expense of coining to be 
 estimated at i2-d. per pound ? 
 
 Ex. 28. In the year 1794, 43,250,746 yards of Irish linen were ex- 
 ported from Ireland : how many packages did they make, e;ich package 
 containing 20 pieces, and each piece S6 yards ? How many shirts 
 would this linen make, at the rate of 3^ yards per shirt ? 
 
 Ex. eg. The circumference of the earth is estimated at 24,912 miles : 
 how many barley-corns, (three of which make an inch), would fill up 
 this space ? 
 
 Ex. 30. The territory of the United States of America contains a 
 million of square miles, or 640 millions of square acres; of these, 
 about 56 millions are water: what number of acres, roods, and perches 
 of land, do the United States contain, and how many inhabitants will 
 they support, allowing to each 4| acres ? 
 
 Ex. 31. There are now in England, Scotland, and Wales, 23 mil- 
 lions of acres of waste land : how many farms might -these be divided 
 into, allowing to each 75 acres : and allowing 5 persons to each farm, 
 how many souls would these waste acres support ?* 
 
 NOTE. 
 
 * "England and Wales contain 73,334,400 acres, and 8,873,oooJ 
 inhabitants, Scotland has 1,600,000, and Ireland about 4.250,000 in- 
 habitants. England and Wales have 152 inhabitants for each square 
 mile: Scotland 55, and Ireland 146. 
 
 " England contains 7^1 millions of acres: its rents are estimated. 
 
MISCELLANfeOUS QUESTIONS. 91 
 
 Ex.32. Between the 5th of July, 1810, and the same day, 1811, there 
 were brewed, by 12 brewers only, 939,900 barrels of porter : how much 
 would this quantity sell for when retailed out at 5d, per quart ? 
 
 Ex.33. How many hours, minutes, and seconds have elapsed since 
 the birth of Christ, which is 1808 years, supposing 365^ days in a year? 
 
 Ex. 34. It is said the Small-pox carries off in London, by death, 50 
 persons a week ? how many (if the disease is not checked) will it destroy 
 in ten years ? 
 
 Ex.35. There are about 10,540 tons of cheese imported into London 
 annually : how much do they sell for at the average price of 7-Jd.per Ib. 
 
 Ex. 36. It is computed that there are 50,000 tons of butter annually 
 consumed in London : what is the expense, supposing the average price 
 JOjd. perlb. ? 
 
 Ex. 37. About 120,000 persons are employed in the cotton trade : 
 if of these one-fourth are men, who earn 3-s. 6d. a day, and one-fourth 
 women, who earn is. id. a day, and the rest children, who earn, each, 
 3s. per week, how much is earned by manual labour in the cotton ma- 
 nufacture every year ? 
 
 Ex. 38. There have been 20,000,000 Ibs. of tea imported in a single 
 year from China : what was the value of it, supposing the average price 
 4s. gd. per Ib. ? 
 
 Ex.39. The consumption of tobacco in this country is about 1 69,000 
 cwt. : how much is expended on this article at id. per oz. ? 
 
 Ex.40. Sir R. Phillips, (the publisher of this Arithmetic), caused to 
 be printed, for various books, between the years 1798 and 1 808, as many 
 sheets of paper as would, if joined together, extend round the world. 
 Considering each sheet as 21 inches in length, how many reams of 
 paper did he use in that time ? and what was the value of the paper, 
 reckoning it at thirty shillings per ream? See Ex.29. 
 
 x. 41 . The consumption of milk is not less than 6,980,000 gallons 
 annually, in London : how much is expended on this article at %d. per 
 pint ? 
 
 Ex. 12. In London alone, 630,000 chaldron of coals are burnt : what 
 is the cost at 4f d. per peck? 
 
 Ex.43. The iron rails round St. Paul's cost ll,202Z. os.Cd., and they 
 weighed 200 tons and 81 Ibs. : what was the iron charged per Ib. ? 
 
 Ex.44. Westminster-bridge cost 389, 500/, in building: how soon 
 \voald it have been paid for by foot passengers, at a halfpenny each, 
 supposing 2420 went over each day ? 
 
 NOTE. 
 
 at about 29 millions, but are in reality 50. The stock on the land is es-. 
 timated at 145 millions; the money in the country 50 ; the shipping 
 190 : merchandize and manufactures 60: of the land 13 millions of 
 acres are inclosed, 1 1 arable ; 6$ waste in England, ij in Wales, 14 Jin 
 Scotland. For eight millions of inhabitants, the country produces 11 
 ounces of wheat, and 7 * of meat per day." See Middleton's Survey of 
 Middlesex. The above estimate was taken in 1793. 
 
PROPORTION, 
 
 OR 
 
 THE RULE OF THREE. 
 
 This Rule is called the Rule of Three, because, by 
 numbers being given we find a fourth ; and it is either tkn 
 Rule of Three Direct or Inverse. 
 
 THE RULE OF THREE DIRECT 
 
 teaches, from three given numbers, to find a fourth, which 
 shall have the same proportion to the second, as the thi'rd 
 has to the first; that is, if the first be greater than the 
 third, the second will be greater than the fourth; and, if 
 the first be less than the third, the second will be less than 
 the fourth, 
 
 RULE I; STATE THE QUESTION : that is, place the 
 given numbers so that the first and third may be of the same 
 kind, and the second the same as the number required. 
 
 2. Bring the first and third numbers into the same deno- 
 mination, and the second into the lowest denomination 
 mentioned. 
 
 3. Multiply the second and third numbers together, and 
 divide the product by the first, and the quotient will be the 
 answer, * in the same denomination as that in which the 
 
 second number ivas left. 
 
 / 
 
 NOTE. 
 
 * If there be a remainder after division, it is always of the same de- 
 nomination as that of the middle number, and must be brought into 
 the next lower denomination, and then divide by .the first number as 
 before. 
 
KULE OF THREE DIRECT, 9S 
 
 Ex. 1. What is the value of a pipe of wine, if 5 ^allons 
 cost 4. 17*. ? 
 
 jal. . s. pipe. Instating the question, I first consider what "is 
 5:4 17 : ' 1 known, viz. that 5 galls, cost 4/.17-?., and the demand 
 20 2 is, what a pipe will cost at the same rate ? I theieibre 
 
 Q- ~~^ '-say, if 5 galls; cost 4 /. 1 75., what cost l pipe ? for such 
 
 63 is the meaning of the -statement: 5 gal. : 4l. 17$. : : i 
 -- pipe. The first term is gallons ; I must accordingly 
 126 bring the third term, or the pipe into gallons : the se- 
 9? cond, or middle terro, is a mixed number ;"!'bring it 
 882 therefore to its lowest denomination, or shillings, 
 1134 "and then multiply the 126 galls, by 97 shillings, and 
 5^1022 divide the product by the first term 5, and the answer 
 is 2444 shillings, because the middle number is 
 2,0)244,4 2 shillings, and there is a remainder of 2 : this I 
 122 4 ___ 12 bring into pence, and divide again by 5 ; there is 
 5)24 now 4 remaining, this I bring into farthings, and 
 " ~ _ . divide again by 5, and the answer is 2444s. 
 4%d. -?, or by bringing the shillings into 
 _ pounds, 122 1. . 4s, 4|d.-f. 
 
 Ex:2. If'l can buy 27lb. of sugar for 1. 13s. how much 
 L I purchase for thirty guineas ? 
 
 , lb. guineas. In this example, it is known that iL 135. will 
 3 : 27 : : 30 purchase ^7 lb., these will therefore be the first 
 
 21 and second terms; and as the demand is, how 
 
 many pounds can be purchased for so guineas, 
 the second, or middle term, must be' pounds. 
 _ - Having stated the question, I bring the first and 
 4410 third terms into the same denomination, shillings, 
 
 1360 }bs. and then multiply the second and third terms to- 
 33)] 7010(515 gather, and divide by the 'first ; the quotient, or 
 165 answer, 5s515lb.j but there being a remainder 
 
 of 15, I multiply this by 16, because 16 ounces 
 make a lb. ; dividing again, the quotient is 7 oz., 
 3 '._ with a remainder of 9, which 1 might bring into 
 
 180 drams; but sugar is never bought or sold with 
 
 165 such accuracy. The answer is, therefore, 51 5lb. 
 
 ~~~^ 7 oz. -^, or by bringing the Ibs. into cwts. and 
 
 3 6 ^ rs "j *^ e answer is, 4 cwt. 2 qrs. J 1 lb. 7 o 
 
 33)240(7 
 231 
 
94? RULE OF THREE DIRECT. 
 
 Ex. 3. What is the value of 28 ells of cloth, if 4 ell 
 
 cost 18s. ? 
 
 ells, shill. ells. All questions of this kind, in which the first and 
 
 4 : 18 : : 28 | third terms are of the same denomination, and 
 l 8 either of them is a unit, may be solved by Multi- 
 plication only. Thus if lib. cost od., what cost 
 28lb. ? I multiply the 28 by 9, and the answer is 
 found in pence. 
 
 It often happens that the first or third terms may- 
 be reduced to a unit, by dividing both by a com- 
 mon number, and then the question is solved by 
 Multiplication only. In the example before us, 
 it is instantly seen, that 4 will divide 4 and 28 : then the statement is, 
 l : l 8 : : 7, and 1 8 multiplied by 7, gives us 126 shillings, or 6/. 65. for 
 the answer as before. 
 
 Ex. 4. If six yards of cloth cost 24 shillings, what will 81 yards 
 cost?* 
 
 Ex. 3. If 8 bushels of coals cost gs. 6d. t what is the value of 35 
 chaldrons ? 
 
 Ex. 6. If sib. of potatoes cost 4d., what is the worth of l cwt. on the 
 same terms ? 
 
 Ex. 7. If .5lb. of potatoes cost 3d., how many can I buy for 405. ?f 
 Ex. 8. If 10 ells or cloth cost 2/. 105., what is the value of 5 pieces, 
 each containing 26 yards? 
 
 Ex. 9. If -JO yards of muslin cost 10 guineas, how many ells can I 
 buy for 45/. ? 
 
 Ex. 10. If I can purchase 25 books for il. 85., how many can I have 
 for a io/. note? 
 
 Ex. 11. If a servant's wages be 25 guineas a-year, how much has he 
 to receive for 87 day's service ? 
 
 NOTES. 
 
 * In this example, the first and second terms may be divided by 6, 
 and thea it becomes a question in Multiplication : trie original state- 
 ment is, 6 yds. : 24 shil. : : 81 yds. j but, by Division, it is l yd. : 4s. 
 : i 8|i yds., and the answer is, 324 shillings, or 16/.4S. The general 
 rule therefore is, " Divide the first, and either the second or third term, 
 but noi loth, by some commo?i measure, that is, by some number that 
 will divide the two without leaving a remainder, and use the results in- 
 stead of the original terms. 
 
 f In the statement to this example, viz., 3d. : 5lb. : : 40 shillings, 
 iKiiher the second nor third terms are divisible by 3 : but when the 
 third, or 40 shillings, is reduced to pence, then it is divisible by 3, and 
 the statement, 3d : 5lb. : : 480(2., or as l : 5 : : 160, and the question 
 is answered by. multiplying 160 by 5, which gives 800 Ibs., or 7 cwti 
 D qrs. iClb. for the answer. 
 
 From these hints the pupil will frequently sec that the labour of thij 
 operation may be very much shortened. 
 
RULE OF THREE DIRECT. 9o 
 
 Ex. 12. If a servant receive three guineas and a half for 20 weeks 
 service : how long ought he to remain in his place for 12 guineas ? 
 
 Ex. 13. If I pav half-a-crown for 4 Ib. of cheese: how much can I 
 have for three crowns and nine- pence ? 
 
 Ex.14. It 2lb. 4oz. of honey cost 3s. gd. : what is the value of 28lb.?* 
 
 Ex. 15. It is estimated that tvvelv? millions of sheep ar Ld in this 
 country : now, if 1 1 sheep produce 28lb. of wool every year, how much 
 wo-->l will there he- from the whole number? 
 
 Ex. ,6 If a dozen of wine glasses cost 10$. Gd. : what is the valoc 
 of 500 ? - 
 
 Ex. 17. If I can buy 3 pair of shoes for 1 1. 4s. gd. : what must I pay 
 for 17 pair? 
 
 Ex. 18. If a cwt. of tobacco cost 8 guineas : what is the value of 
 7,000,000 oflbs ? 
 
 Ex. 19. If 6 Ib. of different kinds of soap cost 5s. gd.: what is the 
 value of a cwt. in the same propoition ? 
 
 Ex. 20. If I pay 39 shillings per cwt. for lead: how much will it cost 
 to cover the roof of a builJing with lead that weighs 5505 Ib. ? 
 
 Ex. 21.1 want to know how much I have to pay for a cistern ooolbs., 
 at the rate of 2/. is. per cwt., the plumber agreeing to allow me at the 
 rate of ll. 14.?, per cwt. for the old lead, which weighs 458 ib. ? 
 
 Ex.22. If four journeymen ayers can earn 5 1. ils. in six days : how 
 much will their master have to pay them for 305 days, at the same rate, 
 and how much will each man's income be ? 
 
 Ex. 23. The brazen statue of Apollo, that was erected by Chafes, 
 at Rhodes, weighed 720,000 Ibs. : how much did the olfl biass sell tor 
 at four guineas per cwt ? 
 
 Ex. 24. If I pay ll. 7s. for 18 gallons of porter: bow much shall I 
 expend in that article in a year, if my family drink nine gallons of it every 
 week ? 
 
 Ex. 25 If I buy, at the Custom-house sale, 14 gallons of brandy for 
 18Z. : how much must I pay, at the same rate, for four hogsheads, each 
 containing 63 gallons ? 
 
 Ex. 26. If I buy, at Sheffield, 6 razors for 85. 6d. : how much shall 
 L have to pay for twelve dozen, at the same rate ? And, how much can 
 [ sell them for, so as to gain by the bargain 2 %d. each razor ? 
 
 Ex. 27. In building an out house 5050 bricks have been used : how 
 Imuch do they come to at 4$. Crf. per hundred? 
 
 * The operation in this example may be shortened thus : the state- 
 ^ nent is, 2 lb.4oz. : 3s. gd. : : 28lb. Instead of bringing the first terra 
 ^ nto ounces, I bring it into quarters of a Ib., by multiplying by 4, and 
 ofl aking in the 4 oz. as one, then the statement becomes as 9 qrs : 45d. : : 
 qrs.; but the first and second terms are divisible by 0, and the 
 tatement is, 1 : 5 : : 112, and the answer is, 5 X 112* or 5<)0 jienrc, 
 /. 6s. 8d. 
 
!n> RULE OF THREE DIRECT. 
 
 Ex. 28. It requires 32 bricks to pave 9 square feet: how many bricks 
 will be wanted for the pavement of a cellar 24 feet long,and 19 feet 
 wide?* 
 
 Ex. 29. It requires 144 Dutch clinkers to pave 9 square feet: how 
 many will be wanted for a court 35 feet long, and 29 feet wide, and 
 how much will they come to at 5s. 6d per hundred? 
 
 Ex. 'so. It requires sixty persons six days to manufacture a pack of 
 \vool into cloth : how much wool will they work up in a year, supposing 
 they work 5 days in each week ? 
 
 Ex. 3 1 . Six children of different ages will earn in five days, at spin- 
 ning wool, 5s.gd.i and the mother will earn ls.4d. per day : how much 
 will they all earn in a year, allowing that they work, one week with 
 another, 5 J days per week? 
 
 Ex. 32. "At same large iron foundeiies, they can run oflT60GO Ibs. of 
 iron in twenty-four hours : how many tons weight will they cast in a 
 year, allowing them to work 298 days, and 16 hours each day ? 
 
 Ex. 33. By a patent machine for making combs, the teeth of two combs 
 can be cut in three minutes: how many can be manufactured in 28 
 days, if the machine is worked at the rate of eight hours a day ? 
 
 Ex. 34. What is the price ofacarpet that measures 15 feet each way, 
 at ;.v. Cd. for every 9 feet ? f 
 
 Ex. 35. 'If 13 cwt. of fine Lisbon sugar costme 58?. 105. : 'how much 
 must I pay for l 5 casks of the same, each cask weighing -icwt. 2qr.i2lb.? 
 Ex. 36. How much hay can I purchase for 355 guineas, at 3/. los. 
 per 1 jad ? 
 
 Ex. 3 7. 'if candles sell for 113. Gil. per dozen : how much will 250lb. 
 cost ? 
 
 Ex.38. If mould candles cost 125. 6d. per dozen : how many pounds 
 can I purchase for fifty guineas? 
 
 Ex. 39. The best mottled soap is bought at 4/. 6s. per cwt. : for hew 
 much must it be sold per lb., so as to allow a profu of one penny on 
 each pound ? 
 
 Ex. 40. If I buy 6-J yards of Irish cloth for l/. 3s. lod. : how much 
 must I pay for eight pieces, each containing 26 yards ? 
 
 Ex.4l. If 40 yards of Irish cloth will make 12 shirts: how many 
 may be made out of 4 pieces, each containing 26 yards ? 
 
 Ex. 42. If 12 gallons of brandy pay 3/. 18s. duty at the Custom-Louse : 
 how much will be paid for 65,873 gallons, which were imported last 
 week ? 
 
 Ex. 43. The average price of sugar, exclusive of duty was, Aug. 21, 
 1805, -2l. 1 u. g^d. per cwt. : I demand the value of the 9>99>'36o Ibs. 
 ihat were imported into London the preceding week ? 
 
 * To find the number of square feet in the cellar, multiply the length 
 by the breadth. 
 
 f The size of a carpet, 01 the number of square feet that it contain.??,, 
 is found by multiplying the 15 by itself, thus 15 X i? ~ 225. 
 
RULE OF 1 IIRKE DIRECT, ()/ 
 
 lx.44 The average price of tallow was, oji the same day, 4s. id. 
 per stone of sib. : what is the worth of 276 tons, imported the pie- 
 reding week ? 
 
 Ex. 45. What will 31218 gallons of Port wine, imported last week, 
 sell for, at 2/. 135. 6d. per dozen, supposing each dozen to contain 
 3 gallons ? 
 
 Ex. 46. What is the value of 1 1 5 seal-skins, at 35. 6d. per Ib. supposing 
 the skins to weigh, one with the other, 9 ounces each ? 
 
 Ex. 47. Ox hides, fit for tanning, were sold on Friday, the 23d of 
 August, at 3s. gd. per stone : what did 50 of them fetch, supposing each 
 weighed 96 Ib. ? 
 
 Ex. 48. How much brown Holland can I buy for ten guineas, if I pay 
 53. Qd. for four yards and a quarter ? 
 
 Ex. 49. Suppose a person save, out of his income 5s. 6d. per week: 
 how long will he be in laying by loo/. ? 
 
 Ex. 50. I want to know the height of a tree, by means of the length, 
 of its shadow ; I set up a straight stick that measures, above the ground 
 3 feet 4 inches'; the shadow of this is 5 feet 2 inches, and the shadow 
 of the tree, at the same moment I find to be 79 feet ] o inches ? * 
 
 Ex. 51. What is the height of a steeple, whose shadow is 148 feet 4 
 inches, when a shadow 5 feet 3 inches long is projected from a staff 6 
 feet 4 inches? 
 
 Ex. 52. If I pay 4-5. g<L fora hundred of pens : how many shall I get 
 for loot. ? 
 
 Ex. 53. If my income is 450/. per ann. : how much may I spend in 
 /O days, supposing I mean to lay by 50 guineas at the year's end ? 
 
 Ex. 54. What is the value of 57 yards of muslin, at the rate of 
 32s. 6d. per ell? 
 
 Ex. 55. How many ells of Holland can be bought for 35J., at the 
 rnte of 6s. Gd, per yard ? 
 
 Kx. 56. I have a tankard that weighs 2lb. 8oz. that cost loL. 2s. Gd. -. 
 how much, at the same rate, -will a service of plate cost that weighs 
 I25lb. 9 oz.? 
 
 Ex. 57. Hew much must be paid for 130,544 bushels of wheat, at 
 4l. 8s. per quarter? f 
 
 Ex. 58. What was paid for 2 2,2 7 5 bushels of flour, at 4l.Js. the sack 
 of five bushels? 
 
 Ex. 59. What is the value of 6 casic of raisins, each weighing 3 cwr. 
 9 qrs. 14 Ib. at 5/. 105. 6d. perrvvt. ? 
 
 Ex. co. How much must \ give for a gold snuff-box that weigh* 
 8 oz. 9 dwts., at the rate of 4 1. 3s. gd, per oz. ? 
 
 NOTES. 
 
 * The question is, if f> feet 2 inches of shadow is cast from a stick oi 
 .1 feet 4 inches, what will be the length of an object whose shadow is 
 79 feet 10 inches ? 
 
 f- This was the price of wheat, and the quantity sold pt Mark-lane, 
 from the 5th to the loth of August, 1805 ; the same may be savl o; 
 -flour in the next question. 
 
 F 
 
yS RULE OF THREE DIRECT. 
 
 Ex. 61. What must I pay to the property- tax for 5861. per annum, 
 at the rate of 6j per cent. ? 
 
 Ex. 62. A bankrupt has but 1020^. to pay debts to the amount of 
 S^s. 1 }/. : how much can he pay in the pound ? 
 
 Ex. 03. A merchant failing, his assignees find effects and good debts 
 to the amount of 3335^. ; but he owes 4225/. ; the expenses attending 
 his bankruptcy will be 212/. 95. : how much, therefore, will he pay in 
 the pound ? 
 
 Ex. 64. An honest tradesman, through unforeseen misfortunes, is 
 obliged to call his creditors together ; he finds his debts to be 43261. 
 and he can pay 14s. 6d. in the pound: how much has he still left? 
 
 Ex. 65. Hops are remaikably cheap, and I have lool. to spare : how 
 many can I purchase at 3l. 1 5s. 6d. percwt. ? 
 
 Ex. 66. If 1 2lbs. of tea are worth gl. 6s. : how much of the same sort 
 can I purchase for 70 guineas? 
 
 Ex. 67. "What must 1 pay for the carriage by the canal, from Man- 
 chester to Etruria, of 705 tons, :>cwt. of goods, at 1 5s. per ton ; and what 
 is the difference between this and the land-carriage, at 2/. 15s. per ton ? 
 
 Ex. 68. What weight of goods can be carried on the canal between 
 Manchester and Birmingham tor S5/. at the rate of ] /. 10s. per ton : and 
 bow much can be carried the same distance, by land-carriage, at 5l. 
 per ton ? 
 
 Ex. 69. The clothing of a regiment of 760 men comes to ao.so/. : 
 how much is that per man ? 
 
 Ex. 70. What may a man spend per week, whose income is 2000/. 
 per annum, supposing 32 weeks in a year ? 
 
 Ex. 71. If, by selling fine Irish cloth at 5s. per ell, I gain 8l percent., 
 what will be the rate of my profits if I sell it at 6s. 3d. per ell ? 
 
 Ex. 72. If sugar, that cost gd. per lb., be sold at alb. for 2s.gd. y 
 what is the profit per cent? 
 
 Ex. 73. 1 purchased :> pieces of Holland , each containing 36 yards, at 
 45. 9d. per yard : how much shall I gain by selling it at 6s. <2d. per ell ? * 
 
 Ex. 74. Two persons part at the same time from the same place, the 
 one travels north 24 miles a day, and the other 21 miles a day south : 
 when will they be 1000 miles asunder ? f 
 
 Ex. 7. r >. If a pack of wool weighs 3 cwt. 2 qrs. 7 lb., what is it worth 
 at 215. Gd. per tod of 14 Ibs. ? 
 
 Ex. 76. The rents of a parisi amount to 17SO/., and a rate for the 
 poor is wanted of 65l. 7s. 6d. : what is that per pound ? J 
 
 NOTES. 
 
 * The proper method of working this example by the Rule of Three 
 is, by two statings; by the first we find how much the Holland cost, 
 and by the second what it sold for; deduct the former from the latter, 
 and the result is the answer. 
 
 J- Here it is evident, that the distance they both travel in one day 
 must make the first term in the question, and we say, if 45 miles are 
 passed in one day : how many days will it take to pass 1000 miles ? 
 
 J The Author advises the student to pass over the two next rules for 
 the present and pii^s at once to Vulgar and Decimal Fractions 
 
99 
 
 THE RULE OF THREE INVERSE. 
 
 This rule, like the last, teaches, from three given numbers, 
 to find a fourth, which fourth number shall bear the same 
 proportion to the second, as the first has to the third. Thus, 
 if the question be, If 10 men can mow a certain field in (> 
 days, how soon can it be done by 20 men ? The answer will 
 evidently be in 3 days, because double the number of men 
 will certainly do the same work in half the time : the pro- 
 portion will therefore stand,. 10 men : 6 days : : 20 men: 3 
 days; and 3 bears the same proportion to C, that 10 does 
 to 20 ; that is, the fourth number bears the same proportion 
 to the second, that the first does to the third, 
 
 RULE. State the question^ and, when necessary > reduce 
 the terms as before. Multiply the first and second terms 
 together, and divide the product by the third term; the 
 quotient is the answer in the same denomination as the se~ 
 
 cond term ; thus in the foregoing 1 example, \ zr 3 days* 
 
 Ex. 1. If 15 reapers can cut down a field of corn in 4 
 
 days, in how long time will the same work be performed 
 by 40 men ? 
 
 1 5 : 4 : : 40 The answer is a day and a half, and the 
 
 4 reason of the thing is self-evident, because 
 
 ~~ 4(l men niust d trie sa rae job in much Uss 
 
 time than 15 men,* 
 day. 
 
 * The best method of proving questions in this rule is, to reverse the 
 operation. Thus, in this example, I now say, if 40 men : 1 J day : : ] 5 
 men, and the -answer is 4 days, which shews the truth of the former 
 operation. 
 
 The precepor will therefore direct his pupil to prove the truth of each 
 example, which, in fact, is the same thing as giving double the numU j r 
 
 Thus, in the second example, I first consider what is- know; 
 r a 
 
100 RULE OF THREE INVERSE. 
 
 Ex. 2. If the penny loaf weighs 4 ounces when flour is 4s. per peck, 
 how much must it weigh when flour is 5s. 4d. per peck ? 
 
 Ex. 3. A person lent me 240/. for 8 months : in return for his kind- 
 ness, how much ought I to lend him for eighteen months ? 
 
 Ex. 4. How many men must be employed to finish a canal in 12 
 days, which 5 could perform in six werks, or 36 days ? 
 
 Ex.5. If 24 pioneers can make a trench in 12 days, what length of 
 time would the same work employ 9 men ? 
 
 Ex. 6. The floor of a chapel 96 feet in length and 70 feet in breadth, 
 is to be covered with matting 2 feet six inches broad: how many yards 
 will it require? 
 
 Ex. 7. If a person travel 12 hours a day, and finish his journey in 
 three weeks : how long would the same journey take him, if he travelled 
 only 9 hours a day at the same rate ? 
 
 Ex. 8. If the town and garrison of Bhurtpoor, containing 22,400 
 persons, have provisions to last three weeks^ how many inhabitants must 
 Holkar send away, so as to make the same provisions last 7 weeks, 
 which is as long as Geneial Lake can carry on the siege ?* 
 
 Ex. g. If a besieged garrison have 4 months provisions, at the rate of 
 18 ounces per man per day: how long will they be able to hold out, if 
 each man is allowed only 12 ounces per day? 
 
 Ex. 10. If there are in a garrison provisions sufficient for 1500 men 
 10 weeks, which, on account of the rains, is seven weeks longer than 
 the siege can last : how many soldiers may be brought in to defend the 
 place for three weeks, without lessening the quantity of food to any in- 
 dividual ? f 
 
 Ex. 11. If 9 plasterers can finish the inside of a chapel in 10 days: 
 how long will it take 4 men, supposing the other 5 sent away to a new 
 job? 
 
 Ex. 12. If 3 j yards of broad cloth, ij wide, will make a suit of 
 clothes : how much will be necessary of cloth only J wide ? 
 
 NOTES. 
 
 that when flour is 4s. a peck, the loaf weighs 4 ounces ; these are the 
 first and second terms, and the question is, hoiu much it will weigh 
 when tiour is 5*. 4d. per peel;. The answer, or unknown quantity, is 
 weight. I therefore state it thus : 45. : 4 oz. : : 55. 4d. : 3 oz.zzthe an- 
 swer. To prove ttuTtrut h of it I say, if I have 3 oz. of bread when flour 
 is 5-?. 4d. per peck, how much shall I have when it is 4-s. per peck : thus 
 5s. 4d. : 3 oz. : : 45- and I find the answer is 4 ounces. 
 
 * The answer to this question is the number of people to be sent 
 away; therefore, when I have found how many the provisions will 
 support for 7 weeks, I subtract this number from that given, and the 
 remainder shews what number are to be dismissed, which in this case 
 will be found to be 1-2,800. 
 
 f Having found the number of men that may be supported three 
 weeks, subtract from that number the 1 500 already in the garrison, and 
 the remainder is the true answer. 
 
RULE OF THREE INVERSE. 101 
 
 Ex, 13. If 52 clerks in the Bank are sufficient to make up the books 
 in a certain office in 1 5 days, how many clerks would be required to do 
 the same work in 6 days? 
 
 Ex. 14. If the carriage of 15^-cwt., for 60 miles, came to Js. gd: 
 how far can I have carried 3 a. cwt. for the same sum ? 
 
 Ex. 15. The apartment in which the late Duke of Gloucester lay in 
 state previously to his funeral, was 50 feet long, 40 feet broad, and 24 
 feet in height : how many yards of black cloth, l J yards wide, were 
 used in covering the walls, and how much did it cost at l si. per yard.* 
 
 Ex. 16. If 12 inches in length, and 12 inches in breadth, make a 
 square foot: what length of board, 8 inches broad, will be equal to the 
 same measure ? 
 
 Ex. 17. If 220 yards in length, and 22 in breadth, make an acre * 
 what must be the breadth when the length is 121 yards? 
 
 Ex. 18. If 5 horses can be maintained when oats are I 8s. per quarter : 
 how many can be supported at the same cost, when they are 30 shillings 
 per quarter ? 
 
 Ex. 19. If 250?. gain iaZ. 10s. at interest, in 12 months, what prin- 
 cipal will gain an equal sum in 5 months ? 
 
 Ex.20. There are two rooms, in the floors of which there are an equal 
 number of square feet ; the length of the one is 50 feet, and its breadth 
 is 42 ; but the breadth of the other is 48 feet : what is its length ? 
 
 Ex. 21. The cock to a large water-tub will empty it in 36 minutes J 
 how many such cocks will empty it in 4 minutes? 
 
 Ex. 22. The sides of a room are found to measure 1 38 feet in length,, 
 and the height is 14 feet 6 inches: how much pamper, 2 feet 3 inches 
 wide, will cover it; and what is the value of it at gd. per yard? 
 
 Ex. 23. If 50 cows can be kept in a field 17 days: how long will the 
 same pasture feed 70 cows ? 
 
 Ex. 24. How many Venetian ducats, at 4s. 4d, each, must I take in 
 payment for 560 English crowns? 
 
 * As there are four sides to the room, add the length to the breadth, 
 and multiply by 2, which gives the length ot the sides : then say, as the 
 height of rhe room is to the length of the sides found, so is the breadth 
 of the cloth to the quantity used. The value of the cloth is found after 
 wards by the Rule of Three Direct, 
 
10-2 
 
 THE DOUBLE RULE OF THREE. 
 
 The Double Rule of Three teaches, from five given num- 
 bers to find a sixth. Three of the numbers contain the 
 suppositions, and the remaining- two are terms of demand. 
 
 RULE (1.) Put the terms of supposition one above ano- 
 ther in the first place, except that which is of the same na- 
 ture with the term sought, which put in the second place. 
 
 (2.) Place the ttirms of demand one above another in the 
 third place, in the same order as the terms of the supposi- 
 tion were put in the first place, 
 
 (3.) The first and third term in every row will be of the 
 same nature, and must be reduced to one denomination ; 
 and the middle term must be brought to the lowest deno- 
 itioii mentioned. 
 
 (I.) Examine each stating separatelv, iisin^- the middle 
 term as common to both, in order to know if the propor- 
 tion be direct or iur< rse. When it is direct mark th j iiist 
 teim with an asterisk, and when it is incew, mark the third 
 term with an asterisk. 
 
 (5.) Multiply the numbers together which are marked 
 for a -divisor, and those which are not marked for a dividend, 
 and the quotient will be the answer. 
 
 Ex. 1. If 12 persons spend .100 in 4 months: how 
 much will ?32 persons expend in 8 months ? 
 
 persons. . persons. The terms of supposition are, that 12 
 
 *12 : 160 : : 3? persons spend, in 4 months, 3 (30/. ; the 12 
 
 months. months. and 4 are therefore the first terras; .and as 
 
 * 4 : : : . 8 the answer will be in money, the 1GO/. is 
 
 or, the middle term. The terms of demand 
 
 1-2X4 : 160 : : 3Q X 8 are, How much a 2 persons will expend in 
 
 8 months ; these are accordingly the third 
 
 terms It is evident, from inspection, that both the .stating* in this ex- 
 ample, are in direct proportion, because the fourth terms will be greater 
 
DOUBLE RULE OF THR&H. 103 
 
 (.him the second, thatls, 32 persons will expend more than 12, am-l 
 months expenditure will be greater than 4 months. * 
 
 32 X 8 X 160 
 
 = 8532. 6,9. g^. 
 
 12 X 4 
 
 Ex. 2. If a garrison of 600 men have provisions for 5 weeks, allow 
 ing each man 12 ounces per day : how many men can be maintained 
 10 weeks by the same quantity, if each man is limited to 8 ounces 
 a day ? 
 
 weeks, men. weeks. In this example the statings are inverse, 
 
 5 : 600 :: 10* for in theirs/, if the same quantity of pro*- 
 
 oz. oz. visions is to serve 10 weeks, there must be 
 
 12 : . : 8* a smaller number of men : in the second, 
 
 or, when each man's proportion is reduced 
 
 5 X 12 : 600 : : 10 X 8 from 12 to 8 ounces the same provisions 
 
 will maintain a greater number, 
 
 5 X 32 X 600 
 
 . ~450 men, the Answer. 
 
 10 X 8 
 
 Ex. 3. If 15 pecks of wheat will last a family of 9 persons 22 days: 
 m how many days will six persons consume 20 pecks ? 
 pecks, days. pecks. In the first stating of this example, the 
 *15 : 22 : : 20 proportion is diiecf, because a greater quan- 
 peispns. - persons, tity of wheat will last a greater number or 
 
 9 : ; : 6* days. In the second stating the proportion 
 
 or, is inverse t because a smaller number of peo- 
 
 9 X 22 X 20 pie will require more days to eat the same 
 
 i ' X "is"'""" 14 " a y s * quantity of wheat. The divisors are there- 
 fore 1 5 and 6, and the answer is 44 days. 
 
 Ex. 4. If 6 pioneers can dig a ditch 34 yards long in lo days : how 
 many yards may be dug by 20 men in 15 days ? 
 
 Ex. 5. If 1050 soldiers consume 250 quarters of corn in 6 months : 
 how many soldiers will 960 quarters serve 4 months ? 
 
 * Examples in .the Double Rule of Three may be worked by two 
 statements in the Single method ; and this will be a good method of 
 proving the truth of the sums. Thus, in the foregoing example, I say, 
 
 men. $ (? men. ? <? 
 
 ; lSt Statement, 12 : 160 O O : : 32 : 426 13 4 
 
 months . months. 
 
 Qd Statement, 4 : 426 13 4 : : 8 : 853 6 8 
 
 Thus the fourth term in the proportion, found by the first stating, 
 becomes the middle term of the second stating. 
 
DOUBLE RULE OF THREE. 
 
 Ex. 6. If a cask of beer last 8 persons 14 days: how many cask 
 serve 2 persons 365 days ? 
 
 Ex. 7- K 10 men in six weeks earn got. : how many weeks must i 5 
 men work to earn 150/. ? 
 
 Ex. 8. Suppose I walk 66 miles in 4 days, of eight hours each day : 
 how many days, of 14 hours each, shall I be in going from London to 
 York, or 196 miles ? 
 
 Ex. 9. If 3 boats take GOOO herrings in 8 days : how long will 600 
 boats be in taking 20,000 barrels, each containing 700 herrings? f 
 
 Ex. 10. If, against a general mourning, 6 tailors can make 10 suits 
 of clothes in 4 days : how many suits can Goo men make in the 7 days 
 which occur before the mourning is wanted ? 
 
 Ex. 11. If 12 mantua makers can make 27 mourning dresses in 4 
 days : how many persons would be required to make 1 89 dresses in 3 
 days ? 
 
 Ex. 12. If 3000 copies of a History of America, each containing 11 
 sheets, require 66 reams of paper : how much paper will 5000 take, if 
 the work be extended to 12 sheets? 
 
 . Ex.13. As 12 inches in length, 12 in breadth, and 12 in thickness, 
 make a solid foot : what length of plank, which is 7 inches broad and 
 3 inches thick, will make the same ? f 
 
 Ex. 14. If 450 tiles, each 12- inches square, will pave my cellar : how 
 many tiles must 1 have, if the tiles are 9 inches long and 8 broad? 
 
 Ex.15. If the expense of 3 persons on a tour for five months be 123 /.8s.: 
 what will 2 persons spend in 9 months ? 
 
 Ex. 16. If 12 ounces of wool make 2 yards of very fine cloth, 6 
 quarters wide : how much wool would be required to 150 yards, 4 
 quarters broad? 
 
 Ex. 17. If 300/. gain 9 per cent interest in a year, in what time will 
 900/. gain 150/. ? 
 
 Ex. 18. If an iron bar 4 feet long, 3 inches broad, and l inch thick, 
 weigh 36 Ibs. : how much w:l! a bar weigh that is 6 feet long, 4 inches 
 broad, and 2 inches thick ? J 
 
 NOTES. 
 
 * It is asserted, that this number of herrings have been caught in a 
 single season in Loch Fyne, a salt lake communicating with the sea. 
 The fact may be readily credited, when it is added, that at one single haul 
 1-2,000 makarel v/eie drawn on shore at Sidmouch, a few years since. 
 
 f The statements m this example are, 
 broad, long. broad. It is evident these are both inverse pro- 
 
 12 : 12 :: 1* portions, because 7 in breadth will require 
 
 thick. thick, more length than 12 inches to make an 
 
 12 : :: 3* equal^surface. The same may be said of 
 
 the -3 inches in thickness. 
 
 The statements are direct in this example . 
 ifaus, * 4 : 36 : t 6 \ 36 X 6 X 4 X 2 ' 
 
 *a x i* 4 x fo 4x ax ir- = O5lb " 
 
MISCELLANEOUS QUESTIONS. 105 
 
 Miscellaneous Questions on all the foregoing 
 Rules. 
 
 Ex. i. What three numbers are those, the first of which is 105* 
 the second |ds of the first, and the third 67 less than the first and 
 second together ?* 
 
 Ex. 2. A gentleman left his eldest daughter 1000 guineas more than 
 the youngest, - and to three other daughters he left 7000 guineas between 
 them, which was equal to the sum left to the youngest and eldest to- 
 gether : what was each child's fortune ? 
 
 Ex. 3. What is the difference in value between five times five and 
 twenty guineas, and five times twenty-five guineas ? 
 
 Ex. -i. What was the value of a prize taken by 25 sailors, besides 
 officers, so that each sailor received -ig/. gs. Qd., and the officers re- 
 ceived as much as the sailors ? 
 
 Ex. 5. A prize valued at 1 3,177?. 10$., after the officers have had 
 their share, is to be divided among 5-25 sailors : what would each man 
 have to take ? 
 
 Ex. 6. What is a fourth proportional to the numbers 6, 9, and 
 
 24 ?f 
 
 Ex. 7. What is the value of 4 packs of cloth, each pack containing 
 4 parcels, each parcel 10 pieces, and each piece 26 yards, at the rate 
 of 2/, 85. for 3 yards ? 
 
 Ex. 8. How many yards of paper, 3 quarters wide, will be sufficient 
 for a room 48 yards round, and four yards high: and what is the value 
 of the paper, at the rate of 1 ss. per piece of 24 yards ? 
 
 Ez. 9. If loo/, gain 5t. in J2months, what will 7*Z. gain in nine 
 months ? 
 
 Ex. 10. If 48 cannon crmsume, in 3 days, 288 barrels of powder, 
 how much will be spent in 15 days, when 144 cannon are 10 be sup- 
 plied? 
 
 Ex. ll. Fifteen people joined to purchase a lottery ticket, for wlrch 
 they gave three shillings less than eighteen guineas: if it came up a 
 prize of 30 3 000 guineas, what did each man receive, and what was his 
 gain ? 
 
 Ex. 12. A tobacconist bought two parcels of tobacco, which 
 weighed 9 cwt. 2 qrs., for a hundred guineas, the difference of the 
 
 NOTES. 
 * To find |ds of any number, multiply the number by two, and 
 
 divide by three: thus, |ds of a guinea is - - 14 -hillings. 
 
 3 
 
 'f A fourth proportional is found, as is evident, rule, p. 92, by mul- 
 tiplying the second and third numbers together, and dividing by the 
 first,; thus a fourth proportional to the iiumb^r b, 12, and iC, i 
 
 12 X 16 
 
 ~ 24, or 8 : 12 
 
106 MISCELLANEOUS QUESTIONS. 
 
 parcels in weight was 3 qrs. 12lb., and in value eight guineas : what 
 was their weight and values ? 
 
 Ex. 13. The clothing of lOOcharity children came to -21 1 /., of which 
 13 5/. was expended on boys: \\hat was paid for the 40 giiis, -and 
 how much did the cloaths of each child cost ? 
 
 Ex. 14. A great grazier left to his fou f sons 220 oxen and 1200 
 sheep : I demand the value of each son's legacy, supposing the oxen 
 worth l 8 guineas each, and the sheep 39 shillings each ? 
 
 Ex. 15. What number is that which, multiplied by 3S4, will give a 
 
 product of ;3, 010, '24 S ? 
 
 Ex. 16. What is gained by the sale of 436 yards of broad cloth, that 
 was bought at the rate of 5 yards for 4l. ls. y and sold at the rate of 
 61. 45. for 6 yards ? 
 
 Ex. 17, If <} printers can set up the New Testament in 22 days, i-n 
 what time could it be done if 1 3 were employed ? 
 
 Ex. 1 s. If 8 pressmen will, on the average, earn 14.1. 8s. in six days, 
 how much can 1 6 men earn m -27 days ? 
 
 Ex. 19. When the quotient is JOSJ, and the divisor 533, what is the 
 dividend, if there be ^.remainder of 79 ? 
 
 Ex. 20. There are aboiU SOO new books and pamphlets published 
 cveiy year in London, the value for one copy of each work is estimated 
 *t Q.JO/. : what is the a\erae value of each book ? 
 
 .'1. If each of the soo publications con rain on an averap 
 n.s, and of each there be 12.">0 copic^ printed, I demand how much 
 paper is used in the business, and its v:iiue, allowing to each ream of 
 paper 500 .-hr* ts, and the price of it at 2 .><?. tfd. per ream ?* 
 
 Ex, 2'2. Th-' silk mill ?.t Derby winds off 73, 726 yards of silk every 
 time the great whed goes round, which is thrice in a minute: how 
 many yards will it wind in - a year, allowing that it works -every day, 
 except Sundays, 15 'hours, and how many skeins will be made, sup- 
 posing 900 yards gc to the skein? 
 
 x. ->;i. iu the partition of some -waste l^ndivin the west of England, 
 A hud Si) 1 acres, B J0~ acres, C lioacr. 2 r. 12 per., D 1,5 acres, and 
 E ;}o aci.-o-r. 12 per., but these, taken torre-ther, were but one-fifth of 
 the whole : how many acres were divided, and what was the value of 
 the whole, supj'osing each acre worth'-* ~>l. QS. 6(2. ? 
 
 Ex. *24. Aii island in the West Indies contains -1C parishes, and ev ry 
 '.parish 76 hoascs, and each house at the rate of 5' ; white persons ; bo 
 'sides these ; the re were 65 negroes to each of 54 plantations : how many 
 .people were there on the -whole island ? 
 
 Ex. TJ. r >. In the ^liib mentioned in the Spectator (No. o), there were 
 ;a, r > persons, weighing together 3 tons: how many pounds, ounces, and 
 'drams, Avoirdupois, did each man weigh? 
 
 Ex. 26. The British possession im Hindostan. contain 212,400 square 
 miles, and the population is estimated at fourteen .millions.: "how many 
 rr.habitunts -are there .to a square -mile.? 
 
 'NOTE. 
 
 * This, arid the preceding question, do-not include the newedrticaac 
 mf books. 
 
M'i S C K L L A X K ( ) [J S Q T E S T IONS. 107 
 
 Ex. 27. If lb. of tea cost 3/. 7*. 7<J., what is the worth of four 
 chests, each \\eighing one hundred and a halt ? 
 
 Ex. 2tf. What shall I give for a farm containing 256 acres and 
 a half, for which 1 am to pay at the rate of 95 guineas for three 
 acres ? ' 
 
 Ex. 29. What will it cost a young man to come into a farm, for 
 the lease of which he is to pay 1000 guineas ; for 22 hordes he is to 
 pay at the rate of 18 guineas each; for ciops in the ground ,'354 /.; 
 for 210 bushels of wheat he, is to p-.*y i/. 105. per quarter ; the house- 
 hold furniture is appraised to him at 2ys guineas, and for farming 
 utensils of all kinds "he-is to:pay 196^.? 
 
 Ex.30. The revenue -collected in Hindostaii, by the British, is 
 reckoned at a, 400,000*'.., how much is that from each inhabitant, sup- 
 posing they amount to 14 millions? 
 
 Ex. 3] . The.number of nergoes in" Jamaica is estimated at -2 50, GOO, 
 ' .nil oi'Avhitos at 20,000, how many slaves are there to a single white 
 n-:an, and what do the planters reckon their property worth in the 
 article of slaves only, supposing each to be worth 93 guineas? 
 
 Ex.32. The population in the United States is estimated at six 
 millions and a half, and the number of slaves still existing in that 
 fice country is reckoned to be 697,697, how many free people are 
 there to one slave ? 
 
 Ex.33. The extent of China Proper. isxqua! to ,3 ,3fr7, 999 square 
 miles, and the population is estimated at 333,000,000, how many in- 
 habitants are there to a square nu'e ? 
 
 Ex. 31. In Spain each person pays 10 shillings to government for 
 protection; in France, under the old government, each paid 20.?. for 
 protection; and in England we pay full three guineas each for the 
 same advantages, how much is the revenue of the three governments, 
 supposing the population of Spairxfo be io millions; of France, at 
 the period referred to, 25 millions; and of .-England and Wales 
 9,343,173 ? 
 
 Ex. 35. The population of London, Westminster, and Southwark, 
 is 86i,S6.s, that of Paris 547,756, how much does the population pf 
 London exrced that of Paris ? 
 
 Ex. 30. How many minutes and seconds .-have elapsed since the 
 birth of Christ, or 1-00-8 years ? 
 
 Ex. 3;. How long would it require -to count five hundred millions 
 sterling, supposing a person were to.isckon 150/. in a minute, and 
 were to be employed 10 hours each day, and six days a week, till 
 he had finished the job ? 
 
 Ex. 38. How many barley-corns will reach round the earth, sup- 
 posing the length to be 25, 200 miles ? 
 
 Ex. 39. How many seven-shilling pieces are there in a thousand 
 pounds? 
 
 Ex. 40. A French franc is worth lod., how many francs are there 
 in loo/. ? 
 
 Ex. 41. If 8 men can mow 18 acres in 4 days, how many men 
 will be required to mow 50 acres in six days.? 
 JEx. 42. A balloon has moved at the rate of 6492 feet in a minute^ 
 
108 MISCELLANEOUS QUESTIONS. 
 
 how long would it have been sailing round the earth at the same rate, 
 supposing the circumference of the earth to be 25,200 miles ? 
 
 Ex. 43. How much ottener will the small wheel of a coach turn 
 than the large one, between London and Bristol, or 120 miles, if the 
 former be 10 feet 8 inches in circumference, and the latter 18 feet 4 
 inches ? 
 
 Ex. 44. If my income be 250/. per annum, and I have foolishly ex- 
 pended is*, per day, how much shall I be in debt at the year's end, 
 and what may I expend per day the following year, so as to have ten 
 guineas in hand at the conclusion of it ? 
 
 Ex. 45. It is said the impositions of hackney-coachmen, by over- 
 charges, are equal to one-fourth of what they earn : now, if they earn 
 each on an average 185. per day, and there be 1100 employed 313 days 
 in a year, I demand the amount of their overcharges in a year ? 
 
 Ex. 46. There were at Vauxhail gardens on the Prince of Wales'a 
 birth-day, 1805, 10,059 persons : the admission money was 3s. each ; 
 now, supposing each person to spend 3s. more, the half of which was 
 profit to the proprietor, what would he clear by the night, allowing that 
 the incidental expenses were 250/. ? 
 
 Ex. 47. How much lime, in the course of 30 years, does a person 
 gain, who rises at five o'clock in the morning, over another who lies 
 till eig.ht, supposing both go to bed at half past ten at night, and sup- 
 posing the year to consist ot 365]- (lays ? 
 
 Ex.48. If S\ yards of clo-th will make a shirt, how much, of the 
 same stuff will be wanted to make two shirts for each man pf a regi- 
 ment, consisting of 855 men ? 
 
 Ex. 49. In November, 1800, 276,3;J4 five-pound bank notes were 
 issued; in December 2,626,700; and in the January following 
 2,769,160: what was the nominal value of the notes issued in these 
 three months ; and what was the cost of white rags, from which they 
 were made, supposing each ounce of rag might be manufactured into 
 twenty five -pound notes, and the rags to be worth 8<L per Ib. ? 
 
 Ex. 50. Two persons d part from London for York on the same day; 
 the one walks 19 miles a day, the other only 15^ miles: how far dis- 
 tant will they be from one another after ten days travelling, arid when 
 will each get to York, which is 1 Q7 miles from London? 
 
 Ex, 61. The population of the world is estimated at a thousand mil- 
 lions of human beings : if the face of the earth be re-peopled every 33 
 years, how many persons are born and die in a year, week, day, and 
 minute ?* 
 
 Ex. 52. The field opposite my house will serve 50 cows forty days : 
 how long will it afford 220 with equal feed ? 
 
 Ex. 53. If 10 persons expend ^50/. in 4 months : how much ought 
 3 persons to expend in 12 months ? 
 
 NOTE. 
 
 * The number of persons that are born and die in a year 
 1,000,000,000 divided by 33, 
 
109 
 
 FRACTIONS. 
 
 A Fraction is the part, or parts of a whole, or of any 
 whole quantity expressed by unity, and is expressed by two 
 figures, tvith a line drawn between them, as \, f, . 
 
 The upper figure of a fraction is called the numerator, 
 and the under one the denominator. 
 
 The denominator shews how many parts the unit is di- 
 vided into, and the numerator how many of these parts are 
 to be taken; thus ,or three-fourths, shews that -the whole 
 is divided into four parts, and that three of those pa-rts are 
 to be taken: and A, or five-eighths, shew that the whole is 
 divided into eight paits, and that, five of these parts are 
 taken. 
 
 There are four sorts of fractions, simple and compound, 
 proper and improper. 
 
 A simple fraction has only one numerator and deno-mi- 
 nator, as -}-, or J-. 
 
 A cow pound fraction consists of two or more parts, and 
 is known by the word of placed between them, as J of 6; 
 or f off of T V 
 
 A proper fraction is, when the numerator is less than the 
 denominator. 
 
 An improper fraction is, when the numerator is equal to, 
 or greater than, the denominator. 
 
 A mixed number is formed from an integer and a frac- 
 tion joined together, as 8f. 
 
 A complex fraction is one that has a fraction or a mixed 
 number for its numerator, or denominator, or 
 
SEDUCTION OF FRACTIONS. 
 
 THE method of changing fractions from one form to ano- 
 ther, without altering their value, is called Reduction ; 
 -J-J =4J -- y'V F =z i' Reduction serves to prepare frac- 
 tions for Addition, Subtraction, Multiplication, .;md Divi- 
 sion. 
 
 CASE I. To reduce fractions to their least term*. 
 
 RULE. Divide the terms of the gicen fraction by any 
 number which will divide them both without a remainder, 
 and the quotients will be the terms of a new fraction, equal 
 iti value to the given fraction. Repeat the operation, till 
 the terms of (he, reduced fraction aie divisible only by 1. 
 
 Kx. Reduce -} J 5 to ^ ts lowest terms. 
 
 \:>i3f> 392 >3y2 49 \49 7 
 
 8 ) zz and, 8 ) - ,and7 . 
 
 /35h4 448 ' 4 4 8 50 /56" * 
 
 Reduce the following fractions to their lowest terms. 
 
 Kx. i. 
 
 Ex. 2. 
 
 Ex. 3. 
 
 Ex. 4. 
 
 Ex. 5. Ex. 6. 
 
 Ex.7. 
 
 Ex. 8, 
 
 32 
 
 208 
 
 136 
 
 156 
 
 360 384 
 
 5184 
 
 2 4 7 5 
 
 120' 
 
 681:' 
 
 ~72 ' 
 
 336* 
 
 70S 5 a -us 2* 
 
 -6012* 
 
 288.0* 
 
 
 
 Ex. 
 
 9. 
 
 Ex.io 
 
 
 
 
 
 40 
 
 30 
 
 3105 
 
 
 
 Reduce --'to its lowest terms. 
 
 3 X 4 X 7 X 
 
 In all compound fractions, if there be 
 X 3 X 4 X 5 10 5 the same figure in the denominator as 
 
 ~"T7TTr7~ 7ft ^a' there is in the numerator, they mav he 
 JX^A/AO oo zo . ' j * 
 
 omitted in the work ; thus \ve have no 
 
 further concern with the 3 and 4, be- 
 cause they occur in both terms of the fraction. 
 
 3X8X9X2 
 
 Reduce- to the lowest terms. 
 
 4 x a X 14 X 36 
 
 3X8XQ.X2_ 3X2X4X0X2 2 I 
 
 4X3X14X536 4X3X2X7X4X9 28 14 
 
 3X4X15X4 10 X 27 X 30 X 32. 
 
 **' T 11 -i ^Y 1 ^ . 
 
 5 X 6 X 24 X 3" " 15 X 9 X 55 X ac. 
 
FRACTIONS, HI 
 
 CASE II, To find the greatest common measure of a 
 i raction. 
 
 UIM.K. Divide the greater icrm hi/ the /r.v.?, and this di- 
 sor hi/ the j'cmauider, then the last divisor Mill he the 
 reatest common measure of both terms of the fraction. * 
 
 Ex. What is the greatest common measure of the frac- 
 
 9l'S)l99S(2 Here 54 being the last divisor, it is 
 
 1 8 a 6 the greatest common measure of both 
 
 "Tii^Qi S 5 terms in the "fraction ; and to reduce 
 
 "''Jjio'" the said fraction to the lowest terms, 
 
 the-numerator and denominator are to 
 
 J0b 162(1 be divided by 54, the common mea- 
 
 sure, thus : 
 54)208(2 r ^ 918 _ 17 
 
 108 /19<)S~~37* 
 
 What is the greatest common measure of the following fractions ? 
 
 K.X. 1. 
 
 Ex. -2. 
 
 Ex.3. 
 
 Ex.4. 
 
 Ex. 5. 
 
 270 
 
 108 U 
 
 720 
 
 3:j 6 
 
 3108 
 
 ; ,ioo ' 
 
 1-2-2 i' 
 
 1786* 
 
 868* 
 
 355-2* 
 
 Ex, 
 
 S). 
 
 Ex. 7. 
 
 Ex. 
 
 8. 
 
 4125 
 1-^320* dSOO* 
 
 ( 'ASE III. To reduce an improper fraction to an < 
 
 whole, or mixed number. 
 RTJLK. Divide the numerator />>/ the denominator , and 
 the quotient will be the integer > or mixed number required: 
 thus V 4J, and 4 / :n5. 
 
 Reduce the "following improper fractions to their proper terms. 
 
 Ex. i. Ex. 2. Ex. 3. Ex, -i. Ex. 5. Ex. 6. Ex. 7. 
 2Q :,; 69 7^> 96 101 850 
 
 -8* 7 ' 8 ' 12* 16' 13* 24 " 
 
 Ex. 8. Ex. 9-. Ex. 10. 
 
 97^4 ..564J 889 
 
 556 450 3 
 
 * A number, ending with an even figure, or a cypher, ~.an be divided 
 by '2, without a remainder. 
 
 A number ending with , r , or 0, is divisible by 5. 
 
 if a fraction has a cypher, or cyphers, at the right-han.d of both its 
 -, it may be abbreYiated by cutting off the cyphers. 
 
112 REDUCTION OF 
 
 CASE IV. To reduce a mixed number to an equivalent im- 
 proper fraction. 
 
 RULE. Multiply the whole number by the denominator of 
 the fraction, to the product add the numerator, for a new 
 numerator, under which place the denominator; 
 
 Thus 4| = 5 8 5 , and 2961 = 8 | 9 . 
 
 Reduce the following mixed numbers to their equivalent improper 
 fractions. 
 
 Ex. 1. 3*. Ex, 2. 8*. Ex. 3. 6^. Ex. 4. 7^. 
 
 Ex. 5. 16 5 7 . Ex. 6. 435U. Ex. 7. 37S. Ex. 8. 499 T V, 
 
 Ex. 9. 5-JJL. Ex. 10, 6/if * 
 
 CASE V. To reduce a compound fraction to an equivalent 
 simple one. 
 
 RULE (1). If any of the proposed quantities be integers, 
 cr mixed numbers, reduce them to their proper terms. 
 
 (2). Multiply all the numerators together for anew nu~ 
 merator, and all the denominators for a new denominator, 
 <wd then reduce the fraction to its lowest terms. 
 Reduce of 3 of 7 to a simple fraction. 
 4 3 47__2 X 2 X 3 X 47 __^ 
 Operation ~ x f X y TxT X~2~xT~"~7' t 
 
 The fraction "^ is already in its lowest teims, because no figure higher 
 than unit will divide both terms of the fraction without a remainder. 
 
 Ex. 1. I of of 5 of |. Ex. 2, | of 4 of 5* . 
 
 Ex. 3. T * T of 8 of 7f of 12. Ex. 4. .'_. of ^ of 12 of 9-J. 
 
 Ex. 5. VG of ]0 of V 2 of 18*- Ex. (). j of j. of f of T %. 
 
 * Under this ease a whole number may be reduced to an equivalent 
 fraction, having any given denominator : thus, 5 }. For a whole 
 number maybe converted into a fraction, by placing under it an unit ; 
 and therefore it may be reduced to an improper fraction with any given 
 denominator, by multiplying it by the denominator, and the product will 
 be the numerator required. Thus, to reduce 1 5 to an equivalent frao 
 
 15X6 00 
 
 ticn, having the denominator 6, 1 5ay, :n . 
 
 6 6 
 
 f Here, as has already been observed, 2 and 3 being found in the nu- 
 merator and denominator, ure dropped in the work. 
 
FRACTIONS. 115 
 
 CASE VI. To reduce fractions of different denominators to 
 others of equal value, having- a common denominator. 
 
 RULE. (1). Multiply each numerator into all the deno- 
 minators 9 except its own, for a new numerator, and all ike 
 denominators for a common denominator. 
 
 Reduce J, J, 3f , and 3, to a common denominator. 
 Operation, f, , y, f. 
 
 New riu- I multiply the numerator of the first 
 
 tncrators. fraction 3, by 9, and 3, and l : and 
 
 3X9X3Xld"81 then 7, the numerator of the second 
 
 7X5X3Xlrz 105 fraction, by 5, and 3, and 1, and so of 
 
 HX5X9X1HI 495 the others ; and for a new denominator. 
 
 3X5X9X3 405 I multiply the 5, and 9, and 3, and l, 
 
 New denom. together. 
 5 X 9 X 3 X 1 135 
 
 Answer, -&V, *H> H*> If*- 
 Ex. 1. Reduce -*-, -J, and |, to a common denominator.* 
 
 2. Reduce |- , |-, f , and -J-, to a common denominator. 
 
 3. Reduce | , , f , and 3, to a common denominator.! 
 
 4. Reduce T \, y 1 ^, 8, and 11|, to a common denominator. 
 
 5. Reduce T 3 T , , f , 4, and 2}, to a common denominator. 
 
 6. Reduce |, J, |, and ^, to a common denominator. 
 
 7. Reduce |, J, J, |-, and 7, to a common denominator. 
 
 8. Reduce T 4 > |-, y, and T 9 ^, to a common denominator. 
 
 9. Reduce ^, | , y , and T 4 T of 9, to a common denominator. 
 
 (2). To find the least common denominator. 
 
 Set down the denominators of the given fractions in a 
 line, and divide as many of them as possible, by any number 
 which wty leave no remainder, and set down the quotients, 
 
 * If the products of the denominators are divided by their greatest 
 common measure, the answer will be in the least common denomina- 
 tor, as in this example. See also next rule. 
 
 New numerators. Here 4 being common to each new numera- 
 
 2X4X4 tor, and to the denominator, may be omitted, 
 
 5X3X4 and tjie answer will be 
 3X3X4 
 
 Denominator. , IA, ^ ~ to the given fractions |, J, and . 
 
 3X4X4 
 |- In the work there is no need to put down the units as multipliers, 
 
1 14- REDUCTION OF 
 
 find the undivided numbers below. Repeat the operation 
 till there be no two numbers which can be divided without 
 a remainder. Then the product of all the divisors, and 
 the quotients in the last lines will give the least common de- 
 nominator. Divide this least common denominator by each 
 of the given denominators separately, and multiply the 
 quotients by their several numerators, their products will 
 be the new numerators. 
 
 Reduce , J, y , -}-, to the least common denominator. 
 
 3)5, g, 3, i 
 
 then 3X5X3X1X1 45, is the common deno 
 
 5,3, 1,1 
 
 minator, and 45 divided by the given denominators, 6, Q, 3, l, give 
 9) 5, 15, 45 ; these multiplied by the given numerators, give 37, 35, 
 1C5, 135, for new numerators, and the fractions will stand |*-, ^, 'A 5 * 
 
 Reduce --, -J., |- , , and -J-, to the least common denomi- 
 nator. 
 
 3)3, 4, 5, 6, 8 The least denominator is, accordingly, 
 
 4)774717^ 3X4X2X5- 120 ; 
 
 1-20 ~ o, 4, 5, 6, 8 r= 40, 30, 24, '20, 15. 
 
 2)1, 1, 5, 2, 4 -10 X 2 ; 30 X 3 ; 24 X 2; 20 X 4 ; 15 X 3, 
 
 l, ], 5, i, l for new numeiators ; therefore the fractions re- 
 
 required are T $, v v;,, 
 
 The learner may now reduce to the least common denominator, 
 the ten examples given under the first part of the Rule.* 
 
 NOTE. 
 
 * To find the least common multiple of two or more given numbers. 
 
 RULE. Find the greatest common measure, ly inspei-tinn 9 O/TWO of 
 the numbers t and divide, the product of them ly the common measure 
 so found ; multiply this quotient ly the third numL-cr y d7id divide the 
 product ly the common measure of the multiplier and multiplicand* 
 and so proceed to the last number ; the last quotient u-HL le the least 
 commo7i multiple. 
 
 Ex. Find the least number that can be divided by 2, 3, 4, 5, 6, and 
 7, without remainders. 
 2 X 3_ 
 
 l The greatest common measure of 2 end 3 is i, 
 
 6 X 4 _ ( and 2 ^X 3, divided by l is C : the greatest com- 
 
 ^ 12 mon measure of the 6 just found, and 4, the next 
 
 ]0 ^ ^ given number, is 2, and 6 X 4 divided by 2 rz 12 : 
 
 Co the greatest common measure of 12 so found, and 
 
 5, the next figure, is l ; and 60 divided by l n: 60, 
 ^._^ 6 -~ G0 and so on. It is found that 4-20 is the least num- 
 
 r> ber that can be divided by 2, 3, 4, 5, 6, and 7. 
 
 Xo x 7 
 
 Ex. i. What 
 
FK ACTIONS. I 1J 
 
 C/'ASE VII. To reduce a fraction of one denomination to the 
 
 fraction of another denomination of equal value. 
 RULE. (1). When it is from the less to a greater denomi- 
 nation, " Multiply the denominator by all the denomina- 
 tions from that given to the and sought." 
 
 / o o 
 
 Thus, to reduce ^ of a penny to a fraction- t>f a pound, the answer 
 
 will be rz 
 
 4 X 1-2 X 20 9GO 
 
 (2). When it is from a greater to a lets denomination, 
 " Multiply the numerator by all the denominations, from 
 that given to the one song /it." 
 
 Thus, to reduce f of a pound to the fraction of a farthing, 
 
 C X 20 X 12 X 4 5760 
 
 _ .__. 
 
 Ex. 1. Reduce 2 J 9 of a tarthing to the fraction of a pound. 
 
 2. Reduce 
 
 3. Reduce 
 
 4. Reduce 
 
 5. Reduce 
 (>. Reduce 
 7. Reduce 
 
 ^8. Reduce 
 9. Reduce 
 
 10. Reduce 
 
 11. Reduce 
 
 12. Reduce 
 
 of a penny to the fraction of a shilling-. 
 
 of a pound to the fraction of a farthing-. 
 V of a pound to the fraction of a penny. 
 
 J of a pound to the traction of a farthing-. 
 3 shilling's to the fraction of a pound. 
 
 of a d\vt. to the fraction of a Ib. Troy. 
 
 of a cwt. to the fraction of an ounce. 
 
 of a week to the fraction of an hour. 
 
 of a mile to the fraction of a yard. 
 
 of a pipe to the fraction of a gall on.. 
 
 a pint to the fraction of a had. of ale. 
 
 CASE VIII. To find the value of u fraction in numbers of 
 inferior denomination. 
 
 RULE. Multiply the integer, or ifs valve in the next 
 tower denomination, by the numerator, and divide by the 
 denominator : 
 
 Thus, the value of | of a pound is equal to ~ 12 shillings, 
 
 2 y j *2 
 
 and | of a shilling is equal to ~ 8 pence. 
 
 Ex. 1. What is the least number that can be divided by 4, 6, and 
 JO. without a remainder ? 
 
 t2. \Vtiat is the least number that can be divided by 3, 5, 8, ajid 10, 
 without a remainder ? 
 
 3. What is the least number that can be divided, without a remain- 
 tier, by a, -i, 8, ]0, and 16 ? 
 
1 16 ADDITION OF 
 
 Ex. 1. What is the value of -|- of a pound ?* 
 
 2. What is the value of f of a shilling ? 
 
 3. What is the value of 7% of half a crown ? 
 
 4. What is the value of ^ of a Ib. Troy ? 
 
 5. What is the value of T P T of a cvvt. ? 
 0. What is the value of f of a mile ? 
 
 7. What is the value of f of a barrel of beer ? 
 
 8. What is the value of T % of a chaldron of coals ? 
 i). What is the value of | of a hogshead of wine ? 
 
 CASE IX. To reduce a complex fraction to an equivalent 
 simple traction. 
 
 RULE. If the numerator, or denominator, or both, be 
 whole or mixed numbers, reduce them to improper frac- 
 tions ; and multiply the denominator of the lower fraction 
 into the numerator of the upper, for anew numerator, and 
 the denominator of the upper fraction into the numerator 
 of the lower, for a new denominator. 
 
 Thus - - -L - 4 X 8 - *L And ^ - A - _L 
 
 ' f- . "" 7~>n ~~ T' IT " " ~i~ ~~ 50 * 
 
 5| V 47 ,9 $ 63 
 
 And ~ - - ~ . And = = . And 
 
 8.4 64 3* V 23 
 
 51 s_j 147 
 
 again - -~- iz - - . No other varieties can happen. 
 
 3J 
 
 Ex. 1. Reduce to a simple fraction. 
 
 2. Reduce to a simple fraction. 
 
 f 
 
 3. Reduce - to a simple fraction. 
 
 4. Reduce -^ to a simple fraction. 
 
 53 
 
 NOTE. 
 
 * Where there is a remainder we proceed as in Compound Division ; 
 
 thus 4 of a pound T: ~ rr Us. lief. I. See p. 83. 
 
 9 9 
 
FRACTIONS. 117 
 
 Ex. 5. Reduce to a simple fraction. 
 *^ 
 
 71 
 
 6. Reduce to a simple fraction. 
 
 02 
 
 j-jf 
 
 /^ 
 
 7. Reduce to a simple fraction. 
 
 8. Reduce ^~ to a simple fraction. 
 
 ADDITION OF FRACTIONS. 
 
 RULE. Reduce mixed numbers to improper fractions, 
 and compound or complex fractions to simple oncs^ and 
 bring them all to their least common denominator. Add 
 all the nemerators together, and ivrite the sum over the 
 common denominator. 
 
 Ex. Add f, -|, 5|, and J together; which is thus per- 
 formed: |, |, v, >. 
 
 3X3X2X4 72 ^ _.,, 
 
 2 x 5 x 2 x 4 so ) Therefore -^ + ^ + -ffg- 
 
 11 X. 5 X 3 X 4 = 660 ^ -f T \o = fU = 7 T *T = 7 & 9 
 
 iX5X3xa= 30 / which is the answer. 
 
 5X3X2X4ZT 120 * 
 
 This may be performed by bringing the given fractions to the least 
 common denominator: See p. 213. 
 
 Thus, J, f , y , J, then "---:, and the new deno- 
 5, 3, 1, 2 
 
 minator = 60; the fractions will be ^f 4- -J--J- 4- Vo + is 
 4_?_i 7_\ 
 
 " o o ' ti "o 
 
 Ex. 1. Add |-, -J, and -- together. 
 
 2. Add |, |, and f together. 
 
 3. What is th^ sura of 2, y, and 4^ ? 
 ' 4. Axld toother 3^ 4^', and |-.* 
 
 NOTE; 
 
 * When there are two or more mixed number?, a? in the 4th exam- 
 ple, the fraaions may be first added, and join these ro the sum of the 
 whole numbers thus, 1 add % ~, ;ind ' tr.-cther which are ~ * '. *-^ 
 siul the answer 3 4-4-1- i^ 
 
118 SUBTRACTION OF 
 
 Ex. 5. Add -*-, j, -23, and 5| together. 
 (). What is the sum of 7f, 3*, and f ? 
 
 7. What is the sum of 2- of a guinea, -f- of a shilling, 
 and of a penny ?* 
 
 8. What is the sum of f of a pound, 4 of a shilling, 
 and T 7 T of a penny ? 
 
 9. What is the sum of i of a guinea, -f- of a shilling, 
 arid -jSj- of a penny ? 
 
 10. If I have of a coasting vessel, and purchase an- 
 other share of T 9 f , what part of her will belong to me ? 
 
 11. Add -*- of a yard, and | of a mile together. 
 
 12. What is the sum of $ of a yard, \ of a foot, and f 
 of an inch? 
 
 13. Add -I- of a Ib. troy to J- of an ounce. 
 
 14. What is the sum of I of a hhd. of beer, and | of a 
 barrel ? 
 
 15. Add of a chaldron to -5 of a bushel * 
 
 SUBTRACTION OF FRACTIONS. 
 
 RULE. Reduce the given fractions to the same denomi- 
 nator, as in Addition, iltcn subtract the lesser numerator 
 from tJic greater, and under the difference place the com* 
 won denominator. 
 
 Ex. Take jj- from - t -\- : and r " ;7 , from -J--J-. 
 :> X 9") 
 
 'i v i i I 45 3f> 9 ] 
 
 j * i- v Ti lere f ore ._ Answer. 
 
 ( 108 108 12 
 
 9 X 12j 
 
 * To add fractions of different integers, find their respective values 
 by Case VII,, and proceed as in Compound Addition : thus, 
 
 | of a guinea m zr O 9 
 
 3 X 12 
 
 I of a shilling zi ----- -z: o 
 
 8 
 5X4 
 
 of a penny n: - ~ o o 
 
 .0 
 
 .0 9 
 
il X If} 
 
 9 X 16 I 
 
 15 X IOJ 
 
 FRACTIONS 
 
 165 144 
 
 Therefore : 
 
 Ex. 1. From J take |. * 
 
 2. From -f take f . 
 
 3. From if take T V 
 
 4. From V s take -J-. 
 
 5. From 9| take 4f.* 
 
 6. From 12 i take f. of 17. 
 
 7. From of a shilling take ^ of a pound. 
 
 8. From I of a pound take T 7 T of a pound. 
 
 9. From 1 take T Vt 
 
 10. From 1 take f- of j-., 
 
 11. From 12 take A. J 
 
 12. From 10/. take f of a pound. 
 
 13. From of a pound take -/^ of a pound* 
 
 14. From |- of a pound take ^ of of a shilling. 
 
 15. From f of 6 Ib. avoirdupoise take f of 51b. 
 10. Subtract T \ of a ton from 8,} of a ton. 
 
 * In mixed numbers, the subtraction may frequently be performed 
 without reducing them to improper fractions. After the fractions are 
 brought to a common denominator, subtract the numerator of the lower 
 fraction from the common denominator; to the remainder add the 
 numerator of the upper fraction, and carry one to the lower whole 
 number : thus, 9f 4| rz 9| 4| rr 4|. 
 
 Here Jand being brought to a common denominator, as 
 3 X 8 ) ~ 24 28 G 7 9| 
 
 H v- i f ~ n: 5 therefore ./ 
 
 _i_ J 4 x ^ 8 SL 
 
 4X8 4-J 
 
 *f* T subtract a proper fraction from an unit : Subtract ike numera- 
 tor from the denominator ; the remainder leing placed over the deno- 
 minator, gives the atisiuer required: thus, take ^ from 1, answer r. 
 
 J To subtract a proper fraction from any whole number: Subtract 
 the. numerator fro*n the denominator , and the remainder ft laced over 
 th* denominator, gives the fraction wkiih is to be annexed to the 
 icho-'c number made less by i : thus, take | from 11, the answer 10|. 
 
 To subtract fractions jof different integers: Find their respective 
 calnes, ard proceed as in. C<K]>outtd Subtraction : See p. Q8. 
 From i of a pound, take >- of a shilling. 
 
 3 X 20 60 , 5 X 12 
 
 2z-- n: 6$, Sd. ; and -~7-v/. ; fherefote thr answer 
 
 9 17 8 
 
 will be 65. 0-W. 
 
120 
 
 MULTIPLICATION OF FRACTIONS. 
 
 RULE. Reduce mixed numbers to improper fractions , 
 and compound fractions to simple ones ; multiply all the 
 numerators together for anew numerator; and all the de- 
 nominatorsfor a common denominator. 
 
 Ex. Multiply 3, I, and of 8 together. 
 
 29 3 5 8 29 X 3 X 5 X 8* 29 X 5 145 
 
 , _ \/ __ \ _ \* ~ - 
 
 * 
 
 8 4 6 1 ~ 8X4X3X2 "4X2" 8 
 
 1Z. 18 1, the answer. 
 
 Ex. 1. Multiply. T " T by f ; and -J- by -yV 
 
 2. What is the product of , A, and 3 V? 
 
 3. What is the product of 57 by ^ T ? 
 
 4. What is the product of 7f multiplied by 35 ? 
 
 5. What is the product of -* , f , 12^ and f of 10 ? 
 
 0. \Vhat is the continued product of , -}, 5, and G ? 
 
 7. What is the product off of -, -$ of|J ? 
 
 8. What is the product off, |, f, A, jj, | and W ? 
 
 9. How many yards are there in 5J pieces of Irish, 
 oach containing 26| ? 
 
 10. How many pouads are tliere in 8* cheeses, ecich 
 containing 25 lb. 
 
 NOTE. 
 
 Fractions must be abbreviated, when it can be done, thus we 
 strike out 8 and 3, because they are found in the upper and under line, 
 Note. The learner will, perhaps, be struck with the difference be- 
 tween common multiplication and the multiplication of fractions. 
 By the former, the product of any two numbers above unir is greater 
 than either ; but in what is called multiplication of fractions, the 
 product is less than the numbers multiplied together ; thus X J rr -J-% 
 and i X ~ m |: and, in general, whatever part of unity the multiplici 
 >s, the product will be the same part of the multiplicand : thus, l is 
 one third of l, and I is one third part of J 5 : again, J X f ~ ": 
 here -J is three fourths of l, and J^, or , is three- fourths of two-thirds: 
 thus, 4 of 4 of a shillings e^ual of a shilling, or sixpence, the tiuth 
 of wlm-ij is f\ident. fc: jf Jfrfij|iil]i!..g is 8., and |- of Sd. is 6rf, 
 
12 i 
 
 DIVISION OF FRACTIONS. 
 
 RULE. Reduce the fractions, as in Multiplication; then 
 invert the divisor, and proceed as in Multiplication : thus, 
 -I- to be divided by -*. 
 
 A ^_ 3. .3. x 9 1 7. Jt 
 s ~ 9 ft * 5 is s* 
 
 Ex. Divide J of 4f by | of J. 
 
 3 23 3 1 3 X 23 3 3 X 23 4X7 
 
 ~- X r X or j- = X 
 
 8 5 7 48X5 4X7 4X2X5 3 
 
 ~ \-V =z 16 T V 5 the answer. 
 
 EXAMPLES. 
 
 Ex. 1. Divide -ff of 12 by -J. 
 
 2. Divide W of 8 by T \. 
 
 3. Divide V b y ri- 
 
 4. Divide | of 54 by f. 
 
 5. Divide f of 12 by 3|. 
 
 6. Divide 4 of 36 by 3|. 
 
 7. Divide 4 of 4 by of 2. 
 
 8. Divide llf by | off. 
 
 9. Divide |- of f of 5 by | of . 
 
 10. Divide ^ of f by T * T of 5. 
 
 11. What number multiplied by f will give 9 ?* 
 
 12. What part of 56 is T y of 3 ? 
 
 13. W r hat number multiplied by f of f of 15, will 
 produce of 4 ? 
 
 14. From 5 subtract I of f of 4, and dhide the re- 
 mainder by 4. 
 
 15. What is a person's share of a prize of . 20,000, 
 -J-ths of which is to be divided among 13 persons ? 
 
 NOTE. 
 
 * The answer to this will be the quotieot of 9 j divided by f . 
 
 G 
 
122 
 
 PRACTICE. 
 
 PRACTICE* is a method of finding the value of any 
 quantity of goods, from the price of an integer being given. 
 
 ALIQUOT PARTS of any number or quantity, are such 
 as will exactly divide it without leaving a remainder : thus 
 7 and 4 are aliquot parts of 28, 4 pence is an aliquot part 
 of a shilling, and 5 shillings is an aliquot part of a pound. 
 
 TABLES OF ALIQUOT PARTS. 
 
 Aliquot parts of a . Parts of a shilling. Parts of 3 pence. 
 
 3. tl. 
 
 d. 
 
 f- 
 
 10 ~ $ 
 
 6 = i 
 
 3 _L 
 
 6 8 = $ 
 
 4 = i 
 
 a ~~~ 6 
 
 ,50= i 
 
 3 = J 
 
 i = TV 
 
 40=} 
 
 2 x = t 
 
 
 3 4 = i 
 
 
 Parts of a penny. 
 
 2 6 =z i 
 
 I = T v 
 
 i* =' I' 
 
 2 = T V 
 
 
 i = J 
 
 18 = T V 
 
 Parts of sixpence. 
 
 
 1 4 = T v 
 
 * = -j- 
 
 
 13 = ^, 
 
 \ "Z^ T T T 
 
 10=^, 
 
 
 I. When the price is less than a penny. 
 
 RULE. Divide the quantity by the aliquot parts in a 
 penny, and the quotient by 12 and 20. 
 
 * PRACTICE has its name from its general use in business, as it 
 teaches the best and most compendious methods of answering almost 
 all questions that occur in trade and mercantile transactions, and is 
 to be preferred to Compound Multiplication, and to the Rule of 
 Three, whenever the first term is unity. 
 
PRACTICE. 123 
 
 Ex. What is the value of 7853 yards of tape, at J per 
 
 yard ? 
 
 f J 7853 In this example I say, is the half of a 
 
 . penny, and a J is the half of a halfpenny. I 
 
 39204 first divide the number of yards by 2, ai.'d 
 
 i 963- the answer is 3926J pence, or the value of 
 
 7 8 53 yards, at J per yard; I then divide this 
 
 12) 5889J sum by 2, which gives 1963^, or the value of 
 the tape had it been only ~ per yard. To 
 
 2.0)49.0 9f find the value at f per yard, I add these two 
 
 sums together, and 5889f pence is the value 
 
 Answer, .24 10 9f of the tape at f per yard ; I then divide this 
 
 surn by 12, to bring the pence into shillings, 
 
 afterwards by 20, to bring the shillings into pounds. 
 
 EXAMPLES. 
 
 Ex. l. 4567 at 4: per yd. 2. 6784 at J per Ib. 3. 8976 at % 
 
 4. 7655 at per yd. 5. 7486 at J per Ib. 6. 9984 at 
 
 7. 6327 at f per yd. 8. 5934 at % per Ib. 9. 7585 at -| 
 
 10. 4767 at \ per yd. 11. 6493 at J per Wb. 12. 5388 at | 
 
 II. When the pi^jflb an aliquot part of a shilling. 
 RULE. Divide tlie^fha number by the aliquot part, 
 and this quotient by 20 fTBff answer will be in pounds. 
 
 Ex. What is the value of 2785 Ibs. of salt, at 4>d. per Ib ? 
 
 2785- Four pence being l of a shilling, I divide 
 
 the given number by 3, and the answer is 
 
 2,0)92,8 4 928 shillings and l over, that is 928s. 4d.,be- 
 
 cause each pound of salt is worth 4d. ; I then 
 
 Answer, . 46 8 4 divide by 20 to bring the shillings into 
 
 pounds. 
 
 Ex. 1. 3764 at 2d. 2. 5943 at ad. 3. 4953 at i{rf. 
 
 4. 5943 at 4d. 5. 3987 at Qd. 6. 5964 at itL 
 
 7. 5684 at 4d. ' 8. 2705 at 2d. 9. 3456 at 2d. 
 
 10. 5924 at lrf. 11. 5964 at <2d. 32. 5215 at 4d. 
 
 III. When the price is pence and farthings, and no aliquot 
 part of a shilling. 
 
 RULE. (1) Find what aliquot part of a shilling is nearest 
 to the given price, and divide the proposed number by it. 
 (2) Consider what part the remainder is of this aliquot 
 part of the given price 9 and divide the former quotient by 
 it, fyc. (3) Add the several quotients together, and the 
 answer will be in shillings, which divide by 20 to bring 
 into pounds. 
 
 G2 
 
134 
 
 PE ACTIGK. 
 
 Ex. What is the value of 4277 yards, at 10' d. per yard ? 
 
 In this example, I frst divide by 2, be- 
 use 6d. is the 3 of a shilling; then I take 
 parts for the 4|d., and say 3d. is the J of 6 ; 
 of 3d., and | is the of 1^ : and 
 of course I divide the first answer by 2, and 
 this quotient by 2, then that last found by 
 and having-, added the four quotients to- 
 gether, the answer is 38315. sfoL ; which, di 
 vided by 20, gives igl/. 115. 
 
 & 
 
 J 
 
 4277 
 
 
 
 In i 
 
 
 
 . 
 
 
 
 cause 
 
 3 
 
 i 
 
 2138 
 
 6 
 
 
 parts f 
 
 3 i 
 
 I 
 
 1069 
 
 3 
 
 
 lj is 
 
 i 
 
 I 
 
 534 
 
 7f 
 
 
 of cou 
 
 
 
 89 
 
 If 
 
 
 this q 
 6 * anc 
 
 2,0)38-3,1 
 
 ** 
 
 
 gether, 
 
 
 
 
 
 
 
 Ans. . 
 
 191 11 
 
 5 | 
 
 
 
 
 
 
 
 d. 
 
 
 Ex 
 
 . i. 
 
 4784 
 
 at 
 
 l i 
 
 2. 
 
 
 4. 
 
 176^5 
 
 at 
 
 n 
 
 5. 
 
 
 7. 
 
 7641 
 
 at 
 
 2 2 
 
 8. 
 
 
 10. 
 
 3592 
 
 at 
 
 3^ 
 
 11. 
 
 
 13. 
 
 8764 
 
 at 
 
 ! 
 
 14. 
 
 
 16. 
 
 9714 
 
 at 
 
 44 
 
 17. 
 
 
 19. 
 
 8934 
 
 at 
 
 5^ 
 
 ao. 
 
 
 22. 
 
 5687 
 
 at 
 
 5 i 
 
 23. 
 
 
 5. 
 
 5943 
 
 at 
 
 9^ 
 
 26. 
 
 
 28. 
 
 1956 
 
 at 
 
 s| 
 
 29. 
 
 
 il. 
 
 2748 
 
 at 11 
 
 32. 
 
 
 34. 
 
 1594 
 
 *t 
 
 3?r 
 
 35. 
 
 5964 at 
 4305 at 
 9875 at 
 3046 at 
 5921 at 
 17. 5643 at 
 20. 248 at 
 
 1435 at 
 
 1876 at 
 29. 4235 at 
 
 i at 
 
 d. 
 
 if 
 
 2| 
 
 4 
 
 d. 
 
 3. 4659 at 2| 
 
 6. 3694 at 3;J 
 
 9. 5476 at lof 
 
 12. 3214 at 11 
 
 15. 5178 at 
 
 18. 4932 at 10^ 
 
 21. 8764 at 1 }|- 
 
 24. 5842- at 7 J- 
 
 27. 4316 at 7| 
 
 SO. 1327 at 9 
 
 33. 4285 at ll{ 
 
 36. 1114 at 54 
 
 IV, 
 
 When the price is more than one shilling 1 , and less 
 
 than two. 
 
 RUL'F. Let the given number stand for shillings, and 
 ivorkjor the pence and farthings as before. 
 
 Ex- What is the value of 1187, quartern loaves, at 
 Is. lid. each-? 
 
 Here I take parts for the l%d. 9 that is, ij 
 is the |th of a shilling, and a % is i of i J, 
 I first divide the number by 8, and that quo- 
 tient by 0, and add the two quotients thus 
 found, to the given number which stood for 
 shillings : and the sum thus found, divided 
 by 20, gives the answer in pounds. 
 
 s. 
 
 2. 4876 at i 
 
 5. 4092 at 
 
 8. 3724 at 1 
 11. 5928 at 
 .14. 4371 at 
 17. 5629 at 
 20. 0271 at 
 23. 5928 at 
 
 1 
 
 i 
 
 1187 
 
 
 * : 
 
 x 
 
 148 4 
 
 f 
 
 
 
 24 f 
 
 f 
 
 
 2 t C 
 
 >)136.O 1 
 
 
 Ans 
 
 . i 
 
 ,. 68 1 
 
 i 
 
 
 
 
 . d. 
 
 Ex. 
 
 1. 
 
 3456 at 
 
 21 
 
 
 4. 
 
 2632 at 
 
 3| 
 
 
 7. 
 
 4735 at 
 
 4| 
 
 1 
 
 0. 
 
 7321 at 
 
 7| 
 
 ] 
 
 3. 
 
 8465 at 
 
 Ql 
 
 1 
 
 6. 
 
 1234 at 
 
 11 
 
 1 
 
 9. 
 
 5678 at 
 
 2| 
 
 i 
 
 2. 
 
 8234 at 
 
 ' *J 
 
 S. 
 
 d. 
 
 
 
 
 d. 
 
 1 
 1 
 
 H 
 
 3. 
 6. 
 
 5792 
 2596 
 
 at 
 at 
 
 H 
 
 10 
 
 1 
 
 9? 
 
 9. 
 
 3451 
 
 at 
 
 6 4 
 
 1 
 
 11 
 
 12. 
 
 6542 
 
 at 
 
 8 A 
 
 1 
 
 3i 
 
 15. 
 
 8937 
 
 at 
 
 3 
 
 1 
 
 lj 
 
 18. 
 
 4516 
 
 at 
 
 I 2 
 
 3 
 
 4 4 
 
 21. 
 
 5461 
 
 at ] 
 
 t 7 
 
 1 
 
 10} 
 
 34. 
 
 8750 
 
 at] 
 
 I 5 
 
PRACTICE. 
 
 125 
 
 V. When the price is any number of shillings tinder 20. 
 
 RULE. (1) If the price is an even number, multiply the 
 given quantity by.hqlfpf it, doubling the first figure to the 
 right-hatuMor shillings, and the rest are pounds. (2) ff 
 the price is an odd number, find for the greatest even num- 
 ber, as before^ to ivhich add the ^th of the given number 
 _jor the odd shilling, and the sum is the answer. 
 
 Ex. What is the value of 3456 yards of cloth, at 18s. 
 per yard ? 
 
 3456 
 9 
 
 Ans. ' 3110 8 
 
 I multiply the given quantity by 9, and 
 the first product 4, I double for shillings, car- 
 rying the 5 to the next figure. 
 
 Ex. What is the value of 2592 yards of second clotb, 
 at 1L>\ per yard ? 
 
 I multiply by 5, as before, which gives the 
 value of the cloth at los. per yard: I then 
 divide the quantity by 20, and adding this 
 quotient to the last found gives the, answer. 
 
 EXAMPLES. 
 
 Ans. . 14425 12 
 
 Ex. i. 5975 at 25. 
 4. 7591 at 55. 
 7. 5734 at 105. 
 
 10. 2935 at 135. 
 
 13. 4917 at 185. 
 16. 2514 at 165. 
 
 2. 4374 at 35 
 5. 6743 at 65 
 8. 5946 at 115 
 11. 4392 at 145 
 14. 3271 at 9' 
 17. 1392 at 105 
 
 3. 5916 at 45. 
 6. y4-JO at 85. 
 9. 3004 at 75. 
 12. 5931 at 19?. 
 15. 9315 at 175. 
 18. 5432 at 195. 
 
 VI. When the price is shillings and pence. 
 
 RULE. (1) If they are an aliquot part of a pound, di- 
 vide the quantity by that part, and the quotient is the ?i- 
 sircr. (2) If they are mot an aliquot part, multiply by the 
 shillings, and take parts for the pence. 
 
 Ex. What is the value of 2769 yards of Irish, at 35. 4rf. 
 per yard ? 
 
 35. id. ] | 2769 35. 4_d. being Ith of a pound, I divide 
 
 by 6, and the quotient is the answer, 
 
 
 
 481 105. 
 
126 
 
 PR ACT1CJ 
 
 Ex. What is the value of 3756 yaids of muslin, ar 
 12s. 9d. per yard ? 
 
 I multiply by 12 for the shillings, and 6d. 
 being J of a shilling, I dividj^the given 
 quantity by 2 ; then 3d. bemg ^ or 6t/., I di- 
 vide the last quotient by 2, ~and add the 
 three sums together, which gives the answe* 
 in shillings. 
 
 6 
 
 
 
 3756 
 
 Ans 
 
 2.0 
 
 45072 
 1878 
 939 
 
 )4788.0 
 
 ,2394 QS. 
 
 Ex. 1. 8943 at 2 
 
 4. 2524 at 3 
 
 7. 3764 at 10 
 
 10. 8756 at 15 
 
 13. 5642 at 18 
 
 16. 9143 at 6 
 
 d. 
 
 
 * 
 
 
 10 
 
 *J 
 
 8 
 
 *. 
 
 3. 8671 at 7 
 
 6. 546O at 9 9; 
 
 9. 3745 at 9 11 
 
 12. 2475'at 16 8 
 
 15. 5931 at 17 6 
 
 18. 4604 at 19 <3 
 
 d, 
 6 
 
 EXAMPLES. 
 
 s. d. 
 
 2. 3532 at 4 
 
 5. 5971 at 5 10 
 
 8. 5638 at 8 11 
 11. 3942 at 4 5 
 14. 1764 at 5 8 
 17. 7189 at 3 7 
 
 VII. When the price is pounds and shillings, or pounds, 
 shillings, pence, and farthings. 
 
 RULE. Multiply the quantity by the pounds, and ivork 
 the rest by the foregoing rules. 
 
 Ex. What is the value of 5428 hogsheads of ale, at 
 41. 12*. per hogshead ? 
 
 5428 I multiply first by 4, for the pounds; 
 
 4 12 then 12 being an even number, I mul- 
 
 1 tiply by the half, or 6, according to 
 
 21712 Case V., and add the two sums together 
 
 3256 16 for the answer. 
 
 Answer, . 24968 165. 
 
 Ex. What is the value of 2714 cwt. of sugar, at 3/. 
 12*. 9|d. per cwt. ? 
 
 Having multiplied by 3 for the 
 pounds, I take the aliquot parts for 
 12s. 9d., that is, los. is J, 25. 6d. 
 is the 4th of that, 3d. is the one- 
 tenth of that, and is the one-sixth 
 of 3d.; then, adding the several 
 sums together, I obtain the answer, 
 
 Answer, . 9877 16 7 
 
 IDS. 
 
 i 
 
 2714 
 
 
 
 
 
 3 
 
 
 
 
 
 8142 
 
 
 
 s. 6d. 
 
 1 
 
 1357 
 
 
 
 3d. 
 
 fo 
 
 339 
 
 5 
 
 
 
 i 
 
 
 33 
 
 38 
 
 6 
 
 
 
 5 
 
 13 
 
 1 
 
PRACTICE, 
 
 127 
 
 
 
 
 . 
 
 5\ 
 
 d. 
 
 
 
 
 . 
 
 A'. 
 
 d. 
 
 Ex. i. 
 
 5674 
 
 a.t 
 
 5 
 
 1? 
 
 6 
 
 2. 
 
 6431 
 
 at 
 
 4 
 
 8 
 
 4 
 
 3. 
 
 3416 
 
 at 
 
 5 
 
 11 
 
 <* 
 
 4. 
 
 4931 
 
 at 
 
 9 
 
 4 
 
 
 
 5. 
 
 3146 
 
 at 
 
 10 
 
 12 
 
 9 
 
 6. 
 
 4316 
 
 at 
 
 10 
 
 19 
 
 6| 
 
 7. 
 
 5648 
 
 at 
 
 12 
 
 13 
 
 
 
 8. 
 
 1436 
 
 at 
 
 10 
 
 10 
 
 6 
 
 9. 
 
 1340 
 
 at 
 
 3 
 
 13 
 
 4 
 
 10. 
 
 2714 
 
 at 
 
 18 
 
 9 
 
 
 
 11. 
 
 0614 
 
 at 
 
 4 
 
 14 
 
 6 
 
 12. 
 
 5789 
 
 at 
 
 7 
 
 7 
 
 
 
 13. 
 
 1590 
 
 at 
 
 12 
 
 12 
 
 
 
 14. 
 
 6341 
 
 at 
 
 8 
 
 18 
 
 6 
 
 15. 
 
 4803 
 
 at 
 
 9 
 
 9 
 
 a* 
 
 16. 
 
 3405 
 
 at 
 
 a 
 
 15 
 
 O 
 
 17. 
 
 7182 
 
 at 
 
 11 
 
 12 
 
 10 
 
 18. 
 
 1604 
 
 at 
 
 4 
 
 11 
 
 10 
 
 VIII. If there be a fraction in the given quantity. 
 
 RULE. Work for the whole number , according to the 
 preceding rules, to which add J, J, |, , <Jrc. of the price, 
 according to the nature of the question. 
 
 Ex. What is the value of 5354| cwt. of soap, at 4/. 4s. 8d. 
 per cwt. 
 
 
 
 
 
 
 
 
 
 
 
 
 . s. d. 
 
 
 45. 
 
 1 
 
 5 
 
 5354 f { 
 
 a 
 
 4 
 
 4 
 
 S I multiply by the 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 % 
 
 } 
 
 9 
 
 2 
 
 4 parts tor the 45. 8c/.; 
 
 
 
 
 21416 
 
 
 1 
 
 1 
 
 2 viz. 45. *is the one- 
 
 
 8ci. 
 
 1 
 
 ft 
 
 10 
 
 *0 
 
 16 
 
 
 
 
 
 
 fifth of a pound, and 
 
 
 
 
 1 
 
 ;s 
 
 9 
 
 4 
 
 .3 
 
 3 
 
 6 8c/. is the one-sixth 
 
 
 
 
 
 3 
 
 3 
 
 6 
 
 of 45. ; then, to find the value of the f , 
 
 
 
 
 
 
 
 
 I t 
 
 /i^ 
 
 riy 
 
 rtc r 
 
 
 Answer, 
 
 
 
 , 226*1 
 
 8 
 
 10 
 
 ing the 3/. 35. 6rf. thus found, to the 
 
 
 
 
 
 
 
 
 oil 
 
 
 fVxr 
 
 
 
 
 
 
 
 
 
 
 S. 
 
 
 
 . 5. d. 
 
 Ex.1. 
 3. 
 
 4562J 
 26^41 
 
 at 
 at 
 
 a 
 
 7 
 
 15 
 15 
 
 4 
 
 
 
 r> 
 
 4. 
 
 6744J at 9 9 ioj 
 7394| at 12 8 sj 
 
 
 5. 
 
 46514: 
 
 at 
 
 5 
 
 12 
 
 10 
 
 
 
 6. 
 
 3749J at 16 5 
 
 
 7. 
 
 3875 
 
 at 
 
 8 
 
 18 
 
 64 
 
 
 
 8. 
 
 4365| at 11 11 11 
 
 
 9. 
 
 97244: 
 
 at 
 
 6 
 
 10 
 
 4 3 
 
 
 
 10. 
 
 364 8 at 4 4 6| 
 
 TABLES 
 
 OF ALIQUOT 
 
 PARTS. 
 
 Aliquot parts of 
 
 Aliquot parts 
 
 Aliquot parts Aliquot pts. 
 
 
 a 
 
 ton. 
 
 of a cwt. 
 
 of a qr. of cwt. of a Ib. 
 
 cwt 
 
 qr. 
 
 Ib 
 
 . 
 
 
 
 qrs 
 
 . Ib. 
 
 
 
 
 Ib. oz. 
 
 10 
 
 
 
 6 
 
 ~ 
 
 i 
 
 
 2 
 
 : 
 
 
 i 
 
 
 14 = J 8=| 
 
 5 
 
 
 
 
 
 
 
 ] 
 
 
 1 
 
 - 
 
 
 I 
 
 
 7 = i 4= J- 
 
 4 
 
 
 
 
 
 
 
 j. 
 
 
 
 16 2 
 
 
 f 
 
 
 4 = f 2 zi -j- 
 
 2 
 
 3 
 
 12 
 
 _n: 
 
 4,. 
 
 
 
 14 : 
 
 
 -i- 
 
 
 3 1 j- i _JL. 
 
 2 
 
 2 
 
 = 
 
 i 
 
 V 
 
 
 
 8 : 
 
 14 
 
 2 = T V 
 
 2 
 
 
 
 = 
 
 iV 
 
 
 
 7 : 
 
 
 iV 
 
 
 H rV 
 
 ' 1 
 
 
 
 n: -fa 1 iz aV 
 
128 
 
 PRACTICE. 
 
 IX. When the given quantity is of several denominations. 
 
 RULE. Multiply the given price by the highest dcno-mina- 
 
 -tion, as in Compotibd Multiplication, attd take parts of the 
 
 price for the inferior denominations of the given quantity. 
 
 Ex. What is the value of 22 cwt. 3 qi . 21 Ib. of hops, 
 at 4/. 18s. 6d. per cwt. ? 
 . s. d. 
 
 2qr. 4 is 
 
 6 
 
 
 Here, 
 
 for the 22 cwt., 
 
 I 
 
 multiply by ll 
 
 
 
 ] 1 
 
 and by 
 
 2 ; then 
 
 I take parts for the 3 qrs. 
 
 _. __; 21 Ib., according to 
 
 the p 
 
 receding table. 
 
 54 
 
 a 
 
 jC 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 108 
 
 7 
 
 O 
 
 ~ value of 2 2 
 
 cwt. 
 
 
 
 
 
 
 1 qr. 2 
 
 9 
 
 8 
 
 .m 
 
 ditto 
 
 2 
 
 qrs. 
 
 
 
 
 
 
 14lb. -J 1 
 
 4 
 
 /- 1 . 
 
 /S 
 
 ditto 
 
 l 
 
 qr. 
 
 
 
 
 
 
 7 Ib, J 12 
 
 
 ditto 
 
 14 
 
 Ib. 
 
 
 
 
 
 
 
 e 
 
 M 
 
 f ~ 
 
 ditto 
 
 7 
 
 Ib. 
 
 
 
 
 
 
 Ans. . 112 19 
 
 4 
 
 l 
 
 
 
 
 
 
 
 
 
 cwt. 
 
 qr. Ib. 
 
 . 
 
 s. 
 
 d. 
 
 
 
 
 
 Ex, l. 
 
 
 8 
 
 2 12 at 
 
 4 
 
 12 
 
 7 per cwt. 
 
 2. 
 
 16 
 
 1 21 at 
 
 3 
 
 13 
 
 9 per cwt. 
 
 3. 
 
 37 
 
 3 22 at 
 
 12 
 
 11 
 
 7 per cwt. 
 
 4. 
 
 73 
 
 2 loj at 
 
 3 
 
 16 
 
 9 per cwt. 
 
 5. 
 
 38 
 
 1 16 at 
 
 2 
 
 12 
 
 6 per cwt. 
 
 6. 
 
 33 
 
 2 8 at 
 
 39 
 
 3 
 
 8 per cwt. 
 
 7. 
 
 84 
 
 3 14 at 
 
 12 
 
 11 
 
 8 per cwt. 
 
 
 
 
 
 
 
 
 
 
 s. 
 
 d. 
 
 Ex. 8. 56 tons, 
 
 4 
 
 cwt. 
 
 2 qrs. 
 
 o Ib. at 
 
 58 7 
 
 6 
 
 per ton. 
 
 g. 39 tons, 
 
 1'2 
 
 cwt. 
 
 1 qr. 
 
 14 Ib. at 
 
 25 
 
 12 
 
 8 
 
 per 
 
 ton. 
 
 10. 124 tons, 
 
 16 
 
 cwt. 
 
 2 qr. 
 
 16 Ib. at 
 
 12 
 
 18 
 
 7 
 
 per 
 
 ton. 
 
 31. 16 Ib. 
 
 8 
 
 OZ 
 
 . 12 
 
 dr. 
 
 - - 
 
 at 
 
 4 
 
 3 
 
 6 
 
 per Ib. 
 
 12. 25 Ib. 
 
 ]2 
 
 OZ 
 
 4 
 
 dr. 
 
 - - 
 
 at 
 
 8 
 
 12 
 
 6 
 
 per 
 
 Ib. 
 
 13. 35 Ib. 
 
 4 
 
 OZ 
 
 . 12 
 
 dwt. 
 
 - - 
 
 at 
 
 11 
 
 9 
 
 9 
 
 per 
 
 Ib.* 
 
 34. 48 Ib. 
 
 S 
 
 OZ 
 
 . 16 
 
 dwt. 
 
 . 
 
 at 
 
 14 
 
 4 
 
 4 
 
 per 
 
 Ib. 
 
 15. 25 Ib. 
 
 6 
 
 OZ 
 
 5 
 
 dwt. 
 
 - - 
 
 at 
 
 15 
 
 3 
 
 9 
 
 per 
 
 Ib. 
 
 16. 38yds 
 
 . 2 
 
 qr, 
 
 3 
 
 nails 
 
 - - 
 
 at 
 
 
 
 16 
 
 8 
 
 per 
 
 yard. 
 
 17. 55 yds 
 
 . 2 
 
 qr. 
 
 a 
 
 nails 
 
 - - 
 
 at 
 
 J 
 
 3 
 
 9 
 
 per yard. 
 
 18. 15 acr. 3 
 
 rd. 
 
 24 
 
 per. 
 
 - - 
 
 at 
 
 38 
 
 3 
 
 r> 
 
 per 
 
 acre. 
 
 19. 25 acr. l 
 
 rd. 
 
 4 
 
 per. 
 
 - - 
 
 at 
 
 46 
 
 1 
 
 
 
 per 
 
 acre. 
 
 20. 39 acr. 2 
 
 id. 
 
 IS 
 
 per. 
 
 - - 
 
 at 
 
 5 
 
 5 
 
 
 
 .per 
 
 acre. 
 
 NOTE. 
 
 * The aliquot parts of Ibs., yards, ells, acres, &c., are easily 
 found, by dividing the integer, or any part of it, by the quantity, 
 the aliquot part of which is required, and the quotienf } jf there be 
 KO remainder, will be the part sought. 
 
129 
 
 TARE AND TRET. 
 
 TARE AND TRET are a set of practical rules for deduct- 
 ing- certain allowances, made by wholesale dealers in sell- 
 ing their goods by weight. 
 
 GROSS WEIGHT is the whole weight of goods, including 
 package, or whatever contains them. 
 
 NEAT WEIGHT is what remains after all allowances are 
 made. 
 
 TARE is an allowance to the buyjer. for the weight of 
 the package, and is either at so much per. barrel, chest, 
 &.C., or at so much per cwt., or at so much for the whole. 
 
 TRET is an allowance of 4 Ib. in every 104 Ib. for 
 waste, dust, &c. or the ^th part of the whole. 
 
 CLOFF is an allowance, after Tare and Tret are de- 
 ducted, of 2 Ib. upon every 3-cwt* that the weight -may 
 hold good when sold by retail. 
 
 SUTTLE is when only, part of the allowance is deducted 
 from the gross. Thus,. after the tare is deducted from4he 
 gross, the remainder is calkd tare suttle. 
 
 CASE I. When the tare is at so much for the whole. 
 
 RULE. From the gross weight subtract the tare, and the 
 remainder will be the neat weight required. 
 
 Ex. What is the neat weight of 25 barrels of indigo, 
 weighing 1.16- cwt, 2 qr/ 14 Ib., allowing 2 cwt. 3 qr. 12 Ib 
 tare ? 
 
 cwt. qr. Ib. 
 
 116 2 14 
 
 2 -3 12 
 
 Answer, - 133 3 2 neat weight. 
 
 Ex, 1. What is the neat weight of 55 barrels of figs, weighing 
 35 cwt. 2 qr. islb., tare being allowed at 1 cwt. i qr. 24 Ib. ? 
 
 Ex. 2. What is the neat weight of 20 casks of Russian tallow, 
 ; weighing 74 cwt,, tare being allowed at 2 cwt. 2 qr. 5 Ib ? 
 
 G5 
 
130 TARE AND TRET. 
 
 CASE II. When the tare is at so much per barrel, chest, &c. 
 
 RULE. Multiply the tare by the number of hogsheads, 
 barrels, chests, fyc., subtract the product from the gross, 
 and the remainder ivill be the neat weight required, 
 
 Ex. What is the neat weight of 8 hhds. of tobacco, each 
 weighing- 4 cwt. 2 qr. 24 Ib. gross, tare being allowed at 
 2 qrs. 4 Ib. per hhd. ? 
 
 cwt. qr. Ib. qr. Ib. 
 
 4 2 24 24 
 
 8 8 
 
 Gross weight 37 2 24 414 Tare. 
 
 414 
 
 Answer, - 33 l 20 neat weight. 
 
 Ex. 1. What is the neat weight of 25 frails of Malaga raisins, each 
 weighing 2 cwt. 3 qrs. 12 Ib., when the tare upon each frail is 17 Ib. ? 
 
 Ex. 2. In 79 barrels of figs, each weighing i cwt. and 12 Ib., and 
 tare gib. per barrel, what is the neat weight? 
 
 Ex. 3. What is the neat weight of 24 hhds. of tobacco, the weight of 
 each being 4 J cwt., and tare 67 Ib.'per hhd. ? 
 
 " Ex. 4. In 1 8 casks of currants, each weighing 6 cwt. l qr. 12 Ib., 
 and tare 61 Ib. per cask, what is the'neat weight? 
 
 CASE III. When the tare is at so much per cwt. 
 
 Ru LE. Take the aliquot part or parts of the whole gross 
 weight that the tare is of a cwt., as in Practice, and sub- 
 tract the result from the gross iveight. 
 
 Ex. What is the neat weight of 24 barrels of figs, each 
 weighihg 3 cwt. 2 qrs. 12 Ib., and tare 12 Ib. per cwt. ? 
 
 cwt. qr. Ib. 
 3 2 32X24IZ6X4 
 
 6 
 
 -- Ib. cwt. qr. Ib. 
 
 21 2 16 8 | fL | 86 2 8 
 
 4 
 
 4 | J- j 6 o 20 
 
 Gross weight 86 2 8 oz. -j ) 3 o 10 
 
 Tare, - 91 2 13 j 
 
 9 i 
 
 Answer, - 77 l 5 2f neat weight. 
 
 Ex. l. What is. the neat weight of 21 bairels of pot-ash, each barrel 
 weighing \ cwt. 3 qr. 8 Ib., tare being 10 Ib. per cwt. ? 
 
TARE AND TRET. 131 
 
 Ex. 2. What is the neat weight of 35 barrels of anchovies, each, 
 weighing l qr. 12 lt>., tare at 14 lb. per cwt. ? 
 
 Ex. 3, Required the neat weight of 15 hhds. of tobacco, each weigh- 
 ing 4 cwt. 2 qrs. 12 lb., tare at 20 lb. per cwt. 
 
 Ex. 4. What is the value of 26 hogsheads of tobacco, at 8/. 8s. per 
 cwt., each hogshead weighing 4| cwt., and the .allowance for tare being 
 13 lb. per cwt. ? 
 
 CASE IV. When there is an allowance both of tare and tret. 
 
 RULE. Find the tare by the last rule, subtract it from 
 the gross weight, the remainder, or suttle, divided by 26, 
 gives the tret, which faing subtracted from the suttle, 
 gives the answer. 
 
 Ex. What is the neat weight of 15 casks of tallow, each 
 weighing 6 cwt. 2 qr. 12 lb., tare being 12 lb. per cwt., 
 and tret as usual ? 
 
 cwt. 
 6 
 
 qr. 
 2 
 
 lb. 
 12 X 15 ~ 5 X 
 5 
 
 3. 
 
 cwt. qr. lb 
 5 99 12 
 
 70 8 1* 
 32 46 
 
 
 
 38 
 
 
 
 4 Ib- 
 
 3 8 i J 
 
 Gross weight 
 Tare 
 
 99 
 10 
 
 
 
 2 
 
 I* 4 \ 
 12 
 
 
 26) 
 
 88 
 3 
 
 2 
 1 
 
 
 
 176 
 
 10 2 12 18 
 
 Answer, 
 
 85 
 
 
 
 10 neat weight.* 
 
 Ex. 1. In 18 cwt. l qr. 6 lb. gross, tare 63 lb., and tret as usual., 
 how much neat ? 
 
 Ex. 2. In 14 casks of raisins, each 2 cwt. 14 lb. grogs, tar is lb. 
 per cwt., and tret as usual, what is the neat weight ? 
 
 Ex. 3. What is the neat weight of 9 cwt. 2 qr. 17 lb. gross, tare 
 39 lb., and tret as usual ? 
 
 Ex. 4. In 9 chests of sugar, each weighing 8 cwt. 2 qr. 10 lb., tare 
 14 lb. per cwt , and tret as usual, what is the neat weight? 
 
 CASE V. When*Cloff is allowed. 
 
 RULE. Subtract the tare from the gross, and the tret 
 from the tare suttle ; then divide the tret suttle by 168, and 
 
 NOTE. 
 
 * No account is taken of the remainders; tallow in quantities lik*. 
 this never being weighed to a greater nicety than a pound, 
 
132 TARE AND TRET- 
 
 the result ivill be the Cloff, which being subtracted from 
 the last suttle, gives the neat weight required.* 
 
 Ex. What is the neat weight of 19 cwt. 1 qr. 2 Ib. gross, 
 tare 3 cwt. 3 qr. 22 Ib., and tret and clotf at the usual 
 rate ? 
 
 ewt. qr. Ib. cwt. qr. Ib. 
 
 Gross - 19 1 2 4)14 2 -26 -r- 168 =Z 4 X 6 X 7 
 
 Tare - 3 3 22 ~-" 80Z. 
 
 26)15 
 Tret - 
 
 l 
 
 2 
 
 8 
 10 
 
 
 7)2 12 
 
 12 
 
 9 
 
 13^ 
 
 Tret suttle 14 
 Cloff 
 
 2 
 
 26 
 9 
 
 13 oz. 
 
 Answer, cwt. 14 
 
 2 
 
 16 
 
 3 neat weight. 
 
 Ex. 1. What is the neat weight of 224 cwt. 3 qr. 20 Ib. of tobacco, 
 tare being 25 cwt. 3 qr., tret and cloffas usual ? 
 
 Ex. 2. In 14 hhds. of tobacco, each weighing 5 cwt. 3 qr. 17 Ib. 
 gross, tare 1 1 Ib. per cwt;, and tret and doff as usual, what is the neat 
 weight * 
 
 Ex. 3. What is the neat weight of 15 casks of currants, each weigh- 
 n 5.} cwt. gross, tare 35 Ib. per cask, tret and cloff as usual ? 
 
 Ex. 4. In 9 chests of sugar, each containing 7 cwt. 2 qr. 12 Ib. gross, 
 tare 13 Ib. per cwt., tret and cloff as usual, what is' the neat weight, 
 and what is the value of it at Qd. per Ib. ? 
 
 NOTE. 
 
 * In general, the allowance for cloff is 2 Ib. for 3 cwt., according, to 
 the foregoing definition (p. 129) ; or, what is the same thing, 1 Ib. for 
 every 168 Ib. : but other allowances of cloff are made in different 
 places. " At t]e Custom -house, on goods imported, 1 Ib. is allowed 
 Itpon goods weighing less than 1 cwt. ; 2 Ib. if they weigh from 1 to 
 8 cwt. ; 3 Ib. from 2 to 3 cwt. ; 4 Ib. from 3 to 18 cwt. ; and 9 Ib. for 
 ^11 higher weights. 
 
 There are cases in which an allowance is made for damage, that is, 
 so much in the whole for any part of the merchandize which may have 
 received injury. 
 
 f- If an unit of any kind, as one pound, or one hogshead, be divided 
 into 100 equal parts, then 65 represents sixty-five of those parts. If 
 a decimal consists of four figures, one or unity is supposed to be divided 
 into 10,000 parts, of which the decimal represents as many as the num- 
 ber expresses: thus, .o625 is so many parts of an unit divided into ten 
 thousand parts : in this case the is placed before the 6, to shew that 
 the unit is divided into 10,000 ; otherwise, if it stood *625, it would 
 appear that it was divided in 1000 parts only. 
 
133 
 DECIMAL FRACTIONS. 
 
 1. DECIMAL, or DECIMATED FRACTIONS, are such as 
 always have 1 wnh one or more cyphers for their denomi- 
 nators. The denominators are rrever expressed, being jm- 
 derstood to be 10/100, 1000, &c., according as the nume- 
 rators consist of 1, 2, or 3 figures : thus, instead of T %, -jVo j 
 rVo5 tl ]e numerators only are written, with a dot or in>- 
 verted comma before them, as .2; .24; .211. 
 
 2. If a decimal consists of only one figure, one is sup- 
 posed to 'be divided into ten eqi/al parts, and the decimal 
 represents as many of those parts as the decimal figure ex- 
 presses; thus, .7 means seven-tenths of an wit : If it con- 
 sist of two figures, one is supposed to be divided into 100 
 equal parts, of which the decimal represents as many as the 
 figure expresses; thus, .65 means sixty-five hundredths of 
 an unit, (f See Note to this on the opposite page.) 
 
 TABLE. 
 
 M '** "^ -S ^ 
 
 ' 2 . ^ .JT S $ 
 
 2 8 4 I , J 
 
 S "V J9 T3 . rj- ' ' 
 
 g . S. I f | | . -s . 
 
 I 1 1 S ' I S'Ig1 
 
 HHEHP 'HSHH^K 
 
 - 77777, 7 7 7 7 -7, &c. 
 
 WHOLE NUMBERS. DECIMALS. 
 
 By this table the relative values of whole numbers and decimals are 
 at once seen : thus, take the three first figures of each ; in ichole num- 
 bers they express seven hundred and seventy seven ; in decimals they 
 express seven hundred and seventy-seven parts of an unit. 
 
 3. Cyphers to the right-hand of decimals cause no "dif- 
 ference in their Tarue, for .5; .50; .500, are decimals of 
 the same value, being each equal to f ;nhafis, .5 = T V ; 
 .50 =-. fVV * '^00 = yVyV but if the cyphers are placed 
 on the^ left-hand of decimals, they diminish their value in 
 a ten-fold proportion, thus ',3 ; .03"; .'003, are -O-teuths* 
 
134 REDUCTION OF DECIMALS. 
 
 3-hundredths ; 3-thousaudths ; and answer to the vulgar 
 fractions T V, T o-> rtfW respectively. 
 
 4. A whole number and decimal is thus expressed, 85.74 
 
 which is equal to 85 T ^V = TT^r an( * 85.04 = 85 T T = 
 
 100 
 
 , &c. 
 
 , 
 
 REDUCTION OF DECIMALS. 
 
 CASE I. To reduce a vulgar fraction to a decimal of an 
 
 equal value. 
 
 RULE. Divide the numerator of the fraction, increased 
 by a cypher, or cyphers, by the denominator, and the quo- 
 tient ivill be the decimal sought. 
 
 Reduce \, \, ^ T y, to decimals of the same value, 
 i.o' i.oo i.ooo 
 
 $ * Z: .5. i ZI ZZ .25. J = = .125. 
 
 a 4 8 
 
 1.0000 
 
 A = = -625. 
 
 16 
 
 The cyphers added to the numerators are separated from the original 
 figur.e&by a dot, to shew that they are borrowed for the sake of forming 
 the decimal. 
 
 Ex. 1. What decimal expressions answer to the follow- 
 ing vulgar fractions, f , -{-, f , | 2 |i ? 
 
 Ex.2. Required the equivalent decimals of the fractions 
 
 T 5 T> T\ 4> TT' T 9 ^ 
 
 Ex* 3. What is the. decimal that answers to F V r 
 
 1.000000 
 Jj ZZ Z= .015625.* 
 
 Ex. 4. What are the decimals answering to the fractions 
 yls, VA> and ^ T ; 
 
 Ex. 5. What decimal expressions answer to ^, -fg, and 
 'aTT $ ee notc at page 141. 
 
 NOTE. 
 
 * As in this example, the numerator requires two cyphers before 
 it is equal to, or larger than, the denominator: a cypher must be pre- 
 fixed to tl e figures in ihe quotient; for in all cases the number of 
 figures in the quotient must be equal to the number of cyphers made 
 use of in the division. 
 
REDUCTION OF DECIMALS, 135 
 
 CASE II. To reduce numbers of different denominations to 
 their equivalent decimal values. 
 
 RULE. (1) Write the given numbers under each other 
 for dividends, proceeding from the least to the greatest. 
 (2) Place on the left side of each dividend, for a divisor, 
 'the number that will bring it to the next superior denomi- 
 nation. (3) Begin ivith the uppermost number, and set 
 down the quotierit of each division, as decimal parts, on 
 the right-hand of the dividend next below it, and so pro- 
 ceed to the last quotient, which is the decimal required. 
 
 Ex. Reduce 12s. 3^d. to the decimal of a pound. 
 
 , Qqrs. I divide the J by 4, supplying cy- 
 
 3d. .7 5 phers to the a by the imagination; 
 
 125. .3125 the quotient is .75, which is placed 
 
 by the side of the 3d., and then di- 
 
 .615625 decimal of a. vide the 3.75 by 12 ; the quotient, 
 .31-25, I set by the side of the 325., 
 
 and divide by 20, which gives .615625 for the answer : that is, if a 
 pound were divided into 1 ,000,000 parts, the 125. 3 %d. would be 61. *> 62 5 
 such parts, in the same manner as if a penny were divided into 100 
 parts, J would be equal to .7* such parts. * 
 
 Ex. l. Reduce 85, 4^d. to the decimal of a pound. 
 
 Ex. 2. What decimal of a pound are 155. 5%d. ?* 
 
 Ex. 3. What decimal of a pound are 45. G^d. ? 
 
 Ex. 4. Reduce 18$. 6//., 85. 2</., and 55. to decimals of a pound. 
 
 Ex.5. Reduce' 5 oz. 6 dwts. sgr.- troy, to the decimal of a Ib. 
 
 Ex. 6. Reduc. 3 qrs. 7 Ib. 8 oz. avoirdupois, to the decimal of a cwt. 
 
 Ex. 7. Reduce 2 qrs. l n. to the decir/ial of a yard. 
 
 Ex. 8. Reduce 32 bush. 3 p. to the decimal of a chaldron. 
 
 CASE III. To find the value of any given decimal in terms 
 of the integer. This is the reverse of the last case. 
 
 RULE. Multiply the decimal by the number of parts in 
 
 NOTE. 
 
 * Here the learner will find, that in dividing by 12 there will be no 
 termination to the quotient ; it may be carried to any extent, so as to 
 bring the answer as near the truth as he pleases, but it can never be the 
 exact answer. The same thing will occur in the next example. The 
 two sums together, as they s'and in the questions, make a pound ; but 
 the two decimals, viz. .77395833 -f- .22604)66, will not, when-added 
 together, be a pound, bur .99999999 of a pound, which wants the 
 hundred millionth part of a pound only; and by carrying on the di- 
 vision in both examples, the answers may be brought indefinitely near 
 the truth. 
 
136 ADDITION OF DECIMALS. 
 
 the next less denomination, and cut off as many places to 
 the right-hand, us there are places in the given decimal, 
 and so proceed through each denomination, 
 
 fe. What is tfei'e value of .615025 of a pound ? 
 
 .615625 
 
 20 It maybe observed, that as cyphers to the 
 
 right do not alter the value in decimals, they 
 
 12. 31 >2 500 are omitted in each step of the operation. 
 
 3.7500 
 4 
 
 Answer, - 125. 3$d, 
 - 3.00 
 
 Ex. l. What is the value of .625 of a shilling ? 
 Ett. 2. What is the value of .1275 of a pound? 
 Ex, 3. What is tbe valup~of .575 of a cwt. ? 
 "Ex, 4. W"hat is the value of .875 of a chaldron of coals ?* 
 
 ADDITION OF DECIMALS. 
 
 RULE. (1) Arrange the numbers under each other, ac- 
 'cording to their -several values. (2) Find the sum as >in 
 Addition of whole numbers, and cut off', for decimals, -as 
 many figures to the right as there are decimals in any one 
 of the given numbers. 
 
 " Ex. What is the sum of 23.45, 7.849, 543.2, 8.62&1, 
 and 253,004? 
 
 23.45 
 " "7-S49 
 543.2 
 
 8.6234 
 253.001 
 
 Answer, 836.1264 
 
 Ex. i.^What is the sum of 37.035, 4.26, 598.034, 9.3076, 4.321 3 
 and 5 ? 
 
 Ex. 2. 'Find the value of 89,33, 4.2056, .98735, 46.287> 3.7491, 
 and 3.004. 
 
 NOTE. 
 
 * To these examples may be added the answers of the questions in 
 Case II., as in example 3, we say, what is the value of ,22604166 of 
 pound ? 
 
 
MULTIPLICATION OF DECIMALS, 
 
 SUBTRACTION OF DECIMALS. 
 
 RULE. Arrange the numbers according to their value ; 
 subtract, as in whole numbers, and cut ojf\ for decimals,, 
 *&'in Addition. 
 
 Ex. Subtract 35.87043 from 132.005. 
 
 132.005 
 35.87043 
 
 Answer, 96.13457 
 
 Ex. 1. What is the difference between 104.326 and 74.05 ? 
 Ex. 2. Find the, difference between 394. S32 and 148.0070. 
 Ex. 3. From 372.971 take 270.30041. 
 
 MULTIPLICATION OF DECIMALS. 
 
 RULE. Multiply "as in whole numbers, and cut off ^ 
 many figures from the product as there are decimals in the 
 multiplier and multiplicand. 
 
 Ex. Multiply .025 by .045 : also 4.82 by 3.53. 
 .025 4.82 In the first instance, there be- 
 
 .045 3.58 '"'ing but four 'figures in the pro 
 
 ToT 1416 duct, and six decimals in the 
 
 ~~ * ' multiplier and multiplicand, two 
 
 14 46 cyphers must be added to the 
 
 .001125 ... . . left hand of the product. 
 
 17.0146 
 
 ''Ex.' "l. Multiply 70.43 by .875 : also .897 by .452. 
 
 Ex. 2. Multiply 324.004 by .7872. 
 
 Ex. 3. What is the product of 9.57 and .074 ? 
 
 Ex. 4. Multiply .643 by .389.* 
 
 NOTE. 
 
 * When the number of decimals in the multiplicand is large, and it 
 is not wished to carry 'th^ operation to more than a certain number of 
 decimals in the product, it is done by the following Rule, which 1 shall 
 illustrate by an example. 
 
 RULE. Having arranged the multiplicand, count, as many figures 
 from the decimal point, as you intend to keep decimals in the product, 
 
138 MULTIPLICATION OF DECIMALS. 
 
 and make a * over the last of these, under which, after you have in- 
 verted the multiplier, place the units figure of the multiplier thus, in- 
 verted, and the others in their proper order. Then multiply each 
 figure of the inverted multiplier, beginning, as usual, at the right-hand, 
 and set down the respective products, so that the right-hand figures 
 may fall in a strait line under one another. In multiplying, no atten- 
 tion is to be paid to the figures on the right-hand of that which you 
 multiply by, unless it be with the two preceding figures, to find what 
 nun her should be carried. 
 
 Ex. Required the product of 1.570796, multiplied by 26.3719, with 
 four places of decimals in the product. This, in the usual\ method, 
 would yield ten places of decimals: by contraction it is thus performed. 
 
 * 
 
 1.570796 
 9.17362_ 
 3.14159 zz product with 2 regard being had to 2 X 6 
 
 94247 = 6 6X9 
 
 4712 3 3X7 
 
 _14J= 9 9 X 57 
 
 41.4240 
 
 We will now work the example in the common way. 
 
 3.570796 
 26.3719 From this it will appear plain, why 
 
 _ _ in the contracted form the multiplier 
 
 14 137164 is inverted : the last product here be- 
 
 15|70796 ing the first there. In the contracted 
 
 1099 5572 form, the units place is 6 ; it would 
 
 4 1214 118 however be 8, if th*i 2 were carried 
 
 94247 76 from the 27, obtained in the next line 
 
 31415Q 2 by Addition. 
 
 41.4248750324 
 
 Ex. 2. Multiply 128.678 by 38.24, so as to have but one place of 
 decimals. 
 
 Common method. Contracted method. 
 
 
 
 128.678 128.678 
 
 38.24 42.83 
 
 514712 38603 
 
 257356 10294 
 
 1029424 257 
 
 386034 51 
 
 4920.64672 4920.5 
 
DIVISION OF DECIMALS.* 
 
 RULE. (1) Divide, as in whole numbers, and cut off as 
 many figures in the quotient, as the decimal places in the 
 dividend exceed those of the divisor. (2) If there be not 
 figures enough in the quotient, the deficiency must be , w/>- 
 plied by prefixing cyphers: (3) If there be a remainder, 
 : or there be more decimal places in the divisor than in the 
 dividend, cyphers may be affixed to the dividend^ and the 
 quotient carried on to any extent* 
 
 Divide 1.7154 by 1.5; and .37046 by 16. 
 
 1,5)1.7154 16) .37046 In the first example, by sup- 
 
 plying a single cypher there is 
 
 1.1136 .02315375 no remainder left; but in the 
 
 second I must supply three cy- 
 phers to obtain an eteu answer ; and I find the quotient has one 
 figure less than there are decimals in the dividend so supplied, I must 
 therefore prefix a cypher to the quotient found. 
 
 Ex. 1. Divide 25.64 by 3.645. 
 
 Ex. G. Divide 4752 by .9587. 
 
 Ex. 3. Divide .865439 by .155. 
 
 Ex. 4. Divide 79 by 3965. 
 
 Ex. 5. Divide 3J.S4472 by .882. 
 
 Ex. 6. Divide .218 by 7.435. 
 
 Ex. 7. Divide 76.42 by 58. 
 
 Ex. 8. Divide 88 by .88. 
 
 To find, by inspection, the value of any decimal of a pound 
 sterling 1 . 
 
 RULE. Double the first figure for shillings, and if the se- 
 cond figure be 5, or more than 5, add one shilling for that : 
 then reckon the remaining figures in the second and third 
 
 NOTE. 
 
 * The Contracted method of Division may be thus performed. 
 
 RULE. Having determined how many places of whole numbers 
 there will be in the quotient, if any, which is easily known by in- 
 spection ; if there are none, then consider of what value the first 
 figure in the quotient will be, and proceed as in common Division, 
 only omitting one figure of the divisor at each operation ; viz., for 
 every figure of the quotient dot off one in the divisor, remembering 
 
140 'DIVISION OF DECIMALS. 
 
 places so maqy farthings, deducting one when the farthings 
 are above 12, and two when they are more thati 37. 
 
 Ex, 1 and 2. What are the values of .125/. 3 and ,983/. ? 
 
 1st Example. 2d Example. 
 
 ' is. Od. ~ double 1 385. od. ~ double of 9, 
 
 f24 farthings, one _ (for the 5 out of 8 in 
 
 O 6 ""-< being taken as "" \ the place of tenths. 
 
 (^ by rule. ( 32 faithings, one be- 
 
 i O 8 m < ing deducted -as by 
 O 2 6 . -- trule, 
 
 .019 8 
 
 EXAMPLES. 
 
 * Ex. o, 4 3 5, and 6. Kequired the value of *375i*.-; .708/.; -494/. ; 
 and ,396/. 
 
 "Ex. 7. Find the value of the -following decimals by inspection, 
 and their sum : viz. .567/. -fr- .804/. -f- .sgbl. + .o6l/. 4~ .oao/. *^ 
 
 .009/. 
 
 Ex. 8. What is the value of the following decimals and their dif- 
 ference, .7Q4/. and .17 5/. ? 
 
 Ex. 9. What is the value of the tenth part of .366.? 
 
 to carry for the increase of the figures cut off, as was done, in Multi- 
 plication. 
 
 Ex. Let it be required to divide 23.41 by 7.9863. 
 
 Contracted method. Common method. 
 
 .7.9863)23.410002.9312 | 7-9863)23.410 
 ..... 15.9/26 15.9726 
 
 H^re it must be ob- .74374 .743740 
 
 served, that in each of 71876 71876.7 
 the subtractions, ex- ' ' | 
 
 cept the first, unit .2497 .2497 J30 
 
 must be carried to the 2395 23Q5i89 
 
 first figure, as would 1< 
 
 be the. case in the usual .101 .lOJ ! 4io 
 
 .course. 79 79J863 
 
 .21 .21:5470 
 
 15 159726 
 
DIVISION OF DECIMALS, 141 
 
 To reduce, by inspection, shillings, pence, and farthings, 
 to an equivalent decimal of a pound. 
 
 RULE. When the number of shillings are even, write half 
 ths number for the first decimal ; and when the number of 
 shillings are odd, for the remaining shilling write 5 in the 
 second place. Reduce the pence and farthings to far*- 
 things and write them in the second and third places, ob- 
 serving to increase the figure in the third place by 1, when 
 the farthings are 12, or more than 12, and by 2 when they 
 arc 36 and upwards. 
 
 Ex. 1. Reduce Us. (\\d., and 17s. -11 Jf., to the decimals 
 of a pound ? 
 
 14 o ir .7 16 o zz .8 
 
 6 ~ .026 1 O .05 
 
 excess of 12 m .001 on|zz .047 
 
 excess of 36 .002 
 
 .727 
 
 .899 
 
 Ex, 2. Reduce the following suras to decimals- of a pound, by in- 
 spection, viz. 16s. ; 135. (k/. ; 155. g^d. ; lls. 8%d. ; 6s. sd. ; 75. 6d. ; 
 85. 4%d. ; and 195. llfrf. 
 
 NOTE. 
 
 f* The answers to the fractions, Ex. 5, p. 134, will be .333, &c., 
 and .0202Q2, &c., and .123 123123, c. In the first example, one 
 figure, or the .3 is repeated; in the second there are two figures re- 
 peated, and in the third there are three figures repeated. These are 
 called circulating decimal?; and the circulating figures are called re- 
 petends ; if oe figure only repeats, it is called a single repetend ; if 
 more than one repeats, it is a compound repetendv There are many 
 other varieties, but they are of too little importance to be introduced 
 into a work which excludes every thing that is not practically use- 
 ful. Repetends are reduced to fractions by making the repe- 
 tend the numerator, and for the denominator put as many 0's as 
 
 there are figures in the repetend : thus .333 ~ zz i ; .02 &c 
 
 999 
 
 2 123 41 
 
 : and .123, &c. zz zz 
 
 99 999 333 
 
142 
 INVOLUTION. 
 
 INVOLUTION is a method of raising numbers to higher 
 powers.* 
 
 A. power is the product arising from-multiplyingany given 
 number into itself once, or oftener : thus, 3 X 3 IT 9 is 
 the second power of 3, and it is denoted in this manner 3". 
 
 The number denoting the power is called the index, or 
 exponent of that power : thus, in 3* the 2 is the index or 
 exponent. 
 
 The third power of 4 is 4 s =z 4 X 4 X 4 = 64. 
 
 The fourth power of 3 is 3 4 ~ 3 X 3 X 3 X 3 n 81. 
 
 The sixth power of 5 is5 6 z:5X5X5X5X5X 5r:.15625. 
 
 The third power of | is J) 3 X X i =2 fo 
 
 The fifth power of .03 is .03 5 n: .03 X .03 X .0.3 X .03 X .03 
 ~ .0000000243. 
 
 EXAMPLES. 
 
 Ex. 1 . What is the sixth power of 6 ? 
 Ex. 2. What is the eighth power of 7 ? 
 Ex. 3. What is the fourth power of | ? 
 
 Note. The 2d power is 
 called the square ; the 
 3d is called the cube: 
 
 Ex. 4. What is the fifth power of | ? j the 4th is called the 
 
 Ex. ,5. What is the third power of .25 ? 
 Ex. 6. What is the fourth power of .05 ? 
 Ex. 7. What is the third power of .3Q5 ? 
 Ex. . What is the ninth power of .9 ? 
 
 quadrate ; the rest are 
 generally denominated by 
 the numbers, as thejifth, 
 sixth, seventh, &c. 
 
 Ex. 9. What are the squares of 3 and 6 ; 5 and 10 ; 6 and 12 ; 2, 
 4, 8, and I6?f 
 
 Ex. ic. What are the cubes of 3 and 6 ; 5 and 10 ; 6 and 12 ; 2, 
 4, 8, and 16 ? 
 
 * This rule, when the powers are high, is performed bes*- by means 
 of logarithms, which see further on. 
 
 f The solution of these several examples, will lead the pupil to the 
 knowledge of this fact, viz. that the square of any number is four times 
 cs great as the square of half that number: thus, the square of 6 is 
 36, which is four times 9, the square of 3 : and the square of 10 rs 
 1$0 ; but the square of 5 is only 25. 
 
 Here we find that the cube of any numler is eight times as great 
 as {he cube of half that number* 
 
143 
 
 EVOLUTION. 
 
 EVOLUTION is the method of extracting roots.* 
 The root of any number, or power, is such a number, 
 as being multiplied into itself once, or oftener, produces 
 that power: thus 3 is the square root of 9, because 3 mul- 
 tiplied into itsell gives 9 : 4 is the cube root of 64, because 
 4 multiplied into itself twice, gives 64. The roots are de- 
 noted by indices, or exponents, in this manner: 
 
 The cube root of 125 is -/~ 5. 
 
 The square root of 81 is j/^ _- Qf 
 The fifth foot O f 243 is /^ 3 .f 
 
 Ex. 1 . What are the square roots of 49 and 64 ? 
 Ex. 2. What are the ctube roots of 216, 343, 5 12, and 729 ? 
 Ex. 3. What are the fourth roots of 625, 2401, and 409f> ? 
 Ex. 4. What are the fifth roots of 3125 and 3-2768 ? 
 
 To extract the square root. 
 RULE. (1) Divide the given number into periods of two 
 
 figures each) by placing a dot over units, another over 
 hundreds, and so on. (2) Find the greatest square in the 
 
 first period, and set its root on the right-hand, as a quotient 
 
 figure in division. (3) Subtract the square thus found, and 
 to the remainder annex the succeeding period for .a new di- 
 vidend. (4) Double the root for a divisor, and examine 
 how often it is contained in the dividend, exclusive of the 
 
 place of units, and put the result into the quotient and in 
 the units place of the divisor. (5) Multiply the divisor 
 tfius increased by the new quotient figure, and subtract the 
 
 * This is, better done by means of logarithms, \vhich sec further on. 
 
 f* It is evident from these examples, and others that follow, that 
 the root of a power of a given number may be found exactly ; but 
 there are many numbers, the roots of which can never be accurately 
 determined, as the square-root of 2, 3, 5, &c., because no numbers 
 multiplied into themselves will give 2, 3, 5 ; but by the help of deci- 
 'rnals, the roots of these and of any others may be attained to any degree 
 of exactness. 
 
144 
 
 EVOLUTION, 
 
 product from the dividend. (0) Bringdown the next period 
 Jind a divisor as before, by doubling the figures already in 
 the root, and proceed as before. 
 
 The rule will be rendered clear by the following ex- 
 amples : 
 
 What are the square roots of 167772 16 and 45046721 > 
 
 . ... In the first example, having pointed off the 
 
 periods, I find that the first period of 16 is 
 the square of 4 ; the 4 I put in the quotient, 
 and its square under the period ; as there is no 
 remainder, I bfing down the second period 
 77, and for a new divisor I double the root, 
 4, found ; but the 8 being greater than 7, I 
 bring down the next period, and put a cy- 
 pher in the quotient; the double of the root 
 is now 80, and this will go 9 times in 777, 
 the 9 I place in the quotient, and also in the 
 divisor, which is now soo ; this multiplied 
 by the 9, and the product subtracted from 
 the dividend, leaves 491, to which I bring the 
 other period, and double the root 4 09 for a 
 new divisor, which is contained 6 times in the 
 four first figures of the dividend, and proceed 
 as before. . 
 
 16777216(4096 
 16 
 
 809)* '7772 
 7281 
 
 8186)40116 
 49116 
 
 43046721(6561 
 36 
 
 125)704 
 
 1306)7967 
 7836 
 
 13121)13121 
 13121 
 
 The mode of working the second example 
 is evident. 
 
 EXAMPLES. 
 
 Ex. 1 . What is the square root of 1 1 7649 ? 
 Ex. 2. What is the square root of 262144 ? 
 Ex. 3. What is the square root of 531441 ? 
 Ex. 4. "What is the square root, of 1679616 ? 
 Ex. 5. What is the square root of 5764801 ? 
 Ex. 6. What is the square root of 1073741824 ? 
 Ex. 7. Vvshat is the square root of 119550669121 ? 
 Ex. 8. What is the square root of 20 ?* 
 Ex. 9. What is the square root of 300 ? 
 Ex. 10. What is the square root of 1000 ? 
 Ex. 1 1 . What are the square roots of -| ; ^L ; ^_ ? 
 Ex. 12. What is the square root of .25 ? 
 
 NOTE. 
 
 * This and the following examples require additional periods of 
 cyphers, in order to be carried to any degree of accuracy : for 4> 
 is Jess than the root of 50, and 5 is greater than the root, the square 
 
EVOLUTION. 145 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. l. A gentleman desirous of making his kitchen garden, which 
 is to contain 4 acres, a complete, square, I demand what will be the 
 length of the side of the garden? 
 
 Ex. 2. Six acres of ground are to be allotted to a square garden; 
 but for the sake of more wall for fruit, there is to- be a smaller square 
 within the larger, which is to contain 3 acres. I demand the length of 
 the sides of each square ? 
 
 Ex, 3. What is the mean proportional between 12 and 75?* 
 
 Ex. 4. How long must a ladder be to reach a window 30 feet 
 high, when the bottom stands 12 feet from the house P-f 
 
 To extract the cube root. 
 
 I. RULE. (1) Find, by trials, the nearest cube to the 
 given number, and call it the assumed cube. * (2) Say, as . 
 twice the assumed cube added to the given number, is to 
 twice the number added to the assumed cube, so is the root 
 of the assumed cube to the root required nearly. 
 
 What is the cube root of 27455 ? 
 
 Here the nearest root that is a whole number is 30, the cube of 
 which is 27000 : therefore I say, 
 
 As 27000 X 2 + 27455 : 27455 X 2 + 27000 : : 30 
 or 81455 : 81910 : : 30 : 30.1675. 
 
 NOTES. 
 
 of 4 being 16, and the square of 5 beiug 25 : it must therefore be 
 worked thus : 
 
 20(1.4721 6rc. 
 
 16 
 
 84)400 
 336 
 
 687)6400 
 6209 
 
 6942)10100 
 17884 
 
 8944) 121600 
 
 By multiplying the root 4.4721 into itself, rtic answer will be 
 19.99996, &c., which is very nearly, though not quite, equal to 20 ; 
 and by carrying the operation still further, greater accuracy would be 
 obtained. 
 
 * This is found by multiplying the given numbers together, and 
 taking the square root of the product. 
 
 f Square the given numbers, and take the square root of their 
 sum. 
 
 H 
 
346 EVOLUTION. 
 
 It is evident that the true root, omitting the last two figures, is 
 somewhere between 30.16 and 30.17, the former being too little, 
 the latter something too large. By taking the root thus found 30.16, 
 as the assumed cube, and repeating the operation, the root will be 
 had to a still greater degree of exactness. 
 
 Ex. l. What is the cube root of 15625 ? 
 
 Ex. 2. Whrt is the cube root of 140608 ? 
 
 Ex. 3. What is the cube root of 444194947 ? 
 
 Ex. 4. What is the cube root of the difference between 140608 and 
 J4025? 
 
 II. RULE 1. Separate the given number into periods of 
 three figures each, beginning from units place ; then from 
 the first period subtract the greatest cube it contains 9 
 put the root as a quotient, and to the remainder bring 
 down the next period for a dividend. 2. Find a divisor 
 by multiplying the square of the root by 300, sec how often 
 it is contained in the dividend, and the answer gives the 
 next figure in the root. 3. Multiply the divisor by the 
 last figure in the root. Multiply all the figures in the root 
 l>y 30, except the last, and that product by the square of 
 the last. Cube the last figure in the root. Add these 
 three last found numbers together-) and subtract this sum 
 jfrom the dividend ; to the remainder bring down the next 
 period for a new dividend, and proceed as before. 
 
 Ex. 5, What is the cube root of 444194947 ? 
 
 444194947(763 Answer. 
 
 .343 
 
 7 X 7 X 300 =r 14790)101194 
 95976 
 
 *6 X 76 X 300~ 1732800)5218947 1732SOO n divisor 
 ^4700 divisor 5218937 3_ 
 
 ?L ....... 5198400 
 
 88200 20520 = 76X30X0 
 
 7560 z: 7 X 30 X '30 27 =: 3X3X3 
 
 JU6_IZ6 X 6 X 6 5218947 
 
 95976 ' 
 
 Ex. 6. What is the cube root of 40<556 ? 
 Ex. 7. What is the cube root of 65939264 ? 
 Ex. 8. What is the cube root of 3, carried to 2 places cf decimals? 
 Ex. 9. What is the cube root of ? f, ? 
 Ex. 10. What is the cube root of ^ s a ? 
 Ex. 1 1 . What is the cube root of .729 ? 
 JLs, 13, What is the cube root of .003325 ? 
 
ARITHMETICAL PROGRESSION. 147 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. 1. A corn-factor requires a cubical bin that shall hold 840 
 bushels of wheat : I demand the inside length of one of ks sides. See 
 Table, p. 49, note. 
 
 Ex. 2. In a cubical building that measures 2744 feet, what is the 
 length of a side ? 
 
 ARITHMETICAL PROGRESSION, 
 
 When a scries of numbers increases or decreases by some 
 common excess, or common difference, it is said to be in 
 arithmetical progression, such as 1, 3, 5, 7, 9, fyc., and 
 12, 10, 8, 6, 4, $c. 
 
 The numbers which form the series are called the terms 
 ef the progression; of these the jirst and last are called the 
 extremes. 
 
 The first term is called . . * 
 The last term is called . . % 
 The number of terms is called . n 
 The common difference is called d 
 
 The sum of all the terms is called s.* 
 Any three of these terms being given, the others may be easily found. 
 
 I. When the first term a, and the last term z, and th 
 number of terms w, are given, to find the sum of all the 
 terms, 6". 
 
 RULE. Multiply the sum of the extremes by the number 
 of ie tint, and divide by 2, the quotient is the answer : or 
 
 . n 
 
 a + * X s. 
 
 Ex. i . What is the sum of the terms of an arithmetical scries, whose 
 first term is 5, last term 29, and the number of terms 7 ? 
 
 238 
 
 Here $~ 54- 29 X = 34 X n: rr no, the answer. 
 
 2 22 
 
 NOTE. 
 
 * These symbols will be easily remembered ; a and a being the first 
 and last letters of the alphabet, may properly represent the first and 
 last terms of ap.y series ; <h$ reason of the other letters ;s sufficiently 
 obvious, 
 
 H2 
 
148 ARITHMETICAL PROGRESSION. ,' 
 
 Ex. 2. The first and last terms of a series are 8 and ill, and the 
 number of terms 37 : what is the sum ? 
 
 Ex. 3. How many strokes do the clocks of Venice strike in 24 hours, 
 where they strike from l to 24 ? 
 
 Ex. 4. The first and last terms of a series are l and 1000, and the 
 number of terms loo : required the sum. 
 
 Ex. 5. If 100 stones are placed in 3 right line, exactly a yard asun- 
 der, and the first one yard from a basket, what length of ground will 
 a man go over, who gathers them up, orse by one, returning .with 
 each to the basket? 
 
 Ex. 6. What must a man give for 54 timber trees, for which he pays 
 5 shillings for the first, and 2o. for the last, and the prices of the 
 others being in arithmetical progression ? 
 
 Ex. 7 . A butcher buys a drove of oxen, consisting of 32 ; for the first 
 he pays 15s., and for the others he is to pay in arithmetical progres- 
 sion, so that for the last he is to pay 38/. : what will they all come to? 
 
 Ex. 8. Ahorse-dealer sends to a fair 63 hoises. of various kinds and 
 worth, which he is willing to dispose of according to the principles at' 
 arithmetical progression, demanding 3/. only for the fiist, provided he 
 had 5'3l. for the last: how much did he receive for the whole, and 
 what wasthe average value of each hoise ? 
 
 II. The first and last terms, a and z, and number of terms 
 being given, to find the common difference, d. 
 
 RULE. The difference of the extreme terms divided by 
 the number of terms, less 1, will be the common difference 
 sought: 
 
 Ex. i. What is the common difference of an arithmetical progres- 
 sion, whose extremes are 8 and 200, and ihe number of terms 17 ? 
 8 ** 200 200 8 __ 192 __ 
 
 16 16 16 ~~ 
 
 Ex. 2. When the extremes of an arithmetical progression are 6 and 
 57, and the number of terms 18, what is the common difference ? 
 
 Ex. 3. A gentleman gives at Christmas, among his 25 poor neigh- 
 bours, a sum of money in arithmetical progression : to the least needy 
 he gives 5 shillings, and to the poorest, with a very large family, he 
 gives five guineas : what was the common difference ? 
 
 NOTE. 
 
 * This mark *<*, is put for the difference of any two numbers, which 
 ever of them is largest ; thus, suppose a be 10, and z 4, then a & % 
 ~ fy & it may be, a is 4, and z 1 5 ; then a * z ~ \l 
 
ARITHMETICAL PROGRESSION. 149 
 
 Ex. 4. A traveller is out on his journey a month, of which he travels 
 2.5 days ; on the first he rides 7 miles, and on the last, having little to 
 do, he comes 43 miles : how much was the daily increase of his tra- 
 velling, and how many miles did he ride in the whole ? 
 
 III. The extreme terms a and z, and common difference d 
 being given, to find the number of terms n. 
 
 RULE. Divide the difference of the extremes by the 
 common difference, and the quotient increased by unity is 
 
 the ^number sought : or ** + i =: n. 
 
 Ex. i. When the extremes are 4 and 106, and the common differ- 
 ence is 3, what is the number of terms ? 
 
 4u106 106 4 102 
 
 . i. i + l = |- 1 ZI 34 + 1 r= 35 = . 
 
 3 30 
 
 Ex. 2. If the least term be 6, the greatest 216, and the common 
 difference 5, what is the number of terms ? 
 
 Ex. 3, What debt canJ^paid, and in what time, supposing I agree 
 to lay by 35. the first w^^V5. the next, 1 is. the third, and so on in 
 arithmetical progressionjMPnhe last saving be four guineas ? 
 
 Ex. 4. I set out for Hastings, which is 69 miles from this place, and 
 I walk the first day 4 miles, the second 7, increasing every day by 3 
 miles, and on the last 19 miles : how many days will the journey take? 
 
 In addition to the above, the learner may commit 'to memory the 
 following facts on the subject : 
 
 1. If three numbers are in arithmetical progression, the 
 sum of the extremes is equal to double the mean term ; as 
 6, 9, 12, where 6 + 12 = 2 X T = 18. 
 
 2. If four numbers be in arithmetical progression, the 
 sum of the two extremes is equal to the sum of the means ; 
 as 5, 8, 11, 14, where 5 + 14 = 8 + 11 = 19. 
 
 3. When the number of terms is odd, the double of the 
 middle term will be equal to the sum of the extremes ; or 
 of any other two means equally distant from the middle 
 term; as 3, 8, 13, 18, 23, 28, 33', where 3 + 33 =: 2 X 18 
 = 13 + 23 = 8 4- 28. 
 
160 
 
 GEOMETRICAL PROGRESSION!. 
 
 A GEOMETRICAL PROGRESSION is a series of numbers* 
 the terms of which gradually increase or decrease by the 
 constant multiplication or division of some particular 
 number ; as 1, 3, 9, 27 ; 81, 243, &c, or 64, 32, 16, 8, 4, 
 2, 1, |, &c. 
 
 In the first case, the series is increasing by the constant 
 multiplication of 3; in the second, it is a decreasing series 
 by the constant division of 2. It is evident that both series 
 may be carried on for ever. 
 
 The number by which the series is constantly increased 
 r diminished is called the ratio. 
 
 The first term is called 
 
 The last term is called 
 
 The number of terms is can 
 
 The ratio is called r 
 
 The sum of ail the terms is sailed .?. 
 
 jAfc 
 
 4lf 
 canea 
 
 Any three of these terms being given or known, the ethers may fee 
 determined. 
 
 I. Given the first term a, the last term z, and the common 
 ratio r y to find the sum s. 
 
 RULE. Multiply the last term by the ratio, (and from the 
 product subtract the first term, and the remainder divided 
 by the ratio, less one 9 will give the sum of the series ; r 
 
 2 X r a 
 
 ~ 5, 
 
 r i 
 
 Ex. i . The first term of a series in geometrical progression is 5, the 
 last term is 3645, and the ratio is 3 : what is the sum ? 
 
 3645 X 3 5 10935 5 10930 
 
 Here 5 ~ zz z: 5465. 
 
 31 2 2 
 
 For the terms are 5, 15, 45, 135, 435, 1215, and3S45j which> 
 being added together, make 5465. 
 
 Ex. 2. The first and last terms of a geometrical series are 4 and 
 32Q4i7^and the common ratio is 7 : what is the sum? 
 
 Ex. 3,'r'he first and last terms of a geometrical progression are 4 and 
 262.144, and the ratio 4 : what is the sum ? 
 
GEOMETRICAL PROGRESSION. .1*51 
 
 II. Given the first term a, the number of terms w, and the 
 ratio r 9 to find the last term z. 
 
 The last term may be obtained by continual multiplication ^ but a:; 
 that, in a long scries, is a tedious process, we shall give the following 
 rule: i 
 
 1. When the first or least -term is equal to ratio. 
 
 RULE. Write down some of the leading terms, of thcgco~ 
 metrical series, over which place the arithmetical series 
 
 1, 2, 3, 4, fyc. 9 as indices;* find what figures of these in* 
 dices added together. will gicc the index of the term wanted 
 in the geometrical series; then multiply the number s* 
 standing under such indices, into each other, and their 
 product will be the term sought. 
 
 Ex. i. What is the last term of a geometrical series having 13 terms, 
 of which the first is 2, and the ratio 2 ? 
 
 Here the series, with their indices, will stand thus : 
 2', 4% S 3 , 16*, 32*, C4 (i , &c. 
 
 The number of terms being 33, the index to the last term will be 
 13 equal to the indices ||+ 5 + 6, which figures standing over 4, 32, 
 and 64, shew that these?ast are to be multiplied together, and the pro- 
 duct is the term sought ; thus 4 X 32 X 64 zz 3192. 
 
 Ex. 2. What is the last term of the series having 9 terms, of which 
 the first is 3, and the ratio 3 ? 
 
 Ex. 3. What did the last of 12 oxen cost, the first of which was sold 
 for 3s,, the second for gs., and so on ? 
 
 2. When the first term a, of the series, is not equal to the 
 
 ratio r. 
 
 RULE. Write down the leading! terms of the series, and 
 place their indices over them, beginning with a cypher, add 
 together the most convenient indices to make an index less 
 one than the number expressing the place of the term 
 sought; then multiply the numbers standing under such 
 
 * When the natural numbers 1,2,3,4,5, &c., are set over a geo- 
 metrical series, they are called indices, or exponents, and they shew 
 the distance of any term from unity, or from the first term : thus, iu 
 theseries2 1 , 4% 8 3 , 16 4 , 64 5 , 1Q8 G , &c., 1, 2, 3, &c. are the indices, 
 and shew the distance of any term of the series from the first term ; 
 t,he index 5, for instance, shews that 64 is the fifth term in the series. 
 
GEOMETRICAL PROGRESSION, 
 
 indices, into each other, dividing the product of every two 
 by the first term in the geometrical series ; the last quotient 
 is the term required. 
 
 Ex. l . What is the last term of the series, whose first term is 4, 
 ratio 3, and number of terms 15? 
 
 4, 12 1 , 36% 108 s , 324 4 , 97 2 5 , 291 6 6 , &C. 
 
 The number of terms being 15, the index sought must be 14 equal 
 to 6 + 5 + 3, under which stand the terms 2916, 972, and 108, then 
 2916 X 972 , 708588 X 108 
 
 z: 708588, and :z 19131876 =z z zz last 
 
 4 4 
 
 term. 
 
 . Ex. 2. The first term of a geometrical series is 2, the number of 
 terms 12, and the ratio 5, required the last term ? 
 
 Ex. 3. The first term of a geometrical series is i, the ratio 2, and the 
 number of terms 25, what is the last term, and also the sum of all the 
 terms ? 
 
 Ex. 4. The first term of a series is 5, the ratio 3, and the number of 
 terms 16, what is the last term, and the sum of the terms ? 
 
 Ex. 5. A hosier sold 12 pair of stockings, the first pair at 3d,, the 
 second gd., and so on in geometrical progression ; for what did he sell 
 the last pair, and how much had he for the whole ? 
 
 Ex. 6. What would a horse fetch, supposir^it was sold on condition 
 of receiving for it one farthing for the first nail in his shoes, a halfpenny 
 for the second, one penny for the third, and so on, doubling the price 
 of every nail to 32, the number in his four shoes ? 
 
 Ex. 7. A husbandman agreed to serve his master during hay -time 
 and harvest, or five-and-forty clear days> provided he would give him 
 a barley-corn only for the first day's work, 3 for the second, o for tht 
 third, and so on in geometrical proportion ; what would he have to 
 receive in money for his labours, supposing there were half a million 
 of grains in a bushel, and each bushel was worth 4s. ? 
 
 The following facts may be committed to memory : 
 
 1. If three numbers are in geometrical progression, the 
 product of the extremes is equal to the square of the means; 
 as 3, 9, 27, here 3 X 27 = 9 X 9 zz 81. 
 
 2. If four numbers are in geometrical progression, the 
 product of the extremes is equal to the product of the means ; 
 as 2, 4, 8, 16; here 2 X 76 = 4 x 8 = 32. 
 
 3. If the series contain an odd number of terms, the 
 square of the middle term is equal to the product of the 
 adjoining extremes, or of any two terms equally distant 
 
 from them; as 3, 9, 27, 81, 243 ; here 27* a* 3 x 243 sat 
 9 X 81. 
 
153 
 LOGARITHMS. 
 
 LOGARITHMS are artificial numbers, invented for the 
 purpose of facilitating certain tedious arithmetical opera- 
 tions. 
 
 If any series of numbers in arithmetical progression be- 
 ginning with 0, be taken, and a corresponding series of 
 geometrical numbers beginning with 1, the former series 
 will be logarithms to the corresponding numbers in the 
 latter; thus, 
 
 0, 1, 2, 3, 4, 5, 6, 7, s, 9 logarithms. 
 . 1, 2, 4, 8, 16, ^2, 61, 12-8,256, 512 numbers. 
 
 HereO, 1, 2, &c. are the logarithms; of 1, 2, 4, &c., and 
 it will be seen at once, 1. That Addition in logarithms an- 
 swers to Multiplication in common numbers : 
 
 Thus, if the logarithms '2 arid 6 are added together, the sum is 8 
 which answers to the logarithm of 25&, the number that is obtained by 
 the multiplication of 4 and 64, which are the numbers standing under 
 the logarithms 2 and 6. By adding the logarithms 4 and 5 we have 
 9, which stands over 51-2, the number obtained by multiplying toge- 
 ther 16 and 32. Hence the addition of logarithms answers to multi- 
 plication in common numbers. 
 
 2. Subtraction in logarithms answers to division of com- 
 mon numbers. 
 
 Divide 256 by 8, and you have 32, over which stands 5 n 8 3 ; 
 the logarithms standing above. 
 
 3. Multiplication in logarithms answers to involution of 
 common numbers. 
 
 Ex. The square of 8 is 64 ; now 3 is the logarithm answering to 
 8, and 3X2, (because 2 is the index of the square,) is equal to 6. 
 which is the logarithm of 64. 
 
 4. Division in logarithms answers to evolution in com." 
 mon arithmetic. 
 
 Ex. l. The square root of 256 is 16, over which stands the logarithm 
 4 ; which answers to 8 -~ 2, 8 being the logarithm of 256.* 
 
 Ex. 2. The cube root of 512 is 8 ; and o, which is the logarithm 5 
 of 512, divided by 3, the sign of the cube, gives 3, which js the. 
 logarithm of 8. 
 
 The same indices will serve for any geometric series ; but the 
 115 
 
154 LOGARITHMS. 
 
 logarithms generally made use of are those which increase in a tenfoM 
 properties, as 
 
 0. 1. 2. 3. 4. 5. 6. &C, 
 
 1. 10. 100. 1000. 10000. 100000. 1000000. 
 
 Here it is evident, that the logarithms of numbers between i and 
 10, are greater than o, and less than one, as will be seen in the table 
 at the end of the volume, thus the logarithms of 2, 6, 8, &c. are, 
 .3010300, .7781513, .9030900, &C. 
 
 The logarithms of the numbers between 10 and 100, are greater 
 than l, and less than 2 ; thus the logarithm of 15 is 1.1/609 13, and- 
 the logarithm of 95 is 1.9777236. 
 
 The logarithms of numbers between loo and 1000, are greater than 
 2, and less than 3 ; thus the logarithm of 1C5 is 2.2174889, and of 984 
 is 2.992Q951. 
 
 The logarithms between looo and 10000, must be somewhere be- 
 tween sand 4 ; between 10.000 and 100.00O they must be between 
 4 and 5 ; and so on. 
 
 The logarithms in the above series are called indices, which are 
 frequently neglected, the decimal parr only being put down ; thus, 
 if it be required to find the logarithm of 248, it will be sufficient to 
 put down .3944517, and the number being between 100 and 1000, 
 J know the index is 2. Therefore the rule for finding the index is 
 his : 
 
 The index is always one less than the number of figures 
 in the whole number ; or the figures in the whole number 
 must be always one more than the index. 
 
 The logarithm of 248 is 2.3944517 
 
 -*- 2480 3.3944517 
 
 24800 4.3944517 
 
 24.8 1.8944517 
 
 2.48 0.3944517 
 
 .248 1.3944517 
 
 .O248 2.3944517 
 
 .00248 3.3944517 
 
 Here the decimal figures remain the same ; and the only different* 
 re in the indices, which are increased or diminished by unit for every 
 ten- fold increase or decrease of the whole number. It will be observed, 
 that where there is but one whole number, the index will be o ; 
 but if the figures be decimals, as .249, the index is minus one, or l ; 
 if by the prefixing o to the decimal figure, their value is diminished 
 in a ten-fold proportion, then the index is 2, or minus two : if 
 there are two cyphers on the left of the decimal, then the index is 3, 
 minus three, and so on. 
 
 We shall now proceed to shew the use of Logarithms and the mannex 
 of working by them 5 and give seme instances of their application. 
 
 
LOGARITHMS. 16$ 
 
 First. To FIND THE LOGARITHMS OF NUMBERS. 
 
 I. To find the logarithm of .any whole number less than 
 
 100. 
 
 RULE. Look for the given number in the first page of 
 the table under No., then directly against it you have its 
 logarithm with the proper index : 
 
 Thus, if I want the logarithms of 26, 58, and 87, I find under 
 No. by the side of 26 the logarithm 1.4149733 
 
 _ 58 1.7634280 
 
 . 87 1.93Q5193 
 
 Ex. "Write down the logarithms of 6, 18, 30, 54, 72, 90, and 99. 
 
 II. To find the logarithm of any whole number between 
 
 100 and 1000. 
 
 RULE. Seek in the table for the given number in the 
 left-hand column, opposite to which is the logarithm 
 sought, tvith its proper index 2. 
 
 If I want the logarithms of 176, 375, 684, I find 
 
 Opposite to the number 176 the logarithm 2.2455127 
 
 375 , 2.5/40313 
 
 . 084 2.3350561 
 
 Ex. Write down the logarithms answering to the numbers 486, 5>3 , 
 671, 756, 843, 921. 
 
 III. To find the logarithm of any whole number between 
 
 1000 and 10000. 
 
 RULE. Find, as before, the logarithm belonging to the 
 first three figures ; take, from the margin, the difference 
 between this and the next logarithm, which multiply by 
 the fourth figure of the said number ; add the product to 
 the logarithm first found, prefix the index 3, and it will be 
 the logarithm required very nearly. 
 
 Ex. 1. To find the logarithm of 4528. 
 
 In the table, against 452, is the decimal . .6551384 
 The difference marked in the margin is 959, 
 
 which multiplied by 8, the fourth figure, m 7672 
 
 Answer . . 3.6559050 
 
 If the fourth figure had been o instead of 8, then the decimal 
 .6551384, with the index 3, would have been the logarithm sought. 
 
156 LOGARITHMS. 
 
 Ex. 2. To find the logarithm of 8884- 
 
 Here the decimal against 888 ~ .9484130 
 
 Difference is 488, which multiplied by 4, gives 1952 
 
 Answer - - 3.9486082 
 
 Ex. 3. Find the logarithms answering to 2465, 4265, 6425, 5387, 
 3420, 58/6, 8464, and 4932. 
 
 IV. To find the logarithm answering to a decimal. 
 
 RULE. Work precisely as if it were a whole number, re- 
 membering that the index mnst always be one less than the 
 figures counted in the integral part ; that is, one less than 
 the figure before the decimal point : 
 
 Thus, if instead of the foregoing examples, the numbers be 452.8 
 and 88.84, tfeen the logarithm of 452,8 2.6559056, and 
 88.84 1.9486O82 
 
 But if they were of still less value, by removing the point forwarder, 
 then the logarithm of 4.528 :z: 0.6559056 
 .8884 ~ 1.9486082 
 
 If the given number were a fraction, as |, or , then the fractions 
 must be reduced to decimals, and the logarithms are to be found 
 as before thus (See the Tables) 
 
 the logarithm jof | zz .875 rz 1.9420081 
 
 , i ~ 
 
 1.9719711 
 
 NOTE. . 
 
 * This is the usual method ; but, as is evident, " The logarithm of 
 a fraction may le found ly subtracting the logarithm of the dejiominalor 
 from the logarithm of the numerator" Thus, 
 
 The logarithm of 7 is .8450980 
 
 The logarithm of 8 is . .9030900 
 
 1.94-20080 
 
 And, The logarithm of 15 is . . 1.1760913 
 The logarithm of 16 is . . 1.2041200 
 
 1.9719713 
 
 In these examples, as one is borrowed in the subtraction, we must 
 put down 1, or minus in the answer. 
 
LOGARITHMS. 157 
 
 Secondly. To FIND THE NATURAL NuMBEk OF ANY 
 GIVEN LOGARITHM. 
 
 I. To find the natural number answering to any given 
 
 logarithm, to 2 or 3 places of figures. 
 
 RULE. Look in the table for the given logarithm, and 
 if it be exactly found, or nearly so, the natural number 
 stands against it in the left-hand column : 
 
 Thus, the natural numbers answering to 1.8617278, 2.1216834, 
 and 2.5312Q93, are 23, 132, and 339. The first is found in the table ; 
 the others are not to be met with accurately, and therefore we take 
 the natural number belonging to the next nearest logarithm. 
 
 Ex. What numbers answer to the following logarithms 2.4650000 
 2.8148132; 2. 5346780 5 1.9684829; 2.2454674; and 2.8819765 ? 
 
 II. When great accuracy is required, and the logarithm is 
 
 not to bo found very nearly in the table. 
 
 RULE. Seek in the table the difference between the next 
 greater and next less logarithms, and say, 
 As this difference 
 Is to unity or 1, 
 So is the difference between the given logarithm 
 
 and the next less logarithm 
 To a fourth number. 
 
 This fourth number is to be added to the natural number of 
 the less tabular logarithm, which gives the number sought. 
 
 Ex. What is the natural number of 2.4723564 ; 
 
 By the table, I find the number is between 296 and 297, 
 next less . . 2.4 7 12917J 14647 - diff . 
 next greater . 2. 4727554 J 
 
 given logarithm ^^ } 10647 - 2ddirT. 
 
 As 14647 : 1 : : 10647 : .73 nearly, which added to 296, give 
 for the true answer 296.73.* 
 
 * For all common purposes there will be no need of working the 
 numbers: thus it will be seen at once that 10657 is nearly f of 14053, 
 and we know that the decimal of J is .75, of course ./5 might be 
 substituted, which would be sufficiently near. When, however, the 
 index is more than 3, it becomes necessary to be as exact as possible, 
 because the answer is then in whole numbers. 
 
153 LOGARITHMS, 
 
 If the index of the given logarithm had been 3 instead of 2, then 
 the answer would have been 2967. 3 : and if the index had been 4, 
 the answer would have been altogether a whole number, as -29673. 
 
 Ex. What are the natural numbers answering to 2.7896453; 
 3,5648750, and4.2l6S435? 
 
 MULTIPLICATION BY LOGARITHMS. 
 
 To find the product of two given whole, or mixed numbers. 
 RULE. Find the logarithm of each given number , and 
 their sum will be the logarithm of the product , whose cor- 
 responding number in the tables is the answer. 
 
 Ex. i. Multiply 84 zz log, 1.9242703- 
 By - 25 log. 1.39; 9400 
 
 Product 2100 3.322:2193, which answers to 210 in 
 
 the table; but the index being 3, there must be .four figures in the 
 answer, or 2-100. 
 
 Ex.2. Multiply 41.5 ~ log. 1.6180481 
 By - 7-24 iz log. 0.8597386 
 
 Product 300.4 zr 2.47 7 7 867* 
 
 EXAMPLES. 
 Multiply 59 b> 35 : 14 by 4 : 76.3 by 3.24 : and 2.76 by 345. 
 
 DIVISION BY "LOGARITHMS. 
 
 To divide a whole or mixed number, by a less whole or 
 mixed number. 
 
 RULE. From the logarithm of the dividend subtract the 
 logarithm of the divisor, and the remainder is the logarithm 
 of the quotient. 
 
 NOTE. 
 
 * To prove the truth of this, I look into the table, and find the 
 next greater . 
 
 As 14452 : 1 :: 6054 : .4, which added to 300, gives 30Q.4. 
 The same method is applicable in all other cases. 
 
LOGARITHMS. 159 
 
 Ex. 1. Divide 624 ZZ log. 2.795184'S 
 By 26 ZZ log. 1.4149733 
 
 Quotient 24 ~ log. 1.3802113 
 
 Ex. 2. Divide 1221 =: log. 3.0867143 
 By - 81 .4 n: log. 1.9106244* 
 
 15 rz log. 1.1760899 
 
 Ex. 3. Divide 34. S6 log. 1.542326S 
 By - 8.3 n log. .9190781 
 
 4.2 m log. .6232487 
 Examples. Divide 56 by 4 ; 8650 by 2.5 ; and 1870 by 55* 
 
 PROPORTION, OR THE RULE OF THREE BY 
 LOGARITHMS. 
 
 RULE. Add the logarithms of the second and third terms 
 together ', and from the sum subtract I he logarithm of the 
 first ; ike remainder is the logarithm of the fourth term, 
 
 Take the 3d, 4th, and 5th examples in the Rule of Three, p. 94. 
 
 E. shiil. E. 
 
 If 4 : 18 : : 23 
 
 ,6020600 : 1,2'5S2725 :: 1.4471580 
 1.2552725 
 
 2.7024305 
 .6020600 
 
 2.1003705, this is found by 
 the table to answer to the number 126 shillings, or 6l. 6s* 
 
 * I find the lo-g. 122 2.0863598 ; the difference between this 
 
 and the next is 3545 -1 Q 
 
 which is co be added, J *"I 
 
 2.0867143; but the index is now 3. 
 
 This is not quite accurate, as will be observed by the answer j bujt 
 it is sufficiently so for common purposes ; and in working loga- 
 rithms to great nicety, where there are more tfean 3 or 4 figures, 
 larger tables are required than this work admits of. The learner will 
 see the use of them by this specimen, and will, after this, find no 
 difficulty of pursuing the study on a more enlarged scale, by the aid 
 of Button's Tables, 
 
160 
 
 LOGARITHMS. 
 
 yards. 
 Ex.4. If 6 
 
 .7781513 
 
 shillings. 
 
 24 
 1.3802112 
 
 bush. 
 Ex. 5. If 8 
 
 .9030900 
 
 shill. 
 
 9.5 
 
 .977/230 
 
 yards. 
 
 81 
 
 1.9084850 
 1.3802112 
 
 3.2886962 
 .7781513 
 
 2.5105449 ~ 324 shil- 
 ling?, or 161. 45. 
 
 bush. 
 
 35 X 36 
 
 3.1003705 
 
 .9777236 
 
 4.0780941 
 .9030900* 
 
 3.1750041 n 1496.25 
 74/, 165. 3d. 
 
 INVOLUTION BY LOGARITHMS. 
 
 RULE. Multiply the logarithm of the root by the index of 
 the power ; thus, to square any number, multiply its loga- 
 rithm by 2 ; and to cube a number, multiply its logarithm 
 by 3, and so on. 
 
 Ex. 1. What is the square of 25 ; and the 5th power of 2 ? 
 
 25 i= log. 1.8979400 
 25 2 
 
 2 5 . = .301030 X 5 =Z 
 1.505150 ~32. 
 
 625 Z= log. 2.7958800 
 
 Ex. 2. What is the third power of ? See p. 142. 
 
 I bring | to a decimal zz .875, the logarithm, of this is 1.9420008 ;* 
 this multiplied by 3 gives 1.826024, which answers to the decimal 
 .67. And *it will be found that | X J X ~ fff =z -67. This ad- 
 
 * By the table, I find that the number answering to this logarithm 
 is more than 1490 : by the rules already given, I put down 
 
 :lllllfz } 29 5 first diff - :ns?863 } isi78 second diff " therefore 
 
 As 2905.0 : l : : 18178 : 625, which, is to be annexed to 149 ir 
 1499.25, because the index is 3. 
 
LOGARITHMS. 161 
 
 mHb of other proofs : thus the log. 7 .845098 
 
 * 8 r: .903090 
 
 therefore | 1.942008, this multiplied 
 
 ^_ .645098 
 
 by 8, gives 1.826024, as above.* Or f 3 n X 3 ~ 
 
 2.535294 .903090 
 
 , and. subtracting the denominator from the numerator, we 
 
 2.709270 
 
 have 1.826024 IZ .67, 
 
 EVOLUTION BY LOGARITHMS. 
 
 To extract the root of any number. 
 
 RULE. Divide the logarithm of the number by the pro- 
 posed index, and the number answering to the quotient is 
 the required root. 
 
 Ex. 1. What is the square root of 225 ? 
 
 Log. of 2*25 iz 2.3521825, which ~- by 2, gives 
 1.1760912, which is the log. of 15. 
 
 Ex. 2. What is the square root of 6561 ? 
 
 Logarithm of 6561 = 
 
 2)3.8169694 
 
 Logarithm of 1024 rr \ 3 * 
 
 1.9084847, which answers to 81 5 
 of course the square root 61" 6561 is found to be 81* 
 
 Ex. 3. What is the 5th root of 1024 ? See p. 143. 
 
 1.0086002 
 16948 
 
 5)3.0102950f 
 
 .0020590, which answers very 
 nearly to 4, the fifth root of 1024. 
 
 Ex. 4. What is the square root of .25 ? See p. 144. 
 
 The log. of .25 zz 1.39794, which -~ 2, gives 
 , 69897 ~ .5 Answer. 
 
 Ex. 5. What is the square root of 144 ? 
 Ex. 6. What is the cube root of 1728 ? 
 
 NOTES. 
 
 * The learner may in this way work all the examples in p. 142. 
 
 ff It may not occur at first sight to the reader, how 1.942008 
 X 3 should give 1.826024; but if he divide the expression into 
 two sums, which he may, as 1 and .942008, and multiply each 
 by 3, he gets 3 and 2.826024. The plus 2 in the latter expression 
 will destroy 2 of the former ; the remainder will of course be 
 1.826024. 
 
I6'2 
 INTEREST. 
 
 INTEREST, is the sum of money paid, or allowed for the 
 loan or use of same other sum, lent for a certain time, ac- 
 cording to a fixed rate. 
 
 The sum lent) and on which the interest is reckoned, is 
 called the PRINCIPAL. 
 
 The sum per Cent, agreed <w as interest) is called the 
 RATE. 
 
 The principal an J interest added together, is called ihc 
 AMOUNT. 
 
 Interest is distinguished into SIMPLE and COMPOUND. 
 
 SIMPLE INTEREST, is that which is reckoned on the prin- 
 cipal only, at a certain rate for a year, and at a propor- 
 tionately greater or less sum, fvr a greater or less term : 
 thus, if 5/. is the rate of interest of 100/. for one year, 
 10/. is the interest for two years, 20/. for four years, and 
 so on. 
 
 RULE (1). Multiply the principal by the rate, and divide 
 the product by 100, and the quotient is the interest for one 
 year : 
 
 250 5^ 5 
 
 Thus the interest of 250/.at 5 percent, is ~ 12J, los. 
 
 100 
 
 (2). Multiply the interest for one year by the number oj 
 years, and the product is the interest for the same : 
 
 Thus the interest of 250/. for 7 years is 12/. 10*. X 7 ~ 87/. 10*. 
 
 (3). If parts of a year be given, they must he worked for 
 by the aliquot parts of a year, as in Practice, or by the 
 Rule of Three Direct. 
 
 Ex. 1. What is the interest of 8531. 10*. for 4 years and 
 8 months, at 5 per cent, per annum ?* 
 
 * By law, more than 5 per cent, cannot be received as interest oi 
 money in this country ; though at various periods of our history 
 different rates of interest have been ailowed, as will be evident from 
 the following table : 
 
INTEREST. 
 
 163 
 
 . s. 
 
 53 10 6 
 5 2 
 
 . s. d. 
 
 J 42 13 6 rz 
 
 i> 4 
 
 interest for one year. 
 
 Here I multiply the in- 
 terest for one year by 4, 
 for the number of years, 
 and take parts for the 9 
 months, by saying 6 months* 
 is the half, and 2 months 
 the third of that, as in 
 Practice. 
 
 .42,67 10 
 20 
 
 shill. 13.50 
 pence 6.00 
 
 170 14 
 21 6 9 
 723 
 
 .199 3 
 
 Answer, - 190Z. S*. od. 
 
 To fnd the amount t I must add the principal to the interest. In 
 
 this example, the amount is equal to 853/, 105. + 100/. 3$, :n 
 
 Ex. 2. What is the amount of 142/. 10*. for four years 
 and 52 days, at 4-J per cent ?* 
 
 . s. d. 
 
 6 8 3 interest for one year. 
 
 i 
 
 25 13 z: interest for four years, 
 To find the interest for the 52 days, I say> 
 
 In 1255 ~ 
 
 JSOTE. 
 
 . s. 
 
 50 
 
 d. 
 
 o per cent, per annum was 
 given as interest. 
 
 1270 to 1307 45 O 
 
 1422 to 1470 15-0 
 
 1545 it was restricted to 10 o 
 16-25 reduced to 80 
 
 1645 to 1600 6 O 
 
 1660 to 16cjO 766 
 
 1690 to 1697 7 10 O 
 
 1697 to 1706 600 
 
 ]/l4 to the present time 500 ~^ 
 
 In many parts of the world a much higher rate of interest & 
 given, and also in the colonies belonging to this country. In India, 
 for instance, 12 per cent, is the legal interest lor money : and in 
 the English settlements in New South Wales, the rate of interest is 
 fixed at 8 per cent. 
 
 * In the courts of law, interest is always computed in years, 
 and days; but in computing the interest on the public 
 
164 INTEREST. 
 
 days. . .5. d. days. 
 If 365 : 6 8 3 : : 52 
 
 20 
 
 . s. d- 
 
 25 13 128 
 18 3^: 12 
 
 26 11 3 4: IT. interest 1539 
 142 10 o zz principal 52 
 
 169 l 3^; amount. 3078 
 76y5 
 
 12 
 
 365) 8002 8 (2 l 
 
 18 3;f: interest for 52 days. 
 
 Ex. 3. What is the interest of 46 1 /. at 4 per cent, for 5 years ? 
 Ex. 4. What is the interest of 230/. 155. for 6j years, at 5 per cent, 
 per annum ? 
 
 Ex. 5. What is the amount of 225Z. for ^ years, ajt 3 per cent, per 
 annum. 
 
 Ex. 6. How much shall I have to receive at the end of 5 years for 
 350. supposing 4^ per cent, be allowed as interest? 
 
 In most computations relating to simple interest, the work is short- 
 ened, if the interest of ll. for a given term is known, as the interest 
 of any other sum for the same term will then be found by only mulr 
 tiplying by the given sum, 
 
 The interest of ll. fora year must be in the same proportion as 
 the interest of loo/, to its principal; therefore, at 5 per cent., W 
 say, asiopZ. : bl. : : ll. : ,05/. Hence the interest of ll. for one year 
 
 . . 
 
 At 3 per cent, is - - - ,03 
 3j - - - - ,035 
 
 4 ----- ,04 
 4j ,045 
 
 5 ----- ,05 
 
 Ex. 7. What sum will one penny amount to in 1808 years, at s 
 per cent. ?* 
 
 bonds of the South-Sea and East-India companies, the time generally 
 taken is in calendar months and days ; and on Exchequer- bills, in 
 quarters of a year and days. 
 
 * Here the sum is fal. zi .004166 ; this multiplied by 1808, and 
 the product multiplied by .05, gives something more than 7s. 6%d. 
 See COMPOUND INTEREST, where the difference between Simple and 
 Compound Interest will be put in a most striking point of view by this 
 same question. 
 
INTEREST. 
 
 The Interest of One Pound for any number of Years. 
 
 165 
 
 
 Years. 
 
 3 per 
 Cent. 
 
 3 P cr 
 Cent. 
 
 4 per 
 
 Cent. 
 
 4^ per 
 Cent. 
 
 5 per 
 Cent. 
 
 
 
 10 
 
 ,3 
 
 ,35 
 
 ,4 
 
 ,45 
 
 ,5 
 
 
 
 20 
 
 ,6 
 
 ,7 
 
 >8 
 
 ,9 
 
 1,0 
 
 
 
 30 
 
 ,9 
 
 *1,05 
 
 1,2 
 
 1,35 
 
 1,5 
 
 
 
 40 
 
 1,2 
 
 1,4 
 
 1,6 
 
 1,8 
 
 2,0 
 
 
 
 50 
 
 1,5 
 
 1,75 
 
 2,O 
 
 2, '2 5 
 
 2,5 
 
 
 
 60 
 
 1,8 
 
 2,1 
 
 2,4 
 
 2,7 
 
 3,0 
 
 
 
 70 
 
 2,1 
 
 '2,45 
 
 2,8 
 
 3,15 
 
 3,5 
 
 
 
 80 
 
 2,4 
 
 2,8 
 
 3,2 
 
 3,6 
 
 4,0 
 
 
 
 90 
 
 2,7 
 
 3~,15 
 
 3,6 
 
 4,05 
 
 4,5 
 
 
 
 100 
 
 3,0 
 
 3,5 
 
 4,0 
 
 4,5 
 
 5,0 
 
 
 The 365th part of the yearly interest is always considered as the 
 proper inteiest for a day, and its multiples as the interest for any 
 number of days : thus, at 5 per cent., the Irgal rate, the interest 
 
 for a. day is -^ .0001369; and the interest for 12 days, at the 
 
 365 
 
 same rate, is .0001369 X 12 0016428. Hence, by means of the 
 following table, all calculations at 5 per cent. Simple Interest are 
 easily performed, for any number of days. 
 
 days 
 
 Amount. 
 
 days 
 
 Amount. 
 
 days f Amount. 
 
 lays 
 
 Amount. 
 
 1 
 
 ,0001369 
 
 26 
 
 ,0035616 
 
 51 
 
 ,0069863 
 
 .76 
 
 ,0104109 
 
 2 
 
 ,0002739 
 
 27 
 
 ,0036986 
 
 52 
 
 ,0071232 
 
 77 
 
 ,0105479 
 
 3 
 
 ,0004109 
 
 28 
 
 ,0038356 
 
 53 
 
 ,0072602 
 
 78 
 
 ,0106849 
 
 4 
 
 ,0005479 
 
 29 
 
 ,0039726 
 
 54 
 
 ,0073972 
 
 79 
 
 ,0108219 
 
 5 
 
 ,0006849 
 
 30 
 
 ,0041095 
 
 55 
 
 ,007^,2 
 
 8O 
 
 ,0109589 
 
 6 
 
 ,0008219 
 
 31 
 
 ,0042465 
 
 56 
 
 ,0076712 
 
 81 
 
 ,0110958 
 
 7 
 
 ,0009589 
 
 32 
 
 ,0043835 
 
 57 
 
 ,00780*9 
 
 82 
 
 ,0112328 
 
 8 
 
 ,0010958 
 
 33 
 
 ,0045205 
 
 68 
 
 ,0079452- 
 
 83 
 
 ,01 13698 
 
 9 
 
 ,0012328 
 
 34 
 
 ,0040375 
 
 59 
 
 ,0080821 
 
 84 
 
 ,0115068 
 
 10 
 
 ,0013698 
 
 35 
 
 ,0047945 
 
 60 
 
 ,0082191 
 
 85 
 
 ,0116438 
 
 11 
 
 ,0015068 
 
 36 
 
 ,0049315 
 
 61 
 
 ,0083561 
 
 86 
 
 ,0117808 
 
 12 
 
 ,001* t^H 
 
 37 
 
 ,00>0634 
 
 02 
 
 ,0084931 
 
 7 
 
 ,0119178 
 
 13 
 
 ,0017808 
 
 38 
 
 ,0052054 
 
 63 ,0086301 
 
 *>s 
 
 ,0120547 
 
 14 
 
 ,00 1 9 1 7 
 
 39 
 
 -0053424 
 
 64 ,0087671 
 
 89 
 
 ,0121917 
 
 15 ! ,002U54/ 
 
 40 
 
 ,0054794 
 
 65 ,0089041 
 
 90 
 
 ,0123287 
 
 16 _ ,OO>.10i; 
 
 41 
 
 ,0^56164 
 
 66 ,0090411 
 
 01 
 
 ,0124657 
 
 17 : ,00 -..;;.. -7 
 
 42 
 
 ,0057 34 
 
 67 ,0091^780 
 
 92 
 
 ,0126027 
 
 18 ,00246i7 
 
 4.J 
 
 ,00589 4 
 
 68 ,0093150 
 
 93 
 
 ,0127397 
 
 19 ,0026u27 
 
 44 
 
 ,0060-274 
 
 69 ,0034520 
 
 94 
 
 ,0128767' 
 
 20 jOt9-7397 
 
 45 
 
 ,0001 6 '13 
 
 70 ,0095890 
 
 95 
 
 ,0130137 
 
 21 ,OJ '8767 
 
 46 
 
 ,0063013 
 
 71 ,0097260 
 
 96 
 
 ,0131506 
 
 22 
 
 ,OO>,0 1 3/ 
 
 4" 
 
 ,0064383 
 
 72 ,0098630 
 
 97 
 
 ,0132876 
 
 23 
 
 ,0031506 
 
 48 > ,0. .-65753 
 
 73 ,0100000 
 
 98 
 
 ,0134216 
 
 24 
 
 ,0032876 
 
 *y , ,JuG/ .123 
 
 74 ,0101369 
 
 99 
 
 ,0135616 
 
 5 
 
 } 0034246 
 
 50 1 ,0068493 | 75 ,0102739 llOO 
 
 ,01 3< 
 
166 COMMISSION AND BROKERAGE. 
 
 RULE. Multiply the figures corresponding with the nuw- 
 ber of days by the sum : 
 
 Thus, if the interest of J5l. for 61 clays be required : I find opposite 
 to 61, the number .0083561, which multiplied by 75, gives .626705 
 of a pound, which reduced, is equal to I2.s-. 6%d.* 
 
 Ex, 1. What is the interest of 155/. for 49 days; for 76 days, for 
 184-f- days, and for 198 days ? 
 
 Ex. 2. How much do I lose by suffering 37 5 1. to lie at my banker* 
 9V days, instead of laying it out in Exchequer bills or India Bonds ? 
 
 COMMISSION AND BROKERAGE. 
 
 COMMISSION is an allowance of a certain sum per cent, 
 to a correspondent or agent, for buying vnd selling goods 
 for his employer , or to a banker for. drawing bills and 
 managing accounts. 
 
 BROKERAGE, though of a different name, is of the same 
 nature as COMMISSION. 
 
 Ex. 1. A salesman at Smithfield, in the course of a 
 year, sells for his correspondents 1120 loads of hay, at the 
 
 * Though it is the most convenient in common practice to make 
 use of tables for finding the interest for days, yet the same may be found 
 by the following 
 
 RULE. Multiply the given sum ly the numler of days, and divide' 
 ly 7300. Ex. What is the interest of 7 >2J. for 56 days? 
 
 712 X 56 
 
 ZZ bl, 95. 2f d, 
 
 7300 
 
 f The interest for any greater number of days than are contained 
 in the table, is easily found by means of it: thus, to find the interest 
 of loot, for 165 days; by the table, the interest for 100 days is 
 .0136986 X 100 zz 1.36986, and for 65 days it is .0089041 X 100 
 m 89041 ; these sums added together give 2.26027 2/. 5s. 2jd. for 
 the interest required. 
 
 J See the next section but one, p. 168, fire. 
 
 This and the following rules of INSURANCE, BUYING and SELL- 
 ING of STOCKS, are all worked in a similar manner to the rule of 
 Simple Interest. If, therefore, the pupil is ready in the examples 
 already given, he will find no difficulty in what fellows. 
 
COMMISSION AND BROKERAGE. 167 
 
 average price of 57. 10$. per load ; and 620 loads of straw, 
 at 55$. per load : 1 wish to know the commission money, 
 at 2 J per cent ? 
 
 .620 
 
 124O 
 310 
 , 155 
 
 '.6160 what the hay sold for. 
 
 1705 =: what the straw sold for. 
 
 .Answer, 176^, igs. 
 
 3.00 
 
 Ex.2. A Manchester manufacturer allows his agent in London 
 4j per cent, for goods sold by him ; in the course of the year 1807 
 be sold to the amount of 15,400/., what was his commission for that 
 year, and how much was the agent's clear gains, supposing his losses 
 on the year's account, by bad deb^s, amounted to 2-2 5/. 105. 6d. ? 
 
 Ex. 3. A Liverpool merchant sells goods in a year, for his American 
 correspondents to the amount of 144,454/. 105., on which he reckons 
 his clear gains at th rate of | per cent., what is his income on this one 
 concern ? 
 
 Ex. 4. "What is the commission of 1206/. los. 6rf. at 3f per cent. ? 
 
 Ex. 5. A bookseller in London aHows his agent in America 5 per 
 cent, commission ; what does he pay him for the remittance of 
 8540/. 155. gd. ? 
 
 Ex. 6. What is the brokerage of 1210Z., at J per cent. ? 
 
 Ex. 7. What is the claim of a broker, at 3| per cent, on 1550J. 
 ios. lod. ? 
 
 Ex. 8. What is the commission on 1000 guineas, at | per cent. ? 
 
 Ex. 9. What have I to pay my broker for the sale of goods to the 
 amount of 9950^. 95., at i per cent. ? 
 
 Ex. 10. What will the commission of a country banker amount 
 to on 123141. as. gd., at i per cent. ? 
 
 Ex. u> What is the brokerage of 1526*. IQS. Qd,> at if per'cerit. ? 
 
168 
 BUYING AND SELLING STOCKS.' 
 
 STOCK is a general name for the capitals of our trading 
 companies. It also signifies any sum of money which has 
 been lent to Government, on condition of receiving a certain 
 interest till the money is repaid. 
 
 The price of stocks, or rates per cent, are the several 
 sums for which .100 of those respective stocks sell at any 
 given time. 
 
 Thus, on the 2d of March, 1808, I bought tool. Consols, at the 
 rate of 64/. per cent. ; of course the purchase cost 320/. But I paid 
 the broker ith, or 25. 6d.* per cent, for the purchasing, that is, 
 125. 6^., so that my purchase cost me 32oZ. 125. 6d.; for which I 
 shall receive interest 15/. per annum, so long as I keep the same.*!' 
 
 NOTES. 
 
 * The brokerage is 2s. 6d. (or 1) per cent, on the capital pur- 
 chased : on TERMINABLE Annuities it is 2s. 6d. per cent, on the sum 
 laid out. See . 6. of this note. 
 
 f* I shall in this note give the price of stocks for one day, and an 
 explanation, so as to render the information, on this head, contained 
 in the papers, intelligible to the youngest reader. 
 
 PRICE OF STOCKS. FEB. 20. 
 
 Bank Stock 226 
 
 India Stock 
 
 3 per Cent. Red. 62| 63| 63 
 
 3 per Cent. Cons. 62||1 
 
 4 per Cent. Cons. 80| 81 1 
 
 5 per Cent. Navy 95| 96! 96 
 Bank Long Ann. i;| is 
 
 Omnium l{ 
 India Bonds 25. dis. 
 ( Imp. Ann. 8 1-16 
 *Ex. Bills 15. dis. 15. pre. 
 3 per Cent. Imp. 62^ i 
 Lottery Tickets ISL 
 Cons, for Feb. 25. 6>2i 
 
 1. Bunk Stock 226 ; that is, 226J. must be be given on that day to 
 purchase loo/, of that Stock, the annual interest of this is about 10 or 
 11 per cent. 
 
 2. India Stock ; none of this stock was sold on the day 
 
 3. 3 per Cent. Red. 62f, 63^, 63. The price of this stock fluctuated 
 in the course of the day; it began at 62|, or Gj/.l. ijs. 6d. ; it rose to 
 631, or 63/. 2,9. 6d. ; and when the market, FS it is called, closed, the 
 value of looZ. in the 3 per Cent. Reduced w^ 63/. exactly. 
 
 4. 3 per Cent. Cons. 62|, |, J. This stock fluctuated as the last, 
 viz. from 62/. 7-*. 6</. to f><2l. ] 25. 6d., and then back to 62/. 10$. The 
 reason of this stock being of less value than the 3 per Cent. Reduced 
 is, that more interest was due upon the former than on the Utter ; that 
 is, half a year's interest is due at Lady Day on the Reduced, but 
 the half yeaYs interest on the Consols is net due till' Midsummer. 
 
STOCKS. 169 
 
 Ex. i. "What will 5QOl. 3 per cent. Consols cost, at 61 J per cent. ? 
 500u at 61/. 105. 307 10 * j 
 
 Brokerage - o 12 6 9? 
 
 308 2 6 
 
 Ex, '2. What will iar>Z. 4 per cent Annuities cost, at 69 J per cent. ? 
 
 Ex. 3. What will 1128/. 6s. sd. Reduced 3 per cents, cost, at 61-J 
 per cent. ? 
 
 Ex. 4. A person sells 1000/. 3 per cent. Consols, at 70^ per cent., 
 and purchases 635t. 175. (jd. Navy 5 per cents, at 94f per cent., what 
 additional interest docs he get? " ^ 
 
 NOTE. 
 
 5. 4 per cent. Cons., 5 per cent. Navy, and 3 per cent. Imp., will 
 be understood from what has been said. 
 
 6. Bank Long Ann. 172 to '18. This refers to certain annuities 
 granted for a term of years ; the market price of which on this day was 
 from 17 J to is years, that is, if I wish to purchase 50/. per ann. of 
 these annuities, I must at the lowest price pay 501.X 17for 8Q3/. 15s., 
 and at the highest 50 X 18, 01 QOOl. : and for this 893/. 155., or 900^ , 
 I should be entitled to 50l. per annum for about 52 years ; the time 
 when these annuities terminate. Hence these are called terminable 
 annuities. Imp. Ann. 8- 1-16, or S-.L, is of the same kind, but worth 
 only 8-^ yeais purchase, because they terminate so much sooner; that 
 50/. per annum in these might be purchased for 403?. 2s. C>d. 
 
 7. Omnium, ij- pic. This is a woul that refers to the several sorte 
 of stocks in which a new loan is made : for instance, if government 
 borrow 20 millions, arid give to each lender, for every lOQl. so pur- 
 chased, ioo/. 3 per cent. Consols, sal. in the Reduced, and the rest in 
 Long Annuities ; then this stock, the moment it is subscribed, is sale- 
 able, and while the different articles are sold together, it is styled om- 
 nium ; and l premium mentis, that a person, to purchase 1007. of 
 this loan, must pay l^-, or il. 55. more than the original lender; had 
 it been l$ discount, then the purchase would have been ll. bs. less 
 than the original cost, or gs/. 135. 
 
 8. India-Bonds, as. dis.: this phrase shews, that the bonds of looi 
 given by the East India Company art at 2.?. each discount; that is\ 
 to purchase 9 of these I must pay 899'. 2s. instead of QOO/. 
 
 9. Ex. Bills, 15. dis., is. pre., shews that Exchequer bills of IOQL 
 each, fluctuated in value from is. discount to is. premium ; atone; 
 part of the day 10 of them would have been purchased for 10 shillings 
 less than looo/., and at the close of the market 10 shillings more than 
 3000/. must have been given for them. It may be observed, that 
 India-Bonds and Exchequer-bills aie convenient stocks to lay . 
 
 out in, because they may be sold at any time, and d 
 seldom more thai: a l>:w shillings per cent. 
 
 10. Lottery Tickets, ixl., shews the price o: 
 time being. 
 
 1-1. Consols for Feb. 25, 6-2-?r, shev that. r:nnie persons V.ad i 
 stock in anticipation, and agreed to give foi 
 the rate of ii?.l. 10^. p':r cent. 
 
170 STOCKS. 
 
 Ex. 5. What will so/, per annum Long Annuities cost, at I7^yeais 
 purchase ? 
 
 Ex. ^ VfczJ shall I receive for 450^Bar.k Stock, at 213 percent.? 
 
 Ex. flf How much 3 per cent. Consols must I purchase, to produce 
 an income of 120Z. per annum ? 
 
 Ex. 8. How much reduced 3 per cents, can I purchase for 500/., the 
 price being 60| per cent. ? 
 
 Ex. 9. f What will 2 197^. I3s 4 4d. 4 per cents, cost, at 7 sf percent.? 
 
 Ex. 10. Whgt isrthe difference on 1200J. 3 percent. Consols bought 
 at 59|, and sold'at 6l| ? 
 
 Ex. 11. A person has J50l. to invest in the 3 per cent. Conosls, 
 which are at 60^; what sum must he give an order to his broker fjr, 
 so that including the commission it may cost exactly the sum he has 
 to lay out? 
 
 Ex. 12. How much a year in the Long Annuities can I purchase for 
 1000/., the price being 17^ years purchase ? 
 
 A general Method for finding the Value of any Quan- 
 tity of Stock sold or purchased, which is i^uch 
 used. 
 
 RULE. Multiply the price of the stock by the quantity, 
 observing, insteadof the fractional part of the price, if any, 
 to affix the respective fgures agreeing thereto, in the fol- 
 lowing scale, with a point on the left side ; then, for the 
 Perpetual Annuities, as 3, 4, or 5 per cents., fyc., point 
 off two more figures from the product than were affixed 
 for the decimal part of the price, and the figures on the left 
 side of the point will be pounds* the remaining fgures, 
 being multiplied by 20, and the same number again pointed 
 off, will be the shillings, if 'any ; and the remaining figures 
 being multiplied by 12, and pointed off, as before, will be 
 the pence, if any ; which pounds, shillings, and pence, 
 make up the ivhole of the purchase- money. 
 
 For the Terminable Annuities, as Long, Short, Impe- 
 rial, fyc. 9 mark off so many figures only, in the products, 
 were affixed for the fractional part of the price. 
 
 SCALE. 
 
 T V ,0625 A ,5625 
 
 i ,125 i ,625 
 
 -ft ,1875 H >GS75 
 
 J ,25 i ,75 
 
 T V ,3125 |f ,8125 
 
 % ,375 i ,875 
 
 T V ,4375 -H >W5 
 
 ,5 
 
STOCKS. 
 
 EXAMPLES. 
 
 What is the purchase of 816, three per cent, Consols, 
 it 72| per cent. ? 
 
 Look in the scale, and | is found equal ,625 ; therefore 
 72,625 X 816, 
 
 4,8 Answer, 592 125. 4d. 
 
 What is the value of 58, Long Annuity, at 21 T % years 
 purchase ? T S F n ,1875 ; therefore, 
 
 21,18/5 
 58 
 
 1695000 
 1059375 
 
 1228,8750 
 20 
 
 17,500O 
 12 
 
 6 ; oooo Answer, 122 8 17*. 6d, 
 
 W r hat is the purchase of 2470, five per cent* of 1797, 
 at 104 i per cent. ? 
 
 2470* 
 
 104,5 t 
 
 1235O 
 9880 
 2470 
 
 2581,150 
 20 
 
 3,000 Answer, 258! 3$. od. 
 
 NOTE, 
 
 * It is immaterial whether the price or the quantity of stock is made 
 thr multiplier, 
 
 I 2 
 
l/^ INSURANCE FROM FIRE. 
 
 What is the value of 20, Short Annuity, at 5|-J years 
 purchase ? 
 
 5,9375 
 
 26 
 
 356250 
 118750 
 
 154,3750 
 20 
 
 7,5000 
 12 
 
 6,0000 Answer, i54 7$. 6d. 
 
 INSURANCE, 
 
 INSURANCE is a security given in consideration of a pre- 
 mium of so much per cent, paid by the proprietors of 
 Broods, &c., to the insurers, for which they engage to an- 
 swer for the damage of houses, ships, goods, Sec., by fires, 
 dangers of the sea, and other accidents. 
 
 To find what premium must be given for an insurance 
 of property, at any rate per cent. 
 
 RULE. Multiply the value of the properly to be insured 
 by the rate, and the product divided by 10025 the premium. 
 
 Ex. 1. I insure my house, and goods for 1700, for 
 which I pay a premium of -^th per cent, to the Phoenix 
 Cilice, and jth per cent, is paid to government for duty, 
 what do I pay annually ? 
 
 1700 1700.0 
 
 A* . s. d, 4- 
 
 A ns. l 14 o Insurance. 2 1 
 
 1.70 226 Duty. 
 
 OQ 
 
 ( 
 
 Ans. 3 16 6 ~ sum annually paid. 2.500 
 
 14.00 __ L 2 __ 
 
 6.000 
 
 jL'.x. ?. How much must be paid for the insurance of hazardous pro- 
 perty, v<:iue 54001. 9 at the rate of 11 percent., and duty 1 per cent. ? 
 
 NOTE. 
 
 * To multiply by a fraction, whose numerator is unity, or one, is, 
 as we have seen, to divide the sum to be multiplied by the denomina- 
 tor of the fraction. ' 
 
173 
 SEA INSURANCES. 
 
 THE premium is a per centage on the sum insured, and 
 is usually taken in guineas, but sometimes in pounds. AH 
 Sea Insurances pay a duty of 5s. per cent, for foreign voy- 
 ages, and 25, 6d. per cent, for coasting voyages to and 
 from any part of the united kingdom. 
 
 Ex. i. A merchant in London has consigned to him from Jamaica 
 1 00 hogsheads of sugar, valued at 20/. per hogshead, which he insures 
 -for the voyage at 6 guineas per cent. - 
 
 2000 at 6/. 6s, is l26 
 Duty, - s 
 
 131 
 
 Ex. 2. What will the insurance of 1600J. come to, from Embden to 
 London, at 4 - guineas per cent. ? 
 
 Ex. 3. What will the insurance come to of 500 casks of butter, 
 valued at 280o/.,from Waterford to London, at 2^ guineas per cent. ? 
 
 Ex. 4. What will the insurance of 700J., from London to Baltimore } 
 come to, at 2|/. per cent. ? 
 
 Ex. 5. What will the insurance come to of 10,OOOZ., from Rio Ja- 
 neiro to the Cape of Good Hope, and from thence to Calcutta, at 
 4 gumeas per cent. ? 
 
 Most persons who have occasion to make insurances of 
 this kind employ a broker, who receives a shilling for each 
 guinea or pound of the premium for his commission. 
 
 Ex. 6. What does an underwriter receive for insuring 700/. from 
 Hamburg to London, at 2^ guineas per cent. ? 
 
 .700 at .2 7s. 3 d. per cent, is .16 10 9 
 Duty 1 15 o 
 
 1859 
 Brokerage, is. per guinea o 15 9 
 
 17 10 o 
 
 Ex. 7. What does an underwriter receive for insuring 5-00/. from 
 Newcastle to Southampton, at 2/. percent.? 
 
 In time of war, insurances are generally done at a much 
 higher premium, with a condition to return a certain part 
 of it, if the ship sails with convoy, and arrives at her 
 destined port. 
 
 Ex. 8. What is to be paid for insurance of 2000?. from Jamaica to 
 London, at 12 guineas percent., with an agreement to return five 
 guineas per cent, if the ship depart with convoy for the voyage and 
 .arrive. The ship having had convoy for the voyage and arrived? 
 2000 at 12 125, per cent is 252 o o 
 Duty - 500 
 
 257 o o 
 Jlsturn for Convoy - 105 Q .p 
 
 152 0^ 
 
174 SEA INSURANCES. 
 
 Ex. 9. What is to be paid for insurance of 1200/. from Stockholm 
 to Plymouth, at 6 guineas per cent., to return 2 guineas per cent, if 
 the ship departs from the Sound with convoy for the Downs, and l 
 guinea per cent, more, it with convoy from thence for the voyage and 
 arrive ? The ship having had convoy for the whole voyage and ar- 
 rived. 
 
 Ex. 10. What is an underwriter to receive from a broker for insur- 
 ing looo/. from Liverpool toDantzic, at 10 guineas per cent., to return 
 two pounds per cent, if the ship depart from the place of rendezvous 
 with convoy for the voyage and arrive ? The ship having had swch 
 convoy and arrived. 
 
 Ex. 1 1 . What is an underwriter to receive from a broker for insuring 
 300/. from London to Buenos Ayres, at loj guineas per cent., to return 
 4/. per cent, if the ship depart with convoy for the voyage and arrive, 
 or a J pounds per cent, if with convoy for St. Helena, and arrive ? The 
 vessel having had convoy to St. Helena, and being arrived. 
 
 In case it appears that the value of the goods actually 
 shipped is less than the sum ordered to be insured, a re- 
 turn of premium is made on the short interest, deducting 
 10s. per cent. 
 
 A general average does not affect the stipulated returns 
 for sailing with convoy ; but in case of a particular average, 
 the returns for convoy are not allowed on such part of the 
 sum insured as is claimed for- the average. 
 
 Ex. 12. 1500 is insured on sugar, valued at 30/. per hogshead, 
 from Grenada to London, at 12 guineas per cent., to return 6 guineas 
 per cent, if the ship sails with convoy for the voyage and arrives. The 
 ship had convoy for the voyage and arrived ; but it appeared that only 
 45 hogsheads had been shipped ; the insured is therefore entitled to a 
 return of premium for short interest on 150/., and likewise claims a 
 general average, amounting to ll. 7s. 5d. per cent. What balance has 
 he to pay on this insurance ? 
 
 1500 at 12 12$. percent, is 189 O o 
 
 Duty - 3 15 
 
 Return for Short Interest 18 18 O "192 15 o 
 Deduct J per cent. o 15 o 
 
 18 3 o 
 
 Return for Convoy on 1350^. 85 l 
 General average on ditto 18 10 l J 
 
 -- 121 14 ij 
 
 Ex. 13. 900 is insured from Hull to Tonningen, at 5 guineas 
 per cent., to return 2 guineas per cent, if the ship sails with convoy 
 and arrives. The ship had convoy for the voyage and arrived ; but 
 having met with bad weather, a general average is adjusted, amounting 
 to 17/. 6s. 6d., and a particular average at 29/. 5s. What is the sum 
 to be paid or received ? / 
 
1/0 
 
 DISCOUNT. 
 
 DISCOUNT is an allowance made for advancing money 
 on securities before they are due. Thus I receive a note 
 of 60, which is payable at the end of two months, but 
 having- immediate occasion for the money, I must pay any 
 person who will give me cash, as much as the legal inte- 
 rest at 5 per cent, on 60 for two months. 
 
 I. In business, it is usical to calculate after the rate of 
 one penny per pound per month.* 
 
 NOTE. 
 
 * This is, at the rate of 5 per cent, per annum, the legal interest of 
 the country ; for, if 100 pounds yield 1 OO shillings in a year, one pound 
 will yield one shilling, or 12 pence in a year, or one penny per month, 
 
 The Rule of Discount is, in fact, the same thing as what we had 
 under that of Simple Interest ; but the rule given in the text is that 
 which is in general use, and admitted by custom, and in practice, though 
 it is at a rate somewhat higher than 5 per cent. 
 
 Perfect accuracy would require us to find the present worth of the 
 bill or bills to be discounted. Thus the present worth of \ool. due one 
 year hence, at 5 per cent., is not 95^., but g5/. 4s. 9%(L nearly; and 
 therefore he who allows 5l. for the discount, allows 45. g^d. too much ; 
 it is, however, on the true principles that Smart's tables are calcu- 
 lated, and those are chiefly in use by persons in the habit of discounting 
 bills. To find the present worth of any sum payable at any time, table* 
 are given which are calculated thus : 
 
 As the amount of loot, for the given rate and time, 
 Is to the interest of lool. for that time ; 
 So is the given sum 
 To the discount required : arid 
 The discount subtracted from the sum is the present worth, 
 
 Upon this principle: To find the discount of lool. for one year, at 5 
 per cent, interest, we say, as 10,5 : 5 : : 100 : 4/. 15s. 2^d, 
 
 To find the discount of 6ol. for 3 months, at 5 per cent., we say*, 
 as lOiJ. 55. : iZ. 55. : : 6ol. : 145. g$d. ; hut, by the common rule 
 in the text, it would be reckoned 155,, or 2^rf. too much. 
 
 When the sum is large, and the time long, the difference between 
 these modes of calculation is an object deserving of attention* 
 
 Ex. Suppose I have a bill of loool. payable two years hence, then, 
 by the rule in the text, I must pay lool. for discounting the same 3 
 but, by this last rule, I say, as 
 
 100/. : 10/. : : lOOOJ. 
 10 
 
 no) 10000 (go/. 185. 2d., which is the true 
 discount, which makes a difference of 9/, 15. iQtl. 
 
li'O DISCOUNT. 
 
 In this case the discount will be twice 60 pence, that 
 is, 120 pence, or 10 shillings. 
 
 Ex. 1. I have just received two bills of .70 each, the 
 one is payable at two months, and the other at four, how 
 much must I pay for discounting them ? 
 
 . cl . s. d. 
 
 Discount of 70 for 2 months ~70X2r=Oli 8 
 70 4 ^ 70 x 4 ~ i 3 4 
 
 Answer - i 15 o 
 
 Ex. 2. I have in my possession the following bills, which 
 1 wish to get discounted, what shall I have to pay the per- 
 son who will give me cash for them ? 
 
 A bill of 540/. 05. due 3 months hence. 
 A do. 20/. 05. ^- 6 ditto. 
 A do. 35 5/. 05. 5 ditto. 
 A do. 85*. 10>?. 6 ditto. 
 
 II. To fold the discount of a sum of money foi* any number 
 
 of days. 
 
 RULE. Multiply the number of pounds by the number 
 of days, and divide by 365, the answer in in shillings, be- 
 the interest -of one pound is one shilling for a year. 
 
 Ex. 1. What is the discount of .1000 for 25 days ? 
 
 1000 X 25 
 
 IZ 3/. 85. 6a, nearly. 
 
 365 
 
 Ex. 2. What is the discount of a bill of .87 10s. thai 
 has 28 days to run ? 
 
 8JL 103. X 28 _ 87.5 X 28 __ ^ , 
 
 365 365 ~~ ' -X '*"" 
 
 Ex. 3. How much must I pay for discounting .the fol- 
 lowing bills ? 
 lOol. at 7'5 days; 245/. at 42 days ; 987 Z. at 68 days ; $il. at no days. 
 
 Ex. 4. On the first of May I want to discount a bill of lioJ. due the 
 10th of July ; how much must I pay for the same ? 
 
 Ex. 5. What is the discount of 33oZ. for 95 days ? 
 
 Ex. 6. How much ought I to pay for discounting four bills of 75/. 
 each, at 2 and 4 months, six weeks, and 7 5 days ? 
 
 III. To find the time at which several bills payable at dif- 
 ferent times, may be paid at once, or exchanged for one 
 bill, without loss either to the holder or receiver. 
 
DISCOUNT. 177 
 
 HOLE, Multtpty each payment bij the time at which it 
 becomes due, and divide the sum of the products by the sum 
 of the payments, and the quotient will he the medium tinie 
 required. 
 
 Ex. 1. I have to receive 967 in notes, as follow ; viz. 
 135 in 3 months, 473 in 5 months, 167 in 6 months, 
 (59 in 9 months, and 123 in 15 months; but as it is 
 more convenient to have the whole in one note, for what 
 time must it be given ? 
 
 135 X 3 n 405 
 
 473 X 5 2365 
 
 167 X 6 IGO'2 
 
 69 X 9 621 
 
 123 X 15 18.45 
 
 967 6238, and 5?S 6 or 6 mont h s an d . 13 days. 
 967 
 
 Ex. 2. I have to pay .250 12*-. at three payments, viz. 
 .50 10*. Gd. at 2 months, .90 9s. 9c/. at 4 months, and 
 the remainder at six ; what length of time must a single 
 note be, to pay the whole at once?* 
 
 . s. d. 
 
 50 10 6 
 90 9 9 
 109 11 9 
 
 . s. 
 
 (due at 2 months) x 2 n 101 i 
 ( 4 ) x. 4 ~ 361 19 
 ( 6 -r J x 6 n 657 10 
 
 ft 7 . 
 O 
 
 o 
 
 G 
 
 250 12 
 20 
 
 5012 
 12 
 
 -CO 1.4 4 
 
 1120 10 
 20 
 
 fj 
 
 n.days. 
 14 nearly. 
 
 22410 
 12 mo; 
 
 60144)208926( 4 
 240576 
 
 .2S350 
 30 
 
 60144)850500(14 
 
 * If the times of payment, or debts, are of different denominations, 
 as days, weeks, or months, and pounds, shillings, or pence, they must 
 be reduced to the same denomination befor^ the several multiplications 
 take place. 
 
 J5 
 
178 DISCOUNT. 
 
 Ex. 3. At how long date must I have one bill of .906 
 4s. 9d. for the following notes ; viz. 426 56'. 3d. payable 
 at 55 days, 229 12s. 2d. payable at 99 days, and the re- 
 mainder at 135 days ? 
 
 . 
 
 S. 
 
 d. 
 
 . 
 
 s. 
 
 d. 
 
 416 
 
 5 
 
 3 X 55 
 
 IZ 23444 
 
 8 
 
 9 
 
 229 
 
 12 
 
 2 X 99 
 
 rr 22731 
 
 4 
 
 6 
 
 250 
 
 7 
 
 4 X 135 
 
 = 33799 
 
 10 
 
 O 
 
 906 
 
 4 
 
 9 
 
 79975 
 
 3 
 
 3 
 
 20 
 
 
 
 20 
 
 
 
 18124 
 
 
 
 1599503 
 
 
 
 4 
 
 
 
 4 
 
 
 
 72499* 72499)6398013(88 days. 
 
 Answer - 88 days. 
 
 Ex. 4. On the 5th of April, I have in my possession one bill for l&5l. 
 105. due August loth; one for 236J. due Sept. 1 ; one for 95 J. Is. 6d. 
 due Sept. 8th ; another 723/. 3s. gd. due Oct. i ; one for 83/. due 
 Oct. 14 ; and one for 1 122/. 7$. 5rf. due Nov. 12 : at what date ought 
 I to take one bill for the same ? 
 
 Ex. 5. I go on the 13th of December to a country banker with the 
 follow.ng bills, viz, one for 68/. 0*. 4d. due Feb. l, one for 17 l/. 4s. 2d. 
 due Feb. 17, one for 238*. due Jan. l,one for333/> 7s. lod. due Jan. 
 28th, and one for 210/. 2s. 4d. at two months after date j and he gives 
 me 350J. in cash, and his bill on London at two months for 600/., for 
 which bill he charges me an eighth per cent, commission, and 75. 6d. 
 for #ie stamp : how much will remain due to me or to the banker ? 
 
 Ex. a. A traveller in the wine trade has received in local notes, 13 
 of 5 guineas each, 3 of $L each, 17 of 2/. each, and 28 of one guinea 
 each ; also a 40'. bill due in 23 days, a 73 /. bill due in six weeks, ano- 
 ther for 217^. 85. sd. due in 47 days, and another for loal. due in 56 
 days. At what date ought a country banker to give him a bill on Lon- 
 don for the same, supposing him to charge for drawing such bill an 
 eighth per cent commission, and 75. (3d. for the stamp ? 
 
 * In both cases the money is brought into threepences, which 
 somewhat; easier than to bring it into pence, 
 
179 
 PROFIT AND LOSS, 
 
 Is a rule that discovers what is gained or lost on the 
 prime cost in the pur chase and sale of goods y and it teaches 
 persons how to fix the price of their goods so as to gain so 
 much per cent. 
 
 Questions in this rule are performed by the Rule of 
 Three Direct, upon this principle, that quantities, or sums 
 of money, ^hich gain or lose at the same rate, are to one 
 another as their gains or losses. 
 
 Ex. 1. A tallow chandler has this day purchased mottled 
 soap, at 102.?. 6d. per cwt., at how much per Ib. must he 
 retail it out to gahi 10 per cent, profit ? 
 . . . d. 
 
 100 : 110 : : 102 6 -~ 112 
 
 20 10-2 6 
 
 3000 220 
 1100 
 55 
 
 2.000)11.275 
 
 ~ and 
 
 _ A= , Ji0 j A nearly _ 
 
 Ex. 2. How much per cent, is gained at the rate of 2d. in a shilling? 
 
 Ex. 3. If 35. is gained, in selling at a guinea, at what rate per cent. 
 is that ? 
 
 Ex. 4. Three pounds of tobacco are bought at 5s. Qd. and sold for 
 ?s. 6d. 9 what is the gain upon the sale of what cost loot. ? 
 
 Ex. 5. Bought cheese at 3Z. 3s. per cwt., and sold it again at lOjcL 
 per Ib. : what is the gain per cent,, supposing the loss in weight to~be 
 4 Ib. per cwt. 
 
 Ex. 6. Bought silk stockings at 125. gd. per pair : what must they be 
 sold for to gain 20 per cent, profit ? 
 
 Ex. 7. If 375 yards of cloth be sold for ago/., and there be 20 per 
 cent, profit, what did it cost per yard ? 
 
 Ex. 8. Sold i cwl. of hops, at 6l. 155., at the rate of 25 per cent. 
 profit : what would have been the gain per cent, if I had sold them for 
 8 guineas per cwt. ? 
 
 Ex. 9. If '90 ells of cambric cost 120^., for how much must I sell it 
 per yard to gain 1 8 per cent. ? 
 
 Ex. 10. A plumber sold 5 fother of lead for 102Z. 25. 6d. (the fother 
 being I9cwt.), and gained after the rate of izl. 105. per cent. : what 
 did it cost him per cwt, ? 
 
 Ex. 11. Bought 218 yards of cloth, at the rate of 85. 6d per yard, 
 and sold it for 105. 4d. per yard : what was the gain of the whole ? 
 
 Ex. 12. Paid 69/. for one ton of steel, which is retailed at B 
 Ib., what is the profit or loss by the sale of 12 tons ? 
 
ISO 
 PARTNERS 11 IF 
 
 Is a general rule, by which merchants, &c., trading 1 in 
 company with a joint stock, are enabled to ascertain each 
 person's particular share of the gain or loss, in proportion 
 to his share in the stock.* 
 
 This rule divides itself into two parts, viz. 1. Partnership 
 without regard to time; and 2. Partnership with time. 
 
 I. PARTNERSHIP WITHOUT TIME. 
 
 RULE. "As the whole stock is to the whole gain or loss, 
 so is each man's share in the stock to his share of the gain 
 or loss." 
 
 Ex. 1. Tw r o merchants embark in business, the one puts 
 in as capital .5550, and the other .3420, and they gain 
 in the first year .1200, what is each man's gain ? 
 
 . 5550 
 3420 
 
 8970 ~ joint stock. 
 
 BQTO/. : 12602. : : 5550/. : 779/. 12s. nearly; of course the profits 
 of the other are I2t?o/. 779/. 125. 480/. 8s. 
 
 Ex.2. Three persons trade together: A puts in loot. ; B 150/. ; 
 'C 200/. ; and they gain 900! : what is each man's gain ? 
 
 Ex. 3. A, B, and C, enter into partnership; A puts in 364Ql' 9 B, 
 *4'S207., and 'C '50002., -and they gained 8670/. : what is each man's 
 share in proportion to his stock ? 
 
 Ex. 4. P'our merchants, B, C, D, and E, make a stock; B put in 
 22/0/., C 3490/., D 1150>. and E 4390J.; in trading they gained 
 4280/. : I demand each merchant's share of the gain ? 
 
 Ex, 5. Three persons, D, E, and F, join in company; D's stock 
 was 3750Z., E's2800/., and F's 25001., and at the end of 12 months 
 'they gained 3420/. ; what is- each man's particular share of the gain ? 
 
 II, PARTNERSHIP WITH TIME. 
 
 RULE. As the sum of the product of each man" s money 
 and time is to the who< gam or /oss, -so is each man's pro- 
 duct to the share of the gain and loss. 
 
 NOTE. 
 
 * This rule is of grea~t use in various concerns ; by it a bankrupt's 
 estate may be accurately dhnded among his creditors. Legacies are also 
 -adjusred'by it, when there is --not money enough left to answer all- tin, 
 hclega^": 
 
ALLIG-ATiON, 181 
 
 Ex. 1. Two persons lay out 1500/. in trade, ii? the pro- 
 portion of 3 to 2 : that is, A put in 900/., and B GOO/. ; 
 A leaves his money in the concern 9 months, and B does 
 not want his for 12 months : what profits belong* to each, 
 supposing- they gain 250/. ? 
 
 . 900 x o rz 8ioo 
 
 600 X 12 ZZ 7-200 
 
 15200 
 
 250 : : SI 00 
 250 
 
 153,00)202 
 
 \ver, A's share of profit .13-2 7 o 
 B's - - - - - 117 13 o 
 
 50 O 
 
 Ex. 2. A puts into a concern 2080/. for 2 months, B 97 o. for i 
 months, and C 400/. for 15 months ; they gain among them 650/,; 
 *vhat must each receive for his share of profit ? 
 
 Ex. 3. Three merchants join in company for 18 months: D put 
 in 500/., and at 5 months end took out 200/. ; at 10 months end put 
 in 500/., and at the end of 14 months takes out ISO/. E puts in 400/., 
 >:;d at the end of 3 months 27o/. more; at g months he takes out 
 ) iO/., but puts in lOOl. at the end of ui months, and withdraws 0o/., 
 at the end of 15 months. F put in 900/., and at-6 months took out 
 I 200l. ; at the end of 11 months put in doott., but takes out that and 
 100^. more at the end of 13 months. They gained 200/. I desire to 
 , each man's share of the gain ? 
 
 ALLIGATION 
 
 Teaches to mix things of different values, so as <to as- 
 certain the price of the mixture. There are two cases -in 
 this rule. 
 
 1. To find the mean value of a mixture composed of se- 
 veral quantities of different values. 
 
 RULE. Multiply each quantity ly its respective value, 
 and divide the sum of the products by the sum of the quan- 
 tities. 
 
 Tvx. L A tea-dealer mixes 3J c\vt, of tea. at "9$, per IK-., 
 
182 ALLIGATION, 
 
 with 2 cwt. at 7s., and 4* cwt. at 5s. 6d.> at how much 
 perlb. can he sell the whole mixture? 
 
 3 X 112 z: 392^) ("392 X 9 == 3528 
 
 2 X 112 224 > and <J 224 X 7 = 1568 
 4| X 112 476 J (.476 X 5.6~2618 
 
 10Q2 )7 7 14(7 5 
 
 7644 
 
 Answer - 7*. O^rf. . -70 
 
 Ex. 2. What is a Ib. ofsugar worth which is compounded of 3 cwt. 
 
 at 46s. ; 2 cwt. at 595. 5 ij ewt. at 84s. ; and 56 Ib. at 605. ? 
 
 Ex. 3. What is the average earnings of workmen, 48 of whom earn 
 
 zl. each per week ; 80 earn 45.?. each ; 120 in inferior business will 
 
 get only 25s. each ? 
 
 Ex. 4. A tobaccor-iist mixes 80 Ib. of tobacco at 20d. per Ib. ; 1 50 Ib. 
 
 at 25 3d. per Ib. ; and 40 Ib. at 3J. JOd. per Ib. ; what will be the 
 
 \alue of the mixture per oz. ? 
 
 II, To find how much of different things of different va- 
 lues, must be taken, in order to make a mixture of a 
 certain mean value. 
 
 RULE (1). Set down the names of the things to be mixed, 
 together with their prices ; then, finding the difference be- 
 tween each of these, and the proposed price of the mixture ; 
 place these differences in an alternate order, and they will 
 shew the proportion of the ingredients. 
 
 Ex. 1. Orange wine,* at 95. per gallon, is to he mixed 
 with raisin wine at @s. per gallon; what will he the pro- 
 portions, so as to sell the mixture at 7s, per gallon ? 
 
 Proposed 
 
 Orange, - 95."] price, f l ~J A mixture therefore of these wines 
 > 7 s. 3 >in the proportion of l orange to 
 Raisin 65. J (^2 J two raisin, will be the answer. 
 
 Ex. 2. A spirit at 16 shillings, and another at 12 shil- 
 lings per gallon, are to be mixed with low wines at 6 and 
 5 shillings, in order to produce a mixture worth 9 shillings 
 per gallon ; what must the quantities of each be ? 
 
 Spirit, '16^ 3 The answer is, 3 gallons at 16s., 4 
 
 Ditto, " r 12 j Q 4 at 125. ; 7 at 6s.; and 3 at 5s.; will 
 Wine, " I 6 7 make a mixture that may be sold foe 
 
 Ditto, -^-5 3 9 shillings per gallon : for 
 
ALLIGATION. 1 S3 
 
 3 X 16 =r 48 
 
 4 x 12 zr 48 
 
 7 X 6~42 
 
 3 X 5 zz is 
 
 153 
 
 17 153 and 95. Proof. 
 
 Ex. 3. A tea-dealer would mix four sorts of tea together, viz. at 
 4*., 45. 6d., 55. 6d. t 6s., and 7s. per Ib. ; in order that he may sell the 
 whole mixture at 55. 6d. per Ib., what proportion of each will he use ? 
 
 Ex. 4. How much snuff, at 4.?., 3s. 6c/., 25. 3d., and 25. per Ib.^ will 
 compose a mixture worth 25. 6d. per Ib.? 
 
 III. When the prices of all the things to be mixed are 
 given, likewise where the quantity of one, and the mean 
 rate are also given, to find the several quantities of the 
 others. 
 
 RULE. (1) Take the difference between each price and 
 the mean rate as before. (2) As the difference of that 
 thing, ivhose quantity is given, is to the rest of the diffe- 
 rences severally ; so is the quantity given to the several 
 quantities required. 
 
 Ex. 1. A rectifier of compounds has 200 gallons of spirit 
 that he can sell for 12s. 6d. per gallon, but he means to 
 mix it with three other kinds of spirit at 13$. 4d.. at 15$., 
 and 18$. 4d, per gallon, in or<}er that he may sell the whole 
 at 14$. 2d. per gallon ; how much must he use of each ? 
 
 I reduce the several prices to pence, which stand as follow : 
 
 150-. 50 
 
 16<X 10 50 : 10 : : 200 : 40 
 
 7 18CT 10 50 : 10 : : 200 : 4O 
 
 220-^ 20 50 : 20 : : 200 : 80 
 
 The answer is; to 200 gallons, at 125. 6d., must be added 40 at 
 135. 4d.y 40 at 155., and 80 at 185. 4d. j the truth of which is proved 
 thus; 
 
 $. d. . $. d. 
 
 200 at 12 6 ZZ 125 
 
 40 at 13 4 rz 26 13 4 
 
 40 at 15 ZZ 30 
 
 80 at 18 4 ZZ 73 6 8 255 
 
 " -- and zr 145. 2(2. Proof. 
 
 360 255 O 
 
 360 
 
 Ex.2. A grocer has 100 Ib. of tea worth 45. per Ib., which he 
 means to mix with others at 125. 3d., 105. , and 65. per Ib. ; in order to 
 fell the whole at 85., how much of each must be used ? 
 
POSITION. 
 
 IV. When the price of each thing- is given, also thequan* 
 tity and the mean rate, to find how much of each sort 
 will make that quantity. 
 
 HULK. (1) Take the difference between each price and 
 the mean rate as before: then (2) As the sum of the dif- 
 ferences is to each particular difference, so is the quan- 
 tity given to the quantity required. 
 
 Ex. 1. A wine merchant, means to mix 800 o-allons of 
 wine to sell for 8s. a gallon, out of other wines that he al- 
 ready sells for 12s., 9s., (5s. , and 5s. per gallon, how much 
 jRust he fake of each ? 
 
 3 10 : 3 
 2 10 : 2 
 1 10 : 1 
 
 4 10 : 4 
 
 : 8,00 : 258 
 
 : 860 : 1/2 
 
 : 8(>0 : 8(> 
 
 : bGO : 344' 
 
 'Sum of differences 10 8fiO 
 
 The answer is 258 gallons at 125. ; 172 at 9s.; 86 at 6s. ; and 314 
 
 at 5s. per gallon, may be mixed and sold at Bs per gallon. 
 
 Ex. 2. A goldsmith has four sorts of gold, viz. of 24, 10, 18, and 
 
 15 carats fine, wishes 126 oz. of the fineness of 17 carats, how much 
 
 will he want of each sort ? 
 
 Ex.3. A drug grinder has bark worth 165. per lb., some at I.QS. 
 
 and some at 4s. ; but he is desiraus of making up two parcels, viz. 
 
 one containing a cwt. at <)$., and the other 84 lb. at 12>'.^ what pro 
 
 portions of each must be used? 
 
 POSITION. 
 
 'PosiTTON, or as it is sometimes called, the RULE of 
 FALSE, is a rule, that by means of any supposed numbers, 
 others that are true, and that answer to the terms of the 
 question, are found. There -are two kinds of Position, 
 viz. Single and Double. 
 
 SINGLE POSITION is performed, by using 1 a supposed 
 number, and working with it as the true one, till the real 
 mimber is found. 
 
 RULE. Take <.wy number and perform the work -with it.. 
 
POSITION. 
 
 as if it were the 3 right number : then sat/, As the result of 
 this work is to the position, so is the result in the question 
 to the number required. 
 
 Ex. 1. A person counting- some guineas, being asked 
 how many he had, replied: " If you had as many, rind as 
 many more, and half as many, and one quarter as many, 
 you would have 264." How many had the person who wus 
 counting his gold ? 
 
 By way of supposition, I take 80 as the number ; then, by -the 
 terms of the question, it will be 
 
 80 C5 
 
 As many more, so 220 : 264 :: 80 96 
 
 Half as many, 40 80 48 
 
 4th as many, 20 24 
 
 220)21120(06 Answ. 
 
 220 264 Proof. 
 
 Ex. 2. A person, after spending , |, and th of his 
 money, iinds he had 500/. left, what was his original pro- 
 perty ? 
 
 I take a number divisible by 2, 4, and 6, for the supposition^ 
 viz. 60, 
 
 Suppose Go Co 55 rr: 5, therefore Proof. 
 
 - As 5 : 60 : : 500 J rz 3000 
 
 - SO 60 ZZ 1500 t 
 
 15 i ~ 1000 
 
 v ' ; 10 5)30.000 
 
 5500 
 
 55 Answer, 6.000 .* 500 rern. 
 
 Ex. 3. Three persons bought goods at Manchester which cost 
 noo/. The first person was .to have a third part more than the second, 
 and the third a fourth part more than the first ; what \vas each man's 
 share? 
 
 Ex. 4. In a leaky vessel there were three pumps of different capaci- 
 ties ; the first would empty the hold of the ship in 20 minutes, the se- 
 cond would require double that time, and the third would not perform 
 the business in less than an hour j how long would all three together 
 take in doing it? 
 
 NOTE. 
 
 * Any other number, as 12 for instance, would have answered -th 
 same purpose : then it would have been 12 11 zr. l> arvi 
 1 : 12 :: SOO : 6QOQ/. 
 
186 
 DOUBLE POSITION. 
 
 QUESTIONS in this rule are resolved by making suppo- 
 sitions of two numbers, which may both prove false ; in 
 that case the errors are made to correct each other. 
 
 RULE. (1) Place each error against its respective posi- 
 tion, and multiply them cross ways. (2) If the errors are 
 alike, that is, both greater or both less than the given 
 number, take their difference for a divisor, and the differ* 
 ence of their products for a dividend. But if unlike, 
 take their sum for a divisor, and the sum of their products 
 for a dividend) the quotient will lie the answer. 
 
 Ex. 1. Three persons have obtained the 20,OQO/. prize 
 in the lottery, and it is to be so divided, that the second is 
 to have 600/. more than the first, and the third 800Z. more 
 than the second, what is each person's share ? 
 
 Suppose the first had 5000 Suppose the first had 5600 
 Then the second had 5600 The second had 6200 
 
 and the third had 6400 The third had 7000 . 
 
 17000 too little by 3000 18800- 
 
 3000 5000^1 fSOOO X 5600 16800000 
 
 X > that \s,< 
 
 1200 5600 J ( 1200 X 5000 m 600000 
 
 Diff. of Products, - 1 o 800000 zr dividend, 
 3000 1200 n: isoo (diff. of errors) for a divisor. 
 
 ,, ie. soo.ooo eooo 
 
 Therefore, _____ ~ eooo . 66oo 
 
 7400 
 
 . 20.000 Proof. 
 
 Ex. 2. A gentleman, at Christmas, wished to give several poor fa- 
 milies > shillings each, but he found he had 16*. sd. too little; he 
 then gave them 3s. od. each, and found he had 4s. 4d. left, how many 
 families were there ? 
 
 Ex.3. A person purchased a house and land, together with a car- 
 riage and horses, for 1500/. ; he paid 4 times the price of the carriage 
 and horses for the land, and 5 times the price of the land for the house, 
 what was the value of each separately ? 
 
187 
 COMPOUND LNTEREST AND ANNUITIES, 
 
 COMPOUND INTEREST, or interest upon interest, is that 
 which is paid not only for the use of the money lent, but 
 also for the use of the interest as it becomes due.* 
 
 There are three methods of working 1 Problems in this 
 Rule, viz. by Common Arithmetic ; by Decimals ; and by 
 Logarithms : I shall give examples under each. 
 
 I. By Common Arithmetic. 
 
 RULE 1. Find the amount of the given principal for the 
 time of the first payment by simple interest. 2. Consider 
 this amount as the principal for the second payment, the 
 amount of which is to be calculated as before, and so on 
 through all the payments to the last, still reckoning the 
 last amount as the principal for the next payment. 
 
 Ex. 1. What is the amount of 550/. for three years, at 
 5 per cent, compound interest ? 
 
 20)550 O given principal. 
 
 27 10 o first year's interest. 
 
 20)577 10 second year's principal. 
 
 28 17 6 second year's interest. 
 
 20)606 7 6 third year's principal. 
 30 6 4j third year's interest. 
 
 Answer 636 13 lo| 
 
 Ex. 2. What is the amount of 40oZ. for four year's, at 5 per cent,, 
 compound interest ? 
 
 Ex. 3. What is the compound interest of 6001. for five years, at 5 
 per cent, compound interest ?-j* 
 
 * It is not lawful to lend money at compound interest : but in 
 granting or purchasing annuities, leases, or reversions, compound 
 interest for money is allowed. 
 
 f Here, when the amount is found, the principal must be taken 
 from it, and the remainder is the compound interest. Thus, in the 
 first example, the compound interest is 6361. iss. lojd. 550^., or 
 96/. 135. lo%d. In short periods compound interest differs but little 
 from simple interest ; in this case, for instance, the simple interest 
 
I SB COMPOUND INTEREST 
 
 II. By Decimals. 
 
 RULE 1. Find the amount of 11. for a year, at thcqicen 
 rate per cent. 2. Involve* the amount thus found, to such. 
 a power as is denoted by the number of years. . 3. Multiply 
 this power by the principal, or given sum, and the product 
 will be the amount required. A* Subtract the principal 
 from the amount, <and the remainder icill be the interest. 
 
 Ex. 1. What is the compound interest of55Q/. for 3 
 years, at 5 per cent, per annum ? 
 
 3.05 ~ amount of I/, for a year, at 5 per cent. ; 
 Then 1.05 X 1.05 X 1.05 1.1576-25, and 
 
 1.15/6-25 X 550 n: 636.69375 zi amount, 
 636.69375 550 ZI 86.60875 ~ 66/. 135. lO^f/. 
 Ex. 2. What is the amount of 400/. for -i years, at 5 per cent, per 
 annum ? 
 
 Ex. 3. What is the compound interest of 62 o/. for 5 years, at 5 per 
 !;ent.? 
 
 111. By Logarithms. 
 
 RULE. Multiply the logarithm of the amount of II. for 
 a year, by the number of years, and to the product add 
 the logarithm of the principal, and the answer is the amount 
 required. 
 
 Ex. 1. What is the amount of 5507. for 3 years, at 5 per 
 cent, per ami. compound interest ? 
 
 0.021 18Q3 m log. Of 1.05 
 
 _ 3 
 
 0. 0635679 =T log. of 1.05 X 3 
 
 .2.74036-27 =Z log. Of 550 
 
 2.80393o6f zr 636^. 135. loJrL, as before, n amount. 
 
 NOTES. 
 
 would be S2Z. 105. When, however, the interest is suffered to accu 
 mulate for many ages, the difference between simple and compound 
 interest is almost beyond belief. To give an example, we have seen., 
 p. 164, that a penny put out to simple interest at the birth of Christ 
 would, at the end ot the year 1800, only amount to about 75. 6'/. ; 
 whereas the s-ame sum, for the same period, at Compound Interest, J 
 would have amounted to a sum greater than could be contained in si-x 
 hundred millions of globes, each equal to the earth in magnitude, and 
 all of solid gold. 
 
 * See p. 142. 
 
 -f- The nearest logarithm to 2.8039306 is 5,8034571, which ? 
 to 036 the difference between the logarithms is 4 7 =":>, > 
 
AND ANNUITIES. 
 
 EJC. 2. What is the amount of 400Z. for four years, at 3 per cent, pe- 
 annum compound interest?* 
 
 Ex. 3. What is the compound interest of 609^. for 5 years, at 5 per 
 cent, per annum ? 
 
 Ex.4. ^V/iat is the amount of 845/. for 14 years, at 5 per c?nL 
 com; cst? 
 
 ^ v proceed to consider this subject more generally by 
 
 the . . :'p of tables, the construction of which will be shewn in the nete 
 belo i 
 
 NTE. 
 
 multiply by 20 to bring into shillings, and divide by the common dif- 
 ference found at the margin of the table ; thus, 
 4735 
 
 20 
 
 (5S23)94700(ia 
 88699 
 
 6001 
 12 
 
 6323)72012(10 
 68230 
 
 37S2 
 
 '- v v 
 13646 
 
 1482 
 
 * We have purposely given the same examples in all three methods, 
 in order that the pupil may compare the difference in working. 
 
 j- Although we have intentionally abstained from the use of symbols 
 or letters, yet it seems necessary to give some account of the tables by 
 which Compound Interest, Life Annuities, Reversio-ns, &c. are calcu- 
 lated. This will be perfectly intelligible to these who have made 
 themselves masters of what is gone before; and in a work on Algebra, 
 preparing for the press, will be given a demonstration of the rules of 
 common arithmetic : 
 
 Let r ~ the amount of I/, for one year, 
 n ~ the number of years, 
 p ~ principal, 
 a zz amount, or principal and interest of the given sum for the 
 
 time required. 
 
 By proportion we say, as I/, piincipal is to the amount of l/. for 
 one yea so is the amounfof li. for one year, to the amount for two 
 
 years, 
 
 d so on : 
 
 : ?^ : : r : r 2 ~ amount or I/, in 2 years. 
 
 : r : : r 2 : ? >s n amount of ll in 3 years. 
 
 : r : : r 3 : T* ~ amount ol l/. in 4 years. 
 
 : r : : r 1 : r A arn'-uii'. t l,'. in b vea-rs. 
 
190 COMPOUND INTEREST 
 
 NOTES. 
 
 Hence is seen the reason of the foregoing rules ; for, by involving 
 the amount of ll. for a yesr, to such a power as is denoted by the 
 number of years, we get the amount ; that is, the principal and in- 
 terest of ll. fof the given -umber of years ; that is r , or r raised 
 to the power whose exponent is the number of years, will be the 
 amount of ll. in those years ; and as 
 
 1 : r : : p : a zr the amount of any given principal in the 
 
 same time. 
 
 We have seen in Proportion, or the Rule of Three, that when three 
 of the terms are given, we readily find the fourth ; hence the following 
 Theorems. 
 
 Theo. I. The principal, time, and rate of interest are given, to find 
 the amount ? 
 
 Since 1 : r n : : p : a, therefore p X r rz a rz amount. 
 Ex. Suppose the amount of 520/. for 15 years be required, at 5 per 
 cent, compound interest ; then ^ 
 
 500 X nT-TV 5 iz a. 
 
 Theo. II. The amount, time, and rate arc given, to find the princi- 
 pal? then 
 
 n p n piincipal. 
 
 n 
 r 
 
 Ex. What is the principal that will amount to 6361. 13$. lo%d. in 
 3 years, at 5 per cent, compound interest ? 
 636/. 135 
 
 i.o 5 r 
 
 Theo. HI. The principal, amount, and time, are given, to find the 
 rate ? 
 
 nf~~a~ 
 
 V/ zr r zz amount of ll. for one year, and r i zr rate. 
 
 P 
 
 Ex. At what rate per cent, will 55oJ. amount to 635/. 135. iod. in 
 3 years? 
 
 3 / 636J. 135. lod. 
 
 V IT* ~ T - 
 
 Theo. IV. The principal, amount, and rate, are given, to find the 
 time ? 
 
 _*L = r n f Therefore being divided by r till nothing remains, 
 
 P ~ j P 
 
 t the number of divisions will zi n. 
 
 The foregoing theorems are more easily expressed by logarithm* ; 
 
 Thus, I. Log. p 4- n X log. r. zi log. a. 
 II. Log. a 7i x log. r. zi log, p. 
 
 IV. -- 
 
 Log, r 
 
AND ANNUITIES. 191 
 
 TABLE' I. 
 
 Shewing the Sum to which ll. Principal will increase at 5 per cent. 
 Compound Interest, in any number of years not exceeding a 
 hundred. 
 
 Yrs. 
 
 Amount. 
 
 Yrs. 
 
 Amount. 
 
 Yrs. 
 
 Amount. 
 
 Yrs. 
 
 Amount. 
 
 1 
 
 1.05 
 
 
 3.555672 
 
 51 
 
 12.040769 
 
 76 
 
 40.774320 
 
 2 
 
 1.1025 
 
 27 
 
 3.733456 
 
 52 
 
 12.642808 
 
 77 
 
 42.813036 
 
 3 
 
 1.157625 
 
 28 
 
 3. "920 129 
 
 53 
 
 13.274948 
 
 78 
 
 44\953688 
 
 4 
 
 1.215506 
 
 29 
 
 4.116135 
 
 54 
 
 13.938696 
 
 79 
 
 47.201372 
 
 b 
 
 1.276281 
 
 30 
 
 4.321942 
 
 55 
 
 14.635630 
 
 80 
 
 49.561441 
 
 6 
 
 1.340095 
 
 31 
 
 4.538039 
 
 56 
 
 15.367412 
 
 81 
 
 52.039513 
 
 7 
 
 1.407100 
 
 32 
 
 4.764941 
 
 57 
 
 16.135783 
 
 82 
 
 54.641488 
 
 8 
 
 3.477455 
 
 33 
 
 5.003188 
 
 58 
 
 16.942572 
 
 83 
 
 57.373563 
 
 9 
 
 1.551328 
 
 34 
 
 5.253347 
 
 59 
 
 17-789700 
 
 84 
 
 60.242241 
 
 10 
 
 1.628894 
 
 35 
 
 5.516015 
 
 60 
 
 18.679185 
 
 85 
 
 63.254353 
 
 11 
 
 1.730339 
 
 36 
 
 5./91816 
 
 61 
 
 19-613145 
 
 86 
 
 66. 417071 
 
 12 
 
 1.795856 
 
 37 
 
 6.081406 
 
 62 
 
 20.5g3802 
 
 87 
 
 69.737924 
 
 13 
 
 1.885649 
 
 38 
 
 6.385477 
 
 63 
 
 21.623492 
 
 88 
 
 73.Q24820 
 
 14 
 
 1.979931 
 
 39 
 
 6.704751 
 
 64 
 
 22.704667 
 
 89 
 
 76.886061 
 
 15 
 
 2. 0/ 8Q2 8 
 
 40 
 
 7.039988 
 
 65 
 
 23.839900 
 
 90 
 
 80.730365 
 
 16 
 
 2.182S74 
 
 41 
 
 7-391988 
 
 66 
 
 25.031895 
 
 91 
 
 84-766883 
 
 17 
 
 2.292018 
 
 4-2 
 
 7.761587 
 
 67 
 
 26.283490 
 
 92 
 
 89.005227 
 
 18 
 
 2.406619 
 
 43 
 
 8.149666 
 
 68 
 
 27-597664 
 
 93 
 
 93.455488 
 
 19 
 
 2.526950 
 
 44 
 
 8.557150 
 
 69 
 
 28.977548 
 
 94' 
 
 98.128263 
 
 -20 
 
 2.623297 
 
 45 
 
 8.985007 
 
 70 
 
 30.426425 
 
 95 
 
 103.034676 
 
 21 
 
 2.785962 
 
 46 
 
 9.434258 
 
 71 
 
 31.947746 
 
 96 
 
 108.186410 
 
 2 ^ 
 
 2.925260 
 
 47 
 
 9.905971 
 
 72 
 
 33.545134 
 
 97 
 
 113.595730 
 
 23 
 
 3.071523 
 
 48 
 
 10.401269 
 
 73 
 
 35.222390 
 
 98 
 
 119.275517 
 
 24 
 
 3.225099 
 
 49 
 
 10.921333 
 
 74 
 
 36.983510 
 
 99 
 
 125.239293 
 
 25 
 
 3.386354 
 
 50 
 
 ] 1.467399 
 
 75 
 
 38.832085 
 
 100 
 
 131.501257 
 
 I. To find by means of the table what any sum will amount 
 to in a given number of years. 
 
 RULE. Multiply the number in the table, opposite to the 
 term of years, by the sum, and the product will be the answer, 
 
 Ex. I. To what sum will 500/. amount in 44 years, at 
 5 per cent, compound interest ?* 
 
 NOTES. 
 
 On these principles all tables of Compound Interest are calculated ; 
 of which we shall give those that are the most useful and applicable to 
 other subjects, of which we shall have occasion to treat. 
 
 * The legal interest of the country being 5 per cent., we have re- 
 stricted our tables and reasoning to this. The pupil will, however, 
 readily apply this and any questions, either to higher or lower rates, of 
 which tables are to be fourJd in divers bocks. See PRICE'S Annuities, 
 
COMPOUND INTEREST, 
 
 Opposite to 4-1 in the table I find 8.557150, this I multiply by 
 and the answer is 427 s/. us. 6d. 
 
 x. 2. What will 350/. amount to in 25 years, at 5 per cent, com- 
 pound interest ? 
 
 Ex. 3. A prudent young man marries at the age of 2 a ; the fortune 
 which he has with his wife is 2500/., half of which he readily gives 
 into the hands of trustees to be accumulated at 5 per cent, compound 
 interest ; what will it amount to, supposing he lives 32 years, which 
 he may reasonably expect ?* 
 
 Ex. 4. The year 1808, is that in which the late Mr. Pitt calculated 
 there would be four millions surplus to be applied to the payment of 
 the national debt ; I demand to how much this single four millions 
 will accumulate in half a centujy, at 5 percent, compound interest ?f 
 [See other questions on this subject after the. next table.] 
 
 II. To find the number of years in which a given sum will 
 increase to another given sum, in consequence of being 
 improved at Compound Interest. 
 
 RULE. Divide the latter sum by the former, and the 
 linn in the table which is nearest to the quotient will shew 
 the term required. 
 
 * We- shall, page 201, grve a table on this subject, and the- reasons 
 upon which it is founded ; as a general rule, and which will answer 
 in all ages except the very young, and the very old, we may give this 
 as the mode of estimating the expectation of life : u Subtract your own 
 age from 86, and divide ly -2 : thus the young man 0/22, taking that 
 nun'-'-rfrom 80", on' JituUng the remainder ly %, finds his expecta- 
 tion nf life 32 years.'" This is not perfectly accurate ; by the table it 
 will be found th:-it his expectation is -32.39, that is 32 years, four months, 
 and a fortnight. Take another example ; I am now 45 ; by the rule 
 
 HO 45 41 
 just given my expectation of life is m ~ 20^ ; and by 
 
 the table to be given hereafter (p. 201) it will be found that a person 
 of 45 may expect 20.52; in this case the difference is but a week. 
 
 f It will be seen by the table, that one pound put out to compound 
 interest, will be about o-ij years in accumulating to lool. principal, 
 It is frequently said, that money doubles itself at 5 per cent, compound 
 interest, in 14 years, this is by no means accurate, as is evident from 
 the table ; if money doubled itself in 14 years, then 1 1. in 98 years 
 would produce 3 28/., whereas in that time it only produces HQl. 5s. $, 
 But an annuity of I/, suffered to accumulate at 5 per cent, compound 
 interest, will produce loot, in somewhat less than .; years. *< .S 
 Table in the opposite page/' 
 
AND ANNUITIES. 193 
 
 Ex. 1. In what time will 200/. increase fo 1500/., if 
 improved at 5 per cent, compound interest ? 
 
 m 7.5. The nearest number in the table to 7.5 is 7.391988, op- 
 200 
 
 posite to which is 41, the number of years. Of course 200/. in a 
 little more than forty-one years would, by being accumulated at com- 
 pound interest, at 5 per cent., amount to 1500/. 
 
 Ex. 2. In what time will loo/, increase to 500Z. at the same rate of 
 interest ? 
 
 Ex. 3. In what time will 86oZ. increase to o,oooZ. ? 
 
 Ex. 4. In how long would five millions be in paying the national 
 debt, which in January, 1806, was upwards of 580 millions? 
 
 Ex. 5. Admiral Rainier has left, this month, May, 1808, 25,000f. 
 towards paying off the national debt, when will it have accumulated 
 to a million at 5 per cent, compound interest ? 
 
 TABLE II. 
 
 Shewing the sum to which ll. per annum will increase at 5 per cent. 
 Compound Interest, in any number of years not exceeding a hun- 
 dred. 
 
 Yrs 
 
 Amount. 
 
 Yrs. 
 
 Amount. 
 
 YJS. 
 
 Amount. 
 
 Yrs. 
 
 Amount. 
 
 i 
 
 1,0000 
 
 26 
 
 51,1135 
 
 51 
 
 220,8154 
 
 76 
 
 795,4864 
 
 2 
 
 2,0500 
 
 27 
 
 54,6691 
 
 52 
 
 232,8562 
 
 77 
 
 836,2607 
 
 3 
 
 3,1525 
 
 28 
 
 58,4026 
 
 53 
 
 245,4990 
 
 78 
 
 879,0738 
 
 4 
 
 4,3101 
 
 29 
 
 62,3227 
 
 54 
 
 258,7739 
 
 79 
 
 924,0274 
 
 5 
 
 5,5256 
 
 30 
 
 66,4388 
 
 55 
 
 272,712(3 
 
 80 
 
 971,2288 
 
 6 
 
 6,8019 
 
 31 
 
 70,7608 
 
 56 
 
 287,3482 
 
 81 
 
 1020,7903 
 
 7 
 
 8,1420 
 
 32 
 
 75,2988 
 
 57 
 
 302,7157 
 
 82 
 
 1072,8298 
 
 8 
 
 9,5491 
 
 33 
 
 80,0638 
 
 58 
 
 318,8514 
 
 83 
 
 1127,4713 
 
 9 
 
 11,0266 
 
 34 
 
 85,0670 
 
 59 
 
 335,7940 
 
 84 
 
 1184,8-448 
 
 .30 
 
 12,5779 
 
 35 
 
 90,3203 
 
 60 
 
 353,5837 
 
 85 
 
 1245,0871 
 
 11 
 
 14,2068 
 
 36 
 
 95,8363 
 
 6! 
 
 372,2629 
 
 ft 6 
 
 1008,3414 
 
 32 
 
 15,9171 
 
 37 
 
 101.6281 
 
 62 
 
 391,8760 
 
 87 
 
 1374,758V> 
 
 13 
 
 17,7130 
 
 38 
 
 107,7095 
 
 63 
 
 412,4698 
 
 88 
 
 1444,4964 
 
 14 
 
 19,5986 
 
 39 
 
 114,0950 
 
 64 
 
 434,0933 
 
 89 
 
 1517,7212 
 
 15 
 
 21,5786 
 
 40 
 
 120,7998 
 
 65 
 
 456,7980 
 
 go 
 
 15Q4,6073 
 
 16 
 
 23,6575 
 
 41 
 
 127,8398 
 
 66 
 
 480,6379 
 
 91 
 
 1675,3377 
 
 17 
 
 25,8404 
 
 42 
 
 135,2317 
 
 67 
 
 505,6698 
 
 92 
 
 1760,1045 
 
 38 
 
 28,1328 
 
 43 
 
 142,9933 
 
 68 
 
 531,9533 
 
 93 
 
 1849,1098 
 
 19 
 
 30,5390 
 
 44 
 
 151,1430 
 
 69 
 
 559,5510 
 
 94 
 
 1942,5653 
 
 20 
 
 33,0659 
 
 45 
 
 259,7002 
 
 70 
 
 588,5285 
 
 95 
 
 2040,6935, 
 
 '21 
 
 35,7192 
 
 46 
 
 168,6852 
 
 71 
 
 618,9549 
 
 96 
 
 2143,7282 
 
 22 
 
 38,5052 
 
 47 
 
 178,1194 
 
 72 
 
 650,9027 
 
 97 
 
 2251,9146 
 
 23 
 
 41,4305 
 
 48 
 
 188,0254 
 
 73 
 
 684,4478 
 
 98 
 
 2365,6103 
 
 24 
 
 44 ; r >020 
 
 49 
 
 193,4267 
 
 74 
 
 719,6702 
 
 99 
 
 2434,7859 
 
 5^ 5 
 
 47,7271 
 
 50 ! 
 
 ^.00,3480 
 
 75 
 
 7 56,6. W 
 
 10O 
 
 2610,0252 
 
 K 
 
194 COMPOUN 7 D INTEREST. 
 
 1. To find in what time a given annuity will amount to a 
 given sum at compound interest. 
 
 RULE. Divide the given sum by the given annuity, and 
 the number in the table nearest to the quotient ivill be the 
 answer. 
 
 Ex. 1. A person owes 10007. and resolves to appro- 
 priate 207. per annum, to be accumulated at 5 per cent, 
 per ann. compound interest, in how many years will the 
 debt be paid ? 
 
 1000 
 
 z 50. The nearest number in' the table to 50 is 51.1135, 
 
 and the number answering to this is 26, so that in less than 26 years 
 a debt of loool. would be extinguished by laying by, and accumu- 
 lating at compound interest, annually 20/. per annum. If the rate 
 of interest had been 6 per cent. 24 years would have paid the debt, 
 kut at 4 per cent, it would have taken between 28 and 29 years. 
 
 Ex. 2. How long will 75 guineas a year be in accumulating to 
 300o/., at the same rate ? 
 
 Ex. 3. In what time will an annuity of 25J. amount to 357 5Z., at 
 the same rate ? 
 
 Ex. 4. How long will the national debt, left at the time of Mr. 
 Pitt's death, viz. 581 millions, be in paying off, supposing five mil- 
 fions annually be appropriated for that purpose, and the rate of com- 
 pound interest 5 per cent. ? 
 
 Ex. 5. The national dtbt was, at Midsummer ISO/, 756* millions 
 of pounds, out of which the commissioners had redeemed l ] 7 millions 
 and a half, how long would the remainder take in paying off, if eight 
 millions be applied annually, at the rate of 5 per cent, compound in- 
 terest for the purpose ?f 
 
 * This includes a debt of nearly 65 millions, the debt of Ireland-; 
 the whole is of course the debt of the United Kingdom. 
 
 f- The fund applied to reducing the national debt was in (isos) 
 millions, which, as has been shewn in this question, will, in 33 
 years, reduce a debt of 639 millions to nothing. It cannot however 
 be expected that 5 per cent, can be obtained through the whole pro- 
 gress, still the operation of this sum, at 4 per cent, even, would be 
 almost omnipotent. In the year 1806, the amount, to which 8 mil- 
 lions per annum, would accumulate, at different periods during the 
 present century, if improved at 4 and 5 per cent, compound interest, 
 as stated in round numbers as follows 
 
 In the 
 
A N AN N U *T I K S . 
 
 Ex. 6. Duiing the quarter, between Lady-day and Midsummer 
 1809, upwards of 2,800,ooo/. was appropriated towards paying off 
 the national debt : supposing as much set apart every quarter, or 
 1 l,200,000/. annually, in how long time will such an annuity pay oft 
 1,000, 000, ooo., a sum to which the debt must accumulate in a fevr 
 years ? 
 
 II. To find how much a given annuity will amount to in a 
 given term, at 5 per cent, compound interest 
 
 RULE. Multiply the given annuity by the number in the 
 table standing opposite to the given term of years. 
 
 Ex. 1. I can lay by 507. per annum with its interest; 
 that is, I can appropriate 507. a year to be accumulated at 
 5 per cent, compound interest, how much shall I have 
 saved if I live 21 years ? 
 
 Opposite to 21 years I find 35.719, which multiplied by 50, gives 
 1785.962550. Answer, 1785/. 19s. 2<J. 
 
 Ex. 2. How much will an annuity of 35Z. amount to in 83 years ? 
 
 Ex. 3. To what sum will an annuity of 100 guineas amount in 1-9 
 years, at 5 percent, compound interest? 
 
 Ex. 4. To what sum will 6o/, per annum amount to in 25 years ? 
 
 I III. The PRESENT VALUE of an annuity is that sum 
 which, if improved at compound interest, would be suffi- 
 cient to pay the annuity. For, Ujtis tlje following table is 
 adapted. 
 
 4 per cent. 5 p?r cent. 
 
 Millions. Millions., 
 
 In the year 1810, or in 4 years 33 04 
 
 1820 146 156 
 
 1830 312 35$ 
 
 1840 358 690 
 
 IS 50 $23 1209 
 
 1860 1463 2070 
 
 1900 77S2 15540 
 
 The reader will see at ones, in what manner the sums standing 
 under 5 per cent, are obtained, from the table in p. 193. The calcula- 
 tion was made in 1S06, of course, in 1810, the sum will have bee* 
 SOUR years accumulating ; multiply therefore, 4,3101, which is op- 
 posite 4 in the table, by 8 millions, and we ?et for the answer 34.40% 
 r almost 04 millions and a half; and so of fhe others, 
 
 K2 
 
196 
 
 ANNUITIES. 
 
 TABLE III. 
 
 Shewing the present Value of an Annuity of iL for any number oi 
 Years not exceeding 100, at 5 per cent, per annum, Compound 
 Interest. 
 
 Yrs. 
 
 Value. 
 
 Yrs. 
 
 Value. 
 
 Yrs. 
 
 Value. 
 
 Yrs. 
 
 Value. 
 
 1 
 
 ,952381 
 
 26 
 
 14,375185 
 
 51 
 
 18,338977 
 
 76 
 
 19,509495 
 
 2 
 
 1,859410 
 
 27 
 
 14,643034 
 
 52 
 
 18,418073 
 
 77 
 
 19,532853 
 
 3 
 
 2,723248 
 
 28 
 
 14,898127 
 
 53 
 
 18,493403 
 
 78 
 
 19,555098 
 
 ' 4 
 
 3,545950 
 
 29 
 
 15,141074 
 
 54 
 
 18,565146 
 
 70 
 
 19,576284 
 
 5 
 
 4,329477 
 
 30 
 
 15,372451 
 
 55 
 
 18,6^33472 
 
 80 
 
 19,596460 
 
 6 
 
 5,075692 
 
 31 
 
 15,592810 
 
 56 
 
 18,698545 
 
 81 
 
 19,615677 
 
 7 
 
 5,786373 
 
 32 
 
 15,802677 
 
 57 
 
 18,760519 
 
 82 
 
 19,633978 
 
 8 
 
 6,463213 
 
 33 
 
 16,002549 
 
 58 
 
 18,819542 
 
 83 
 
 19,651407 
 
 9 
 
 7,107822 
 
 34 
 
 16,192904 
 
 59 
 
 18,8/5754 
 
 84 
 
 19,668007 
 
 10 
 
 7,721735 
 
 35 
 
 16,374194 
 
 60 
 
 18,929290 
 
 85 
 
 19,633816 
 
 11 
 
 8,306414 
 
 36 
 
 16,546852 
 
 61 
 
 18,980276 
 
 86 
 
 19,698873 
 
 12 
 
 8,863252 
 
 37 
 
 16,711287 
 
 62 
 
 19,028834 
 
 87 
 
 19,713212 
 
 13 
 
 9,393573 
 
 38 
 
 16,867893 
 
 63 
 
 19,075080 
 
 88 
 
 19,726869 
 
 14 
 
 9,898641 
 
 39 
 
 17,017041 
 
 64 
 
 19>119124 
 
 89 
 
 19,739875 
 
 15 
 
 10,379658 
 
 40 
 
 17,159086 
 
 65 
 
 19,161070 
 
 90 
 
 19,752262 
 
 16 
 
 10,837770 
 
 41 
 
 17,294368 
 
 66 
 
 19,201019 
 
 91 
 
 19,764009 
 
 17 
 
 11,274066 
 
 42 
 
 17,423208 
 
 67 
 
 19,239066 
 
 92 
 
 19,775294 
 
 18 
 
 11,689587 
 
 43 
 
 17,545912 
 
 68 
 
 19,275301 
 
 93 
 
 19,785994 
 
 19 
 
 12,085321 
 
 44 
 
 17,662773 
 
 69 
 
 19,309810 
 
 94 
 
 19,796185 
 
 20 
 
 12,462210 
 
 45 
 
 17,774070 
 
 .70 
 
 19,34'2f>77 
 
 95 
 
 19,805891 
 
 21 
 
 12,821153 
 
 46 
 
 17,880066 
 
 71 
 
 19,373978 
 
 96 
 
 19,815134 
 
 22 
 
 13,163003 
 
 47 
 
 17,981016 
 
 72 
 
 19,403788 
 
 97 
 
 19,823937 
 
 23 
 
 13,488574 
 
 48 
 
 18,077158 
 
 73 
 
 19,432179 
 
 98 
 
 19,832321 
 
 2.4. 
 
 13,798643 
 
 49 
 
 18,168722 
 
 74 
 
 19,459218 
 
 99 
 
 19,840306 
 
 25 
 
 14,093945 
 
 50 
 
 18,255925 
 
 75 
 
 19,484970 
 
 100 
 
 19,84/910 
 
 To find the present value of an annuity for a term of years. 
 RULE. Multiply the number in the table opposite to the 
 given term of years, by the sum, and the product is the 
 answer. 
 
 Ex. 1. What is the present value of an annuity of I26/. 
 for 21 years ? 
 
 In the table opposite 21 is 12.821153; this multiplied by 126, gives 
 1615.465278 ZZ I6lbl.gs.3d, 
 
 Ex. 2. What is the present value of an annuity of 75! for 12 years, 
 at 5 per cent. ? 
 
 Ex. 3. What present sum is equivalent to a nett rent of 45/. per 
 annum for 84 years, allowing interest of money at 5 per cent. ?* 
 
 * As purchasers of leases generally expect to make more than 
 
197 
 CHANCES.* 
 
 Question I* Suppose a counter, having 1 a black and a 
 white face, be thrown up, to see which will be uppermost, 
 after the counter has fallen to the ground, and if the white 
 face appear uppermost, a person is to have 5 shillings, 
 what is the chance, or probability, that he will be entitled 
 to the five shillings ? 
 
 Solution* Since either the black or the white face must be upper- 
 most, there is an equal chance for the appearance of either face, of 
 course the chance, or the probability, may be expressed by J, or a 
 bystander ought to give him 2*'. 6d. for his chance of getting the five 
 shillings. 
 
 Question //.-^-Suppose there are three counters put into 
 a bag, one red, another white, and a third black ; out of 
 which, if a person blindfolded take the red he is to have 5 
 shillings, I demand the value of the chance, or what is the 
 probability of his drawing the red counter : 
 
 Solution. He has evidently one chance out of three, and therefore 
 the probability may be valued at i, and another person inclining to 
 purchase his chance, ought to give for it the |d of 5 shillings, or 
 is. Sd. 
 
 NOTES. 
 
 5 per cent, of their money, we shall treat more at large on this subject 
 further on. We may, however, observe in this place, that freehold 
 estates are usually valued at so many years purchase, that is, so many 
 years rent : 
 
 If 30 years purchase be given for a freehold 
 estate, it is equivalent to putting the 
 money out at a little more than - 3 J per cent. 
 
 <25 - - 4 
 
 20 5 
 
 16| - . . . 6 
 
 * It is meant only to give so much of the doctrine of chances as 
 shall enable the pupil to understand upon what ground the doctrine of 
 Annuities, &c. depends. To illustrate this part of the subject, recourse 
 will be had to some familiar instances, which may seem, at first sight, 
 to lead to gaming ; but it is believed, that the facts adduced must, if 
 properly considered, deter young persons from this pernicious and de- 
 structive vice, which is too much encouraged by the almost perpetual 
 drawing of state lotteries. 
 
CHA'tfCRS. 
 
 In the former ca^c, the chances for the event's happening ar; ' 
 % ing are equal, and each being equal to -J, the certainty is reckoned as 
 1, or unity. % 
 
 in this last case, there is one chance for the event's happening, and 
 two for its failing ; in other words, the chance for its happening is >d, 
 and for its failing are fds : here, again, the chances for the happening 
 'knd failing are equal to unity, because ] -J- f 3 i. 
 
 Question III. Suppose there are five counters, two 
 white and three black, out of which, when mixed, a per- 
 son blindfolded is to draw one of the white, and in that 
 case is to be entitled to 5*., what is his chance for so 
 tloing 1 , and what is his expectation worth ? 
 
 Solution. It is plain here are five chances in ihe whole, of which 
 \hcre are two" only oat of five for taking a white couTiter, and the other 
 three for taking a black one ; therefore the probability of winning may 
 be expressed by the fraction f, and of missing |, and he might sell his 
 expectation of the five shillings for |ths of that sum, that is, Tor two 
 
 Ex. l. At the conclusion of the last state lottery, when there were 
 Only 'five tickets left in the wheel, there were two prizes of 5ol. each, 
 and three blanks, what was the value of one of tho>e tickets ? 
 
 Ex. 2. What is the value of one ticket, when only five arc- left in 
 one wheel, and in the other there is one prize of loo/, and fctfr 
 blanks ? 
 
 Ex.3. What chance has the holder of a single lottery ticket of a 
 prize, when there are-three blanks to a prize?* 
 
 * In general, it is held out as a lure to the thoughtless multitude, 
 that there are only two blanks to a prize, 'or not two, or not three 
 blanks to a prize ; y.et, how few buy tickets with the hope of gaining 
 the small prizes, of which the number of prizes is almost wholly made 
 up : All hope for the 20,000 or 30,0bo/. ; but to point out 'the folly of 
 such expectations, we shall quote a fact or two deduced from this 
 subject by One of the ablest mathematicians of the age. 
 
 1. Supposing a lottery to consist of 25,000 tickets, of which 20 are 
 prizes of 1000/. and upwards; a person to have an equal chance of 
 only one ot those prizes must purchase S?0 tickets: these, at I8t 7 . 
 each, (and tickets are seldom so low as this) would cost 15,660/. 
 
 2. Supposing there are three prizes of 20, ooo/., and three of io,000/. 
 and a person out of 25,000 tickets has purchased 3000, in hopes of 
 gaining one of each of these capital prizes, still, though he has laid 
 out, at T8L a ticket, 54, OOO/., the chances against such an expecta- 
 tion willi)e as 12 to 1. See article CHANCES, Rees's Cyclopedia, said 
 to have been written by William Moigan, Ksq. 
 
CHARGES. 199 
 
 Question JF. What is the probability of throwing an 
 ice with a single die, in one trial ? 
 
 Sdlu'tio?i. There are six faces to a die, of which one only is the ace, 
 therefore the probability of throwing an ace with a single die in one 
 trial is expressed by I ; and the probability of not throwing an ace 
 is i : here, as before, the chances for not throwing the ace, and that 
 for throwing, are together equal to unity. 
 
 Question F. What is the probability of throwing an 
 ace in four throws ? 
 
 Solution* We must consider the probability of failing in the four 
 throws. The. probability of missing the first time will be ; so it is 
 the second, thkd, and fourth times ;, therefore the probability of miss- 
 
 5 5 5 5 625 
 
 mg in all four throws will be X X X == ~ 
 
 6666 1296 > 
 
 1296625 vf>7i 
 
 which, subtracted from unity or 1. gives n: ,whiCii 
 
 1296 1296 
 
 is the probability of throwing it once or oftener in four turns ; 
 therefore the odds of throwing an ace in four times, is as 671 
 or rather more than an even chance. 
 
 The probability in three throws will be 
 
 .'> 5 5 125 216 125 Ql 
 
 i . x X ml rz 
 
 666 216 216 216 
 
 odds is against throwing the ace in three throws, as 91 is less than 
 125. 
 
 Qttcslion VI. In two heaps of cards, one containing 1 
 the 18 diamonds, the other the 13 spades, placed promis- 
 cuously, what is the probability that, taking one card at a 
 venture, out of each heap, I shall take the two aces ? 
 
 Solution The probability of taking the ace out of the first .heap is 
 JL; the probability of taking the ace out of the second heap is also 
 j'r, therefore the probability of taking out both aces is <fe X ^, 
 
 or T, which, subtracted from i, eives , of course the chances 
 
 169' 169 
 
 against me are as 168 to 1 : in other words-, I may expect to do this 
 onceiri 169 attempts, 
 
 On, similar principles the expectation of life is found, i It is known 
 by accurate observation, that of 46 persons aged 40 yejirs, one will 
 die eyry year, till they are all dead in 46 years; therefore half 46, 
 or 23 years, will be the expectation of life of a person 40 years of age. 
 That is, the, number of years enjoyed by them all, will be just the 
 saiTie as if tvery one of them had lived '23 years, and then died. The 
 same reasoning applies to all other ages, which leads us to a more 
 uJar consideration of the subject. 
 
200 
 
 EXPECTATION OF LIFE. 
 
 From the Bills of Mortality in different places, tables have beefi 
 constructed which shew how many persons, upon an average, out 
 of a certain number born, are left at the end of each year to the 
 extremity of life. From such tables, which, as we have seen, are 
 founded on the doctrine of Chances, the probability of the continu- 
 ance of a life, of any proposed age is known. 
 
 TABLE I. 
 
 Shewing the Probabilities of the Duration of Human Life, deduced 
 from tne Register of Mortality at Northampton. 
 
 
 Persons 
 
 Decrem 
 
 
 Persons 
 
 Decrem 
 
 
 Person: 
 
 Decrem. 
 
 Age. 
 
 livmg. 
 
 of Life. 
 
 Age 
 
 living 
 
 of Life. 
 
 Age 
 
 living. 
 
 of Life. 
 
 
 
 11650 
 
 3000 
 
 33 
 
 4160 
 
 75 
 
 66 
 
 1552 
 
 80 
 
 1 
 
 8650 
 
 1367 
 
 34 
 
 4085 
 
 75 
 
 67 
 
 1472 
 
 80 
 
 2 
 
 7283 
 
 502 
 
 35 
 
 4010 
 
 75 
 
 68 
 
 1392 
 
 80 
 
 3 
 
 6781 
 
 335 
 
 36 
 
 3935 
 
 75 
 
 69 
 
 1312 
 
 80 
 
 4 
 
 6446 
 
 ' 197 
 
 37 
 
 3860 
 
 75 
 
 70 
 
 1232 
 
 80 
 
 5 
 
 6249 
 
 184 
 
 38 
 
 3785 
 
 75 
 
 71 
 
 1152 
 
 80 
 
 6 
 
 6065 
 
 140 
 
 39 
 
 3710 
 
 75 
 
 72 
 
 1072 
 
 80 
 
 7 
 
 5925 
 
 110 
 
 40 
 
 3635 
 
 76 
 
 73 
 
 992 
 
 8O 
 
 6 
 
 5815 
 
 80 
 
 41 
 
 3559 
 
 77 
 
 74 
 
 912 
 
 80 
 
 9 
 
 5735 
 
 60 
 
 42 
 
 3482 
 
 78 
 
 75 
 
 832 
 
 80 
 
 10 
 
 5675 
 
 52 
 
 43 
 
 3404 
 
 78 
 
 76 
 
 752 
 
 77 
 
 31 
 
 5523 
 
 50 
 
 44 
 
 3326 
 
 78 
 
 77 
 
 675 
 
 73 
 
 12 
 
 5573 
 
 50 
 
 45 
 
 3248 
 
 78 
 
 78 
 
 602 
 
 68 
 
 13 
 
 5523 
 
 50 
 
 46 
 
 3170 
 
 78 
 
 79 
 
 534 
 
 65 
 
 14 
 
 5473 
 
 50 
 
 47 
 
 3092 
 
 78 
 
 80 
 
 469 
 
 63 
 
 15 
 
 5423 
 
 50 
 
 48 
 
 3014 
 
 78 
 
 81 
 
 406 
 
 60 
 
 16 
 
 5373 
 
 53 
 
 49 
 
 2936 
 
 79 
 
 82 
 
 346 
 
 57 
 
 17 
 
 5320 
 
 58 
 
 50 
 
 2857 
 
 81 
 
 83 
 
 289 
 
 55 
 
 18 
 
 5262 
 
 63 
 
 51 
 
 2776 
 
 82 
 
 84 
 
 234 
 
 48 
 
 19 
 
 5199 
 
 67 
 
 52 
 
 2694 
 
 82 
 
 85 
 
 186 
 
 41 
 
 20 
 
 5132 
 
 72 
 
 53 
 
 2612 
 
 82 
 
 86 
 
 145 
 
 34 
 
 21 
 
 5000 
 
 75 
 
 54 
 
 2530 
 
 82 
 
 87 
 
 111 
 
 2S 
 
 22 
 
 4985 
 
 75 
 
 55 
 
 2448 
 
 82 
 
 88 
 
 83 
 
 21 
 
 23 
 
 4910 
 
 75 
 
 56 
 
 2366 
 
 82 
 
 89 
 
 62 
 
 16 
 
 24 
 
 4835 
 
 75 
 
 57 
 
 2284 
 
 82 
 
 90 
 
 46 
 
 12 
 
 25 
 
 4760 
 
 75 
 
 58 
 
 2202 
 
 82 
 
 91 
 
 34 
 
 10 
 
 26 
 
 4685 
 
 75 
 
 59 
 
 2120 
 
 82 
 
 92 
 
 24 
 
 8 
 
 27 
 
 4610 
 
 7*> 
 
 60 
 
 2038 
 
 82 
 
 93 
 
 16 
 
 7 
 
 28 
 
 4535 
 
 75 
 
 6J 
 
 1956 
 
 82 
 
 94 
 
 9 
 
 5 
 
 29 
 
 4460 
 
 75 
 
 62 
 
 1874 
 
 81 
 
 95 
 
 4 
 
 3 
 
 30 
 
 4385 
 
 75 
 
 63 
 
 1793 
 
 81 
 
 96 
 
 1 
 
 1 
 
 31 
 
 4310 
 
 75 
 
 64 
 
 1712 
 
 80 
 
 
 
 
 32 
 
 4235 
 
 75 
 
 65 
 
 1632 
 
 80 
 
 
 
 
 Note. 1 . Here it must be observed that, of 1 1 650 infants born, 3000 
 will die in the first year. Of the 8650 who live to be one year old. 
 
EXPECT ATION OF LIFE. 201 
 
 CASE I. To find, by this Table, the expectation of any 
 
 single life. 
 
 RULE. Divide the sum of all the living in the table, at 
 the age whose expectation is required, and at all greater 
 ages, by the sum of all that die annually at that age, and 
 above it, or, which is the same thing, by the number in the 
 table of the living at that age, and half unity, or .5 sub" 
 traded from the quotient will lie the expectation required. 
 Ex. 1. What is the expectation of a life at 60 ? 
 
 The sum of the living at the age of 60 and upwards, by the table, 
 is 27947,* which divided by 2038, tfie number of living at that age, 
 gives 13.71, from which subtract'. 5, and the expectation of a life at 
 60 is equal to 13.21, or 13 years, 11 weeks nearly .-f' 
 
 Ex. 2. What is the expectation of a life 70 years of age, one of 80, 
 and one of 90 ? 
 
 CASE II. To find the probability that a given life shall con- 
 tinue any number of years, or attain a given age. 
 
 RULE. Make the number in the table, opposite to the 
 proposed age, the numerator of the fraction, and for the 
 denominator take the number opposite the present age. 
 
 Ex. What is the probability that I, who am 45, shall 
 live to GO ? 
 
 The number against 60 zi 2038") Therefore the chances in my 
 
 > favour are 20 : 12 nearly, 
 
 The number against 45 zz 3248 J or as - 5 : 3. 
 
 For, since the probability of living is equal to , the chance of 
 
 2038 3248 2038 1210 
 
 dying during that period is 1 zz . The 
 
 J 3248 3248 3248 
 
 NOTES. 
 
 1367 will die in the course of the second year. Therefore, of the 
 11650 new born infants, the chance of living to the end of the year, 
 is to that of dying within that period, as 8650 : 300, or almost a 
 to l. 
 
 Again, the chance which an infant, just born, has of .living two 
 years, is as the number of living at the end of two yea s, is to the 
 number that have died in that time, or as 7233 to (3000 + 1367) 
 4367, or nearly 2 to 1. 
 
 * This number is found by adding all the numbers up from 2038 to 
 1 inclusive. 
 
 that a set or lives* as 100, a;;ed 60, wih, on with another, enjoy 13 
 years 11 weeks ench of existence, some of them enjoying a. duration, 
 as mucti longer as others fall short of it. 
 
 K5 
 
202 
 
 EXPECTATION OF LIFT 
 
 denominators being the same, the- chance of life is to the probability 
 of dying as 2038 to 1210, or as 20 to 12, or as 5 to 3 nearly. 
 
 Ex. 2. What is the probability that a person aged 21, shall -attain 
 to 54 ? 
 
 Ex. 3. What is the probability that a 'person aged 15 should live 
 till 70 ? 
 
 Ex. 4. What chance has a person aged 70 of living 10 years longer ? 
 
 From the foregoing- table is formed 
 
 TABLE II. 
 
 Shewing the Expectation of Human Life at every Age, according to 
 the Probabilities found by Table I. 
 
 Age. 
 
 Espectu- 
 t'on. 
 
 Age. 
 
 Expecta- 
 tion. 
 
 ! 
 
 Age. 
 
 Expecta- 
 tion. 
 
 Age. 
 
 Expecta- 
 tion. 
 
 
 
 25,13 
 
 25 
 
 30,85 
 
 50 
 
 17,99 
 
 75 
 
 6,54 
 
 i 
 
 32,74 
 
 26 
 
 0,33 
 
 51 
 
 17,50 
 
 76 
 
 6,18 
 
 2 
 
 37,79 ' 
 
 27 
 
 20-, 8 2 
 
 52 
 
 17>02 
 
 77 
 
 5,53 
 
 3 
 
 39,55 
 
 28 
 
 2y,so 
 
 53 
 
 16,54 
 
 78 
 
 5,ti 
 
 4 
 
 40,58 
 
 29 
 
 28,79 
 
 
 16,06 
 
 79 
 
 5,11 
 
 '> 
 
 40,84 
 
 30 
 
 28,27 
 
 55 
 
 15,58 
 
 80 
 
 4,75 
 
 6 
 
 42,07 
 
 -31 
 
 27,76 
 
 55 
 
 J5,10 
 
 81 
 
 4,41 
 
 7 
 
 4i,o;3 
 
 32 
 
 27,24 
 
 57 
 
 14,63 
 
 ; 82 
 
 4,00 
 
 8 
 
 40,79 
 
 33 
 
 26,72 
 
 58 
 
 34,15 
 
 S3 
 
 3, SO 
 
 9 
 
 40,36 
 
 34 
 
 26,20 
 
 59 
 
 13,68 
 
 : 84 
 
 3 , ) H 
 
 10 
 
 30,78 
 
 35 
 
 25,68 
 
 60 
 
 13,21 
 
 
 3,'7 
 
 i) 
 
 '39,14 
 
 36 
 
 25,16 
 
 61 
 
 1^,75 
 
 j 86 
 
 3,19 
 
 11 
 
 38,49 
 
 37 
 
 24,64 
 
 62 
 
 12,28 
 
 87 
 
 3,O1 
 
 13 
 
 37,83 
 
 38 
 
 24,12 
 
 63 
 
 11, si 
 
 SS 
 
 2.8(5 
 
 14 
 
 S7.17 
 
 39 
 
 23,60 
 
 64 
 
 11, -3 5 
 
 89 
 
 2,66 
 
 15 
 
 36 ; 51 
 
 40 
 
 29,08 
 
 65 
 
 10,88 
 
 00 
 
 2,11 
 
 56 
 
 34>,S5 
 
 41 
 
 22,56 
 
 66 
 
 10,42 
 
 91 
 
 2,09 
 
 1-7 
 
 33,20 
 
 42 
 
 22,04 
 
 67 
 
 9,96 
 
 92 
 
 1,75 
 
 18 
 
 34,1/8 
 
 43 
 
 21,54 
 
 68 
 
 9,50 
 
 93 
 
 1,37 
 
 19 
 
 33,99 
 
 44 
 
 2 1 ,03 
 
 69 
 
 9,05 
 
 94 
 
 1,05 
 
 '.'0 
 
 33.43 
 
 45 
 
 20,52 
 
 70 
 
 8,60 
 
 95 
 
 0..75 
 
 21 
 
 <Z-2 
 
 3^,90 
 
 ;;_,. >9 
 
 46 
 
 47 
 
 20,02 
 19,51 
 
 71 
 
 1 /- 
 
 ' 8,17 
 7,74 
 
 96 
 
 0,60 
 
 " '23 
 
 3T,S8 
 
 '48 
 
 19,00 j! 73 
 
 7,33 
 
 
 
 24 
 
 81j36 
 
 49 
 
 18,49 74 
 
 6,92 
 
 
 
 r l ( o find the expectation of any given life. 
 
 RULE. Seek in the table the given age, and opposite to 
 a i,-? the expectation, 
 
 Thus, the chance of life to an infant just born is $5.18, or rather 
 tt'.ore than ? year?; to a person ot 45 years ot age 20.52, as we 
 have found before., see ip, !<)2 : and to a -person of 59, just 9 years: 
 
203 
 *Upon t'uese tables is founded the doctrine of 
 
 LIFE ANNUITIES. 
 
 LIFE ANNUITIES are annual payments to continue dur- 
 ing any life or lives. These are generally purchased or 
 sold for a present sum of money. 
 
 " The present value of a life annuity" is the sum that 
 would be sufficient ( allowing for^ the chance of life failing, 
 which has been considered in the preceding pages J to pay 
 the annuity without loss. 
 
 If money bore no interest, the value of an asnuity of 17. 
 would be equal to the expectation of life. Thus, Table II. 
 p. 202, the value of an annuity for a life of 20 years of age, 
 if money bore no interest, would be equal to nearly 33 
 years and a half purchase ; that is, 33/. 10*. in hand for 
 each life, would be sufficient to pay to any number of such 
 ; lives \L per annum, 
 
 If money is capable of being 1 improved by being put out 
 to interest, the sum just mentioned would be more than 
 the value, because it would be more than sufficient to pay 
 the annuity ; and it will be as much more than sufficient 
 as the interest is greater. As, an example, 
 
 If money can be improved at 5 per cent, compound interest, the 
 halfof :ja/. lo.f., or 16/. 155., will, as we have seen, p. 102, in little 
 more than 14 years, produce the 33 /. 105. required. 
 
 It must not however be supposed, that 16/. 155. is the true value of 
 an annuity of l /. during a life of- 20. The value of an annuity certain 
 for a term equal to the expectation, always exceeds the true value 
 because, in a number of life annuities, man)'- of the payments would 
 not be to be made till a much more remote period than the term eoual 
 to the expectation. 
 
 Upon this principle the following table, is computed, 
 from which it appears that the present value of an annuity 
 of I/, on a life of 20 years of age, is equal to' 14/. and a 
 small fraction only; that is, 14/. in hand for each life, 
 jmproved at compound interest, will be sufficient to pay to 
 .any number, of such lives I/, per annum. 
 
204 
 
 LIFE ANNUITIES. 
 
 TABLE I. 
 
 Shewing the Value of an Annuity of ll. on a Single Life, at every 
 Age, according to the probabilities of the Duration of Human Life 
 at Northampton, reckoning interest at 5 per cent. 
 
 Age. 
 
 Value. 
 
 (Age. 
 
 Value. 
 
 Age. 
 
 Value. 
 
 Age. 
 
 Value. 
 
 Birth 
 
 8.863 
 
 25 
 
 13.567 
 
 50 
 
 10-269 
 
 75 
 
 4.744 
 
 lyear 
 
 11.563 
 
 26 
 
 13.473 
 
 51 
 
 10.097 
 
 76 
 
 4.511 
 
 2 
 
 ]3.420 
 
 27 
 
 13.377 
 
 52 
 
 9.925 
 
 77 
 
 4.277 
 
 3 
 
 14.135 
 
 28 
 
 13 278 
 
 53 
 
 9-748 
 
 78. 
 
 4.035 
 
 4 
 
 14.613 
 
 29 
 
 13.177 
 
 54 
 
 9.567 
 
 79 
 
 3.776 
 
 5 
 
 14.827 
 
 30 
 
 13 072 
 
 55 
 
 9.382 
 
 80 
 
 3.515 
 
 6 
 
 15 041 
 
 31 
 
 12.965 
 
 56 
 
 9J93 
 
 81 
 
 3.263 
 
 7 
 
 15.166 
 
 32 
 
 12.854 
 
 57 
 
 8.999 
 
 82 
 
 3.020 
 
 8 
 
 15.226 
 
 33 
 
 12.740 
 
 58 
 
 8.801 
 
 83 
 
 2.797 
 
 9 
 
 15.210 
 
 34 
 
 12.623 
 
 59 
 
 8.599 
 
 84 
 
 2.627 
 
 10 
 
 15.139 
 
 35 
 
 12.502 
 
 60 
 
 8.392 
 
 85 
 
 2.471 
 
 11 
 
 15.043 
 
 36 
 
 12.377 
 
 61 
 
 8.181 
 
 86 
 
 2.328 
 
 12 
 
 14.937 
 
 37 
 
 12.249 
 
 62 
 
 7-966 
 
 87 
 
 2.193 
 
 13 
 
 14.826 
 
 38 
 
 12.116 
 
 63 
 
 7.742 
 
 88 
 
 2.080 
 
 14 
 
 14.710 
 
 39 
 
 11.C79 
 
 64 
 
 7.514 
 
 89 
 
 1.924 
 
 15 
 
 14.588 
 
 40 
 
 11.837 
 
 65 
 
 7.276 
 
 90 
 
 1.723 
 
 36. 
 
 J4.460 
 
 "41 
 
 Il.f95 
 
 66 
 
 7.034 
 
 91 
 
 1.447 
 
 17 
 
 14.334 
 
 42 
 
 11.551 
 
 67 
 
 6.787 
 
 92 
 
 1.153 
 
 18 
 
 14.217 
 
 43 
 
 11.407 
 
 68 
 
 6.536 
 
 93 
 
 0.816 
 
 19 
 
 14.108 
 
 44 
 
 11.258 
 
 69 
 
 6.281 
 
 94 
 
 0.524 
 
 20 
 
 14.007 
 
 45 
 
 11.105 j 
 
 70 
 
 6.023 
 
 95 
 
 238 
 
 21 
 
 13.917 
 
 46 
 
 10.947 
 
 71 
 
 5,764 
 
 9b 
 
 0.000 
 
 22 
 
 13.833 
 
 47 
 
 10.784 
 
 72 
 
 5.504 
 
 
 
 23 
 
 13.746 
 
 48 I 10.616 
 
 73 
 
 5.245 
 
 
 
 24 
 
 13.658 
 
 40 l 10.443 
 
 74 
 
 4.990 
 
 
 
 To find the value of an annuity for a person of any given 
 age. 
 
 Ru LE. Multiply the number in the table against the 
 given age, by the sum, and the product is the answer. 
 
 Ex. 1. What should a person, aged 45, give to purchase 
 an annuity of 607. per annum during life, interest being 
 reckoned 5 per cent ? 
 
 The value in the table against 45 years is 11.105, and this multi- 
 plied by 60 gives the answer, 666Z. 6s. 
 
 Ex. 2. A person aged 69 years would purchase an annuity of 200Z. 
 for life, what must he pay for it in ready money at the same rate of 
 interest ? 
 
LIFE ANNUITIES. 205 
 
 Ex. 3. A merchant marries a lady aged 28, whose fortune for life is 
 300^. per annum, being desirous of converting the same into money, 
 what ought he to have for it, allowing interest 5 per cent. ? 
 
 Ex. 4. What is the value of an annuity of 200/. during the life of a 
 person aged 25 years? 
 
 Ex. 5. What is the value of 50l. per annum, payable during the life 
 of a person aged 41 years ? 
 
 Ex. 6. What is the value of a clear annuity of 75^. during the life of 
 an old man aged 76 ? 
 
 Ex. 7. What is the value of a landed estate during the life of a person 
 aged 38, producing nett 30/. gs. per annum? 
 
 Ex. 8. What is the life interest of a person aged 5S, in 1250J. 3 per 
 cent. Consols worth ? 
 
 Ex. 9. A gentleman aged 60, who receives an annuity of isol. per 
 annum, for life, out of a freehold estate, wishes to exchange his life 
 for that of his wife, aged 32 : what ought to be required of him for so 
 doing ? 
 
 Ex. lo. A person having an annuity of loo/, during a life of 37 
 years, agrees to exchange it for an equivalent annuity during^ a life of 
 45 ; what annuity should be granted him ? 
 
 Ex. 11. What annuity will loo/, purchase during the life of a person 
 aged 28 ? 
 
 Ex. 12. A parish means to raise a sum of money for building a 
 workhouse, by life annuities ; at what ages should they grant 7, 8,.and 
 9 per cent.?* 
 
 Ex. 13. What is the difference in value between an annuity of 40L 
 during a life of 36, and an annuity certain for 20 years ?f 
 
 Ex. 14. A person aged 27 is possessed of 6o/. per annum in the 
 government long annuities, which have 51 years to run, and which he 
 is willing to relinquish for an annuity during his life j what should 
 the equivalent annuity be ? 
 
 Ex. 15. What annuity should be granted to a person aged 57 during 
 his life, for 2,000^. five per cent, stock, which is now at 99| ? 
 
 * Questions of this sort are answered by dividing 100/. by the rafes 
 per cent., and opposite to the numbers in the table that aie nearest t >e 
 quotient, are the required ages : thus, to find at what age a iife an- 
 
 100 
 
 nuity of 9 per cent, should be granted, - ~ 1 1 .1 1 1 , t 
 
 number in the table is 11.105, by the side of which is 45, hence, to 
 ages of 4!?, an annuity of 9 per cent, may be granted. 
 f See Tables, p. 204 and 196. 
 
206 ITIFE ANNUITIES. 
 
 TABLE II. 
 
 Shewing the Value of an Annuity during the joint continuance of 
 Two Lives, according to the probabilities 'of Life at Northampton, 
 reckoning interest at 5 per cent. 
 
 Ages. 
 
 Value. 
 
 Ages. 
 
 Value. 
 
 Ages. 
 
 Value. 
 
 Ages. 
 
 'Value. 
 
 5-5 
 
 11-984 
 
 15-35 
 
 10.655 
 
 30-30 
 
 10.255 
 
 45-70 
 
 5.1 93 
 
 5-10 
 
 12.315 
 
 15-40 
 
 10.205 
 
 30-35 
 
 9.954 
 
 45 75 
 
 4.206 
 
 5-15 
 
 11.954 
 
 15-45 
 
 9.690 
 
 30-40 
 
 9.576 
 
 45-80 
 
 3.197 
 
 5-20 
 
 11.561 
 
 15 50 
 
 9-076 
 
 30-45 
 
 9.135 
 
 50-50 
 
 7.522 
 
 5-25 
 
 11.281 
 
 15-55 
 
 8.403 
 
 30-50 
 
 8. .->96 
 
 50-55 
 
 7.098 
 
 5-30 
 
 10.959 
 
 1 5-60 
 
 7.622 
 
 30-55 
 
 7-099 
 
 50-60 
 
 6.563 
 
 5-35 
 
 10.572 
 
 15-65 
 
 6.705 
 
 30-60 
 
 7-292 
 
 50-65 
 
 5.897 
 
 5-40 
 
 10. 202 
 
 15-70 
 
 5.631 
 
 30-65 
 
 6.447 
 
 50-70 
 
 5.054 
 
 5 45 
 
 0.57J 
 
 15-75 
 
 4.495 
 
 30-70 
 
 5.442 
 
 50-/5 
 
 4.112 
 
 5-50 
 
 8.Q41 
 
 15-80 
 
 3.372 
 
 30-75 
 
 4.365 
 
 50-80 
 
 3.140 
 
 5-55 
 
 .'256 
 
 20-20 
 
 11.232 
 
 30-80 
 
 3.290 
 
 55-55 
 
 6.735 
 
 5-60 
 
 7.466 
 
 20-25 
 
 10.980 
 
 35-35 
 
 9.680 
 
 55-60 
 
 6.272 
 
 5-65 
 
 6.546 
 
 20-30 
 
 10.707 
 
 35-40 
 
 9.331 
 
 55-65 
 
 5.671 
 
 5-70 
 
 .472 
 
 2O-35 
 
 10.363 
 
 35-45 
 
 8.921 
 
 55-70 
 
 4-. 8 93 
 
 5-75 
 
 4.362 
 
 20-40 
 
 9.937 
 
 35-50 
 
 8.415 
 
 55-75 
 
 4.006 
 
 5-80 
 
 3.238 
 
 20-45 
 
 9.448 
 
 35-55 
 
 7.649 
 
 55-80 
 
 3.076 
 
 .10-10 
 
 12.665 i 
 
 20-50 
 
 8.861 
 
 35-60 
 
 7.174 
 
 60^60 
 
 5.888 
 
 10-15 
 
 12.302 
 
 20-55 
 
 8,216 
 
 35-fi5 
 
 6.360 
 
 60-65 
 
 5.372 
 
 10-20 
 
 11.906 
 
 20-60 
 
 7.463 
 
 35-70 
 
 5.382 
 
 60-70 
 
 4.680 
 
 "10-25 
 
 11.027 
 
 20 65 
 
 6.576 
 
 35-75 
 
 4.327 
 
 60-75 
 
 3.866 
 
 "10-30 
 
 11.304 
 
 20-70 
 
 5.532 
 
 35-80 
 
 3.268 
 
 60-80 
 
 2.992 
 
 10-35 
 
 10.916 
 
 20-75 
 
 4.424 
 
 40-40 
 
 9-016 
 
 65 65 
 
 4.960 
 
 10-40 
 
 10.442 
 
 20-80 
 
 3.325 
 
 40-45 
 
 8.643 
 
 65-/0 
 
 4.3/8 
 
 10-45 
 
 9.900 
 
 25-25 
 
 10.764 
 
 40-50 
 
 8.171 
 
 65-75 
 
 3.665 
 
 1O-50 
 
 9.260 
 
 25-30 
 
 10.499 
 
 40-55 
 
 7.654 
 
 65-80 
 
 2.873 
 
 10-55 
 
 ',560 
 
 25-35 
 
 10.175 : 
 
 40-6.0 
 
 7.015 
 
 70-70 
 
 3.930 
 
 4 0-60 
 
 7.750 
 
 25-40 
 
 9.771 i 
 
 40-65 
 
 6.240 
 
 70-75 
 
 3.347 
 
 10-65 
 
 6.803 
 
 25-45 
 
 9.301 
 
 40-70 
 
 5.298 
 
 70-80 
 
 2.675 
 
 10-70 
 
 5.700 
 
 25-50 
 
 8.739 
 
 40-75 
 
 4-.272 
 
 75-75 
 
 2.917 
 
 *10-75 
 
 4.522 
 
 25-55 
 
 8.116 
 
 40-80 
 
 3.236 
 
 75-80 
 
 2.381 
 
 10-80 
 
 3.395 
 
 25-60 
 
 7.363 
 
 45-45 
 
 8.312 
 
 80-80 
 
 2. CIS 
 
 '55-15 
 
 11,960 
 
 25-65 
 
 6.515 
 
 45-50 
 
 7-891 
 
 85-85 
 
 1.256 
 
 15-20 
 
 11.585 
 
 25-70 
 
 5.489 ! 
 
 45-55 
 
 7.411 
 
 90-QO 
 
 0.909 
 
 -15-25 
 
 11.324 
 
 25-75 
 
 4.396 
 
 45-60 
 
 6.822 
 
 
 
 :15-30. 
 
 11.021 
 
 25-80 
 
 3.308 
 
 45-65 
 
 6.094 
 
 
 
 CASE 1. To 'find the value of an annuity on the longest of 
 two single lives. 
 
 RULE. From the sum of the values of the single lives 
 subtract the value of their joint continuance, and the rf- 
 ^^ai nttcr will give the value of the longest of the livM* 
 
LIFE ANNUITIE-S. 07 
 
 '"Ex. 1. What is the value of the longest of two lives 
 aged 10 and 15-? 
 
 , . , r ( The value of a life at - - - - - 10 in 15.139 
 
 5 l ' I _-_->-- 15 - 14.588 
 
 20.727 
 
 Table II. The value of the joint Continuance of two 
 
 lives of - - - - -- - - 10 and 13 m 12.302 
 
 Value of the longest of the (wo lives - - - 17.425 
 Therefore an annuity of loo/, a year upon the longest of two lives, 
 6ne 10 and the other 15, woruld be worth nearly 17 years and a half 
 purchase, or more accurately, 1742/. los. 
 
 Ex. 2, What is the value of an annuity On the longest of two lives 
 whose ages are thirty and forty. 
 
 CASE II. To find the value of an annuity on three joint 
 lives. 
 
 RULE. Take the value of the two elder, and fold the 
 age of a single life equal to that ; then find the value of 
 the joint lives of this now found, and the youngest. 
 Ex. 1. Let the three lives be 20, 30, and 40. 
 
 The value of the joint continuance of the two eldest ; viz of 30 
 and 40 (by Table II.) is equal to 9-576, which answers to a single life 
 (by Table -I.) of -54. ' Now, the value of ihe joint lives of 20 and .'>4 
 by Table II., or the ages which come nearest, viz. 20 and 55, is 
 8.216 * for the value sought : hence an annuity of 40/. on three joint 
 lives would be worth about 3 2*8 /. 125. 
 
 Ex. 2. To find the value of 3 joint lives of the -ages 15, 30, and 45. 
 
 Ex. 3. -What is the value of an annuity of 1 50/, on the joint con- 
 tinuance-of -three lives of the ages 50, 60, and 70 ? 
 CASE HI. To find the value of the longest of any three 
 lives. 
 
 RULE. From the sum of the values of all the single lives, 
 subtract the sum of the values of all the joint lives, com- 
 bined two and two. To the remainder add the value of 
 the three joint lives > and the sum tvill be the value of the 
 longest of the three lives. 
 
 Ex. 1. What is the value of the longest of three lives, 
 whose ages are 20, 30, arid 40 ? 
 
 fValue of a life-of - 20 :n 14.007 
 Table I. < so ~ 13.072 
 
 (^ 40 ZI 11.837 
 
 38.91ft 
 
 * The numbers 9.576 and 8.216, are not quite accurate, because 
 the limits of this book do -not admit of a table giving the combination* 
 of all ages. 
 
908 LIFE ANNUITIES. 
 
 Value of two joinf lives of 20 and so rz 10.707 
 20 and 40 9.937 
 
 so and 40 n. 9-576 
 
 38.916 
 
 30.220 
 
 8.696 + 8.216 (the value of the joint lives found in Ex. i. 
 Case II.) iz 16.912 the value of the longest of the three lives. 
 
 Ex. 2. What is the value of the longest of three lives, whose ages 
 are 15, 30, and 45 ? 
 
 Ex. 3. What is the value of an annuity on the longest of three lives, 
 whose ages are 50, 60, and 70 ? 
 
 EXAMPLES FOR PRACTICE. 
 
 Ex. l. What is the present value of an annuity of 50L, on the joint 
 lives of two persons, each 30 years of age ? 
 
 Ex. 2. What is the present value of an annuity of 65Z., during the 
 joint lives and the life of the survivor, of a man aged 45, and his wife 
 aged 35 ? 
 
 Ex. 3. What is the value of a lease producing 27^. 13s. per annum, 
 on the longest of two lives aged 60 and 45 ? 
 
 Ex. 4. What is the value of an annuity of 40/. on two joint lives of 
 70 and 5 years? 
 
 Ex. 5. What is the value of an annuity of 50Z. on the longest of two 
 lives of 70 and 5 years ? 
 
 CASE IV. To find the value of an annuity on a given life 
 for any number of years. 
 
 , RULE. Find the value of a life as many years older than 
 
 the given life as are equal to the term for which the annuity 
 
 is proposed. Multiply this value by \L payable at the end 
 
 of this term, and also by the probability that the life will 
 
 continue so long. Subtract the product from the present 
 
 ' value of the given life, and the remainder multiplied by 
 
 the annuity will be the answer. 
 
 Ex. 1. What is the value of an annuity of 50/. per ann. 
 for 14 years, on a life of 35 ? 35 + 14 = 49. 
 
 The value of a life of 49 (l 4 years older than the given 
 
 life, by Table I.) - ... 10.443 
 
 The value of l /. payable at the end of 1 4 years (Table, 
 
 p. 209) -_-_-- - - 28 .505063 
 
 The probability that a life of 35 will continue 14 1 _ 2936 
 years (Table, p. 200, and the 2d Case in p. 201 .) J 401o 
 
 30.443 X .505068 x[ J -7322 zz 3.861, which, subtracted from 
 
 \4010/ 
 
 3 2. 502, the value of a life of 35, Table I. gives 8.641 ; and 8.641 X 50 
 ~432/. 15. 
 
 Ex. 2. -What is the value of an annuity of SO/, per annum for 20 
 years, provided a person aged 45 live so long ? 
 
 
LIFE ANNUITIES. 
 
 209 
 
 TABLE, 
 
 Shewing the present Value of 1 /. to be received at the end of any num- 
 ber of years, not exceeding loo ; discounting at 3 per Cent. Com- 
 pound Interest. 
 
 Yrs. 
 
 Value. 
 
 Yrs. 
 
 Value. 
 
 Yrs. 
 
 Value. 
 
 Yrs. 
 
 Valuft. 
 
 1 
 
 .952381 
 
 26 
 
 .281241 
 
 51 
 
 .083051 
 
 76 
 
 .024525 
 
 2 
 
 .9070-29 
 
 27 
 
 .267848 
 
 52 
 
 .079096 
 
 77 
 
 .023357 
 
 3 
 
 .863838 
 
 28 
 
 .255094 
 
 53 
 
 .075330 
 
 78 
 
 .022245 
 
 4 
 
 .822702 
 
 29 
 
 .242946 
 
 54 
 
 .075743 
 
 79 
 
 .021; 86 
 
 5 
 
 .783526 
 
 30 
 
 .231377 
 
 55 
 
 .068326 
 
 8O 
 
 .020177 
 
 6 
 
 .746215 
 
 31 
 
 .220359 
 
 56 
 
 .065073 
 
 81 
 
 .019216 
 
 7 
 
 .710681 
 
 32 
 
 .209866 
 
 57 
 
 .061974 
 
 8-2 
 
 .018301 
 
 8 
 
 .676839 
 
 33 
 
 .199873 
 
 58 
 
 .059O23 
 
 83 
 
 .017430 
 
 9 
 
 .644609 
 
 34 
 
 .190355 
 
 59 
 
 .056212 
 
 84 
 
 01660O 
 
 10 
 
 .613913 
 
 35 
 
 .181290 
 
 60 
 
 .053536 
 
 85 
 
 .015809 
 
 11 
 
 .584679 
 
 36 
 
 .172657 
 
 61 
 
 .O50986 
 
 86 
 
 .015056 
 
 12 
 
 .556837 
 
 37 
 
 .164436 
 
 62 
 
 .048558 
 
 87 
 
 .014339 
 
 13 
 
 .530321 
 
 38 
 
 .156605 
 
 63 
 
 .046246 
 
 88 
 
 .013657 
 
 14 
 
 .505068 
 
 39 
 
 .149148 
 
 64 
 
 .044044 
 
 89 
 
 .013006 
 
 15 
 
 .481017 
 
 40 
 
 .142046 
 
 65 
 
 .041946 
 
 90 
 
 .O12387 
 
 16 
 
 .458112 
 
 41 
 
 .135282 
 
 66 
 
 .039949 
 
 91 
 
 01J7C7 
 
 17 
 
 .436297 
 
 42 
 
 .128840 
 
 67 
 
 .038047 
 
 92 
 
 .011235 
 
 18 
 
 .415521 
 
 43 
 
 .122704 
 
 68 
 
 .036235 
 
 93 
 
 .010700 
 
 19 
 
 .395734 
 
 44 
 
 .116861 
 
 69 
 
 .034509 
 
 94 
 
 010191 
 
 20 
 
 .376889 
 
 45 
 
 .111297 
 
 70 
 
 .032866 
 
 95 
 
 009705 
 
 21 
 
 .358942 
 
 46 
 
 .105997 
 
 71 
 
 .031301 
 
 96 
 
 .009243 
 
 22 
 
 .341850 
 
 47 
 
 .100949 
 
 72 
 
 .029811 
 
 97 
 
 .008803 
 
 23 
 
 .325571 
 
 48 
 
 .096142 
 
 73 
 
 .028391 
 
 98 
 
 .008384 
 
 24 
 
 .310068 
 
 49 
 
 .091564 
 
 74 
 
 .027039 
 
 99 
 
 007985 
 
 25 
 
 .295303 
 
 50 
 
 .087204 
 
 75 
 
 .025753 
 
 100 
 
 .007604 
 
 In order to find the present worth of any sum which is to be re- 
 ceived at the end of a certain number of years Multiply the number 
 in the table opposite to the term of years, by the sum, and the product 
 will be the answer. 
 
 Ex. 1. What is the present value of 750^., to be received at the ex~ 
 piration of 9 years ? 
 
 The number in the table even with 9 years is .644609, which is 
 to be multiplied by 750. .644609 
 
 750 
 
 Answer, - 483/, 95. i 
 
 2.48 
 
'$''10 LIFE ANNUITIES. 
 
 Ex. 2. What is the present value of 57-ll. 10s. 6d., to be received 
 15 years hence? 
 
 CASE V. To find the value of a given sum payable at the 
 decease of a person, whenever that shall happen. That 
 is, to find the value of an assurance of any given sum on 
 the whole duration of life. 
 
 RULE. Subtract the value of the life from the perpetuity,* 
 Multiply the remainder by the product of the given sum into 
 the rate, and this last product divided by 100/. increased 
 by its interest for a year, will give an answer in a single 
 present payment. This payment divided by the value of 
 the life, will give the answer in annual payments during 
 the continuance of life. 
 
 Ex. 1, What ought I 5 who am now 45, to pay, to assure 
 on my life 1000 ; thtft is, what ought I to pay annually, 
 to insure to my children at my decease 1000, allowing 
 money at 5 per cent. ? 
 
 The value of a life of 45, by Table, p. 204, is 11.105, and the 
 
 100 
 perpetuity is ~ 20. Therefore, by the rule ; 
 
 20 11.105 8.895, which, multiplied by 5000, gives 44475; 
 
 this, divided by 105, or 4252.' iis. bd., equal the answer 
 
 305 
 
 423?. 115. art. 
 in a single present payment. Therefore : ~3Sl. is. lOd. 
 
 nearly, in annual payments continued during life.f 
 
 Ex. 2. Let the life be 30 : the sum 100, and the rate 
 5 per cent. ? 
 
 1 The value '6f a life of 30 is, by Table, p. 20 -I, equal to 13.072, 
 and the perpetuity 20. Therefore, 20 ,13.072 rz 6.928, which, 
 
 3464 
 multiplied by 500, gives 3464, which, divided by >I 05, or 1H' 
 
 33/. nearly, being the sum to be paid in "a single payment; and 
 
 -NOTES. 
 
 * See Note, p. 21 5, for an explanation. of the word Perpetuity. 
 f Something- more than this will be demanded at the most respect- 
 able offices, as the Royal Exchange, and Equitable Insurance Offices, 
 because, in all their calculations, they do not suppose that 5 per cent. 
 can atall times be made of money. The difference between 4 and ,5 
 ^percent, will be seen in the next question. 
 
,1/1 FE ANNUITIES. 211 
 
 - r 2/. 105. 6d. nearly, in annual payments continued during 
 13.072 
 fife; 
 
 If the interest of money be supposed 4 psr cent., then the value of 
 
 100 
 a life of 30 is equal 14.68,* and the perpetuity is equal zz 25. 
 
 4 
 
 Therefore 25 14.08 zz 10.3-2. This multiplied b'y 400Z. zz 4128. 
 
 /4T28 , 39'. 145. 
 
 And zz s/. 14-5. nearly ; and ZZ 2Z. 145. 
 
 104 34.63 
 
 .Hence it appears, that when the values are required in a single 
 payment, the difference in the rate per cent, is considerable, though 
 but trifling when made in annual payments during life. In this ques'- 
 tion, if money be improved at 5 per cent,, the value of the single 
 payment would be 33/-. ; bu-t -at-4--percent.it would be Qgl. 14s., 
 which is one fifth more in the latter -case than in the former : but, 
 when the value is paid in annual sums during life ; at 5 per cent., 
 eich payment is 1l. los. 6c/., and at 4 "per cent, it is 2/. 145., making 
 a difference of 3 5, 6rf. per annum, being an increase of less than one- 
 fourteenth. 
 
 If the first of the annual payments is to be made immediately, 
 then the single payment is fo be divided by the value of the life, 
 
 T3 
 
 with unity added to it, so that at a per cent, it will be : - ZZ 2/. 
 
 6s. lid. nearly ; and at 4 per cent, it will be zz 2/. 9*. 4jtf. 
 
 15.68 
 
 Ex. 3. Let the 'life be 25, the sum 1000/., and the rate 5 per cent. 
 Ex. 4. Let the life be Go,' the sum looo/., and the rate 5 per cent. 
 
 CASE VI. To determine the value of an annuity certain on 
 -a -given life for any number of years. 
 
 RUfcE. Find the value of a life as many years older 
 than the given life as are equal to the term for which the 
 annuity is proposal. Multiply this value by \L payable 
 at the end of this term, and aho by the probability that 
 t!t>.$ life will continue so long. Subtract the product from 
 the present value of the- given life, and the remainder mul- 
 tiplied by the annuity will be the answer. 
 
 Ex. 1 . Let the annuity be 50/., the age of the given life 30 
 years, and the term proposed 15 years; interest 5 percent. 
 
 The value of a life of 45, or 15 years older than the given life, by 
 Table, p. 204, zz 11.105. The value of iL payable at the end of 15 
 years is, by table, p. 209, rz .481 ; and the probability that the life of 
 
 K o r P. . 
 
 * This is taken from a table not in this book. See Price's Rever- 
 sionary Payments, and Morgs;- - of Annuities, &c. 
 
LIFE ANNUITIES. 
 00 will exist so long, is by Table, p. 200 n .74 nearly. The;** 
 
 4385 
 
 fore 11.105 X .481 X .74 zi 3. 9^3. And the present value of the 
 given life, by Table, p. 204, zi 1-.072 ; therefore 13.072 3.958 ir 
 p. 119, and this multiplied by 50 zr 455Z. 19$. 
 
 Had the interest been only 4 per cent, the value would have been 
 about 490/. ; that is, in the one case 455/. 19s., and in the other 4gol. 9 
 by a person who would insure an annuity of 50Z. per ann. for 15 years 
 certain, which depends on the contingency of the life of a person 
 aged 30. 
 
 Ex. 2. Let the annuity be 40Z,, the age of the given life 40} and the 
 term proposed 20 years. 
 
 ASE VIT. To find the value of a given sum payable at the 
 decease of a person, should that happen within a given 
 term. In other words : What ought a person to give for 
 having his life assured to him for a certain term ? 
 
 RULE. From the value of an annuity certain for the 
 given term> subtract the value of the life for the same 
 term, and reserve the remainder. Multiply the value' of 
 I/, due at the end of the given term, by the perpetuity,* 
 and also by the probability that the given life shall FAIL 
 in the given term. The product is to be added to the re- 
 served remainder, and the sum multiplied by the given 
 sum : this last product divided by the perpetuity increased 
 by unity, gives the value in one present payment. 
 
 Ex. 1. A merchant at Liverpool, aged 30, expects to 
 realize a considerable property in the next 15 years ; but 
 as he may die before he can accomplish his views, he is 
 willing to insure on his life, during that period, the sum 
 of 5000/., what must he pay for the same. 
 
 The value of an annuity certain for 15 years, by Table, p. 196, is 
 equal to 10,379; and by example, p. 2 il, the value of an annuity 
 certain for 15 years on a life of 30 ~ 9.119 ; therefore 10.379 9. 112 
 ~ J .26 zr reserved remainder. 
 
 The value of iL to be received at the end of 15 years, by Table, 
 p. 209, m 481 ; and the probability that a life of 30 shall fail in 15 
 1142 100 
 
 years, is .26 :f and the perpetuity is 20. Therefore, 
 
 4385 5 
 
 NOTES. 
 
 * For the meaning of the word perpetuity, see note to p. 21 5. 
 f The probability of a life's failing, is always equal to the proba- 
 bility of its continuing, subtracted from unity. Thus the probability 
 
LIFE ANNUITIES. 213 
 
 .481 X .26 X 20 zz 2.5, and this added to the reserved remainder 
 1.26 zr 3.76, which multiplied by 5000, the given sum, and divided 
 by 21 (the perpetuity increased by unity) is equal 895!. 5s. nearly, 
 the value required in a single payment. That is, a person of 30 must 
 give 8951. 5s. to secure to his heirs 50001. supposing he dies within 15 
 years. Or he must pay annually during the 15 years, if he live so 
 long, 985l. 5s. divided by Q.Aig, or gsl. 3s. 4d.* for the same security. 
 
 If money can be improved at 4 per cent, only, then the sum to be 
 paid at once will be 929!. 4s> 2d., and the annual payments will be 
 lOll. nearly. 
 
 Ex. 2. It I live 7 years, I shall receive 20001. ; what must I give to 
 insure my life forthat period, being now 46 years of age ? 
 
 CASE VIII. To explain, by examples, the mode of granting- 
 annuities by the British Government established in the 
 year 1808. 
 
 [The following examples are deduced from the tables printed and 
 circulated by Government, and which may be had, gratis, at the 
 Office, Bank Buildings, Royal Exchange, London.] 
 
 Ex. i. By the tables it appears, that for every lool. stock in the 
 3 per cent, consolidated annuities, will be given annually for life, 
 to a person of 46 years, si. lu.-f* If, therefore, a person of that age 
 transfer loool. stock, he will receive an annuity for life of 55/. los. 
 But he will receive interest 3ol. and keep his capital ; and to insure 
 6601. at the Equitable, or Royal Exchange Offices, he must pay 
 rather niore than 4 per cent. ; that is, he must pay between 26 and 
 27!. annually, during life, to insure to his heirs at his dea. uie 
 6601., which he transfers to Government : he will of course be a 
 loser, by the transfer, of between one and two pounds per annum, 
 
 of a life of 30 continuing 15 years, is by table, p. 200, zz m .74,, 
 
 4385 
 
 3248 44S5--3248 1137 
 
 and the probability of its failing ml ~ 
 
 4385 4385 4385 
 
 zz .26. See Chances, p. 198 and 199. 
 
 * The payments are supposed to be made at the eml of every year. 
 But in all assurances, the first premium is paid immediately, and the 
 remaining ones at the beginning of every year after ; hence the proper 
 divisor will be the value of the life for one year less than the given 
 term added to unity, or, in this case, the value of a life for 14 years. 
 And generally: the divisor for determining the annual payments must 
 be increased by unity, whenever it is proposed that the first payment 
 should be made immediately. See p. 211. 
 
 f Supposing stocks to be at 66, which they are at present. 
 
214 REVERSION'S. 
 
 It is therefore obvious, that no one, when stocks are at 66, can j^ntt 
 in the plan held out by Government, who is not willing to give, up 
 his capital. 
 
 Ex. 2. When stocks are at 60, he will receive for loool. stock, 
 52/. 105. ; and to insure 600/. must pay more than 241. to insure his 
 life, and will of course be a loser of il. 1 05. per annum. 
 
 Ex. 3. When stocks are at 80, as they may be, he will receive for 
 the lo.ool. stock 6.2!.; but to insure 8001., he must pay annually 
 rather more than 321.; in this case there will be his interest left, and 
 he will be neither gainer nor loser. 
 
 These examples will suffice for the whole. 
 
 REVERSIONS. 
 
 REVERSIONS, or Reversionary Annuities, are those which 
 do not commence till after a certain number of years, or 
 till the decease of a person, or sonve other future event has 
 happened. 
 
 CASE I. To find the present value of an annuity for a term 
 of years, which is. not to commence till the expiration of 
 a certain period. 
 
 HULK. Subtract from the value of an annuity for the 
 whole period, the vahie of an annuity to the ti?ne when the 
 reversionary annuity is to commence. 
 
 Ex. 1. "What is the present value, at 5 per cent, com- 
 pound interest, of 80/. per annum for 24 years, com- 
 mencing at the end of 8 years ? 24 + 8 = 32. 
 
 The present value of an annuity (Table, p. 196,) for 32 years, is 
 15. 80267 7, and the value of one for 8 years is 6.46321.3, therefore 
 15.802677 
 0.463213. 
 
 9.339464 X 80. 747-15732 ~ 747/. 35. ijd. 
 
 Ex. -2* What is the present v.aiue of an annuity of 55/. for 15 years., 
 Co commence at the end of 15 years ? 
 
 Ex. 3. What is tfce present value of an annuity for 49 years, to 
 commence at the end of 47 years ? 
 
 CASE II. To find the value of an annuity certain for a given 
 term, after the extinction of any life or lives. 
 
 I^E. Subtract the value of the life or lives from the per- 
 
REVERSIONS* 
 
 petuity,* and reserve the remainder. Then say, as thepef- 
 petuity is to the present value of the annuity certain, so is 
 the reserved remainder, to the number of years purchase 
 required. 
 
 Ex. 1. What is the value of an annuity certain for 14 
 years, to commence at the death of a person aged 35, al- 
 lowing: 5 per cent. ? 
 
 The value of a life of 35 (Table, p. 204) zz 12.502 ; this subtracted 
 from 20, the perpetuity, leaves 7.498zi reserved remainder. Then, as 
 20 : 9.898f :: 7.498 : 3. 7107 ~ number of years purchase. 
 
 Ex. 2. A and his heirs are entitled to an annuity certain for 2p 
 years, to commence at the death of a cousin aged 45 years ; what can 
 A sell his interest in this annuity for ? 
 
 NOTES. 
 
 * PERPETUITY, is the number of years purchase to be given for 
 an annuity which is to continue forever; and it is found by di- 
 viding 100/. by the rate of interest; thus, allowing 5 per cent., the 
 
 10O 
 perpetuity is 20 years, or ~ 20 ; and at the rates most usually 
 
 adopted, the perpetuity is as follows : 
 100 
 
 At 3 per cent. ~ 33.33, &c. 
 
 3 
 
 100 
 
 ditto 
 
 28.57 &C 
 
 
 3.5 
 
 ditto 
 
 100 
 
 
 4 
 
 ditto 
 
 100 
 HI 22,22, &C S 
 
 
 4.5 
 
 (5-itto 
 
 100 
 
 
 20 
 5 
 
 ditto 
 
 100 
 n 16.66, &c. 
 
 
 6 
 
 ditto 
 
 100 
 ~ 14.28, &C. 
 
 
 7 
 
 ditto 
 
 100 
 =12.5 
 
 These are the number of years purchase to be given for a perpe- 
 tual annuity, on the supposition that it is receivable yearly : but, as 
 annuities are more commonly received half-yearly, and the interest 
 of money likewise paid half-yearly ; in this case the perpetuity will be 
 somewhat greater or less than the above, as the periods at which the 
 annuity is payable are more or less frequent than those at which the 
 rate of interest is here supposed payable. 
 
 ^ Tbe value of an annuity certain foi M years, Table, p. 196, 
 
216 REVERSIONS. 
 
 CASE III. To find the value of an annuity for a term cer- 
 tain ; and also for what may happen to remain of a given 
 life after the expiration of this term. 
 
 RULE. Find the value of a life as many years older than 
 the given life, as are equal to the term for which the an- 
 nuity certain is proposed. Multiply this value by I/, pay- 
 able at the end of the given term, and also by the probabi- 
 lity that the given life tvill continue so long. Add the 
 product to the value of the annuity certain for the given 
 term, and the sum will be the answer. 
 
 Ex. 1. What is the value of an annuity of GO/, for 14 
 years, and also for the remainder of a life now aged 35, 
 after the expiration of that term ? 35 + 14 = 49. 
 
 The value of a life aged 49 (Table l. p. 204) - - rz 10.443 
 The value of ll. payable at the end of 14 years 
 
 (Table, p. 2 09.) = .505068 
 
 The probability that the life will exist so long, } 2936 
 
 (Table, p. 200 and 201) - - - - 5 "~~Mo7o" 
 
 2936 
 Therefore, 10,443 X .505068 X 3.861 ; this added to 9. 898 % 
 
 4010 
 
 the value of an annuity certain for 14 years, (see Table, p. 195) zr 
 13.759, the number of years purchase; and 13.759 X 60 iz 825/. 
 105. 9|rf. 
 
 Ex. 2. What is the value of an annuity of 75/. for 10 years, and 
 also the remainder of a life now aged 24, after the expiration of that 
 term ? 
 
 CASE IV. To find what annuity can be purchased for a 
 given sum, during the joint lives of two persons of given 
 ages, and also during the life of the survivor, on condi- 
 tion that the annuity shall be reduced one-half at the 
 extinction of the joint lives. 
 
 RULE. Divide twice the given sum by the sum of the 
 value of the two single lives, and the quotient will give the 
 annuity to be paid during the joint lives, one-half of which 
 is therefore the annuity to be paid during the remainder of 
 ike surviving life. 
 
 Ex. 1. A man and his wife, aged 35 and 27, are desir- 
 ous of sinking 2000/., in order to receive an annuity during' 
 their joint lives, and also another annuity of half the value 
 cluring the remainder of the servivmg life : what annuities 
 ought to be granted them ? 
 
REVERSIONS. 
 
 o_f,, J Tab!e M , 204 
 
 25.879 The.e- 
 
 4000 (twice the sum) 
 fore, >; zr 154*. Us. 3a. rz annuity during their 
 
 25.879 
 joint lives: and ?.fl. 5s. 7 Jt/. annuity during the life of the survivor. 
 
 Ex. 2. A single man, aged 60, possessed of 1500/. is desirous of pur- 
 chasing with it an annuity for himself and his sister, aged 40, during 
 their joint lives, with one of half the value, during the remainder of the 
 life of the survivor, at the death of either ; what will be the value of 
 the annuities ? 
 
 Ex: 3. A man, possessed of lOOOL, which he will sink in the same 
 way, and for the same purposes, during the joint lives of himself and 
 father; the age of the one is 55, of the other 80 : what annuities can 
 be given for it ? 
 
 V. To find the value of the expectation of a perpetual an- 
 nuity, provided one person of a given age survives ano- 
 therof a given age. 
 
 (i). IF THE EXPECTANT BE THE ELDER. 
 RULE. Find the value of an annuity on two equal joint 
 lives, whose common age is equal to the age of the oldest 
 of the two proposed lives i subtract this value from the 
 perpetuity) and take half the remainder: then say, 
 
 As the expectation of the duration of life of the younger, 
 Is to that of the elder ; 
 
 So is the half remainder to a fourth- proportional: 
 which will be the number of years purchase, if the expect- 
 ant is the older. 
 
 (-2). IF THE EXPECTANT BE THE YOUNGER.. 
 
 Add the value found, as above* to that of the joint live# 9 
 and let the sum be subtracted from the perpetuity , and the 
 remainder is the answer. 
 
 Ex. 1. What is the value of B's expectation, (aged 30), 
 of an estate of 507. per annum, provided he survive A, 
 aged 20? 
 
 Value of two joint lives, aged 3O-, (Table II. p. 2O6) zz 10.255, the 
 difference between which and 20, (the perpetuity), is 9-745, the half <>f 
 which is 4.872 : 'therefore, 
 
 on } 
 
 4,119 205Z. IQs. 
 
 L 
 
218 REVERSIONS, 
 
 Ex. 2. What is the value as above, when B is 20, and 
 A 30? 
 
 Then, to 4. lig just found, add 
 
 10. 707, Value of the joint lives (Table II. p. 206.) 
 
 14.826 ; this subtracted from 20, the perpetuity, and 
 the remainder, 5.174 X 50IZ258/. 14s. is the true answer. 
 
 EXAMPLES FOR PRACTICE. 
 
 Ex. l. What is the additional value of an annuity certain for the term 
 of 28 years, if extended to the longest of 3 lives that may survive that 
 term, each now 30 years of age? 
 
 Ex. 2. What is the additional value of an annuity certain for the term 
 of 40 years, for the longest Of 3 lives that may survive that term, each 
 now 32 years of age ? 
 
 Ex. 3. What is the difference in the value of an annuity of 20/. cer- 
 tain for 30 years, and an annuity of the same amount on the longest of 
 tyvo lives, aged 25 and 40 ? 
 
 Ex.4. What is the value of an estate of isoZ. per annum, held on the 
 longest of two lives, aged 40 and 50, subject to the payment of an an- 
 nuity of ]4/. tea life of 62, and another annuity of 18 /.to a life of 65? 
 
 Ex. 5. What is the present worth of 2000/. to be received at the de- 
 cease of a person aged 65 ? 
 
 Ex. 6. What is the present value of 362. a-year, being the third part 
 of a farm in Essex, after the death of a person aged 54 years? 
 
 Ex. 7. What is the present value of a reveisionary annuity of 259/. 
 3s. Sd. curing the life oi a person aged 24, in case he survives his bro- 
 ther, aged 34? 
 
 Ex. 8. What should be the consideration to be paid at the death of a 
 person aged 80, for loool. now advanced to a person aged 25, in case 
 the latter survives the former? 
 
 Ex. p. What is the reversion worth of 2000Z. 3 per cent, consols, if a 
 life of 3*7 survives one of 53 ? 
 
 Ex. 10. What is the present worth of the reversion of 3075/. 3 per 
 cent, consols, if a life of 37 survives two lives, one aged 58 years j and 
 the other 30 ? 
 
 Ex. 1 1 . What is the value of the reversion of 91 1. per annum for ever, 
 after the death of a person aged 53 ? 
 
 Ex. 12. A person aged 52, is entitled to 800/. at the death of another 
 aged 76, provided the former survives the latter : what is its present 
 worth ? 
 
 Ex. 13. What is the present value of an annuity on the longest of 
 two lives, now aged 25 and 30, the annuity not to commence till 14 
 years hence ? 
 
 Ex. 14. What is the value of an annuity of 135/., secured. by, and 
 payable out of, the dividends on 3 percent. Reduced Bank Annuities, to 
 which the purchaser will become entitled on the decease of a lady in 
 her 72d year, and to enjoy the same during the life of a healthy gentle- 
 man, now in his 40th year, rf he is the survivcr? 
 
219 
 
 LEASES. 
 
 A LEASE is a conveyance of any lands and tenements, 
 iwade, in consideration of rent, or of a present sum of mo- 
 ney, for life, or for a term of years. 
 
 The purchaser of a Lease may be considered as the pur- 
 chaser of an annuity equal to the rack-rent of the estate ; 
 its value must therefore be calculated on the same princi- 
 ples as that of an annuity. 
 
 The sum paid down for the grant of a lease is so much, 
 as being- put out to interest will enable the landlord to re- 
 pay himself the rack-rent of the estate, or the yearly value 
 of his interest therein. 
 
 The value of the lease depends on the length of the term, 
 and the rate of interest which the landlord can make of his 
 money. 
 
 The value of leases at 5 per cent, compound interest r 
 be found from Table, p. 19(?. 
 
 Thus, the value of a lease for 14 years, of a farm worth 150Z. per am.. 
 is, by that table 9.898641 X 150 = 14841. 155. lid. 
 
 Ex. 1. What ought to be given for a lease of 26 years of 
 ah estate of 18/. per annum clear annual rent, * in order that 
 a purchaser may mak,e 5 per cent, of his money ? 
 
 Ex. 2k A friend has just purchased the lease of a houpe 
 for 54 years, for which he gave 550/., and he is to pay a 
 ground-rent of I/, per annum: how much ought the house 
 to let for, allowing 5 per cent, interest only ? 
 
 Leases are generally calculated at a higher rate of interest; 
 we shall therefore insert the following 
 
 TABLE, f 
 
 Shewing the Number of Years Purchase that ought to be 
 given for a Lease, for any Number of Years not exceed- 
 ing 100, at 6, 7, and 8 per cent, interest. 
 
 * That is, the next surplus rent, after deducting the reserved rent, 
 if any, and all taxes and other annual charges. 
 
 t See Gregory's Dictionary of Arts and Sciences : also Bail : 
 Leases. L 2 
 
220 
 
 LEASES. 
 
 Yis. 
 
 6 per C. 
 
 7 per C. 
 
 8 per C. 
 
 Yrs. 
 
 6 per C. 
 
 7 per C. 
 
 8 per C. 
 
 1 
 
 .9433 
 
 .9345 
 
 .9259 
 
 51 
 
 5.8130 
 
 13.8324 
 
 I-2.U 1 i 
 
 2 
 
 1.8333 
 
 1.8080 
 
 1-7832 
 
 52 
 
 5.8613 
 
 13.8621 
 
 12.2715 
 
 3 
 
 2.6730 
 
 2.6243 
 
 2.5770 
 
 c >3 
 
 5 9009 
 
 13,8898 
 
 i2 28b4r 
 
 4 
 
 3.4651 
 
 3.3872 
 
 3.3121 
 
 54 
 
 5.9499 
 
 13.9157 
 
 12.3041 
 
 5 
 
 4.2123 ' 
 
 4.1001 
 
 3.99^7 
 
 55 
 
 15-9905 
 
 13.9399 
 
 12.3186 
 
 6 
 
 4.9173 
 
 4.7665 
 
 4.6228 
 
 56 
 
 16.0288 
 
 13 9265 
 
 12,3320 
 
 7 
 
 5.5823 
 
 5.3892 
 
 5 2063 
 
 57 
 
 16.0649 
 
 13.5837 
 
 12.3444 
 
 8 
 
 6.2097 
 
 5.9712 
 
 5./466 
 
 58 
 
 16.0989 
 
 14.0034 
 
 12. 3:> 60 
 
 V 
 
 6.8016 
 
 6.5152 
 
 6.2463 
 
 59 
 
 16.1311 
 
 14.0219 
 
 ' 12.3669 
 
 10 
 
 7.3600 
 
 7.0235 
 
 6-7100 
 
 60 
 
 16.1614 
 
 14.0391 
 
 12.3765 
 
 11 
 
 .7.8868 
 
 7.4986 
 
 7.1389 
 
 61 
 
 16.1900 
 
 14.0553 
 
 12.3856 
 
 12 
 
 8.3838 
 
 7 9426 
 
 7-5360 
 
 62 
 
 I6.2i;0 
 
 14.0703 
 
 12.3941 
 
 13 
 
 8.8526 
 
 8.3576 
 
 7-9037 
 
 63 
 
 16.2424 
 
 14.0844 
 
 12.4020 
 
 14 
 
 9 2949" 
 
 8.7454 
 
 8.2442 
 
 64 
 
 16.2664 
 
 ' 14.0976 
 
 12.4092 
 
 15 
 
 9.7122 
 
 9.1079 
 
 8.5594 
 
 65 
 
 16.2891 
 
 14.1099 
 
 12.4159 
 
 16 
 
 10.1058 
 
 9.4466 
 
 8.8313 
 
 66 
 
 lp.3104 
 
 14.1214 
 
 12.4222 
 
 17 
 
 10 -4 7 7 a 
 
 9. /632 
 
 9.1216 
 
 67 
 
 16.3306 
 
 14.1321 
 
 12.4279 
 
 18 
 
 10.8276 
 
 10.0590 
 
 9-3718 
 
 68 
 
 16 3496 
 
 14.1422 
 
 12.4333 
 
 19 
 
 11.1581 
 
 10.3855 
 
 9-6035 
 
 69 
 
 16.3676 
 
 14.1516 
 
 12.4382 
 
 20 
 
 11.4699 
 
 10.5940 
 
 9.8181 
 
 70 
 
 16.3845 
 
 14.1603 
 
 12.4428 
 
 21 
 
 11.7-40 
 
 10.8355 
 
 10.0168 
 
 71 
 
 16 4005 
 
 14.l68\ r > 
 
 12.4470 
 
 22 
 
 12.0415 
 
 11.061-2 
 
 1.0.2007 
 
 72 
 
 16.4155 
 
 14.1762 
 
 12.4509 
 
 23 
 
 12.3033 
 
 il.2721 
 
 10.3710 
 
 70 
 
 16.4297 
 
 14.1834 
 
 12.4546 
 
 24 
 
 12.5503 
 
 11.4693 
 
 10.5287 
 
 74 
 
 16.4431 
 
 14.1901 
 
 12.4579 
 
 25 
 
 12.7833 
 
 11.6535 
 
 10.6747 
 
 75 
 
 16.455^8 
 
 14.1903 
 
 12.4610 
 
 26 
 
 13.0031 
 
 11.8257 
 
 10.809-J 
 
 76 
 
 16.4677 
 
 14.2022 
 
 ; 12.4639 
 
 27 
 
 13/2105 
 
 11.9867 
 
 1O.93 51 
 
 77 
 
 16.4790 
 
 14.2076 
 
 12.4666 
 
 28 
 
 13.4061 
 
 12.1371 
 
 11.0510 
 
 78 
 
 16.4896 
 
 ' 14.2127 
 
 12.4691 
 
 29 
 
 3.5907 
 
 12. '2776 
 
 li.1584 
 
 79 
 
 16.4996 
 
 14-2175 
 
 12.4713 
 
 80 
 
 13.7648 
 
 1^.4090 
 
 11.2577 
 
 80 
 
 16.5091 
 
 14.2220 
 
 12.4735 
 
 31 
 
 13.9290 
 
 12.5318 
 
 11.3497 
 
 81 
 
 16.5180 
 
 14.2261 
 
 12.4754 
 
 32 
 
 14.0840 
 
 12.6465 
 
 H-4349 
 
 82 
 
 16.5264 
 
 14.2300 
 
 12.4772 
 
 33 
 
 14.2302 
 
 12.7537 
 
 U.5138 
 
 83 
 
 16.5343 
 
 14.2337 
 
 12f4789 
 
 04 
 
 14.3681 
 
 12.8540 
 
 11.5869 
 
 84 
 
 16-5418 
 
 14.2371 
 
 12.4805 
 
 35 
 
 14.4982 
 
 12.0476 
 
 11.6545 
 
 85 
 
 16 5489 
 
 14/2402 
 
 12.4819 
 
 35 
 
 14 6209 
 
 13.0352 
 
 H.7171 
 
 86 
 
 16.5556 
 
 14.2432 
 
 12.4833 
 
 37 
 
 14.7367 
 
 13.1170 
 
 11-7731 
 
 87 
 
 16.6618 
 
 14.2460 
 
 12.4845 
 
 38 
 
 14.8460- 
 
 13.1934 
 
 H.8288 
 
 88 
 
 16.5678 
 
 14.2486 
 
 12.4856 
 
 39 
 
 1 i.9490 
 
 13.2640 
 
 11.8785 
 
 89 
 
 16-5734 
 
 , 14.2510 
 
 12.4867 
 
 40 
 
 15.0462 
 
 13.3317 
 
 H.9246 
 
 90 
 
 16.5787 
 
 14.2533 
 
 12.4377 
 
 41 
 
 15.1380 
 
 13.3941 
 
 11.9672 
 
 91 
 
 16.5836 
 
 14.2554 
 
 12.4886 
 
 42 
 
 15.2245 
 
 13.45-34 
 
 12.0066 
 
 92 
 
 16.5883 
 
 14.2574 
 
 12.4894 
 
 43 
 
 15.3061 
 
 13.5069 
 
 12.0432 
 
 93 
 
 16.59-28 
 
 14.2592 
 
 12.4902 
 
 44 
 
 ^5.3831 
 
 13.5579 
 
 1-2.0770 
 
 94 
 
 16.5&69 
 
 14.2610 
 
 12.490g 
 
 45 
 
 15.4558 
 
 13.6055 
 
 12.1084 
 
 95 
 
 U\6009 
 
 14.262/3 
 
 12.4916 
 
 46 
 
 15,5-243 
 
 13.6500 
 
 12.1374 
 
 96 
 
 16.6046 
 
 14.2641 
 
 12. 4 9-2 '2 
 
 47 
 
 15.5890 
 
 13.6916 
 
 12.1642 
 
 97. 
 
 16.6081 
 
 14/2655 
 
 12.4928 
 
 48 
 
 15.650(> 
 
 13.7004 
 
 '12.1 891 
 
 98 
 
 16.61 14 
 
 14/2668 
 
 12.4933 
 
 49 
 
 15.7075 
 
 13.7667 
 
 T2.-2121 
 
 99 
 
 16.6145 
 
 14.2680- 
 
 12.493S 
 
 50 15.7618 
 
 13.8007 
 
 12.2334 
 
 100 
 
 16.6175 
 
 14/2692 
 
 12.4942 
 
LEASES. 
 
 I. To find the sum that ought to be given for a lease. 
 
 RULE. Look in the table against the number of years for 
 whidh the lease is to continue, and on the line even with it, 
 under the given rate of interest, is the number of years 
 purchase that ought to be given for the same. 
 
 Ex. 1. What sum ought to be given for the lease of aa 
 estate ef 17 years, of the clear annual rent of 75/. allow- 
 ing the purchaser to make 7 per cent, interest of his 
 money ? 
 
 Answer, 9.7632 X 75 ~ 732.24 = 7-32/.4s. 9^-.* 
 
 Ex. 2. What must be given for a lease of 21 years, at the clear an- 
 nual rent of 50 guineas, allowing 8 per cent, for money ? 
 
 Ex. 3. .What is the worth of a lease of 83 years of an estate of 79k 
 per annum, interest being 6 per cent. ? 
 
 Ex. 4. What sum ought to be given for a lease of 69 years, of a farm 
 of 150/. per annum, the purchaser being allowed 6 per cent, for his 
 money? . 
 
 Ex. 5. What sum ought to be given for the lease of 46 years, of an 
 estate estimated at 2<)0/., but which is charged with the payment of a 
 reserved rent of 70/. 15.9., besides taxes and incidental expenses to the 
 amount of 4g/. 12s. annually; allowing the purchaser 6 per cent. 
 merest for his money ? 
 
 Ex.6. What sum ought to be given for the ground rent of a 
 house of 15^. per annum, for 18 yeais, allowing the purchaser 8 per 
 cent. ? 
 
 CASE II. To find the annual rent corresponding to any 
 given sum paid for a lease, 
 
 Ru LE. Divide the sum^ paid for the lease jy the number 
 of years purchase that are found against the given term, 
 and under the rate of interest intended to be made of the 
 pur chase money, the quotient will be the annual rent re- 
 quired. 
 
 * This sum of 7-32Z. 45. 9-Jt/., put out to compouHd interest at the 
 rate of 7 per cent., will produce a clear income of 7 5^. per annum 1 for 
 17 years ; consequently, if it be agreed that 7 per cent, is the proper* 
 interest, then the landlord has a just equivalent for his grant. 
 
 f The purchaser ought to include in this sum, the money>paid 
 down for the lease, and every expense that may be incurred previously 
 to entering upon it. 
 
.FREEHOLDS. 
 
 Ex. I. I am asked 15007. for a 40 years* lease, to what 
 annual rent is that equivalent, allowing 6 per cent, for 
 money ? 
 
 1500 
 
 Answer, rz 99/. 13s. lid. 
 
 15,046 
 
 Ex. -2. If I sell the lease of my house, which has 81 years to run, 
 for 800 guineas, at what rent will the purchaser stand, who will have 
 a ground rent of 5/. 55. per annum to pay likewise, allowing 7 per 
 cent. ? 
 
 CASE III. To find the number of years purchase; given for 
 a lease that cost a certain sum of money. 
 
 RULE. Divide the sum paid for the lease by the clear 
 annual rent of the estate for which it is given, and the quo- 
 tient will be the number of years purchase required. 
 
 Ex. 1. The lease of a house, at the clear annual rent of 
 116/. was sold for 1630/., what number of years purchase 
 was given for it ? 
 
 1630 
 
 : ~ 14 yrs. o mo. 2 weeks, 4 days. 
 116 
 
 Ex. 2. How many years purchase did the lease of a house sell for 
 which cost sool.y and the rent was eo guineas ? 
 
 For the Renewal of Leases, see p. 224, 
 
 FREEHOLDS. 
 
 CASE I. To find the gross sum which ought to be paid for 
 
 a freehold estate.* 
 
 RU'LE. (1) " Multiply the number of years purchase by 
 the annual rent." Or, (2) " Multiply the annual rent by 
 100, and divide the product by the rate of interest which 
 it is proposed to make of money ; the quotient will be the 
 sum required." 
 
 * We have already shewn, (Note, p. 215,) the number of years 
 purchase that ought to be given for the perpetuity of a freehold, accord- 
 in? o -he several rates of interest which the purchaser may make of 
 his money. 
 
FREEHOLDS. 223 
 
 Ex. What ought I to give for a freehold, the rent of 
 which is 75/. per annum, supposing I mean to make 4 per 
 cent, of my money ? 
 
 By the 1st Rule, the answer is 25 X 75 rr 1875Z. 
 
 75 X 100 
 Bythe2d. - 
 
 If I had wanted 5 per cent, for my money, the answer would 
 have been - - 1st. 20 X 75 n 1500. 
 
 75 X 100 
 
 2d. -- .1=1 500/. 
 
 5 
 
 But if I were contented with 3 per cent, then I might afford to 
 give for it 2500^. very nearly, for 
 
 1st. 33.333 &c. X 75 ZI 2499^. 19$. lid. 
 
 75 X 100 
 
 2d. --- - %49gl. 19S. lid. 
 
 3 
 
 CASE II. To find the clear annual rent which a freehold 
 ought to produce, so as to allow the purchaser a given 
 rate of interest for his money ? 
 
 RULE, Multiply the sum paid for the same, by the given 
 rate per cent., and divide by 100, the quotient will be the 
 annualjrent required. 
 
 Ex. A person has given 3000 guineas for a freeholdestate, 
 and wishes to let it so as to have 4| per .cent, for his money, 
 what must be the annual rent ? 
 
 3150 X 4-J 
 Answer, - - -- -- - 141Z. iss. 
 
 100 
 
 CASE III. To find the value of a freehold, to be entered 
 upon after a certain term. 
 
 RULE. Subtract the value of that certain term, from the 
 value of the perpetuity, and the difference will be the true 
 value. 
 
 Ex. 1. What sum should be given for the reversion of a 
 freehold after 14 years, allowing interest 6 per cent., and 
 the clear annual lent 857. 
 
 Value of a lease of 14 years, Table, p. 220, zr 9.295 ; which, sub- 
 tracted from 16.667, the perpetuity, leaves 7,872 ; and this multiplied 
 by Q5i. gives the value 626Z. 12s. 4fd. 
 
 Ex. 2. What ought I to give for the reversion of a freehold worth 
 120^. per annum , but a lease of which is sold for 5 years to come sup- 
 posing interest 5 percent, 
 
RENEWAL OF LEASES. 
 
 rVvsr I* To ascertain what fine should be given for the re- 
 newal of any number of years lapsed in a lease originally 
 granted for 21 years, See Bailey on Leases. 
 
 Tvifi.E. This is clone by means of the following 
 
 TABLE, 
 
 / 
 
 $F&s Renewing any Number of Years lapsed in a lease for 
 Twerity-on6 Years. 
 
 Years 
 
 3 per Ct.. 
 
 t per Ct. 
 
 5 per Ct. 
 
 6 per Ct. 
 
 8 per Ct. 
 
 .1 1.564 
 
 per Cent. 
 
 3 
 
 .538 
 
 .439 
 
 .359 
 
 .^94 
 
 .1<>9 
 
 .100 
 
 2 
 
 1.091 
 
 895 
 
 .736 
 
 .606 
 
 .413 
 
 .213 
 
 a 
 
 1.661 
 
 1.370 
 
 1.132 
 
 .9-36 
 
 .645 
 
 .338 
 
 '4 
 
 2.24Q 
 
 1.863 
 
 1.547 
 
 1.287 
 
 .895 
 
 .477 
 
 5 
 
 2.854 
 
 2.377 
 
 1.983 
 
 1.658 
 
 1.165 
 
 ^633 
 
 6 
 
 3.477 
 
 2.93 1 
 
 2.441 
 
 2.052 
 
 1.457 
 
 .806 
 
 7 
 
 4.119 
 
 3.466 
 
 2.922 
 
 2.469 
 
 1-773 
 
 1.000 
 
 8 
 
 4.780 
 
 4.043 
 
 3.428 
 
 2.911 
 
 2.113 
 
 1.216 
 
 9 
 
 5.461 
 
 4.644 
 
 3.958 
 
 3.380 
 
 2.481 
 
 1.457 
 
 10 
 
 6.132 
 
 5.:-69 
 
 4.515 
 
 3.877 
 
 2:878 
 
 1.726 
 
 11 
 
 6.885 
 
 5. Q18 
 
 5.099 
 
 4.104 
 
 3.307 
 
 2.026 
 
 19 
 
 7-629 
 
 6.594 
 
 5.713 
 
 4.962 
 
 3.770 
 
 2.361 
 
 *13 
 
 6.1*95 
 
 7-290 
 
 6.358 
 
 5.551 
 
 4.270 
 
 2.734 
 
 14 
 
 9.185 . 
 
 8.027 
 
 7.035 
 
 6.182 
 
 4.810 
 
 3.151 
 
 ? 15 
 
 9.998 
 
 8.787 
 
 f.745 
 
 6.847 
 
 5.394 
 
 3.616 
 
 16 
 
 10.835 
 
 9.577 
 
 8.492 
 
 7.552 
 
 6.024 
 
 4.135 
 
 17 
 
 11.69*j 
 
 10.399 
 
 9*275 
 
 8.299 
 
 6-705 
 
 4.713 
 
 IS 
 
 12.386 
 
 11.254 
 
 10.098 
 
 9.091 
 
 7.440 
 
 5.359 
 
 19 
 
 10.502 
 
 12.143 
 
 10:962 
 
 9-931 
 
 8.234 
 
 6.079 
 
 20 
 
 14.444 
 
 13.068 
 
 11.86-9 
 
 10.821 
 
 9.091 
 
 6.882 
 
 total 
 
 15.415 
 
 14.029 
 
 12.821 
 
 11.764 
 
 10.017 
 
 7-779 
 
 Ex. 1. What ought to be given as a fine for the renewal 
 of 15 years lapsed, or expired in a lease for 21 years, al- 
 lowing the tenant 5 per-cent. interest, and estimating the 
 clear and improved rent at 60 guineas per annum. 
 
RENEWAL OF LEASES. 
 
 Against 15 in the table, and under 5 per cent., is 7.745, and this mul- 
 tiplied by 63/. gives 487.935 ~487^. 18$. 8%d.* 
 
 If the interest agreed on had been 6 or 8 per cent., the answers would 
 have been 
 
 6.847 X 63 n: 431/. 7s. 2d. 
 Or, 5.394 X 63 339/. 165. $d. 
 
 Ex. 2. What ought to be given to a landlord for adding seven year* to 
 a lease, of which fourteen years are unexpired, allowing the tenant 6 
 per cent, interest for his money, and the improved f rent to be 6ol. per 
 
 annum ? 
 
 
 
 CASE II. To ascertain the value of the 'fine which ought 
 to be paid for renewing: a given number of years in any 
 lease. 
 
 RULE. The value for renewing an additional term, or 
 for adding any number of years to ike unexpired part of 
 an old lease, is equal to the difference between the value of 
 the lease for the whole term, and the value of the unexpired 
 part. 
 
 Ex. 1. What ought to be given for the atlditio.n of sever* 
 ye.ars to a lease, of which 13 are unexpired ; allowing 6 
 per cent.' for money ? 
 
 The whole term for which the .new- lease is to be granted is .20 years ; 
 therefore, Table p. 220, under 6 per cent., and 
 
 against 20 is 11. -169, and 
 
 against 13 is 8.852; therefore this last subtracted frorja the 
 former, will leave 2.617 fqr the number of years' purchase whicji 
 ought to be given for the renewal. 
 
 Ex. 2. What should be giveri for the completing a 60 years' lease, of 
 Tvhich a tenant has an unexpired term of 15 years, allowing him 7 per 
 Cent, for his money ? 
 
 * This is the sum which, put put to. interest at s per cent, would, 
 tifter the n-ext six years, the remainder of the lease, produce a clear an- 
 nual income- of 63/. for 1 5 years ; and therefore is the true sum that 
 ought to be given for the advance of these 15 yearly rents ot 63/. each, 
 and which the landlord would not .otherwise receive till the end of the 
 seventh and 14 following years. 
 
 f It often happens, that when a tenant applies for the renewal of 
 s the years lapsed in a lease, the estate has incseased in value since it has 
 been in his possession j and in such cases the landlord usually demands 
 a fine in proportion to what he conceives the rent ought to.Joe -frpriMts 
 value, .This is called the improvement. 
 
226 RENEWAL OF LEASES. 
 
 Ex. 3. I have a house for a lease of 48 years, but I wish to extend the 
 lease to gj years : how much must I pay for it, supposing the house 
 worth 5oi. per annum, and the interest 8 pei cent? 
 
 It will be seen, by working Ex. 2, of Case i, by this rule, that the an- 
 swer will be precisely the same by both methods : for the whole term for 
 which the new lease is granted is 2 1 years ; the value of a lease for this 
 term is, by Table, p. 220, 11.764, and the value of the 14 years' lease 
 yet to come is 9/295; this, subtracted from the other, gives 2.469, as be- 
 fore, which, multiplied by 60, and the answer is 148/. 25. g%d. 
 
 The value of leases or estates for single or joint lives, or 
 for the longest of two or three lives, is found by the same 
 rules that have been given, p. 2048, forfinding the value 
 of annuities for the same terms. 
 
 When estates are held on two or three lives, and one of 
 the lives nominated in the lease becomes extinct, the tenant 
 is often desirous of replacing such life, or of putting in a new 
 one, in order that the estate may continue to be held on the 
 same number of lives in being, and thereby his interest in 
 the same may be prolonged. In such cases it is customary, 
 if the estate has improved in value since the original grant 
 of the lease, for the landlord to demand a fine proportionate 
 to such improved value; and to the age of the person in- 
 tended to be added to those already in possession. 
 
 The tenant will, as rt is his interest, add one of the best 
 lives he can find, that is, a life which has the greatest ex- 
 pectation of living, according to the best tables of mortality ; 
 and such a life will be about eight or ten years : at any 
 rale few persons will be disposed to put in a life above the 
 age of twenty. 
 
 The following table will comprehend the cases that most 
 frequently occur at the rate of 5 and 6 per cent. 
 
RENEWAL OF LEASES. 
 
 227 
 
 TABLE. 
 
 For Renewing, with One Life, the Lease 
 Lives. 
 
 of an Estate held on Three 
 
 Life 
 
 Age of 
 
 
 
 Life 
 
 Age of 
 
 I 
 
 put 
 
 lives in 
 
 5 per Ct. 
 
 6perCt, 
 
 put 
 
 lives in 
 
 5perCt.6perCt. 
 
 in. 
 
 possession. 
 
 
 
 in. 
 
 possession. 
 
 1 
 
 
 30 30 
 
 1.741 
 
 1.305 
 
 
 4075 
 
 3.943 
 
 3.076 
 
 
 30 40 
 
 2.035 
 
 1.521 
 
 
 5050 
 
 3.289 
 
 2.536 
 
 
 3050 
 
 2.431 
 
 1.832 
 
 
 5060 
 
 3.910 
 
 3.039 
 
 
 30 60 
 
 2.839 
 
 2.160 
 
 
 5070 
 
 4.546 
 
 3.579 
 
 
 3070 
 
 3.277 
 
 2.535 
 
 15 
 
 5075 
 
 4.816 
 
 3.819 
 
 
 30 75 
 
 3.4(32 
 
 2.571 
 
 
 6060 
 
 ..692 
 
 3.678 
 
 
 40 4O 
 
 2.397 
 
 1-792 
 
 
 6070 
 
 5.780 
 
 4.627 
 
 
 4050 
 
 2.916 
 
 2.204 
 
 
 60 75 
 
 6.034 
 
 4.849 
 
 
 40 6O, 
 
 3.451 
 
 2.637 
 
 
 7070 
 
 7.125 
 
 5.805 
 
 10 
 
 40 70 
 
 3.914 
 
 3.032 
 
 
 30 30 
 
 1.404 
 
 1.079 
 
 "- 
 
 4070 
 
 50 50 
 50 60 
 
 4.264 
 3.563 
 4.206 
 
 3.273 
 2.723 
 3.242 
 
 
 3O 40 
 30 50 
 3060 
 
 1.673 
 2.019 
 2.363 
 
 1.284 
 1.557 
 1.831 
 
 
 50 70 
 
 4.8/3 
 
 3.819 
 
 
 3070 
 
 2.813 
 
 2.218 
 
 
 50 75 
 60 60 
 6070 
 
 5.174 
 5.023 
 6 161 
 
 4.062 
 3.911 
 4.917 
 
 
 3075 
 4040 
 40 50 
 
 2.845 
 2.027 
 2.467 
 
 2.241 
 1.558 
 1.908 
 
 
 6075 
 
 6.452 
 
 5.142 
 
 
 40 60 2.043 
 
 2.293 
 
 
 7070 
 
 7-556 
 
 6.124 
 
 20 
 
 40 70 
 
 3.358 
 
 2.641 
 
 
 30 30 
 
 1.572 
 
 1 191 
 
 
 4075 
 
 3.6.5 
 
 2 8/3 
 
 
 30 40 
 
 1,857 
 
 1.407 
 
 
 50 50 3 010 
 
 2.341 
 
 
 3050 
 
 2.227 
 
 1.699 
 
 
 5060 
 
 9.^07 
 
 2.828 
 
 
 30 60 
 
 2. GOO 
 
 1.996 
 
 
 5070 
 
 4.208 
 
 3.337 
 
 15 
 
 30 -70 
 
 3.05-2 
 
 2.381 
 
 
 .5075 
 
 4.474 
 
 3.576 
 
 
 3075 
 
 3.127 
 
 2.408 
 
 
 60 60 4.347 
 
 3.433 
 
 
 40 40 
 
 2.224 
 
 1.687 
 
 
 60 7O i 5.386 
 
 4.338 
 
 
 40 JO 
 4060 
 
 2.701 
 3.205 
 
 2.067 
 2.474 
 
 
 60 75 ! 5 636 
 70 7O 6.695 
 
 4.558 
 
 5.489 
 
 
 40 70 
 
 3.641 
 
 '2.839 
 
 
 
 
 RULE. The years'* purchase in the table, my It. . ;V by ike 
 improped annual value of the estate , -:;ef/ond the re -t pay- 
 able uiider the lease, gives the fine to be paid for pitting- 
 in the new life* 
 
 Ex. What must be given to put in a life o<~ 10 years, 
 when the ages of those in possession are 40 and 50, allow^ 
 ing 6 per cent, for money ? 
 
 Answer, 2.204, or uot quite 2-.{ years' purchase, 
 
 If the life to be added be 15 yeais, the answer would be 2.067, or 
 very little more than 2 years' purchase. And, 
 
 If the life to be added be 20 years, the answer would be 1.908, or 
 less than 2 years' purchase. 
 
22S 
 PERMUTATIONS AND COMBINATIONS. 
 
 THE PERMUTATION of quantities is the changing <*r 
 varying the order of things. 
 
 The COMBINATION of quantities is the-shewing how 
 
 ; often a less-number of things can be taken out of a greater, 
 
 fcnd combined together, without considering their .places, 
 
 or the order in which they stand. 
 
 CASE I. To find the number of changes that can be made 
 
 of any given num.ber of things* all different from each 
 
 other. 
 
 RULE. Multiply all the terms one into another, and the 
 last product will be the number of changes required. 
 
 Ex. 1. How many changes can be rung on 12 bells ? 
 
 1X2X3X4X5X6X7X8X9X1'OX 11X12479,001,600. 
 Ex.2. How many days- can eight persons be placed in a different 
 position at a -dinner table? 
 
 CASE II. Any number of different things being given, to 
 find how many changes may be made out of them, by 
 taking a given number of quantities at a time. 
 
 RULE. Multiply the number of things given by itself less 
 
 1, and that product by the same number less 2, : diminishing 
 
 each succeeding multiplier by an unit, till there arc as mtfny 
 
 products, except one, as there are things taken at a time 
 
 "the last product ivill be the answer. 
 
 Ex. How many changes can be rung with 4 bells out 
 of 12 ? 
 
 12 X 12 1 X 12 2 X 12 3 12 X 11 X 10' X 9 ~ 11880. 
 
 Ex. 2. How many changes can be rung with 5 bells out of 10 ? 
 
 Ex. a. Whaf 'niimber of words, containing each 6 letters, can be 
 formed out of the 24 letters in the alphabet supposing any 6 to form a 
 word ? 
 
 -CASE III. Ta find the combinations of a less mimber of 
 things out of a greater, all different. 
 
 RULE. Take the series 1, 2, 3, 4, fyc. up to the less num- 
 ber of things, cuid multiply them continually together for a 
 divisor : then take a series of as many term?, decreasing, 
 
PERMUTATIONS AND COM BIN ATIONS. 229 
 
 each by an unity from the greater number of things, and 
 multiply them continually together for a dividend. Divide 
 the hitter product by the former, and the quotient icill be 
 the answer. * __ 
 
 Ex. 1. HoW'many combinations can be made of 10 things 
 put of 100 ? 
 
 IX 2X3X4X5X6X7X8X9X10 
 
 (the number to be taken at a time) 3,628,800. 
 
 100 X 99 X 98 X 97 X 96 X 95 X 94 X 93 X 92 X 91 
 
 (the same number of terms taken from loo) 
 
 rz 62,91 5,650,955,529,472,000. 
 
 , 62815650955529472000 
 
 and 17310309456440.* 
 
 ^ 3628800 
 
 Ex. 2. How many combinations can be made of 3 letters out of the 
 S4 letters in the alphabet ? 
 
 Ex. 3. A club of 21 persons agreed to meet weekly, five at a time, 
 so long as they could, without the same five persons meeting together, 
 how long would the club exist ? 
 
 CASE IV. To find the coin positions of any number, in sets 
 of equal numbers, the things or persons themselves be- 
 ing different* 
 
 RULE. Multiply the number of things in every set con- 
 tinually together, and the product is the answer. 
 
 Ex. 1. There are three parties of crick etters, in each 
 eleven men, in how many ways can 11 of them be chosen, 
 one out of each ? 
 
 Answer, ] r x 11 X 11 rz 1331. 
 
 Ex. 2. In how many ways can the four suits of cards be taken, four 
 at a time ? 
 
 Ex.3. There are four parties of whi'st-players ; in one there are 6, 
 in the second 5, in the third 4, and hi the fourth 3 persons, how often 
 can the set differ with these persons? 
 
 NOTE. 
 
 * Operations of this sort^re shortened by'the following mode : 
 
 100 X 99 X 08 X 97 X 96- X 95 X 94 X 93 X 92 X 91 
 
 10 X 9 X 2 X 6 X 8 X 5 X IX 3 X 4 X 7 
 
 n 10 X 11 X 49 X 12 X 19 X 31 -X 23 iX 13 X 94 X * as 
 
 6 
 
 above ; and dividing the 12 by 6, Tve place 2 anicng the -numaeratox^ 
 and get rid of all the- denomlRAto*s a 
 
230 
 EXCHANGE. 
 
 BY Exchange is meant the bartering 1 , or exchanging, the 
 money of one place for that of another, by means of an 
 instrument in writing, called a bill of exchange. 
 
 Exchanges are carried on by merchants and bankers all 
 over Europe, and are transacted on the Royal Exchange of 
 London, the Royal Exchange of Dublin, the Exchange of 
 Amsterdam, and those of the, principal cities of the con- 
 tinent. 
 
 When an exchange is mentioned between two places, 
 one place gives a determined price, to receive an undeter- 
 mined one. 
 
 The determined price is called certain ; thus, 
 
 London gives a pound sterling, which is a certain price, to receive 
 from Paris a number of francs, more or kss, to be paid or received 
 there. Again, London gives loo/., which is a certain price, to Dublin 
 and other parts of Ireland, for an uncertain number of pounds, shil- 
 lings, and pence Irish, to be paid or received there, viz. from 105/. to 
 l \bl. Irish, as the exchange may be. 
 
 The undetermined price is called uncertain, because it is 
 always subject to variation ; for instance, 
 
 London pays an uncertain price to Spain, as a number of pence ster- 
 ling, to receive a dollar which is certain in exchange. 
 
 The real money of a state signifies one piece or more, of 
 any kind of metal coined, and made current by public au- 
 thority, as guineas, shilling's, &c. "of England. 
 
 The imaginary money is chiefly usedin keeping accounts, 
 as pounds sterling, for which there is no coin to answer. 
 
 The par of exchange is the quantity of the money, whe- 
 ther real or imaginary, of one country, which is equal in 
 value to a certain quantity of the money of another ; thus, 
 
 . 100 sterling is equal in value to 108/. 6,?. 8rf. Irish : and 100/. 
 sterling is worth 140^ of the currency in the West Indies, and equal 
 to I66l. 13?. 4d. currency of the United States.* 
 
 NOTE. 
 
 * In the following note is subjoined the par of exchange between 
 London and some of the principal commercial and other places in 
 Europe. 
 
EXCHANGE. 231 
 
 The course of exchange is the value agreed upon by mer- 
 chants and others, and is continually fluctuating above or 
 below the par of exchange, according as the demand for 
 bills is greater or less.* 
 
 Par in Sterling. . s. d. 
 
 Rome - - - i crown - zz 6 if 
 
 Naples - 
 Florence 
 Sicily - 
 Vienna - 
 Franckfort 
 Bremen - 
 
 ducat - zz 3 4^ 
 
 crown - zz o 5 4| 
 
 crown - zz 5 o 
 
 rix-dollar - zz 4 8 
 
 florin - zz 3 O 
 
 rix-dollar - zz o 3 6 
 
 Berlin - - - 1 ditto - zz 4 O 
 
 Embden - - 1 ditto - zz 3 6 
 
 Dantzic - - 13 florins - TOO 
 Stockholm- - 3 4f dollars - zz i O 
 Petersburgh, and other parts 
 
 of Russia - 1 ruble - ZZ 4 5 
 
 Turkey - - - 1 asper - zz o 4 6 
 
 In addition to the above, we may observe, that in Switzerland, at 
 Nuremberg, Leipsic, Dresden, many parts of Poland, Denmark, and 
 Norway ; at Riga, Revel, &c. the rix-dollar is, at ,the par of exchange, 
 45. 6d. ; and as it is worth more or less than this in exchange with 
 other places, the course of exchange is said to be for or against these 
 places. 
 
 * The demand for bills depends upon what is called the lalince of 
 trade, which is for or against a country, according as more or less 
 goods are exported or imported by that country, in comparison of 
 some other. Thus, if London ships to Hamburgh goods to the 
 amount of 500,000^, and Hamburgh in the same time sends to 
 London goods only to the amount of 400,ooo/., the balance of trade 
 is said to be in favour of London ; and as Hamburgh can discharge 
 only to the amount of 400,000/. by bills of exchanged the way of 
 trade, that is, to the amount of the value of goods sent to London, 
 there is a balance against her of 1OO,OOO/., which she must pay by 
 bills of exchange procured elsewhere, and for these she must pay a 
 premium. If, in this case, Hamburgh pays l/. percent, for bills, 
 she will, to liquidate the debt, have to pay looo/. premium ; in this 
 way the balance of trade affects the fluctuation of exchanges. An 
 unfavourable state of exchange furnishes a motive for exportation. 
 The merchant can, in such case, afford to sell his commodities as 
 much cheaper, as the premium which he is obliged to pay for a bill 
 of exchange amounts. Hence the course of exchange always tends 
 to an equilibrium ; and it can never exceed the expense of sending 
 gold or silver bullion to the place upon which the bill is drawn, 
 
2o2 EXCHANGE/ 
 
 Agio denotes the difference in Amsterdam and other 
 places, between current money, and the exchange or bank- 
 money, the latter being finer than the former. 
 
 Usance is a certain space of time allowed by one coun- 
 try to another for the payment of bills of exchange.* Bills 
 are either payable at sight, or at a certain number of days 
 after sight : at usance, double usance,. or half usance. At 
 one, two, &-c usaruce. means at one, two, &c. months' date. 
 Half usance is 15 days, be the month what it may. 
 
 Days of grace are a certain number of days allowed for 
 the payment of bills of exchange, after the expiration of 
 the term specified in such bills, and are variable in differ- 
 ent countries. In England three days are allowed, f 
 
 RULES for finding what quantity of the money of one 
 country will be equal to a given quantity of the- money of 
 another, according to a given course of exchange. 
 
 CASE I. "."When the course of exchange is given, how much 
 money of one. country answers to a certain sum of ano- 
 ther, a* of Great Britain ? 
 
 NOTh'E. 
 
 since this is the money of the commercial-world, and will every where 
 be accepted in payment. 
 
 The principal exchanges of Europe are governed by those of Lon- 
 don, Amsterdam, and Venice ; and the exchanges from foreign coun- 
 tries are to be only had from the merchants and bankers residing 
 abroad . 7 
 
 When England remks an uncertain price tc any other kingdom, as 
 Spain or Portugal, the lower the price of exchange, the more is it to the 
 advantage of England, -as giving 35 pewce to Spain for a dollar, instead 
 of 36 or 3 8 pence ; OT,^ giving 60 pence to. Portugal for a milreis, in- 
 stead of 65 pence. See page 230. 
 
 When England remits the-certain price, the higher the exchange is 
 ihe better, as for instance, giving ;'o Fiance ll. for 16 francs, is better 
 than for 24 or 25 ; or loo/, sterling, for 112J. Irish, is better than for 
 ac8i 65. 8^., which is the par of Exchange. Seep. 23,5. 
 
 * This space of time varies according to the custom of countries, and 
 frequently in proportion ..to the distance -of the .places from each 
 other. 
 
 f On the third day' they -must be paid ; that is, a bill drawn at two 
 5 nenths,on the 1 4th of July, must be paid on the 17th of September, 
 xBills due on Sunday must be -paid >n Saturday j for those at .sight nc 
 .days .of grace, are allowed. 
 
EXCHANGE. 
 
 RULE. As the given course of exchange, is to one pound 
 sterling, so is the given sum in foreign money, to its cor- 
 responding value in sterling monei/. 
 
 Ex. 1. How much sterling 1 money can I have for 2035 
 Flemish shillings, when the course of exchage is 37 shil- 
 lings for IL ? 
 
 , Here I say, As 37 : 1 : : 2035 : 55 z pounds sterling. 
 
 Ex. 2. How much sterling money can 1 get for 4086 
 florins, 4 stivers, 6 penings banco, supposing I/. is worth 
 08 schillings and *2 grotes ?* 
 
 schil gr. . florins st. p. 
 
 38 2 : 1 : : 40S6 4 G 
 12 40 
 
 458 163440 grotes 
 
 8 grotes zz 4 stivers 
 ^ of a grote ~ 6 penings 
 
 458)l63448.J(356Z. 17*. 6<Z. Answer, 
 
 Ex. 3. What sterling money will 2937. 10s. Gd. Irisk 
 fetch, when the exchange is 114/. Irish for 100/. sterling ? 
 
 114^. : 100*. :: 293/. 10s. 6(1. : 25?L 9s.6*d. 
 
 Ex. 4. Dublin remits to London 826/. 335., what must be received 
 there, exchange being no I. percent. ? 
 
 Ex. 5. Jamaica remits to London 287^. Os. lOjc?. currency, v/hal 
 must be received for it, exchange being 135/. per cent. ? 
 
 CASE II. Given the course of exchange, to "bring any 
 quantity of sterling' money j into the money of another 
 count iy. 
 
 RULE. As I/, sterling is to the course of exchange, so i# 
 the given sum, in sterling money, to its corresponding value 
 in foreign m&ney. 
 
 Ex. 1. How much Flemish money will 233/. 6s. Sd. 
 
 
 
 NOTE. 
 
 
 
 
 
 
 
 
 
 s. 
 
 d 
 
 
 * 8 
 
 -penings make 
 
 grote, or 
 
 penny 
 
 rz o 
 
 
 
 54 
 
 2 
 
 grotes 
 
 stiver 
 
 . _ . 
 
 ZI 
 
 i 
 
 .09 
 
 12 
 
 grotes " 
 
 schilling 
 
 - - - 
 
 zi o 
 
 6 
 
 .36 
 
 2O 
 
 schillings 
 
 pouml M 
 
 cmish - 
 
 ~ 10 
 
 1 1 
 
 .IB v 
 
 40 
 
 grotes '. . 
 
 guilder, 
 
 ar. florin 
 
 .3= i 
 
 9 
 
 ^ 
 
34 EXCHANGE. 
 
 sterling be worth, when the exchange is 34s. per I/, ster- 
 ling ? 
 
 I/. : 34J. : : 233/. 65. 8d. '. 3Q3Z. 13s. 4cZ. Answer. 
 
 Ex. Q. How much Flemish money must be given for 62b/. 105. 
 sterling, when the exchange is 335. sd. per . sterling. 
 
 CASE III. To reduce the currency of any state into bank 
 or exchange money. 
 
 RULE. As 100, with the agio added to it, is to 100, so 
 is any given sum current to its indue in bank money. 
 
 Ex. 1. How much bank money can a merchant in Am- 
 sterdam have for 5550 guilders, when the agio is 4| per 
 cent. ? 
 
 104 : 100 :: 5530 : 5311 Answer. 
 
 104.5 
 
 Ex.2. How many florins bank \villaooo currency purchase, agio 
 being 63 per cent. ? 
 
 CASE IV. To reduce bank money into currency. 
 
 RULE. As 100 is to 100, with the agio added to it, so 
 is the bank money to the currency. 
 
 Ex. 1. How much currency can I have in Venice for 
 1500 ducats bank, when the agio is 15 per cent. ? 
 
 100 ; 115 : : 1500 : 1725. 
 
 Ex. 2. How much currency can I have for 5000 bank florins; agio 
 being 8 per ent. ? 
 
 IRELAND. 
 
 ACCOUNTS are kept in Ireland as in England, viz. in 
 pounds, shillings, and pence. 
 
 The par of exchange in Ireland is 108/.-6.?. 8d. ; that is, 
 IOS/. 6s. Sd. Irish, is equal in value to 1007. sterling; or 
 Is. Id. Irish, is equal to one shilling English. 
 
 The course of exchange varies from 105/. to 1157. ac- 
 cording to the balance of trade. See p. 230. 
 
 Ex. 1. London remits to Dublin 300Z. sterling, what 
 must be received for it, exchange being 10G/. Irish per 
 cent., and also when it is 112 per cent. ? 
 
 Here I say, As 100 : 106 : : 300 : 318/. Answer, 
 and JOO : 112 : r 300 : 336/. Answer, 
 
EXCHANGE, 
 
 Here it is evident, that when England remits the certain price to 
 another country, the higher the exchange, the greater advantage is 
 derived by England ; for when the exchange is 106, she will receive 
 318/. for her 300^., and when it is 112/., she will receive 336/. for 
 the same sum. See p. 230. 
 
 Ex. 2. Dublin remits to London 7007. Irish, what is it 
 equal to when the exchange is 108/., and also when it is 
 
 no/. ? 
 
 Here Dublin remits the certain, and London gives the uncertain 
 price, and I say, As 108 : 100 : : 700 : 648/. 25. ii^,*. Answer. 
 110 : 100 : : 700 : 636/. 75. 3d. Answer. 
 
 II- c Dublin is gainer when the exchange is low, because, in that 
 case, 700/. purchases 648Z. 25. llJcZ., and in the other it purchases 
 only 636/. 7s. 3d. 
 
 Ex. 3. London remits to Dublin 545Z. 305. sterling, for how much 
 Irish must London be credited, exchange bein? 110 ? 
 
 Ex.4. Dublin remits to London 900. 15s., how much sterling 
 must be received, exchange being 112^. ? 
 
 Ex.5. I purchase sundry books in Dublin, for wh ch 1 give as 
 follows ; for the first - ' - .096} 
 second - - - o 18 o * 
 third - - - o 5 6 
 fourth - - 1 5 0^ 
 
 what are they worth in English money ? 
 
 AMERICA AND THE WEST INDIES. 
 
 ACCOUNTS are kept in these places as in England, in 
 pounds, shillings, and pence. 
 
 In America the par of exchange is 166/. 13$. 4d. ; in 
 the West Indies it is 140/. 
 
 Ex. 1. London remits to Barbadoes 945?. 17$. sterling, -'how much 
 currency will this amount to, when the exchange is 140 currency ? 
 
 Ex. 7. Sir Francis Baring writes word, that he has received for .me 
 a remittance of one quarter's dividend on 4000 dollars, at 5j per cent, 
 interest, and the exchange is 164 per cent., what has he to pay me? . 
 
 In this case the regular inteiest is 55 dollars, which, at 45, 6cl. each, 
 when exchange is at par, or at 166Z. 135. 4d., would be 12/. 175. 6c?.> 
 but the exchange is 164 ; therefore I say, 
 
 As 164 : 1661. 135. 4d. : : 12^. 75. 6d. : 12/. 115. 6d. Answer. 
 
 Or by decimals, 
 164 : 166.66 &c. : : 55 : 55.899 dollars ~ 1 2^. 1 1 s. 6d.* 
 
 NOTE. 
 
 * Exchange being lower than the par, I am a gainer of 45. ; for 
 a person who takes a bill of exchange is always benefitted by a law 
 course of exchange, 
 
36 EXCHANGE. 
 
 -The following is a TABLE of the COURSE OF EXCHANGE, 
 taken, with slight variations, from the Monthly Maga- 
 zine for the 1st of May, 1808. 
 
 COURSE OF EXCHANGE. 
 April 5. April 12. 
 
 Hamburgh gives 34.5 34.6 for iL 
 
 Altona - - gives 34.7 34.7 for do. 
 
 Amsterdam gives 35<5.2U.- 35.4.2U. do. 
 
 Ditto, sight gives 34.9 34.8 for do. 
 
 Paris, l.d. gives 23.13 l. d. 24.0 fordo. 
 
 Leghorn receives 4.9^ pence 49^ for l pezza of 8 rials 
 
 Naples - - rec. 42 ditto 42 for l ducat 
 
 Genoa . - - rec. 45 ditto 45^ for l pezza 
 
 Lisbon - - rec. 60 ditto ' 60 ) r 
 
 Oporto - - rec. 65 ditto 65 } f r 1 milrea 
 
 Madrid - - rec. 3s| do. Eff. for l dollar 
 
 Palermo-- - rec. 92 per ozf 92 perez. 
 
 Dublin - - rec. lio$i. 110 for 100 
 
 This table, in addition to what is gone before, will afford an oppor- 
 tunity of explaining every thing that a man of business will wish to be 
 acquainted with. 
 
 On the 5th of April, the exchange between Hamburgli and Lon- 
 don was at the rate of 34 schillings, 5 grotes', .for a pound sterling ; 
 that is, if a merchant in London sell a bill on Hamburgh for 500/., 
 he would be paid for it 34.5 X 500 17208 schi). 4 gro. ; but on 
 the 12th, such a bill would have fetched 34.6 X 500 ~ 17250 shil- 
 lings. Here, the higher the exchange the greater the advantage to 
 England ; for the merchant, in this instance, gains 41 schil. 8 gro. by 
 *he rise in the exchange. 
 
 For Altona, the course of exchange is the same on both days, viz. 
 the . is worth 37 schil. 7 gro. : and for Amsterdam, the course of 
 exchange falling, the merchant in London would be a loser, who put 
 off his market from the 5th to the 12th. 
 
 In this case 35.5. 2U. means, that a pound sterling is worth, on the 
 
 ' 5th, 35 schil. 5 gro,, allowing it to be payable at two months' date : 
 
 buj: if it is payable at sight, it is then worth only 34 schiJ. 9 gr. This 
 
 Difference, which on a bill of loo/, is equal to 34 schil. 4 gr., is instead 
 
 of the interest of money for the interval. 
 
 The course of exchange rose between London and Paris from the 
 5th to the 12th of April. On the first of these days l/. was at l d., 
 that is, at one day's sight, worth 23.13, or 23 francs, and 13 cents. ; 
 but on the 12th its value was 24 francs. 
 
 Leghorn receives 49j pence for l pezza of 8 rial<, that is, a bill 
 4t exchange of -5000 pezza would be wonh 4$. ifd. multiplied by 
 
EXCHANGES. 
 
 0, OP 1036Z. 95. 3d. A Naples ducat was worth 35. 6d, : a Gene* 
 pezzK3s. gd. : a milrea of Lisbon 5 shillings, arid one of Oporto 
 55. 5d. 
 
 Madrid receives 38|rf. Eff\ for l piastre of 8 rials,* that is, a Spa- 
 nish piastre o f ' exchange was worth 3> 2frf. 
 
 A species of r'^yer money .. denominated vales rials y is circulated in 
 Spain, the value o f whu:ii independently of interest on them, is this : 
 Vales rial- fp boo dcr-an are worth 9035 rials, 10 maravedis of 
 vellon^ that is, (as 34 rn.travedies is equal to one rial) l dollar pay- 
 able in this sort of pa^T is worth 15 rials, 2 maravedies. The paper 
 is trans Unable by indorsement ; and,- by law, should be received in 
 paymen; according o ihe nominal value ; but as it experiences depre- 
 ciation, 5t is nece.s';ar_y in drawing oii Spain for effective money, to in- 
 sert the words tf payable in effective" in the body of the bill, which 
 might otherwise be pavable in vales rials : hence the word Eff. in the 
 table, which is an abridgment of " in effective" 
 
 Palermo 92 pence per oz. In Sicily exchanges are made per onza 
 by the ounce of silver,- for which, on the day referred, to Palcuno re- 
 ceived 9-2 pence, or 75. 8d.$ 
 
 Dublin noj: for loo/., that is, at the date of the table there would 
 nave been given on the exchange of London a bill on Dublin tor no/. 
 55. for loot, pound sterling. See page 234. 
 
 By the agio of the Bank of Holland is meant, as we have seen, p. 
 232, the difference between cash and bank money, which, by the table, 
 is on the 5th of April, 6-J, or 6/. 10^. per cent.; that is, 106/. 105. 
 currency must be given for lOOl. bank, and so in proportion, 
 
 NOTES. 
 
 * In some parts of Spain they reckon by silver money, which is of 
 two kinds, viz. old and new plate,, the former is the most valuable : 
 thus the piastre of exchange consists of 8 rials old plate, or of 10 
 rials new plate, the rial behrgat the par of exchange worth little more 
 than 5-|</. 
 
 -f- The copper money of Spain is called vellon. 
 In Madrid, and the principal places of Spain, accounts are kept in 
 piastres (culled also ddlars) rials, and maravedies ; and sometimes in 
 chacats. 
 
 TABLE. 5. d-. 
 
 34 maravedies *} C i rial - ~ o 5| 
 
 8 rials - > make ^ l piastre zz 3 7 
 375 maravedies j (_ l ducat n 4 nj{ 
 
 Hence the piastre at par is 35. ;//., and the ducat at par 45. ll^d. 
 but the course ot\exch an ge of the piastre varies from 35 to 45 pence. 
 
 + The Sicilian ounce is Coo grains, and the monies are regulated by 
 the following Table : 
 
 10 grains - ma':e - l cailin, 
 
 2 carlins - make - 1 tar-in, 
 
 30 tarins - fGoog.r) - l ounce. 
 
 A crown (seudo) is equal 240 grs., \herefcre 5 crowns ~ 2 ounces. 
 
238 EXCHANGES. 
 
 Exchange between London and other Places in this 
 Country* 
 
 The several cities, towns, &c. in Great Britain, exchange 
 with London for a small premium in favour of London, as 
 from 4 to 1, or I| percent. The premium is more or less, 
 according to the greater or less distance, and according to 
 the demand for bills. 
 
 Ex. York draws on London for 560/. I0s, exchange 
 being f per cent. ; how much money must be paid at 
 York for the bill ? 
 
 560 10 
 2 16 
 
 1 8 
 
 . 564 14 Of 
 
 To avoid paying the premium, which, in some cases, would not be 
 just, it is the usual practice to take the bill payable a certain number 
 of days after date. On this principle, interest being 5 per cent. 78 
 
 365 
 
 days are equivalent to I/, per cent, because -~ 73. 
 
 Ex. A friend at Exeter has received for me 68 guineas, in which 
 he is no ways interested, and having no means of sending the money 
 but by a bill of exchange, he agrees \vith his banker to draw it 30 
 days after date, rather than pay the premium of J per cent. , 13 my 
 friend, or the banker, the gainer, allowing 5 per cent. ? 
 
 EXAMPLES FOR PRACTICE. 
 
 Ex. i. How much currency will 6630 guilders, bank- money, be 
 worth in Holland, agio being 8^ per cent. ? 
 
 Ex. 2. What is the agio of 3310 guilders, at 6| per cent. ?* 
 
 Ex. 3. What is the agio of 5000 dollars, at 4| per cent,, and how 
 much bank money will the 5000 currency purchase? 
 
 Ex. 4. A London merchant draws on Amsterdam for l 564/. ster- 
 ling ; how many pounds Flemish, and how many guilders will that 
 amount to, exchange being 34 schil. 8 gro. per . sterling.-f See 
 Table, p. 233. 
 
 NOTES. 
 
 * If the agio only be required, say, as 100 : a^io per cent. : : so is 
 the given sum to the agio required : here, as 100 : 6^ : : 3310 : to 
 the required sum. 
 
 f The money in Holland is sometimes reckoned in guilders and 
 stivers, as well as in schillings and grotes. To reduce Flemish 
 pounds and schillings into guilders and stivers, multiply by 6 ; and 
 if there be any pence multiply them by 8 for penings : or divide 
 
EXCHANGES. 239 
 
 Ex'. 3. How much sterling money will pay a Portuguese bill of ex- 
 change of 1654*^372 millreas; that is, of 1654 millreas and 372 reas, 
 exchange being 65 J pence sterling per millrea? * 
 
 Ex. 6. How many Portuguese reas will 750/. sterling amount to, ex- 
 change being 64| per millrea? 
 
 Ex. 7. A Spanish merchant imports from Seville goods to the value 
 of 1081 piastres, 6 rials: how much sterling money will this amount 
 -to, exchange being, on the day of payment, 41^ pence per piastre? See 
 Table, p. 236-7? 
 
 Ex. 8. I want to purchase gobds at Cadiz, and for this purpose pay 
 into a Spanish house 1000Z. : how much value, in piastres, may I ex- 
 pect, exchange being 3s. 6 Jd. per piastre ? 
 
 ARBITRATION OF EXCHANGES. 
 
 The course of exchange, between nation and nation, na* 
 turally rises or falls, as we have seen, according as the cir- 
 eum.stances and balance of trade may happen to vary. To 
 draw upon, and to remit money to foreign places, in this 
 fluctuating state of exchange, in the way that will turn out 
 most profitable, is the design of arbitration. 
 
 Arbitration of Exchange, then, is a method of finding 
 such a rate of exchange between any two places, as shall 
 
 the Flemish pence by 40, and the quotient will be guilders ; and half 
 the remainder, if theie be any, will be stivers : thus, to bring 3882. 175. 
 4g-., or 81328 Flemish pence into guilders: 
 . s. gr.~ 
 
 338 17 4 4 4 0)8132 4 8 
 
 guild, st'iV". 
 
 guild, stiv, 2033 8 n 2038 4 
 
 2033 4 203-J 4 
 
 * In Portugal accounts arc kept in rear, and millreas, the latter being 
 equal to 1 000 of the former ; and they are distinguished from each other 
 by some such mark as that in the question. 
 
 The millrea, in exchange with this country, is at par 6; sterling, 01 
 5s. 7g&'., and the course usually runs. from 5s. 3rf. to 5s. sd. 
 
 TABLE Par in sterling. 
 
 s. d. f. 
 
 l rea GO 0.27 
 
 400 reas 1 , f 1 crusade 2 3 
 
 1000 reas J n " \ 1 millrea = 5 7 
 The reas being the thousandth parts of the millreas, are annexed to 
 *?!ie integer, and the work proceeds as in decimals. 
 
2'4O EXCHANGES. 
 
 be in proportion with the rates assigned between each of 
 them and a thirdplace. 
 
 By comparing the par of exchange thus found, with the 
 present course of exchange, a person is enabled to find which 
 way to draw bills, or remit the same to most advantage. 
 
 [Questions in this rule are performed by the Rule of 
 Three.] 
 
 Arbitration of exchange, is either simple or compound. 
 
 In simple arbitration, the rates of exchange from one 
 place to two others are given, by which is found the cor- 
 respondent price between the said two places, called the 
 arbitrated price. 
 
 An example or two will make the subject clear. 
 Ex. 1. If exchange between London and Amsterdam be 
 34 schil. grotes per . sterling, and if exchange between 
 London and Genoa be 45 pence per pezza (see Table, p. 
 236,) what is the par of arbitration between Amsterdam 
 and Genoa: 
 
 Here ll. n: 240 pence : therefore, as 
 
 240d.|.' 34s. 9gr. : : 4bd. : 78 5 r. 
 Answer, 78 Flemish grotes, or pence per pezza Genoa. 
 
 , Ex. 2. If exchange from London to Amsterdam be 33s, yd. per . 
 and if exchange from London to Paris be 32(/. per crown, what must 
 be the rate of exchange from Amsterdam to Paris ? 
 
 Ex. 3. If exchange from Paris to London be did. per cro\vn, and if 
 exchange from Paris to Amsterdam be 5id. Flemish per crown, what 
 must be the .rate of exchange between London and Amsterdam, in 
 order to be on a par with the other two ? 
 
 Ex. 4. Amsterdam exchanges on London, at 35 schil. 5 gro. per . 
 sterling ; and the exchange between London and Lisbon is 6O 
 pence per milrea, what is the exchange between Amsterdam and 
 Lisbon ? 
 
 The course of exchange being given, and the par of arbitration found, 
 we obtain a method of drawing and remitting to advantage. 
 
 4 v 
 
 Ex. 5. If exchange from London to Paris be 32 pence 
 sterling per crown , 'and to Amsterdam 405 Flemish per ., 
 and if I learn that the course of exchange between Paris 
 and Amsterdam is fallen to 52 pence Flemish per crowu ; 
 
COMPOUND ARBITRATION. 
 
 what may be gained per cent., by drawing on Paris and 
 remitting to Amsterdam ? 
 
 By Ex. 2, the par of arbitration between Paris and Amsterdam is 54c/ 
 Flemish per crown : then 
 
 d; cr. . cr. 
 
 32 : 1 :: loa : 750 drawn, at Pa ris^ 
 
 cr. d. Fl. cr. d. Fl. 
 
 1 : 52 : : 750 : 3QOOO credit at Amsterdam. 
 d. Fl. . d.Fl. . s. d. 
 
 405 : 1 :: 39000 : 96 5 1 1 to be remitted ; 
 therefore 100/. Q6L 5s. lid. 3. 145. id. gain per cent. 
 
 If the course of LV.hange between Paris and Amsterdam be at 50 
 Flemish per crown, instead of 52 ; and if I would gain by the nego- 
 tiation, I must draw on Amstefdam and remit to Paris : thus 
 
 . d.Fl. . d.Fl. 
 
 1 : 405 :: loo : 40500 drawn at Amsterdam. 
 d.Fl. cr. d.Fl. cr. 
 
 56 : 1 :: 40500 : 723 credit at Paris. 
 
 cr. d. cr. ' s. 
 
 1 : 32 : : 72$ : $6 8 
 therefore loo/. 96^. ss. rr 3. 125. gain per cent. 
 
 In these cases, credit at one foreign place pays the debt at the other. 
 
 We might carry the subject of Exchanges to almost any length ; 
 but we have sard enough to render the theory and practice easy ; and 
 from what the pupil has seen he will be able to apply the foregoing prin- 
 ciples and rules to the practice of any merchant's counting-house in 
 which he may be situated. We shall, however, give an example in 
 Compound Arbitration. 
 
 COMPOUND ARBITRATION. 
 
 IN Compound Arbitration, the rate of exchange between 
 three or more places is given, to find how much a remit- 
 tance passing through them all will amount to at the last 
 place : or to find the arbitrated price 3 or par of arbitration, 
 between the first and last place. 
 
 Examples of this kind may be worked by several suc- 
 cessive statings in the Rule of Three, or according to the 
 following Rules* 
 
 (1) Distinguish the given rates, or prices, into ante- 
 cedents and consequents, placing the antecedents in one 
 
 M 
 
242 COMPOUND ARBITRATION. 
 
 column, and the consequents in another, ivith the tign of 
 equality between them. 
 
 (2) The first antecedent, and the last consequent to 
 which an antecedent is required, must be of the same kind. 
 
 (3) The second antecedent must be of the same kind with 
 the first consequent, and the third antecedent of the same 
 kind with the second consequent, fyc. 
 
 (4) Multiply the antecedents together for a divisor, and 
 the consequents together for a dividend, and 'the quotient 
 will be the answer required 
 
 Ex. If a merchant in London remit 5007. sterling 1 to 
 Spain by way of Holland, at 35 shillings Flemish per pound 
 sterling, thence to France at 58 pence per crown, thence 
 to Venice at 10 crowns for 6 ducats, and thence to Spain 
 at 360 mervadies per ducat ; how many piastres of 272 
 mervadies will" the 500/. amount to in Spain ? 
 
 lZ. rz 35*. or 420d. Fl. 
 
 5$d. m l crown 
 
 10 cr. zz 6 ducats 
 
 1 due. zr 360 mervadies 
 
 272 mer. zr l piastre 
 
 How many piastres ~ 500/.? 
 
 ~ . . , 420 X 6 X 360 X 500 
 
 Omitting the units, we have by the rule, . > 
 
 58 X 10 X 272 
 and this fraction, reduced to its lowest terms, gives 
 
 21 X 3 X 45 X 500 1417500 
 
 - = 2875} piastres, which is the 
 
 29 X 17 493 
 
 answer. 
 
 By the Rule of three we should have said, 
 
 U. 
 
 . 420eJ. : 
 
 : 5001. 
 
 210000(2. 
 
 5Qd. 
 
 : 1 cr. 
 
 : 210000d. : 
 
 3C20 cr.* 
 
 10 cr. 
 
 : 6 due. : 
 
 : 3620 cr. : 
 
 2172 due. 
 
 l due. 
 
 : 36o mer. : 
 
 : 2172 due. : 
 
 781920 rner. 
 
 272 mer. 
 
 1 pias. : 
 
 : 781Q20 mer. : 
 
 2875J piastres. 
 
 If the course of direct exchange to Spain were 42 pence sterling, 
 then the 500/. remitted would only amount to 2823J piastres, of 
 course 2875J - 2823^, gives 52, which is the number of piastres 
 gained by the negotiation. 
 
 * The fractions are omitted, and on that account the answer by this 
 method will not be quite accurate. 
 
243 
 DUODECIMALS. 
 
 DUODECIMALS, or Cross Multiplication, is made use of 
 by artificers in measuring- their several works, and is per- 
 formed by means of the following table : 
 
 12'"' fourths - make 1 third. 
 
 12'" thirds l second. 
 
 12" seconds - - 1 inch. 
 
 12' inches 1 foot. 
 
 Glaziers, Masons, and others, measure by thejsquare foot. Paint- 
 ers, Paviors, Plasterers, &c., by the square yard. Slating, tiling, floor- 
 ing, &c., by the square of 100 feet. Brickwork is measured by the 
 rod of 16 feet, the square of which is 27 2|> See p. 42. 
 
 RULE. (1) Arrange the terms of the multiplier under 
 the. same denominations of the multiplicand. (2) Multiply 
 each term in the multiplicand, beginning at the lowest 9 * 
 by the feet in the multiplier, aud write the result of each 
 under its respective term, observing to carry one for every 
 twelve. (3) Multiply , in the, same manner, by the inches* 
 and set the result of each term one place removed to the 
 right-hand of those in the multiplicand.^ (4) Multiply 
 then by the seconds, setting the result of each term tivo 
 places removed to the right hand of those in the multiplicand. 
 
 Multiply 9 ft. 4 in. 8 sec. by 5 ft. 8 in. C sec. 
 ft. in. sec. 
 
 948 I multiply by 5, saying 5 times 8 are 
 
 586 40, 4 and carry 3 ; 5 times 4 are 20 and 
 
 -, 3 are 23, 1 1 and carry l ; 5 times nine 
 
 46 11 4 are 45 and l are 46. For the second 
 
 631 4 /tf line I say, 8 times 8 are 64, 4 andcairy 
 
 484 Q 1 "' 5, but the 4 over are thirds ; and so of 
 
 , , the rest. 
 
 53 7 . 
 
 NOTES. 
 
 * Hence the origin of the term cross multiplication, the operation 
 being crossways, compared with multiplication in the common way. 
 It is called Duodecimals, because the feet, inches, c. are divided into 
 twelve parts, whereas in decimals the unit is divided into tenths. 
 ^ Feet multiplied into feet give feet. 
 Feet multiplied into inches give inches. 
 Feet multiplied into seconds give seconds. 
 Inches multiplied into inches give seconds. 
 Inches multiplied into seconds give thirds. 
 Seconds multiplied into seconds give fourths. 
 M2 
 
244 DUODECIMALS. 
 
 Ex. 1. How much must I pay for a slab of marble 7 ft. 4 in. longy 
 and 2 ft. i in. 6 sec. broad, at the rate of 7 5. per square foot ? 
 
 Ex. 2. What will be the expense of glass for a window that mea- 
 sures, in the clear, 10 ft. 6in. in height, and 4 ft. 9 in. in width, at 
 is. gd. per foot? 
 
 Ex. 3. How much will a room cost in painting, at g^d. per yard ; 
 the sides are 18 ft. 10 in. by 10 ft. 3 in., and the two ends are 16 ft. 
 6 in. by 10 ft. 3 in. ? 
 
 Ex. 4. What shall 1 have to pay for statuary marble about my fire- 
 place, at 14s. per foot ; the hearth measures 6 ft. 4 in. by a fu 3 in., 
 the three fronts are each 4 ft. 2 in. by 8 in., and the mantle-piece slab 
 is 6ft. by 9 in.? 
 
 Ex. 5. What will the paving of a court-yard come to, at is. id. per 
 foot, the yard being 74 feet long, and 56 ft. 8 in. wide ? 
 
 Ex. 6. How much shall I have to pay for slating a house, consisting 
 of two sloping sides, each measuring 24 ft. 5 in. by l 5 ft. 9 in. at the . 
 rate of 415. per square of l oo feet ? 
 
 Ex. 7. What will the tiling of 10 houses come to, the roof of each 
 house consisting of two sides, each 18 feet by 14, and the price of 
 tiling at 28s. per square ? 
 
 Ex. 8. How many square rods are there in a brick wall 44 ft. 6 in. 
 long, and 7 ft. 4 in. high, and 2- bricks thick?* 
 
 Ex. 9. If an oblong garden bs 254 ft. 6 in. long, and 184 ft. -8 in. 
 wide, what will a wall cost 10 ft. 6 in. high, and 2^ bricks thick, at 
 15/. 155. per square rod ? 
 
 i Ex. 10. How much shall I have to pay for the plate-glass of four 
 windows ; each window consists of 16 panes, and each pane measures 
 20^ inches by I5f inches at Qs. Gd. per foot? 
 
 Ex. 11. How Kiany solid feet of fir are there in a piece of timber 
 35 ft. 4 in. long, and 13-J inches by 14^ inches ?f 
 
 Ex. 12. How many solid feet of oak are there in a piece 14 feet 
 3j inches long, and 2 feet 10^ by 2 feet 2 inches ? 
 
 Ex. 13. How many solid feet of fir are there in 46 joists, each 14 
 feet 3 J inches long, 7 J inches deep, by 3^ inches broad ? 
 
 * Bricklayers value their work at the rate of a brick and a half, 
 or three half bricks thick ; and if the wall be more or less than this, it 
 must be reduced to that thickness by the following rule : *.' Multiply 
 the measure found by the number of half bricks, and divide by three :" 
 thus, if the wall be 2- bricks thick, I multiply by 5, and divide the pro- 
 duct by 3. 
 
 Ex. If the wall be 50 feet long, and 9 high, and 2 bricks thick, it 
 
 will be 50 X 9 X zz 600 feet; and zz 2 isqu. rods nearly. 
 
 3 4 27-24- 
 
 'f- Carpenters' rules aie divided into eighths 9 so that in these cases 
 the eighths must be reduced to twelfths, or the whole must be worked 
 by decimals. In this and the following questions, the length, breadth, 
 and thickness, must be multiplied into one another. 
 
245 
 GEOGRAPHICAL CALCULATIONS. 
 
 3 feet - - - make i yard, 
 1760 yards - i mile, 
 69^: miles, or 60 geographical miles 1 degree.* 
 
 The degree is usually reckoned in round numbers, at 69^ miles ; 
 but if accuracy be attended to, the number in the table is too large : 
 the real length of a degree is 365184 English feet, or 69 miles 288 
 yards: this has been ascertained by actual measurement, so that the 
 circumference of the earth is equal to 69 miles, 288 yards X 360 (be- 
 cause in every circle there are 360 degrees) z: 25000 miles nearly. 
 
 Geographers reckon on the globe two kinds of degrees, viz. degrees 
 of latitude, and degrees of longitude.* The degrees of latitude, which, 
 are measured, from north to south, on the meridian, are all of one 
 -length, as above. But the degrees of longitude, or the circles which 
 pass round the earth in each parallel of latitude, continually diminish 
 in proceeding from the equator towards the Poles, but at the equator 
 they are of the same length as those of latitude. The following is a 
 Table of the length of the degrees of longitude, carried to three places 
 of decimals, in every 5 degrees of latitude. 
 
 TABLE. 
 
 Lat. 
 
 Eng. miles, j 
 
 Lat.lEng. miles. 
 
 Lat. 
 
 Eng. miles. 
 
 Lar. 
 
 Eng. nu 
 
 
 
 6Q.200 
 
 23 
 
 62.716 * 
 
 50 
 
 44.481 
 
 75 
 
 17.910 
 
 5 
 
 68.93(5 
 
 30 
 
 59-929 
 
 55 
 
 39.691 
 
 80 
 
 12.016 
 
 10 
 
 63.149 
 
 35 
 
 56.685 
 
 60 
 
 34.600 
 
 85 
 
 6.031 
 
 15 
 
 66.842 
 
 40 
 
 53.010 
 
 65 
 
 29.245 
 
 90 
 
 o.oop 
 
 20 
 
 65.026 
 
 45 
 
 48.931 
 
 70 
 
 23.667 
 
 
 
 Here it is evident, that at latitude 40, the degree is little more than 
 53 miles in length ; at 70 it is only 23 miles ; and^t the pole, or 90, 
 it comes to nothing, it being supposed to be a point. 
 
 I. To find the distance, in miles, "between airy two places, 
 having- the same degree of latitude. 
 
 RULE. Having found the distance between the places^ 
 in degrees, multiply the number so found by the number in 
 the table opposite the given degree of latitifde. 
 
 Ex. How many miles distant is Madrid, in Spain, from 
 Bursa, in Natolia; the latitude of both is 40 N., but the 
 long-, of Madrid is about 3 W. ; and that of Bursa 29 E. ? 
 
 * Longitude expresses the distance of meridians, or circles, which 
 are supposed to pass over the head from north to south ; and latitude 
 expresses the distance of a place north and south from the equator. 
 
S46 GEOGRAPHY. 
 
 The difference in longitude is 3 -f 29 rz 32*, this multiplied by 
 53.01, the number of miles in a degree at the given latitudes, gives 
 1696 for the miles between Madrid and Bursa. 
 
 II. To find the distance between any two places, having 
 
 the same degree of longitude. 
 
 RULE. Multiply the number of degrees between the 
 places by 69.2, and the answer is in miles. 
 
 Ex. How far is London from Mount Atlas in Africa, the 
 former is 51 ^ N. L., the latter 31 1 N. L. ? 
 
 Here the distance is -20, and 20 X 69.2 ~ 1384 miles. 
 
 TIME is measured by the revolution of the earth about 
 its axis : every revolution is completed in 24 hours; and as 
 
 there are 360 in the great circle cf the earth, so =z 
 
 15 Q = 1 hour of time. Hence this TABLE : 
 
 15 of motion answers to 60' in time, or 1 hour. 
 l 4' 
 
 I. To convert time into motion. 
 
 RULE. Multiply the hours by 15, and divide the minutes 
 hy 4, and the answer is in degrees, fyc. 
 
 Thus 4 h. 20 min. in time, answer to 65 in motion. 
 
 II. To convert motion into time. 
 
 RULE. Divide the given number of degrees of motion 
 by 15, and the answer i? in time. 
 
 Thus 65 of motion answer "to -i b. 20 min. in time : 15)65(4 
 
 60 
 
 5 
 
 60 
 
 15)300(20 
 300 
 
 Ex. 1. What o'clock is it at Athens, which is 23 57' 
 east longitude of London, now it is 12 at the metropolis ? 
 
 Athens being east of London, the clocks there will be before the 
 clocks here. 15)23 bl'(\ hour. 
 
 15 Answer. When it is 12 o'clock at 
 
 g London, it will be 36 min. past l at 
 60 Athens. 
 
 15)537(36 min. nearly. 
 
 Ex. 2. What o'clock is it at Philadelphia in America, 
 now it is 12 at London ? 
 
 Philadelphia is 75 s'west longitude of London, of course the clocks 
 there are behind those here. 
 
GEOGRAPHY. 247 
 
 15)75 s'(5 ho. omiri. 32 sec. 
 75 
 
 T~ In this case the answer is. 
 
 co 12 h. 5h. om. 32 s. zz 6h. 59m. 28s. 
 
 or very nearly 7 in/ the morning. 
 15)480(32 
 
 In many maps the longitude is reckoned from Ferro, one 
 ef the Canary Islands, which is 17 45' west of London. 
 
 III. To reduce the longitude of Ferro to that of London. 
 
 RULE 1. If the place be EAST of London, subtract from 
 it 17 30', and the remainder is tht longitude taut uf London. 
 
 Thus, from Ferro, Constantinople is 46 44' ; to reduce this to the 
 longitude reckoned from the meridian of London, we say, 
 46* 44' 17 45' IZ 28 59'. 
 
 2. If the place be WEST from Ferro, add to the given 
 longitude 17 45'. 
 
 Thus, Boston is 52 48 ; west of Ferro, but it is west of London 
 
 52 4&' + 17 45' = 70 33'. 
 
 3. If the place lies between Ferro and London, its lon- 
 gitude will be obtained by subtracting its longitude east 
 of Ferro from 17 45'. 
 
 Thus, Lisbon is 8 4o' east of Ferro, and it is west of London 17 45' 
 8 Q 40' =. 9 5'. 
 
 By a reverse method may be reduced the longitude from London to 
 that of Ferro. 
 
 The earth being globular, it is a useful problem to as* 
 certain the extent of the visible horizon : or 
 
 IV. To find the distance to which a person can see at any 
 given height of the eye. 
 
 RULE. Multiply the square-root of the height of the eye, 
 infect, by 1.2247, and the product i the distance in miles 
 to which we can see from that height. See p. 143. 
 
 Ex. 1. How far can a sailor see, standing at the top- 
 mast of a ship, 144 feet high ? 
 
 The square-root of 144 is 12 ; therefore 1.2247 X 12 ~ 14.7 miles. 
 Thus, in this situation, a sailor might, on a very clear day, descry land 
 at the distance of 5 leagues, nearly ; and he might see the top-mast of 
 another ship at a still greater distance. 
 
 Ex. 2. To what distance could a perjson see from the 
 top of St. Paul's, which is 340 feet high ? 
 
 - 340 x 1.2247 =: 18.44 X 1.2247 ~ 22.58 miles, or something 
 more than 22-| miles. 
 
'248 
 A TABLE 
 
 OF THE 
 
 LOGARITHMS OF NUMBERS, 
 
 FROM i TO looo. 
 (See also page 155.y 
 
 No. 
 
 Logarithms. 
 
 No. 
 
 Logarithms. 
 
 No. 
 
 Logarithms. 
 
 1 
 
 2 
 
 
 
 O.ooooooo 
 
 0.3010300 
 04771 21 3 
 
 34 
 35 
 
 1.53147RO 
 1.5440680 
 
 67 
 68 
 f\n 
 
 1.8260748 
 1.8325089 
 
 4 
 5 
 
 0.6020600 
 60807 00 
 
 36 
 
 #7 
 
 1.5563025 
 1 56820 1 7 
 
 70 
 
 1.8450980 
 
 
 
 38 
 
 1 57 97 863 
 
 7 1 
 
 1 8 5 1 ^ 5 S3 
 
 6 
 
 7 
 
 0.7781513 
 0.84509SO 
 
 39 
 40 
 
 1.591 0646 
 1.6020600 
 
 72 
 
 73 
 
 7/1 
 
 l.S57<3u25 
 1.8683229 
 
 g 
 10 
 
 0.9542425 
 1 OOOOOOC 
 
 41 
 42 
 
 1.6127SQ 
 1 623 4 Q3 
 
 75 
 
 1.8750613 
 
 
 
 43 
 
 1 6334685 
 
 7fi 
 
 1 8 80S 136 
 
 11 
 
 12 
 
 1.0413027 
 1.0791812 
 
 1 I 1 1 fl 4 '1 A 
 
 44 
 45 
 
 1.6434527 
 1.6532125 
 
 77 
 
 78 
 
 1.8864907 
 1.8920946 
 
 14 
 
 1 5 
 
 1.1461280 
 117 6001 3 
 
 46 
 
 4 7 
 
 1.6627578 
 Ifi 7 ofto7 o 
 
 80 
 
 1.9030900 
 
 
 
 4 8 
 
 1 681241^ 
 
 8 1 
 
 
 16 
 17 
 
 1.2041200 
 1.2304489 " 
 
 49 
 50 
 
 J. 6 00 1961 
 1.6989700 
 
 82 
 83 
 
 1.9138139 
 1.Q1Q07 81 
 
 19 
 
 Of) 
 
 1.2787536 
 1 3010300 
 
 51 
 
 5^ 
 
 1.7075702 
 171 60C33 
 
 85 - 
 
 1.9294189 
 
 
 
 53 
 
 1 7 9 407^0 
 
 86 
 
 1 9344Q85 
 
 21 
 22 
 o ^ 
 
 1.32221Q3 
 1.3424227 
 
 Iqfi i 7 o " ft 
 
 54 
 55 
 
 1.7323938 
 1.7403627 
 
 87 
 88 
 
 C A 
 
 1.9395193 
 1.Q444827 
 1 9493900 
 
 24 
 
 1.3802112 
 
 56 
 
 1.7481880 
 
 90 
 
 1.9542425 
 
 
 
 5 8 
 
 1 7 634280 
 
 Ql 
 
 1 95904 1 4 
 
 26 
 
 27 
 
 1.4349733 
 1.4315688 
 
 59 
 60 
 
 1.7708520 
 1.7781513 
 
 92 
 93 
 
 Q4- 
 
 1.9637878 
 1.9684829 
 i Q 7 A 1 > 7 O 
 
 29' 
 
 1.4623980 
 
 61 
 fio 
 
 1.7853298 
 
 95 
 
 1.9777236 
 
 
 
 63 
 
 1 7QQ3405 
 
 06 
 
 1 98^^712 
 
 31 
 32 
 33 
 
 1.4913617 
 1.5051500 
 
 It I O t. I Of> 
 
 64 
 65 
 
 1.8061800 
 1.8129134 
 
 97 
 98 
 99 
 
 1.986/717 
 1.9912261 . 
 1 9Q56352 
 
 
 * 
 
 66 ; 
 
 Ic8105439 
 
 100 
 
 2.00OOOOO , 
 
LOGARITHMS. 
 
 249 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 101 
 
 2.0043214 
 
 
 146 
 
 2.1643529 
 
 
 102 
 
 2.0086002 
 
 4278 
 
 147 
 
 2.1673173 
 
 2964- 
 
 103 
 
 2.0128372 
 
 4237 
 
 148 
 
 2.1702617 
 
 2944 
 
 104 
 
 2.0170333 
 
 4196 
 
 149 
 
 2.1731863 
 
 2954 
 
 10.5 
 
 2.0211893 
 
 4156 
 
 150 
 
 2.1760913 
 
 2905 
 2885 
 
 106 
 
 2.0253059 
 
 4116 
 
 151 
 
 2.1789769 
 
 2866 
 
 107 
 
 2.0293838 
 
 4077 
 
 152 
 
 2.181 8436 
 
 2847 
 
 108 
 
 2.0334238 
 
 4039 
 
 163 
 
 2.1846914 
 
 
 109 
 
 2.0374265 
 
 4002 
 
 154 
 
 , 2.1875207 
 
 2820 
 
 110 
 
 2.0413927 
 
 3966 
 
 155 
 
 2.1903317 
 
 2811 
 2792 
 
 111 
 112 
 113 
 114 
 115 
 
 2.0453230 
 2.0492180 
 2.0530784 
 2.0569049 
 2.0606978 
 
 393O 
 3895 
 880 
 3826 
 3793 
 
 157 
 
 158 
 159 
 160 
 
 2.1931240 
 2.1958997 
 2.1986571 
 2.2013Q71 
 2.2041200 
 
 2775 
 
 2757 
 
 2740 
 2722 
 2705 
 
 116 
 
 117 
 
 2.0644580 
 2.0681859 
 
 3727 
 
 161 
 162 
 
 2.2068259 
 2.2095150 
 
 2689 
 . 2672 
 
 us 
 
 119 
 120 
 
 2.0718820 
 2.0755470 
 2.0791812 
 
 3665 
 3634 
 36o4 
 
 163 
 164 
 165 
 
 2.2121876 
 2.2 14 843 8 
 2.2174839 
 
 2656 
 2640 
 2624 
 
 121 
 
 2.0827854 
 
 
 166 
 
 2.2201081 
 
 2601 
 
 122 
 
 2.0363598 
 
 3574 
 
 O E ,1 e 
 
 167 
 
 2.22-27165 
 
 2592 
 
 123 
 
 2.0899051 
 
 O O-iD 
 O e -i c 
 
 168 
 
 2.2253093 
 
 2577 
 
 124 
 
 2.0934217 
 
 O O \ O 
 
 0400 
 
 169 
 
 2.227SS67 
 
 2562 
 
 125 
 
 2.0969100 
 
 oto O 
 
 3460 
 
 170 
 
 2.2304489 
 
 2548 
 
 126 
 
 2. 1003705 
 
 3433 
 
 171 
 
 2.2329961 
 
 2532 
 
 127 
 
 2.1038037 
 
 
 172 
 
 2.2355284 
 
 2517 
 
 128 
 
 2.10721* o 
 
 S4Q6 
 
 173 
 
 2.2380461 
 
 250-3 
 
 129 
 
 2.1105897 
 
 3379 
 
 174 
 
 2.2405492 
 
 24SS 
 
 130 
 
 2-1139434 
 
 3328 
 
 175 
 
 2.2430380 
 
 2474 
 
 131 
 
 2.1172713 
 
 
 176 
 
 2.2455127 
 
 2460 
 
 132 
 
 2.1205739 
 
 3302 
 
 1/7 
 
 2.2479733 
 
 2446 
 
 133 
 
 2.1238516 
 
 3277 
 
 178 
 
 2.2504200 
 
 243^ 
 
 134 
 
 2.1271048 
 
 3253 
 
 179 
 
 2.2528530 
 
 2419 
 
 135 
 
 2.1303338 
 
 3229 
 3205 
 
 180 
 
 2.2552725 
 
 2406 
 
 136 
 
 2.1335389 
 
 3181 
 
 181 
 
 2.2576786 
 
 2392 
 
 137 
 
 2.1367206 
 
 3 1 58 
 
 182 
 
 3.2600714 
 
 2379 
 
 138 
 
 2.1398791 
 
 3135 
 
 183 
 
 2.2624511 
 
 2365 
 
 139 
 
 2.1430J48 
 
 3113 
 
 184 
 
 2. 2648178 
 
 2353 
 
 140 
 
 2.1461280 
 
 3091 
 
 185 
 
 2.2671717 
 
 2341 
 
 141 
 
 2.1492191 
 
 
 186 
 
 2.2695129 
 
 2328 
 
 142 
 
 2.1522883 
 
 3069 
 
 187 
 
 2.2718416 
 
 2316 
 
 143 
 
 2.1553360 
 
 3047 
 3026 
 
 18S 
 
 2.2741578 
 
 2304 
 
 144 
 
 2.1583625 
 
 3005 
 
 189 
 
 2.2/64618 
 
 2201 
 
 145 
 
 2.16136-80 
 
 2984 
 
 190 
 
 2.2787536 
 
 227-0 
 
 j 
 
 
 f 
 
 
 
 
250 
 
 LOGARITHMS, 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 ! No. 
 
 Logarithms. 
 
 DifF. 
 
 191 
 
 2.2810334 
 
 
 i 236 . 
 
 2.3729120 
 
 
 192 
 
 2.2833012 
 
 2267 
 
 , 237 
 
 2.3747483 
 
 1836 
 
 193 
 
 2.2855573 
 
 2256 
 
 i 238 
 
 2,3765770 
 
 1828 
 
 194 
 
 2.2878017 
 
 2244 
 
 1 239 
 
 2.3783979 
 
 1820 
 
 195 
 
 2.2900346 
 
 2232 
 
 j 240 
 
 2.3802112 
 
 1813 
 
 196 
 
 2.2922561 
 
 
 ! 241 
 
 2.3820170 
 
 
 197 
 
 2.29446fi2 
 
 2210 
 
 242 
 
 2.3838154 
 
 1798 
 
 198 
 
 2.2966652 
 
 2199 
 
 i 243 
 
 2.3856063 
 
 1790 
 
 199 
 
 2.2988531 
 
 2187 
 
 | 244 
 
 2.3873898 
 
 1783 
 
 200 
 
 2.3010300 
 
 2176 
 
 ^166 
 
 i 245 
 
 2.3891661 
 
 1776 
 
 1 7 fiO 
 
 201 
 
 2.3031961 
 
 
 246 
 
 2.3909351 
 
 
 202 
 
 2.3053514 
 
 2155 
 
 247 
 
 2.3926Q69 
 
 1761 
 
 203 
 
 2.3074960 
 
 2144 
 
 248 
 
 2.3944517 
 
 1754 
 
 204 
 
 2.3096302 
 
 2134 
 
 249 
 
 2.3961993 
 
 1747 
 
 205 
 
 2.31 17539 
 
 2123 
 2113 
 
 250 
 
 2.3979400 
 
 174O 
 1 733 
 
 206 
 207 
 208 
 
 2.3138672 
 2.3159703 
 2.3180633 
 
 2103 
 2093 
 
 251 
 252 
 253 
 
 2.3996737 
 2.4014005 
 2.4031205 
 
 1726 
 1720 
 
 209 
 
 2.3-201463 
 
 2083 
 
 254 
 
 2.4048337 
 
 1713 
 
 210 
 
 2.3222193 
 
 2073 
 063 
 
 255 
 
 2.4065402 
 
 1706 
 1 690 
 
 211 
 212 
 213 
 
 2.3242825 
 2.3263359 
 2.3283796 
 
 2053 
 2043 
 2034 
 
 256 
 257 
 258 
 
 2.4082400 
 2.4099331 
 2.4116197 
 
 1693 
 1686 
 1680 
 
 214 
 
 215 
 
 2.3304138 
 2.3324385 
 
 2024" 
 2015 
 
 259 
 
 260 
 
 2.4132998 
 2.4149733 
 
 1673 
 166? 
 
 216 
 
 2.3344538 
 
 2005 
 
 261 
 
 2.4166405 
 
 166O 
 
 217 
 
 2.3364597 
 
 1996 
 
 262 
 
 2.4183013 
 
 1654 
 
 218 
 
 2.3384565 
 
 1987 
 
 263 
 
 2.4199557 
 
 164S 
 
 219 
 
 2.3404441 
 
 1978 
 
 264 
 
 2.4216039 
 
 1642 
 
 220 
 
 2.3424227 
 
 1959 
 
 265 
 
 2.4232459 
 
 1635 
 
 221 
 
 2.3443923 
 
 I960 
 
 266 
 
 2.4248816 
 
 1629 
 
 222 
 
 2.3463530 
 
 1951 
 
 267 
 
 2.4265113 
 
 1623 
 
 323 
 
 2.3483049 
 
 1943 
 
 268 
 
 2.4281348 
 
 1617 
 
 224 
 
 2.3502480 
 
 1934 
 
 269 
 
 2.4297523 
 
 1611 
 
 225 
 
 2.3521825 
 
 1925 
 
 270 
 
 2.4313638 
 
 1605 
 
 226 
 
 2.3541084 
 
 1 Q1 7 
 
 271 
 
 2.4329693 
 
 1599 
 
 227 
 
 2.3560259 
 
 
 272 
 
 2.4345689 
 
 1593 
 
 228 
 
 2.35/9348 
 
 1 900 
 
 273 
 
 2.4361626 
 
 1588 
 
 229 
 
 2.3598355 
 
 189 
 
 274 
 
 2.4377506 
 
 1582 
 
 230 
 
 2.3617278 
 
 1884 
 
 275 
 
 2.4393327 
 
 1576 
 
 231 
 
 232 
 
 2.3636120 
 2.3654880 
 
 1876 
 1867 
 
 276 
 
 277 
 
 2.4409091 
 2.4424798 
 
 J570 
 1565 
 
 233 
 
 2.3673559 
 
 1 860 
 
 278 
 
 2.4440448 
 
 1559 
 
 234 
 
 2.3692159 
 
 1 852 
 
 279 
 
 2.4456042 
 
 1553 
 
 235 
 
 2.3710679 
 
 1844 
 
 280 
 
 2.4471580 
 
 1543 
 
LOGARITHMS. 
 
 No. 
 
 Logarithms- 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 281 
 282 
 283 
 284 
 285 
 
 2.4487O63 
 2 4502491 
 2.4517864 
 2,4533183 
 2.4548449 
 
 1542 
 1537 
 1531 
 1526 
 1521 
 
 I 326 
 327 
 328 
 329 
 330 
 
 2.5132176 
 2.5145478 
 2.5158738 
 2.5171959 
 2.5185139 
 
 1330 
 1326 
 1322 
 1318 
 1314 
 
 286 
 287 
 288 
 289 
 290 
 
 '2. 4 568660 
 2.4578819 
 2.4593Q25 
 2.4608978 
 2.462398O 
 
 1515 
 1510 
 1505 
 1500 
 1495 
 
 ! 331 
 
 332 
 333 
 334 
 335 
 
 2. 51932.80 
 2.5211381, 
 2.5224442 
 2.5237465 
 2.5250448 
 
 1310 
 1306 
 1302 
 1298 
 1294 
 
 291 
 292 
 293 
 294 
 295 
 
 2.4638930 
 2.4653829 
 2.4668676 
 2.4683473 
 2.4698220 
 
 1489 
 1484 
 1479 
 1474 
 1469 
 
 336 
 337 
 338 
 339 
 340 
 
 2.5263393 
 2.5276299 
 2.528916/ 
 2.5301997 
 2.5314789 
 
 129O 
 1286 
 1283 
 1279 
 
 1 O 7 \ 
 
 296 
 297 
 298 
 299 
 300 
 
 2.4712917 
 2.47-27564 
 2.4742163 
 2.4756712 
 2.4771213 
 
 1464 
 1459 
 1454 
 1450 
 1445 
 
 341 
 342 
 343 
 344 
 345 
 
 2.5327544 
 2.5340261 
 2.53ML941 
 2.5365584 
 2.5378191 
 
 1271 
 
 1268 
 1264 
 1260 
 1 9 "> 7 
 
 301 
 302 
 303 
 304 
 305 
 
 2.4785665 
 2.4800069 
 2.4814426 
 2.4828736 
 2.4842998 
 
 1440 
 1435 
 1431 
 1426 
 1421 
 
 346 
 347 
 348 
 349 
 350 
 
 2.5390761 
 2.54O3295 
 2.5415792 
 2.5428254 
 2.5440680 
 
 1253 
 1249 
 1246 
 1242 
 1239 
 
 306 
 307 
 308 
 309 
 310 
 
 2.4857214 
 2.4871384 
 2.4885507 
 '2.4899585 
 2.4913617 
 
 1417 
 1412 
 
 1407 
 
 1403 
 1398 
 
 351 
 352 
 353 
 354 
 355 
 
 2.5453071 
 2.5465427 
 2.5477747 
 2.5490033 
 2.5502283 
 
 1235 
 1232 
 1228 
 1225 
 12! 
 
 311 
 312 
 313 
 314 
 315 
 
 2.4927604 
 2.4941546 
 2.4955443 
 2.4969296 
 2.4983106 
 
 1394. 
 1389 
 1385 
 1381 
 1376 
 
 356 
 357 
 358 
 359 
 360 
 
 2.5514500 
 2.5526682 
 2.5538830 
 2.5550944 
 2.5563025 
 
 121S 
 1214 
 1211 
 1208 
 1204 
 
 316 
 317 
 318 
 319 
 320 
 
 2 4996871 
 2.5010593 
 2.502427J 
 2.5037907 
 2, 505 J 500 
 
 1372 
 1367 
 1363 
 1359 
 1355 
 
 361 
 362 
 363 
 364 
 365 
 
 2.5575072 
 2.558/086 
 2.5599066 
 2.5611014 
 2.5622929 
 
 1201 
 1198 
 1194 
 1191 
 1188 
 
 321 
 322 
 323 
 324 
 325 
 
 2.5065050 
 2.5078559 
 2.5092025 
 2.5105450 
 2.5118834 
 
 1350 
 1346 
 1342 
 1338 
 1331 
 
 366 
 367 
 368 
 369 
 370 
 
 2.5634811 
 2.5.646061 
 2.5658478 
 2.5670264 
 2.5682017 
 1 
 
 1185 
 1181 
 1178 
 1175 
 117* 
 
LOGARITHMS. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 371 
 372 
 373 
 374 
 375 
 
 2,5693739 
 2.5705429 
 2.5717088 
 2.5728716 
 2.5740313 
 
 1169 
 1165 
 116'J 
 1159 
 
 416 
 417 
 418 
 419 
 420 
 
 2.6190933 
 ' 2.6201361 
 2.6211763 
 2.6222140 
 2.6232493 
 
 1042 
 1040 
 1037 
 1035 
 
 376 
 
 377 
 378 
 379 
 380 
 
 2.5751878 
 2.5763414 
 2.5774918 
 2,5786392 
 2.5797836 
 
 1 156 
 
 1153 
 1150 
 1147 
 1114 
 
 421 
 422 
 423 
 424 
 425 
 
 2.6242821 
 2.6253124 
 2.6263404 
 2.6273659 
 
 2.6283889 
 
 1032 
 1030 
 1028 
 1025 
 102-3 
 
 381 
 382 
 383 
 384 
 385 
 
 2.5809250 
 2.5820634 
 2.5831988 
 2.5813312 
 2.5854607 
 
 1138 
 1135 
 1132 
 1129 
 
 426 
 427 
 428 
 420 
 430 
 
 2.6294096 
 2.6304279 
 2.6314438 
 2.6324573 
 2.6334685 ' 
 
 1018 
 1013 
 1013 
 1011 
 
 386 
 387 
 388 
 389 
 390 
 
 2.5865873 
 2.587/110 
 2.5888317 
 2.5899496 
 2.5910646 
 
 1123 
 1120 
 1117 
 1115 
 1 ' 12 
 
 431 
 432 
 433 
 434 
 435 
 
 2.6344773 
 2.6354837 
 2.6364879 
 2.6371897 
 '2.6384893 
 
 1000 
 1004 
 1001 
 999 
 
 QO 7 
 
 391 
 392 
 393 
 334 
 395 
 
 2.5Q21768 
 2.5932861 
 2.5943926 
 2.5954962 
 2.5965971 
 
 1109 
 1106 
 1103 
 MOO 
 1098 
 
 436 
 
 437 
 438 
 439 
 440 
 
 2.6394865 
 2.6404814 * 
 2.6414741 
 2. 64240,} 5 
 2.6434527 
 
 994 
 992 
 990 
 
 988 
 
 Oc; 
 
 396 
 397 
 3Q8 
 3Q9 
 400 
 
 2.59/6952 
 2.59&7905 
 2.5998831 
 2.6009729 
 2.602.0600 
 
 1095 
 1092 
 1089 
 10S7 
 1 084 
 
 441 
 
 442 
 443 
 414 
 445 
 
 2 6444386 
 2.6454223 
 2.6464037 
 2.6473830 
 2.d483600 
 
 982 
 981 
 979 
 977 
 
 974. 
 
 401 
 402 
 403 
 404 
 406 
 
 2.6O31444 
 2.6042261 
 2.6053050 
 2.6063814 
 2.6074550 
 
 1081 
 1078 
 1076 
 
 1*073 
 
 1071 
 
 446 
 447 
 448 
 449 
 450 
 
 2.64Q3349 
 2.6503075 
 2.6512780 
 2.6522463 
 2.6532125 
 
 972 
 970 
 968 
 966 
 964 
 
 406 
 407 
 408 
 409 
 410 
 
 2.6085260 
 2.6095944 
 2. 6106502 
 2.6117233 
 2.6127839 
 
 1068 
 1065 
 1063 
 1060 
 
 1057 
 
 451 
 452 
 453 
 454 
 455 
 
 2.6541765 
 2.6551384 
 2.6560982 
 2.65/0559 
 2.6580114 
 
 961 
 959 
 957 
 955 
 
 953 
 
 411 
 
 412 
 413 
 414 
 415 
 
 2.6138418 . 
 2.614S9/2 
 2.6159501 
 2.6170003 
 2.6180481 
 
 1055 
 1052 
 1050 
 1047 
 
 1045 , 
 
 456 
 
 457 
 458 
 459 
 460 
 
 2.6589648 
 2. 65991^2 
 2.6608655 
 2.6618127 
 2.66275/8 
 
 951 
 94,9 
 
 947 
 945 
 
 :&*. 
 
LOGARITHMS. 
 
 253 
 
 No. 
 
 Logarithms. 
 
 Dift. 
 
 No. 
 
 Logarithms, 
 
 Ditf. 
 
 461 
 46-2 
 463 
 464 
 465 
 
 2.6637009 
 2.6646420 
 2.6655810 
 2 6665180 
 2.6674530 
 
 941 
 939 
 937 
 935 
 932 
 
 506 
 507 
 508 
 509 
 510 
 
 2-7041505 
 2.7050080 
 2.7058637 
 2.7067178 
 2.7075702 
 
 857 
 855 
 854 
 852 
 8 r iO 
 
 466 
 467 
 468 
 469 
 470 
 
 2.6683859 
 2.6693169 
 2.6702459 
 
 2.6711723 
 2.6720979 
 
 931 
 929 
 926 
 925 
 923 
 
 511 
 
 512 
 513 
 514 
 515 
 
 2.7084209 
 2.709270O 
 2.7101174 
 2,7109631 
 2.7H8072 
 
 849 
 847 
 845 
 844 
 842 
 
 471 
 4/2- 
 473 
 474 
 475 
 
 2.6730209 
 2.6739420 
 2.6748611 
 2.6757783 
 2.6766936 
 
 921 
 919 
 917 
 915 
 913 
 
 516 
 
 517 
 518 
 519 
 520 
 
 2.7126497 
 2.7134905 
 2.7143298 
 2.7151674 
 2.7160033 
 
 840 
 839 
 83'/ 
 835 
 834 
 
 476 " 
 477 
 478 
 47Q s 
 
 480 
 
 2.6776069 
 2.67851 84 
 2.6794279 
 2.6803355 
 2.6812412 
 
 911 
 909 
 607 
 905 
 903 
 
 521 
 522 
 523 
 
 524 
 525 ( 
 
 '2.7168377 
 2.7176705 
 2.7185017 
 2.7193313 
 2.7201593 
 
 832 
 831 
 
 829 
 828 
 820 
 
 481 
 482 
 483 
 484 
 
 485 
 
 2.6821451 
 2.6630470 
 2.6839471 
 2.6848454 
 
 2.6857417 
 
 901 
 900 
 898 
 S96 
 894 
 
 526 
 
 527 
 528 
 529 
 530 
 
 2.7209857 
 2.7218106 
 2.7226339 
 2.7234557 
 2.7242759 
 
 824 
 823 
 821 
 820 
 818 
 
 486 
 
 487 
 488 
 489 
 490 
 
 2.686(53p3 
 2.6875290 
 2.6884198 
 2.6893089 
 2.6901961 
 
 892 
 890 
 
 889 
 887 
 885 
 
 531 
 532 
 533 
 534 
 535 
 
 2.7250945 
 '2.7259116 
 2.7267272 
 2.7275413 
 2,7283538 
 
 817 
 815 
 814 
 612 
 811 
 
 491 
 
 492 
 493 
 494 
 495 
 
 2.6910815 
 2.fi919651 
 2.6928469 
 2.6937269 
 2.6946052 
 
 883 
 881 
 880 
 878 
 876 
 
 5S6 
 537 
 538 
 539 
 54O 
 
 2.7291648 
 2.7299743 
 2.7307823 
 2.7315888 
 2.7323938 
 
 809 
 808 
 806 
 805 
 
 496 
 497 
 498 
 499 
 500 
 
 2.6954817 
 2.^963564 
 2.6972293 
 2.6981005 
 2.6989700 
 
 874 
 872 
 871 
 869 
 867 
 
 541 
 542 
 543 
 544 
 
 545 
 
 2.7331973 
 2.733Q993 
 2.7347998 
 2.7355989 
 2.7363965 
 
 802 
 80O 
 799 
 797 
 
 501 
 502 
 503 
 504 
 505 
 
 2 6998377 
 2.7007037 
 2.7015680 
 2.7024305 
 2.7032914 
 
 866 
 864 
 862 
 860 
 
 859 
 
 546 
 547 
 548 
 549 
 550 
 
 2.7371926 
 2-7379873 
 2.738>806 
 2.7393723 
 2.7403627 
 
 794 
 793 
 791 
 79O 
 
 788 
 
2,54 
 
 LOG ARITHMS. 
 
 No. 
 
 Logarithms. 
 
 Dift. 
 
 No. 
 
 Logarithms. 
 
 Difir. 
 
 551 
 
 552 
 553 
 554 
 555 
 
 2.7411516 
 2.7419391 
 2.7427251 
 2.7435098 
 2.7442930 
 
 787 
 786 
 784 
 783 
 781 
 
 596 
 597 
 598 
 599 
 600 
 
 2.7752463 
 2.7759743 
 2.7767012 
 2.7774268 
 2.7/81513 
 
 728 
 727 
 725 . 
 724 
 723 
 
 556 
 557 
 558 
 550 
 560 
 
 2.7450748 
 2.7458552 
 2.7466342 
 2.7474118 
 2.7481880 
 
 780 
 779 
 777 
 776 
 
 601 
 602 
 603 
 604 
 605 
 
 2.7788745 
 2.7795965 
 2.7803173 
 2.7810369 
 2.7817554 
 
 722 
 720 
 719 
 718 
 717 
 
 561 
 562 
 563 
 564 
 565 
 
 2.7489629 
 2.7497363 
 2.75O5084 
 2.7512791 
 2.7520484 
 
 773 
 772 
 770 
 769 
 768 
 
 606 
 607 
 608 
 * 609 
 610 
 
 2 7824726 
 2.7831887 
 2.7839036 
 2.7846173 
 2.7853298 
 
 716 
 
 714 
 713 
 712 
 7J l 
 
 566 
 567 
 $68 
 
 :>69 
 570 # 
 
 2.7528164 
 2.7535831 
 2.7543483 
 2. 7551123 
 2.7558749 
 
 766 
 765 
 763 
 762* 
 761 
 
 611 
 612 
 613 
 f>14 
 615 
 
 2.7860412 
 2-7867514 
 2.7874605 
 2.7881684 
 2.7888751 
 
 710 
 709 
 707 
 706 
 
 705 
 
 571 
 
 572 
 
 573 
 
 574 
 
 575 
 
 2.7566361 
 2.7573960 
 2.7581546 
 2.7589119 
 2.7596678 
 
 760 
 
 758 
 
 757 
 
 755 
 
 h K.A 
 
 616 
 617 
 618 
 619 
 620 
 
 2.7895807 
 2.7902852 
 2-7909885 
 2-7916906 
 2.7923917 
 
 704 
 703 
 702 
 
 701 
 
 fiQQ 
 
 576 
 
 577 
 
 578 
 579 
 580 
 
 2.7604225 
 2.7011758 
 2.7619278 
 2.7626786 
 2.7634280 
 
 753 
 752 
 750 
 749 
 
 y 4fi 
 
 621 
 622 
 623 
 C24 
 625 
 
 2. /930916 
 2.7937904 
 2.7944880 
 2.7951816 
 2.7958800 
 
 698 
 697 
 696 
 695 
 694 
 
 581 
 582 
 583 
 584 
 585 
 
 2.7641761 
 2.7649230 
 2.7656686 
 2.7664128 
 2-7671559 
 
 746 
 745 
 744 
 743 
 941 
 
 626 
 627 
 628 
 629 
 630 
 
 2-7965743 
 2.79/2675 
 2.7979596 
 2.7986500 
 2.7993405 
 
 693 
 692 
 691 
 689 
 688 
 
 586 
 587 
 588 
 58Q 
 590 
 
 2.7678976 
 2.7686381 
 2.7693773 
 2.7/01153 
 2.7708520 
 
 74O 
 739 
 
 738 
 736 
 
 7 q c 
 
 631 
 632 
 633 
 634 
 635 
 
 2.8000294 
 2.8007171 
 2-8014037 
 2.8020893 
 2.8027737 
 
 687 
 C86 
 685 
 684 
 
 683 
 
 591 
 592 
 593 
 594 
 595 
 
 2.7715870 
 2-7723217 
 2./730547 
 2.773/864 
 2.7745170 
 
 734 
 733 
 731 
 730 
 
 729 
 
 (536 
 
 637 
 C38 
 639 
 640 
 
 2.8034571 
 2.8041394 
 2.8048207 
 2.8055009 
 2.8091800 
 
 692 
 <J81 
 68O 
 679 
 678 
 
LOGARITHMS, 
 
 "No. 
 
 Logarithms. 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 641 
 642 
 643 
 644 
 645 
 
 2.8068580 
 2.80/5350 
 2.8082110 
 2.8088859 
 2.8095597 
 
 677 
 676 
 674 
 673 
 
 686 
 687 
 688 
 689 
 690 
 
 2.8363241 
 2.8369567 
 
 2.8375884 
 2.8382192 
 2.8388491 
 
 632 
 631 
 630 
 629 
 
 646 
 647 
 648 
 649 
 650 
 
 2.8102325 
 2.8109043 
 2.8115750 
 2.8122447 
 2.8129134 
 
 672 
 
 671 
 670 
 669 
 668 
 
 AT * 
 
 691 
 692 
 693 
 694 
 
 695 
 
 2.8394780 
 2.8401061 
 2.8407332 
 2.8413595 
 2.8419848 
 
 628 
 
 628 
 627 
 625 
 625 
 
 651 
 652 
 653 
 654 
 655 
 
 2.8135810 
 2.8142476 
 2.8149332 
 2.8155777 
 2.8162413 
 
 667 
 666 
 665 
 664 
 
 663 
 #;/:>> 
 
 696 
 697 
 698 
 699 
 700 
 
 2.8426092 
 2.8432328 
 2.8438554 
 Q. 84447/2 
 2.8450980 
 
 624 
 623 
 
 622 
 621 
 
 620 
 
 656 
 657 
 658 
 659 
 660 
 
 2.8169038 
 2.8175654 
 2.8182-259 
 2.8188854 
 2.8395439 
 
 661 
 660 
 659 
 658 
 
 fi CM 
 
 701 
 702 
 703 
 704 
 
 705 
 
 2.8457180 
 2.8463371 
 2.8469553 
 2.8475727 
 2.8481891 
 
 620 
 
 619 
 618 
 617 
 616 
 
 661 
 662 
 663 
 664 
 665 
 
 2.8202015 
 2.8208580 
 2.8215135 
 2.8221681 
 2.8228216 
 
 O57 
 
 656 
 655 
 654 
 653 
 
 f. r. n 
 
 706 
 
 707 
 
 708 
 709 
 710 
 
 2.8488047 
 2,8494194 
 2.8500333 
 2.8506462 
 2.8512583 
 
 615 
 
 614 
 613 
 612 
 612 
 
 666 
 667 
 668 
 669 
 670 
 
 2.8234742 
 2.8241258 
 2.8247765 
 2.8254261 
 2.8260748 
 
 651 
 650 
 649 
 648 
 64. 1 
 
 711 
 
 712 
 713 
 714 
 715 
 
 2.8538696 
 2.8524800 
 2.8530895 
 2.8536982 
 2.8543060 
 
 6 J 1 
 
 610 
 60Q 
 
 608 
 607 
 
 671 
 672 
 673 
 674 
 675 
 
 2.8267225 
 2.8273693 
 2.8280151 
 2.8286599 
 2.8293038 
 
 646 
 645 
 644 
 643 
 642 
 
 716 
 717 
 
 718 
 
 71Q 
 
 720 
 
 2.8549130 
 2.8555192 
 2.8561244 
 2.8567289 
 2.85/3325 
 
 607 
 606 
 605 
 604 
 603 
 
 676 
 677 
 678 
 679 
 680 
 
 2.8299467 
 2.8305887 
 2.8312297 
 2.8318698 
 2.8325089 
 
 642 
 641 
 640 
 639 
 638 
 
 721 
 
 722 
 
 723 
 
 7-24 
 725 
 
 2.85/9353 
 2.8585372 
 2.8591383 
 2.8597386 
 2.8603380 
 
 (501 
 601 
 600 
 599 
 
 598 
 
 681 
 682 
 683 
 684 
 685 
 
 2.8331471 
 2.8337844 
 2.8344207 
 2.8350561 
 2.8356906 
 
 637 
 636 
 635 
 634 
 633 
 
 726 
 
 727 
 
 728 
 729 
 730 
 
 2.8609366 
 2.8615344 
 2.8621314 
 2.8627275 
 2.8633229 
 
 597 
 597 
 596 
 595 
 594 
 
$56 
 
 LOGARITHMS. 
 
 .No. 
 
 Logarithms 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 731 
 732 
 733 
 734 
 735 
 
 2.8639174 
 2.6645111 
 2.8651046 
 2.8656Q61 
 2.8662873 
 
 593 
 592 
 592 
 591 
 
 776 
 777 
 778 
 779 
 780 
 
 2.8898617 
 2.8904^10 
 2;8909796 
 2.8915375 
 2.8920946 
 
 559 
 558 
 557 
 557 
 
 736 
 737 
 738 
 739 
 740 
 
 2.8636778 
 2.8674675 
 2.8680564 
 2.8686444 
 ' 2.8692317 
 
 590 
 
 589 
 588 
 588 
 
 587 
 
 781 
 
 782 
 783 
 784 
 785 
 
 2.8026510 
 2.8932008 
 1 2.8937618 
 2.8943161 
 2.8948697 
 
 55(5 
 555 
 555 
 554 
 553 
 
 741 
 
 742 
 743 
 744 
 745 
 
 2.869S182 
 2.8704039 
 2.8709888 
 2.8715729 
 2.8721563 
 
 586 
 585 
 584 
 584 
 583 
 
 786 
 
 787 
 
 ' 788 
 789 
 790 
 
 1 ^2.8954225 
 2.8959747 
 2.8965262 
 2.8970770 
 2.8976271 
 
 552 
 552 
 551 
 550 
 
 550 
 
 746 
 
 747 
 743 
 749 
 730 
 
 2.8727388 
 2.8733206 
 2.8739016 
 2.8744818 
 2.8750613 
 
 581 
 581 
 580 
 579 
 
 791 
 
 792 
 793 
 794 
 795 ' 
 
 2.8981765 
 2.8987252 
 2.8992732 
 2.8998205 
 2.9003671 
 
 548 
 543 
 
 5'47 
 546 
 
 C. A f\ 
 
 751 
 752 
 753 
 754 
 755 
 
 2.8756399 
 2.8762178 
 2.8/67950 
 2.8773713 
 2.8779469 
 
 0/8 
 
 377 
 577 
 576 
 
 5/5 
 
 3* i 
 
 796 
 797 
 798 
 79P 
 800 
 
 2.9009131 
 2.9014583 
 2.9020029 
 2 9025468 
 2.9030900 
 
 545 
 544 
 543 
 543 
 542 
 
 736 
 757 
 
 758 
 759 
 760 
 
 2.8785218 
 2. 879"9>9 
 2.8796692 
 2.8802418 
 2.8808136 
 
 574 
 573 
 5/2 
 571 
 r i7 1 
 
 801 
 
 802 
 803 
 804 
 805 
 
 2.9036325 
 2.Q041744 
 ' 2.9047155 
 2.9052560 
 2.9057960 
 
 541 
 541 
 440 
 540 
 539 
 
 761 
 762 
 763 
 764 
 765 
 
 2.8813847 
 2.88195.50 
 2.88252-15 
 2.8830934 
 2.8836614 
 
 570 
 
 539 
 568 
 568 
 567 
 
 806 . 
 
 807 
 
 808 
 809 
 810 
 
 2,9063350 
 2.9068735 
 2.9074114 
 2.9079485 
 2.9084850 
 
 538 
 537 
 537 
 536 
 535 
 
 766 
 767 
 768 
 769 
 770 
 
 2.8842288 
 2.8847954 
 2. 88536 I 2 
 2.8859263 
 2.8864007 
 
 566 
 565 
 565 
 564 
 563 
 
 811 
 812 
 813 
 814 
 815 
 
 2.9090209 
 2.9095560 
 2,9100905 
 2.9106244 
 , 2.9111576 
 
 '535 
 534 
 533 
 533 
 
 532 
 
 771 
 
 772 
 773 
 774 
 
 775 j 
 
 2.8870544 
 2.88761/3 
 2.6881795 
 2.8887410 
 2.8893017 
 
 562 
 562 
 561 
 560 
 560 
 
 816 
 
 817 
 818 
 819 
 820 
 
 2,9116902 
 2.9122220 
 2.9127533 
 2.9132839 
 2.9138339 
 
 531 
 531 
 530 
 530 
 529 
 
LOGARITHMS. 
 
 257 
 
 No. ' 
 
 Logarithms. 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 821 
 822 
 823 
 824 
 
 825 
 
 2.9143432 
 2.9148718 
 2.9153998 
 2.9159272 
 2.9164539 
 
 528 
 528 
 527 
 526 
 526 
 
 866 
 867 
 868 
 869 
 870 
 
 2.9375179 
 2.9380191 
 2.9385197 
 2.9390198 
 2.9395193 
 
 501 
 50O 
 500 
 499 
 498 
 
 826 
 827 
 828 
 829 
 830 
 
 2.9169800 
 2.9175055 
 2.9180303 
 '2.9185945 
 2.9190781 
 
 525 
 524 
 524 
 523 
 522 
 
 871 
 872 
 873 
 874 
 875 
 
 2.9400182 
 2.9405165 
 2.9410142 
 2.9415114 
 2.9420081 
 
 498 
 497 
 497 
 496 
 496 
 
 831 
 832 
 833 
 834 
 835 
 
 2.9196010 
 2.9201233 
 2.9206450 
 2.9211661 
 2.9216865 
 
 522 
 521 
 521 
 520 
 519 
 
 876 
 
 877 
 878 
 879 
 880 
 
 2.Q425041 
 2.9429996 
 2.9434945 
 2.9439889 
 2.9448427 
 
 495 
 494 
 494 
 493 
 493 
 
 836 
 837 
 838 
 839 
 840 
 
 2.9222063 
 2.9227255 
 2.9232440 
 2.Q237620 
 2.9242793 
 
 519 
 518 
 518 
 517 
 516 
 
 881 
 
 882 
 883 
 884 
 '885 
 
 2.9449759 
 2.9454686 
 2.9459607 
 2.9464523 
 2.9469433 
 
 492 
 493 
 491 
 491 
 490 
 
 841 
 842 
 843 
 844 
 845 
 
 2.9247960 
 2.9253121 
 2.9258276 
 2.9263424 
 2.9268567 
 
 516 
 515 
 514 
 514 
 
 513 
 
 886 
 
 887 
 888 
 889 
 890 
 
 2.9474337 
 2.9479236 
 2.9484130 
 2.9489018 
 2.9493900 
 
 489 
 
 489 
 488. 
 488 
 487- 
 
 840 
 847 
 848 
 849 
 850 
 
 2.9273704 
 2.9278834 
 2.9283959 
 2.9289077 
 2.9294189 
 
 513 
 512 
 511 
 51] 
 510 
 
 891 
 892 
 393 
 894 
 895 
 
 2.9498777 
 2.9503649 
 2.9508515 
 2.9513375 
 2.9518230 
 
 487 
 486 
 486 
 485 
 485 
 
 851 
 852 
 853 
 854 
 855 
 
 2.9299296 
 2.9304396 
 2.9309490 
 2.9314579 
 2.9319661 
 
 510 
 509 
 508 
 508 
 
 507 
 
 896 
 897 
 898 
 899 
 900 
 
 2.9523080 
 2.9527924 
 2.9432763 
 2.9537597 
 2.9542425 
 
 484 
 483 
 483 
 
 482 
 482 
 
 856 
 
 857 
 868 
 859 
 860 
 
 2.9324738 
 2.9329808 
 2.9334873 
 2.9339932 
 2-9344985 
 
 507 
 506 
 505 
 505 
 504 
 
 901 
 902 
 903 
 904 
 905 
 
 2.9547248 
 2.9552065 
 2.9556878 
 2.9561684 
 2.95664S6 
 
 481 
 4St 
 48O 
 480 
 479 
 
 861 
 862 
 863 
 864 
 865 
 
 2.9350032 
 2.9355073 
 2.9360108 
 2.9365137 
 2.9370161 
 
 504 
 503 
 502 
 502 
 .sm 
 
 906 
 907 
 908 
 90^ 
 910 
 
 2-9571282 
 2.9576073 
 2.9580858 
 2.9585639 
 2.9590414 
 
 479 
 
 478 
 4/8 
 477 
 
 .ATI 
 
LOGARITHMS. 
 
 "No. 
 
 Logarithms. 
 
 Diff. 
 
 No. 
 
 Logarithms. 
 
 Diff. 
 
 911 
 912 
 913 
 914 
 915 
 
 2. 95951 S4 
 2.9399948 
 2.9604/08 
 2.9609462 
 . 2.9614211 
 
 476 
 476 
 4/5 
 474 
 4/4 
 
 956 
 957 
 958 
 959 
 960 
 
 2.980457Q 
 2.9809H9 
 2.9813655 
 2.9818186 
 2.Q8-22712 
 
 454 
 453 
 453 
 452 
 
 916 
 17 
 918 
 919 
 920 
 
 2.9618Q55 
 2.06-23693 
 
 2.9628427 
 2.9633155 
 
 2.9637878 
 
 473 
 473 
 
 472 
 472 
 
 471 
 
 961 
 962 
 963 
 964 
 965 
 
 2.9827234 
 2.9831751 
 2.9836263 
 2.9840770 
 2.9845273 
 
 451 
 451 
 
 450 
 450 
 
 44O 
 
 9-21 
 922 
 9-23 
 924 
 925 
 
 2.9642596 
 2.9647309 
 2.9652017 
 2.9656720 
 2.9661417 
 
 471 
 470 
 470 
 
 469 
 469 
 
 969 
 967 
 968 
 969 
 9/0 
 
 2-9849771 
 2.9854265 
 2.9858754 
 2.9862238 
 2.9867717 
 
 4iy 
 
 449 
 448 
 448 
 447 
 447 
 
 926 
 927 
 928 
 92Q 
 930 
 
 2.9660110 
 2.9670/97 
 2.9675480 
 2.9680J57 
 2.9684829 
 
 468 
 468 
 467 
 467 
 46^> 
 
 971 
 972 
 973 
 974 
 975 
 
 2.9872192 
 2.9876663 
 2 988] 128 
 2.9885590 
 2.9890046 
 
 447 
 '446 
 446 
 445 
 445 
 
 931 
 
 932 
 933 
 934 
 935 
 
 2.96S9497 
 2.9dy4159 
 2.96Q8S16 
 2.9/03469 
 2.97081 1(5 
 
 466 
 
 4f>3 
 465. 
 464 
 464 
 
 976 
 977 
 978 
 979 
 986 
 
 2.9894498 
 2.9898946 
 2.9903389 
 2.9907827 
 2.9912261 
 
 444 
 444- 
 443 
 443 
 
 443 
 
 006 
 937 
 938 
 939 
 940 
 
 2.9712758 
 2.9717396 
 2.9/22028 
 2.9726656 
 2.9731279 
 
 463 
 463 
 462 
 462 
 
 461 
 
 981 
 982 
 9S3 
 984 
 985 
 
 2.9916690 
 2.0Q21115 
 2.9925535 
 2 9929951 
 2.9934362 
 
 442 
 44-2 
 
 411 
 441 
 440 
 
 <J41 
 94:2 
 943 
 944 
 945 
 
 2.9735896 
 2.9740509 
 2.9/45117 
 2.9749/20 
 2.9754318 
 
 461 
 460 
 460- 
 459 
 459 
 
 986 
 987 
 988 
 9*9 
 91)0 
 
 2. 993*769 
 2.9943172 
 2.934/560 
 2.9951963 
 2.9956352 
 
 44 O 
 439 
 439 
 438 
 438 
 
 946 
 
 f)4/ 
 948 
 
 949 
 
 950 
 
 2.9758911 
 2.9763500 
 2.9768O83 
 2.9772662 
 
 2.9777236 
 
 458 
 
 458 
 457 
 457 
 45*6 
 
 991 
 
 992 
 993 
 994 
 995 
 
 2.9960737 
 2.9.965117 
 2.9969492 
 2.997^864 
 2.9978231 
 
 .438 
 437 
 437 
 436 
 436 
 
 951 
 952 
 
 953 
 9.Y4 
 955 
 
 2.9781805 
 2.9786369 
 2.079092Q 
 2.9795484 
 2.9800034 
 
 456 
 455 
 455 
 454 
 454 
 
 996 
 997 
 998 
 999 
 1000 
 
 2.9982593 
 2.9986952 
 2.9991305 
 2.9995655 
 3.0000000 
 
 435 
 435 
 435 
 434 
 
 
 
 F I 1 
 
 sT I S. 
 
 
 
 James Gillet, Printer, Crown-court, Fleet-street, London. 
 

14 DAY USE 
 
 RETURN TO DESK FROM WHICH BORROWED 
 
 LOAN DEPT. 
 
 This book is due on the last date stamped below, or 
 
 on the date to which renewed. 
 Renewed books are subject to immediate recall. 
 
 L.n 
 
 -=* 
 
 JAN b 
 
 PHOTQCOPy APR 1 
 
 
 LD 21A-60m-3,'65 
 (F2336slO)476B 
 
 General Library 
 University of Calif orr 
 Berkeley 
 
tA 02413 
 
 M507409 
 
 ft- 
 

 
 i