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 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 . . 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 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 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 an54) 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. 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 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/. !:>$. 6^., of B 97^. 145. 9^-, of .C 175/. 105.., of JD 99/, 45, and of E 139/. 12s. 4d. 13. A person borrows of several friends the following sums of money ; of the first 5OO/. ; of the second 225/. 125. ; of the third fifty guineas; of the fourth seventy guineas and 2-2 crowns; of the -fifth he had 150/. 75. Qd. ; how much will he have to pay interest, for ? 14. A man borrowed a sum of money, and paid at different times 75 guineas, .but he still owed ssl.Qt. 9^.: what WAS the ordinal xiebt? COMPOUND SUBTRACTION Is the method of finding the difference between two given compound numbers. RULE I. Having arranged the numbers so that the smaller may stand under the greater, subtract eat ber in the lower line from that which stands above it, and ' down the remainders. 2. When any of the lower denominations are greater than the upper, increase the upper number by as many as m&ke cue of the next superior denomination, from which take the figure, in the lower line, set down the difference, and c7 16 Ex, 22. Q09 Ex.20. 764 15 4j Ex. 21, 674 6 oj 800 19 U'f 98 12 11 249 19 9$ 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. 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 $\ll. IQs. Qd. 2. Proof. The method of proof is by Com - 2 j the last remainder 6, by the first di- 5 9 H;f 6 ) visor 3, and take in the first remainder 3, and then place under it the common divisor 27. The answer is 5/. QS. 3 l%d.*g. See p. 27. 4 s. d. Ex. 2. 769 4. 594 7 6 6. 333 10 10J 8. 498 9 10. 596 12. 465 14. 564 16. 678 18. 999 20. 564 22. 248 24. 505 23. 25. 27. &)l350 10 11 The answer is 5/. QS. d. >7 12 6 2 ~ - 14 9 15 8^ - - 15 J6 9 9 - - 16 J7 f " - 18 9 5 - 24 9 18 7 - 27 7 9 9| - - 30 >7 4 6 - - 33 4 3 6} - - 42 7 14 4 - - 48 7 12 11 - - 72 5 13 3j - - 88 4 18 8^ - 108 5 3 3 - - 1,32 9 8i 26. 564 28. 8b8 11 3^ 8 6 20 25 28 32 36 44 49 54 56 63 84 99 120 144 jEx. 3. 167 3. 339 o. 486 7. 987 9. 439 11. 379 1-3. 487 15. 17- 19- 327 21. 3 5 6 4 When there are three component parts. Ex. 1350. 10^. lid. -r- 240 =5 x 6 x S. , s. d. The division in this example fol- lows the same rule as before, but there are three remainders, to find 6)270 22 4 C the true value of which, I multiply the third l , by the second divisor 6, 8)45 4^ 2") arid take in the second remainder, ( 8 ^ 2 4 4 4 o 2 > tilat is > once ^ is 6, and 2 are 8 ; 5 12 '6j 1 J L then this product, 8, I multiply by 5, the first divisor, and take in the first remainder, that is, 8 times 5 are 40, and 4- are 44 r so that the true answer Is 5/. 125. 6|t/. 4 5 , as will be proved in the next page by Long Division. See also p. 27 and 28. * The arithmetician-will easily perceive that the fractions | -f- -| -f- 5 are equal'^g -f- ^ 4- -^ ~ (because now they have each a com- mon denominator) COMPOUND DIVISION. 35 Ex. 1. 5527. 10$. 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 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. ? 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 .>;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*. 7i3f> 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, 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. Couttd 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, |, , 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$. 6//., 85. 22 500 are omitted in each step of the operation. 3.7500 4 Answer, - 125. 3$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 ; 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 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<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 at how much perlb. can he sell the whole mixture? 3 X 112 z: 392^) ("392 X 9 == 3528 2 X 112 224 > and 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$. 2d. per gallon ; how much must he use of each ? I reduce the several prices to pence, which stand as follow : 150-. 50 160 : 8(> : bGO : 344' 'Sum of differences 10 8fiO The answer is 258 gallons at 125. ; 172 at 9s.; 86 at 6s. ; and 314 at 5s. per gallon, may be mixed and sold at Bs per gallon. Ex. 2. A goldsmith has four sorts of gold, viz. of 24, 10, 18, and 15 carats fine, wishes 126 oz. of the fineness of 17 carats, how much will he want of each sort ? Ex.3. A drug grinder has bark worth 165. per lb., some at I.QS. and some at 4s. ; but he is desiraus of making up two parcels, viz. one containing a cwt. at <)$., 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, 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. 2119124 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 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 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 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-| 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 > 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