I LLINOI S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Urbana-Champaign Library Brittle Books Project, 2012. COPYRIGHT NOTIFICATION In Public Domain. Published prior to 1923. This digital copy was made from the printed version held by the University of Illinois at Urbana-Champaign. It was made in compliance with copyright law. Prepared for the Brittle Books Project, Main Library, University of Illinois at Urbana-Champaign by Northern Micrographics Brookhaven Bindery La Crosse, Wisconsin 2012 :{11A . . . y .. i t 2?>. f ;:lsl s: iil:iata ,".,"j+j<{ 3. -}] .:t:.:. .... +t: ?z3. lt)f 11 t"F . rl i 2;4 ;7Ytit91z/3 1 j}?Riii ;fit f: ! P,. I;r E 1 . 3 r :IlTI l: f #}i itl t tw 1r11Nsl r1 3 i MORGAN l nr. lot, ti i t - - iOE - -- t - - r_7 71 7 t - _ _ ~8~ttii~n~ 'LIB RA RY OF THE U N IVERS ITY Of ILLINOIS D39e 1--1111~1111, - -- I THE ELEMENTS ARITHMETIC. BY AUGUSTUS DE MORGAN, OF TRINITY COLLEGE, CAMBRIDGE; FELLOW OF THE ROYAL ASTRONOMICALSOCIETY, AND OFTHE CAMBRIDGE PHILOSOPHICAL SOCIETY; AND PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. "Hominis studiosi est intelligere, quas utilitates proprie afferat arithmetieaWt si solidam et perfectam doctrinam in caeteris philosophise partibus explicant. Quod enim vulgo dicunt, principium esse dimidium totius, id vel maxlme in philosophise partibus conspicitur."--MELANCTHON. "Ce n'est point par la routine qu'on s'instruit, c'est par sa propre rtflexion; et ii est essentiel de contracter l'habtude de se rendre raison de ce qu'on fait; cette habitude s'acquiert plus facilement qu'on ne pense; et une fois acquise, elle ne se perd plus."--CONDILLAC. THIRTEENTH THOUSAND. THIRTEENT'H THOUSAND. LONDON: TAYLOR, WALTON, AND MABERLY, BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE, UPPER GOWER STREET, AND IVY LANE, PATERNOSTER ROW. M.DCCC.LI. 1 $3 THE ELEMENTS OF ARITHMETIC. LONDON: PRINTED BY J. WERTHEIMER AND CO. CIRCUS PLACE, FINSBURY CIRCUS. PREFACE. THE preceding editions of this work were published in 1830, This fifth edition differs from the 1832, 1835, and 1840. three preceding, as to the body of the work, in nothing which need prevent the four, or any two of them, from being used together in a class. But it is considerably aug- mented by the addition of eleven new Appendixes,* relating to matters on which it is most desirable that the advanced student should possess information. The first Appendix, on Computation, and the sixth,. on Decimal Money, should be read and practised by every student with as much attention as any part of the work. The mastery of the rules for in- stantaneous conversion of the usual fractions of a pound sterling into decimal fractions, gives the possessor the greater part of the advantage which he would derive from the introduction of a decimal coinage. At the time when this work was first published, the importance of establishing arithmetic in the young mind upon reason and demonstration, was not admitted by many. The case is now altered: schools exist in which rational Some separate copies of these Appendixes are printed, for those who may desire to add them to the former editions. 4 ":ki~i PREFACE. vi arithmetic is taught, and mere rules are made to do no more than their proper duty. There is no necessity to advocate a change which is actually in progress, as the works which are published every day sufficiently shew. And my principal reason for alluding to the subject here, is merely to warn those who want nothing but routine, that this is not the book for their purpose. A. DE MORGAN. London, May 1, 1846. TABLE OF CONTENTS. BOOK I. PAGE SECTION I. Numeration...... II. Addition and Subtraction 14 III. Multiplication IV. Division . V. Fractions . . . 24 34 51 VI. Decimal Fractions VII. Square Root VIII. Proportion. ..... IX. Permutations and Combinations S . . . . . 65 . 89 S.100 . . 118 BOOK II. I. Weights and Measures, &c. IT. Rule of Three . III. Interest, &c. . . . . 124 . 144 . 150 ..... APPENDIX. I. On the mode of computing . . . . . II. On verification by casting out nines and elevens . III. On scales of notation . . . . . . .. IV. On the definition of fractions . . . . V. On characteristics . . ...... VI On decimal money . . . . . . VII. On the main principle of book-keeping . . . VIII. On the reduction of fractions to others of nearly equal IX. On some general properties of numbers X On combinations . ........ XI. On Horner's method of solving equations . . XII. Rules for the application of arithmetic to geometry . . . 161 . 166 . 168 . . . . . . 171 174 176 180 value . 190 . 193 201 . . 210 . 217 ELEMENTS OF ARITHMETIC. BOOK I. PRINCIPLES OF ARITHM1ETIC. SECTION I. NUMERATION. 1. IMAGINE a multitude of objects of the same kind assembled together; for example, a company of horsemen. One of the first things that must strike a spectator, although unused to counting, is, that to each man there is a horse. Now, though men and horses are things perfectly unlike, yet, because there is one of the first kind to every one of the second, one man to every horse, a new notion will he formed in the mind of the observer, which we express in words by saying that there is the same number of men as of horses. A savage, who had no other way of counting, might remember this number by taking a pebble for each man. Out of a method as rude as this has sprung our system of calculation, by the steps which are pointed out in the following articles. Suppose that there are two companies of horse- men, and a person wishes to know in which of them is the greater number, and also to be able to recollect how many there are in each. 2. Suppose that while the first company passes by, he drops a pebble into a basket for each man whom he sees. There is no connexion between the pebbles and the horsemen but this, that for every horseman B 2 PRINCIPLES OF ARITHMETIC. § 2-4. there is a pebble; that is, in common language, the number of pebbles and of horsemen is the same. Suppose that while the second company passes, he drops a pebble for each man into a second basket; he will then have two baskets of pebbles, by which he will be able to convey to any other person a notion of how many horsemen there were in each company. When he wishes to know which company was the larger, or contained most horsemen, he will take a pebble out of each basket, and put them aside. He will go on doing this as often as he can, that is, until one of the baskets is emptied. Then, if he also find the other basket empty, he says that both companies contained the same number of horsemen; if the second basket still contain some pebbles, he can tell by them how many more were in the second than in the first. 3. In this way a savage could keep an account of any numbers it which he was interested. He could thus register his children, his cattle, or the number of summers and winters which he had seen, by means of pebbles, or any other small objects which could be got in large numbers. Something of this sort is the practice of savage nations at this day, and it has in some places lasted even after the invention of better methods of reckoning. At Rome, in the time of the republic, the prretor, one of the magistrates, used to go every year in great pomp, and drive a nail into the door of the temple of Jupiter; a way of remembering the number of years which the city had been built, which probably took its rise before the introduction of writing. 4. In process of time, names would be given to those collections of pebbles which are met with most frequently. But as long as small numbers only were required, the most convenient way of reckoning them would be by means of the fingers. Any person could make with his two hands the little calculations which would be necessary for his purposes, and would name all the different collections of the fingers. He would thus get words in his own language answering to one, two, three, four, five, six, seven, eight, nine, and ten. As his wants in- creased, he would find it necessary to give names to larger numbers; but here he would be stopped by the immense quantity of words which S4-5. NUMERATION. he must have, in order to express all the numbers which he would be He must, then, after giving a separate name obliged to make use of. to a few of the first numbers, manage to express all other numbers by means of those names. 5. I now shew how this has been done in our own language. The English names of numbers have been formed from the Saxon: and in the following table each number after ten is written down in one column, while another shews its connexion with those which have preceded it. One eleven ten and one* two twelve ten and two three thirteen ten and three four fourteen ten and four five fifteen ten and five six sixteen ten and six seven seventeen ten and seven eight eighteen ten and eight. nine nineteen ten and nine ten twenty two tens five tens twenty-one two tens and one fifty twenty-two two tens and two sixty six tens seventy seven tenIS eighty eight ten S ninety nine tens S a hundred ten tens &c. &c. thirty &c. forty &c. &c. &c. three tens &c. four tens &c. a hundred and one &c. ten tens and one &c. a thousand ten hundreds ten thousand * It has been supposed that eleven and twelve are derived from the Saxon for one lef and two left (meaning, after ten is removed); but there seems better reason to think that leven is a word meaning ten, and connected with decenm. § 5-8. PRINCIPLES OF ARITHMETIC. a hundred thousand a million ten hundred thousand or one thousand thousand ten millions a hundred millions &c. 6. Words, written down in ordinary language, would very soon be too long for such continual repetition as takes place in calculation. Short signs would then be substituted for words; but it would be impossible to have a distinct sign for every number: so that when some few signs had been chosen, it would be convenient to invent others for the rest out of those already made. The signs which we use are as follow: o I a 3 4 5 6 7 8 9 nought one two three four five six seven eight nine I now proceed to explain the way in which these signs are made to represent other numbers. 7. Suppose a man first to hold up one finger, then two, and so on, until be has held up every finger, and suppose a number of men to do the same thing. It is plain that we may thus distinguish one number from another, by causing two different sets of persons to hold up each a certain number of fingers, and that we may do this in many different ways. For example, the number fifteen might be indicated either by fifteen men each holding up one finger, or by four men each holding up two fingers and a fifth holding up seven, and so on. The question is, of all these contrivances for expressing the number, which is the most convenient ? In the choice which is made for this purpose consists what is called the method of numeration. 8. I have used the foregoing explanation because it is very probable that our system of numeration, and almost every other which is used in the world, sprung from the practice of reckoning on the fingers, which children usually follow when first they begin to count. The method which I have described is the rudest possible; but, by a little g NUMERATION. 8-Izo. alteration, a system may be formed which will enable us to express enormous numbers with great ease. 9. Suppose that you are going to count some large number, for example, to measure a number of yards of cloth. Opposite to yourself suppose a man to be placed, who keeps his eye upon you, and holds up a finger for every yard which he sees you measure. When ten yards have been measured he will have held up ten fingers, and will not be able to count any further unless he begin again, holding up one finger at the eleventh yard, two at the twelfth, and so on. But to know how many have been counted, you must know, not only how many fingers he holds up, but also how many times he has begun again. You may keep this in view by placing another man on the right of the former, who directs his eye towards his companion, and holds up one finger the moment he perceives him ready to begin again, that is, as soon as ten yards have been measured. Each finger of the first man stands only for one yard, but each finger of the second stands for as many as all the fingers of the first together, that is, for ten. In this way a hundred may be counted, because the first may now reckon his ten fingers once for each finger of the second man, that is, ten times in all, and ten tens is one hundred (5).* Now place a third man at the right of the second, who shall hold up a finger whenever he perceives the second ready to begin again. One finger of the third man counts as many as all the ten fingers of the second, that is, counts one hundred. In this way we may proceed until the third has all his fingers extended, which will signify that ten hundred or one thousand have been counted (5). A fourth man would enable us to count as far as ten thousand, a fifth as far as one hundred thousand, a sixth as far as a million, and so on. 10. Each new person placed himself towards your left in the rank opposite to you. Now rule columns as in the next page, and to the right of them all place in words the number which you wish to represent; in the first column on the right, place the number of fingers * The references are to the preceding articles. B2 PRINCIPLES OF ARITHMETIC. § 10-12i. which the first man will be holding up when that number of yards has In the next column, place the fingers which the second been measured. man will then be holding up ; and so on. 5 I. 7 fifty-seven. o 4 one hundred and four. II. I III. I o one hundred and ten. 2 3 8 two thousand three hundred and fortyeight. IV. I 5 9 o 6 fifteen thousand nine hundred and six. I 8 7 o o 4 one hundred and eighty-seven thousand and four. VII. 3 6 9 7 z 8 5 three million, six hundred and ninetyseven thousand, two hundred and eighty-five. V. VI. 11. In I. the number fifty-seven is expressed. tens and seven. The first has therefore Thiis means (5) five counted all his fingers five times, and has counted seven fingers more. This is shewn by five fingers of the second man being held up, and seven of the first. the number one hundred and four is represented. ten tens and four. In II. This number is (5) The second person has therefore just reckoned all his fingers once, which is denoted by the third person holding up one finger; but he has not yet begun again, because he does not hold up a finger until the first has counted ten, of which ten only four are completed. When all the last-mentioned ten have been counted, he then holds up one finger, and the first being ready to begin again, has no fingers extended, and the number obtained is eleven tens, or ten tens and one ten, or one hundred and ten. This is the case in III. You will now find no difficulty with the other numbers in the table. 12. In all these numbers a figure in the first column stands for only as many yards as are written under that figure in (6). A figure in the second column stands, not for as many yards, but for as many tens of yards ; a figure in the third column stands for as many hundreds of yards ; in the fourth column for as many thousands of yards; and so on : that is, if we suppose a figure to move from any column to the one NUMERATION. I2-15. I on its left, it stands for ten times as many yards as before. Recollect this, and you may cease to draw the lines between the columns, because each figure will be sufficiently well known by the place in which it is; that is, by the number of figures which come upon the right hand of it. 13. It is important to recollect that this way of writing numbers, which has become so familiar as to seem the natural method, is not more natural than any other. For example, we might agree to signify one ten by the figure of one with an accent, thus, I'; twenty or two tens by z'; and so on: one hundred or ten tens by I"; two hundred by 2"; one thousand by I"'; and so on: putting Roman figures for accents when they become too many to write with convenience. The fourth number in the table would thea be written z"' 3" 4' 8, which might also be expressed by 8 4' 3" z"'", 4' 8 3" 2"'; or the order of the figures might be changed in any way, because their meaning depends upon the accents which are attached to them, and not upon the place in which they stand. Hence, a cipher would never be necessary; for o04would be distinguished from the second I'4. 14 by writing for the first I"4, and for The common method is preferred, not because it is more exact than this, but because it is more simple. 14. The distinction between our method of numeration and that of the ancients, is in the meaning of each figure depending partly upon the place in which it stands. Thus, in 44444 each four stands for four of something; but in the first column on the right it signifies only four of the pebbles which are counted ; in the second, it means four collections of ten pebbles each; in the third, four of one hundred each; and so on. 15. The things measured in (11) were yards of cloth. In this case one yard of cloth is called the unit. The first figure on the right is said to be in the units' place, because it only stands for so many units as are in the number that is written under it in (6). The second figure is said to be in the tens' place, because it stands for a number of tens of units. The third, fourth, and fifth figures are in the places of the hundreds, thousands, and tens of thousands, for a similar reason. § 16-19. PRINCIPLES OF ARITHMETIC. 16. If the quantity measured had been acres of land, an acre of land would have been called the unit, for the unit is one of the things which are measured. Quantities are of two sorts ; those which contain an exact number of units, as 47 yards, and those which do not, as 47 yards and a half. Of these, for the present, we only consider the first. 17. In most parts of arithmetic, all quantities must have the same unit. You cannot say that 2 yards and 3 feet make 5 yards or 5 feet, because z and 3 make 5; yet you may say that z yards and 3 yards make 5 yards, and that z feet and 3 feet make 5 feet. It would be absurd to try to measure a quantity of one kind with a unit which is a quantity of another kind; for example, to attempt to tell how many yards there are in a gallon, or how many bushels of corn there are in a barrel of wine. 18. All things which are true of some numbers of one unit are true of the same numbers of any other unit. together make zz pebbles; and so on. that Thus, 15 pebbles and 7 pebbles 15 acres and 7 acres together make zz acres, From this we come to say that I 5 and 7 make 22, meaning 15 things of the same kind, and 7 more of the same kind as the first, together make 22 of that kind, whether the kind mentioned be pebbles, horsemen, acres of land, or any other. For these it is but necessary to say, once for all, that 15 and 7 make 22. Therefore, in future, on this part of the subject I shall cease to talk of any particular units, such as pebbles or acres, and speak of numbers only. A number, considered without intending to allude to any particular things, is called an abstract number: and it then merely signifies repetitions of a unit, or the number of times a unit is repeated. 19. I will now repeat the principal things which have been mentioned in this chapter. I. Ten signs are used, one to stand for nothing, the rest for the first nine numbers. They are o, I, 2, 3, 4, 5, 6, 7, 8, 9. The first of these is called a cipher. II. Higher numbers have not signs for themselves, but are signified by placing the signs already mentioned by the side of each other, and agreeing that the first figure on the right hand shall keep the value z9-2I. 1 NUMERATION. which it has when it stands alone; that the second on the right hand shall mean ten times as many as it does when it stands alone; that the third figure shall mean one hundred times as many as it does when it stands alone ; the fourth, one thousand times as many ; and so on. III. The right-hand figure is said to be in the units' place, the next to that in the tens' place, the third in the hundreds' place, and so on. I V. When a number is itself an exact number of tens, hundreds, or thousands, &c., as many ciphers must be placed on the right of it as will bring the number into the place which is intended for it. The following are examples : Fifty, or five tens, 50: seven hundred, 700. Five hundred and twenty-eight thousand, 528000. If it were not for the ciphers, these numbers would be mistaken for 5, 7, and 528. V. A cipher in the middle of a number becomes necessary when any one of the denominations, units, tens, &c. is wanting. Thus, twenty thousand and six is z00o6, two hundred and six is zo6. Ciphers might be placed at the beginning of a number, but they would have no meaning. Thus 026 is the same as 26, since the cipher merely shews that there are no hundreds, which is evident from the number itself. 20. If we take out of a number, as 16785, any of those figures which come together, as 67, and ask, what does this sixty-seven mean? of what is it sixty-seven ? the answer is, sixty-seven of the same collections as the 7, when it was in the number; that is, 67 hundreds. For the 6 is 6 thousands, or 6 ten hundreds, or sixty hundreds; which, with the 7, or 7 hundreds, is 67 hundreds: similarly, the 678 is 678 tens. This number may then be expressed either as I ten-thousand 6 thousands 7 hundreds 8 tens and 5 ; or I6 thousands 78 tens and 5; or i ten thousand 678 tens and 5 ; or i67 hundreds 8 tens and 5 ; or I678 tens and 5, and so on. 21. EXERCISES. I. Write down the signs for ;-four hundred and seventy-six; two 10 PRINCIPLES OF ARITHMETIC. § 21-5. thousand and ninety-seven ; sixty-four thousand three hundred and fifty; two millions seven hundred and four; five hundred and seventyeight millions of millions. II. Write at full length 53, i8o5, 1830, 66707, 180917324, 66713721, 90976390, 25000000. III. What alteration takes place in a number made up entirely of nines, such as 99999, by adding one to it ? IV. Shew that a number which has five figures in it must be greater than one which has four, though the first have none but small figures in it, and the second none but large ones. For example, that o1011 is greater than 9879. 22. You now see that the convenience of our method of numeration arises from a few simple signs being made to change their value as they change the column in which they are placed. The same advantage arises from counting in a similar way all the articles which are used in every-day life. For example, we count money by dividing it into pounds, shillings, and pence, of which a shilling is i2 pence, and a pound zo shillings, or 240 pence. We write a number of pounds, shillings, and pence in three columns, generally placing points between the columns. Thus, 263 pence would not be written as 263, but as £1. I. II, where £ shews that the I in the first column is a pound. Here is a system of numeration in which a number in the second column on the right means 12 times as much as the same number in the first ; and one in the third column is twenty times as great as the same in the second, or 240 times as great as the same in the first. In each of the tables of measures which you will hereafter meet with, you will see a separate system of numeration, but the methods of calculation for all will be the same. 23. In order to make the language of arithmetic shorter, some other signs are used. They are as follow: I. 15+38 means that 38 is to be added to 15, and is the same thing as 53. This is the sum of 15 and 38, and is read fifteen plus thirty- eight (plus is the Latin for more). II. 64-12 means that iz is to be taken away from 64, and is the 11 NUMERATION. V23-24. same thing as 52. This is the diference of 64 and I2, and is read sixty- four minus twelve (minus is the Latin for less). III. 9x8 means that 8 is to be taken 9 times, and is the same thing as 72. IV. This is the product of 9 and 8, and is read nine into eight. ac8 6 means that Io8 is to be divided by 6, or that you must find out how many sixes there are in Io8; and is the same thing as IS. This is the quotient of io8 and 6; and is read a hundred and eight by six. V. When two numbers, or collections of numbers, with the foregoing signs, are the same, the sign = is put between them. Thus, that 7 and 5 make 12, is written in this way, 7+5=12. This is called an equation, and is read, seven plus five equals twelve. It is plain that we may construct as many equations as we please. 7+9-3=12+1 ; -- I -+ Thus: 3x2=II, and so on. 24. It often becomes necessary to speak of something which is true not of any one number only, but of all numbers. Io and 7 ; their sum* is 17, their difference is 3. For example, take If this sum and difference be added together, we get 20, which is twice the greater of the two numbers first chosen. If from 17 we take 3, we get 14, which is twice the less of the two numbers. The same thing will be found to hold good of any two numbers, which gives this general proposition,If the sum and difference of two numbers be added together, the result is twice the greater of the two; if the difference be taken from the sum, the result is twice the lesser of the two, If, then, we take any numbers, and call them the first number and the second number, and let the first number be the greater ; we have (1st No.+2d No.)+(ist No.-2d No.)=twice 1st No. (Ist No.+2d No.)--(st No.-zd No.)=twice zd No. The brackets here enclose the things which must be first done, be- fore the signs which join the brackets are made use of. Thus, * Any little computations which occur in the rest of this section may be made on the fingers, or with counters. 12 PRINCIPLES OF ARITHMETIC. ,§ 24-25. 8-(z2+)x(i+I) signifies that z+i must be taken I+I times, and the product must be subtracted from 8. In the same manner, any re- sult made from two or more numbers, which is true whatever numbers are taken, may be represented by using first No., second No., &c., to stand for them, and by the signs in (23). But this may be much shortened; for as first No., second No., &c., may mean any numbers, the letters a and b may be used instead of these words; and it must now be recollected that a and b stand for two numbers, provided only that a is greater than b. b by zb. Let twice a be represented by za, and twice The equations then become (a+b)+(a-b)=za,and (a+b)-(a-b)=23. This may be explained still further, as follows : 25. Suppose a number of sealed packets, marked a, b, c, d, &c., on the outside, each of which contains a distinct but unknown number of counters. As long as we do not know how many counters each contains, we can make the letter which belongs to each stand for its number, so as to talk of the number a, instead of the number in the packet marked a.' And because we do not know the numbers, it does not therefore follow that we know nothing whatever about them; for there are some connexions which exist between all numbers, which we call general properties of numbers. For example, take any number, multiply it by itself, and subtract one from the result ; and then subtract one from the number itself. The first of these will always contain the second exactly as many times as the original number increased by one. Take the number 6 ; this multiplied by itself is 36, which dimi- nished by one is 35: again, 6 diminished by I is 5; and 35 contains 5, 7 times, that is, 6+i times. This will be found to be true of any number, and, when proved, may be said to be true of the number contained in the packet marked a, or of the number a. If we represent a multiplied by itself by aa,* we have, by (23) aa=--a+ a--I I. * This should be (23) aXa, but the sign x is unnecessary here. It is used with numbers, as in 2X 7, to prevent confounding this, which is 14, with 27. 2 26-27. 13 NUMERATION 26. When, therefore, we wish to talk of a number without specifying any one in particular, we use a letter to represent it. Thus: Suppose we wish to reason upon what will follow from dividing a number into three parts, without considering what the number is, or what are the parts into which it is divided. Let a stand for the number, and 6, c, and d, for the parts into which it is divided. Then, by our supposition, a = b+c+d. On this we can reason, and produce results which do not belong to any particular number, but are true of all. Thus, if one part be taken away from the number, the other two will remain, or a-b = c+d. If each part be doubled, the whole number will be doubled, or za = 2b+2c+2d. If we diminish one of the parts, as d, by a number x, we diminish the whole number just as much, or a-x = b+c+(d-x). 27. EXERCISES. 12, b= What is a+zb-c, where a= What is 18, c= 7 ?-Answer, 41. aa-bb , where a=6 and b=2 ?-Ans. 8. a-b What is the difference between (a+b) (c+d) and a+bc+d, for the following values of a, b, c, and d ? a b c d I 2 3 4 10 2 12 7 I 25 I I I I Ans. I 14 PRINCIPLES OF ARITHMETIC. § 28-29. SECTION II. ADDITION AND SUBTRACTION. 28. There is no process in arithmetic which does not consist entirely in the increase or diminution of numbers. There is then nothing which might not be done with collections of pebbles. these or the fingers were used. Probably, at first, either Our word calculation is derived from the Latin word calculus, which means a pebble. Shorter ways of counting have been invented, by which many calculations, which would require long and tedious reckoning if pebbles were used, are made at once with very little trouble. The four great methods are, Addition, Subtraction, Multiplication, and Division; of which, the last two are only ways of doing several of the first and second at once. 29. When one number is increased by others, the number which is as large as all the numbers together is called their sum. The process of finding the sum of two or more numbers is called ADDITION, and, as was said before, is denoted by placing a cross (+) between the numbers which are to be added together. Suppose it required to find the sum of 1834 and 2799. In order to add these numbers, take them to pieces, dividing each into its units, tens, hundreds, and thousands: 1834 is I thous. 8 hund. 3 tens and 4; 2799 is 2 thous. 7 hund. 9 tens and 9. Each number is thus broken up into four parts. If to each part of the first you add the part of the second which is under it, and then put together what you get from these additions, you will have added 1834 and 2799. In the first number are 4 units, and in the second 9 : these will, when the numbers are added together, contribute 13 units to the sum. Again, the 3 tens in the first and the 9 tens in the second will contribute 12 tens to the sum. The 8 hundreds in the first and the 7 hundreds in the second will add I5 hundreds to the sum ; and the thousand in $ 29-31. 15 ADDITION AND SUBTRACTION. the first with the z thousands in the second will contribute 3 thousands to the sum ; therefore the sum required is 3 thousands, 15 hundreds, I2 tens, and 13 units. To simplify this result, you must recollect that13 units are. x2 tens ........ ten and 3 units. are ..... I hund. and z tens. 15 hund. are I thous. and 5 hund. 3 thous. are 3 thous. Now collect the numbers on the right-hand side together, as was done before, and this will give, as the sum of 1834 and z799, 4 thousands, 6 hundreds, 3 tens, and 3 units, which (19) is written 4633. 30. The former process, written with the signs of (23) is as follows : 1834= xxooo+ 8xloo+ 3xIo+4 2799= zx ooo+ 7xoo+ 9X10+ 9 Therefore, 1834+ 2799 = 3X10ooo+ But 15xoo0+ 13= I2X 15x 10= zoo= IxIooo+ 3x oo000 = 3x1000 zxxo + 13 IxIo+ IXKIOO+ 3 ZXIO 5xoo0 Therefore, 8 1 34+2799=4x1000ooo+ 6xIoo+ 3x10+ 3 = 4633. 31. The same process is to be followed in all cases, but not at the same length. In order to be able to go through it, you must know how to add together the simple numbers. This can only be done by memory; and to help the memory you should make the following table three or four times for yourself: 16 PRINCIPLES OF ARITHMETIC. I 3 4 2 I 2 3 4 § 31-33. 5 5 6 7 6 7 8 9 '0 II 8 9) I 5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 o10 II 12 4 5 6 7 8 9 10 12 13 5 6 7 8 9 10 II 12 13 14 6 7 9 10 o II 12 13 14 15 7 8 10 II x2 13 14 15 6 8 9 15 I4 6 17 16 17 18 9 zo I0 9 13 zI 12 13 12 I 14 15 II The use of this table is as follows: Suppose you want to find the sum of 8 and 7. Look in the left-hand column for either of them, 8, for example; and look in the top column for 7. On the same line as 8, and underneath 7, you find 15, their sum. 32. When this table has been thoroughly committed to memory, so that you can tell at once the sum of any two numbers, neither of which exceeds 9, you should exercise yourself in adding and subtracting two numbers, one of which is greater than 9 and the other less. You should write down a great number of such sentences as the following, which will exercise you at the same time in addition, and in the use of the signs mentioned in (23). xz+6 = - 8 22+6 = 28 63 56+7 = 63 100-9 = 91 27-8 =.19 54+9 19+8 = 27 22+8 = 30 44-6 = 38, &c. 33. When the last two articles have been thoroughly studied, you will be able to find the sum of any numbers by the following process,* which is the same as that in (29). * In this and all other processes, the student is strongly recommended to look at and follow the first Appendix. 17 ADDITION. § 33* RULE I. Place the numbers under one another, units under units, tens under tens, and so on. II. Add together the units of all, and part the whole number thus obtained into units and tens. Thus, if 85 be the number, part it into 8 tens and 5 units; if I36 be the number, part it into 13 tens and 6 units (20). III. Write down the units of this number under the units of the rest, and keep in memory the number of tens. IV. Add together all the numbers in the column of tens, remembering to take in (or carry, as it is called) the tens which you were told to recollect in III., and divide this number of tens into tens and hundreds. Thus, if 335 tens be the number obtained, part this into 33 hundreds and 5 tens. V. Place the number of tens under the tens, and remember the number of hundreds. VI. Proceed in this way through every column, and at the last column, instead of separating the number you obtain into two parts, write it all down before the rest. EXAMPLE.-What is 1805+36+19727+3+1474+2008 80o5 36 The addition of the units' line, or 8+4+3+7+6+5, gives 33, that is, 3 tens and 3 units. Put 3 in the units' place, and 19727 add together the line of tens, taking in at the beginning the 3 3 tens which were created by the addition of the units' line. 1474 That is, find 3+0+7+2+3+0, 2008 of tens; that is, I hundred and 5 tens. which gives I 5 for the number Add the line of hun- 25053 dreds together, taking care to add the I hundred which arose in the addition of the line of tens; that is, find 1+0+4+7+8, which gives exactly o20 hundreds, or 2 thousands and no hundreds. Put a cipher in the hundreds' place (because, if you do not, the next figure will be taken for hundreds instead of thousands), and add the figures in the thousands' line together, remembering the 2 thousands which arose from the hundreds' line; that is, find 2+2+1+9+1, c2 which gives 15 18 PRINCIPLES OF ARITHMETIC. thousands, or I ten thousand and 5 thousand. § 33.36. Write 5 under the line of thousands, and collect the figures in the line of tens of thousands, remembering the ten thousand which arose out of the thousands' line; Write 2 under the ten thousands' that is, find x+i, or 2 ten thousands. line, and the operation is completed. 34. As an exercise in addition, you may satisfy yourself that what I now say of the following square is correct. The numbers in every row, whether reckoned upright, or from right to left, or from corner to corner, when added together give the number 24156. zo201642121(656385212963492 252 20524248 2448 1116 324 720252 2916 1548 3348 1584 19803780 2 124 9723168 216 612 2412 4320 1764 3960 1404 3204 044 2844 1440 3240 o8o 360 z160o 4356 1800 3600 756 2556 3312 1 522952 3744 3888 13323528 5762772 288 zo88 4284 1728 3924 1368 3564 zoo8 28o8 648 6842484 2880 1692 93631321 792 3962196 3996 1836 3636 1476 3276 2592 3622z32 4032 1872 3672 1512 1188 2988 432 2628 72z2268 4068 1908 3708 3384 828 3024 468 2664 1224 3420 864 3060 zo8 230441041944 504 2700 41761620 3816 12603456 9003096 144 2340 4140 8 5402736 1 01237 6 35. If two numbers must be added together, it will not alter the sum if you take away a part of one, provided you put on as much to the other. It is plain that you will not alter the whole number of a collection of pebbles in two baskets by taking any number out of one, and putting them into the other. Thus, 15+7 is the same as 12+10o, since 12 is 3 less than x 5, and io is three more than 7. This was the principle upon which the whole of the process in (29) was conducted. 36. Let a and b stand for two numbers, as in (24). It is impossible to tell what their sum will be until the numbers themselves are known. In the mean while a+b stands for this sum. To say, in algebraical 19 ADDITION. § 36-40. language, that the sum of a and b is not altered by adding c to a, provided we take away c from b, we have the following equation: (a+c)+(b--c) = a+b ; which may be written without brackets, thus, a+c+b-c= a+b. For the meaning of these two equations will appear to be the same, on consideration. 37. If a be taken twice, three times, &c., the results are represented in algebra by za, 3a, 4a, &c. The sum of any two of this series may be expressed in a shorter form than by writing the sign + between them; for though we do not know what number a stands for, we know that, be it what it may, za+za=4a, 3a+za=5Sa,4a+9a=13a ; and generally, if a taken m times be added to a taken n times, the result is a taken m+n times, or ma+na = (m+n)a. 38. The use of the brackets must here be noticed. They mean, that the expression contained inside them must be used exactly as a single letter would be used in the same place. Thus, pa signifies that a is taken p times, and (m+n)a, that a is taken m+n times. It is, therefore, a different thing from m+na, which means that a, after being taken n times, is added to m. Thus (3+4)xz is 7x2 or 14 ; while 3 + 4xz is 3+8, or I I. 39. When one number is taken away from another, the number which is left is called the difference or remainder. The process of finding the difference is called SUBTRACTION. The number which is to be taken away must be of course the lesser of the two. 40. The process of subtraction depends upon these two principles. I. The difference of two numbers is not altered by adding a number to the first, if you add the same number to the second; or by subtracting a number from the first, if you subtract the same number from the second. Conceive two baskets with pebbles in them, in the first of which are ioo pebbles more than in the second. If I put 50 more 20 PRINCIPLES OF ARITHMETIC. § 40-41. pebbles into each of them, there are still only ioo more in the first than in the second, and the same if I take 50 from each. Therefore, in finding the difference of two numbers, if it should be convenient, I may add any number I please to both of them, because, though I alter the numbers themselves by so doing, I do not alter their difference. II. Since 6 exceeds 4 by 2, and 3 exceeds z by i, and 12 exceeds 5 by 7, 6, 3, and 12 together, or z I, exceed 4, 2, and 5 together, or 11, by 2, I, and 7 together, or o0: the same thing may be said of any other numbers. 41. If a, b, and c be three numbers, of which a is greater than b (40), I. leads to the following, (a+c)-(b+c)= a-b. Again, if c be less than a and b, (a-c)-(b-c) = a-b. The brackets cannot be here removed as in (36). is not the same thing as p-q-r. That is, p-(q-r) For, in the first, the difference of q and r is subtracted from p; but in the second, first q and then r are subtracted from p, which is the same as subtracting as much as q and r together, or q+r. Therefore p-q-r is p-(q+r). In order to shew how to remove the brackets from p-(q-r) without altering the value of the result, let us take the simple instance Iz-(8-5). 8 from 12, or form If we subtract 2--8, we subtract too much; because it is not 8 which is to be taken away, but as much of 8 as is left after diminishing it by 5. In forming I2-8 we have therefore subtracted 5 too much. This must be set right by adding 5 to the result, which gives 12-8+ for the value of 12-(8-5). 5 The same reasoning applies to every case, and we have therefore, p -- (q+r) =p-q-r. p-(q-r) = p-q+r. By the same kind of reasoning, 21 SUBTRACTION. § 41-43* a-(b+c-d-e)= a-b--c+d+e. 2a+3b-(a--zb) = za+3b-a+zb = a+5b. 4+g--(x7-g9y) = 4m+g-17+9!y = y-3x. I 42. I want to find the difference of the numbers 57762 and 34631. Take these to pieces as in (29) and 57762 is 5 ten-th. 7 th. 7 hund. 6 tens and z units. 34631 is 3 ten-th. 4 th. 6 hund. 3 tens and I unit. Now 2 units exceed . . . 6 tens ....... 7 hundreds I unit ..... 3 tens . . by i unit. .. 6 hundreds . 3 tens. . . . . . I hundred. 4 thousands . 7 thousands.... 5 ten-thousands . . . 3 thousands. 3 ten-thous. ... z ten-thous. Therefore, by (40, Principle II.) all the first column together exceeds all the second column by all the third column, that is, by 2 ten-th. which is 23131. or 57762-3463 3 th. x hund. 3 tens and I unit, Therefore the difference of 57762 and 34631 is 23131, = 23131. 43. Suppose I want to find the difference between 61274 and 39628. Write them at length, and 61274 is 6 ten-th. a th. 2 hund. 7 tens and 4 units. 39628 is 3 ten-th. 9 th. 6 hund. 2 tens and 8 units. If we attempt to do the same as in the last article, there is a difficulty immediately, since 8, being greater than 4, cannot be taken from it. But from (40) it appears that we shall not alter the difference of two numbers if we add the same number to both of them. Add ten to the first number, that is, let there be 14 units instead of four, and add ten also to the second number, but instead of adding ten to the number of units, add one to the number of tens, which is the same thing. numbers will then stand thus, 6 ten-thous. I thous. z hund. 7 tens and 14 units.* 3 ten-thous. 9 thous. 6 hund. 3 tens and 8 units. * Those numbers which have been altered are put in italics. The 22 PRINCIPLES OF ARITHMETIC. §43-45* You now see that the units and tens in the lower can be subtracted from those in the upper line, but that the hundreds cannot. To remedy this, add one thousand or Io hundred to both numbers, which will not alter their difference, and remember to increase the hundreds in the upper line by Io, and the thousands in the lower line by I, which are the same things. And since the thousands in the lower cannot be subtracted from the thousands in the upper line, add I ten thousand or io thousand to both numbers, and increase the thousands in the upper line by io, and the ten thousands in the lower line by I, which are the same things; and at the close the numbers which we get will be, 6 ten-thous. Ix thous. Iz hund. 7 tens and 4 ten-thous. Io thous. 14 units. 6 hund. 3 tens and 8 units. These numbers are not, it is true, the same as those given at the beginning of this article, but their difference is the same, by (40). With the last-mentioned numbers proceed in the same way as in (42), which will give, as their difference, z ten-thous. I thous. 6 hund. 4 tens, and 6 units, which is 21646. 44. From this we deduce the following rules for subtraction : I. Write the number which is to be subtracted (which is, of course, the lesser of the two, and is called the subtrahend) under the other, so that its units shall fall under the units of the other, and so on. II. Subtract each figure of the lower line from the one above it, if that can be done. Where that cannot be done, add ten to the upper figure, and then subtract the lower figure; but recollect in this case always to increase the next figure in the lower line by x, before you begin to subtract it from the upper one. 45. If there should not be as many figures in the lower line as in the upper one, proceed as if there were as many ciphers at the beginring of the lower line as will make the number of figures equal. do not alter a number by placing ciphers at the beginning of it. example, oo8 i8 is the same number as 8 18, for it means o ten-thous. o thous. 8 hunds. I ten and 8 units; You For § 45-46. 23 SUBTRACTION. the first two signs are nothing, and the rest is 8 hundreds, I ten, and 8 units, or 8 8. The second does not differ from the first, except in its being said that there are no thousands and no tens of thousands in the number, which may be known without their being mentioned at all. You may ask, perhaps, why this does not apply to a cipher placed in the middle of a number, or at the right of it, as, for example, in 28007 and 39700 ? But you must recollect, that if it were not for the two ciphers in the first, the 8 would be taken for 8 tens, instead of 8 thousands; and if it were not for the ciphers in the second, the 7 would be taken for 7 units, instead of 7 hundreds. EXAMPLE. 46. What is the difference between 3708291640030174 and Difference 30813649276188 3677477990753986 EXERCISES. I. What is 18337+149263200-6472902 ?-Answer 142808635What is 1000-464+3279-646 ?-Ans. 3169. II. Subtract 64+76+144-18 from 33 -2+1ooo37.--Ans. 99802. III. What shorter rule might be made for subtraction when all the figures in the upper line are ciphers except the first? for example, in finding 10000000-2731634. IV. Find 18362+2469 and 18362-2469, add the second result to the first, and then subtract 18362; subtract the second from the first, and then subtract 246 9 .- Answer 18362 and 2469. V. There are four places on the same line in the order A, B, C, and D. From A to D it is 1463 miles; from A to c it is 728 miles; and from B to D it is 1317 miles. How far is it from A to B, from B to C, and from c to D ?-Answer. From A to B 146, from B to C 582, and from c to D 735 miles. 24 PRINCIPLES OF ARITHMETIC. § 46-49* VI. In the following table subtract B from A, and B from the reFind how mainder, and so on until s can be no longer subtracted. many times B can be subtracted from A, and what is the last remainder. B A 23604 . . . 9999 . . 209961 No. of times. . 37173 2 5 . . Remainder. . 3606 24096 74712 6792 11 o 4802469 654321 7 222222 18849747 3141592 6 195 987654321 123456789 8 9 SECTION III. MULTIPLICATION. 47. I have said that all questions in arithmetic require nothing but addition and subtraction. I do not mean by this that no rule should ever be used except those given in the last section, but that all other rules only shew shbrter ways of finding what might be found, if we pleased, by the methods there deduced. Even the last two rules themselves are only short and convenient ways of doing what may be done with a number of pebbles or counters. 48. I want to know the sum of five seventeens, or I ask the 17 following question: There are five heaps of pebbles, and seven- 17 teen pebbles in each heap; how many are there in all ? 17 five seventeens in a column, and make the addition, which gives 17 85. 17 process of finding the product is called MULTIPLICATION, which 85 gives nothing more than the addition of a number of the same quantities. Write In this case 85 is called the product of 5 and 17, and the Here 17 is called the multiplicand, and 5 is called the multiplier. 49. If no question harder than this were ever proposed, there would be no occasion for a shorter way than the one here followed. But if 25 MULTIPLICATION. § 49-51. there were 2367 heaps of pebbles, and 429 in each heap, the whole number is then 1367 times 429, or 429 multiplied by 1367. I should have to write 429 1367 times, and then to make an addition of enorTo avoid this, a shorter rule is necessary, which I now mous length. proceed to explain. 50. The student must first make himself acquainted with the products of all numbers as far as io times io by means of the following table,* which must be committed to memory. i i 1 5 7 9 II I2 22 IO 24 2 3 4 4 6 8 I0 I2 3 6 9 12 15 18 21 24. 27 30 33 4 8 15 16 20 24 28 32 36 40 44 45 50 55 6o 30 I 2 6 14 8 IS i6 36 48 5 I0 15 20 25 6 xz 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 x6 24 32 40 48 56 64 72 80 88 96 9 x8 27 36 45 54 63 7'$ 8I 90 99 Io8 20 30 10 II 12 35 40 40 50 60 70 80 90 100 110 120 22 33 44 55 1 -- 66 77 88 99 Io 1z 132 24 36 48 60 72 84 96 Io8 ;: " -- 120 232 I44 If from this table you wish to know what is 7 times 6, look in the first upright column on the left for either of them; 6 for example. Proceed to the right until you come into the column marked 7 at the top. You there find 42, which is the product of 6 and 7. 51. You may find, in this way, either 6 times 7, or 7 times 6, and for both you find 42. That is, six sevens is the same number as seven * As it is usual to learn the product of numbers up to 12 times 12, I have extended the table thus far. In my opinion, all pupils who shew a tolerable capacity should slowly commit the products to memory as far as 20 times 20, in the course of their progress through this work. 26 PRINCIPLES OF ARITHMETIC. sixes. § 51-54. This may be shewn as follows: Place seven counters in a line, and repeat that line in all six times. The number of counters in the whole is 6 times 7, or six sevens, if I reckon the rows from the top to . . . the bottom; but if I count the rows that stand . . row, the whole number of which is 7 times 6, or side by side, I find seven of them, and six in each . . ,seven And the whole number is 42, which- sixes. ever way I count. . may be The same method applied to any other two numbers. .. If the signs of (23) were used, it would be said that 7x6= 6x7. 52. To take any quantity a number of times, it will be enough to take every one of its parts the same number of times. Thus, a sack of corn will be increased fifty-fold, if each bushel which it contains be replaced by 50 bushels. A country will be doubled by doubling every acre of land, or every county, which it contains. Simple as this may appear, it is necessary to state it, because it is one of the principles on which the rule of multiplication depends. 53. In order to multiply by any number, you may multiply separately by any parts into which you choose to divide that number, and add the results. For example, 4 and z make 6. To multiply 7 by 6, first multiply 7 by 4, and then by z, and add the products. give 42, which is the product of 7 and 6. This will Again, since 57 is made up of 32 and 25, 57 times 50 is made up of 32 times 50o and 25 times 50, and so on. If the signs were used, these would be written thus : 7x6 = 7x4 +7x2. 50x32+50x25. 50x57 = 54. The principles in the last two articles may be expressed thus: If a be made up of the parts x, y, and z, ma is made up of my, mae, and mz ; or, if a = x +y +z. ma = mx+my+mz. or, m(x+y+z) = mx+my+mz. A similar result may be obtained if a, instead of being made up of 27 MULTIPLICATION. 1 54-55. X, y, and z, is made by combined additions and subtractions, such as x+y-z, x-y+z, x-y-z, &c. To take the first as an instance: Let a=, then ma = +y -. mx+my--rz. For, if a had been x+y, ma would have been mx+my. But since a is less than x+y by z, too much by z has been repeated every time that x+y has been repeated;-that is, mz too much has been taken; consequently, ma is not mr+my, but mx+my-m. Similar reason- ing may be applied to other cases, and the following results may be obtained : m(a+b+o--d) ma+mb+mo-md. a(a-b) = aa-ab. 7a(7+zb) = 49a+I4ab. b(a-b) = ba-bb. (aa+a+x)a = aaa+aa+a. 3(za-4b) -- 6a-zb. (3ab-zc)4abo = xzaabbc-8abcc. 55. There is another way in which two numbers may be multiplied together. Since 8 is 4 times 2, 7 times 8 may be made by multiplying 7 and 4, and then multiplying that product by z. To shew this, place 7 counters in a line, and repeat that line in all 8 times, as in figures I. and II. II. I. • i • , . B B • . , . * . . ° . . . • " • • . • , . • • . ° , , . • The number of counters in all is 8 times 7, or 56. But (as in fig. I.) enclose each four rows in oblong figures, such as A and B. The num- 28 PRINCIPLES OF ARITHMETIC. § 55-57* ber in each oblong is 4 times 7, or 28, and there are two of those oblongs; so that in the whole the number of counters is twice 28, or z8xz, or 7 first multipled by 4, and that product multiplied by 2. In figure II. it is shewn that 7 multiplied by 8 is also 7 first multiplied by 2, and that product multiplied by 4. applied to other numbers. The same method may be Thus, since 8o is 8 times o10, 256 times 80o is 256 multiplied by 8, and that product multiplied by xo. If we use the signs, the foregoing assertions are made thus : 7x8 = 7xx4. 7x4xz = 256x80 = z56x8xIo = 256xxox8. EXERCISES. Shew that 2x3x4x5 - 2x4x3x5 = 5x4x23, &c. Shew that 18x1oo = iSx57+18x43. 56. Articles (51) and (55) may be expressed in the following way, where by ab we mean a taken b times; by abc, a taken b times, and the result taken c times. ab -= ba. abc = acb = bca = bac, &c. abc = ax(bc) = bx(ca) = ex (ab). If we would say that the same results are produced by multiplying by b, c, and d, one after the other, and by the product bed at once, we write the following: axbxcxd = axbcd. The fact is, that if any numbers are to be multiplied together, the product of any two or more may be formed, and substituted instead of those two or more; thus, the product abcdef may be formed by multiplying ab cde f abf de c abc def 57. In order to multiply by the multiplicand. &c. 1o, annex a cipher to the right hand of Thus, Io times 2356 is 23560. 2356 at length, which is To shew this, write 29 MULTIPLICATION. § 57-58. 2 thousands, 3 hundreds, 5 tens, and 6 units. Take each of these parts ten times, which, by (52), is the same as multiplying the whole number by xo, and it will then become 2 tens of thou. 3 tens of hun. 5 tens of tens, and 6 tens, which is 2 ten-thou. 3 thous. 5 hun. and 6 tens. This must be written 23560, because 6 is not to be 6 units, but 6 tens. Therefore 2356xo0 = 23560. In the same way you may shew, that in order to multiply by ioo you must affix two ciphers to the right; to multiply by iooo you must affix three ciphers, and so on. The rule will be best caught from the following table: 130 13X 10 = 13X 100= 142X 1000= 142000 1300 23700x 10= 237000 3040X 1000= 3040000 zOOoxIC O = oo00 400po0co 1 13X 1000= 13000 13xo10000= 130000 1 58. I now shew how to multiply by one of the numbers, 2, 3, 4, 5, I do not include because multiplying by i, or taking x, 6, 7, 8, or 9. the number once, is what is meant by simply writing down the number. I want to multiply 1368 by 8. Write the first number at full length, which is I thousand, 3 hundreds, 6 tens, and 8 units. To multiply this by 8, multiply each of these parts by 8 (50) and (52), which will give 8 thousands, 24 hundreds, 48 tens, and 64 units. Now 64 units are written thus 48 tens . 24 hundreds 8 thousands . .. ....... 64 . . ...... . . 480 2400 . . . . 8000 Add these together, which gives 10944 as the product of 1368 and 8, or 1368x8= o0944. By working a few examples in this way you will see the following rule. D2 § 59-62. PRINCIPLES OF ARITHMETIC. 30 59. I. Multiply the first figure of the multiplicand by the multiplier, write down the units' figure, and reserve the tens. II. Do the same with the second figure of the multiplicand, and add to the product the number of tens from the first ; put down the units' figure of this, and reserve the tens. III. Proceed in this way till you come to the last figure, and then write down the whole number obtained from that figure. IV. If there be a cipher in the multiplicand, treat it as if it were x= o, ox2 = o, &c. a number, observing that oxI 60. In a similar way a number can be multiplied by a figure which is accompanied by ciphers, as, for example, 8oo0. For 800o is 8xc000o, and therefore (55) you must first multiply by 8 and then by Ioco, which last operation (57) is done by placing 3 ciphers on the right. Hence the rule in this case is, Multiply by the simple number, and place the number of ciphers which follow it at the right of the product. EXAMPLE. Multiply 1679423800872 6oooo by 100765428052320000 61. What is EXERCISES. I007360x7 ? Answer, 7051520. 123456789x9+IO and 123x9+4 ?-Ans. III1IIIIII and 1111. What is 136x3+I29x4+147x8+27x3000 ?-Ans. 83100. An army is made up of 33 regiments of infantry, each containing 8o0 men; 14 of cavalry, each containing 6oo men; and 2 of artillery, each containing 300 men. The enemy has 6 more regiments of infantry, each containing Too more men; 3 more regiments of cavalry, each containing Ioo men less; and 4 corps of artillery of the same magnitude as those of the first: two" regiments of cavalry and one of infantry desert from the former to the latter. How many men has the second army more than the first ?-Answer, 13400. 62. Suppose it required to multiply 23707 by 4567. Since 4567 is 31 MULTIPLICATION. S62-64. made up of 4000, 500, 60, and 7, by (53) we must multiply 23707 by each of these, and add the products. Now (58) 23707x 7 is 165949 (60) 23707x 60 is 1422420 500 is 11853500 23707x4000 is 94828000 x 23707 o108269869 The sum of these is which is the product required. It will do as well if, instead of writing the ciphers at the end of each line, we keep the other figures in their places without them. If we take away the ciphers, the second line is one place to the left of the first, the third one place to the left of the second, and so on. Write the multiplier and the multiplicand over these lines, and the process will stand thus : 23707 63. There is one more case to be noticed; that is, 4567 where there is a cipher in the middle of the multiplier. 165949 142242 The following example will shew that in this case nothing more is necessary than to keep the first figure 118535 of each line in the column under the figure of the 94828 multiplier from which that line arises. quired to multiply 365 by xozool. 108269869 made up of xooooo, 1ooo and 1. Suppose it re- The multiplier is Proceed as before, and is 365 is 365000 365x100000 is 36500000 The sum of which is 36865365 365x1 (57) 365/1000 and the whole process with the ciphers struck off is: 365 o00oo0 365 365 64. The following is the rule in all cases: I. Place the multiplier under the multiplicand, so that the units of one may be under those of the other. II. Multiply the whole multiplicand by each figure 365 of the multiplier (59), and place the unit of each line in 36865365 the column under the figure of the multiplier from which it came. 32 PRINCIPLES OF ARITHMETIC. § 64-66. III. Add together the lines obtained by II. column by column. 65. When the multiplier or multiplicand, or both, have ciphers on the right hand, multiply the two together without the ciphers, and then place on the right of the product all the ciphers that are on the right both of the multiplier and multiplicand. x13000? For example, what is 3200 First, 3200 is 32X100, or one hundred times as great as 32. Again, 32x13000 is 32x13, with three ciphers affixed, that is 416, with three ciphers affixed, or 416000. 1oo But the product required must be times as great as this, or must have two ciphers affixed. It is therefore 41600000, having as many ciphers as are in both multiplier and multiplicand. 66. When any number is multiplied by itself any number of times, the result is called a power of that number. Thus: 6 is called the first power of 6 6x6 . . second power of 6 6x6x6 . third 6x6x6x6 . fourth power of 6 power of 6 &c. &c. The second and third powers are usually called the square and cube, which are incorrect names, derived from certain connexions of the second and third power with the square and cube in geometry. cises in multiplication, the following powers are to be found. Number pro Dposed. Cube. Square. 918330048 xoo8 944784 1016064 1024192512 3142 9872164 31018339288 3163 X0004569 31644451747 5555 6789 30858025 171416328875 46090521 312908547069 972 The fifth power of 36 ... fourth 50 ... fourth . . . . 277 ... ro8 ... fourth is 60466176 6250000 136048896 ... 5887339441 As exer- 33 MULTIPLICATION. § 67-68. 67. It is required to multiply a+b by c+d, that is, to take a+b as many times as there are units in c+d. By (53) a+b must be taken c times, and d times, or the product required is (a+b)c+(a+b)d. But (52) (a+b)o is ac+bo, and (a+b)d is ad+bd; whence the product required is ac+bc+ad+bd;or, (a+b)(c+d) = ac+bc+ad+bd. By similar reasoning (a-b)(c+d) is (a-b)c+(a--b)d; or, (a-b)(c+d) = ac-bc+ad-bd. To multiply a-b by c-d, first take a-b c times, which gives ac-bc. This is not correct; for in taking it c times instead of c-d times, we have taken it d times too many ; or have made a result which is (a-b)d too great. The real result is therefore ac--bc-(a-b)d. But (a-b)d is ad-bd, and therefore (a-b)(c-d) = ac-bc-(ad-bd) = ac--b-ad+bd (41) From these three examples may be collected the following rule for the multiplication of algebraic quantities: Multiply each term of the multiplicand by each term of the multiplier; when the two terms have both + or both - before them, put + before their product; when one has + and the other -, put - before their product. In using the first terms, which have no sign, apply the rule as if they had the sign +. 68. For example, (a+b)(a+b) gives aa+ab+ab+bb. But ab+ab is nab; hence the square of a+b is aa+zab+bb. Again (a-b)(a-b) gives aa-ab-ab+bb. But two subtractions of ab are equivalent to subtracting zab ; hence the square of a-b is aa-zab+bb. Again, (a+b)(a-b) gives aa+ab-ab-bb. But the addition and subtraction of ab makes no change; hence the product of a+b and a-b is aa-bb. Again, the square of a+b+c+d or (a+b+c+d)(a+b+c+d)will be found to be aa+zab+zac+2ad+bb+zbc+zbd+cc+zcd+dd; the rule for squaring or such a quantity is: Square the first term, and multiply all that come after by twice that term; do the same with the second, and so on to the end. 34 § 69- 7 I. PRINCIPLES OF ARITHMETIC. SECTION IV. DIVISION. 69. Suppose I ask whether I56 can be divided into a number of parts each of which is 13, or how many thirteens 156 contains; I propose a question, the solution of which is called DIVISION. In this case, 156 is called the dividend, 13 the divisor, and the number of parts required is the quotient; and when I find the quotient, I am said to divide 156 by 13. 70. The simplest method of doing this is to subtract I3 from i56, and then to subtract I3 from the remainder, and so on; or, in common language, to tell of I56 by thirteens. A similar process has already occurred in the exercises on subtraction, Art. (46). Do this, and mark one for every subtraction that is made, to remind you that each subtraction takes 13 once from 156 13 I 143 13 I 130 I3 1 117 13 i 104 13 156, which operations will stand as follows: Begin by subtracting I3 from I56, which leaves 130; 143. Sub- 13 tract 13 from 143, which leaves and so on. At last only remains, from which when 13 is subtracted, there remains nothing. Upon counting the number of times which you have subtracted 13, you find that this number is Iz; or I56 contains twelve thirteens, or contains 13 twelve times. I This method is the most simple possible, and might be done 3 78 13 I 65 13 I with pebbles. Of these you would first count 156. You would then take 13 from the heap, and put them into one heap by 53 there were none left. 39 13 I heaps, which you would find to be I2. 26 themselves. You would then take another 13 from the heap, and place them in another heap by themselves; and so on until You would then count the number of 13 71. Division is the opposite of multiplication. In multiplication you have a number of heaps, with the same number 13 3 o of pebbles in each, and you want to know how many pebbles there are in all. In division you know how many there are there are in all. In division you know how many there are -. 35 DIVISION. § 71-74 in all, and how many there are to be in each heap, and you want to know how many heaps there are. 72. In the last example a number was taken which contains an exact number of thirteens. ber. But this does not happen with every numFollow the process of (70), and it will Take, for example, 159. appear that after having subtracted I3 twelve times, there remains 3, from which 13 cannot be subtracted. We may say then that 159 con- tains twelve thirteens and 3 over; or that I59, when divided by 13, gives a quotient 12, and a remainder 3. 159 = If we use signs, 13X12+3. EXERCISES. 146 = 24x6+2 , or 146 contains six twenty-fours and 2 over. 146 = 6 x24+2, or 146 contains twenty-four sixes and 2 over. 300 = 42x7+6, or 300 contains seven forty-twos and 6 over. 39624 = 7277x5+3239. 73. If a contain b q times with a remainder r, a must be greater than bq by r; that is, a = bq+r. If there be no remainder, a = bq. Here a is the dividend, b the divisor, q the quotient, and r the remainder. In order to say that a contains b q times, we write, b= q, or a b = q, which in old books is often found written thus: a+b = q. 74. If I divide 156 into several parts, and find how often 13 is contained in each of them, it is plain that i56 contains 13 as often as all its parts together. For example, 156 is made up of 91, 39, and 26. Of these, 9 contains 13 7 times, 39 contains 13 3 times, 26 contains 13 2 times; therefore 91+39+26 contains 13 7+3+2 times, or 12 times. Again, 156 is made up of 100, 50, and 6. 36 PRINCIPLES OF ARITHMETIC. Now Ioo contains 13 7 times and 9 over, 50 contains 13 3 times and 6 contains Therefore 13 ii over, o times* and 6 over. xoo+5o+6 contains 13 7+3+0 times and 9+11+6 over; or But 26 is itself 2 thirteens; 156 contains 13 Io times and z6 over. therefore S74-76. 156 contains Io thirteens and z thirteens, or iz thirteens. 75. The result of the last article is expressed by saying, that if a b e d a=b+c+d, then - = -- +-+-- m m m m 76. In the first example I did not take away 13 more than once at a time, in order that the method might be as simple as possible. But if I know what is twice 13, 3 times 13, &c., I can take away as many thirteens at a time as I please, if I take care to mark at each step how many I take away. For example, take away 13 ten times at once from 156, that is, take away I30, and afterwards take away 13 twice, or take away z6, and the process is as follows : 156 130 zo times 13. 26 26 2 times 13. Therefore 156 contains 13 10+2, or 12 times. Again, to divide 3096 by 1800 8. Therefore 3096 contains 3096 zoo times 18. 8 oo00+50+20o+, or 172 times. 77. You will now understand the following 1296 900oo So times i8. sentences, and be able to make similar assertions 396 of other numbers. 360 zo times 18. 36 450 is 75x6; it therefore contains any number, as 5, 6 times as often as 75 contains it. 36 2 times 1S. * To speak always in the same way, instead of saying that 6 does not contain 13, I say that it contains it 0 times and 6 over, which is merely saying that 6 is 6 more than nothing. § 77-78. 37 DIVISION. 135 Twice i35 26 times; therefore, 3 52 or 3 . a twice 26 Io times 135 3 26o or Io times 26 135 3 1300 or 50 times 26 50 times 47z contains 18 .- 21 times; therefore, more than 4720 contains i8 more than zzo times, 47200 contains I8 more than 00oo times, 472000 contains 18 more than 2100ooo times, 112 32 320 3 z 3200 30 20o I2 oo00 00o 300 ( 6 Iz 2000 3000 &c. 32000 &c. &c. 78. The foregoing articles contain the principles of division. The question now is, to apply them in the shortest and most convenient way. Suppose it required to divide 4068 by 18, or to find4068 (23). i8 If we divide 4068 into any number of parts, we may, by the process followed in (74), find how many times 18 is contained in each of these parts, and from thence how many times it is contained in the whole. Now, what separation of 4068 into parts will be most convenient ? Observe that 4, the first figure of 4068, does not contain 18; but that 40, the first and second figures together, does contain 18 more than twice, but less than three times." But 4068 (20) is made up of 40 hundreds, and 68 ; of which, 40 hundreds (77) contains 18 more than zoo times, and less than 300 times. Therefore, 4068 also contains more than z00 times i8, since it must contain 18 more times than 4000 does. It also contains I8 less than 300 times, because 300 times 18 is 5400, a greater number than 4068. Subtract IS zoo times from 4068; that is, subtract 3600, and there remains 468. Therefore, 4c68 contains 18 zoo times, and as many more times as 468 contains 18. It remains, then, to find how many times 468 contains 18. Proceed * If you have any doubt as to this expression, recollect that it means "contains more than two eighteens, but not so much as three." E 38 PRINCIPLES OF ARITHMETIC. exactly as before. Observe that 46 contains i8 more than twice, and S78, less than 3 times; therefore, 460 contains it more than 20, and less than 30 times (77) ; as does also 468. Subtract I8 20 times from 468, that is, subtract 360; the remainder is Io8. 18 Therefore, 468 contains zo times, and as many more as io8 contains it. Now, io8 is found to contain IS 6 times exactly ; therefore, 468 contains it zo+6 times, and 4068 contains it 20o+2o+6 times, or 226 times. If we write down the process that has been followed, without any explanation, putting the divisor, dividend, and quotient, in a line separated by parentheses, it will stand, as in example (A). Let it be required to divide 36326599 by 1342 (B). B. 1342)36326599(20000+7000+60+9 26840000 A. 9486599 9394000 18)4068(200+20+6 3600 92599 468 80520 360 1zo79 io8 o08 12078 I 0 As in the previous example, 36326599 is separated into 36320000 and 6599; the first four figures 3632 being separated from the rest, because it takes four figures from the left of the dividend to make a number which is greater than the divisor. Again, 36320000 is found to contain 1342 more than 20000, and less than 30000 times; and 1342 20000 is subtracted from the dividend, after which the remainder is 9486599. The same operation is repeated again and again, and the result is found to be, that there is a quotient 2oooo+7000+60+9, or 27069, and a remainder I. Before you proceed, you should now repeat the foregoing article at length in the solution of the following questions. 10093874 3207 66779922 What are 2718218 1114433 13352 39 DIVISION. § 78-8o, the quotients of which are 3147, 583, 203; and the remainders 1445, 65483, 7762. 79. In the examples of the last article, observe, ist, that it is useless to write down the ciphers which are on the right of each subtrahend, provided that without them you keep each of the other figures in its proper place: zd, that it is useless to put down the right-hand figures of the dividend so long as they fall over ciphers, because they do not begin to have any share in the making of the quotient until, by continuing the process, they cease to have ciphers under them: 3d, that the quotient is only a number written at length, instead of the usual way. For example, the first quotient is is 20000+7000+60+9, or 27069. 2oo+2o+6, or 226; the second Strike out, therefore, all the ciphers and the numbers which come above them, except those in the first line, and put the quotient in one line; and the two examples of the last article will stand thus: 18)4068(226 36 1342)36326599(27069 2684 9486 46 36 9394 io8 108 0 9259 8052 I2079 12078 80. Hence the following rule is deduced: I. Write the divisor and dividend in one line, and place parentheses on each side of the dividend. II. Take off from the left hand of the dividend the least number of figures which make a number greater than the divisor ; find what number of times the divisor is contained in these, and write this number as the first figure of the quotient. III. Multiply the divisor by the last-mentioned figure, and subtract the product from the number which was taken off at the left of the dividend. 40 § 8-8 I. PRINCIPLES OF ARITHMETIC. IV. On the right of the remainder place the figure of the dividend which comes next after those already separated in II.: if the remainder thus increased be greater than the divisor, find how many times the divisor is contained in it; put this number at the right of the first figure of the quotient, and repeat the process: if not, on the right place the next figure of the dividend, and the next, and so on until it is greater ; but remember to place a cipher in the quotient for every figure of the dividend which you are obliged to take, except the first. V. Proceed in this way until all the figures of the dividend are exhausted. In judging how often one large number is contained in another, a first and rough guess may be made by striking off the same number of figures from both, and using the results instead of the numbers themselves. Thus, 4,732 is contained in 14,379 about the same number of times that 4 is contained in 14, or about 3 times. The reason is, that 4 being contained in 14 as often as 4000 is in 14000, and these last only differing from the proposed numbers by lower denominations, viz. hundreds, &c. we may expect that there will not be much difference between the number of times which 14000 contains 4000, and that which 14379 contains 4732: and it generally happens so. But if the second figure of the divisor be 5, or greater than 5, it will be more accurate to increase the first figure of the divisor by x, before trying the method just explained. Nothing but practice can give facility in this sort of guess-work. 81. This process may be made more simple when the divisor is not greater than iz, if you have sufficient knowledge of the multiplication table (50). For example, I want to divide 132976 by 4. the process stands thus : At full length § 8, 41 DIVISION. But you will recollect, without the necessity of 4)132976(33244 I2 writing it down, that 13 contains 4 three times with I2 a remainder x; this i you will place before 2, the 12 next figure of the dividend, and you know that iz 9 8 contains 4 3 times exactly, and so on. It will be more convenient to write down the quotient thus : 17 4)132976 33244 I6 i6 While on this part of the subject, we may mention, that the shortest way to multiply by 5 is to o annex a cipher and divide by 2, which is equivalent to taking the half of io times, or 5 times. To divide by 5, multiply by 2 and strike off the last figure, which leaves. the quotient ; half the last figure is the remainder. divide by 4. To multiply by 25, annex two ciphers and To divide by 25, multiply by 4 and strike off the last two figures, which leaves the quotient; one fourth of the last two figures, taken as one number, is the remainder. To multiply a number by 9, annex a cipher, and subtract the number, which is equivalent to taking the number ten times, and then subtracting it once. To mul- tiply by 99, annex two ciphers and subtract the number, &c. In order that a number may be divisible by 2 without remainder, its units' figure must be an even number.* That it may be divisible by 4, its last two figures must be divisible by 4. Take the example 1236: this is composed of Iz hundreds and 36, the first part of which, being hundreds, is divisible by 4, and gives Iz twenty-fives; it depends then upon 36, the last two figures, whether 1236 is divisible by 4 or not. A number is divisible by 8 if the last three figures are divisible by 8 ; for every digit, except the last three, is a number of thousands, and looo is divisible by 8; whether therefore the whole shall be divisible by 8 or not depends on the last three figures: thus, divisible by 8, since 946 is not so. only when the sum of its digits is divisible by 3 or 9. 1234 ; this is * Among the even figures we include 0. F2 127946 is not A number is divisible by 3 or 9 Take for example 42 PRINCIPLES OF ARITHMETIC. § 81-83. 999 and x I thousand, or 2 hundred, or 3 tens, or three times 9 and 3 and 4 or . twice 99 and 2 . ...... 4 Now 9, 99, 999, &c. are all obviously divisible by 9 and by 3, and so will be any number made by the repetition of all or any of them any number of times. digits, whether It therefore depends on 1+2+3+4, or the sum of the z234 shall be divisible by 9 or 3, or not. From the above we gather, that a number is divisible by 6 when it is even, and when the sum of its digits is divisible by 3. Lastly, a number is divi- sible by 5 only when the last figure is o or 5. 82. Where the divisor is unity followed by ciphers, the rule becomes extremely simple, as you will see by the following examples: This is, then, the rule: Cut off as many 100)33429(334 figures from the right hand of the dividend 300 342 300 429 as there are ciphers. These figures will be the remainder, and the rest of the dividend will be the quotient. 400 Or we may prove these results thus: from 29 (20), 2717316 is 271731 tens and 6; of which the first contains 1o 271731 times, and the second not at all; the quotient is therefore Io)27173x6 271731 and rem. 6. 271731, and the remainder 6 (72). Again (20), 33429 is 334 hundreds and 29; of which the first contains 1oo 334 times, and the second not at all; the quotient is therefore 334, and the remainder 29. 83. The following examples will shew how the rule may be shortened when there are ciphers in the divisor. With each example is placed another containing the same process, all unnecessary figures being removed; and from the comparison of the two, the rule at the end of this article is derived. S83-84. I. 43 DIVISION. 1782000)6424700000(3605 1782)6424700(3605 5346000 5346 10787000 10787 10692000 10692 9500000 8910000 9500 8910 590000 II. 590000 1o300000)476189300oo(3428 123)421761(3428 36900000 369 52761893 527 49200000 492 356 35618930 24600000 246 110189300 98400000 984 11789300 11789300 The rule, then, is : Strike out as many figures* from the right of the dividend as there are ciphers at the right of the divisor. Strike out all the ciphers from the divisor, and divide in the usual way; but at the end of the process place on the right of the remainder all those figures which were struck out of the dividend. 84. EXERCISES. Divisor. Dividend. 9694 • 47 Quotient. 206 56 Remainder. I2 175618 3136 23796484 130000 183 6484 14002564 1871 7484 0 310314420 7878 39390 0 3939040647 6889 571787 4 22876792454961 43046721 531441 0 * Including both ciphers and others. 2 44 § 84-87. PRINCIPLES OF ARITHMETIC. Shew that I. 0ooxooxI0oo-43x43x43 100-43 -- ooxIoo+ zoox3+4343x43 ooxooxx00+43x43x43 II. 1oo+43 33 IOOX Ioo-100oox43+43x43. III. 76xx7+2x76x52+5x52 76+5 76+52 I2 XI2XI2XI--I IV. I+12+I2X12+12XI2XIz = x I2-I What is the nearest number to 1376429 which can be divided by 36300 without remainder ?-Answer, 1379400. If 36 oxen can eat 216 acres of grass in one year, and if a sheep eat half as much as an ox, how long will it take 49 oxen and 136 sheep together to eat 17550 acres ?-Answer, 25 years. 85. Take any two numbers, one of which divides the other without remainder; for example, 32 and 4. Multiply both these numbers by any other number ; for example, 6. The products will be 192 and 24. Now, 192 contains 24 just as often as 32 contains 4. Suppose 6 baskets, each containing 32 pebbles, the whole number of which will be 192. Take 4 from one basket, time after time, until that basket is empty. It is plain that if, instead of taking 4 from that basket, I take 4 from each, the whole 6 will be emptied together : that is, 6 times 32 contains 6 times 4 just as often as 32 contains 4. The same reasoning applies to other numbers, and therefore we do not alter the quotient if we multiply the dividend and divisor by the same number. 86. Again, suppose that 200 is to be divided by 50. Divide both the dividend and divisor by the same number; for example, 5. 200 is 5 times 40, and 50 is 5 times io. Then, But by (85), 40 divided by zo gives the same quotient as 5 times 40 divided by 5 times io, and therefore the quotient of two numbers is not altered by dividing both the dividend and divisor by the same number. 87. From (55), if a number be multiplied successively by two others, it is multiplied by their product. Thus, 27, first multiplied by 5, and the product multiplied by 3, is the same as 27 multiplied by 5 times 3, or 15s Also, if a number be divided by any number, and the quotient 45 DIVISION. § 87-9. be divided by another, it is the same as if the first number had been divided by the product of the other two. For example, divide 6o by 4, which gives 15, and the quotient by 3, which gives 5. It is plain, that if each of the four fifteens of which 6o is composed be divided into three equal parts, there are twelve equal parts in all; or, a division by 4, and then by 3, is equivalent to a division by 4x3, or i2. 88. The following rules will be better understood by stating them If 32 be multiplied by 24 and divided by 6, the result in an example. is the same as if 32 had been multiplied by the quotient of 24 divided by 6, that is, by 4; for the sixth part of 24 being.4, the sixth part of any number repeated 24 times is that number repeated 4 times; or, multiplying by 24 and dividing by 6 is equivalent to multiplying by 4. 89. Again, if 48 be multiplied by 4, and that product be divided by z4, it is the same thing as if 48 were divided at once by the quotient of 24 divided by 4, that is, by 6. For, every unit which is repeated 6 times in 48 is repeated 4 times as often, or 24 times, in 4 times 48, or the quotient of 48 and 6 is the same as the quotient of 48x4 and 6x4. 90. The results of the last five articles may be algebraically expressed thus : ma mb a bb (85) b If n divide a and b without remainder, n=a b (86) (87) c b be n ab -= b ax- (88) cc - a b (89) It must be recollected, however, that these have only been proved in the case where all the divisions are without remainder. 91. When one number divides another without leaving any remainder, or is contained an exact number of times in it, it is said to be a measure of that number, or to measure it. Thus, 4 is a measure of 136, or measures 136; but it does not measure 137. The reason for 46 PRINCIPLES OF ARITHMETIC. § 91-95* using the word measure is this: Suppose you have a rod 4 feet long, with nothing marked upon it, with which you want to measure some length; for example, the. length of a street. happen to be 136 If that street should feet in length, you will be able to measure it with the rod, because, since I36 contains 4 34 times, you will find that the street is exactly 34 times the length of the rod. But if the street should happen to be 137 feet long, you cannot measure it with the rod ; for when you have measured 34 of the rods, you will find a remainder, whose length you cannot tell without some shorter measure. is said to measure 136, but not to measure 137. Hence 4 A measure, then, is a divisor which leaves no remainder. 92. When one number is a measure of two others, it is called a common measure of the two. and 75. Thus, 15 is a common measure of I8o Two numbers may have several common measures. example, 360 and several others. 168 For have the common measures 2, 3, 4, 6, 24, and Now, this question may be asked: Of all the common measures of 360 and i68, which is the greatest? The answer to this question is derived from a rule of arithmetic, called the rule for finding the GREATEST COMMON MEASURE, which we proceed to consider. 93. If one quantity measure two others, it measures their sum and difference. Thus, 7 measures 21 and 56. and 56-21, or 77 and 35. It therefore measures 56+z1 This is only another way of saying what was said in (74). 94. If one number measure a second, it measures every number 15 measures 30, 45, 60, 75, &c.; all which numbers are measured by 5. It is plain which the second measures. Thus, 5 measures I5, and that if 15 contains 5 30, or 3 times, I5+15 contains 5 3+3 times, or 6 times, 45, or 15+15+15 contains 5 3+3+3 or 9 times; and so on. 95. Every number which measures both the dividend and divisor measures the remainder also. To shew this, divide 360 by quotient is 3, and the remainder 24, 12. The that is (72) 360 is three times 112 47 § 95-98. DIVISION., and 24, or 360 = II2x3+24. From this it follows, that 24 is the differ- ence between 360 and 3 times Ixz, or 24 = 360-II2x3. Take any num- ber which measures both 360 and Ixz; for example, 4. Then 4 measures 360, 4 measures II2, and therefore (94) measures 112x3, or I12+12+x12. Therefore (93) it measures 360-Iz2x3, which is the remainder 24. The same reasoning may be applied to all other measures of 360 and xlz; and the result is, that every quantity which measures both -the dividend and divisor also measures the remainder. Hence, every common measure of a dividend and divisor is also a common measure of the divisor and remainder. 96. Every common measure of the divisor and remainder is also a common measure of the dividend and divisor. and recollect that 360 = I12x3+24. Take the same example, Take any common measure of the remainder 24 and the divisor I 12 ; for example, 8. Then 8 measures 24 ; and 8 measures 112, and therefore (94) measures i12x3 . Therefore (93) 8 measures I xx3+z4, or measures the dividend 360. Then every common measure of the remainder and divisor is also a common measure of the divisor and dividend, or there is no common measure of the remainder and divisor which is not also a common measure of the divisor and dividend. 97. I. It is proved in (95) that the remainder and divisor have all the common measures which are in the dividend and divisor. II. It is proved in (96) that they have no others. It therefore follows, that the greatest of the common measures of the first two is the greatest of those of the second two, which shews how to find the greatest common measure of any two numbers,* as follows: 98. Take the preceding example, and let it be required to find the g. c. m. of 360 and II2, and observe that * For shortness, I abbreviate the words greatest common measure into tieir initial letters, g. c. m. 48 PRINCIPLES OF ARITHMETIC. 360 divided by 112 § 98-99. xz gives the remainder 24, divided by 24 gives the remainder I6, 24 divided by I6 gives the remainder 16 divided by 8 gives no remainder. 8, Now, since 8 divides x6 without remainder, and since it also divides itself without remainder, 8 is the g. c. m. of 8 and I6, because it is impossible to divide 8 by any number greater than 8; so that, even if I6 had a greater measure than 8, it could not be common to I6 and 8. Therefore 8 is g. c. m. of i6 and (97) g.c. m.of I6 and 8isg. c.m. of 24.and 8, I6, g. c. m. of 24 and x6 is g. c. m. of II2 and 24, g. c. m. of I1x and 24 is g. c. m. of 360 and 112, Therefore 8 is g. c. m. of 360 and 112. The process carried on may be written down in either of the following ways: 112)360(3 The rule for finding the greatest common mea- 336 sure of two numbers is, 24)z12(4 I. Divide the greater of the two by the less. 96 II. Make the remainder a divisor, and the divisor a dividend, and find another remainder. III. Proceed in this way until there is no 16)24(1 8)x6(z remainder, and the last divisor is the greatest common measure required. o 99. You may perhaps ask how the rule is to I12 360 3 96 336 4 measure. i6 24 1 speaking, no such numbers, because all numbers 16 I6 2 are measured by I; that is, contain an exact o 8 shew when the two numbers have no common The fact is, that there are, strictly number of units, and therefore I is a common measure of every two numbers. sure, the last divisor will be 1, as If they have no other common meain the following example, where the greatest common measure of 87 and 25 is found. 49 DIVISION. 1 99-oz. EXERCISES. 25)87(3 75 g. c. m. Numbers. 6197 9521 1 58363 2602 I 5547 628I 147008443 1849 2 28915 31495 5 o 1509 300309 3 xz)zs(2 24 )12(z2 3260o4 571 What are 36x36+2x36x72+72x72 and 36x36x36+72x72x72; and what is their greatest common measure ?-Answer, 11664. 100. If two numbers be divisible by a third, and if the quotients be again divisible by a fourth, that third is not the greatest common measure. For example, 360 and 504 are both divisible by 4. are 90 and Iz6. Now 90 and The quotients 126 are both divisible by 9, the quotients of which division are xo and 14. By (87), dividing a number by 4, and then dividing the quotient by 9, is the same thing as dividing the number itself by 4x9, or by 36. Then, since 36 is a common measure of 360 and 504, and is greater than 4, 4 is not the greatest common measure. Again, since zo and 14 are both divisible by 2, 36 is not the greatest common measure. It therefore follows, that when two numbers are divided by their greatest common measure, the quotients have no common measure except I (99). Otherwise, the number which was called the greatest common measure in the last sentence is not so in reality. 101. To find the greatest common measure of three numbers, find the g. c. m. of the first and second, and of this and the third. For since all common divisors of the first and second are contained in their g. c. m., and no others, whatever is common to the first, second, and third, is common also to the third and the g. c. m. of the first and second, and no others. Similarly, to find the g. c. m. of four numbers, find the g. c. m. of the first, second, and third, and of that and the fourth. 102. When a first number contains a second, or is divisible by it without remainder, the first is called a multiple of the second. The words mult ple and measure are thus connected: Since 4 is a measure F 50 PRINCIPLES OF ARITHMETIC. of 24, 24 is a multiple of 4. 48, and several others. 12, 24, 48, &c. The number 96 is a multiple of 8, 12, 103. 24, It is therefore called a common multiple of 8, The product of any two numbers is evidently a common multiple of both. 8. SI02- Thus, 36x8, or 288, is a common multiple of 36 and But there are common multiples of 36 and 8 less than 288; and because it is convenient, when a common multiple of two quantities is wanted, to use the least of them, I now shew how to find the least common multiple of two numbers. 103. Take, for example, 36 and 8. Find their greatest common measure, which is 4, and observe that 36 is 9x4, and 8 is zx4. The quotients of 36 and 8, when divided by their greatest common measure, are therefore 9 and 2. Multiply these quotients together, and multiply the product by the greatest common measure, 4, which gives 9x2x4, or 72. This is a multiple of 8, or of 4x2, by (55) ; and also of 36 or of 4x9. It is also the least common multiple; but this cannot be proved to you, because the demonstration cannot be thoroughly understood without more practice in the use of letters to stand for numbers. But you may satisfy yourself that it is the least in this case, and that the same process will give the least common multiple in any other case which you may take. it is the least. It is not even necessary that you should know Whenever a common multiple is to be used, any one will do as well as the least. It is only to avoid large numbers that the least is used in preference to any other. When the greatest common measure is x, the least common multiple of the two numbers is their product. The rule then is: To find the least common multiple of two numbers, find their greatest common measure, and multiply one of the numbers by the quotient which the other gives when divided by the greatest common measure. To find the least common multiple of three num- bers, find the least common multiple of the first two, and find the least common multiple of that multiple and the third, and so on. 51 FRACTIONS. § Io3-1o4. EXERCISES. Least common multiple. Numbers proposed. 21 14, I6, I, 2, 3, 4, 6, 8, 5, 42 24 240 5, 6, 7, 8, 9, 10o I I, 6, 2520 20 2640 876, 864 63072 868, 854 52948 A convenient mode of finding the least common multiple of several numbers is as follows, when the common measures are easily visible: Pick out a number of common measures of two or more, which have Write them as divisors, and themselves no divisors greater than unity. divide every number which will divide by one or more of them. Bring down the quotients, and also the numbers which will not divide by any of them. Repeat the process with the results, and so on until the num- bers brought down have no two of them any common measure except unity. Then, for the least common multiple, multiply all the divisors For instance, let it be required by all the numbers last brought down. to find the least common multiple of all the numbers from Ix to 21. 2, 2, 3, 5, 7)11 II 19 20 2o 3 19 I x 22z 3 14 x5 16 17 18 x13 I I 4 17 There are now no common measures left in the row, and the least common multiple required is the product of 2, 2, 3, 5, 7, i , 13, 4, 17, 3, and :19 ; or 232792560. SECTION V. FRACTIONS. 104. Suppose it required to divide 49 yards into five equal parts, or, as it is called, to find the fifth part of 49 yards, If we divide 49 by 5, the quotient is 9, and the remainder is 4 ; that is (72), 49 is made up of 5 times 9 and 4. Let the line A B represent 49 yards: UNIVERSITY * ILLINOIS LIBRAR' 52 PRINCIPLES OF ARITHMETIC. S10o4-10o5. A B C IK-------- D E- - L-- F M- .G N- IKLMN H r-ri--r, Take 5 lines, c, D, E, p, and G, each 9 yards in length, and the line H, 4 yards in length. Then, since 49 is 5 nines and 4, c, D, E, F, G, and H, are together equal to A B. Divide H, which is 4 yards, into five equal parts, I, K, L, M, and N, and place one of these parts opposite to each of the lines, c, D, E, F, and G. It follows that the ten lines, c, D, E, F, C, I, K, L, M, N, are together equal to A B, or 49 yards. Now D and K together are of the same length as c and I together, and so are E and L, F and M, and G and N. Therefore, c and I together, repeated 5 times, will be 49 yards; that is, c and I together make up the fifth part of 49 yards. 105. c is a certain number of yards, viz. 9; but I is a new sort of quantity, to which hitherto we have never come. It is not an exact number of yards, for it arises from dividing 4 yards into 5 parts, and taking one of those parts. a FRACTION of a yard. It is the fifth part of 4 yards, and is called It is written thus, - (23), and is what we must 5 add to 9 yards in order to make up the fifth part of 49 yards. The same reasoning would apply to dividing 49 bushels of corn, or 49 acres of land, into 5 equal parts. We should find for the fifth part of the first, 9 bushels and the fifth part of 4 bushels ; and for the second, 9 acres and the fifth part of 4 acres. We say, then, once for all, that the fifth part of 49 is 9 and 4, or 9+4; which is usually written 9-, or if we use signs, 5 945. 5 5 EXERCISES. What is the seventeenth part of 1237 ?--Answer, 72.. 17 I o5-ro7. 53 FRACTIONS. What are 10032 663819 and22773399P hat 197 424 and 371 Answer, 5 16z 1974 23649 , 237o' 23710 2343 9394-. 2424. 106. By the term fraction is understood a part of any number, or the sum of any of the equal parts into which a number is divided. Thus, 49 4 20, are fractions. The term fraction 49 4 20 even includes whole num5 '5 7 17 34 51 bers :"* example, 17 is -, -, for -, &c. The upper number is called the numerator, the lower number is called the denominator, and both of these are called terms of the fraction. As long as the numerator is less than the denominator, the fraction is less than a unit: thus, - 17 is less than a unit; for 6 divided into 6 parts gives I for each part, and must give less when divided into 17 parts. Similarly, the fraction is equal to a unit when the numerator and denominator are equal, and greater than a unit when the numerator is greater than the denominator. 107. By - is meant the third part of z. 3 the third part of I. This is the same as twice To prove this, let A B be two yards, and divide each of the yards AC and c B into three equal parts. F A D E C T F G B Then, because A E, E F, and F B, are all equal to one another, AE is the third part of z. It is therefore -. But A E is twice A D, and A D 2 x is the third part of one yard, or -; therefore - is twice-; that is, in 3 3 2 order to get the length -, it makes no difference whether we divide two 13 yards at once into three parts, and take one of them, or whether we divide one yard into three parts, and take two of them. By the same reasoning, 8 may be found either by dividing 5 into 8 parts, and taking one of them, or by dividing I into 8 parts, and taking five of them. In future, of these two meanings I shall use that which is most convenient at the time, as it is proved that they are the same thing. This prin- * Numbers which contain an exact number of units, such as 5, 7, 100, &c., are called whole numbers or integers, when we wish to distinguish them from fractions. F2 54 PRINCIPLES OF ARITHMETIC. § IO7-109. ciple is the same as the following: The third part of any number may be obtained by adding together the thirds of all the units of which it consists. Thus, the third part of z, or of two units, is made by taking one-third out of each of the units, that is, 2 I 3 3 This meaning appears ambiguous when the numerator is greater than the denominator: thus, 5 would mean that I is to be divided into 7 parts, and 15 of them are to be taken. We should here let as many units be each divided into 7 parts as will give more than 15 of those 7 parts, and take 15 of them. 108. The value of a fraction is not altered by multiplying the numerator and denominator by the same quantity. Take the fraction 3, mul- tiply its numerator and denominator by 5, and it becomes same thing as 3; that is, one-twentieth 4- 5 5, which is the 20 part of 15 yards is the same 4 thing as one-fourth of 3 yards: or, if our second meaning of the word fraction be used, you get the same length by dividing a yard into o20 parts and takinrg 15 of theim,,as you get by dividing it into 4 parts and taking 3 of them. I To prove this, C- A C I I D E I 1 B let a B represent a yard ; divide it into 4 equal parts, A c, c D, D E, and E B, and divide each of these parts into 5 equal parts. Then A E is 3 4 But the second division cuts the line into o20 equal parts, of which A contains 15. 5 It is therefore -. Therefore, 20zo Again, since 3 is made from 1 4 o and 3 are the same thing. 4 by dividing both the numerator and 20 denominator by 5, the value of a fraction is not altered by dividing both its numerator and denominator by the same quantity. This principle, which is of so much importance in every part of arithmetic, is often used in common language, as when we say that 14 out of 21 is 2 out of 3, &c. 109. Though the two fractions 3 and -5 are the same in value, and 4 20 § og-x10o. 55 FRACTIONS. either of them may be used for the other without error, yet the first is more convenient than the second, not only because you have a clearer idea of the fourth of three yards than of the twentieth part of fifteen yards, but because the numbers in the first being smaller, are more convenient for multiplication and division. It is therefore useful, when a fraction is given, to find out whether its numerator and denominator have any common divisors or common measures. In (98) was given a rule for finding the greatest common measure of any two numbers; and it was shewn that when the two numbers are divided by their greatest common measure, the quotients have no common measure except 1. Find the greatest common measure of the terms of the fraction, and divide them by that number. The fraction is then said to be reduced to its lowest terms, and is in the state in which the best notion can be formed of its magnitude. EXERCISES. With each fraction is written the same reduced to its lowest terms. 2794 22X27 22 2921 23x127 23 2788 17x164 4920 30x 93208 764xx22 13786 113X22 888800 40359600 _ 17 i64 22x40400 999x40400 30 _ 764 11 3 22 999 95469 121x789 121 359784 456x789 456 110. When the terms of the fraction given are already in factors,* any one factor in the numerator may be divided by a number, provided some one factor in the denominator is divided by the same. lows from (88) and (108). This fol- In the following examples the figures altered by division are accented. * A factor of a number is a number which divides it without remainder: thus, x3, 4, 6, 8, are factors of 24, and 6 X4, 8x 2 x 2X2X 3, are several ways of decomposing 24 into factors. 56 § II0-II. PRINCIPLES OF ARITHMETIC. 12XIIXIO 2X 3X 3'XIIXIO 18XISXI3 I'XIX5' 2 X 4 I'XI' XI 3X I' 2ox54x52 ' I'XI' XI' _'X3'XI' 4'x 6'x4.' 2'x2'' I i6 3'x4' 3'x2' 6 I'xzIO' 27X28 9X70 I'X5' 5 111. As we can, by (108), multiply the numerator and denominator of a fraction any number, without altering its value, we can now by readily reduce two fractions to two others, which shall have the same value as the first two, and which shall have the same denominator. Take, for example, Z and ±; multiply both terms of Z by 7, and 7 terms of 4 by 3. It 3 4 then appears that both 3 7 2 - 3 4 - 7 2X7 14 s or 3x7 21 4X3 12 or -. 7x3 21 is -- 14 12 2 4 Here are then two fractions - and -, equal to - and -, and 21 21 2 4 3 7 having the same denominator, 21; in this case, and - are said to be 7 3 reduced to a common denominator. 15 7 It is required to reduce , 6, and - to a common denominator. Multiply both terms of the first by the product of 6 and 9; of the second by the product of Io and 9; and of the third by the product of zo and 6. Then it appears (108) that I . x6x9 ox6x9 54 540 s 5XI109 or 450 540 10 5 -is 6 6xiox9 7 - is 7x1ox6 or-.420 9 9xIox6 540 On looking at these last fractions, we see that all the numerators and the common denominator are divisible by 6, and (108) this division will not alter their values. On dividing the numerators and deno- minators of 54 450, and 420 by 6, the resulting fractions are, o, -, 70 and -. 90 540 540 540 90 90 These are fractions with a common denominator, and which ( 57 FRACTIONS. III-II2, are the same as ' 5 7 ,.6, and -; Io' 6' 9 and therefore these are a more simple answer to the question than the first fractions. Observe also that 540 is one common multiple of zo, 6, and 9, namely, xox6x9, but that 90 is the least common multiple of io, 6, and 9 (103). cess, therefore, is better. To reduce the fractions The following pro- x and -, to others , having the same value and a common denominator, begin by finding the least common multiple of Io, 6, and 9, by the rule in (103), which is go. Observe that Io, 6, and 9 are contained in 90 9, S5, and zo times. Multiply both terms of the first by 9, of the second by 15, and of the third by Io, and the fractions thus produced are 9 9' as before. 90 ,and 7,the same 9o' 90 90 90 If one of the numbers be a whole number, it may be reduced to a fraction having the common denominator of the rest, by (106), EXERCISES. Fractions proposed reduced to a common denominator. 2 1 x 20 6 5 3 5 6 30 30 30 I 2 3 12 3 28 24 I8 48 63 3 7 14 21 4 84 84 84 84 84 3 4 5 6 3000 400 100 oo 1000 00oo 000 o 33 379 28 - 677 50 6 000 1000 2234 106499 256583 256583 112. By reducing two fractions to a common denominator, we are able to compare them; that is, to tell which is the greater and which the less of the two. For example, take I and 2 . These fractions S5 reduced, without alteration of their value, to a common denominator, are 1 30 and . Of these the first must be the greater, because (107) it 30 may be obtained by dividing i into 30 equal parts and taking 15 of them, but the second is made by taking 14 of those parts. It is evident that of two fractions which have the same denominator, the greater has the greater numerator; and also that of two fractions which have the same numerator, the greater has the less denominator. 58 §1Iz-r4. PRINCIPLES OF ARITHMETIC. 8 8 Thus, - is greater than -, since the first is a 7th, and the last only a 7 9 9th part of $. Also, any numerator may be made to belong to as small a fraction as we please, by sufficiently increasing the denominator. IO I 100 I0 Thus, -is-, - I O is -,and 1000 - I00 O is ---- I000000 I (108). I00000 We can now also increase and diminish the first fraction by the second. For the first fraction is made up of 15 of the into which I is divided. The second fraction is 14 30 equal parts of those parts. The sum of the two, therefore, must be 15+14, or 29 of those parts; that is, _+- is 29. The difference of the two must be 15-14, or I of those 2 IS 30 7 I parts ; that is, -I 2 15 30 113. From the last two articles the following rules are obtained : I. To compare, to add, or to subtract fractions, first reduce them to a common denominator. When this has been done, that is the greatest of the fractions which has the greatest numerator. Their sum has the sum of the numerators for its numerator, and the common denominator for its denominator. Their difference has the difference of the numerators for its numerator, and the common denominator for its denominator. EXERCISES I 1 3 4 5 II _+-+- - - 2 8 3 2 +-+._S__300oo + 10 - 4 000 3 8 94 -+t---7 = 2 16 44 . 153 -- 18329 -- 53 . 6o 3 1281 427 I2 2---+-- 1834 = -- 3 7 I000 253 91 13 163 --- 97 93066 -- =so---- 521 z 3 188 881 - 459001 114. Suppose it required to add a whole number to a fraction, for example, 6 to 4. By (106) 6 is 5, and -+- is 5$ that is, 6+ , or as 9 4 58 9 99 9 9 it is usually written, , is-. The rule in this case is: Multiply the 9 9 whole number by the denominator of the fraction, and to the product add the numerator of the fraction; the sum will be the numerator of the result, and the denominator of the fraction will be its denominator. Thus, 3 =,224 that in (105). 4 9 0, o 9 74 55 = 4072. 55 This rule is the opposite of x15-1IS. 59 FRACTIONS. 115. From the last rule it appears that 1723 100000 225is 667225, and 2399 667 is 2300099 1000 000ooo 00000 7230907 907 is 173907 Hence, when a whole number is to be added to a fraction whose denominator is I followed by ciphers, the number of which is not less than the number of figures in the numerator, the rule is : Write the whole number first, and then the numerator of the fraction, with as many ciphers between them as the number of ciphers in the denominator exceeds the number of figures in the numerator. This is the numerator of the result, and the denomi- nator of the fraction is its denominator. If the number of ciphers in the denominator be equal to the number of figures in the numerator, write no ciphers between the whole number and the numerator. EXERCISES. 2457 610 23707 , Reduce the following mixed quantities to fractions: I 299 1207---------299, and 10000000ooooooo 2210100000 23320 0000 - 116. Suppose it required to multiply by 4. This by (48) is taking 2 2 2 2 2 3 8 - four times; that is, finding -+ +-+-. This by (112) is-; so that to 3 3333 3 multiply a fraction by a whole number the rule is: Multiply the numerator by the whole number, and let the denominator remain. 117. If the denominator of the fraction be divisible by the whole number, the rule may be stated thus: Divide the denominator of the fraction by the whole number, and let the numerator remain. For by 6. This (116) is 4, which, since the numerator 367 36 and denominator are now divisible by 6, is (108) the same as 6. It is example, multiply plain that is made from in the manner stated ii the rule. 118. Multiplication has been defined to be the taking as many of Thus, to multiply 12 by 7 is to take as many twelves as there are units in 7, or to take 12 as many one number as there are units in another. times as you must take I in order to make 7. Thus, what is done with I in order to make 7, is done with 12 to make 7 times 12. 7 is 7 times Iz is I+t+I 1-I For example, I+++I. z2+12+12z+2z+2+12+12. When the same thing is done with two fractions, the result is still 60 PRINCIPLES OF ARITHMETIC. § I18-120. called their product, and the process is still called multiplication. There is this difference, that whereas a whole number is made by adding 1 to itself a number of times, a fraction is made by dividing I into a number of equal parts, and adding one of these parts to itself a number of times. This being the meaning of the word multiplication, as applied to fractions, what is 3 multiplied by 7 ? make Whatever is done with I in order to must now be done with 3; but to make 4 parts, and 7 of them are taken. I is divided into 8 ,373 must be di48 43 Therefore, to make 3-x-, vided into 8 parts, and 7 of them must be taken. the same thing as 4. Since 2 32 Now - is,by (108), is made by dividing I into 32 parts, and 32 taking 24 of them, or, which is the same thing, taking 3 of them 8 times, if 4 be divided into 8 equal parts, each of them is 3 ; and if 7 of these parts be taken, the result is (116) : therefore 3 multiplied by is 8 4 32 and the same reasoning may be applied to any other fractions. 2is made from 3and ; 32 But by multiplying the two numerators together for the numerator, and the two denominators for the denominator; which furnishes a rule for the multiplication of fractions. 119. If this product 5 I 32 is to be multiplied by a third fraction, for 5 example, by , the result is, by the same rule, ;05 and so on. The general rule for multiplying any number of fractions together is therefore: Multiply all the numerators together for the numerator of the product, and all the denominators together for its denominator. 120. Suppose it required to multiply together duct may be written thus : 5x8 6x 20 and is I6xxo - and -. 6 The pro- Io ,-which reduced to its lowest 160 terms (109) is 3. This result might have been obtained directly, by 4 observing that 15 and 1o are both measured by 5, and 8 and i6 are both measured by 8, and that the fraction may be written thus: 3x5x8 2x8xzx5 Divide both its numerator and denominator by 5x8 (108) and (87), and the result is at once 3 ; therefore, before proceeding to multiply any 4 number of fractions together, if there be any numerator and any denominator, whether belonging to the same fraction or not, which have a common measure, divide them both by that common measure, and use the quotients instead of the dividends. 61 FRACTIONS. I20-IZI. A whole number may be considered as a fraction whose denominator isi; thus, 16 is 6- and the (106); same rule will apply when one or I more of the quantities are whole numbers. EXERCISES. 136 --- x 36448 3432465 = 7470 919 S2 18224 6864930 268 3 - xx-= 23 4 5 2 I -x 5 17 2 =- 17 45 45 2 x13241 6266 13 601 59 7847' 461 II 7 19 Fraction proposed. 7813 5071 Square. Cube. 701 49 401 344472101 158 24964 3944312 140 19600 2744c00 141 19881 2803221 355 126025 44738875 113 12769 1442897 From zoo acres of ground, two-thirds of them are taken away; 50 acres are then added to the result, and 5 of the whole is taken,; what 7 II number of acres does this produce ?-Answer, 59-. 21 121. In dividing one whole number by another, for example, io8 by 9, this question is asked,-Can we, by the addition of any number of nines, produce 108 ? and if so, how many nines will be sufficient for that purpose ? 2 Suppose we take two fractions, for example, - 4 and 4, and ask, Can 3 we, by dividing 4 into some number of equal parts, and 5 adding a num5 2 ber of these parts together, produce - ? if so, into how many parts must 3 4 we divide 4, and how many of them must we add together? solution of this question is still called the division of - 3 by 4; and The the 5 fraction whose denominator is the number of parts into which 4 is 5 divided, and whose numerator is the number of them which is taken, is called the quotient. The solution of this question is as follows: Reduce both these fractions to a common denominator (111), which does not alter their value (108) ; they then become and 1. 15 15 The 62 PRINCIPLES SIZ1I-IZ3. OF ARITHMETIC. question now is, to divide . into a number of parts, and to produce 15 I 15 by taking a number of these parts. Since - is made by dividing I 15 Is into 15 parts and taking Iz of them, if we divide into Is equal I 15 if we take Io of these parts, the result parts, each of these parts is I; 10 2 15 10 or - (108), we must divide is o. Therefore, in order to produce io 3 15 iz I5 4 - or - into Is parts, and take io of them; that is, the quotient is --. 15 5 12 If we call z the dividend, and 4 the divisor, as before, the quotient in 5 3 this case is derived from the following rule, which the same reasoning will shew to apply to other cases: The numerator of the quotient is the numerator of the dividend multiplied by the denominator of the divisor. The denominator of the quotient is the denominator of the dividend multiplied by the numerator of the divisor. This rule is the reverse of multiplication, as will be In multiplying 4 5 I ask, if out of 4 be taken io parts out of 12, how much of a unit 2 2 4 5 40 by -, I ask is taken, and the answer is -, or-. Again, in dividing 3 5 3 1o 4 what part of 4 is -, the answer to which is -. 12 5 3 122. By taking the following instance, we shall see that this rule seen by comparing what is required in both cases. by I, 16 28 Observe that i6 is Divide 6 by -. 15 33 4x4, and 28 is 4x7 ; 33 is 3x11, and 15 is 3x5 ; therefore the two fractions are 4x4 and 4x7, and their quotient, according to the rule, is can be sometimes simplified. 44X3X5 XII 3x5 4x4x3x5, in which 4x3 is found both in the numerator and denominator. 3xxx4.x7 4x o20 The rule of The fraction is therefore (108) the same as - 5 , or -. IIx7 77 the last article, therefore, admits of this modification : If the two numerators or the two denominators have a common measure, divide by that common measure, and use the quotients instead of the dividends. 123. In dividing a fraction by a whole number, for example, - by S3 The rule gives - as the quotient. 15 as the fraction -. consider 15, I 45 Therefore, to divide a fraction by a whole number, multiply the denominator by that whole number. § 123-Iz4. FRACTIONS. 63 EXERCISES. Dividend. Divisor. 41 63 33 Ij 467 41 189 907 151 Quotient. 101 471567 136957 7813 6o0 13 5071 II 461 z II 2 -x- 5 5 5 What are5 5 5 2 --- x17 17 7 7 2 17 7 8 8 -X-Ix and II II 3 3 II II I 2 8 5 17 II I 3 II Answer, 559, and A can reap a field in 12 1. days, B in 6, and C in 4 days; in what time can they all do it together ?*-Answer, 2 days. In what time would a cistern be filled by cocks which would separately fill it in 12, II, IO, and 9 hours ?-Answer, 24 hours. 753 124. The principal results of this section may be exhibited algebraically as follows ; let a, b, o, &c. stand for any whole numbers. (107) s a - = -xa b (111) (112) a and -='- a b -+a- b c are the same as ma a (108) b ad a+b a b c C C Then ab be and a-b =-b C C c a s ad+bc a c ad-bo (113) b d d b d bd (118) a o ac b x d =b d bd a o 3 (121) - divd. byo or b d c ad ad bo * The method of solving this and the following question may be shewn thus: If the number of days in which each could reap the field is given, the part which each could do in a day by himself can be found, and thence the part which all could do together; this being known, the number of days which it would take all to do the whole can be found. 64 PRINCIPLES OF ARITHMETIC. § 125- 125. These results are true even when the letters themselves rea present fractions. For example, take the fraction -, whose numerator d and denominator are fractional, and multiply its numerator and denoae minator by the fraction e, which (121) f, which gives is aedf ce bfce df which, dividing the numerator and denominator by ef (108), is a But the original fraction itself is ad a ; hence = d ad e T xff . C which core- df sponds to the second formula* in (124). In a similar manner it may be shewn, that the other formulae of the same article are true when the letters there used either represent fractions, or are removed and fractions introduced in their place. All formulae established throughout this work are equally true when fractions are substituted for whole numbers. For example (54), (m+n)a = ma+na. Let m, n, and a be respectively the fractions -, q s and b. c p Then m+n is q +, or s qs , and (m+n)a is ps+qr S -,or (ps+qr)b psb+qrb b or . But this (112) is psb qs a qsc pb rb .psb pb is + , since =-,and qco so qsc qgo Sx-. qsc qsc qrb rS -= qso p Therefore (m+n)a, or so (108). so +r) q qsc which pb 6 rb But b=-x, and qo q a so =xbrxb. so grb qrb In a similar man qgoso ner the same may be proved of any other formula. The following examples may be useful: a ce g b X +f X acfh+bdeg a aedh+bcfg e c g -x- +-x-I b 1 ab+a b I a a+-" I a+ bc+1 bc+x abc+a+c b+* A formula is a name given to any algebraical expression which is commonly used. 8 65 DECIMAL FRACTIONS I25-I28. I I Thus, I _ _57 57 350 6+s 6+ 7+- The rules that have been proved to hold good for all numbers may be applied when the numbers are represented by letters. SECTION VI. DECIMAL FRACTIONS. 126. We have seen (112) (121) the necessity of reducing fractions to a common denominator, in order to compare their magnitudes. We have seen also how much more readily operations are performed upon fractions which have the same, than upon those which have different, denominators. On this account it has long been customary, in all those parts of mathematics where fractions are often required, to use none but such as either have, or can be easily reduced to others having, the same denominators. Now, of all numbers, those which can be most easily managed are such as io, 1ooo, 100, &c., where I is followed by ciphers. These are called DECIMAL NUMBERS ; and a fraction whose denominator is any one of them, is called a DECIMAL FRACTION, Or more commonly, a DECIMAL. 127. A whole number may be reduced to a decimal fraction, or one decimal fraction to another, with the greatest ease. For example, 0 0 0 94 is 940 IO , or 94000 I00 1000 , or 3 30 IO 9400 I00 (106) ; -is , or-, 300 I000 or 3 I0000 (108). The placing of a cipher on the right hand of any number is the same thing as multiplying that number by zo (57), and this may be done as often as we please in the numerator of a fraction, provided it be done as often in the denominator (108). 128. The next question is, How can we reduce a fraction which is not decimal to another which is, without altering its value ? Take, for example, the fraction -, multiply both the numerator and deno- minator successively by 10, 100, 1000, &c., which will give a series of (108), viz.160' 700' 7coo fractions, each of which is equal to 1626o 160o' 16co o2 66 PRINCIPLES OF ARITHMETI. 70000 600 i60000 &c. § I28. The denominator of each of these fractions can be divided without remainder by i6, the quotients of which divisions form the series of decimal numbers Io, Ioo, 1000, 10000, &c. If, therefore, one of the numerators be divisible by I6, the fraction to which that numerator belongs has a numerator and denominator both divisible by I6. When that division has been made, which (108) does not alter the value of the fraction, we shall have a fraction whose denominator is one of the series Io, too, 1oo000, &c., and which is equal in value to . The ques- tion is then reduced to finding the first of the numbers 70, 700, 7000, 70000, &c., which can be divided by 16 without remainder. Divide these numbers, one after the other, by x6, as follows: 16)70(4 16)700(43 64 16)70000(4375 16)7000(437 64 64 64 60 6o 6o 48 48 48 12 120 120 112 6 112 8o 8o 8 It appears, then, that 70000 is the first of the numerators which is divisible by 16. But it is not necessary to write down each of these divisions, since it is plain that the last contains all which came before. It will do, then, to proceed at once as if the number of ciphers were without end, to stop when the remainder is nothing, and then count the number of ciphers which have been used. 6 x4375, 70000 0000which is 16x4375 x43 160000 6xoooo In this case, since 70000 is or - 4375 gives the fraction required. -, oooo0000 Therefore, to reduce a fraction to a decimal fraction, annex ciphers to the numerator, and divide by the denominator until there is no remainder. The quotient will be the numerator of the required fraction, and the denominator will be unity, followed by as many ciphers as were used in obtaining the quotient. § 128-129. 67 DECIMAL FRACTIONS. EXERCISES. Reduce to decimal fractions I 2 I 3927 I 5 453 z50' ' 4' 25' 50' 625 8 25 -, 2 3r416 7248 , IO oo'I00 100'10000 ,and Iooo0000 100 Answer, -- 129. It will happen in most cases that the annexing of ciphers to the numerator will never make it divisible by the denominator without remainder. For example, try to reduce - to a decimal fraction. 7 7)000000000000000000oooooooooooooooooo, &c. 142857x42857142857, &c. The quotient here is a continual repetition of the figures 1, 4, ,, 8, 5, 7, in the same order; therefore - cannot be reduced to a decimal 7 fraction. But, nevertheless, if we take as a numerator any number 142857142857, of figures from the quotient &c., and as a denominator I followed by as many ciphers as were used in making that part of the quotient, we shall get a fraction which differs very little from I, and 7 which will differ still less from it if we put more figures in the numerator and more ciphers in the denominator. is less 10 (than Thus, 7 I 3by which is not so much as 70 b 10 1 14 I 2 I00 7 70 142 oco 0 00 6 1 7000 1000 1 .7 1428 I 4 1 10000 7 70000 10000 14285 I 5 100000 7 700000 142857 I oooo000000 7 &c. &c. I 7000000 &c. 1 100000 x ooooo1000000 &c. In the first column is a series of decimal fractions, which come nearer I and nearer to -, as the third column shews. Therefore, though we can- not find a decimal fraction which is exactly 7, we can find one which differs from it as little as we please. 68 PRINCIPLES OF ARITHMETIC. § I29-131. I This may also be illustrated thus: It is required to reduce - to a 7 decimal fraction without the error of say a millionth of a unit; multiply the numerator and denominator of I by a million, and then divide both 7 by 7; we have then I I000000 1428574 7 7000000 1000000 If we reject the fraction _ in the numerator, what we reject is really 7 the 7th part of the millionth part of a unit ; or less than the millionth part of a unit. Therefore 142857 I000000 is the fraction required. EXERCISES. Make similar tables with? these fractions The recurring qurotient of 3 '7 _3 I7 and I 91' 143' 247 is 329670,329670, &c. ... II8881,I1888i, &c. 143 --... 247 404858299595141700,4048582 &c. 130. The reason for the recurrence of the figures of the quotient in the same order is as follows: If iooo, &c. be divided by the number 247, the remainder at each step of the division is less than 247, being either o, or one of the first 246 numbers. If, then, the remainder never become nothing, by carrying the division far enough, one remainder will occur a second time. If possible, let the first 246 remainders be all different, that is, let them be I, 2, 3, &c., up to 246, variously distributed. As the 247th remainder cannot be so great as 247, it must be one of these which have preceded. From the step where the remainder becomes the same as a former remainder, it is evident that former figures of the quotient must be repeated in the same order. 131. You will here naturally ask, What is the use of decimal fractions, if the greater number of fractions cannot be reduced at all to decimals? The answer is this: The addition, subtraction, multiplica- tion, and division of decimal fractions are much easier than those of 69 DECIMAL FRACTIONS. S131-z132. common fractions; and though we cannot reduce all common fractions to decimals, yet we can find decimal fractions so near to each of them, that the error arising from using the decimal instead of the common fraction will not be perceptible. For example, if we suppose an inch to be divided into ten million of equal parts, one of those parts by itself will not be visible to the eye. Therefore, in finding a length, an error of a ten-millionth part of an inch is of no consequence, even where the finest measurement is necessary. (129), we shall see that Now, by carrying on the table in does not differ from I by I 0000000 oo00000ooco 7 and if these fractions represented parts of an inch, the first might be 1428571 used for the second, since the difference is not perceptible. In applying arithmetic to practice, nothing can be measured so accurately as to be represented in numbers without any error whatever, whether it be length, weight, or any other species of magnitude. It is therefore un- necessary to use any other than decimal fractions, since, by means of them, any quantity may be represented with as much correctness as by any other method. EXERCISES. Find decimal fractions which do not differ from the following fraotions by I . 100000000 I Answer, 0Answer, -- Answer33333333 3 100000000 113 355 100000000 4 57142857 i00000000 355 113 314159292 100000000 7 31830985 132. Every decimal may be immediately reduced to a quantity consisting either of a whole number and more simple decimals, or of more simple decimals alone, having one figure only in each of the numerators. 14736 47326 4326 Take, for example, . By (115) 147326is 147326 ; and since I000 1000 326 is made up of 300, and zo, and 6; by (112) 1000 6 1000 6 . I000 But (108) 300 3 is -, and - 20 is -.2 I000 IO 1000 I00 made up of 147+- --1 IO 00 +-- 1000 . 300+-2 1000 + 1000 147326 Therefore,-4736 1000 is Now, take any number, for example, 147326, and form a number of fractions having for their numerators this number, and for their denominators I, lo, 0oo, 1000, 10000, &c., and 70 PRINCIPLES OF ARITHMETIC. § 132-133. reduce these fractions into numbers and more simple decimals, in the foregoing manner, which will give the table below. DECOMPOSITION OF A DECIMAL FRACTION. 147326 -= 1 47326 6 147326 14732+- 10 147326 100 10 6 2 = 147326 = 147+3++ 10 . 3 14+7 1000 147326= 100 I0 . I00 3 47 10 100 10 4 10000 100 7 - 7IO0000 147326 :4 147326_ =... 1000000 147326 10000000 I-10 I00 -100 1000 6 2 1000 10000 - + I1000 3 +7 _ I000 4 + 1000 +- ---100000 6 + .4 6 2 -- -------__ 10000 2 + 7 + 10000 1000000 100000 10000 3 100000 + 2 1000000 +------- 6 10000000 N.B. The student should write this table himself, and then proceed to make similar tables from the following exercises. EXERCISES. Reduce the following fractions into a series of numbers and more simple fractions : 31415926 10 31415926 100 2700031 2700031 I0 100 2073000 2073000 10 100 3331303 3331303 1000 &C. &c. &c. &c. 10000 133. If, in this table, and others made in the same manner, you look at those fractions which contain a whole number, you will see that they § 33-134 71 DECIMAL FRACTIONS. may be made thus: Mark off, from the right hand of the numerator, as many figures as there are ciphers in the denominator by a point, or any other convenient mark, This will give 14732'6 when the fraction is 147326 IO . . . 47"326 .. . . 1473"26.. .. . . . . 147326 100 . . . . . .. ICOO &c. &c. The figures on the left of the point by themselves make the whole number which the fraction contains. Of those on its right, the first is the numerator of the fraction whose denominator is that whose denominator is zoo, and so on. 1o, the second of We now come to those fractions which do not contain a whole number. i47326 134. The first of these is , in which the number of ciphers in 732 1000000 the denominator is the same as the number of figures in the numerator. If we still follow the same rule, and mark off all the figures, by placing the point before them all, thus, "147326, the observation in (133) still holds good; for, on looking at I 4 -+-10 ICO 7 14 + -100 1000 x 3 + in the table, we find it is 6 + + 1000 +- I0000+ The next fraction is In this, 147326 1000000 +I00000 24-,7326 10000000 , which we 7_ + I0000 3 3 I00000 1000000 find by the table to be 2 1000000 is not divided by xo, but by 1oo0; +----- 6 10000000 if, therefore, we put a point before the whole, the rule is not true, for the first figure on the left of the point has the denominator which, according to the rule, the second ought to have, the second that which the third ought to have, and so on. In order to keep the same rule for this case, we must con- trive to make I the second figure on the right of the point instead of the first. This may be done by placing a cipher between it and the 72 PRINCIPLES OF ARITHMETIC. § 134-135. Here the rule holds good, for by that rule this point, thus, "0147326. fraction is o+ I+ S-I00 IO 4 7 3 I000 10000 100000 + + 2 6 -+----------+ +-000000 10000000 which is the same as the preceding line, since - is o, and need not be IO reckoned. Similarly, when there are two ciphers more in the denominator than there are figures in the numerator, the rule will be true if we place two ciphers between the point and the numerator. The rule, therefore, stated fully, is this : To reduce a decimal fraction to a whole number and more simple decimals, or to more simple decimals alone if it do not contain a whole number, mark off by a point as many figures from the numerator as there are ciphers in the denominator. If the numerator have not places enough for this, write as many ciphers before it as it wants places, and put the point before these ciphers. Then, if there be any figures before the point, they make the whole number which the fraction contains. The first figure after the point with the denominator io, the second with the denominator Ioo, and so on, are the fractions of which the first fraction is composed. 135. Decimal fractions are not usually written at full length. It is more convenient to write the numerator only, and to cut off from the numerator as many figures as there are ciphers in the denominator, when that is possible, by a point. When there are more ciphers in the denominator than figures in the numerator, as many ciphers are placed before the numerator as will supply the deficiency, and the point is placed before the ciphers. Thus, *7 will be used in future to denote 7 ,07 The following tables will give the whole of for io -, and so on. 100 this notation at one view, and will shew its connexion with the decimal notation explained in the first section. You will observe that the numbers on the right of the units' place stand for units divided by 1o, zoo, 1ooo, 1ooo, &C. &c. while those on the left are units multiplied by Io, Ioo, § 73 DECIMAL FRACTIONS. I35. The student is recommended always to write the decimal point in a line with the top of the figures or in the middle, as is done here, and The reason is, that it is usual in the higher never at the bottom. branches of mathematics to use a point placed between two numbers or letters which are multiplied together; thus, I5.16, a.b, a+b.c+d stand for the products of those numbers or letters. - cc o . c. 6 6 o-i 6 0 0 O 0 0 W 0 1. O N +- O S0 O O O Iw 0 00 O o 0 + 0 0 + 0 W O O P 0 . 0 01 O ' -N . Q 0 O N 0 W . " 4 0 00 1 00.4 - . 4 j. 1 w w "r il . 0 w O L. O0 O Ow . N O • + O O - IN .4I N S . O 00o N N N o 00N o +666I+ N 000 O O + 0N 0 O - N + + 01 61 + N 00O1- 0 NN 0 + 00 N + ++ 00 0 000 O J EJ o+ 000 O + 1. + 0 0 00 O + 0 gw O + 0 + + * °4 O + t1- + o , olI o N ot 0o 00 0 JN 0 _ °o 0 + J14 o +a O p O I l N 0 O S II 1( II II I I ON JN w + + ,+ N N + . + O o - 74 PRINCIPLES OF ARITHMETIC. I is 000oo inches 2 is 200 . . 3 is 30 . i35-139. 1 . 4 4 is 5 is IV. In 1234'56789 5 of an inch 5 is inches the . 6 is 6 100 1000 8 is. Sis . 10000 9 I00000 136. The ciphers on the right hand of the decimal point serve the same purpose as the ciphers in (10). They are not counted as any thing themselves, but serve to shew the place in which the accompanying numbers stand. They might be dispensed with by writing the numbers in ruled columns, as in the first section. They are distinguished from the numbers which accompany them by calling the latter significant figures. Thus, 0003747 is a decimal of seven places with four significant figures, '346 is a decimal of three places with three significant figures, &c. 137. The value of a decimal is not altered by putting any number The first (135) of ciphers on its right. Take, for example, *3 and "300. is ~, ,00whichis made from the first by multiplying and the second IO0 1000 both its numerator and denominator by ioo, and (108) is the same quantity. 138. To reduce two decimals to a common denominator, put as many ciphers on the right of that which has the smaller number of places as will make the number of places in both fractions the same. Take, for example, '54 and 4'3297. The first is -L_, and the second 43297 I00 the first by the numerator and denominator of0Multiply I0000 zoo (108), which reduces it to 5400, which has the same denominator as 43 10000 But 5400 is '5400 (135). B oooo is (15) 0000 In whole numbers, the decimal point should § 138-140. 75 DECIMAL FRACTIONS. be placed at the end: thus, z29 should be written I29". It is, however, usual to omit the point; but you must recollect that 129 and 129o000 are of the same value, since the first is 129 and the second 129000 1000 139. The rules which were given in the last chapter for addition, subtraction, multiplication, and division, apply to all fractions, and therefore to decimal fractions among the rest. But the way of writing decimal fractions, which is explained in this chapter, makes the application of these rules more simple. We proceed to the different cases. Suppose it required to add 42"634, 45*2806, 2"oo00, and 54. By (112) these must be reduced to a common denominator, which is done (138) by writing them as follows: 42'6340, 45*2806, 200oo10o, and 54*0000. These are decimal fractions, whose numerators are 426340, 452806, 2ooxo, and 540000, and whose common denominator is ioooo. (112) their sum is 426340+452806+20010+540000 IO000 or 143"9156. By which is 1439156 I0000 The simplest way of doing this is as follows: write the decimals down under one another, so that the decimal points may fall under one another, thus: 42"634 45*2806 54 143'9156 Add the different columns together as in common addition, and place the decimal point under the other decimal points. EXERCISES. What are 1527+64732094+oo003+'ooo0000974 ; 2276*3+'I07+'9+26"37z+567300 and oo 'I1+7*7+39+*OO4z+*8838 Answer, 1593'73341374, ; ? 59035*6252, 9*69912. 140. Suppose it required to subtract 9I*07324 from 137"32z. These fractions when reduced to a common denominator are 9r107324 and Their difference is therefore 13732100-9107324 13732zoo (138). 100000 which is 10000464776 or 46*24776. This may be most simply done as fol0000 76 S140-142z. PRINCIPLES OF ARITHMETIC. lows: write the less number under the greater, so that its decimal point may fall under that of the greater, thus : 137'321 9107324 4624776 Subtract the lower from the upper line, and wherever there is a figure in one line and not in the other, proceed as if there were a cipher in the vacant place. EXERCISES. What is 1z36z--274'22zz7+' 5 ; 9976zo73942-00oo 43976728; and x'z+'03+'oo4--'ooo5 ? Answer, 12088*27893, 9976'20595443272; zoo, 141. The multiplication of a decimal by io, and Izz335. ooo, &c., is per- formed by merely moving the decimal point to the right. Suppose, for example, 13zo079 is to be multiplied by Coo. The decimal is 132079 10000 which multiplied by Ioo is (117) 132079, or 1320'79. Again, I*309x 100 1ooooo is z309 39xooooo, or (116) 1000 130900000 130900000 or 1000 130900. From these and other instances we get the following rule: To multiply a decimal fraction by a decimal number (126), move the decimal point as many places to the right as there are ciphers in the decimal number. When this cannot be done, annex ciphers to the right of the decimal (137) until it can. 142. Suppose it required to multiply 17036 by 4"27. these decimals is 17036 1000 and the second 427. 100 The first of By (118) the product of these fractions has for its numerator the product of 17036 and 427, and for its denominator the product of iooo and Ioo; therefore this product 74 37 72 2 is ooooo' or 72'74372. This may be done more shortly by multiplying the two numbers 17036 and 427, and cutting off by the decimal point as many places as there are decimal places both in 17"036 and 4*27, because the product of two decimal numbers will contain as many ciphers as there are ciphers in both. 77 DECIMAL FRACTIONS. SI143-14.5 143. This question now arises: What if there should not be as many figures in the product as there are decimal places in the multiplier and multiplicand together ? To see what must be done in this case, IOI 172 multiply '172 by 'IOI, or -- 7 by 1000 17372 ----- , or '017372 (135). The product of these two is , 1000 Therefore, when the number of places in 1000000 the product is not sufficient to allow the rule of the last article to be followed, as many ciphers must be placed at the beginning as will make up the deficiency. ADDITIONAL EXAMPLES. "OOIX'OI is '0000I 56x'oooi is '0056. EXERCISES. Shew that 3'002x3'002 = 3x3+2X3x'002+002ox'002 II'5609x5'3291 = 8'z27xo'oozI = 8'44X844-3'2 zo9x3'1209 8xIo+8x'ooz+IoxzI7+'ooix217. Fraction. Square. Cube. 82'92 6875'7264 570135'233088 '0173 '00029929 '000005177717 '009 '000000729 "oooo8I 15'625x 64 = I'5625x '64 = 6 2'924207 'o0449 I'43 6 '01o5 25x'00oo 4 = *64 = ' 1000 '15625x I 15 2'5x'064 = 6 5625000x'064 = 0001 100 ooo0000 144. The division of a decimal by a decimal number, such as Ioo, 1000, io, &c., is performed by moving the decimal point as many places to the left as there are ciphers in the decimal number. If there are not places enough in the dividend to allow of this, annex ciphers to the beginning of it until there are. For example, divide 1734'229 by ooo: the decimal fraction is 734229, which divided by Ioo (123) 1734291000 is 73429 or 734zz 9 . If, in the same way, I'z106 be divided by C1000000 10000, the result is 'ooozIo6. 145. Before proceeding to shorten the rule for the division of one H2 78 PRINCIPLES OF ARITHMETIC. § 145-146. decimal fraction by another, it will be necessary to resume what was said in (128) upon the reduction of any fraction to a decimal fraction. It was there shewn that - 16 As another example, convert is the same fraction as 4375 or '4375. 10000 3 into a decimal fraction. Follow the same process as in (128), thus: 128)300000000000(234375 480 256 384 440 960 384 896 640 640 560 51z 480 o Since 7 ciphers are used, it appears that 30000000 is the first of the series 30, 300, &c., which is divisible by 1z8; and therefore or, which is the same thing (108), 00 is equal to z280000000 o0000000 234375 3 Iz8 or 0234375 (135). From these examples the rule for reducing a fraction to a decimal is: Annex ciphers to the numerator; divide by the denominator, and annex a cipher to each remainder after the figures of the numerator are all used, proceeding exactly as if the numerator had an unlimited number of ciphers annexed to it, and was to be divided by the denominator. Continue this process until there is no remainder, and observe how many ciphers have been used. Place the decimal point in the quotient so as to cut off as many figures as you have used ciphers; and if there be not figures enough for this, annex ciphers to the beginning until there are places enough. 146. From what was shewn in (129), it appears that it is not every fraction which can be reduced to a decimal fraction. It was there shewn, however, that there is no fraction to which we may not find a decimal fraction as near as we please. i4285 Thus, I 14 ._4 142 _ o10I0o0100 1428 10000 00000, &c.,or ., *14, 1I42, "I428, *14285, were shewn to be fractions which approach nearer and nearer to -. To find either of these frac7 tions, the rule is the same as that in the last article, with this exception, 79 DECIMAL FRACTIONS. S146-148. that, I. instead of stopping when there is no remainder, which never happens, stop at any part of the process, and make as many decimal places in the quotient as are equal in number to the number of ciphers which have been used, annexing ciphers to the beginning when this cannot be done, as before. II. Instead of obtaining a fraction which is exactly equal to the fraction from which we set out, we get a fraction which is very near to it, and may get one still nearer, by using more of the quotient. I Thus, '1428 is very near to -, but not so near as 7 *142857; nor is this last, in its turn, so near as 142857142857, &c. 147. If there should be ciphers in the numerator of a fraction, these must not be reckoned with the number of ciphers which are necessary in order to follow the rule for changing it into a decimal fraction. Take, for example, -- ; annex ciphers to the numerator, and divide by the 125 denominator. It appears that Iooo is divisible by 125, and that the quotient is 8. One cipher only has been annexed to the numerator, and therefore zoo divided by 125 is '8. iooo divided by Had the fraction been 25 , since 125 gives 8, and three ciphers would have been annexed to the numerator, the fraction would have been 0oo8. 148. Suppose that the given fraction has ciphers at the right of its denominator; for example, --. 2500 The annexing a cipher to the nu- merator is the same thing as taking one away from the denominator ; for, (108) 310 is the same thing as 2500 31 11, and 310 as 3 3 250 250 The rule, 23 therefore, is in this case : Take away the ciphers from the denominator ; annex ciphers to the numerator; proceed as before; and in counting how many ciphers have been used, reckon not only the ciphers which have been annexed to the numerator, but also those which have been taken away from the denominator. EXERCISES. Reduce the following fractions to decimal fractions: I 800' 36 297 250' 64' -,-and I z28' Answer, *oo2z5, *c288, 4'64o625, and *0078125. Find decimals of 6 places very near to the following fractions: 80 .§148-150.g PRINCIPLES OF ARITHMETIC. 156 27 22 2637 194 I I 9907' 2908 466 3337000' 13 49 3 and277 Answer, "551020, 4"727272, "000594, 14'923076, z266175, *000343, *002145, and "010830. 149. Fiom (121) it appears, that if two fractions have the same denominator, the first may be divided by the second by dividing the numerator of the first by the numerator of the second. Suppose it required to divide x7'762 by 6*25. These fractions (138), when reduced to a common denominator, are 17"762 and 6250, or 17762 and 6250 Their quotient is therefore 1000 17762 I000 7----, which must now be reduced to a 6250 decimal fraction by the last rule. follows: The process at full length is as Leave out the cipher in the denominator, and annex ciphers to the numerator, or, which will do as well, to the remainders, when it becomes necessary, and divide as in (145). 625)17762(284192 1250 5262 Here four ciphers have been annexed to the numerator, and one has been taken from the denominator. 5000 Make five decimal places in the quotient, which then becomes 2.84192, and this is the quotient of x17762 divided by 625. 2620 2500 150. The rule for division of one decimal by Equalise the number of 625 another is as follows: 5750 decimal places in 5625 annexing ciphers to that which has fewest places. 1250 1250 0 common division. the dividend and divisor, by Then, further, annex as many ciphers to the dividend * as it is required to have decimal places, throw away the decimal point, and operate as in Make the required number of decimal places in the quotient. Thus, to divide 6'7 73 by '014 to three decimal places, I first write 6'7173 and o'040, with four places in each. Having to provide for three decimal places, I should annex three ciphers to 6"7173; but, observing * Or remove ciphers the from divisor; or make up the number of ciphers partly by removing from thedivisor annexing to the dividend, ifthere not a suffiand be cient number in the divisor. § 81 DECIMAL FRACTIONS. I50. that the divisor o014o has one cipher, I strike that one out and annex Throwing away the decimal points, then divide two ciphers to 6'7 173. 6717300 by 014 or 14 in the usual way, which gives the quotient 479807 Hence 479*807 is the answer. and the remainder z. The common rule is: Let the quotient contain as many decimal places as there are decimal places in the dividend more than in the divisor. But this rule becomes inoperative except when there are more decimals in the dividend than in the divisor, and a number of ciphers must be annexed to the former. The rule in the text amounts to the same thing, and provides for an assigned number of decimal places. But tie student is recommended to make himself familiar with the rule of the characteristic given in the Appendix, and also to accustom himself to reason out the place of the decimal point. that 26x119+7"2436 Thus, it should be visible, has one figure before the decimal point, and that 26"xI9+724'36 has one cipher after it, preceding all significant figures. Or the following rule may be used: Expunge the decimal point of the divisor, and move that of the dividend as many places to the right as there were places in the divisor, using ciphers if necessary. Then proceed as in common division, making one decimal place in the quotient for every decimal place of the final dividend which is used. Thus 17-314 divided by 6Iz2 is 173'14 divided by 612, and the decimal point must precede the first figure of the quotient. But 17'314 divided by 6617.5 is 1731I4 by 66175; and since three decimal places of 173*14000... must be used before a quotient figure can be found, that quotient figure is the third decimal place, or the quotient is "ooz ..... EXAMPLES. 3I ooz 5 = I240, 00062 '64 = *00096875 EXERCISES. Shew that 15 * ox'o+z'9xz'9xz2'9 0x x 29 2'91 6xShe ' that 006 15= = 2 9X2'9 - 004x'004 5oo002, 2'9X'o0+'oxx'ox. and that 82 PRINCIPLES OF ARITHMETIC. I I What are 314159 of decimals ?--Answer, Calculate ,828 and 3 365 , § 150-151. as far as 6 places 18349 "318310, "367879, and 198920o9221. 78288 io terms of each of the following series, as far as 5 places of decimals. I I I I+-+-+-+2 2X3 2X3X4 -+&c. IIII r1+-+-+-+-+&c. = 2345 8o 81 I'718z4. 2X3X4x5 82 83 -92895. 84 -+-+ - +- + 4+&c. = 9"88286. 81 8z 83 84 85 151. We now enter upon methods by which unnecessary trouble is saved in the computation of decimal quantities. And first, suppose a number of miles has been measured, and found to be 17"846217 miles. If you were asked how many miles there are in this distance, and a rough answer were required which should give miles only, and not parts of miles, you would probably say 17. But this, though the number of whole miles contained in the distance, is not the nearest number of miles; for, since the distance is more than 17 miles and 8 tenths, and therefore more than 17 miles and a half, it is nearer the truth to say, it is 18 miles. This, though too great, is not so much too great as the other was too little, and the error is not so great as half a mile. Again, if the same were required within a tenth of a mile, the correct answer is 178; for though this is too little by "o462I7, yet it is not so much too little as 17'9 is too great; and the error is less than half a tenth, or I --. Again, the same distance, within a hundredth of a mile, is more correctly 17'85 than 17.84, since the last is too little by "oo6217, which is greater than the half of orx; and therefore 1784+oI is nearer the truth than 17"84. Hence this general rule: When a certain number of the decimals given is sufficiently accurate for the purpose, strike off the rest from the right hand, observing, if the first figure struck off be equal to or greater than 5, to increase the last remaining figure by I. The following are examples of a decimal abbreviated by one place at a time. § 151-152. 83 DECIMAL FRACTIONS. 31I4I59, 3'1416, 3*I42, 314, 3*1, 3'0 2'7182818, z'718282, 2z71828, 2'7183, 2'718, 2'72, z'7, 3'0 I9919, I992, I'99, 200oo, 2o0 152. In multiplication and division it is useless to retain more places of decimals in the result than were certainly correct in the multiplier, &c., which gave that result. Suppose, for example, that 9"98 and 8*96 are distances in inches which have been measured correctly to two places of decimals, that is, within half a hundredth of an inch each way. The real value of that which we call 9'98 may be any where between 9"975 and 9"985, and that of 8*96 may be any where between 8*955 and 8'965. The product, therefore, of the numbers which represent the correct distances will lie between 9"975x8*955 and 9*985x8*965, that is, taking three decimal places in the products, between 89"326 and 8951i6. product of the actual numbers given is 89'4208. The It appears, then, that in this case no more than the whole number 89 can be depended upon in the product, or, at most, the first place of decimals. The reason is, that the error made in measuring 8*96, though only in the third place of decimals, is in the multiplication increased at least 9*975, or nearly Io times; and therefore affects the second place. The following simple rule will enable us to judge how far a product is to be depended upon. Let a be the multiplier, and b the multiplicand; if these be true only a+b* of the truth; if to to the first decimal place, the product is within a+b 20 a+b ; and so on. if to three, within places, within -; two decimal 200 2000 Thus, in the above example, we have 9"98 and 8*96, which are true to two decimal places: their sum divided by 2oo is o0947, and their product is 89*4208, which is therefore within o0947 of the truth. If, in fact, we increase and diminish 89'42o8 by "o947, we get 89'5155 and 89'3261, which are very nearly the limits found within which the product must lie. We see, then, that we cannot in this case depend upon the first place of decimals, as (151) an error of *05 cannot exist if this place be correct; and here is a possible error of "09 and upwards. It is hardly necessary to say, that if the numbers given be exact, their product * These are not quite correct, but sufficiently so for every practical purpose. 84 § 152-153. PRINCIPLES OF ARITHMETIC. is exact also, and that this article applies where the numbers given are The rule is: Take correct only to a certain number of decimal places. half the sum of the multiplier and multiplicand, remove the decimal point as many places to the left as there are correct places of decimals in either the multiplier or multiplicand; the result is the quantity within which the product can be depended upon. In division, the rule is: Proceed as in the last rule, putting the dividend and divisor in place of the multiplier and multiplicand, and divide by the square of the divisor; the quotient will be the quantity within which the division of the first dividend and divisor may be depended upon. Thus, if I7*324 be divided by 53*809, both being correct to the third place, their half sum will be 35'566, which, by the last rule, is made '035566, and is to be divided by the square of 53"809, or, which will do as well for our purpose, the square of 50, or 2500. The result is something less than "ooooz, so that the quotient of I7'324 and 53"809 can be depended on to four places of decimals. 153. It is required to multiply two decimal fractions together, sb as to retain in the product only a given number of decimal places, and dispense with the trouble of finding the rest. First, it is evident that we may write the figures of any multiplier in a contrary order (for example, 4321 instead of 1234), provided that in the operation we move each :ine one place to the right instead of to the left, as in the following example: 2221 2221 1234 4321 8884 6663 2221 4442 2221 2740714 4442 6663 8884 2740714 Suppose now we wish to multiply 348'84I4 by 51'30742, reserving only four decimal places in the product. If we reverse the multiplier, and proceed in the manner just pointed out, we have the following: DECIMAL FRACTIONS. § I53. 85 Cut off, by a vertical line, the first four places of decimals, and the columns which produced 3488414 2470315 them. 17442070 It is plain that in forming our abbre- 3488414 10465242 24418 898 1395 3656 69 76828 viated rule, we have to consider only, I. all that 178981522 23188 left of the line, we see 4, 4, 8, 5, 9, of which the is on the left of the vertical line; II. all that is carried from the first column on the right of the line. On looking at the first column to the first 4 comes from 4x1',* the second 4 from ix3', the 8 from 8x7', the 5 from 8x4', and the 9 from 4x2'. If, then, we arrange the multiplicand and the reversed multiplier thus, 3488414 2470315 each figure of the multiplier is placed under the first figure of the multiplicand which is used with it in forming the first four places of decimals. And here observe, that the units' figure in the multiplier 5130742, viz. I, comes under 4, the fourth decimal place in the multi- plicand. If there had been no carrying from the right of the vertical line, the rule would have been: Reverse the multiplier, and place it under the multiplicand, so that the figure which was the units' figure in the multiplier may stand under the last place of decimals in the multiplicand which is to be preserved; place ciphers over those figures of the multiplier which have none of the multiplicand above them, if there be any: proceed to multiply in the usual way, but begin each figure of the multiplier with the figure of the multiplicand which comes above it, taking no account of those on the right: place the first figures of all the lines under one another. To correct this rule, so as to allow for what is carried from the right of the vertical line, observe that this consists of two parts, 1st, what is carried directly in the formation of the different lines, and 2dly, what is carried from the addition of the first column on the right. The first of these may be taken into account by beginning each figure of the multiplier with the one which comes * The 1' here means that the 1 is in the multiplier. 86 PRINCIPLES OF ARITHMETIC. § 153-154, on its right in the multiplicand, and carrying the tens to the next figure as usual, but without writing down the units. But both may be allowed for at once, with sufficient correctness, on the principle of (151), by carrying i from 5 up to 15, 2 from 15 up to 25, &c.; that is, by carrying the nearest ten. Thus, for 37, 4 would be carried, 37 being nearer to 40 than to 30. This will not always give the last place quite correctly, but the error may beeavoided by setting out so as to keep one more place of decimals in the product than is absolutely required to be correct. The rule, then, is as follows: 154. To multiply two decimals together, retaining only n decimal places. I. Reverse the multiplier, strike out the decimal points, and place the multiplier under the multiplicand, so that what was its units' figure shall fall under the n t h decimal place of the multiplicand, placing ciphers, if necessary, so that every place of the multiplier shall have a figure or cipher above it. II. Proceed to multiply as usual, beginning each figure of the multiplier with the one which is in the place to its right in the multiplicand: do not set down this first figure, but carry its nearest ten to the next, and proceed. III. Place the first figures of all the lines under one another; add as usual; and mark off n places from the right for decimals. It is required to multiply I36*4072 by I'30609, retaining 7 decimal places. 1364072000 906031 1364072000 409221600 8184432 122766 178r1600798 § 87 DECIMAL FRACTIONS. 154-155. In the following examples the first two lines are the multiplicand and multiplier; and the number of decimals to be retained will be seen from the results. 33*166248 *4471618 3'7719214 37719214 8161744 I'4142136 • 033166248 3"4641016 1732'508 346410160 63124141 8052371 15087686 3316625 346410160 1508768 1326650 242487112 264034 33166 10392305 3772 13266 692820 2263 663 173205 38 33 2771 30 10 1.6866591 2 600ooI'58373 46"90415 Exercises may be got from article (143). 155. With regard to division, take any two numbers, for example, 16"8o4379z2 and 3"142, and divide the first by the second, as far as any required number of decimal places, for example, five. the following : 3'142) 16*8043791(5'34830 15 710 I 0943 9426 15177 (A) 12568 2609 2 609 9 2514 2513,6 95 96132 94 94126 I 2061 This gives 88 PRINCIPLES OF ARITHMETIC. § 155. Now cut off by a vertical line, as in (153), all the figures which come on the right of the first figure 2, in the last remainder 206r. As in multiplication, we may obtain all that is on the left of the vertical line by an abbreviated method, as represented at (A). After what has been said on multiplication, it is useless to go further into the detail; the following rule will be sufficient: To divide one decimal by another, retaining only n places: Proceed one step in the ordinary division, and determine, by (150), in what place is the quotient so obtained; proceed in the ordinary way, until the number of figures remaining to be found in the quotient is less than the number of figures in the divisor: if this should be already the case, proceed no further in the ordinary way. Instead of annexing a figure or cipher to the remainder, cut off a figure from the divisor, and proceed one step with this curtailed divisor as usual, remembering, however, in multiplying this divisor, to carry the nearest ten, as in (154), from the figure which was struck off; repeat this, striking off another figure of the divisor, and so on, until no figures are left. Since we know from the beginning in what place the first figure of the quotient is, and also how many decimals are required, we can tell from the beginning how many figures there will be ir the whole quotient. If the divisor contain more figures than the quotient, it will be unnecessary to use them : and they may be rejected, the rest being corrected as in (151) : if there be ciphers at the beginning of the divisor, if it be, for example, 'oo003178, since this is '3178, divide by 100 '3178 in the usual way, and afterwards multiply the quotient by Ioo, or remove the decimal point two places to the right. If, therefore, six decimals be required, eight places must be taken in dividing by '3178, for an obvious reason. In finding the last figure of the quotient, the nearest should be taken, as in the second of the subjoined examples. § 155-156. 89 EXTRACTION OF THE SQUARE ROOT. Places required, Divisor, 2 8 *41432 3"1415927 Dividend, 673.1489 271828180 2'51327416 414 32 258 828 20500764 248 592 18849556 10 237* 1651208 8 286 1570796 80412 x951 1 657 62832 294 17580 290 15708 4 1872 4 157 301 0 283 18 19 1624'71 Quotient, "86525596 Examples may be obtained from (143) and (150). SECTION VII. ON THE EXTRACTION OF THE SQUARE ROOT. 156. We have already remarked (66), that a number multiplied by itself produces what is called the square of that number. or x3x13, is the square of 13. Thus, 169, Conversely, 13 is called the square root of 169, and 5 is the square root of 25 ; and any number is the square root of another, which when multiplied by itself will produce that other. The square root is signified by the sign %Vor the square root of 25, or 5; -'; thus, z25 means V 1 6+9 means the square root of and is 5, and must not be confounded with V/16+ / 9, 16+9, which is 4+3, or 7. * This is written 7 instead of 6, because the figure which is abandoned in the dividend is 9 (151). 12 90 PRINCIPLES OF ARITHMETIC. § 157-159* 157. The following equations are evident from the definition: V/ax a = a Vaa = a Vabx ab = ab (Vax /b)x(Vaxb) = Vax /ax^/b whence ax- bxV b = ab = 158. It does not follow that a number has a square root because it has a square; thus, though 5 can be multiplied by itself, there is no number which multiplied by itself will produce 5. It is proved in algebra, that no fraction* multiplied by itself can produce a whole number, which may be found true in any number of instances; therefore 5 has neither a whole nor a fractional square root; that is, it has no square root at all. Nevertheless, there are methods of finding fractions whose squares shall be as near to 5 as we please, though not exactly 5127,whose square, viz. 4 576 , which is equal to it. One of these methods gives 15I27 I5z7 22zz88261z29 6765' 5' differs from 5 by only x6765 or 456 6 6765 6765 45765225 less than oooooo I : hence we are enabled to use V5 in arithmetical 45765225 and algebraical reasoning: but when we come to the practice of any problem, we must substitute for V5 one of the fractions whose square is nearly 5, and on the degree of accuracy we want, depends what For some purposes, 123 may be sufficient, as its 455 square only differs from 5 by 3 ; for others, the fraction first given fraction is to be used. 3025 might be necessary, or one whose square is even nearer to 5. We proceed to shew how to find the square root of a number, when it has one, and from thence how to find fractions whose squares shall be as near as we please to the number, when it has not. We premise, what is sufficiently evident, that of two numbers, the greater has the greater square; and that if one number lie between two others, its square lies between the squares of those others. 159. Let x be a number consisting of any number of parts, for example, four, viz. a, b,c,and d; that is, let * Meaning, of course, a really fractional number, such as - or i, 10 27 though fractional in form, is whole in reality, such as - or -. . not one which, § EXTRACTION OF THE SQUARE ROOT, 159-160. 91 -= a+b+c+d The square of this number, found as in (68), will be aa+za(b+c+d) +bb+zb(c+d) +cc4 zed +dd The rule there found for squaring a number consisting of parts was: Square each part, and multiply all that come after by twice that part, the sum of all the results so obtained will be the square of the whole number. In the expression above obtained, instead of multiplying za by each of the succeeding parts, b, c, and d, and adding the results, we multiplied 2a by the sum of all the succeeding parts, which (52) is the same thing; and as the parts, however disposed, make up the number, we may reverse their order, putting the last first, &c.; and the rule for squaring will be: Square each part, and multiply all that come before by twice that part. Hence a reverse rule for extracting the square root presents itself with more than usual simplicity. It is: To extract the square root of a number N, choose a number A, and see if N will bear the subtraction of the square of A; if so, take the remainder, choose a second number B, and see if the remainder will bear the subtraction of the square of B, and twice B multiplied by the preceding part A : if it will, there is a second remainder. Choose a third number C, and see if the second remainder will bear the subtraction of the square of C, and twice C multiplied by A+B: go on in this way either until there is no remainder, or else until the remainder will not bear the subtraction arising from any new part, even though that part were the least number, which is I. In the first case, the square root is the sum of A, B, C, &c.; in the second, there is no square root. 160. For example, I wish to know if 2025 has a square root. I choose 20 as the first part, and find that 400, the square of zo, subtracted from 2025, gives 1625, the first remainder. I again choose 20, whose square, together with twice itself, multiplied by the preceding part, is zoxzo+2xzox2o, or 1200oo; which subtracted from I625, the 92 ( 160-161. PRINCIPLES OF ARITHMETIC. first remainder, gives 425, the second remainder. I choose 7 for the third part, which appears to be too great, since 7x7, increased by 2x7 multiplied by the sum of the preceding parts zo+2o, gives 609, which I therefore choose 5, which closes the process, since is more than 425. 5x5, together with zx5 multiplied by 20+20, gives exactly 425. The square root of 2025 is therefore 20+20+5, or 45, which will be found, by trial, to be correct; since 45x45 = zoz5. has, or has not, a square root. is xoooo, and the first remainder is 3340. Here Again, I ask if 13340 oo be the first part, whose square Let Let Io be the second part. ox10+2zxoxo100 is 2100, and the second remainder, or 3340- 2100, is 1240. Let 5 be the third part; then 5x5+2x5x(IOO+IO) is 1125, which, subtracted from 1240, leaves 1I5. There is, then, no square root; for a single additional unit will give a subtraction of x +2zxIx(Ioo+Io+5), or 231, which is greater than 115. x5, number proposed had been less by have been 115 less, -I Therefore 13340 - But if the each of the remainders would and the last remainder would have been nothing. 5, or 13225, has the square root 100+10+5, 115; and the answer is, that 13340 has no square root, and that or 13225 is the next number below it which has one, namely, Ix 5 161. It only remains to put the rule in such a shape as will guide us to those parts which it is most convenient to choose. (57) It is evident that any number which terminates with ciphers, as 4000, has double the number of ciphers in its square. Thus, 4000x4000 = 160ooooo0000; therefore, any square number,* as 49, with an even number of ciphers annexed, as 490000, is a square number. 490000 is 700. The roott of This being premised, take any number, for example, 76176; setting out from the right hand towards the left, cut off two figures; then two more, and so on, until one or two figures only are left: thus, 7,61,76. This number is greater than 7,00,00, of which the first figure is not a square number, the nearest square below it being 4. Hence, 4,00,00 is the nearest square number below 7,00,00, which * By square number I mean, a number which has a square root. Thus, 25 is a square number, but 26 is not. t The term ' root' is frequently used as an abbreviation of square root. EXTRACTION OF THE SQUARE ROOT. § 161. has four ciphers, and its square root is zoo. Let this be the first part chosen: its square subtracted from 76176 leaves 36176, the first remainder; and it is evident that we have obtained the highest number of the highest denomination which is to be found in the square root of 76176; for 300 is too great, its square, 9,00,00, being greater than 76176: and any denomination higher than hundreds has a square still greater. It remains, then, to choose a second part, as in the examples of (160), with the remainder 36176. This part cannot be as great as Ioo, by what has just been said; its highest denomination is therefore a number of tens. Let N stand for a number of tens, which is one of the simple numbers I, 2, 3, &c. ; that is, let the new part be ioN, whose square is xoNxIoN, or IooNN, and whose double multiplied by the former part is zoNxzoo, or 4000N; the two together are 4oooN+ iooNN. Now, N must be so taken that this may not be greater than 36176: still more 4000N must not be greater than 36176. We.may therefore try, for N, the number of times which 36176 contains 4000, or that which 36 contains 4. The remark in (81) applies here. Let us try Then, zx9goxzoo+9ox9go, or 4410oo0, is to be subtracted, 9 tens or 90. which is too great, since the whole remainder is 36176. We then try 8 tens or 80, which gives zx8oxzoo+8ox8o, or 38400, which is likewise too great. On trying 7 tens, or 70, we find zx70X200+70X70, or 32900, which subtracted from 36176 gives 3276, the second remainder. rest of the square root can only be units. number of units. The As before, let N be this Then, the sum of the preceding parts being 200+70, or 270, the number to be subtracted is 70oxzN+NN, or 54 oN+NN. Hence, as before, 540N must be less than 3276, or N must not be greater than the number of times which 3276 contains 540, or (81) which 327 contains 54. We therefore try if 6 will do, which gives zx6xz7o+6x6, or 3276, to be subtracted. This being exactly the second remainder, the third remainder is nothing, and the process is finished. The square root required is therefore 200+70+6, or 276. The process of forming the numbers to be subtracted may be shortened thus. Let A be the sum of the parts already found, and N a new part: there must then be subtracted zAN+NN, or (54) 2A+N 94 § 161-163. PRINCIPLES OF ARITHMETIC. multiplied by N. The rule, therefore, for forming it is: Double the sum of all the preceding parts, add the new part, and multiply the result by the new part. 162. The process of the last article is as follows: 7,61,76(200 7,61,76(276 4 00 00 4 70 47)361 400 3,61,76 329 703 29 00 400 140) 6 546)3276 3276 32 76 32 76 - 0 0 In the first of these, the numbers are written at length, as we found them; in the second, as in (79), unnecessary ciphers are struck off, and the periods 6I, 76, are not brought down, until, by the continuance of the process, they cease to have ciphers under them. The following is another example, to which the reasoning of the last article may be applied. 34,86,78,44,01(50000 25 00000000 9000 100000 9000) 986784401 40 18000oo 25 - 10o9) 986 9 81oo000000oo 00ooooc 34,86,78,44,01(59049 981 578 4401 4 72 16 00 11804) 57844 47216 4C/ 0000ooooo 18000 80o 163. 6 28-o6 18089)1o628o0 I 06 28 o1 1062801 0 0 The rule is as follows: To extract the square root of a number ;I. Beginning from the right hand, cut off periods of two figures each, until not more than two are left. II. Find the root of the nearest square number next below the number in the first period. This root is the first figure of the required § 163. EXTRACTION OF THE SQUARE ROOT. 95 root; subtract its square from the first period, which gives the first remainder. III. Annex the second period to the right of the remainder, which gives the first dividend. IV. Double the first figure of the root; see how often this is contained in the number made by cutting one figure from the right of the first dividend, attending to IX., if necessary; use the quotient as the second figure of the root; annex it to the right of the double of the first figure, and call this the first divisor. V. Multiply the first divisor by the second figure of the root ; if the product be greater than the first dividend, use a lower number for the second figure of the root, and for the last figure of the divisor, until the multiplication just mentioned gives the product less than the first dividend; subtract this from the first dividend, which gives the second remainder. VI. Annex the third period to the second remainder, which gives the second dividend. VII. Double the first two figures of the root ;* see how often the result is contained in the number made by cutting one figure from the right of the second dividend; use the quotient as the third figure of the root; annex it to the right of the double of the first two figures, and call this the second divisor. VIII. Get a new remainder, as in V., and repeat the process until all the periods are exhausted; if there be then no remainder, the square root is found; if there be a remainder, the proposed number has no square root, and the number found as its square root is the square root of the proposed number diminished by the remainder. IX. When it happens that the double of the figures of the root is not contained at all in all the dividend except the last figure, or when, being contained once, 1 is found to give more than the dividend, put a cipher in the square root and in the divisor, and bring down the next period; should the same thing still happen, put another cipher in the root and divisor, and bring down another period ; and so on. * Or, more sipply, add the second figure of the root to the first divisor. 96 PRINCIPLES OF ARITHMETIC. § 163-165. EXERCISES. Numbers proposed. 73441 Square roots. 271 2992900 1730 6414247921 80089 903687890625 950625 42420747482776576 205962976 13422659310152401 115856201 164. Since the square of a fraction is obtained by squaring the numerator and the denominator, the square root of a fraction is found 25 5 is , by taking the square root of both. Thus, the square root of since 5x 5 is 25, and 8 x 8 is 64. If the numerator or denominator, or both, be not square numbers, it does not therefore follow that the fraction has no square root; for it may happen that multiplication or division the same number may convert both the numerator and by 27 , which appears at denominator into square numbers (108). Thus, since itis the same as first have no square root, has one in reality, to , whose square root is -. 4 165. We now proceed from (158), where it was stated that any number or fraction being given, a second may be found, whose square is as near to the first as we please. Thus, though we cannot solve the problem, " Find a fraction whose square is 2," we can solve the following, " Find a fraction whose square shall not differ from z by so much as *ooooooox." Instead of this last, a still smaller fraction may be substituted; in fact, any one however small: and in this process we are said to approximate to the square root of 2. This can be done to any extent, as follows : Suppose we wish to find the square root ofz Ia within - of the truth; by which I mean, to find a fraction - whose 57 as is greater than than 2, but such that the square of -+square isless b 57 z. Multiply the numerator and denominator of - by the square of 57, 6498 I or 3249, which gives . On attempting to extract the square root 3249 of the numerator, I find (163) that there is ; remainder 98, and that the square number next below 6498 is 6400, whose root is 8o. the square of 8o is less than 6 4 9 8, while that of 8 Hence, is greater. The § 1 6 5 -I6g. 97 EXTRACTION OF THE SQUARE ROOT. square root of the denominator is of course 57. Hence, the square of 8x 6498 80 is less than 6 . or 2, while that of - is greater, and these two 57 3249 x 57 fractions only differ by -; 57 which was required to be done. 166. In practice, it is usual to find the square root true to a certain number of places of decimals. Thus, 1'4142 is the square root of 2 true to four places of decimals, since the square of I4.142, or 1'99996164, is less than 2, while an increase of only I in the fourth decimal place, giving r4143, gives the square 200'ooo024449, which is greater than 2. To take a more general case: Suppose it required to find the square root of r637 true to four places of decimals. The fraction is 1637 1000 whose square root is to be found within ooo000, or----. Annex ciphers I0000 to the numerator and denominator, until the denominator becomes the square of 1 , which gives 163700000. 100000000 IOCCO extract the square root of the numerator, as in (163), which shews that the square number nearest to it is 163700000 - 13564, whose root is 12794. Hence, 794 or 0000, or 2794, 1794, gives a square less than 1'637, while I2795 gives a square greater. fact, these two squares are I63686436 and In 1'63712025 167. The rule, then, for extracting the square root of a number or decimal to any number of places is : Annex ciphers until there are twice as many places following the units' place as there are to be decimal places in the root; extract the nearest square root of this number, and mark off the given number of decimals. Or, more simply : Divide the number into periods, so that the units' figure shall be the last of a period; proceed in the usual way; and if, when decimals follow the units' place, there is one figure on the right, in a period by itself, annex a cipher in bringing down that period, and afterwards let each new period consist of two ciphers. Place the decimal point after that figure in forming which the period containing the units was used. 168. For example, what is the square root of Ig to five places of decimals? This is (145) over leaf. The second example is the extraction of the root of "o8i to "375, and the process is the first examp!e seven places, the first period being o8, from which the cipher is omitted as useless. 98 § 168-169. PRINCIPLES OF ARITHMETIC. 1,37,5(1'17260 8,1(*2846049 4 48)410 2I 384 227)1650 564) 2600 2256 1589 2342) 6xoo 5686) 34400 4684 34116 23446)141600 140676 23452) 569204) 2840000 2276816 569208) 92400 56318400 *ooooo0413672221x(*oo553599 I 25) 141 125 305) 1636 1525 3103) 11172 9309 31o65) 186322 155325 310709) 3099710 2796381 30332900 169, When more than half the decimals required have been found, the others may be simply found by dividing the dividend by the divisor, as in (155). The extraction of the square root of ix to ten places, which will be found in the next page, is an example. It must, however, be observed in this process, as in all others where decimals are obtained by approximation, that the last place cannot always be depended upon: on which account it is advisable to carry the process so far, that one or even two more decimals shall be obtained than are absolutely required to be correct. g 169. 99 EXTRACTION OF THE SQUARE ROOT. zz2(346410o61513 B 9 692820323o26)537253550831(77545870549 484974226118 64) 300 256 686) 4400 4116 52279324713 4849742261 3781902102 3464101615 6924) 28400 27696 69281) 317800487 277128129 70400 69281 692820 ) 11190000 40672358 6928201 34641016 69282026) 4261799 00 4156921 56 692820321) 104877 4400 69282 0321 6031342 5542562 488780 484974 69282032z5) 35595407900 3806 34641016125 3464 6928203230!) 95 39177500 692 82 3230 342 277 692820323023) 261 5714519900 207 8460969069 65 6z 537253550831 3 which If from any remainder we cut off the ciphers, and all figures line, would come under or on the right of these ciphers, by a vertical we find on the left of that line a contracted division, such as those in Thus, after having found the root as far as 3'464101, we have the remainder 4261799, and the divisor 6928202. The figures on the (165). left of the line are nothing more than the contracted division of this remainder by the divisor, with this difference, however, that we have to begin by striking a figure off the divisor, instead of using the whole divisor once, and then striking off the first figure. By this alone we might have doubled our number of decimal places, and got the addi- 100 § 169-170. PRINOIILES OF ARITHMETIC. tional figures 615137, the last 7 being obtained by carrying the contracted divis:on one step further with the remainder 53. then, this rule: We have, When half the number of decimal places have been obtained, instead of annexing two ciphers to the remainder, strike off a figure from what would be the divisor if the process were continued at length. and divide the remainder by this contracted divisor, as in (155). As an example, let us double the number of decimal places already obtained, which are contained in 3'464o10161513. The remainder is 53725355o831, the divisor 692820323026, and the process is as in (B). Hence the square root of 1z is, 3'4641016151377545870549; which is true to the last figure, and a little too great; but the substitution of 8 instead of 9 on the right hand would make it too small. EXERCISES. Numbers. 001728 Square roots. '0415692194 64'34 8"ozIzzx85 8074 10 89.8554394 1'57 I'2529964086I41667788495 3*16227766 SECTION VIII. ON THE PROPORTION OF NUMBERS. 170. When two numbers are named in any problem, it is usually necessary, in some way or other, to compare the two; that is, by considering the two together, to establish some connexion between them, which may be useful in future operations. The first method which suggests itself, and the most simple, is to observe which is the greater, and by how much it differs from the other. The connexion thus esta- blished between two numbers may also hold good of two other numbers; for example, 8 differs from 19 by II, and 100 differs from 111 by the § . 101 ON PROPORTION OF NUMBERS. 170-172 same number. In this point of view, 8 stands to 19 in the same situation in which oo stands to IxI, the first of both couples differing in the same degree from the second. The four numbers thus noticed, viz. : 8, 19, I00, III, are said to be in arithmetical* proportion. When four numbers are thus placed, the first and last are called the exatremes, and the second and third the means. It is obvious that 1I +8 = 100+19, that is, the sum of the extremes is equal to the sum of the means. And this is not accidental, arising from the particular numbers we have taken, but must be the case in every arithmetical proportion; for in II1+8, by (35), any diminution of III will not affect the sum, provided a corresponding increase be given to 8; and, by the definition just given, one mean is as much less than II as the other is greater than 8. 171. A set or series of numbers is said to be in continued arithmetical proportion, or in arithmetical progression, when the difference between every two succeeding terms of the series is the same. This is the case in the following series: 1, 2, 3, 4, 3, 6, 9, IZ, 2, 2-, I , I I-, &c. 3-, 3, The difference between two succeeding difference. &c. 15, I &c. terms is called the common In the three series just given, the common differences I are, I, 3, and -. 172. If a certain number of terms of any arithmetical series be taken, the sum of the first and last terms is the same as that of any other two terms, provided one is as distant from the beginning of the series as the other is from the end. For example, let there be 7 terms, and let them be, a b c d e f y. * This is a very incorrect name, since the term 'arithmetical' applies equally to every notion in this book. It is necessary, however, that the pupil should use words in the sense in which they will be used in his succeeding studies. 102 PRINCIPLES OF ARITHMETIC. § 172-173. Then, since, by the nature of the series, b is as much above a asf is below g (170), a+g = b+f. Again, since c is as much above b as e is below f(170), b+f = c+e. But a+g = b+f ; therefore a+g = c+e, and so on. Again, twice the middle term, or the term equally distant from the beginning and the end (which exists only when the number of terms is odd), is equal to the sum of the first and last terms; for since c is as much below d as e is above it, we have c+e = d+d = zd. But c+e = a+g; therefore, a+g =- 2d. This will give a short rule for finding the sum of any number of terms of an arithmetical series. 7, viz. those just given. Let there be Since a+g, b+f, and o+e, are the same, their sum is three times (a+g), which with d, the middle term, or half a+g, is three times and a half a+g, or the sum of the first and last terms multiplied by 3-, or _,or half the number of terms. If there had been an even number of terms, for example, six, viz. a, b, c, d, e, and f, we know now that a+f, b+e, and c+d, are the same, whence the sum is three times a+f, or the sum of the first and last terms multiplied by half the number of terms, as before. The rule, then, is: To sum any number of terms of an arithmetical progression, m'ultiply the sum of the first and last terms by half the number of terms. For example, what are 99 term is 99, and the sum terms of the series i, z, 3, &c. ? The is (99+) The sum of 50 terms of the series - ,, , or , 2, &c. is , or or4950. + -, 9 9 th or 17x25, or 425. 173. The first term being given, and also the common difference and number of terms, the last term may be found by adding to the first term the common difference multiplied by one less than the number of terms. For it is evident that the second term differs from the first by the common difference, the third term by twice, the fourth term by three times the common difference; and so on. Or, the passage from the first to the nth term is made by n-I steps, at each of which the common difference is added. 103 ON PROPORTION OF NUMBERS. § 173-175. EXERCISES. To find. Given. No. of terms. Series. I Last term. Sum. 1452 4, 6-, 9, &c. 33 84 I, 3, 5, &c. 28 55 784 100,000 1799984 89999300000 2, 20, 38, &c. 174. The sum being given, the number of terms, and the first term, we can thence find the common difference. Suppose, for example, the first term of a series to be one, the number of terms 1oo, and the sum o0,ooo. Since io,ooo was made by multiplying the sum of the first and 100 -, if we divide by this, we shall recover the sum of the last terms by first and last terms. I 2 first term being I,the last term is 199. 1 to is (12) zoo, and the Now, 10,000divided by We have then to pass from 199, or through 198, by 99 equal steps. Each step is, therefore, 198 or 2, which is the common difference; or the series is 1, 3, 5,&c., 99 -, up to 199. Given. To find. Sum. No. of terms. First term. Last term. 1809025 1345 1 2689 2 44 10 3 29 14 5 45 7075600 1330 4 o10636 Common diff. 8 175. We now return to (170), in which we compared two numbers together by their difference. This, however, is not the method of comparison which we employ in common life, as any single familiar instance will shew. For example, we say of A, who has Iothousand pounds, that he is much richer than B, who has only 3 thousand; but we do not say that C, who has o107 thousand pounds, is much richer than D, who has oo thousand, though the difference of fortune is the same in both cases, viz. 7 thousand pounds. In comparing numbers we take into our reckoning not only the differences, but the numbers themselves. Thus, if B and D both received 7 thousand pounds, B would receive 233 pounds and a third for every ioo pounds which he had before, while D for every ioo pounds would receive only 7 pounds. 104 PRINCIPLES OF ARITHMETIC. § 175-177. And though, in the view taken in (170), 3 is as near to io as Ioo is to Io7, yet, in the light in which we now regard them, 3 is not so near to Io as xoo is to ro7, for 3 differs from io by more than twice itself, while zoo does not differ from 107 by so much as one-fifth of itself. This is expressed in mathematical language by saying, that the ratio or proportion of Io to 3 is greater than the ratio or proportion of Ioo. o107 to We proceed to define these terms more accurately. 176. When we use the term part of a number or fraction in the remainder of this section, we mean, one of the various sets of equal parts into which it may be divided, either the half, the third, the fourth, &c,: the term multiple has been already explained (102). By the term multiple-part of a number we mean, the abbreviation of the words multiple of a part. Thus, z, 2, 3, 4, and 6, are parts of I2 ; - is also a 2 part of I2, being contained in it 24 times; 2, 24, 36, &c., are multiples of I2; and 8, 9, , &c. are multiple parts of iz, being multiples of some of its parts. And when multiple-parts generally are spoken of, the parts themselves are supposed to be included, on the same principle that iz is counted among the multiples of I2, the multiplier being I. The multiples themselves are also included in this term; for 24 is also 48 halves, and is therefore among the multiple parts of I2. Each part is also in various ways a multiple-part; for one-fourth is two-eighths, and three-twelfths, &c. 177. Every number or fraction is a multiple-part of every other number or fraction. If, for example, we ask what part in is of 7, we see that on dividing 7 into 7 parts, and repeating one of these parts 1z times, we obtain Iz ; or, on dividing 7 into 14 parts, each of which is one-half, and repeating one of these parts z times, we obtain 24 4 12 24 36 halves, or 12. Hence, 12 is-, or , or - of 7; andso on. Generally, 14 2 7 a when a and b are two whole numbers, expresses the multiple - part which a is of b, and - that which b is of a. Again, suppose it required a z5 6 to determine what multiple-part is of 3-, orof . These 7 5 7 5 fractions, reduced to a common denominator, are and of which 35 35 the second, divided into I12 parts, gives -,I which repeated 75 times 2- 35 ON PROPORTION OF NUMBERS. § 177-178. gives 7, the first. second is 105 Hence, the multiple-part which the first is of the 12-, being obtained by the rule given in (121), shews which that b, or a divided by b, according to the notion of division there given, expresses the multiple-part which a is of b in every case. 178. When the first of four numbers is the same multiple-part of the second which the third is of the fourth, the four are said to be geometrically* proportional, or simply proportional. This is a word in common use; and it remains to shew that our mathematical definition of it, just given, is, in fact, the common notion attached to it. For example, suppose a picture is copied on a smaller scale, so that a line of two inches long in the original is represented by a line of one inch and a half in the copy; we say that the copy is not correct unless all the parts of the original are reduced in the same proportion, namely, I that of z to I-. Since, on dividing two inches into 4 parts, and taking 3 of them, we get I-, the same must be done with all the lines in the original, that is, the length of any line in the copy must be three parts out of four of its length in the original. Again, interest being at 5 per cent, that is, £5 being given for the use of £ioo, a similar proportion of every other sum would be given; the interest of £70, for example, would be just such a part of £70 as £5 is of £10oo. Since, then, the part which a is of b is expressed by the fraction b, or any other fraction which is equivalent to it, and that which o is of d c a c by d' it follows, that when a, b,c, and d, are proportional, b d. This equation will be the foundation of all our reasoning on proportional quantities; and in considering proportionals, it is necessary to observe not only the quantities themselves, but also the order in which they come. Thus, a, b, c, and d, being proportionals, that is, a being the same multiple-part of b which c is of d, it does not follow that a, d, b, and c are proportionals, that is, that a is the same multiple-part of d * The same remark may be made here as was made in the note on the term arithmetical proportion,' page 101. The word 'geometrical' is, generally speaking, dropped, except when we wish to distinguish between this kind of proportion and that which has been called arithmetical. 106 PRINCIPLES OF ARITHMETIC. which b is of c. 1 178-I82. It is plain that a is greater than, equal to, or less than b,according as c is greater than, equal to, or less than d. 179. Four numbers, a, b, c, and d, being proportional in the order written, a and d are called the extremes, and b and o the means, of the proportion. For convenience, we will call the two extremes, or the two means, similar terms, and an extreme and a mean, dissimilar terms. Thus, a and d are similar, and so are b and c; while a and b, a and o, d and b,d and c, are dissimilar. It is customary to express the proportion by placing dots between the numbers, thus: a: b:: o: d 180. Equal numbers will still remain equal when they have been increased, diminished, multiplied, or divided, by equal quantities. This amounts to saying that if a = b and p = q, a+p = b+q, a-p = b-q, ap = bq, and a b. p q all equal to a. Itis also evident, that a+p-p, a--p+p, p, and axp,are p p 181. The product of the extremes is equal to the product of the ao Let-=-, and multiply these equal numbers by the product a dabd a cbd bd. Then, -xbd= a-(116)=ad, and xbd= d =cb: hence (180), b 6 d d 63 3x7 .3x7 ad = be. Thus, 6, 8, 21, and 28, are proportional, since 6 21 8 4 4x7 = (180); and it appears that 6xz8 = 8xzx, since both products are means. 28 182. If the product of two numbers be equal to the product of two others, these numbers are proportional in any order whatever, provided the numbers in the same product are so placed as to be similar terms; that is, if ab = pq, we have the following proportions :a: p :: q: b a: q:: p: b b : p :: q : a p: a:: b : q p: b::a: q q:a::b:p b:q.:p:a q:b::a:p To prove any one of these, divide both ab and pq by the product of its second and fourth terms ; for example, to shew the truth of a : q : : p: b, divide both ab and pq by bq. Then, ab =,a and = ; hence (180), bq q bq 6' . 182-184. 107 ON PROPORTION OF NUMBERS. a p, or a -= : q :: p : b. The pupil should not fail to prove every one of the eight cases, and to verify them by some simple examples, such as 1x6 = zx3, which gives I : 2 :: 3 : 6, 3 : I :: 6 : 2, &c 183. Hence, if four numbers be proportional, they are also proportional in any other order, provided it be such that similar terms still remain similar. a s For since, when b it follows (181) that ad = bc, all the proportions which follow from ad = be, by the last article, follow da also from a b d184. From (114) it follows that 1+ a b-a Sx-b---,while if a a be greater than a+b a-b b b b be divided by = bI, I, and if be less than I a-b b Also (122), if -1 = a+b the result is . Hence, a, b, c, and d, being a--" proportionals, we may obtain other proportions, thus : Let b = o d Then (114) z+6a +d c+d d or a+b b or a+b : b :: c+d : d That is, the sum of the first and second is to the second as the sum of the third and fourth is to the fourth. For brevity, we shall not state in words any more of these proportions, since the pupil will easily supply what is wanting. Resuming the proportion a : b :: c : d a - or b c - d a c I--- = x--, or b--a b if a be less than I, d-c d that is, b-a : b :: d-c : d or, a-b : b :: c-d : d, ifb be greater than I. PRINCIPLES CF ARITHMETIC. Aga, since a+b b c+d d a-b b beinggreater than x), c-d d a+b dividing the first by the second we have a- § 184-187. c+d =-d' a+b : a-b :: c+d : c-d or and also a+b : b-a :: c+d : d-c, if- be less than I. 185. Many other proportions might be obtained in the same manner. We will, however, content ourselves with writing down a few which can be obtained by combining the preceding articles. :: c+d : o a+b : a a a+c : a-b :: o : -d : a-c :: l+d : b-d. In these and all others it must be observed, that when such expressions as a-b and o-d occur, it is supposed that a is greater than b, and c greater than d. 186. If four numbers be proportional, and any two dissimilar terms be both multiplied, or both divided by the same quantity, the results are proportional. Thus, if a : b :: c : d, and m and n be any two num- bers, we have also the following : ma: a a - ma : nb :: me : nd b :: me : d : mb : md and various others. - c m a n m : mb :: - : md n b a d m" b m m * d m n a :: - n To prove any one of these, recollect that nothing more is necessary to make fdur numbers proportional except that the product of the extremes should be equal to that of the means. Take the third of those just given; the product of its extremes is -xmd, or But since a : b :: : d, mad while that of the means is mbx -, or a-. n n n mad mba . Hence, = by (181) ad = be, whence, by (180), mad = mbe, and --n n a mb, -, and md, are proportionals. no 187. If the terms of one proportion be multiplied by the terms of a if second, the products are proportional; that is, a : b :: c : d, and 109 ON PROPORTION OF NUMBERS. § 187-190. p : q :: r : s, it follows that ap : bq :: cr : ds. For, since ad= be, and ps = qr, by (180) adps = bcqr, or apxds = bqxcr, whence (182) ap : bq :: er : ds. 188. If four numbers be proportional, any similar powers of these numbers are also proportional; that is, if a: b :: c : d aa : Then bb :: cc : dd aaa : bbb :: cce : ddd &c. &c. For, if we write the proportion twice, thus, a: c : d bb :: cc : dd a: But Whence (187) c b :: aa : by (187) b:: a : d b :: c : d aaa : bbb :: ccc : ddd; and so on. '89. An expression is said to be homogeneous with respect to any two or more letters, for instance, a, b, and c, when every term of it contains the same number of letters, counting a, b, and c only. Thus, maab+nabc+rccois homogeneous with respect to a, b, and c; and of the third degree, since in each term there is either a, b, and c, or one of these repeated alone, or with another, so as to make three in all. Thus, 8aaabc, Izabcce, maaaaa, naabbc, are all homogeneous, and of the fifth degree, with respect to a, b, and c only; and any expression made by adding or subtracting these from one another, will be homogeneous and of the fifth degree. Again ma+mnb is homogeneous with respect to a and b, and of the first degree ; but it is not homogeneous with respect to m and n, though it is so with respect to a and n. This being pre- mised, we proceed to a theorem,* which will contain all the results of (184), (185), and (188). 190. If any four numbers be proportional, and if from the first two, * A theorem is a general mathematical fact: thus, that every number is divisible by four when its last two figures are divisible by four, is a theorem; that in every proportion the product of the extremes is equal to the product of the means, is another. 110 o. PRINCIPLES OF ARITHMETIC. § 190 a and b, any two homogeneous expressions of the same degree be formed ; and if from the last two, two other expressions be formed, in precisely the same manner, the four results will be proportional. For example, if a : b :: c : d, and if zaaa+3aaband bbb+abb be chosen, which are both homogeneous with respect to a and b, and both of the third degree; and if the corresponding expressions 2cc+3ccd and ddd+cdd be formed, which are made from a and d precisely in the same manner as the two former ones from a and b, then will zaaa+3aab : bbb+abb : : 2ccc+3ccd : ddd+cdd a To prove this, let b be called x. follows that d = x. a Then, since - = a , and -b c , it But since a divided by b gives x, a multiplied by b will give a, or a = bx. For a similar reason, c = dx. Put be and dx instead ofa and o in the four expressions just given, recollecting that when quantities are multiplied together, the result is the same in whatever order the multiplications are made; that, for example, bbbbx is the same as bbbxx. saaa+3aab = zbxbxbxt-3bxbxb Hence, = which is zbbbxxx+3bbbx6z zxxx+3xx bbb multiplied by or bbb (zxxx+3xx)* Similarly, zccc+3ccd = ddd (zxxx+3xx) Also, bbb+ abb = bbb+bxbb = bbb multiplied by +zx or bbb (I+x) Similarly, ddd +cdd = ddd (x+x) Now, bbb : ddd: : bbb : ddd Whence (186), bbb(txxx+3xx) : ddd(x+i) :: bbb(zxxx+3xx) : ddd (i+x), which, when instead of these expressions their equals just found are substituted, becomes zaaa+3aab : bbb+abb :: zccc+3ccd : ddd+cdd. * If bx be substituted for a in any expression which is homogeneous with respect to a and b, the pupil may easily see that b must occur in every term as often as there are units in the degree of the expression: thus, aa+abbecomes bxbx+bxb or bb(xx+x); aaaS4bbb becomes bxbbxbz+bbb or bbb(xxx+l); and so on. 111 ON PROPORTION OF NUMBERS. § 190-192. The same reasoning may be applied to any other case, and the pupil may in this way prove the following theorems: If a: b:: :d za+3b : b :: c2+3d : d aa+bb : aa-bb :: cc+dd : cc-dd mab : zaa+bb :: mcd : zcc+dd 191. If the two means of a proportion be the same, that is, if a : b :: b : c, the three numbers, a, b, and c, are said to be in continued proportion, or in geometrical progression. The same terms are applied to a series of numbers, of which any three that follow one another are in continued proportion, such as I 2 8 i6 32 64 27 81 243 729 4 3 9 &c. Which are in continued proportion, since 2 2 :4:4:8 2 3 2 9 2 3 2 2 3 2 9 9 27 &c. &c. 192. Let a, b, c, d, &c. be in continued proportion; we have then or c d d e b 6b c c e b : o :: c : d a - ... b : a ::: d ... - d or ... bd=cc ... e ac=L bb ce = dd Each term is formed from the preceding, by multiplying it by the same b 0 a bb o -,-= Thus, b = -xa (180); c = xb; and since number. b or c= -xb. ba a Again, d = -x c, and so on. a d d = -xo, d but b be bb , b' which is = c 6 -;therefore, If, then, - (which is called the common ratio of a the series) be denoted by r, we have b = ar a a c =br -- arr and so on; whence the series d = cr = arrr 112 PRINCIPLES OF ARITHMETIC. § 192-193. a b c d &c. a is ar arr arrr &c. a : e :: Hence a : arr :: aa : aarr (186) :: aa : because, b being ar, bb is arar or aarr. bb Again, a : a : d: arrr :: aaa : aaarrr : aaa : bbb a : e :: aaaa : bbbb, (186) Also and so on; t that is, the first bears to the n h term from the first the same proportion as the n th th power of the first to the n power of the second. 193. A short rule may be found for adding together any number of terms of a continued proportion. Let it be first required to add together the terms i, r, rr, &c. where r is greater than unity. It is evident that we do not alter any expression by adding or subtracting any numbers, For example, provided we afterwards subtract or add the same. p = p-q+q-r+r-s+s Let us take four terms of the series, I, r, rr, &c. or, x+r+rr+rrr It is plain that rrrr- I = rrrr-rrr+rrr-rr+rr-r+r-I rrrr-rrr = rrr (r-x), Now (54), rr-r = r(r-,), rrr-rr = rr(r-I), and the above equation becomes rrrr--I = rrr (r-I) + rr(r--I) + r (r-) +r-x; which is (54) rrr+rr+r+xtaken r-I times. Hence, rrrr-x divided by r-i will give I+r+rr+rrr,the sum of the terms required. In this way may be proved the following series of equations : rr- I r- I+r I = rrr-I+r+rr = r I+r+rr+rrr I+r+rr+rrr+rrrr = = - I rrr-I r-I rrrrr- I T- I 113 ON PROPORTION OF NUMBERS. § 193-194* If r be less than unity, in order to find 1+r+rr+rrr, observe that I-rrrr = I-r+r--rr+rr-rrr+rrr-rrrr = I-rr(I-r)+rr(I-r)+rrr(i-r) ; is whence, by similar reasoning, I+r+rr+rrr found by dividing r-rrrr by I-r ; and equations similar to these just given may be found, which are, --rr I-7 x+r+rr = I-rrr --r I--rrrr i+r+rr+rrr = I-,. z+r+rr+rrr+rrrr = x-rrrrr I-r The rule is: To find the sum of n terms of the series, I+r+rr+&c., divide the difference between I and the (n+x)th term by the difference between I and r. 194. This may be applied to finding the sum of any number of terms of a continued proportion. Let a, b,c, &c. be the terms of which it is required to sum four, that is, to find a+b+-c+d, or (192) a+ar rrrr--I I--rrrr xa, or +arr+arrr, (54) a(i +r+rr+rrr), which (193) is .- * .... or r--I I--7 xa, according as r is greater or less than unity. The first fraction is a-e e-a arrrr-a . The rule, . Similarly, the second is , or (192) r- I I-r r- I therefore, is: To sum n terms of a continued proportion, divide the t h difference of the n+ I and first terms by the difference between unity and the common measure. For example, the sum of to terms of the series I+3+9+27+&c. is required. The eleventh term is 59049, and is 29524. Again, the sum of 18 terms of the series 2+1+ 59049-I 3-I I +-+&c. I I of which the nineteenth term is 1 , is 131072 24 3 1310702 1310o72 EXAMPLES. 9 terms of I+4+ r6+&c. L 2 ane 87381 - 13o072 1 1-- = 114 PRINCIPLES OF ARITHMETIC. 6 12 Io terms of 3+ -+ &c. 7 49 I I 1 20 ...... +-+ z 4 § 194-197. are 847422675 201768035 .. 1048575 1048576 8 +&c . 195. The powers of a number or fraction greater than unity increase; for since 2I is greater than 2 1, 2IX 2 is, is greater than 2-, and so on. 2I is 2- taken more than once, that 2 2 This increase goes on without limit; that is, there is no quantity so great but that some power of 2- is greater. 2 To prove this, observe that every power of z- is made by multiplying the 2 preceding power by , or by 1+I-, 2 that is, by adding to the former 2 power that power itself and its half. added to the xoth There will, therefore, be more power to form the IIth, than was added to the power to form the ioth. 9 th But it is evident that if any given quantity, however small, be continually added to 2-, the result will come in time z to exceed any other quantity that was also given, however great; much more, then, will it do so if the quantity added to 2- be increased at each step, which is the case when the successive powers of z- are formed. It is evident, also, that the powers of x never increase, being always I; thus, Ixi = I, &c. Also, if a be greater than m times b, the square of a is greater than mm times the square of b. Thus, if a = zb+c, where a is greater than zb, the square of a, or aa, which is (68) 46b+4bc+cc is greater than 4bb, and so on. 196. The powers of a fraction less than unity continually decrease; thus, the square of -, or - x , is less than being only two-fifths of it. 5 5 5 This decrease continues without limit; that is, there is no quantity so 2 5 2 1 small but that some power of - is less. For if- 5 = , = , and the powers of 2 5 I are -, and so on. -, zz x 5 Since e' a 5 2 x is greater than I (195), some power of x may be found which shall be greater than a given quantity. Let this be called m ; then 2 is ~mthe corresponding power of 5 ; and a fraction whose denominator can be made as great as we please, can itself be made as small as we please (112). 197. We have, then, in the series I r rr rrr rrrr I. A series of increasing terms, if r be greater than &c. i. II. Of terms § i97. 115 ON PROPORTION OF NUMBERS. having the same value, if r be equal to I. terms, if r be less than x. III. A series of decreasing In the first two cases, the sum I+r+rr+rrr+&c. may evidently be made as great as we please, by sufficiently increasing the number of terms. But in the third this may or may not be the case ; for though something is added at each step, yet, as that augmentation diminishes at every step, we may not certainly say that we can, by any number of such augmentations, make the result as great as we please. To shew the contrary in a simple instance, consider the series, I +-+2 I I +-+ +&c. 8 i6 4 Carry this series to what extent we may, it will always be necessary to add the last term in order to make as much as 2. ( I I+-+- I I +-= I I 1+I I+-+-= 4 24 2 I Thus, = 2 2 I 1+--+ +- 8 24 I I s 2 4+ 8 = z. I +6 6 But in the series, every term is only the half of the preceding; consequently no number of terms, however great, can be made as great as 2 by adding one more. III The sum, therefore, of I, ,4 2 8&c. con- 4 8' tinually approaches to 2, diminishing its distance from 2 at every step, but never reaching it. I I Hence, 2 is called the limit of I+-+-+&c. 24 We are not, therefore, to conclude that every series of decreasing terms has a limit. The contrary may be shewn in the very simple series, III I +-+-+-+&c. which may be written thus: 234 x -+z - + +...up to + +...up to -+ +...up to +&c. We have thus divided all the series, except the first two terms, into lots, each containing half as many terms as there are units in the denominator of its last term. Thus, the fourth lot contains I6 or Each of these lots may be shewn to be greater than -. 3 terms. Take the third, 116 PRINCIPLES OF ARITHMETIC. IIIIIII for example, consisting of , ,o' 9 II'-Iz'' 145 1 4, -,-, II I I0'I I I -6' the last, are greater than -; 197-199. I and --. All except 6 consequently, by substituting I for each of them, the amount of the whole lot would be lessened; and as 8 1 I it would then become 8, or -, the lot itself is greater than i. Now, I I if to I+-,- be continually added, the result will in time exceed any given number. Still more will this be the case if, instead of -, the But it is thus several lots written above be added one after the other. II that the series I +-+-, &c. is composed, which proves what was said, 2 3 that this series has no limit. always has a limit when r is less 198. The series I+r+rr+rrr+&c. To prove this, let the term succeeding that at which we stop than I. I-a be a, whence (194) the sum is - I ,or (112) a I--r 1-r . The terms I-r decrease without limit (196), whence we may take a term so far distant , shall be as small as we from the beginning, that a, and therefore I-r please. I a I-7 I I-- But it is evident that in this case less than I -, may be brought as near to - I-r , though always as we please; that is, the I-r I series I+r+rr+&c. continually approaches to the limit I II . I-Tr I I+-+-+- +&c. where r = -, continually approaches to 2 4 3 I Thus or 2, as I-4 2 was shewn in the last article. EXERCISES. The limit of 2+ - + 3 9 or r 2z(+-+ 3 ........ I - +&c.)is3 9 9 8 +-+-+&c. IO ... 10 100 7 ...... 8+&c. 49 5+-1 +&c.... 199. When the fraction b is not equal to 84 , but greater, . is said to a c have to b a greater ratio than c has to d; and when is said to have to b a less ratio than c has to d. -is less than d, a We propose the fol- 117 ON PROPORTION OF NUMBERS. § I99-zoz. lowing questions as exercises, since they follow very simply from this definition. I. If a be greater than b, and c less than or equal to d, a will have a greater ratio to b than c has to d. II. If a be less than b, and c greater than or equal to d, a has a less ratio to b than c has to d. III. If a be to b as c is to d, and if a have a greater ratio to b than c has to x, d is less than x; and if a have a less ratio to b than c to x, d is greater than x. IV. a has to b a greater ratio than ax to bx+y, and a less ratio than ax to bx-y. 200. If a have to b a greater ratio than c has to d, a+c has to b+d a less ratio than a has to b, but a greater ratio than c has to d; or, in greater than 0 ca+ aa be the greater of the two fractions b and c a+ will be b+d tw' b other words, if than but less a . To shew this, observe that b d mx+ny must m+n lie between x and y, if x and y be unequal : for if x be the less of the msr+n or than x; and if y be the greater two, it is certainly greater than m+n of the two, it is certainly less than my+ny, or than y. It therefore lies bm+n between n and y. Now let -be x, and let - be y: then a = bx, o= dy. 6 as Now bx+dy is something between x and y, d was just proved; therefore b+d a+e is something between a and c Again, since a and a d d. d are respectively cq an siPnQ ap b+d b d b d and -, and since, as has just been proved, ap+cq lies beequal to bp bp+dq dq tween the two last, it also lies between the two first; that is, if p and a a ap+cq lies between b and d. q be any numbers or fractions whatsoever, bp 201. By the last article we may often form some notion of the value I+x of an expression too complicated to be easily calculated. Thus, x+z a_ b I ax+by b a , ax lies between -- and lies between - and -, or I and - ;a bbyy axx x axx+bbyy xx i between and -. And it has been shewn that a+. lies bys a and b,the denominator being considered as z+I. between that is, tha 202. It may also be b a ,a -, and lies among p q r greater than the least. a+b+c+d proved that a fraction such as -------- always p+q+r+s d -, that is, is less than the greatest of them, and s Let these fractions be arranged in order of PRINCIPLES 118 OF S202-20o4. ARITHMETIC. a bb c magnitude; that is, let - be greater than be greater than -, and - d greater than -. 8 QQ P Then by (200) q a+b p+q a p a+b+ o 5 a+b a+b+a+band p+q+r , a p+q p a+b+and a and p+q+r p a+b+c+d p+q+r+s b -and q c r c -and r A d s d s whence the proposition is evident. 203. It is usual to signify " a is greater than b" by a > b, and " a is less than b" by a < b; the opening of A being turned towards the greater quantity. The pupil is recommended to make himself familiar with these signs. SECTION IX. ON PERMUTATIONS AND COMBINATIONS. 204. If a number of counters, distinguished by different letters, be placed on the table, and any number of them, say four, be taken away, the question is, to determine in how many different ways this can be done. Each way of doing it gives what is called a combination of four, but which might with more propriety be called a selection of four. Two combinations or selections are called different, which differ in any way whatever; thus, abed and abce are different, d being in one and e in the other, the remaining parts being the same. Let there be six counters, a, b, c, d, e, and f; the combinations of three which can be made out of them are twenty in number, as follow: - abc ace " bed bef abd acf bee -cde abe ade bcf cdf abf adf bde cef -acd aef bdf def The combinations of four are fifteen in number, namely, 119 PERMUTATIONS AND COMBINATIONS. § zc4-zo6. abed abde acde adef beef abce abdf acdf bcde bdef abcf abef acef bcdf cdef and so on. 205. Each of these combinations may be written in several different orders; thus, abed may be disposed in any of the following ways: abed acdb acbd abde adbc adeb dabs dacb bacd cabd cadb bade bead cbad edab bdae dbac dcab bcda cbda edba bdca dbea deba Each of these is said of which no two are entirely in the same order. to be a distinct permutation of abed. Considered as a combination, they are all the same, as each contains a, b, c, and d. 206. We now proceed to find how many permutations, each containing one given number, can be made from the counters in another If we knew how to find all the per- given number, six, for example. mutations containing four counters, we might make those which contain five thus: Take any one which contains four, for example, abef, in which d and e are omitted ; write d and e successively at the end, which gives abfd, abcfe, and repeat the same process with every other permutation of four; thus, dabc gives dabce and dabef. No permutation of five can escape us if we proceed in this manner, provided only we know those of four; for any given permutation of five, as dbfea, will arise in the course of the process from dbfe, which, according to our rule, furnishes dbfea. Neither will any permutation be repeated twice, for dbfea, if the rule be followed, can only arise from the permutation dbfe. begin in this way to find the permutations of two out of the six, a b c d e each of these gives five; thus, a gives ab ac ad ae af b ...... ba be bd be bf and the whole number is 6x5, or 30. f If we 120 Again, § 2c6-z207. PRINCIPLES OF ARITHMETIC. ab gives abc abd abe abf ac ...... acb acd ace acf and here are 30, or 6x 5 permutations of 2, each of which gives 4 permutations of 3 ; the whole number of the last is therefore 6x 5x4, or I20. Again abc gives abed abce abof abd ...... abdc abde abdf and here are zo, or 6x5x4, permutations of three, each of which gives 3 permutations of four; the whole number of the last is therefore 6x5x4x3, or 360. In the same way, the number of permutations of 5 is 6x5x4x3xz, and the number of permutations of six, or the number of different ways in which the whole six can be arranged, is 6x5x4x3x2x1. The last two results are the same, which must be; for since a permutation of five only omits one, it can only furnish one permutation of six. If instead of six we choose any other number, x, the number of permutations of two will be m(x-I), that of three will be z(-I)(x-z2)(m-3), z(m- I)(x-z), that of four the rule being: Multiply the whole number of counters by the next less number, and the result by the next less, and so on, until as many numbers have been multiplied together as there are to be counters in each permutation: the product will be the whole number of permutations of the sort required. Thus, out of 12 counters, permutations of four may be made to the number of I2zxix1ox9, or 1188o. EXERCISES. 207. In how many different ways can eight persons be arranged on eight seats ? Answer, 40320. In how many ways can eight persons be seated at a round table, so that all shall not have the same neighbours in any two arrangements ?* Answer, 5040. If the hundredth part of a farthing be given for every different * The difference between this problem and the last is left to the ingenuity of the pupil. zo--zo E 9. PERMUTATIONS 121 AND COMBINATIONS. arrangement which can be made of fifteen persons, to how much will the whole amount ? Answer, £1362160o8. Out of seventeen consonants and five vowels, how many words can be made, having two consonants and one vowel in each ? Answer, 4080o. 208. If two or more of the counters have the same letter upon them, the number of distinct permutations is less than that given by the last rule. Let there be a, a, a, b, c, d,and, for a moment, let us distinguish between the three as thus, a, a', a". Then, abca'a"d, and a"bcaa'd are reckoned as distinct permutations in the rule, whereas they would not have been so, had it not been for the accents. To compute the number of distinct permutations, let us make one with b, c, and d, leaving places for the as, thus, ( ) be ( ) ( ) d. If the as had been distinguished as a, a', a", we might have made 3xzxI distinct permutations, by filling up the vacant places in the above, all which six Hence, to deduce are the same when the as are not distinguished. the number of permutations of a, a, a, b, c, d, from that of aa'a"bd, we must divide the latter by 3xzxI, or 6, which gives 6x5x4x3xzxx 3xzx1 or z2o. Similarly, the number of permutations of aaaabbbco is 6 x 9X8x7X X5x4X3X3X 4X3X2XIX3X2XIXIXI EXERCISE. How many variations can be made of the order of the letters in the Answer, Iz6I269oo. word antitrinitarian ? 209. From the number of permutations we can easily deduce the number of combinations. But, in order to form these combinations independently, we will shew a method similar to that in (206). If we know the combinations of two which can be made out of a, b, c, d, e, we can find the combinations of three, by writing successively at the end of each combination of two, the letters which come after the last contained in it. Thus, ab gives abc, abd, abe; ad gives ade only. No combination of three can escape us if we proceed in this manner, provided only we know the combinations of two; for any given combination of three, as acd, will arise in the course of the process from ao, which, according to our rule, furnishes acd. Neither will any combination be repeated M 122 § 209-2I0. PRINCIPLES OF ARITHMETIC. twice, for aed, if the rule be followed, can only arise from ac, since neither ad nor cd furnishes it. If we begin in this way to find the combinations of the five, a a b c d e ac c be ce ...... d ae bd l..... ed cb ad be gives b ...... Of these, de ab gives ac ...... abc abd abe acd ace ad ...... be ade bed ...... cd ae Of these, ce ede ...... be abo and gives bee bde bd ...... de give none. abed abce abd ...... abde acd ...... acde bed bede ...... Those which contain e give none, as before. Of the last, abed gives abode, and the others none, which is evidently true, since only one selection of five can be made out of five things. 210. The rule for calculating the number of combinations is derived directly from that for the number of permutations. Take 7 counters; then, since the number of permutations of two is 7x6, and since two permutations, ba and ab, are in any combination ab, the 7X6 number of combinations is half that 6f the permutations, or -. Since the number of permutations of three is 7x6x5, combination abc has 3xzxx permutations, and as each the number of combina- tions of three is 7x6x5 . Also, since any combination of four, abed, Ixzx3 contains 4x3x2xx permutations, the number of combinations of four is 7x6xx4 and so on. The rule is: To find the number of comIx2x3x4 h biiations, each containing n counters, divide the corresponding number 123 PERMUTATIONS AND COMBINATIONS. § 2o-2II. of permutations by the product of I, 2, 3, &c. up to n. whole number, the number of combinations of two is three is x( ) ix2x3 )( that of four is x( )(3); Ixzx3x4 If x be the IX2 ; that of and so on. 211. The rule may in half the cases be simplified, as follows. Out of ten counters, for every distinct selection of seven which is taken, a distinct combination of 3 is left. Hence, the number of combinations of seven is as many as that of three. We may, therefore, find the combinations of three instead of those of seven; and we must moreover expect, and may even assert, that the two formulae for finding these two numbers of combinations are the same in result, though different in form. And so it proves; for the number of combinations of seven I0x9X8x7x6x5X4 out of ten is ox9xx7x6x4, in which the product 7x6x5x4 occurs xxx3x4x5x6x7 in both terms, and therefore may be removed from both (108), leaving ,iX9gx which is the number of combinations of three out of ten. The Ix2x3 same may be shewn in other cases. EXERCISES, How many combinations of four can be made out of twelve things ? Answer, 495. of combinations can be made of S4 268 6 out of Answer, 330 378 5005 I5 How many combinations can be made of 13 out of 52 ; or how many different hands may a person hold at the game of whist ? Answer, 635013559600. 124 § COMMERCIAL ARITHMETIC. I2'. BOOK II. COMMERCIAL ARITHMETIC. SECTION I. WEIGHTS, MEASURES, &C. 212. IN making the calculations which are necessary in commercial affairs, no more processes are required than those explained in the preceding book. -not which have been But there is still one thing wanted to insure the accuracy of our calculations, but to enable us to compare and judge of their results. We have hitherto made use of a single unit (15), and have treated of other quantities which are made up of a number of units, in Sections II., III., and IV., and of those which contain parts of that unit in Sections V. and VI. Thus, if we are talking of distances, and take a mile as the unit, any other length may be represented,* either by a certain number of miles, or a certain number of parts of a mile, and (z meaning one mile) may be expressed * It is not true, that if we choose any quantity as a unit, any other quantity of the same kind can be exactly represented either by a certain number of units, or of parts of a unit. To understand how this is proved, the pupil would require more knowledge than he can be supposed to have; but we can shew him that, for any thing he knows to the contrary, there may be quantities which are neither units nor parts of the unit. Take a mathematical line of one foot in length, divide it into ten parts, each of those parts into ten parts, and so on continually. If a point A be taken at hazard in the line, it does not appear self-evident that if the decimal division be continued ever so far, one of the points of division must at last fall exactly on A : neither would the same appear necessarily true if the division were made into sevenths, or elevenths, or in any other way. There may then possibly be a part of a foot which is no exact numerical fraction whatever of the foot; and this, in a higher branch of WEIGHTS AND MEASURES. S212. 125 either by a whole number or a fraction. But we can easily see that in many cases inconveniences would arise. Suppose, for example, I say, that the length of one room is I - of a mile, and of another - I of a x80 '74 mile, what idea can we form as to how much the second is longer than the first ? It is necessary to have some smaller measure; and if we divide a mile into 1760 equal parts, and call each of these parts a yard, we shall find that the length of the first room is 9 yards and 7 of a yard, Io 9 and that of the second Io yards and - of a yard. From this we form a much better notion of these different lengths, but still not a very perfect one, on account of the fractions 2 and go. To get a clearer 9 87 idea of these, suppose the yard to be divided into three equal parts, and each of these parts to be called a foot; then 7 of a yard contains 2feet, and - 3 17 3 9 30 io of a yard contains of a foot, or a little more than 87 87 of a foot. x Therefore the length of the first room is now 9 yards, 2 feet, and - of a foot; that of the second is Io yards and a little more I 3 than - of a foot. We see, then, the convenience of having large mea3 sures for large quantities, and smaller measures for small ones; but this is done for convenience only, for it is possible to perform calculations upon any sort of quantity, with one measure alone, as certainly as with more than one; and not only possible, but more convenient, as far as the mere calculation is concerned. The measures which are used in this country are not those which would have been chosen had they been made all at one time, and by a people well acquainted with arithmetic and natural philosophy. We proceed to shew how the results of the latter science are made useful in our system of measures. Whether the circumstances introduced are sufficiently well known to render the following methods exact enough for the recovery of astronomical standards, may be matter of opinion; but no doubt can be entertained of their being amply correct for commercial purposes. mathematics, is found to be the case times without number. What is meant in the words on which this note is written, is, t:at any part of a foot can be represented as nearly as we please by a numerical fraction of it; and this is sufficient for practical purposes. MA2 126 COMMERCIAL ARITHMETIC. § 2I2-2zI3. It is evidently desirable that weights and measures should always continue the same, and that posterity should be able to replace any one of them when the original measure is lost. It is true that a yard, which is now exact, is kept by the public authorities; but if this were burnt by accident,* how are those who shall live 500 years hence to know what was the length which their ancestors called a yard ? To ensure them this knowledge, the measure must be derived from something which cannot be altered by man, either from design or accident. We find such a quantity in the time of the daily revolution of the earth, and also in the length of the year, both of which, as is shewn in astronomy, will remain the same, at least for an enormous number of centuries, unless some great and totally unknown change take place in the solar system. So long as astronomy is cultivated, it is impossible to suppose that either of these will be lost, and it is known that the latter is 365'24zz4 mean solar days, or about 365- of the average interval which 4 elapses between noon and noon, that is, between the times when the sun is highest in the heavens. Our year is made to consist of 365 days, and the odd quarter is allowed for by adding one day to every fourth year, This is the same as adding - of a 4 day to each year, and is rather too much, since the excess of the year which gives what we call leap year. above 365 days is not "25 but "z4224 of a day. The difference is "00776 of a day, which is the quantity by which our average year is too long. This amounts to a day in about centuries. 2x8 years, or to about 3 days in 4 The error is corrected by allowing only one out of four of the years which close the centuries to be leap-years. Thus, A.D. 18oo and are 900oo not leap-years, but zoo0 is so. 213. The day is therefore the first measure obtained, and is divided into 24 parts or hours, each of which is divided into 6o parts or minutes, and each of these again into 60 parts or seconds. thus, * i*,- One second, marked is therefore the 8 6 40 0 th part of a day, and the following is the Since this was first written, the accident has happened. The standard yard was so injured as to be rendered useless by the fire at the Houses of Parliament. + The minute and second are often marked thus, 1', 1": but this notation is now almost entirely appropriated to the minute and second of angular measure. § 213-214. 127 WEIGHTS AND MEASURES. MEASURE OF TIME. 6o seconds I minute . . . . . I m. I hour . are 6o minutes 4 . . . I h. . . . . ... 7 days . . .. I week . . .. I wk. 365 days . . .. I year . . .. I yr. 24 hours . day Id. 214. The second having been obtained, a pendulum can be constructed which shall, when put in motion, perform one vibration in exactly one second, in the latitude of Greenwich.t If we were in- venting measures, it would be convenient to call the length of this pendulum a yard, and make it the standard of all our measures of length. But as there is a yard already established, it will do equally well to tell the length of the pendulum in yards. It was found by commissioners appointed for the purpose, that this pendulum in London was 39"I393 inches, or about one yard, three inches, and - of an inch. 36 The following is the division of the yard. MEASURES OF LENGTH. The lowest measure is a barleycorn.$ 3 barleycorns are z inches ..... . I foot . ..... 3feet I inch . . ft. . . I yard. . I yd. . . . I pole . . 40 poles or zzo yards . I 5- yards . . . . . 8 furlongs or 1760 yards . . I in. I po. I furiony . I I mile . a mi. . fur. * The measures in italics are those which it is most necessary that the student should learn by heart. t The lengths of the pendulums which will vibrate in one second are slightly different in different latitudes. Greenwich is chosen as the station of the Royal Observatory. We may add, that much doubt is now entertained as to the system of standards derived from nature being capable of that extreme accuracy which was once attributed to it. : The inch is said to have been originally obtained by putting togethertlhree grains of barley. 128 COMMERCIAL ARITHMETIC. Also 6 feet ........ 69- miles . I fathom , ..... 3 § 21I4-216. . I degree fth. ° . x deg. or 10. A geographical mile is -th of a degree, and three such miles are one nautical league. In the measurement of cloth or linen the following are also used : i 2- inches 4 4 nails . . . . I quarter (of a yard) . I qr. 3 quarters . . . I Flemish ell . I Fl. e. . . I English ell . . I French ell . are 5 quarters .. 6 quarters . 215. . nail . . . . . . . . I nl. . . E. e. . I Fr. e. . MEASURES OF SURFACE, OR SUPERFICIES. All surfaces are measured by square inches, square feet, &c.; the square inch being a square whose side is an inch in length, and so on. The following measures may be deduced from the last, as will afterwards appear. 144 square inches 9 squarefeet . I 4 are I squarefoot . I sq. ft. I square yard . .. I sq. yd. 30- square yards . . I square pole . z sq. p. 40 square poles . . 1 rood . . 1 rd, . . I acre . . I ac. 4roods . . Thus, the acre contains 8 4 4o . . square yards, which is ten times a square of zz yards in length and breadth. This zz yards is the length which land-surveyors' chains are made to have, and the chain is divided into Ioo links, each "zz of a yard or 7*92 inches. An acre is then io square chains. It may also be noticed that a square whose side is 694 yards is nearly an acre, not exceeding it by I of a square foot. 7 5 216. MEASURES OF SOLIDITY OR CAPACITY.* Cubes are solids having the figure of dice. A cubic inch is a cube each of whose sides is an inch, and so on. * ' Capacity' is a term which cannot be better explained than by its use. When one measure holds more than another, it is said to be more capacious, or to have a greater capacity. Sz216-217. 129 WEIGHTS AND MEASURES. 1728 cubic inches 27 cubic feet are I cubic foot . . I c.ft. . . Icubic yard . . I c. yd. . This measure is not much used, except in purely mathematical questions. In the measurements of different commodities various mea- sures were used, which are now reduced, by act of parliament, to one. This is commonly called the imperial measure, and is as follows : MEASURE OF LIQUIDS AND OF ALL DRY GOODS. 4 gills 1pint are . . . I pt. . . I qt. 2zpints. . . Iquart. 4 quarts . . I gallon . . Igall. .I peck .. . I pk. . . Ibu. quarter . . I qr. . I Ild. 2 gallons . 4 pecks .. I bushel 8 bushels . . 5 quarters . I load . . The gallon in this measure is about 277z274 cubic inches; that is, very nearly 2771 cubic inches.t 217. The smallest weight in use is the grain, which is thus determined. water, A vessel whose interior is a cubic inch, when filled with has its weight increased by 252'458 grains. Of the grains so determined, 7000 are a pound averdupois, and 5760 a pound troy. The * This measure, and those which follow, are used for dry goods only. t Since the publication of the third edition, the heaped measure, which was part of the new system, has been abolished. The following paragraph from the third edition will serve for reference to it: 'The other imperial measure is applied to goods which it is customary to sell by heaped measure, and is as follows : 2 gallons ....... 4 pecks ........... 3 bushels ........ 12 sacks ........ . . Ipeck I bushel 1 sack . I chaldron. The gallon and bushel in this measure hold the same when only just filled, as in the last. The bushel, however, heaped up as directed by the act of parliament, is a little more than one-fourth greater than before." 1 Pure water, cleared from foreign substances by distillation, at a temperature of 620 Fahr. 130 § 217-218. COMMERCIAL ARITHMETIC. first pound is always used, except in weighing precious metals and stones, and also medicines. It is divided as follows : AVERDUPOIS WEIGHT. II 27-32 grains . I6 ounces ....... z28pounds ounce* . . . . I qr. cwt. . I hundred-weight . 4 quarters..... 20 hundred-weight . I oz. i lb. I quarter . .... . ... I pound . dr. ..... I dram are 16 drams, or drachms . I ton ....... x ton. The pound averdupois contains 7000 grains. A cubic foot of water weighs 6'32zxo606 pounds averdupois, or 997'1369691 ounces. For the precious metals and for medicines, the pound troy, containing 5760 grains, is used, but is differently divided in the two cases. The measures are as follow : TROY WEIGHT. 24 grains are I pennyweight zo pennyweights . i zounces ...... . I ounce ... . . . . . dwt. . I oz. 1 pound ....... The pound troy contains 5760 grains. I lb. A cubic foot of water weighs 75'7374 pounds troy, or 908'8488 ounces. APOTHECARIES' WEIGHT. 20 grains are 3 scruples 8 drams . . .. . I scruple . . . . . xz ounces ..... . . dram . . . ounce ..... . . 9 . ..... I pound . . . ltb 218. The standard coins of copper, silver, and gold, are,--the penny, which is ro weight - drams of copper; the shilling, which weighs 3 penny3 15 grains, of which 3 parts out of 40 are alloy, and the rest pure silver; and the sovereign, weighing 5 pennyweights and 3- grains, of which I part out of 12 is copper, and the rest pure gold. * It is more common to divide the ounce into four quarters than into sixteen dra-ms. 131 WEIGHTS AND MEASURES. § 2r8-219. MEASURES OF MONEY. The lowest coin is a farthing, which is marked thus, 4, being one fourth of a penny. 2 farthings 2 halfpence 12 pence are .... . ...... I penny .. I shilling . .. 20 shillings . 2I shillings ..... .... I x halfpenny ..... -d. 2 .... Id. ..... Is. pound* or sovereign . £ guinea.t 219. When any quantity is made up of several others, expressed in different units, such as £x. 14. 6, or z cwt. Iqr, 31bs., it is called a compound quantity. From these tables it is evident that any compound quantity of any substance can be measured in several different ways. For example, the sum of money which we call five pounds four shillings is also Io4 shillings, or 1248 pence, or 4992 farthings. It is easy to reduce any quantity from one of these measurements to another; and the following examples will be sufficient to shew how to apply the same process, usually called REDUCTION, to all sorts of quantities. X18 . xz. 63 ?+ 4 shillings in a pound, there are, in £18, I8x2o, or Since there are zo20 I. How many farthings are there in 360 shillings; therefore, £i8 . i2 is 360+12, or 372 shillings. Since there are xz pence in a shilling, in 372 shillings there are 372x12, or 4464 pence; and, therefore, in £18 . z12. 6 there are 4464+6, or 4470 pence. Since there are 4 farthings in a penny, in 4470 pence there are * The English pound is generally called a pound sterling, which distinguishes it from the weight called a pound, and also from foreign coins. t The coin called a guinea is now no longer in use, but the name is still given, from custom, to 21 shillings. The pound, which was not a coin, but a note promising to pay 20 shillings to the bearer, is also disused for the present, and the sovereign supplies its place; but the name pound is still given to 20 shillings. ( Farthings are never written but as parts of a penny. Thus, three farthings 3 3 2 being - of a penny, is written -, or %. One halfpenny may be written either as or -; the latter is most common. 132 COMMERCIAL ARITHMETIC. § 2i9. 3 and, therefore, in £I8 . Iz2.6 there 4 are 17880+3, or 17883 farthings. The whole of this process may be 4470x4, or 17880 farthings; written as follows: 3 £18. 1z . 6 4 20 360+12 = 372 12 4464+6 = 4470 4 17880+3 = 17883 II. In 17883 farthings, how many pounds, shillings, pence, and farthings are there ? Since 17883, divided by 4,gives the quotient 4470, and the remainder 3,17883 farthings are 4470 pence and 3 farthings (218). Since 4470, divided by I2, gives the quotient 372, and the remainder 6,4470 pence is372 shillings 6 pence. and Since 372, divided by 20, gives the quotient I8, and the remainder 12, 372 shillingsI8 pounds and is 12 shillings. 3 Therefore, 17883 farthings is 4470-d., which is 372s. 6 d., which is 3 - 3 4 +4 £18. 12 .6 . 4 The process may be written as follows: 4)17883 12)4470 ... 3 20)372 ... 6 - £18. Iz . 6 4 EXERCISES. A has £Ioo . 4. 11-, and B has 64392 farthings. If A receive 1492 2I farthings, and B £1 . 2.3-, which will then have the most, and by how much ?-Answer, A will have £33 . 12 . 3 more than B. In the following table the quantities written opposite to each other are the same : each line furnishes two exercises. ( 133 REDUCTION. 219-222. I ii5 lb 1 9 z s 1o 3bs 14 g d wt bu 2 pks 1 6h gall 2 3m 2 farthings. 663072 grains. drams. 1001 dr 9 oz yds 2ft 9 14 9 3m 15302 18 . 9- £15 in 195477 inches. qts 26o pints. 47° 59027 seconds. 220. The same may be done where the number first expressed is For example, how many shillings and pence are there in fractional. 4 4 4 of 0 shillings; of 20 - of a pound ? Now, 4 of a pound is 15 is -5 , or 4 is 4 IS (110), or 3 a shilling is I of 3 15 , or (105) 5- of a shilling. 3 3 12 pence, or 4 pence. Again, I of 3 Therefore, £4 = 5s. 4 d. '5 Also, *23 of a day is "23x24 in hours, or 51''52; and '52 of an hour 6 is *52x6o in minutes, or 3In'2; and *2 of a minute is "zx o in seconds, or I2s; whence '23 of a day is 5 h 31m I2s. Again, suppose it required to find what part of a pound 6s. 8d. is. Since 6s. 8d. is 8o pence, and since the whole pound contains zox12 or 240 pence, 6s. 8d. is made by dividing the pound into 240 parts, and 80 x 80 = I (108); (107), but taking 80 of them. It is therefore ±0£ 240 240 I therefore, 6s. 8d.= £-. 3 3 EXERCISES. -of a day 5 *I2841Iofaday . '257 of a cwt. £'14936 . h is 9 3 6m h 4 m 54 s 6 2 4 . 3 * . . .. 2 31bs 1 2 oz gdr 704 f ' . 2 IId 3 .. * 38 56 221, 222. I have thought it best to refer the mode of converting shillings, pence, and farthings into decimals of a pound to the Appendix (See Appendix On Decimal Money). I should strongly recommend the reader to make himself perfectly familiar with the modes given in When a decimal follows a whole number, the decimal is always of the same unit as the whole number. Thus, 56*5 is five seconds and five-tenths of a second. 0"*5 means five-tenths of a second; 0 h3, three-tenths of an hour. N Thus, 134 S222-224. COMMERCIAL ARITHMETIC. To prevent the subsequent sections from being altered that Appendix. in their numbering, I have numbered this paragraph as above. 223. The rule of addition* of two compound quantities of the same Suppose it required sort will be evident from the following example. to add 192 . 14 . z to £64. The sum of these two is the 13 . i 2 4 whole of that which arises from adding their several parts., Now 3d.+ 4 Iid.+ .= 2 2d. = 13s.+ =d.. 4 to .o .Ix 4 o. 1. I = 27s. I4s. = x3d. = I . 7 . 0 256.o.o = £64 +£192 (219' The sum of all of which is £257 . 8 . z 4 This may be done at once, and written as follows : £192 . 14. 64. 13 . 213 4 I £257. 8. 2- Begin by adding together the farthings, and reduce the result to pence and farthings. Set down the last only, carry the first to the line of pence, and add the pence in both lines to it. Reduce the sum to shillings and pence; set down the last only, and carry the first to the line of shillings, and so on. The same method must be followed when the quantities are of any other sort; and if the tables be kept in memory, the process will be easy. 224. SUBTRACTION is performed on the same principle as in (40), namely, that the difference of two quantities is not altered by adding the same quantity to both. from £24. 5 . 7. I Suppose it required to subtract £ 19. 3 . o103 4 Write these quantities under one another thus : * Before reading this article and the next, aiticles (29) and (42) should be read again carefully. § 224-22:5. 135 SUBTRACTION. £24. 5. 73 10o 19. 13 . 4 3 Since- cannot be taken from- or -, add Id. to both quantities, 4 2 4 which will not alter their difference; or, which is the same thing, add 4 farthings to the first, and id. to the second. in the two lines then stand thus: Now subtract 3 4 4 3 6 The pence and farthings 7-d. and. rIZ3d. 4 from -, and the difference, is -, which must be written under the 4 4 farthings. Again, since IId, cannot be subtracted from 7d., add Is. to both quantities by adding Izd. to the first, and Is. to the second. The pence in the first line are then 19, and in the second I 1, and the difference is 8, which write under the pence. lower line were increased by in the upper one. 1, there Since the shillings in the are now 14s. in the lower, and 5s. Add zos. to the upper and £X to the lower line, and the subtraction of the shillings in the second from those in the first leaves IIs. Again, there are now £20 in the lower, and £24 in the upper line, the difference of which is £4;. therefore the whole difference of the two sums is £4,. I1. 83. If we write down the two sums with 4 all the additions which have been made, the process will stand thus: £24 . 25 20 . Difference 19- . 3 14 . I £4 . I 4 . 8 3 4 225. The same method may be applied to any of the quantities in the tables. The following is another example: From 7 cwt. 2 qrs. .r lbs. 14 oz. Subtract 2 cwt. 3 qrs. 27 lbs. Iz oz. After alterations have been made similar to those in the last article, the question becomes: From 7 cwt. 6 qrs. 49 lbs. 14 oz. Subtract 3 cwt. 4 qrs. 27 lbs. Iz oz. The difference is 4 cwt. 2 qrs. 22 lbs. 2 oz. In this example, and almost every other, the process may be a little 136 COMMERCIAL ARITHMETIC. shortened in the following way. § 225-227. Here we do not subtract 27 lbs. from 21 lbs., which is impossible, but we increase zx lbs. by I qr. or 28 lbs. and then subtract 27 lbs. from the sum. It would be shorter, and lead to the same result, first to subtract 27 lbs. from I qr. or 28 lbs. and add the difference to 21 lbs. 226. EXERCISES. sums to receive: £193 . 14 . II-, I 4 and £49. 14.4,A ,' the following debts A man has the following 3, 6- £6473. £22 . o. 4x to pay: £200oo. 0o o, 19 . 6, 4 305 . 16 . II, z, much will remain after paying the debt: . and £1 9 . 6 . o-. There are four towns, in the eider A, B, C, and D. How 2 Answer, £6190o. 7 . 4- . If a man can f go from A to B in 5 20"' 333, from B to C in 6h 49'" 2 *, and from A to D in 1 9h om 17, how long will he be in going from B to D, and from C to D ? Answer, 13h , h 39m 443 and 6 50 227. In order to perform the process of MIULTIPLICATION, m 425. it must be recollected that, as in (52), if a quantity be divided into several parts, and each of these parts be multiplied by a number, and the products be added, the result is the same as would arise from multiplying the whole quantity by that number. It is required to multiply £7 . . 61 by 13. The first quantity is 4 made up of 7 pounds, 13 shillings, 6 pence, and I farthing. And 13 i x farth. x 13 is 13 farth. or £o. o.3- (219) 6 pence x 13 78 pence, or o.6.6 or 8 .9 .o 13 shill. is x 13 is169 7 pounds x 13 is shill. 91 pounds, or 91 . o . o The sum of all these is £99 . 15. 9- which is therefore £7 . 13 . 6- x 13. 4 This process is usually written as follows : £7 . 13 . £99. 15 6- 4 13 94I 4 137 DIVISION. § 228. tn performed upon the same principle as in (74), 228. DivisroN viz. that if a quantity be divided into any number of parts, and each part be divided by any number, the different quotients added together will make up the quotient of the whole quantity divided by that number. Suppose it required to divide £99 . 15 . 9- by 13, 4 Since 99 divided by 13 gives the quotient 7, and the remainder 8, the quantity is made up The quotient of the first, 13 being of £13x7, or £91, and £8. 15 . 91. Since £8 is the divisor, is £7: it remains to find that of the second. 16os., 175s. 13, 6s. I 9 is 175S. 9 -d., and 175 divided by 13 gives the quotient I 4 4 and the remainder 6; that is, 9 -d. is made up of 169s. and £8 . 15 I 9 -d., 4 the quotient of the first of which is I38., and it remains to find Since 6s. is 7zd., 6s. that of the second. 9 d. is 8 id., and 81 divided 4 4 x by 13 gives the quotient 6 and remainder 3; that is, 8 -d. is 78d. and 4 3 d., of the first of which the quotient is 6d. Again, since 3d. is--, 1 4 4 or I2 farthings, 3-d. is 13 farthings, the quotient of which is I farthing, I 4 or , without remainder. 4 We have then divided £99 . four parts, each of which is divisible by 13, 5. 9- into 4 viz. £91, 1698., 78d., and 13 farthings; so that the thirteenth part of this quantity is £7 . 13 6. 4 The whole process may be written down as follows; and the same sort of process may be applied to the exercises which follow : s. 8. d. £ s. d. 13)99 15 91(7 I3 64 9' 4 4 20 160+15 = 175 13 45 39 6 12 72+9 - 81 78 4 12+1 = 13 13 : 2 138 COMMERCIAL ARITHMETIC. § 228-23o. Here, each of the numbers 99; 175, 81, and 13, is divided by 13 in the usual way, though the divisor is only written before the first of them. EXERCISES. 2 cwt. I qr. 2x lbs. 7 oz. x 2(14h 3 m 27Sx 8x £27.I0. £7 .4.8x 53 = 129 cwt. I qr. 16 lbs. 3 oz. 109 = 2 3 6d 569 = £15666.9."4 123 = £889. 14 8 £166x £40 .4. -= 33 £187.6. 7x 4s. 6d. 2 i6m 3 s IOh x 10- 6 33 2 32 = £5. 12 .4 100 425 I2I = £254. II. - 2 2 4s. 4d. x 4260 = 6s. 6d. x 2840 229. Suppose it required to find how many times is. 4-d. is contained in £3 . 19. Io-3. The way to do this is to find the number of farthings 4 in each. By (219), in the first there are 65, and in the second 3835 farthings. Now, 3835 contains 65 59 times; and therefore the second quantity is 59 times as great as the first. In the case, however, of pounds, shillings, and pence, it would be best to use decimals of a pound, which will give a sufficiently exact answer. Thus Is. 4d. is 3 4 £0o67, and £3 . 19 . is £3'994, and 3'994 divided by "067 is 3994 41 4 by 67, or 59-. This is an extreme case, for the smaller the divisor, 10- the greater the effect of an error in a given place of decimals. EXERCISES. How many times does 6 cwt. 2 qrs. contain I qr. 14 lbs. I oz. ? and Id 2h Om 47s contain 3m 46° ? Answer, 17'30758 and 414367257. If 2 cwt. 3 qrs. I lb. cost £150 . 13 . 10, how much does i lb. cost ? Answer, 9s. 9d. 3. 309 A grocer mixes 2 cwt. 15 lbs. of sugar at IId. per pound with 14 cwt. 3 lbs. at 5d. per pound. At how much per pound must he sell the Answer, d. 3 153. mixture so as not to lose by mixing them ? 4 905 230. There is a convenient method of multiplication called PRACSuppose I ask, How much do 153 tons cost if each ton cost TICE. £2 . 5 72? 2 It is plain that if this sum be multiplied by 153, the product is the price of the whole. buy 139 PRACTICE. § 230. 153 tons at £z . 15.77 I But this is also evident, that, if I each ton, payment may be made by first putting down £2 for each ton, then los. for each, then 5s., then 6d., I I These sums together make up £z . I5 . 7 , and the and then I-d. 2 x 2 reason for this separation of £2 . 15 . 7- into different parts will be soon The process may be carried on as follows: apparent. .... o 0 o o 38 . . £306 76 1. 153 tons, at £z each ton, will cost 5 c 2. Since ros. is £-, 153 tons, at Ios. each, will cost . £153,whichis. .......... 3. Since 5s. is - of zos., 153 tons, at 5s., will cost half as much as the same number at Ios. each, that is,- I of £76. Io, which is ......... of 5s., 153 tons, at 6d. each, will 4. Since 6d. is I 10 cost - of what the same number costs at 5s. 10 each, that is, - Io IO I of £38 . 5, which is . 3 16 6 o 19 12 10 7- ... 5. Since I- or 3 halfpence is - of 6d. or Iz halfpence, 2 x 4 x tons, at x-d. each, will cost - of what the 153 2 4 I same number costs at 6d. each, that is, - of . £3 . x6,6,which is ........... The sum of all these quantities is ... 4 . . . . . . . .425 1 2 which is, therefore, £2 . 15 . 7 x 153. The whole process may be written down as follows: £153 is 2 los. is - £2 306 of £I 76 0 0 38 5 3 16 I 10 2 £I per ton. 0 £i x 5s. is -of 6d. is I 0 2 o 0 0 0 0o o o0 o o' 5 o 6 o 6 0 Ios. of 5s. 0o z 2 4 I I Sum . . . £425 10 7- 2 £2 15 7- 2 140 § 230. COMMERCIAL ARITHMETIC. ANOTHER EXAMPLE. 1735 What do lbs. cost at 9s. io-3d. per lb. ? The price 9s. Io-d. 4 I r 4 is made up of 5s., 4., iod., -d., and -d.; of which 5s. is -of A 1, 2 4 4 4s. is 5 of , is - od. is - of 5s., -d. 6 2 of iod., and -d. is - of -d. 20 4 2 2 Follow the same method as in the last example, which gives the following : £1735 I 5s. is - of £ 4 4s. is - 5 i I 2 15 347 0 0 72 of £i 5 o 20 -d. is -d. 4 is 1 2 of Iod. LI per lb. o 43.3 iod. is 6of 5s. I 0 3 12 o o 8 5 0 o 4 o ro 0o I 3- i 0 0 0- 2 of -d. 2 I r6 by addition ... £858 9 1I 4 34 I 2 o o o o4 Lo 9 103 4 In all cases, the price must first be divided into a number of parts, each of which is a simple fraction* of some one which goes before. No rule can be given for doing this, but practice will enable the student immediately to find out the best method for each case. When that is done, he must find how much the whole quantity would cost if each of these parts were the price, and then add the results together. EXERCISES. What is the cost of 0- 243 cwt. at £14 . I8. 8- per cwt. ?-Answer, £3629 . I . 3 . 4 4 I I 169 bushels at £2z. . 3- per bushel ?-Answer, £348 14 . 94. 4 273 qrs. at 19s. 2d. per quarter ?-Answer, £261 . 12 . 6. 4 2627 sacks at 7s. 84d. per sack ?-Answer, £Io1z . 9. 9-. 2 2 " Any fraction of a unit, whose numerator is unity, is generally called an aliquot part of that unit. Thus, 2s. and 10s. are both aliquot parts of a pound, being £1 and 1 -. 10 § 141 PRACTICE. 231-233* 231. Throughout this section it must be observed, that the rules can be applied to cases where the quantities given are expressed in common or decimal fiactions, instead of the measures in the tables. The following are examples : What is the price of 272'3479 cwt. at £2 . 1. 3- per cwt. ? Answer, £562z849, or £562 . 5 . 8-. S4 66-lbs. at Is. 4 -d. per lb. cost £4 . 11 . 5-. 2 2 4 How many pounds, shillings, and pence, will 279'301 acres let for if each acre lets for £3"1076 ?-Answer, £8679558, or £867 . 19 . I-. I 4 What does I of- 3 of 17 bush. cost at - of of r17. 14 per bushel? 4 13 5 3 1 Answer, £2'3146, or £z . 6. 3-. What is the cost of 19 lbs. 8 oz. I2 dwt. 8 gr. at Answer, £999 . 4 . 4.6 per ounce ?- II 14. I- -. 45 232. It is often required to find to how much a certain sum per day will amount in a year. This may be shortly done, since it happens that the number of days in a year is 240+120+5 ; so that a penny per day is a pound, half a pound, and 5 pence per year. Hence the following rule : To find how much any sum per day amounts to in a year, turn it into pence and fractions of a penny ; to this add the half of itself, and let the pence be pounds, and each farthing five shillings; then add five times the daily sum, and the total is the yearly amount. what does 12s. 3~d. amount to in a year ? For example, This is 147d., and its half 4 5 7 is 738d., which 43 added to 147-d gives zzi-d., which turned into pounds Also, I2s. 3-d.x 5 is £3 . 1. 6-, which added to the 34 4 former sum gives £22 4 . 14. o- for the yearly amount. In the same is £221 . 12 . 6. way the yearly amount of zs. 3 d. is 32 I10. 5 * 33; and that of IId. is £ £41 . i6 . 5s-; that of 6d. is 2 4 6 . 14 . 7. 233. An inverse rule may be formed, sufficiently correct for every purpose, in the following way : If the year consisted of 360 days, or 2 of 240, the subtraction of one-third from any sum per year would give the proportion which belongs to 240 days; and every pound so obtained would be one penny per day. But as the year is not 360, but 365 days, if we divide each day's share into 365 parts, and take 5 away, the whole of the subtracted sum, or 360x5 such parts, will give 360 parts for each 142 COMMERCIAL ARITHMETIC. of the 5 days which we neglected at first. § 233-234. But 360 such parts are left behind for each of the 360 first days; therefore, this additional process divides the whole annual amount equally among the 365 days. Now, 5 parts out of 365 is one out of 73, or the 7 3 d part of the first result must be subtracted from it to produce the true result. Unless the daily sum be very large, the 72d part will do equally well, which, as 72 farthings are 18 pence, is equivalent to subtracting at the rate of one farthing for ISd., or -d. for 3s., or iod. for £3. The rule, then, is as 2 follows: To find how much per day will produce a given sum per year, turn the shillings, &c. in the given sum into decimals of a pound (221); subtract one-third; consider the result as pence; and diminish it by one farthing for every eighteen pence,. or ten pence for every £3. 3 example, how much per day will give £224. 14.o per year? For This 4 is 224'703, and its third is 74*901, which subtracted from 224"703, gives 149802, which, if they be pence, amounts to Izs. 580o2d., in which Is. 6d. is contained 8 times. Subtract 8 farthings, or zd., and we have izs. 3"802d., which differs from the truth only about - of a farthing. the same way, £ioo per year is 5$s. 5 4d. per day. In 20 234. The following connexion between the measures of length and the measures of surface is the foundation of the application of arithmetic to geometry. Suppose an oblong figure, A, B, C, D, as here drawn (which is called a rectangle in geometry), with the side A B 6 inches, and the side A c 4 A a b& d e B E h -c inches. ---l m n o a p D Divide A B and c D (which are equal) each into 6 inches by the points a, b, c, 1, m, &c.; and A c and B D (which are also equal) into 4 inches by the points f, g, h, x, y, and z. Join a and 1, b and m, &c., and f and x, &c. of squares; 143 MEASURES OF LENGTH AND SURFACE. § 234-236. Then, the figure A B C D is divided into a number for a square is a rectangle whose sides are equal, and therefore A a f E is square, since A a is of the same length as Af, both being I inch. There are also four rows of these squares, with six squares in each row; that is, there are 6x4, or 24 squares altogether. Each of these squares has its sides I inch in length, and is what was called in (215) a square inch. By the same reasoning, if one side had contained 6 yards, and the other 4 yards, the surface would have contained 6x4 square yards; and so on. 235. Let us now suppose that the sides of A B C D, instead of being E B A a whole number of inches, contain some inches and a fraction. For example, let A lB be 3-1inches, or (114) 2 9 of an inch, and let A C contain 2- inches, orC 4 2 D of an inch. 4 Draw A E twice as long as A B, and A F four times as long as A c, and complete the rectangle A E F G. The rest of the figure needs no description. Then, since A E is twice A B, or twice F G - inches, it is 7 A F is four times A c, or four times 9 inches, it is 9 inches. 4 Therefore, the whole rectangle inches. And since A E F G contains, by (234), 7x9 or 63 square inches. But the rectangle A E F G contains 8 rectangles, all of the same figure as A B C D; and 63 therefore A B C D is one-eighth part of A E F G, and contains -6 square inches. But 63 7 is made by multiplying 9 and 7 together (118). From 8 4 2 this and the last article it appears, that, whether the sides of a rectangle be a whole or a fractional number of inches, the number of square inches in its surface is the product of the numbers of inches in its sides. The square itself is a rectangle whose sides are all equal, and therefore the number of square inches which a square contains is found by multiplying the number of inches in its side by itself. side is 13 inches in length contains 236. For example, a square whose 13x13 or 169 square inches. EXERCISES. What is the content, in square feet and inches, of a room whose sides are 42 ft. 5 inch. and 31 ft. 9 inch. ? and supposing the piece from 144 § 236-238. COMMERCIAL ARITHMETIC. which its carpet is taken to be three quarters of a yard in breadth, what length of it must be cut off ?-Answer, The content is 1346 square feet 105 square inches, and the length of carpet required is 598 feet 6 - 9 inches. The sides of a rectangular field are 253 yards and a quarter of a mile; how many acres does it contain ?-Answer, 23. What is the difference between S8 square miles, and a square of 18 miles long, or 18 miles square?-Answer, 306 square miles. 237. It is by this rule that the measure in (215) is deduced from that in (214); for it is evident that twelve inches being a foot, the square foot is 12X12 or I44 square inches, and so on. In a similar way it may be shewn that the content in cubic inches of a cube, or parallelepiped,* may be found by multiplying together the number of inches in those three sides which meet in a point. Thus, a cube of 6 inches contains 6x6x6, or 216 cubic inches; a chest whose sides are 6, 8, and 5 feet, contains 6x8x5, or 240 cubic feet. By this rule the measure in (216) was deduced from that in (214). SECTION II. RULE OF THREE. 238. Suppose it required to find what yards cost 17s. 4d. 156 yards will cost, if 22 This quantity, reduced to pence, is 208d.; and if 22 yards cost 20o8d., each yard costs 2d. 22 But i56 yards cost 208 156 x 156 pence, or times the price of one yard, and therefore cost 6 2o8xx5 pence (117). Again, if 25 French francs be 20 shillings 22 2 sterling, how many francs are in £20. 15 ? Since 25- francs are 2o shillings, twice the number of francs must be twice the number of shillings; that is, 51 francs are 40 shillings, and one shilling is the * A parallelepiped, or more properly, a rectangular parallelepiped, is a figure of the form of a brick; its sides, however, may be of any length; thus, the figure of a plank has the same name. is a die. A cube is a parallelepiped with equal sides, such as fortieth part 145 RULE OF THREE § 238-239. of 51 francs, or 51 francs. 40 shillings (219); and since r shilling is - o20 I5s. contain 415 51 francs, 415 shillings is - But r 4 40 francs. x415 francs, or (117) 51X415 40 239. Such questions as the last two belong to the most extensive rule in Commercial Arithmetic, which is called the RULE OF THREE, because in it three quantities are given, and a fourth is required to be found. From both the preceding examples the following rule may be deduced, which the same reasoning will shew to apply to all similar cases. It must be observed, that in these questions there are two quantities which are of the same sort, and a third of another sort, of which last the answer must be. Thus, in the first question there are 22 and i56 yards and zo8 pence, and the thing required to be found is a number of In the second question there are zo and 415 shillings and 25 francs, and what is to be found is a number of francs. Write the three pence. quantities in a line, putting that one last which is the only one of its kind, and that one first which is connected with the last in the question.* Put the third quantity in the middle. In the first question the quantities will be placed thus : 156 22 yds. 17s. yds. 4 d. In the second question they will be placed thus: s £20 15 . 20s. z25 francs. Reduce the first and second quantities, if necessary, to quantities of the same denomination. Thus, in the second question, £0zo be reduced to shillings (219). 15s. must The third quantity may also be reduced to any other denomination, if convenient; or the first and third may be multiplied by any quantity we please, as was done in the second In some cases the * This generally comes in the same member of the sentence. ingenuity of the student must be employed in detecting it. is the best guide. The following may be very often applied. The reasoning of (238) If it be evident that the answer must be less than the given quantity of its kind, multiply that given quantity by the less of the other two; if greater, by the greater. Thus, in the first question, 156 yards must cost more than 22; multiply, therefore, by 156. 146 COMMERCIAL ARITHMETIC. c .§ 39-24 . question; and, on looking at the answer in (238), and at (108), it will be seen that no change is made by that multiplication. Multiply the second and third quantities together, and divide by the first. The result is a quantity of the same sort as the third in the line, and is the answer required. Thus, to the first question so8xx56 22 . 56 pence, or, which is the same thing, 17. the answer is (238) 4d.xx56 7. 4d.x56 22 240. The whole process in the first question is as follows :* yds. yds. 22 156 s. :: d. 17.4 12 20o8 pence. 156 1248 208 3 22)324 4 8(1 4 7 4 d. and , or of a farthing, II 22 . 4 3 or (219) £6 . 2 . - 7 104 88 164 154 xo8 88 20 (228) 4 8o 66 14 The question might have been solved without reducing 17s. 4d. to pence, thus : * It is usual to place points, in the manner here shewn, between the quantities. Those who have read Section VIII. will see that the Rule of Three is no more than the process for finding the fourth term of a proportion from the other three. RULE OF THREE. §-240-24Z, y 4s. 22: yds. 156 147 s. d. ":. 7*4 (227) 156 zz)£35 . 4.o(£6 . ol3 o. (228) 411 132 3x20+4 = 64 44 2= 40 20XI2 220 20o4 = 80 66 14 The student must learn by practice which is the most convenient method for any particular case, as no rule can be given. 241. It may happen that the three given quantities are all of one denomination;, nevertheless it will be found that two of them are of one, and the third of another sort. For example: What must an income of £400 pay towards an income-tax of 4s. 6d. in the pound ? Here the three' given quantities are, £400, 4s. 6d., and LI, which are all of the same species, viz. money. Nevertheless, the first and third are income; the second is a tax,, and the answer is also a tax; and therefore, by (152), the quantities must be placed thus : L £400 : 4s. 6d. 242. The following exercises either depend directly upon this rule, or can be shewn to do so by a little consideration. There are many questions of the sort, which will require some exercise of ingenuity before the method of applying the rule can be found. EXERCISES. If 15 cwt. 2 qrs. cost £198 , 15 . 4, what does I qr. 22 lbs. cost? 3 i85 .85 4 Answer, 5. 144.5 3 h If a horse go 14 m. 3 fur. 27 yds. in 3 26 in going 23 miles? m 4217 12', how long will he be 2462 . Answer, 5h 2 9 m 34S 25327 Two persons, A and B, are bankrupts, and otWe exactly the same 148 COMMERCIAL ARITHMETIC. S242-244. s sum; A can pay i5 . 4-d. in the pound, and B only 7s. 6-d. 2 At the 4 same time A has in his possession £1304. 17 more than B; what do Answer, £3340 . 8 . 33 9 the debts of each amount to ? 1 425 For every Iz- acres which one country contains, a second contains - 56 . 4 2 The second country contains 17,300 square miles. does the first contain? How much Again, for every 3 people in the first, there are 5 in the second; and there are in the first 27 people on every 20 acres. How many are there in each country ?-Answer, The number of square miles in the first is 38444, and its population 3,321,600; and 9 the population of the second is 5,536,000. If 421 yds. of cloth, 18 in. wide, cost £59. 14,. 2, how much will 2 4 118- yds. cost, if the width be i yd. ? Answer, £332 . 5 . 24 4 17 If £9 . 3 . 6 last six weeks, how long will £ioo last ? Answer, 651 45 367 weeks. How much sugar, worth 9-d. a pound, must be given for z cwt. of tea, worth 15 rod. an ounce ? 435 Answer, 32 cwt. 3 qrs. 7 lbs. 35. 39 243. Suppose the following question asked: How long will it take men to do that which 45 men can finish in to days ? It is evident that one man would take 45x10, or 450 days, to do the same thing, and that 15 men would do it in one-fifteenth part of the time which it employs one man, that is, in 450, or 30 days. I5 reasoning the following questions can be solved. By this and similar EXERCISES. If 15 oxen eat an acre of grass in 1z days, how long will it take z6 oxen to eat 14 acres ? Answer, 96 days. 13 If zz masons build a wall 5 feet high in 6 days, how long will it 6 take 43 masons to build Io feet ? Answer, 6 days. 43 244. The questions in the preceding article form part of a more general class of questions, whose solution is called the DOUBLE RULE OF THREE, but which might, with more correctness, be called the Rule of Five, since five quantities are given, and a sixth is to be found. The following is an example : If 5 men can make 30 yards of cloth in 3 days, how long will it take 4 men to make 68 yards? The first 149, RULE OF THREE. thing to be done is to find out, from the first part of the question, the time it will take one man to make one yard. Now, since one man, in 3 days, will do the fifth part of what 5 men can do, he will in 3 days .3 3X5 make k , or 6 yards. He will, therefore, make one yard in 6 or in 30 6 of a day. From this we are to find how long it will take 4 men to make Since one man makes a yard in 3X5 of a day, he will make 68 68 yards. days; and 4 men will do this yards in 3x5x68 days, or (116) in I 30 30 3xx68 in one-fourth of the time, that is (123), in 3 days, orin 8 days. 2 30X4 Again, suppose the question to be : If 5 men can make 30 yards in 3 days, how much can 6 men do in Iz days? Here we must first find the quantity one man can do in one day, which appears, on reasoning yards. Hence, 6 men, similar to that in the last example, to be 6x30 3x5 Izx6x3o in one day, will make --yards, and in iz days will make 5x3 5x3 or I44 yards. From these examples the following rule may be drawn. Write the given quantities in two lines, keeping quantities of the same sort under one another, and those which are connected with each other in the In the two examples above given, the quantities must be same line. written thus: 5 men. 30 yds. 4 men. 68 yds. 3 days. SECOND EXAMPLE. 5 men. 30 yds. 3 days. 12days. 6 men. Draw a curve through the middle of each line, and the extremities of the other. the other. There will be three quantities on one curve and two on Divide the product of the three by the product of the two. and the quotient is the answer to the question. o2 150 § 244-245* COMMERCIAL ARITHMETIC. If necessary, the quantities in each line must be reduced to more simple denominations (219), as was done in the common Rule of Three (238). EXERCISES. If 6 horses can, in 2 days, plough 17 acres, how many acres will Answer, 5928 . 93 horses plough in 4- days ? If 20 men, in 3- days, can dig 7 rectangular fields, the sides of each 4 of which are 40 and 50 yards, how long will 37 men be in digging 53 fields, the sides of each of which are 90 and I25- yards ? 2245 Answer, 75 days. 20720 If the carriage of 6ocwt. through 20 miles cost £i ought to be carried 30 miles for £5 . 8 . 9 ? 4 ios., what weight Answer, I5cwt. If £ioo gain £5 in a year, how much will £85o gain in 3 years and Answer, £i55 . 16 . 8. 8 months ? SECTION III. INTEREST, ETC. 245. In the questions contained in this Section, almost the only process which will be employed is the taking a fractional part of a sum of money, which has been done before in several cases. Suppose it required to take 7 parts out of 40 from £16, that is, to divide £x6 into 16 40 equal parts, and take 7 of them. Each of these parts is and 7 of 40 i6 16x7 hem make -7, or pounds (116). The process may be written 3-, 40 40 as below: £16 7 40)I I2(£2 . 16s. So 32 20 640 40 240 40 0. 151 INTEREST. § 245-246. Suppose it required to take 13 £56 . 13 parts out of a hundred from 7-. 2 I 6 5 . 13 .7- 13 100)736 . 17 . I k(7 * 7 * 41o 700 6 3 X20+17 = 737 700 37X12+I = 445 400 45x4+z - 182 100 8z I Let it be required to take 2- parts out of a hundred from £3 12s. £3 2 The result, by the same rule is Iss.x2I00 , or (123) £3 12s.x5 200 ; so that taking 2- out of a hundred is the same as taking 5 parts out of zoo. EXERCISES. Take 71 parts out of 53 from £I Ios. Answer, 4s. I-d. Take 5 parts out of ioo from £107 138s. 4-d Answer, £5:. 7 . 8 and 3 of a farthing. 20 £56 3s. 2d. is'equally divided among 32 persons. How much does the share of 23 of them exceed that of the rest ? Answer, £24 . I1 . 4- - 22 246. It is usual, in mercantile business, to mention the fraction which one sum is of another, by saying how many parts out of a hundred must be taken from the second in order to make the first. Thus, instead of saying that £16 Its. is the half of £33 48., it is said that the first is 50o per cent of the second. because, if £zoo be divided into Thus, £5 is z ioo 2 per cent of £zoo; parts, 2- of those parts are £5. Also, £13 is 150 per cent of £8 . 13 * 4, since the first is the second and half the second. Suppose it asked, How much per cent is 23 152 COMMERCIAL ARITHMETIC. This, by (238), is 41- I 14 per cent. 6. The question amounts to this: If he who parts out of 56 of any sum ? has £56 gets £oo § 24 for them, how much will he who has 23 receive ? 23xIoo,or 2300 6, x or 41-. 56 56 Hence, 23 out of 56 is 14 i6xIoo Similarly, i6 parts out of i8 is oo18 out of 5 is 5, or 40 per cent. 8 , or 88- per cent, and 2 parts 9 From which the method of reducing other fractions to the rate per cent is evident. Suppose it asked, How much per cent is £6 . Since the first contains x586d., 2. 2 of £1Lx. 3 ? and the second 2 9 16d., the first is 1586 out of 2916 parts of the second; that is, by the last rule, it is 586 2906 1136 or 54-, or £54 . 7 . 9gI per cent, very nearly. The more expeditious way of doing this is to reduce the shillings, &c. to decimals of a pound. Three decimal places will give the rate per cent to the nearest shilling, which is near enough for all practical purposes. For instance, in the last example, which is to find how much £6"608 is of £Iz'2 5, 6*6o8xzoo is 660o8, which divided by Iz'I5 gives £54"38, or £54. 7. Greater correctness may be had, if necessary, as in the Appendix. EXERCISES. How much per cent is 198- out of 233 parts ?-Ans. £85 . r . 8 4 Goods which are bought for £193. 4 12, are sold for £z6 . 13 * 4; how much per cent has been gained by them ? 1. Answer, A little less than £ I8 . 6. A sells goods for B to the amount of £230. 12, and is allowed a commission* of 3 per cent; what does that amount to ? Answer, £6 . I8 4-1 I 425 7 A stockbroker buys £I7oo stock, brokerage being at £I per cent; what does he receive ?-Answer, £2 . 2 . 6. * Commission is what is allowed by one merchant to another for buying or selling goods for him, and is usually a per-centage on the whole sum employed. Brokerage is an allowance similar to commission, under a different name, principally used in the buying and selling of stock in the funds. Insurance is a per-centage paid to those who engage to make good to the payeis g 153 INTEREST. 246-249. A ship whose value is i15,423 is insured at 192 per cent; what 3 IZ does the insurance amount to ?-Answer, £3033 . 3 . 9- 25 247. In reckoning how much a bankrupt is able to pay his creditors, as also to how much a tax or rate amounts, it is usual to find how many shillings in the pound is paid. Thus, if a person who owes £ioo can only pay £50, he is said to pay ios. in the pound. derived from the same reasoning as in (246). of £82 is £82 out of LI, or 5 8 The rule is easily 50oout For example, shillings, or is. pound. 2- in the 4 41 248. INTEREST is money paid for the use of other money, and is always a per-centage upon the sum lent. It may be paid either yearly, half-yearly, or quarterly ; but when it is said that £ioo is lent at 4 per cent, it must be understood to mean 4 per cent per annum; that is, that 4 pounds are paid every year for the use of £o00. The sum lent is called the principal, and the interest upon it is of two kinds. If the borrower pay the interest as soon as, from the agree- ment, it becomes due, it is evident that he has to pay the same sum every year; and that the whole of the interest which he has to pay in any number of years is one year's interest multiplied by the number of years. But if he do not pay the interest at once, but keeps it in his hands until he returns the principal, he will then have more of his creditor's money in his hands every year, and (if it were so agreed) will have to pay interest upon each year's interest for the time during which he keeps it after it becomes due. In the first case, the interest is called simple, and in the second compound. The interest and principal together are called the amount. 249. What is the simple interest of £1049 . 6 . 6 for 6 years and one-third, at 4! per cent ? This interest must be 6 - times the interest 2 3 any loss they may sustain by accidents from fire, or storms, according to the agreement, up to a certain amount which is named, and is a per-centage upon this amount. Tare, tret, and cloff, are allowances made in selling goods by wholesale, for the weight of the boxes or barrels which contain them, waste, &c.; and are usually either the price of a certain number of pounds of the goods for each box or barrel, or a certain allowance on each cwt. 154 COMMERCIAL ARITHMETIC. § 249-250. of the same sum for one year, which (245) is found by multiplying the sum by 4-, and dividing by Ioo. (230) (a) The process is as follows : 6 £1049. 16. ax4 4199. 6. o I ax- 524. I8.3 (82) 4. 3(£47. 4. 100)47,24.- 10 100 20 (228) 4,84* 12 10,I It II (b) £47. 4 . Io -I00 Int. for one yr. bx6 283. 9. bxI 66 0-- 100 I .14. 3 37 100 1 £299. 4. o- -Int. for 6 yrs. 3 100 EXERCISES. What is the interest of £1o 5 . 6 . 2 for 19 years and 7 weeks at 3, Answer, £6o0 . 9, very nearly. per cent? o What is the difference between the interest of £5 . 19 for 7 years at 3 per cent, and for 8 years at 2- per cent ? What is the interest of £157 . 17 . Answer, Ios. 2-d. 6 for one year at 5 per cent ? Answer, £7 . 17. 10o-. Shew that the interest of any sum for 9 years at 4 per cent is the same as that of the same sum for 4 years at 9 per cent ? 250. In order to find the interest of any sum at compound interest, it is necessary to find the amount of the principal and interest at the end of every year; because in this case (248) it is the amount of both * Here the 4s. from the dividend is taken in. t Here the 3d. from the dividend is taken in. ( INTEREST. 250"251. 155 principal and interest at the end of the first year, upon which interest accumulates during the second year. Suppose, for example, it is re- quired to find the interest, for 3 years, on £ioo, at 5 per cent, compound interest. The following is the process : £xoo First principal 5 First year's interest. Amount at the end of the first year. 105 (249) 5. Ito. 5. 5 Interest for the second year on £Io5. 5 Amount at the end of two years. o . 3 Interest due for the third year. r 5 . 15 * 3 Amount at the end of three years. 00oo o. o First principal. . 15 . 15 . 3 Interest gained in the three years. When the number of years is great, and the sum considerable, this process is very troublesome; on which account tables* are constructed to shew the amount of one pound, for different numbers of years, at different rates of interest. To make use of these tables in the present example, look into the column headed " 5 per cent ;" and opposite to the number 3, in the column headed "Number of years," is found 1157625 ; meaning that 1i will become £I1I57625 in 3 years. Now, EIoo must become ioo times as great; and 1157625x100oo is1157625 3d.; therefore the whole amount of (141); but (221) £7625 is 15s. is &£oo £115 . 15 . 3, as before. 251. Suppose that a sum of money has lain at simple interest 4 years, at 5 per cent, and has, with its interest, amounted to £350; itis required to find what the sum was at first. Whatever the sum was, if we suppose it divided into 1oo parts, 5 of those parts were added every year for 4 years, as interest; that is, 20 of those parts have been added to the first sum to make £350. If, therefore, £350 be divided into parts, 100 of those parts are the principal which we want to find, I20zo * Sufficient tables for all common purposes are contained in the article on Interest and on in the Penny Cyclopadia; and ample ones in theTreatise Annuities Reversions, in the Library of Useful Knowledge. 156 COMMERCIAL ARITHMETIC. § 251-253. is t£350X100 and 20 parts are interest upon it; that is, the principal 150 or £291. 13 . 4. 252. Suppose that A was engaged to pay B £350 at the end of four years from this time, and that it is agreed between them that the debt shall be paid immediately; suppose, also, that money can be employed at 5 per cent, simple interest; it is plain that A ought not to pay the whole sum, £350, because, if he did, he would lose 4 years' interest of It is fair, therefore, that he should the money, and B would gain it. only pay to B as much as will, with interest, amount in four years to Therefore, £58 . 6. 8 must be struck £350, that is (251), ;291 . 13 . 4. off the debt in consideration of its being paid before the time. This is called DISCOUNT ;* and £291 . 13 . 4 is called the present value of £350 due four years hence, discount being at 5 per cent. The rule for finding the present value of a sum of money (251) is : Multiply the sum by 0oo, and divide the product by 1oo rate per cent and number of years. increased by the product of the If the time that the debt has yet to run be expressed in years and months, or months only, the months must be reduced to the equivalent fraction of a year. EXERCISES. What is the discount on a bill of £138. discount being at 4- per cent ? 14. 4, due 2 years hence, Answer, £1I . 9 . I. What is the present value of £1031 . 17, due 6 months hence, interest being at 3 per cent ? Answer, £1ox6 . I2. 253. If we multiply by a+b, or by a-b, when we should multiply b b by a, the result is wrong by the fraction a+b, or a-- of itself: being too great in the first case, and too small in the second. Again, if we divide by a+b, where we should have divided by a, the result is too small by the fraction b of itself; while, if we divide by a-b instead a of a, the result is too great by the same fraction of itself. Thus, if we divide by 20 instead of I7, the result is 3 of itself too small; and if 17 * This rule is obsolete in business. When a bill, for instance, of £100. having a year to run, is discounted (as people now say) at 5 per cent, this means that 5 per cent of £100, or £5, is struck off. ( 253-254. 157 INTEREST. we divide by 360 instead of 365, the result is too great by 365 , or 73 of itself. If, then, we wish to find the interest of a sum of money for a portion of a year, and have not the assistance of tables, it will be found convenient to suppose the year to contain only 360 days, in which case its 73d part (the 7 zd part will generally do) must be subtracted from the result, to make the alteration of 360 into 365. The number 360 has so large a number of divisors, that the rule of Practice (230) may always be readily applied. Thus, it is required to find the portion which belongs to 274 days, the yearly interest being £i8 . 9 . Io, or I8"49I. 18"491 274 18o is 2 of 36o 9'z46 94 90 is - of 4is - o180 of 360 4623 *205 90 9)I4"074 8)I'564 "I96 13'878 = £13 . 17 . 7 Answer. But if the nearest farthing be wanted, the best way is to take 2-tenths of'the number of days as a multiplier, and 73 as a divisor; since m+365 is 2m+730, or -- m-73. Thus, in the preceding instance, 1o we multiply by 548 and divide by 73; and 548 x18*49I = 1oi133068, which divided by 73 gives 13*881, very nearly agreeing with the former, and giving £13 . 17 . 7-, which is certainly within a farthing of the truth. 254. Suppose it required to divide £ioo among three persons in such a way that their shares may be as 6, 5, and 9; that is, so that for every £6 which the Frst has, the second may have £5, and the third £9. It is plain that if we divide the £ioo into 6+5+9, or 20 parts, the first must have 6 of those parts, the second 5, and the third 9. Ioox6 oox5 Therefore (245) their shares are respectively, -o, £ , and 20 20 100X9 and £4. r£30, oox , -, or £30 £25;, and £45. 158 § 254-256. COMMERCIAL ARITHIMETIC. EXERCISES. Divide £394. 12 among four persons, so that their shares may be I as i, 6, 7, and I8.-Answer, £iz. 6.7 ; £zz2I. 73 . 19 .9; £86 .6 I 4; 19 . 3. Divide 20zo among 6 persons, so that the share of each may be as much as those of all who come before put together.-Answer, The first two have izs. 6d.; the third £I . 5; the fourth £E2. Io ; the fifth £5; and the sixth £Io. 255. When two or more persons employ their money together, and gain or lose a certain sum, it is evidently not fair that the gain or loss should be equally divided among them all, unless each contributed the same sum. Suppose, for example, A contributes twice as much as B, and they gain £15, A ought to gain twice as much as B; that is, if the whole gain be divided into 3 parts, A ought to have two of them and B one, or A should gain £io and B £5. Suppose that A, B, and C engage in an adventure, in which A embarks £250, B £130, and C £45. They gain £Iooo. How much of it ought each to have? one ought to gain as much for LI as the others. Each Now, since there are 250+130+45, or 425 pounds embarked, which gain LIooo, for each 10 1 Iooxz50 pound there is a gain of £ . Therefore A should gain 425 425 pounds, B should gain I3pounds, and C1000X4 ounds.On 425 425 these principles, by the process in (245), the following questions may be answered. A ship is to be insured, in which A has ventured £1928, and B £4963. The expense of insurance is £474. To. 2. each to pay of it ? How much ought Answer, A must pay £i32. 15 . 2-. A loss of £149 is to be made good by three persons, A, B, and C. Had there been a gain, A would have gained 4 times as much as B, and C as much as A and B together. bear? How much of the loss must each Answer, A pays £59 . I2, B £14 . x8, and C £74. 0o. 256. It may happen that several individuals employ several sums of money together for different times. In such a case, unless there be a special agreement to the contrary, it is right that the more time a sum 159 INTEREST. § 256-257. If, for example, is employed, the more profit should be made upon it. A and B employ the same sum for the same purpose, but A's money is employed twice as long as B's, A ought to gain twice as much as B. The principle is, that one pound employed for one month, or one year, ought to give the same return to each. Suppose, for example, that A employs £3 for 6 months, B £4 for 7 months, and C Liz for z months, and the gain is £ioo; how much ought each to have of it? Now, since A employs £3 for six months, he must gain 6 times as much as if he employed it one month only; that is, as much as if he employed £6x3, or 18, for one month; also, B gains as much as if he had employed £4x7 for one month; and C as if he had employed £izxz for one month. If, then, we divide £Ioo into 6x3+4x7+Izxz, or 70 parts, A must have 6 x3, or i8, izxz, or 24 of those parts. 6x3x1oo £ 6x3+4x7+12x2 4x7X1oo ,£ B must have 4x7, or 28, and C The shares of the three are, therefore, 6x3+4x7+12zx2 ,and £ I2X2XIxoo 6x3+4x7+Izxz" EXERCISES. A, B, and C embark in an undertaking; A placing £3 . 6 for 2 years, B £ioo £4276 . 7 I for i year, and C £xz for i- years. They gain How much must each receive of the gain ? Answer, A £226. 10.4; B £343z2. .I 3; C £617. 15. 5. A, B, and C rent a house together for 2 years, at £150 per annum. A remains in it the whole time, B 16 months, and C 4- months, during the occupancy of B. How much must each pay of the rent ?* Answer, A should pay £190. 12. 6; B £90. 2 . 6; C £18 . 15. 257. These are the principal rules employed in the application of arithmetic to commerce. There are others, which, as no one who under- stands the principles here laid down can fail to see, are virtually contained in those which have been given. Such is what is commonly called the Rule of Exchange, for such questions as the following: * This question does not at first appear to fall under the rule. If A little thought will serve to shew that what probably will be the firt idea of the proper method of solution is erroneous. 160 COMMERCIAL ARITHMETIC. 20 shillings be worth 25- francs, in France, what is £16o worth ? may evidently be done by the Rule of Three. 2§ 57. This The rules here given are those which are most useful in common life; and the student who understands them need not fear that any ordinary question will be above his reach. But no student must imagine that from this or any other book of arithmetic he will learn precisely the modes of operation which are best adapted to the wants of the particular kind of business in which his future life may be passed. There is no such thing as a set of rules which are at once most convenient for a butcher and a banker's clerk, a grocer and an actuary, a farmer and a bill-broker; but a person with a good knowledge of the principles laid down in this work, will be able bo examine and meet his own future wants, or, at worst, to catch with readiness the manner in which those who have gone before him have done so for themselves. APPENDIX TO THE FIFTH EDITION F DE MORGAN'S ELEMENTS OF ARITHMETIC. I. ON THE MODE OF COMPUTING. THE rules in the preceding work are given in the usual form, and the examples are worked in the usual manner. But if the student really wish to become a ready computer, he should strictly follow the methods laid down in this Appendix ; and he may depend upon it that he will thereby save himself trouble in the end, as well as acquire habits of quick and accurate calculation. I. In numeration learn to connect each primary decimal number, io, Ioo, 1000, &c. not with the place in which the unit falls, but with the number of ciphers following. Call ten a one-cipher number, a hundred a two-cipher number, a million a six-cipher number, and so on. If five figures be cut off from a number, those that are left are hundred-thousands; for ioo,ooo is a five-cipher number. Learn to connect tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, &c. with I, 2, 3, 4, 5, 6, &c. in the mind. seventeen-cipher number ? What is a For every 6 in seventeen say million, for the remaining 5 say hundred-thousand: the answer is a hundred thousand millions of millions. If twelve places be cut off from the right of a number, what does the remaining number stand for ?-Answer, As many millions of millions as there are units in it when standing by itself. II. After learning to count forwards and backwards with rapidity, as in 1, 2, 3, 4, &c. or 30, 29, 28, 27, &c., learn to count forwards or backwards by twos, threes, &c. up to nines at least, beginning fro n any number. Thus, beginning from four and proceeding by sevens, we P3 APPENDIX. 162 nave 4, IIx, IS,25, 32, &c., along which series you must learn to go as easily as along the series I, 2, 3, 4, &c.; that is, as quick as you can pronounce the words. The act of addition must be made in the mind without assistance: you must not permit yourself to say, 4 and 7 are Iz, Ix and 7 are i8, &c.; but only 4, i1, 18, &c. And it would be desirable, though not so necessary, that you should go back as readily as forward; by sevens for instance, from sixty, as in 60, 53, 46, 39, &c. III. Seeing a number and another both of one figure, learn to catch instantly the number you must add to the smaller to get the greater. Seeing 3 and 8, learn by practice to think of 5 without the necessity of saying 3 from 8 and there remains 5. And if the second number be the less, as 8 and 3, learn also by practice how to pass up from 8 to the next number which ends with 3 (or I3), and to catch the necessary augmentation, five, without the necessity of formally undertaking in words to subtract 8 from I3. Take rows of numbers, such as 426050 1864 and practise thisrule upon every figure and the next, not permitting yourself in this simple case ever to name the higher one. Thus, say 4 and 8 (4 first, 2 second, 4 from the next number that ends with 2, or I2,leaves 8), 2 and 4, 6 and 4, o and 5, 5 and 5, o and I, I and 7, 8 and 8, 6 and 8. IV. Study the same exercise as the last one with two figures and one. Thus, seeing 27 and 6, pass from 27 up to the next number that ends with 6 (or 36), catch the 9 through which you have to pass, and allow yourself to repeat as much as "27 and 9 are 36." 17729638I09 will give the following practice: Thus, the row of figures 17 and o are 17; 77 and 5 are 82; 72 and 7 are 79; 29 and 7 are 36; 96 and 7 are 103; 63 and o 5 are 68; 38 and 3 are 41; 81 and 9 are 90 ; Io and 9 are 19. V. In a number of two figures, practise writing down the units at the moment that you are keeping the attention fixed upon the tens. In the preceding exercise, for instance, write down the results, repeating the tens with emphasis at the instant of writing down the units. VI. Learn the multiplication-table so well as to name the product 163 ON THE MODE OF COMPUTING. the instant the factors are seen; that is, until 8 and 7, or 7 and 8, suggest 56 at once, without the necessity of saying " 7 times 8 are 56." Thus looking along a row of numbers, as 39706548, learn to name the products of every successive pair of digits as fast as you can repeat them, namely, 27, 63, o, 0, 30, 20, 32. VII. Having thoroughly mastered the last exercise, learn further, on seeing three numbers, to augment the product of the first and second by the third without any repetition of words. Practise until 3, 8, 4, for instance, suggest 3 times 8 and 4, or z8, without the necessity of saying " 3 times 8 are 24, and 4 is z8." Thus, 17923640o8 will suggest the following practice, I6,65, 21, 12, 22, 24, 8. VIII. Now, carry the last still further, as follows: Seeing four figures, as 2, 7, 6, 9, catch up the product of the first and second, increased by the third, as in the last, without a helping word; name the result, and add the next figure, name the whole result, laying emphasis upon the tens. 29." Thus, 2, 7, 6, 9, must immediately suggest " zo and 9 are The row of figures 773698974 will give the instances 52 and 6 are 58; 27 and 9 are 36; 27 and 8 are 35; 62 and 9 are 71; 8I and 7 are 88; 79 and 4 are 83. IX. Having four numbers, as 2, 4, 7, 9, vary the last exercise as follows : Catch the product of the first and second, increased by the third ; but instead of adding the fourth, go up to the next number that ends with the fourth, as in exercise IV. and 4 are 19." Thus, 2, 4, 7, 9, are to suggest" 15 And the row of figures 1723968929 will afford the in- stances 9 and 4 are 13; 17 and 2 are 19; 15 and I are 16; 33 and 5 are 38; 62 and 7 are 69; 57 and 5 are 62; 74 and 5 are 79. X. Learn to find rapidly the number of times a digit is contained in given units and tens, with the remainder. arrive at and repeat " 6 and 5 over." best practice. Thus, seeing 8 and 53, Common short division is the Thus, in dividing 236410792 by 7, 7)236410792 33772970, remainder 2. All that is repeated should be 3 and z; 3 and 5; 7 and 5; 7 and 2; 2 and 6; 9 and 4; 7 and 0o 0 and 2. 164 APPENDIX. In performing the several rules, proceed as follows : ADDITION. Not one word more than repeating the numbers written in the following process: the accented figure is the one to be written down; the doubly accented figure is carried (and don't say " carry 3," but do it). 6, I5, 17, 23, 31, 3"4' ; II, 47063 2598 12, 21, 22, 31, 3"7'; 9, 17, 24, 27, 32, 4"I'; IO, 14, 20, 21, 2"8'; 7, 9, I'3'. 26316 54792 819 6686 In verifying additions, instead of the usual way of omitting one line, adding without it, and then adding the line omitted, verify each column by adding it both 138174 upwards and downwards. SUBTRACTION. The following process is enough. The carriages, being always of one, need not be mentioned. From 79436258190 Take 58645962738 20790295452 8 and 2', 4 and 5', 7 and 4', 3 and 5', 6 and 9', io and z', 6 and o', 4 and 9', 7 and 7', 9 and o', 5 and 2'. It is useless to stop and say, 8 and z make Io; for as soon as the 2 is obtained, there is no occasion to remember what it came from. MULTIPLICATION. The following, put into words, is all that need be repeated in the multiplying part; the addition is then done as usual. The unaccented figures are carried. 670383 9876 4022298 4692681 5363064 6033447 18', 49', 22', 2', 42', 4'O', 21', 58', 26', 2', 49', 4'6', 24', 66', 30', 3', 56', 5 3, 27, 74', 34', 3', 63', 6'o'. 6620702508 Verify each line of the multiplication and the final result by casting out the nines. (Appendix II. p. 166.) It would be almost as easy, for a person who has well practised the 8th exercise, to add each line to the one before in the process, thus: ON THE MODE OF COMPUTING. 165 670383 9876 4022298 8; 21 and 9 are 30'; 59 and 2 50949108 are 61'; 27 and 2 are 29 ; 2 587255508 6620702508 and 2 are 4' ; 49 and o are 49'; 46 and 4 are 5o'. On the right is all the process of forming the second line, which completes the multiplication by 76, as the third line completes that by 876, and the fourth line that by 9876. DIvisioN. Make each multiplication and the following subtraction in one step, by help of the process in the 9th exercise, as follows: 27693)441972809662(15959730 165042 265778 165410 269459 202226 83756 6772 The number of words by which 26577 is obtained from 165402 (the multiplier being 5) is as follows: i5 and 7' are 2"z2; 47 and 7' are 5"4 5 35 and 5' are 4"o; 39 and 6' are 4"5; 14 and 2' are 16. The processes for extracting the square root, and for the solution of equations (Appendix XI.), should be abbreviated in the same manner as the division.* * The teacher will find further remarks on this subject in the Companion to the Almanac for 1844, and in the Supplement to the Penny Cyclopedia, article Computation. 166 APPENDIX. APPENDIX II. ON VERIFICATION BY CASTING OUT NINES AND ELEVENS. THE process of casting out the nines, as it is called, is one which the young computer should learn and practise, as a check upon his computations. It is not a complete check, since if one figure were made too small, and another as much too great, it would not detect this double error; but as it is very unlikely that such a double error should take place, the check furnishes a strong presumption of accuracy. The proposition upon which this method depends is the following: If a, b, c, d be four numbers, such that a = bc+d, and if m be any other number whatsoever, and if a, 6, c, d, severally divided by m, give the remainders p, q, r, s, then p and qr+s give the same remainder when divided by m (and perhaps aie themselves equal). For instance, 334 = 7x1I9+11; divide these four numbers by 7, the remainders are 5, 3, 5, and 4. And 5 and 5x3+4, or 5 and 19, both leave the remainder 5 when divided by 7. Any number, therefore, being used as a divisor, may be made a check upon the correctness of an operation. To provide a check which may be most fit for use, we must take a divisor the remainder to which is most easily found. The most convenient divisors are 3, 9, and I, of which 9 is far the most useful. As to the numbers 3 and 9, the remainder is always the same as that of the sum of the digits. For instance, required the remainder of 246120377 divided by 9. The sum of the digits is z+4+6+1+2+0+3+7+7, or 32, which gives the remainder 5. But the easiest way of proceeding is by throwing out nines as fast as they arise in the sum. Thus, repeat z2, 6 (2+4), I2 (6+6), say 3 (throwing out 9), 4, 6, 9 (throw this away), 7, 14, (or throwing out the 9) 5. This is the remainder required, as would appear by dividing 2461zo377 by 9. A proof may be given thus: It ON VERIFICATION, is obvious that each of the numbers, I, leaves a remainder I, since they are I, 2, 20, 200, &c. leave the remainder z; and so on. 167 &C. o, o100,iooo, &c. divided by 9, 9+1, 99+1, &c. Consequently, 3, 30, 300, the remainder 3; If, then, we divide, say 1764 by 9 in parcels, 1000 will be one more than an exact number of nines, 700 will be seven more, and 6o will be six more. So, then, from I, 7, 6, 4, put together, and the nines taken out, comes the only remainder which can come from 1764. To apply this process to a multiplication : It is asserted, in page 32, 10004569x3163 = 31644451747. that In casting out the nines from the first, all that is necessary to repeat is, one, five, ten, one, seven ; in the second, three, four, ten, one, four ; in the third, three, four, ten, one, five, nine, four, nine, eight, twelve, The remainders then are, 7, 4, I. three, ten, one. Now, 74 is 28, which, casting out the nines, gives I, the same as the product. Again, in page 43, it is asserted that 23796484 = 130000x183+6484. Cast out the nines from 13000, 183, 6484, and,we have 4, 3, and 4. Now, 4x3+4, with the nines cast out, gives 7; and so does 23796484. To avoid having to remember the result of one side of the equation, or to write it down, in order to confront it with the result of the other side, proceed as follows: Having got the remainder of the more com- plicated side, into which two or more numbers enter, subtract it from 9, and carry the remainder into the simple side, in which there is only one number. Then the remainder of that side ought to be o. Thus, having got 7 from the left hand of the preceding, take 2, the rest of 9, forget 7, and carry in 2 as a beginning to the left-hand side, giving 2, 4, 7, 14, 5, I 1, z 6, I4, 5, 9, 0. Practice will enable the student to cast out nines with great rapidity. This process of casting out the nines does not detect any errors in which the remainder to 9 happens to be correct. If a process be tedious, and some additional check be desirable, the method of casting out elevens may be followed after that of casting out the nines. serve that eleven. 0o+1, Ioo-, ooo+, Ob- &c. oooo0000-, are all divisible by From this the following rule for the remainder of division by 168 APPENDIX. I may be deduced, and readily used by those who know the algebraical process of subtraction. For those who have not got so far, it may be doubted whether the rule can be made easier than the actual division by i1. Subtract the first figure from the second, the result from the third, the result from the fourth, and so on. The final result, or the rest of II if the figure be negative, is the remainder required. 1642915 by ii, 4, -1; -i from z, 3 ; 3 from 9, 6; 6 from I, -5 ; -5 from 5, io; and Io is the remainder. gives -5, Thus, to divide and find the remainder, we have I from 6, 5; 5 from But 164 gives -i, and 6 is the remainder. and io is the remainder; 164291 With very little practice these re- mainders may be read as rapidly as the number itself. Thus, for 127619833424 need only be repeated, i, 6, o, i, 8, 0, 3, o, 4, -2, 6, and 6 is the remainder. When a question has been tried both by nines and elevens, there can be no error unless it be one which makes the result wrong by a number of times 99 exactly. APPENDIX III. ON SCALES OF NOTATION. WE are so well accustomed to o, 00oo,&c., as standing for ten, ten tens, &c., that we are not apt to remember that there is no reason why Io might not stand for five, ioo for five fives, &c., or for twelve, twelve Because we invent different columns of numbers, and let twelves, &c. units in the different columns stand for collections of the units in the preceding columns, we are not therefore bound to allow of no collections except in tens. If Io stood for 2, that is, if every column had its unit double of the unit in the column on the right, what we now represent by 1, 2, 3, 4, 5, 6, &c., would be represented by I, Io, I1, 1oor, 1010, IolT, IIo, &c. zoo, 101, IIo, III, 1000, This is the binary scale. If we take the ternary scale, in which io stands for 3, we have 1, 2, io, 11, 12, zo, 169 ON SCALES OF NOTATION. 91, 22, 102, 1Io, &c. oo, :ol, In the quinary scale, in which io is five, 234 stands for 2 twenty-fives, 3 fives, and 4, or sixty-nine. If we take the duodenary scale, in which io is twelve, we must invent new symbols for ten and eleven, because Io and thirteen; use the letters t and e. x now stand for twelve and Then 176 means I twelve-twelves, 7 twelves, and 6, or two hundred and thirty-four; and Ite means two hundred and seventy-five. The number which io stands for is called the radix of the scale of notation. To change a number from one scale into another, divide the number, written as in the first scale, by the number which is to be the radix of the new scale; repeat this division again and again, and the remainders are the digits required. For example, what, in the quinary scale, is that number which, in the decimal scale, is 17036 ? 5)17036 5)3407 Remr. I 5)681 Answer.. 1o02121 Decimal. Quinary. 2 ...... Verification, Ioooooo means 5)136 ...... 5)27 15625 20000 ...... 1250 oo000 ...... 125 1 ...... 100 5)5 5)1 ... ,... 25 20 ...... 2 ...... IO I..... ......o I 10212I ......17036 o ...... I The reason of this rule is easy. Our process of division is nothing but telling off 17036 into 3407 fives and I over; we then find 3407 fives to be 681 fives of fives and 2 fives over. Next we form 68x fives of fives into 136 fives of fives of fives and I five of fives over; and so on. It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, the number carried being always the radix of the system. carry fives instead of tens. and division : Thus, in the quinary system we I now give an example of multiplication 170 APPENDIX. Decimal. Quinary. means 42143 2798 1234 194 324232 1192 25182 232034 2798 134341 42143 542812 1x433z2222 Decimal. Duodecimal. 6 4t9)7 t4eo8(16687 4t19 705)22610744(32071 I460 5074 2814 2546 1394 28te 2546 489 3650 3320 3 3 08 495 Another way of turning a number from one scale into another is as follows: Multiply the first digit by the old radix in the new scale, and add the next digit; multiply the result again by the old radix in the new scale, and take in the next digit, and so on to the end, always using the radix of the scale you want to leave, and the notation of the scale you want to end in. Thus, suppose it required to turn 16687 (duodecimal) into the decimal scale, and I6432 (septenary) into the quaternary scale : 16687 16432 Duodecimals into Decimals. IxX2+6 = 3X7+6 = 18 x12+ 6 222 xI2+8 2672 xI2+7 Answer . . . . 32071 Septenaries into Quaternaries. 31 x7+4 1133 x7+3 22130 x7+2 I02I012 171 ON THE DEFINITION OF FRACTIONS. Owing to our division of a foot into into 12 12 equal parts, the duodecimal Let the square foot be also divided scale often becomes very convenient. parts, each part is Iz square inches, and the Izth of the Izth is one square inch. Suppose, now, that the two sides of an oblong piece of ground are 176 feet 9 inches 7-Izths of an inch, and 65 feet Using the duodecimal scale, and duo- Ix inches 5-Izths of an inch. decimal fractions, these numbers are Iz8'97 and 55"e5. Their product, the number of square feet required, is thus found : Answer, 68e8'I44e (duod.) square feet, or I28'97 55'e5 6 xx66o square feet 16 square inches i7ee It would, however, be exact enough to allow II6095 617ee 2-hundredths of a foot for every quarter of an 617ee 6 ± and 144 12 of a square inch. inch, an additional hundredth for every 3 inches,* and i-hundredth more if there be a 7 inches 12th or z-Izths above the quarter of an inch. Thus, 98e8 i44e should be "76+'o3+'01, or "8o, and I1x would be '95 ; and the preceding I2 might then be found decimally as I76"8x6595 as II659'96 square feet, near enough for every practical purpose. APPENDIX IV. ON THE DEFINITION OF FRACTIONS. THE definition of a fraction given in the text shews that , for instance, is the ninth part of seven, which is shewn to be the same thing as seven-ninths of a unit. But there are various modes of speech under which a fraction may be signified, all of which are more or less in use. 1. In 7 we have the 9 2. 7-9ths of a unit. 9 th part of 7. 3. The fraction which 7 is of 9. * And at discretion one hundredth more for a large fraction of three inches. 172 APPENDIX. 4. The times and parts of a time (in this case part of a time only) which 7 contains 9. 5. The multiplier which turns nines into sevens. 6. The ratio of 7 to 9, or the proportion of 7 to 9. 7. The multiplier which alters a number in the ratio of 9 to 7. 8. The 4th proportional to 9, i, and 7. The first two views are in the text. The third is deduced thus: If we divide 9 into 9 equal parts, each is I, and 7 of the parts are 7; consequently the fraction which 7 is of 9 is Z. The fourth view follows 9 immediately: For a time is only a word used to express one of the repetitions which take place in multiplication, and we allow ourselves, by an easy extension of language, to speak of a portion of a number as being that number taken a part of a time. The fifth view is nothing 7 of its amount 9 has every 9 converted into a 7, and any fraction of a 9 which may more than a change of words : A number reduced to remain over into the corresponding fraction of 7. This is completely 7 of a = 7 times a. The sixth, 9 9 seventh, and eighth views are illustrated in the chapter on proportion. proved when we prove the equation When the student comes to algebra, he will find that, in all the applications of that science, fractions such as b most frequently require that a and b should be themselves supposed to be fractions. It is, therefore, of importance that he should learn to accommodate his views of a fraction to this more complicated case. Suppose we take 4. We shall find that we have, in this case, a better idea of the views from and after the third inclusive, than of the first and second, which are certainly the most simple ways of conceiving 7. We have no notion of the(43 thpart of2, nor of 2- 4 ths of a unit; indeed, we coin a new species of adjective when we talk of the (4)th part of anything. But we can readily imagine that 2- is some fraction of 43; that the first is some part of a time the second; that there must be some multiplier which turns every 4- in a number into 2-; 5 2 and so on. Let us now see whether we can invent a distinct mode of applying the first and second views to such a compound fraction as the above. 173 ON THE DEFINITION OF FRACTIONS. We can easily imagine a fourth part of a length, and a fifth part, meaning the lines of which 4 and 5 make up the length in question; and there is also in existance a length of which four lengths and two-fifths For instance, we of a length make up the original length in question. might say that 6, 6, z is a division of parts, 6, 6, and a third of a part, z. the (z2) th, or 14 into z2 equal parts-z equal 3 So we might agree to say, that (the reader may coin the adjective 2(!)rd, or (2 !st as he pleases) part of 14 If we divide the line A B into eleven is 6. equal parts in c, D, E, &c., we must then say that A c is the IIth part, A A D the (I)th, D C (5I)th, E F G A E the ( 3 2th, AI the H I K A F the (23)th, th, AK the and AB itself the Ist part of A B. Ith, A, the L M B AG the (z-)th, A a the I()th, AM the I Ith, The reader may refuse the language if he likes (though it is not so much in defiance of etymology as talking of multiplying by I); but when A B is called i, he must either call I2 A F -, or make one definition of one class of fractions and another of another. Whatever abbreviations they may choose, all persons will agree that is a direction to find such a fraction as, repeated b times, will give I, and then to take that fraction a times. So, to get 2, the simplest way is to divide the whole unit into 46 4 3 io of these parts, repeated 45 times, give the whole. The parts; I IlI I IH I I II I I I I l 4 th is then -, 4 L 2 3 4 4N 43 I A l ! I I II 45 43 4s and 2 such parts is , or A c. The student should try several examples of this mode of interpreting complex fractions. But what are we to say when the denominator itself is less than unity, as in is it? ? Are we to have a h prt of a unit? and what Had there been a 5 in the denominator, we should have taken 174 APPENDIX. the part of which 5 will make a unit. As there is 2 in the denominator, 2 5 we must take the part of which 2 will be a unit. That part is larger than a unit; it is 2 1 - 2 .5 units; 2- is that of which - is I. The above 2 x 5 fraction then directs us to repeat 2- units 3- times. By extending our 2 2 4 word 'multiplication' to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other. If 2- yards cost 31 shillings, how much does one yard cost ? In 3 2 such a case as this, the student looks at a more simple question. If Io 5 yards cost io shillings, he sees that each yard costs -, or 2 shillings, 5 and, concluding that the same process will give the true result when 3 1 the data are fractional, he forms -3, reduces it by rules to 3 or I-, and 2 2 concludes that I yard costs I8 pence. 2 The answer happens to be correct; but he is not to suppose that this rule of copying for fractions whatever is seen to be true of integers is one which requires no demonstration. In the above question we want money which, repeated 2 times, shall give 31 shillings. 2 If we divide the shilling into 14 equal 1 parts, 6 of these parts repeated 2- times give the shilling. To get 3- 3 times as much by the same repetition, we must take 3- of these 6 parts 21 at each step, or 21 parts. s the price. ~ 2 Hence, -, or i-, is the number of shillings I 2 4 APPENDIX V. ON CHARACTERISTICS. WHEN the student comes to use logarithms, he will find what follows very useful. In the mean while, I give it merely as furnishing a rapid rule for finding the place of a decimal point in the quotient before the division is commenced. When a bar is written over a number, thus, 7, let the number be called negative, and let it be thus used : Let it be augmented by additions of its own species, and diminished by subtractions; thus, 7 and z give 9, and let 7 with z subtracted give 5. But let the addition of a 175 ON CHARACTERISTICS. number without the bar diminish the negative number, and the subtraction increase it. subtracted is 15. Thus, 7 and 4 are 3, 7 and xz make 5, 7 with 8 In fact, consider I, 2, 3, &c., as if they were gains, and I, 2, 3, as if they were losses: let the addition of a gain or the removal of a loss be equivalent things, and also the removal of a gain and the addition of a loss. Thus, when we say that 4 diminished by I- gives 7, we say that a loss of 4 incurred at the moment when a loss of I1 is removed, is, on the whole, equivalent to a gain of 7 ; and saying that 4 diminished by z is 6, we say that a loss of 4, accompanied by the removal of a gain of z, is altogether a loss of 6. By the characteristic of a number understand as follows: When there are places before the decimal point, it is one less than the number of such places. Thus, 32I14, r'oo83, 8 (which is 8o00...) 9"999, all have o for their characteristics. But I7'32, 48, 93II6, all have I; 12z6o3 and 126 have z; 1937264*666 has 7. But when there are no places before the decimal point, look at the first decimal place which is significant, and make the characteristic negative accordingly. 121, Thus, "612, "9004, in all of which significance begins in the first decimal place, have the characteristic I; but i018and -o99 have z; "ooo17 has 4; 0000ooooooo0001 has 9. To find the characteristic of a quotient, subtract the characteristic of the divisor from that of the dividend, carrying one before subtraction if the first significant figures of the divisor are greater than those of the For instance, in dividing I4.6o8 by oo00279. The character- istics are 2 and 3; and 2 with 3 removed would be 5. But on looking, dividend. we see that the first significant figures of the divisor, 27, taken by themselves, and without reference to their local value, mean a larger number than 14, the first two figures of the dividend. Consequently, to 3 we carry I before subtracting, and it then becomes 2, which, taken from 2, gives 4. And this 4 is the characteristic of the quotient, so that the quotient has 5 places before the decimal point. Or, if abcdefbe the first figures of the quotient, the decimal point must be thus placed, abcdef. But if it had been to divide o00279 by 146-08, no carriage would have been required ; and 3 diminished by 2 is 5 ; that is, the first 176 APPENDIX. significant figure of the quotient is in the 5th place. has "oooo before any significant figure. The quotient, then, A few applications of this rule will make it easy to do it in the head, and thus to assign the meaning of the first figure of the quotient even before it is found. APPENDIX VI. ON DECIMAL MONEY. OF all the simplifications of commercial arithmetic, none is comparable to that of expressing shillings, pence, and farthings as decimals of a pound. The rules are thereby put almost upon as good a footing as if the country possessed the advantage of a real decimal coinage. Any fraction of a pound sterling may be decimalised by rules which can be made to give the result at once. Two shillings is £*ioo One shilling is £'050 Sixpence is £'025 One farthing is £'ooi o04 Thus, every pair of shillings is a unit in the first decimal place; an odd shilling is a 50 in the second and third places; a farthing is so nearly the thousandth part of a pound, that to say one farthing is "ooi, two farthings is "002, &c., is so near the truth that it makes no error in the first three decimals till we arrive at sixpence, and then 24 farthings is exactly .o25 or 25 thousandths. But 25 farthings is 'o26, 26 farthings is "o27, &c. Hence the rule for the first three places is One in the firstfor every pair of shillings; 50 in the second and third for the odd shilling, if any ; and I for every farthing additional, with I extrafor sixpence. 177 ON DECIMAL MONEY. 3 -d. = £'o014 zs. 6d. os. 7-d. = £'032 4 zs. 9Id. = #;"139 Is. 2-d. = 38. 23d. Thus, os. £'o6o = £'125 2 4 = £'I6I 3 13s. io d. = £'694 Is. II-d. = £o96 4 In the fourth and fifth places, and those which follow, it is obvious that we have no produce from any farthings except those above sixpence. For at every sixpence, "000046 is converted into "oo, and this has been already accounted for. Consequently, to fill up the fourth and fifth places, Take 4 for every farthing* above the last sixpence, and an additional i for every six farthings, or three halfpence. The remaining places arise altogether from "ooooo6 for every farthing above the last three halfpence; for at every three halfpence complete, "ocooo 0is converted into 'oooo, and has been already accounted for. Consequently, to fill up all the places after the fifth, Let the number offarthings above the last three halfpence be a numerator, 6 a denominator, and annex the figures of the corresponding decimal fraction. .t may be easily remembered that The figures of 6 are 166666... The figures of 6are 666666... 2. S .. 833333... 6...333333... .. 5...5 1 os. 3 d. = 01415813333*** os. 73d. = .032129666... 2S. 6d. 2S. = ... l5 00 000ool d. = 333 92d. -- I3953 ' The student should remember all the multiples of 4 up to 4X25, or 100. 178 APPENDIX. Is. 2-a. = *06014116666... 2 3s. 3 I-d. =.0961875 Is. 2zd.= 4 69479 1666... 13s. Io d. 4 I*614583333... 4 The following examples will shew the use of this rule, if the student will also work them in the common way. To turn pounds, &c., into farthings: Multiply the pounds by 960, iooo or by 10oo00-40, or by roo00(- 4~ ; that is, from subtract 4 per cent of itself. Thus, required the number of farthings in times the pounds £1663 . 11i. 9 4 1 66 3590625xIooo = 1663590*625 4 per cent of this, 66543'625 No. of farthings required, 1597047 What is 471 per cent of £i66. 13 . Io and "6148 of £2 9 71 . I6 . 9 ? 166"691 p. c. 66*6764 5 p. c. z2 p.c. 8*3346 4"1673 40 2971*837 1783Io022 "6 0oI 29'7184 '004 II"8873 0ooo8 2'3775 791783 3 1827"0854 £79. 3. 6 4 1827 . I.82 The inverse rule for turning the decimal of a pound into shillings, pence, and farthings, is obviously as follows: A pair of shillings for every unit in the first place ; an odd shilling for 50 (if there be 50) in the second and third places ; and a farthing for every thousandth left, after abating I if the number of thousandths so left exceed 24. The direct rule (with three places) gives too little, the inverse rule too much, except at the end of a sixpence, when both are accurate. 6 s. 43d. is rather greater than 4 £'319; or when the two do not exactly agree, the common money is the Thus, £X183 is rather less than 3s. 8d., and greatest. But £'z25 and £'35 are exactly 2s. 6d. and 7s. ON DECIMAL MONEY. 179 I 3 9- per cwt. Required the price of 17 cwt. 81 lb. 13 oz. at £3 . II 2' 4. true to the hundredth of a farthing. 3'590625 17 6 z040625 lb. 56 - 6 7 7 8 1"795313 '512946 "224414 c64118 2 I oz. 8 - o016029 4 4 - "co8o15 I I - "002004 4 II 2 2 "001002 £63"664466 I £63 . 13 . 3-- Three men, A, B, C, severally invest £191 . 12 . 71, £6I . I4.. 8. and £122. £5II . iz.6-. How . 91 in an adventure which yields 2 2 ought the proceeds to be divided among them ? A, 19t'6322 9 B, 61'73333 C, 122'o8958 Produce of £1. 375'45520)511'62708(1"3 62686 136 17188 1'362686 z*362686 23 53532 1'362686 33716 oo00801 I 362686 8 17612 25710 362686 1 226417 13627 3183 272537 13627 8176 9538 409 18o 27254 Io9o 409 41 122 27 3 1 2 611346 4 841231 7 92 236191 33 8 980221 663697 180 APPENDIX. £261 . 2. 8 26r'I346 . . . A's share 84I1231 . . . B's ,, . . . 1663697 . . . C's ,, . . . 166 . 4 2.53 84. 4 5II"6274 £51z . 7 . 43 zz. 6 3 4 If ever the fraction of a farthing be wanted, remember that the coinage-result is larger than the decimal of a pound, when we use only three places. From iooo times the decimal take 4 per cent, and we get the exact number of farthings, and we need only look at the decimal then left to set the preceding right. 134"6 Thus, in I23'i 369*7 5'38 4'92 14'79 '2 2 18 '91 we see that (if we use four decimals only) the pence of the above results are nearly 8d. "22 of a farthing, 5d. "i8, and 4-d. 91. 2 2 A man can pay £2376 . 4 . 4-, his debts being £3293 . How much per cent can he pay, and how much in the pound ? I1 . 0-3 4 3293'553)2376"2I80(7214756 70 7309 4 8598 1 5662 2488 183 18 Answer, £72. I 2 . I1- per cent. o . 14 . 1 5- per pound. APPENDIX VII. ON THE MAIN PRINCIPLE OF BOOK-KEEPING. A BRIEF notice of the principle on which accounts are kept (when they are properly kept) may perhaps be useful to students who are learning book-keeping, as the treatises on that subject frequently give too little in the way of explanation. Any person who is engaged in business must desire to know accurately, whenever an investigation of the state of his affairs is made, 181 ON BOOK-KEEPING. 1, What he had at the commencement of the account, or immediately after the last investigation was made; 2, What he has gained and lost in the interval in all the several branches of his business; 3, What he is now worth. the third. From the first two of these things he obviously knows In the interval between two investigations, he may at any one time desire to know how any one account stands. An account is a recital of all that has happened, in reference to any class of dealings, since the last investigation. It can only consist of receipts and expenditures, and so it is said to have two sides, a debtor and a creditor side. All accounts are kept in money. If goods be bought, they are estimated by the money paid for them. If a debtor give a bill of exchange, being a promise to pay a certain sum at a certain time, it is put down as worth that sum of money. All the tools, furniture, horses, &c. used in the business are rated at their value in money. All the actual coin, bank-notes, &c., which are in or come in, being the only money in the books which really is money, is called cash. The accounts are kept as if every different sort of account belonged to a separate person, and had an interest of its own, which every transaction either promotes or injures. If the student find that it helps him, he may imagine a clerk to every account: one to take charge of, and regulate, the actual cash; another for the bills which the house is to receive when due; another for those which it is to pay when due; another for the cloth (if the concern deal in cloth); another for the sugar (if it deal in sugar); one for every person who has an account with the house; one for the profits and losses; and so on. All these clerks (or accounts) belonging to one merchant, must account to him in the end-must either produce all they have taken in charge, or relieve themselves by shewing to whom it went. For all that they have received, for every responsibility they have undertaken to the concern itself, they are bound, or are debtors; for everything which has passed out of charge, or about which they are relieved from answering to the concern, they are unbound, or are creditors. These words must be taken in a very wide sense by any one to whom bookR 182 APPENDIX. keeping is not to be a mystery. Thus, whenever any account assumes responsibility to any parties out of the concern, it must be creditor in the books, and debtor whenever it discharges any other parties of their responsibility. But whenever an account removes responsibility from any other account in the same books it is debtor, and creditor whenever it imposes the same. To whom are all these parties, or accounts, bound, and from whom are they released ? Undoubtedly the merchant himself, or, more properly, the balance-clerk, presently mentioned. But it is customary to say that the accounts are debtors to each other, and creditors by each other. Thus, cash debtor to bills receivable, means that the cash account (or the clerk who keeps it) is bound to answer for a sum which was paid on a bill of exchange due to the house. At full length it would be: " Mr. C (who keeps the cash-box) has received, and is answerable for, this sum which has been paid in by Mr. A, when he paid his bill of exchange." On the other hand, the corresponding entry in the ac- count of bills receivable runs-bills receivable, creditor by cash. length: At fill " Mr. B (who keeps the bills receivable) is freed from all responsibility for Mr. A's bill, which he once held, by handing over to Mr. C, the cash-clerk, the money with which Mr. A took it up." Bills re- ceivable creditor by cash is intelligible, but cash debtor to bills receivable is a misnomer. The cash account is debtor to the merchant by the sum received for the bill, and it should be cash debtor by bill receivable. The fiction of debts, not one of which is ever paid to the party to whom it is said to be owing, though of no consequence in practice, is a stumbling-block to the learner; but he must keep the phrase, and remember its true meaning. The account which is made debtor, or bound, is said to be debited; that which is made creditor, or released, is said to be credited. All who receive must be debited; all who give must be credited. No cancel is ever made. i' cash received be afterwards repaid, the sum paid is not struck off the receipts (or debtor-side of the cash account), but a discharge, or credit, is written on the expenditure (or credit) side. 183 ON BOOK-KEEPING. The book in which the accounts are kept is called a ledger. It has double columns, or else the debtor side is on one page, and the creditor side on the opposite, of each account. always the left. The debtor-side is Other books are used, but they are only to help in keeping the ledger correct. Thus there may be a waste-book, in which all transactions are entered as they occur, in common language; a journal, in which the transactions described in the waste-book entered at stated periods, in the language of the ledger. are The items entered in the journal have references to the pages of the ledger to which they are carried, and the items in the ledger have also references to the pages of the journal from which they come; and by this mode of reference it is easy to make a great deal of abbreviation in the ledger. Thus, when it happens, in making up the journal to a certain date, that several different sums were paid or received at or near the same time, the totals may be entered in the ledger, and the cash account may be made debtor to, or creditor by, sundry accounts, or sundries; the sundry accounts being severally credited or debited fortheir sha'rs of the whole. The only book that need be explained is the ledger. All the other books, and the manner in which they are kept, important as they may be, have nothing to do with the main principle of the method. Let us, then, suppose that all the items are entered at once in the ledger as they arise. twice. It has appeared that every item is entered If A pay on account of B, there is an entry, "A, creditor by B;" and another, " B, debtor to A." This is what is called double- entry ; and the consequence of it is, that the sum of all the debtor items in the whole book is equal to the sum of all the creditor items. For what is the first set but the second with the items in a different order ? If it were convenient, one entry of each sum might be made a double-entry. The multiplication table is called a table of double- entry, because 42, for instance, though it occurs only once, appears in two different aspects, namely, as 6 times 7 and as 7 times 6. Suppose, for example, that there are five accounts, A, B, C, D, E, and that each account has one transaction of its own with every other account; and let the debits be in the columns, the credits in the rows, as follows: 184 APPENDIX. t: $ 4 A B C D z3 19 32 4 6 II 25 10 2 A, Creditor B, Creditor 17 C, Creditor 9 41 D, Creditor 14 28 16 E, Creditor 15 4 6o E 3 I Here the i6 is supposed to appear in D's account as D creditor And to say that the sum by C, and in C's account as C debtor to D. of debtor items is the same as tLat of creditor items, is merely to say that the preceding numbers give the same sum, whether the rows or the columns be first added up. If it be desired to close the ledger when it stands as above, the following is the way the accounts will stand : the lines in italics will presently be explained. A, Creditor. A, Debtor. B, Debtor. I B, Creditor. ToB . . 17 ByB . . 23 To A . . 23By A . . ToC . . 9 ByC . . 19 ToC . . 4I ByC . . To D . . 14 ByD . . 32 ToD . . 28 ByD . . 11 ToE . . 15 ByE . . 4 ToE . . 4 ByE . . 25 By Balance 37 To Balance 23 78j 78 96 17 6 96 ON BOOK-KEEPING. C, Creditor. C, Debtor. ToA ToB D, Debtor. D, Creditor. 9 .. ByA .. 9 To A . . 6 .. ByB .. 41 ToB .. . IByB ByD . o ToC .. . oByC . 2 ToE .. i .. To D . . i6 ToE . . 6o0 ByE . .. By Balance 10 . . To B . . To C . . To D . . . . By B . . 2 ByC . . ByD . . . 0 . I . . 15 To . I4 28 . . 16 3 6r Balance, Cred. 37 By A 39 ByD 6o 3 To B 4 4 z ByE . 7 Balance, Debtor. E, Creditor. ByA ByA 6I 101 E, Debtor. To A To Balance 39 32 By E 76 . . . . 23 7 46 76 To Balance 46 80 80 In all the part of the above which is printed in Roman letters we see nothing but the preceding table repeated. But when all the accounts have been completed, and no more entries are left to be made, there remains the last process, which is termed balancing the ledger. To get an idea of this, suppose a new clerk, who goes round all the accounts, collecting debts and credits, and taking them all upon himself, that he alone may be entitled to claim the debts and to be responsible for the assets of the concern. To this new clerk, whom I will call the balance- clerk, every account gives up what it has, whether the same be debt or credit. The cash-clerk gives up all the cash; the clerks of the two kinds of bills give up all their documents, whether bills receivable or entries of bills payable (remember that any entry against which there is money set down in the books counts as money when given up, that is, as money due or money owing); the clerks of the several accounts of goods give up all their unsold remainders at cost prices; the clerks of the several personal accounts give up vouchers for the sums owing to or from the several parties; and so on. But where more has been paid out than received, the balance-clerk adjusts these accounts by giving R2 186 APPENDIX. instead of receiving; in fact, he so acts as to make the debtor and creditor sides of the accounts he visits equal in amount. For instance, the A account is indebted to the concern 55, while payments or discharges to the amount of 78 have been made by it. The balance-clerk accordingly hands over 23 to that account, for which it becomes debtor, while the balance enters itself as creditor to the same amount. But in the B account there is 96 of receipt, and only 59 of payment or discharge. The balance-clerk then receives 37 from this account, which is therefore credited by balance, while the balance acknowledges as much of debt. The balance account must, of course, exactly balance itself, if the accounts be all right; for of all the equal and opposite entries of which the ledger consists, so far as they do not balance one another, one goes into one side of the balance account, and the other into the other. Thus the balance account becomes a test of the accuracy of one part of the work : if its two sides do not give the same sums, either there have been entries which have not had their corresponding balancing entries correctly made, or else there has been error in the additions. But since the balance account must thus always give the same sum on both sides, and since balance debtor implies what is favourable to the concern, and balance creditor what is unfavourable, does it not appear as if this system could only be applied to cases in which there is neither loss nor gain ? This brings us to the two accounts in which are entered all that the concern began with, and all that it gains or loses--the stock account, and the profit-and-loss account. In order to make all that there was to begin with a matter of double entry, the opening of the ledger supposes the merchant himself to put his several clerks in charge of their several departments. In the stock account, stock, which here stands for the owner of the books, is made creditor by all the property, and debtor by all the liabilities; while the several accounts are made debtors for all they take from the stock, and creditors by all the responsibilities they undertake. Suppose, for instance, there are £5oo in cash at the commencement of the ledger. There will then appear that the merchant has handed over to the cash-box £5oo, and in the stock account will appear, " Stock creditor by cash, £50co;" while, in 187 ON BOOK-KEEPING. the cash account will appear, " Cash debtor to stock, £500." Suppose that at the beginning there is a debt outstanding of £50 to Smith and Co., then there will appear in the stock account, " Stock debtor to Smith and Co. £5o," and in Smith and Co.'s account will appear "Smith and Co. creditors by stock, £5o." Thus there is double entry for all that the concern begins with by this contrivance of the stock account. The account to which everything is placed for which an actual equivalent is not seen in the books is the profit-and-loss account. This profit-and-loss account, or the clerk who keeps it, is made answerable for every loss, and the supposed cause of every gain. This account, then, becomes debtor for every loss, and creditor by every gain. If goods be damaged to the amount of £zo by accident, and a loss to that amount occur in their sale, say they cost £8o and sell for £6o cash, it is clear that there is an entry " Cash debtor to goods £6o," and "Goods creditor by cash £6o." Now, there is an entry of £8o somewhere to the debit of the goods for cash laid out, or bills given, for the whole of the goods. It would affect the accuracy of the accounts to take no notice of this; for when the balance-clerk comes to adjust this account, he would find he receives reckoned upon, would be 0zo less than he might have without any explanation of the reason; and there a failure of the principle of double-entry. convenient that the balance-account Since it is of the goods should merely re- present the stock in hand at the close, the account of goods therefore lays the responsibility of £zo upon the profit-and-loss account, or there is the entry " Goods creditor by profit-and-loss, £zo," and also " Profit-and-loss debtor to goods, £zo." Again, in all pay- ments which are not to bring in a specific return, such as house and trade expenses, wages, &c. these several accounts are supposed to adjust matters with the profit-and-loss account before the balance begins. Thus, suppose the outgoings from the mere premises occupied exceed anything those premises yield by £200, or the debits of the house account exceed its credits by £zoo, the account should be balanced by transferring the responsibility to the profit-and-loss account, under 188 APPENDIX, the entries "House expenses creditor by profit-and-loss, £zoo," " Profitand-loss debtor to house expenses, £zoo." In this way the profit-and- loss account steps in from time to time before the balance account commences its operations, in order that that same balance account may t consist of nothing but the necessary m ters of account for the next year's ledger. This transference of accounts, or transfusion another, requires attentive colsideration. of one account into The receiving account be- comes creditor for the credits, and debtor for the debits, of the transmitting account. The rule, therefore, is : Make the transmitting account balance itself, and, on whichever side it is necessary to enter a balancing sum, make the account debtor or creditor, as the case may be, to the receiving account, and the latter creditor or debtor to the former. Thus, suppose account A is to be transferred to account B, and the latter is to arrange with the balance-account. If the two stand as in Roman letters, the processes in Italic letters will occur before the final close. A, Debtor. A, Creditor. B, Debtor. B, Creditor. To sundries £ ioo00By sundries £500 To sundries £6oo By sundries £400 To B .. .. 400 £500 To Balance 50oo 200 £8oo By A . . .. 400 £8oo And the entry in the balance account will be, "Creditor by B, £200oo," shewing that, on these two accounts, the credits exceed the debits by £zoo. Still, before the balance account is made up, it is desirable that the profit-and-loss account should be transferred to the stock account; for the profit and loss of this year is of no moment as a part of next year's ledger, except in so far as it affects the stock at the commencement of the latter. Let this be done, and the balance account may then be made in the form required. The stock account and the profit-and-loss account, the latter being the only direct channel of alteration for the former, differ in a peculiar manner* from the other preliminary accounts, and the balance account * T.'e treatises on book-keeping have described this difference in as peculiar a 189 ON BOOK-KEEPING. is a species of umpire. They represent the merchant: their interests are his interests; he is solvent upon the excess of their credits over their debits, insolvent upon the excess of their debits over their credits. It is exactly the reverse in all the other accounts. If a malicious person were to get at the ledger, and put on a cipher to the pounds in various items, with a view of making the concern appear worse than it really is, he would make his alterations on the debtor sides of the stock and profit-and-loss accounts, and on the creditor sides of all the others. Accordingly, in the balance account, the net stock, after the incorporation of the profit-and-loss account, appears on the creditor side (if not, it should be called amount of insolvency, not stock), and the debts of the concern appear on the same side. But on the debit side of the balance account appear all the assets of the concern (for which the balance-clerk is debtor to the clerks from whom he has taken them). The young student must endeavour to get the enlarged view of the words debtor and creditor which is requisite, and must then learn by practice (for nothing else will give it) facility in allotting the actual entries in the waste-book to the proper sides of the proper accounts. I do not here pretend to give more than such a view of the subject as may assist him in studying a treatise on book-keeping, which he will probably find to contain little more than examples. manner. They call these accounts the fictitious accounts. Now they represent the merchant himself; their credits are gain to the business, their debits losses or liabilities. If the terms real and fictitious are to be used at all, they are the real accounts, and all the others are as fictitious as the clerks whom we have supposed to keep them. 190 APPENDIX. APPENDIX VIII. ON THE REDUCTION OF FRACTIONS TO OTHERS OF NEARLY EQUAL VALUE. 'THERE is a useful method of finding fractions which shall be nearly equal to a given fraction, and with which the computer ought to be acquainted. Proceed as in the rule for finding the greatest common measure of the numerator and denominator, and bring all the quotients into a line. Then write down, 2d Quot. Ist Quot. x zd Quot.+i Ist Quot. Then take the third quotient, multiply the numerator and denominator of the second by it, and add to the products the preceding numerator and denominator. Form a third fraction with the results for a nume- rator and denominator. Then take the fourth quotient, and proceed with the third and second fractions in the same way; and so on till the quotients are exhausted. For example, let the fraction be 9131 13128 This is the process for finding the 913lI)1328(1, 2 I137 3997(3, I 586(1, 15 551 35(1, 2 greatest common measure 20o 26 8 13128 in 9(, 8 1 of 9131 and its most compact form, and the quotients and fractions are: 1s 3 I 15 I 2 x 8 1 2 7 9 I6 249 265 779 1044 9131 1 3 10 13 23 358 381 0 1501 1 3 1 28 I 11 It will be seen that we have thus a set of fractions ending with the original fraction itself, and formed by the above rule, as follows: I 1st Fraction = Ist zd Fraction Quot. I I zd Quot. 2 dst Quot.x zd Quot.+1 3 2d r Num.x3d Quot.+xst Num . r 3d Fraction =-=-zd Den . x 3d Quot.+ Ist Denr. 2x3+1 31x3+ _ 7 10o 3d Numr.x4th Quot.+2d Numr. 7xI+2 9 r 3d Den . x4th Quot.+zd Denr. Ioxi+3 13' ON and so on. REDUCTION But we have done something more than merely reascend to the original fraction by means of the quotients. I 2 7 191 OF FRACTIONS. 9, The set of fractions, &c. are continually approaching in value to the original I 3' 10 I3 fraction, the first being too great, the second too small, the third too great, and so on alternately, but each one being nearer to the given I 2 Thus, I is too great, and - is fraction than any of those before it. 2 3 too small; but - is not so much too small as - is too great. And 3 I again, -, though too great, is not so much too great as - is too small. IO 3 Moreover, the difference of any of the fractions from the original fraction is never greater than a fraction having unity for its numerator and the product of the denominator and the next denominator for its denominator. i 7 as -, nor - 30 I I 2 Thus, - does not err by so much as -, nor - by so much I 9 353 nor -by so much as -, c. by so much as -I, 3& 130 10 13 299 Lastly, no fraction of a less numerator and denominator can come so near to the given fraction as any one of the fractions in the list. Thus, no fraction with a less numerator than 249, and a less denominator 9131 249 than 358, can come so near to 913- as 249 13128 358 The reader may take any example for himself, and the test of the accuracy of the process is the ultimate return to the fraction begun with. Another test is as follows: The numerator of the difference of any two consecutive approximating fractions ought to be unity. Thus, i6 249 in our instance, we have - and -, which, with a common denominator, 358 23 23x358, have 5728 and 5727 for their numerators. As another example, let us examine this question: The length of the year is 365"z4z24 days, which is called in common life 365- days. 24---224, and proceed as in the rule. Take the fraction ICOOCO 24224)100000(4, 7, I, 4, 9, 2 2496 1 and 757 is '24224 6o8 o 4 3104 64 32 7 8 39 359 757 29 33 161 1482 3125 in its lowest terms. Hence, it appears that the 3125 excess of the year over 365 days amounts to about I day in 4 years, 192 APPENDIX. which is not wrong by so much as I day in I16 years; more accurately, to 7 days in 29 years, which is not wrong by so much as I day in 957 years; more accurately still, to 8 days in 33 years, which is not wrong by so much as I day in 5313 years; and so on. This method may be applied to finding fractions nearly equal to the square roots of integers, in the following manner : Set down the number whose ,4 3 = 6+.... 611 5 4 554 5 1 6 16 6 5 4, &c. 7 3 929 3 6 71 6 76 3, &c. 13 I 15 1 3 1 I 21 1x square root is wanted, say 43. This square root is 6 and a frac- 3, &c. tion. Set down the integer 6 in the first and third row, and I in the second row always. Form the successive rows each from the one before, in the following manner: One row The next row has b', c', formed in this order, a', being thus, a a' = excess of b'c', already formed, over a. b b'= quotient of 43-a c c' = integer in the quotient of 6+a divided by b'. 2 divided by b. Thus the second row is formed from the first, as under: 6 1 = excess of 7xI (both just found) over 6. 6 7 = 43- x6 divided by I. I 6 = integer of 6+6 divided by 7 (just found). The third row is formed from the second, thus: I 5 = excess of Ix6 over I. 7 6 = 43-ixI divided by 7. SI = integer of 6+-1 divided by 6; and so on. In process of time the second column, I, 7, I, occurs again, after which the several columns are repeated in the same order. As a final process, take the set in the lowest line (excluding the first, 6), namely, I, 1, 3, 1, 5,1, 3, &c. and use them by the rule given at the beginning of this article, as follows : I 1 3 I 5 I 3 1 I 1 4 5 29 34 131 165 296 1 2 7 9 52 61 235 296 531 I,&c. 193 GENERAL PROPERTIES OF NUMBERS. is very near the square root of 43, not erring by so much as Hence, 6X 296 r 296x53 If we try it, we shall find 6 3767481 165 1941 to be 1941 the square of which is 296 296 7 8766, 87616 or 43 87616' This rule is of use when it is frequently wanted to use one square root, and therefore desirable to ascertain whether any easy approximation exists by means of a common fraction. /2z 1+.... S iz is often used. O I 70 does not ; consequently, err by I I I For example, Here it appears that I o-x1 70xx69 70 70 I2 2 2 2 2 I 2 5 12 29 In operation, a fair approximation. 2 70 22 is 1'4142857... 7o 7035 1'4142135... fact, -, 2 5 Iz 2 9 70 169 &c. or is, considering the ease of the ixx the truth being The following isan additional example : VJ19 = 4+... 423 3 2 4 135 2 5 3113 I 3 I 2 I 4 II 5 14 14 39 412 I 2 3 42 8 2 I 3 I 2,&c. APPENDIX IX. ON SOME GENERAL PROPERTIES OF NUMBERS. PROP. 1. If a fraction be reduced to its lowest terms, so called,* that is, ifneither numerator nor denominator be divisible by any integer greater than unity, then no fraction of a smaller numerator and denominator can have the same value. Let a be a fraction in which a and b have no common measure greater than unity: and, if possible, let - be a fraction of the same value, d a aa c c being less than a, and d less than b. Now, since b = , we have a b = * This theorem shews that what is called reducing a fraction to its lowest terms (namely, dividing numerator and denominator by their greatest common npeasure), is correctly so called. 194 APPENDIX. let m be the integer quotient of these last fractions (which must exist, since a > c, b > d), and let e and f be the remainders. a mc+e - or b md+f c d Then me md e me mc+e would lie between Hence, e and must be equal, for if not, mf rnd md+f me e a e e Hence, b m- and , instead of being equal to the former. so that if a fraction whose numerator and denominator have no common measure greater than unity, be equal to a fraction of lower numerator and denominator, it is equal to another in which the numea e rator and denominator are still lower. If we proceed with = - in a a f similar manner, we find g where g< e, h