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ELEMENTS OF ARITHMETIC. 
 
^{Xtd. (a^y^^^. 
 
 ELEMENTS 
 
 OF 
 
 ARITHMETIC. 
 
 BY 
 
 AUGUSTUS DE MORGAN, 
 
 OF TBINITT COLLEGE, CAMBRIDGE; 
 
 FELLOW OF '(HK ROYAL A8TR0NOMICAL SOCIETY, AND OP THK CAUBBIDGE f UI LOSOPHICAL gOClBTV; 
 FKOF£SaOR OF MATHEMATICS IN DNITEB8ITX COLLEGE, LOKBON. 
 
 "Hominis studiosi est intelligere, quas utilitates proprie afl'erat aritlimetica 
 his, qui solidam et perfectam doctrinam in cseteris philosopliise partibus explicant. 
 Quod enim vulgo dicunt, principium esse dirnidium totius, id vel maxime in phi- 
 losophise partibus conspicitur."— Melanctuon. 
 
 "Ce u'est point par la routine qu'on e'instruit, c'est par sa propre reflexion; 
 et il est essentiel de contracter I'habitude de se rendre raison de ce qu'on fait: 
 cette habitude s'acquiert plus facilement qu'on nepense; et une foi§ acquise, elle 
 ne se perd plus."— Con dillac. 
 
 SEVENTEENTH THOUSAND. 
 
 LONDON: 
 WALTON AND MABERLY, 
 
 UPPER GOWER STREET, AND IVY LANE, PATERNOSTER ROW. 
 
 M.DCCCLVIH. 
 
goucATioi?! I'iro- 
 
 (Z^^^<=^-<K.'0^ 
 
 LONUOX : 
 
 PRINTED BY J. WEKTHEIMK.H ANP CO. 
 
 CIECUS-PLACK, FINSBUUY-CIHCUS. 
 
 Add to Lib. 
 GIFT 
 
QfWOZ 
 
 JL; h «■ avu 
 
 PREFACE. 
 
 The preceding editions of this work were published in 1830, 
 1832, 1835, and 1840. This fifth edition differs from the 
 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 
 sterUng into decimal fractions, gives the possessor the greater 
 part of the advantage which he would derive from the intro- 
 duction 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. 
 
 41.7 
 
VI ♦ PREFACE. 
 
 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 v^rorks 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. 
 
 SECTION PAQB 
 
 I. Numeration . . 1 
 
 II. Addition and Subtraction . 14 
 
 III. MultipUcation 24 
 
 IV. Division Si 
 
 V. Fractions 51 
 
 VI. Decimal Fractions ... . . . , 65 
 
 VII. Square Root 89 
 
 VIII. Proportion . . 100 
 
 IX. Permutations and Combinations 118 
 
 BOOK II. 
 
 I. Weights and Measures, &c 124 
 
 IT. Rule of Three . 144 
 
 III. Interest, &c 160 
 
 APPENDIX. 
 
 I. On the mode of computing 161 
 
 II. On verification by casting out nines and elevens . , .166 
 
 III. On scales of notation . 168 
 
 IV. On the definition of fractions 171 
 
 V. On characteristics 174 
 
 VI. On decimal money . . 176 
 
 VII. On the main principle of book-keeping . . . 180 
 
 VIII. On the reduction of fractions to others of nearly equal value . 190 
 
 IX. On some general properties of numbers . . . .193 
 
 X. On combinations . , 201 
 
 XI. On Horner's method of solving equations .... 210 
 XII. Rules for the application of arithmetic to geometry . .217 
 
ELEMENTS OF AEITHMETIC, 
 
 BOOK I. 
 
 PRINCIPLES OF ARITHMETIC. 
 
 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, jet, because there is one of the first kind to 
 every one of the second, one man to every horse, a new notion will be 
 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 
 
I PRINCIPLKS 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 r he will 
 then have two baskets of pebbles, by which he \A\\ 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 in 
 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 praetor, 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 
 thera 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 
 
§4-5. 
 
 NUMERATION. 
 
 he must have, in order to express all the numbers which he would be 
 obliged to make use of. He must, then, after giving a separate name 
 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 pre- 
 ceded 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 
 
 eignt 
 
 nine 
 
 
 nineteen 
 
 ten and 
 
 nine 
 
 ten 
 
 
 twenty 
 
 two tens 
 
 » 
 
 twenty-one 
 
 two tens and one 
 
 
 fifty 
 
 five tens 
 
 twenty-two 
 
 two tens and two 
 
 
 sixty 
 
 six tens 
 
 &c. &c. 
 
 &c. &c. 
 
 
 seventy 
 
 seven tens 
 
 thirty 
 
 three tens 
 
 
 eighty 
 
 eight tens 
 
 &c. 
 
 &c. 
 
 
 ninety 
 
 nine tens 
 
 forty 
 
 four tens 
 
 
 a hundred 
 
 ten tens 
 
 &c. 
 
 &c. 
 
 
 
 
 s 
 
 L hundred and one 
 
 ten tens and one 
 
 
 
 &c. &c. 
 
 
 
 
 
 a thousand 
 
 ten hundreds 
 
 
 
 ten thousand 
 
 
 
 
 * It has been supposed that eleven and twelve are derived from the Saxon for one 
 left 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 decern. 
 
4 PRINCIPLES OF ARITHMETIC. § 5-8. 
 
 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 im- 
 possible 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 : 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ught 
 
 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 he hfis 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 
 
§ 8-IO. NUMERATION. O 
 
 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 niunljer 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 fiftn 
 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 woids the number which you wish to repre- 
 ' sent ; in the first column on the right, place the number of fingeis 
 
 ♦ The references are to the preceding articles. 
 
 b2 
 
PRINCIPLES OF ARITHMETIC. 
 
 § IO-12. 
 
 which the first man will be holding up when that number of yards has 
 been measured. In the next column, place the fingers which the second 
 man will then be holding up ; and so on. 
 
 fifty-seven. 
 
 one hundred and four. 
 
 one hundred and ten. 
 
 two thousand three hundred and forty- 
 eight. 
 
 fifteen thousand nine hundred and six. 
 
 one hundred and eighty-seven thousand 
 and four. 
 
 three million, six hundred and ninety- 
 seven thousand, two hundred and 
 eighty-five. 
 
 
 •^ 
 
 S" 
 
 2? 
 
 
 t 
 
 
 to 
 
 a. 
 
 % 
 
 I. 
 
 
 
 
 
 
 5 
 
 7 
 
 II. 
 
 
 
 
 
 I 
 
 o 
 
 4 
 
 III. 
 
 
 
 
 
 I 
 
 I 
 
 o 
 
 IV. 
 
 
 
 
 2 
 
 3 
 
 4 
 
 8 
 
 V. 
 
 
 
 I 
 
 5 
 
 9 
 
 o 
 
 6 
 
 VI. 
 
 
 I 
 
 ' 
 
 7 
 
 o 
 
 o 
 
 4 
 
 VII. 
 
 3 
 
 6 
 
 9 
 
 7 
 
 2 
 
 8 
 
 5 
 
 11. In I. the number fifty-seven is expressed. This means (5) five 
 tens and seven. The first has therefore counted all his fingers five 
 times, and has counted seven fingers more. This is 8he\vn by five 
 fingers of the second man being held up, and seven of the first. In II. 
 the number one hundred and four is represented. This number is (5) 
 ten tens and four. 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 aa many thousands of yards ; and so 
 on : that is, if we suppose a figure to move from any column to the one 
 
§ 12-15. NUMERATION. 7 
 
 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, be- 
 cause 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 2'; and so on: one hundred or ten tens by 1"; two hundred 
 by 1" ; one thousand by 1'" ; and so on : putting Eoman figures for 
 accents when they become too many to write with convenience. The 
 fourth number in the table would then be written 2'" i" 4' 8, which 
 might also be expressed by 8 4' 3" 2'", 4' 8 i" 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 
 104 would be distinguished from 14 by writing for the first \'%^ and for 
 the second 1'/^ 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 /j.^^^.^ each four stands for four 
 oi 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 col- 
 lections of ten pebbles each ; in the third, four of one hundred each ; 
 and 80 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. 
 
6 PRINCIPLES O* ARITHMETIC. § 16-19. 
 
 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 2 and 3 make 5 ; yet you may say that 2 yards and 3 yards 
 make 5 yards, and that 2 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 com 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 miit. Thus, 1 5 pebbles and 7 pebbles 
 together make 22 pebbles; 15 acres and 7 acres together make 22 acres, 
 and so on. From this we come to say that 15 and 7 make 22, meaning 
 that 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 men- 
 tioned in this chapter. 
 
 I. Ten signs are used, one to stand for nothing, the rest for tlie 
 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 tlie signs already mentioned by the side of each other, and 
 agreeing that the first figure on the right hand shall keep the value 
 
§ 19-21. NUMERATION. 51 
 
 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. 
 
 IV. 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 20006, two hundred and six is 206. 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 himdreds. 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 
 
 1 ten-thousand 6 thousands 7 hundreds 8 tens and 5 ; 
 or 16 thousands 78 tens and 5 ; or i ten thousand 678 tens and 5 ; 
 or 167 hundreds 8 tens and 5 ; or 1678 tens and 5, and so on. 
 
 21. £XKRCIS£8. 
 
 I. Write down the signs for ; — four hundred and seventy-six ; two 
 
10 PRINCIPLES OF ARITHMETIC. § 2I-23. 
 
 thousand and ninety-seven ; sixty-four thousand three liundred and 
 fifty ; two millions seven hundred and four ; five hundred and seventy- 
 eight millions of millions. 
 
 II. Write at full length 53, 1805, 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 loiii is 
 greater than 9879. 
 
 22. You now aee 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 12 pence, and a 
 pound 20 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 plut thirty- 
 eight {plus is the Latin for more). 
 
 II. 64—12 means that 12 is to be taken away from 64, and is the 
 
§23-24* NUMERATION. 11 
 
 same thing as 52. This is the difference of 64 and 12, 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. This is the product of 9 and 8, and is read nine into eight. 
 ic8 
 
 IV. -7- means that 108 is to be divided by 6, or that you must 
 
 6 
 
 find out how many sixes there are in 108 ; and is the same thing as 18. 
 This is the quotient of 108 and 6 ; and is read a hundred and eight 
 by six. 
 
 V. When two numbers, or collections of numbers, with the fore- 
 going 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. Thus : 
 
 7+9—3=12+1 ; 1 + 3x2=11, and 80 on. 
 
 24. It often becomes necessary to speak of something which is true 
 not of any one number only, but of all numbers. For example, take 
 10 and 7 ; their sum* is 17, their difference is 3. If this sum and 
 diiference 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 ist No. 
 (ist No.+2d No.)— (ist No.— 2d No.) = twice 2d 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 PEINCIPLES OF ARITHMETIC. § a4-aS* 
 
 8— (24-i)x(i+i) signifies that 2+1 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. Let twice a be represented by 2a, and twice 
 b by 26. The equations then become 
 
 (a+6)+(a— A)=2a, and {a+b')—{a—b)—zh. 
 This may be explained still further, as follows : 
 
 25. Suppose a number of sealed packets, marked a, 6, c, rf, &c., on 
 the outside, each of which contains a distinct but unkno^vn 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 sub- 
 tract 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+1 times. This will be found to be true of any 
 number, and, when proved, may be said to be true of the number con- 
 tained 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— I 
 
 • This should be (23) axa, but the sign x is unnecessary here. It is used with 
 uumbers, as in 2x7, to prevent confounding this, which is 14, with 27. 
 
§ 26-27. 
 
 NUMERATION. 
 
 13 
 
 26. When, therefore, we wish to talk of a number without specify- 
 ing any one in particular, we use a letter to represent it. Thus: 
 Suppose we wish to reason upon what will follow from dividing a num • 
 ber 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 b, 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 rf, by a number ar, we diminish the 
 whole number just as much, or 
 
 a— J? = b+c+{d—j;). 
 
 27. EXERCISES. 
 
 What is a+7.b—c, where 0= 12, b= 18, c=j ? — Answer, 41. 
 
 What is ; — , where a = 6 and 5 = 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 ? 
 
 b 
 
 
 
 d 
 
 Anst 
 
 2 
 
 3 
 
 4 
 
 10 
 
 12 
 
 7 
 
 I 
 
 *5 
 
 1 
 
 1 
 
 1 
 
 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. Probably, at first, either 
 these or the fingers were used. 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 contri- 
 bute 12 tens to the sum. The 8 hundreds in the first and the 7 hundreds 
 in the second will add 15 hundreds to the sum ; and the thousand m 
 
§29-31. ADDITION AND SUBTRACTION. 15 
 
 the first with the 2 thousands in the second will contribute 3 thousands 
 to the sum ; therefore the sum required is 
 
 3 thousands, 15 hundreds, 12 tens, and 13 units. 
 
 To simplify this result, you must recollect that — 
 
 13 units are 1 ten and 3 units. 
 
 12 tens are i hund. and 2 tens. 
 
 15 hund. are 1 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 2799, 
 
 4 thousands, 6 himdreds, 3 tens, and 3 units, 
 which (19) is written 4633. 
 
 30. The former process, written with the signs of (23) is as follows : 
 
 1834= 1x1000+8x100+3x10 + 4 
 *799 = 2x1000 + 7x100 + 9x10 + 9 
 
 Therefore, 
 
 1834+2799 = 3x1000+ 15x100+ 12x10+ 13 
 But 13= 1x10+ 3 
 
 I2X 10= 1x100+ 2x10 
 
 15X 100=1x1000+ 5x100 
 3x1 000 =3x1 000 Therefore, 
 1834+2799=4x1000+ 6x100+ 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. 
 
 § 31-33. 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 H 
 
 •5 
 
 16 
 
 17 
 
 18 
 
 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). 
 
 12+6=18 22+6 = 28 19+8 = 27 
 54+9 = 63 56+7 = 63 22+8 = 30 
 100—9 = 91 27—8=19 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. 
 
§ 33. ADDITION. 17 
 
 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 nimiber, part it into 
 8 tens and 5 units ; if 136 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. 
 
 I V. Add together all the numbers in the column of tens, remember- 
 ing 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. 
 
 PixAMPLK. — "What is 
 
 1805+36+ 19727+3+1474 +2008 
 
 1805 The addition of the units' line, or 8+4+3+7+6+5, gives 
 
 36 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, which gives 15 for the number 
 
 2008 of tens ; that is, i hundred and 5 tens. 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 20 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, which gives 15 
 
 c2 
 
18 
 
 PRINCIPLES OF ARITHMETIC. 
 
 § 33-36. 
 
 thousands, or i ten thousand and 5 thousand. 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 ; 
 that is, find i+i, or 2 ten thousands. Write 2 under the 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 comer to 
 corner, when added together give the number 24156. 
 
 zoi6 
 
 4212 
 
 1656 
 
 3852 
 
 1296 
 
 3492 
 
 936 
 
 3132 
 
 576 
 
 2772 
 
 2,6 
 
 252 
 
 2052 
 
 4248 
 
 1692 
 
 3888 
 
 1332 
 
 35^8 
 
 972 
 
 3168 
 
 612 
 
 2412 
 
 2448 
 
 288 
 
 2088 
 
 4284 
 
 1728 
 
 3924 
 
 1368 
 
 3564 
 
 1008 
 
 2808 
 
 648 
 
 684 
 2880 
 1116 
 
 2484 
 
 720 
 
 2916 
 
 3M 
 
 2520 
 756 
 
 2124 
 
 360 
 
 2556 
 
 4320 
 
 2160 
 
 396 
 
 1764 
 4356 
 2196 
 
 3960 
 1800 
 3996 
 
 1404 
 3600 
 1836 
 
 3204 
 1440 
 3636 
 
 1044 
 3240 
 1476 
 
 2844 
 1080 
 3276 
 
 3312 
 
 1152 
 
 2952 
 
 792 
 
 2592 
 
 36 
 
 2232 
 
 4032 
 
 1872 
 
 3672 
 
 1512 
 
 1548 
 
 3348 
 
 u88 
 
 2988 
 
 432 
 
 2628 
 
 72 
 
 2268 
 
 4068 
 
 1908 
 
 3708 
 
 3744 
 
 1584 
 
 3384 
 
 828 
 
 3024 
 
 468 
 
 2664 
 
 108 
 
 2304 
 
 4104 
 
 1944 
 
 1980 
 4176 
 
 3780 
 1620 
 
 1224 
 3816 
 
 3420 
 1260 
 
 864 
 3456 
 
 3060 
 900 
 
 504 
 3096 
 
 2700 
 540 
 
 144 
 2736 
 
 2340 
 180 
 
 4140 
 2376 
 
 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+10, 
 since 12 is 3 less than 15, and 10 is three more than 7. This was the 
 principle upon which the whole of the process in (29) was conducted. 
 
 36. Let a and h 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+6 stands for this sum. To say, in algebraical 
 
§ 36-40. ADDITION. 19 
 
 language, that the sum of a and i is not altered by adding c to a, pro- 
 vided 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 2a, 3a, 4a, Sec. 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 nrmiber a stands for, we know 
 that, be it what it may, 2a+2a=4a, 3a+2a=5a, 40+90=130 ; and gene- 
 rally, 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)af that o is taken w+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) X2 is 7x2 or 14 ; while 
 3+4x2 is 3+8, or II. 
 
 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 sub- 
 tracting 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 100 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 100 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 dif erence. 
 
 II. Since 6 exceeds 4 by 2, 
 and 3 exceeds 2 by i, 
 and 12 exceeds 5 by 7, 
 
 6, 3, and 12 together, or 21, exceed 4, 2, and 5 together, or 11, by 2, 
 1, and 7 together, or 10: the same thing may be said of any other 
 numbers. 
 
 41. If a, b, and 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 i, 
 
 (a— c)— (ft— c) = a—b. 
 
 The brackets cannot be here removed as in (36). That is, p—{q—r) 
 is not the same thing as p—q—r. For, in the first, the difference of y 
 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 12— (8— 5). If we subtract 
 8 from 12, or form 12—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 12—8 we have therefore subtracted 5 too much. 
 This must be set right by adding 5 to the result, which gives 12—8+5 
 for the value of 12— (8— 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, 
 
§ 41-43' SUBTRACTION. 21 
 
 a-'ijb-i-o—d—e) = a—b—c-i-d+e. 
 
 2a+3J— (a— 2i) = aa+36— a+ai = a+fJ. 
 
 4j?+jr— ( I TX—f^y) = 4J?+y— 1 70^+9^ = i oy— 13a?. 
 
 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 2 units. 
 
 34631 is 3 ten-th. 4 th. 6 hund. 3 tens and i unit. 
 
 Now 2 units exceed ... i unit by 1 unit. 
 
 6 tens 3 tens 3 tens. 
 
 7 hundreds .... 6 hundreds ... i hundred. 
 
 7 thousands .... 4 thousands ... 3 thousands. 
 
 5 ten-thousands . . 3 ten-thous. ... a 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. 3 th. i hund. 3 tens and i unit, 
 which is 23131. Therefore the difference of 57762 and 34631 is 23131, 
 or 57762—34631 = 23131. 
 
 43. Suppose I want to find the difference between 61274 and 39628. 
 Write them at length, and 
 
 61274 is 6 ten-th. 1 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 difii- 
 culty immediately, since 8, being greater than 4, cannot be taken from 
 it. But from (40) it appears that we shaU not alter the difference of 
 two nimibers if we add the same number to loth of them. Add ten tO 
 the first mimber, 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. The 
 numbers will then stand thus, 
 
 6 ten-thous. i thous. a himd. 7 tens and 14 wnite.* 
 3 ten-thous. 9 thous. 6 hund. 3 tens and 8 units. 
 
 • Those numbers which have been altered are put in italics. 
 
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 10 hundred to both numbers, which will not 
 alter their difference, and remember to increase the hundreds in the 
 upper line by 10, 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 10 thousand to both numbers, and increase the thousands in the 
 upper line by 10, 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. 11 thotis. 12 hund. 7 tens and 14 units. 
 
 4 tenrthous. 10 thous. 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, 
 
 a ten-thous. 1 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 i, 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 begin- 
 ning of the lower line as will make the number of figures equal. You 
 do not alter a number by placing ciphers at the beginning of it. For 
 example, 00818 is the same number as 818, for it means 
 
 o ten-thous. o thous. 8 hunds. i ten and 8 units; 
 
§ 45-46. SUBTRACTION. 23 
 
 the first two signs are nothing, and the rest is 
 
 8 hundreds, i ten, and 8 units, or 8i8. 
 
 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 imits, 
 instead of 7 hundreds. 
 
 46, EXAMPLE. 
 
 What is the difference between 370829 16400 30 174 
 and 30813649276188 
 
 Diflterence 3677477990753986 
 
 EXERCISES. 
 
 I. What is 18337+149263200— 6472902? — Answer 142808635. 
 What is 1000—464+3279—646 ? — Ans. 3169. 
 
 II. Subtract 
 
 64+76+144—18 from 33—2+100037. — 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 
 
 looooooo— 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 2469. — 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 OP ARITHMETIC. § 46-49. 
 
 VI. In the following table subtract b from a, and b from the re- 
 mainder, and so on imtil b can be no longer subtracted. Find how 
 many times b can be subtracted from a, and what is the last remainder. 
 
 A 
 
 B 
 
 No. of times. 
 
 Remaii 
 
 23604 . 
 
 • • • 9999 . 
 
 , . 2 . . 
 
 . 3606 
 
 209961 
 
 37173 
 
 5 
 
 24096 
 
 74712 
 
 6792 
 
 II 
 
 
 
 4802469 
 
 654321 
 
 7 
 
 222222 
 
 18849747 
 
 3141592 
 
 6 
 
 195 
 
 987654321 
 
 123456789 
 
 S 
 
 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 shorter 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 sura 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? Write 
 17 five seventeens in a column, and make the addition, which gives 
 17 85. In this case 85 is called the product of 5 and 17, and the 
 17 process of finding the product is called multiplication, which 
 85 gives nothing more than the addition of a number of the same 
 quantities. 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 
 
§49-51' MULTIPLICATION. 25 
 
 there were 1367 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 enor- 
 mous length. To avoid this, a shorter rule is necessary, which I now 
 proceed to explain. 
 
 50. The student must first make himself acquainted with the pro- 
 ducts of all numbers as far as 10 times 10 by means of the following 
 table,* which must be committed to memory. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 3 
 
 6 
 
 9 
 
 12 
 
 x5 
 
 18 
 
 21 
 
 24 
 
 27 
 
 30 
 
 33 
 
 36 
 
 4 
 
 8 
 10 
 
 12 1 16 20 
 
 i 
 
 24 
 30 
 
 28 
 35 
 
 32 
 40 
 
 36 
 
 45 
 
 40 
 50 
 
 55 
 
 48 
 60 
 
 5 
 
 15 
 
 20 25 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 60 
 
 66 
 
 72 
 
 7 
 8 
 
 9 
 
 14 
 16 
 18 
 
 21 
 24 
 27 
 
 28 
 
 32 
 36 
 
 35 
 40 
 
 45 
 
 42 
 48 
 54 
 
 49 
 56 
 63 
 
 56 
 64 
 
 72 
 
 63 
 72 
 81 
 
 70 
 80 
 90 
 
 77 
 8S 
 
 99 
 
 84 
 96 
 
 108 
 
 10 
 
 20 
 22 
 
 30 
 33 
 
 40 
 44 
 
 50 
 55 
 
 60 
 66 
 
 70 
 77 
 
 80 
 88 
 
 90 
 99 
 
 100 
 no 
 
 no 
 121 
 
 120 
 132 
 144 
 
 II 
 
 12 
 
 24 
 
 36 
 
 48 1 60 
 
 72 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 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 ba^'e 
 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. § 5 1- 54. 
 
 sixes. 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 
 
 side by side, I find seven of them, and six in each 
 
 row, the whole number of which is 7 times 6, or 
 
 seven sixes. And the whole number is 42, which- 
 
 ever way I count. The same method may be 
 
 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 
 com 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 sepa- 
 rately by any parts into which you choose to divide that number, and 
 add the results. For example, 4 and 2 make 6. To multiply 7 by 6 
 first multiply 7 by 4, and then by 2, and add the products. This will 
 give 42, which is the product of 7 and 6. Again, since 57 is made up 
 of 32 and 25, 57 times 50 is made up of 32 times 50 and 25 times 50, 
 and so on. If the signs were used, these would be written thus : 
 
 7x6 = 7x4 + 7x2. 
 50x57 = 50x32+50x25. 
 
 64. The principles in the last two articles may be expressed thus : 
 If a be made up of the parts j?, y, and z^ ma is made up of mx^ my, 
 and mz ; or, 
 
 if a = X +1/ +z. 
 
 ma = mx+vit/+mz. 
 or, m{x+y+z) = mx+my+mz. 
 
 A similar result may be obtained if a, instead of being made up of 
 
§ 54-55' MULTIPLICATION. 27 
 
 a*, y, and z, is made by combined additions and subtractions, such as 
 x-^y—z, x—y-VZy x—y—z^ &c. To take the first as an instance : 
 
 Let a = X +y —%. 
 
 then ma = mx+my—mz. 
 
 For, if a had been x+y-, ^na would have been mx-k-my. But since a 
 is less than x-¥y by ar, 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 mx+my^ but mx+my—mz. Similar reason- 
 ing may be applied to other cases, and the following results may be 
 obtained : 
 
 m{a+b+c—d) = ma+mb+mc—md. 
 
 a{a—b) = aa—ab. 70(7+2^) = 490+1406. 
 
 b{a—b) = ba—bb. {aa-^a-\-i)a = aaa+aa-¥a. 
 
 3(20—46) = 6a— 12b. {iab—zc)/yzbc = izaabbc—Sabcc. 
 
 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 prodttct by 2. To shew this, ilace 
 
 7 counters in a line, and repeat that line in all 8 times, as in figures 
 
 I. and II. 
 
 II. 
 
 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 ob- 
 longs; 80 that in the whole the number of counters is twice a8, or 
 28x2, 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. The same method may be 
 applied to other numbers. Thus, since 80 is 8 times 10, 256 times 80 
 is 256 multiplied by 8, and that product multiplied by 10. If we use 
 the signs, the foregoing assertions are made thus : 
 
 7x8 = 7x4x2 = 7x2x4. 
 256x80 = 256x8x10 = 256x10x8. 
 
 EXERCISES. 
 
 Shew that 2x3x4x5 = 2x4x3x5 = 5x4x2x3, &c. 
 Shew that 18x100 = 18x57+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 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 J, 0, and rf, one after the other, and by the product bod at once, we 
 write the following : 
 
 a><bxcyd = 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 abcdefm&y be formed by mul- 
 tiplying 
 
 ab cde f 
 
 abf de o 
 
 abo de/ &c. 
 
 57. In order to multiply by to, annex a cipher to the right hand of 
 the multiplicand. Thus, 10 times 2356 is 23560. To shew this, write 
 2356 at length which is 
 
§ 57-58» MULTIPLICATION-. 29 
 
 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 ntmaber by 10, 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 2356x10 = 23560. 
 
 In the same way you may shew, that in order to multiply by 100 
 you must affix two ciphers to the right; to multiply by 1000 you must 
 affix three ciphers, and so on. The rule will be best caught from the 
 following table : 
 
 13X 10 = 
 
 130 
 
 I4.2X 
 
 1000 = 
 
 142000 
 
 I3X 100 = 
 
 1300 
 
 23700X 
 
 10 = 
 
 237000 
 
 I3X 1000 = 
 
 13000 
 
 3040X 
 
 1000 = 
 
 3040000 
 
 13x10000 = 
 
 130000 
 
 10000x100000 = 
 
 I 000000000 
 
 58. I now shew how to multiply by one of the numbers, 2, 3, 4, 5, 
 ^9 7» ^> or 9. I do not include 1, because multiplying by 1, or taking 
 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 ... 64 
 
 48 tens 480 
 
 24 hundreds 2400 
 
 8 thousands 8000 
 
 Add these together, which gives 10944 as the product of 1368 ;.nd 8, or 
 1368x8 = 10944.. By working a few examples in this way you vv-ill see 
 for following rule. 
 
 d2 
 
30 PRINCIPLES OP ARITHMETIC. § 59-62. 
 
 59. I. Multiply the first figure of the multiplicand by the multiplier, 
 write do^vn 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 
 a number, observing that oxi = o, 0x2 = o, &c. 
 
 60. In a similar way a number can be multiplied by a figure which 
 is accompanied by ciphers, as, for example, 8000. For 8000 is 8x1000, 
 and therefore (55) you must first multiply by 8 and then by 1000, 
 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 
 by 60000 
 
 100765428052320000 
 
 61. EXERCISES. 
 
 What is 1007360x7? Answer, 70^1 ^zo, 
 
 123456789x9+10 and 123x9+4? — Ans. iiiiimii 
 and I III. 
 
 What is 136x3+129x4+147x8+27x3000? — Ans. 83100. 
 
 An army is made up of 33 regiments of infantry, each containing 
 800 men; 14 of cavalry, each containing 600 men; and 2 of artillery, 
 each containing 300 men. The enemy has 6 more regiments of infantry, 
 each containing 100 more men; 3 more regiments of cavalry, each con- 
 taining 100 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 
 
is 
 
 165949 
 
 is 
 
 1422420 
 
 is 
 
 "853500 
 
 is 
 
 94828000 
 
 § 62-64. MULTIPLICATION. 31 
 
 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 
 
 (60) 23707X 60 
 
 23707X 500 
 
 23707x4000 
 
 The sum of these is 108269869 
 
 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 The following example will shew that in this case 
 
 142242 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. Suppose it re- 
 
 10S269869 quired to multiply 365 by 101001. The multiplier is 
 
 made up of icoooo, 1000 and 1. Proceed as before, and 
 
 365x1 is 365 
 
 (57) 365x1000 is 365000 
 
 365x100000 is 36500000 
 
 The sum of which is 36865365 
 
 and the whole process with the ciphers struck off is : 
 
 365 64. The following is the rule in all cases : 
 
 loiooi I. Place the multiplier under the multiplicand, so 
 
 365 that the units of one may be under those of the other. 
 
 365 II. Multiply the whole mvdtiplicand 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 AHITHMETIC, §64-66. 
 
 III. Add together the lines obtained hj 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. For example, what is 3200 
 XI 3000? 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, But the product required must be 
 100 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 power of 6 
 
 6x6x6x6 . fourth 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 se- 
 cond and third power with the square and cube in geometry. As exer- 
 cises in multiplication, the following powers are to be found. 
 
 ttber proposed. Square. 
 
 
 Cube. 
 
 97* 
 
 944784 
 
 
 918330048 
 
 1008 
 
 1016064 
 
 
 1024192512 
 
 3H» 
 
 9872164 
 
 
 31018339288 
 
 3163 
 
 10004569 
 
 
 31644451747 
 
 5555 
 
 30858025 
 
 
 171416328875 
 
 6789 
 
 46090521 
 
 
 312908547069 
 
 
 The fifth power of 
 
 • 36 
 
 is 60466176 
 
 
 ... fourth . . 
 
 50 
 
 6250000 
 
 
 ... fourth . . 
 
 108 
 
 ... 136048896 
 
 
 ... fourth . . 
 
 277 
 
 ... 5887339441 
 
§ 67-68. MULTIPLICATION. 33 
 
 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)c is ac+bc^ 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— rf, first take a—b c times, which gives ac—bc. 
 This is not correct ; for in taking it 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— 6)(c— c?) = ac—bc— {ad— bd) 
 
 = ac— bo— 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 
 zab ; hence the square of a+b is aa+zab+bb. Again (a— i)(a— 6) gives 
 aa—ab—ab+bb. But two subtractions of ab are equivalent to subtract- 
 ing lab ; 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+zad+bb+zbc+zbd+cc+zcd+dd; or the rule for squaring 
 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 PRINCIPLES OF ARITHMETIC. § 69-7 1. 
 
 SECTION IV. 
 
 69. Suppose 1 ask whether 156 can be divided into a number of 
 parts each of which is 13, or how many thirteens 156 contains; I pro- 
 pose 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 re- 
 quired 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 13 from 156, 
 and then to subtract 13 from the remainder, and so on; or, in common 
 language, to tell offi^^ 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 sub- 
 traction takes 13 once from 156, which operations will stand as follows : 
 
 156 Begin by subtracting 13 from 156, which leaves 143. Sub- 
 
 13 I 
 
 -— — tract 13 from 143, which leaves 130; and so on. At last 13 
 
 13 ^ only remains, from which when 13 is subtracted, there remains 
 J.J nothing. Upon counting the number of times which you have 
 
 J 17 subtracted 13, you find that this number is 12 ; or 156 contains 
 
 13 I 
 — - — twelve thirteens, or contains 1 3 twelve times. 
 
 13 ' This method is the most simple possible, and might be done 
 
 J- J with pebbles. Of these you would first coimt 156. You would 
 
 78 then take 13 from the heap, and put them into one heap by 
 
 -77 — themselves. You would then take another 13 from the heap, 
 
 13 ^ and place them in another heap by themselves ; and so on until 
 
 J- J there were none left. You would then count the number of 
 
 39 heaps, which you would find to be 12. 
 13 I 
 
 -r? — 71. Division is the opposite of multiplication. In multi- 
 
 ^3 ' plication you have a number of heaps, with the same number 
 
 ,^ I of pebbles in each, and you want to know how many pebbles 
 
 o there are in all. In division you know how many there are 
 
§ 71-74. Divisioy 35 
 
 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. But this does not happen with every num- 
 ber. Take, for example, 159. Follow the process of (70), and it will 
 appear that after having subtracted 13 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 159, when divided by 13, 
 gives a quotient 12, and a remainder 3. If we use signs, 
 
 159 = 13x12+3. 
 
 EXERCISES. 
 
 146 = 24x6+2, or 146 contains six twenty- fours and 2 over. 
 146 = 6x24+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, 
 
 - = q,ora:b = q^ 
 
 which in old books is often found written thus : 
 
 a-i-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 156 contains 13 as often as 
 
 all its parts together. For example, 156 is made up of 91, 39, and 26. 
 
 Of these 
 
 91 contains 137 times, 
 
 39 contains 133 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 PKINCIPLES OF ARITHMETIC. § 74-76. 
 
 Now 100 contains 137 times and 9 over, 
 50 contains 13 3 times and 11 over, 
 6 contains 130 times* and 6 over. 
 Therefore 100+50+6 contains 13 7+3+0 times and 9+11+6 over; or 
 156 contains 13 10 times and a6 over. But 26 is itself 2 thirteens; 
 therefore 156 contains 10 thirteens and 2 thirteens, or 12 thirteens. 
 
 75. The result of the last article is expressed by saying, that if 
 
 ,,,,,, a b c d 
 a = 6+c+rf, 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 130, and afterwards take away 13 twice, or take 
 away 26, and the process is as follows : 
 
 156 
 
 130 10 times 13. 
 
 ~e 
 
 26 2 times 13. 
 o 
 Therefore 156 contains 13 10+2, or 12 times. 
 Again, to divide 3096 by 18. 
 
 3096 Therefore 3096 contains 18 100+50+20+2, or 
 
 1800 100 times 18. 172 times. 
 
 1296 77. You will now understand the following 
 
 900 50 times 18. sentences, and be able to make similar assertions 
 396 of other nimfibers. 
 
 360 20 times 18. 450 is 75x6; it therefore contains any number, 
 36 as 5, 6 times as often as 75 contains it. 
 
 36 2 times 18. 
 
 o 
 * To speak always in the same way, instead of saying that 6 does not contain IS, 
 I say that it contains it times and 6 over, which is merely saying that 6 is 6 
 more than nothing. 
 
§ 77-78' DIVISION. 37 
 
 135 3 26 times; therefore. 
 
 Twice 135 .1 3 - 5^ or twice 26 » 
 
 10 times 135 o 3 I 260 or 10 times 26 -B 
 
 50 times 135 3 1300 or 50 times 26 
 
 472 contains 18 more than 21 times ; therefore, 
 4720 contains 18 more than 210 times, 
 47200 contains 18 more than 2100 times, 
 472000 contains 18 more than 21000 times. 
 
 32 
 
 
 12 
 
 
 2 
 
 3 
 
 3 
 
 320 
 
 3200 
 
 I 
 
 12 
 12 
 
 1 
 
 20 
 
 200 
 
 1 
 
 1 
 
 30 
 
 300 
 
 [2000 
 
 
 
 12 
 
 a 
 
 20C0 
 
 i 
 
 3000 
 
 &c. 
 
 
 
 
 &c. 
 
 •B 
 
 &c. 
 
 78. The foregoing articles contain the principles of division. The 
 
 question now is, to apply them in the shortest and most convenient way. 
 
 4068 
 Suppose it required to divide 4068 by 18, or to find — — - (23). 
 
 lo 
 
 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 ttoice^ 
 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 200 times, 
 and less than 300 times. Therefore, 4068 also contains more than 200 
 times 18, since it must contain 18 more times than 4000 does. It also 
 contains 18 less than 300 times, because 300 times 18 is 5400, a greater 
 number than 4068. Subtract 18 200 times from 4068 ; that is, subtract 
 3600, and there remains 468. Therefore, 4068 contains 18 200 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." 
 
 B 
 
38 PRINCIPLES OF ARITHMETIC. § 78. 
 
 exactly as before. Observe that 46 contains 18 more than t^nce, and 
 less than 3 times ; therefore, 460 contains it more than 20, and less 
 than 30 times (77) ; as does also 468. Subtract 18 20 times from 468, 
 that is, subtract 360 ; the remainder is 108. Therefore, 468 contains 
 18 20 times, and as many more as 108 contains it. Now, 108 is found 
 to contain 18 6 times exactly ; therefore, 468 contains it 20+6 times, 
 and 4068 contains it 200420+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. 
 
 342)36326599(20000+7000+60+9 
 26840000 
 
 9486599 18)4068(200+20+6 
 
 9394000 
 
 3600 
 
 92599 
 
 80520 
 
 468 
 360 
 
 12079 
 12078 
 
 108 
 
 108 
 
 1 o 
 
 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 1342X 
 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 zoooo+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. What are 
 
 100938 74 66779922 2718218 
 
 3207 ' 1 14433 ' 13352 
 
§ yS-So. DIVISION. 39 
 
 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 : 2d, 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 con- 
 tinuing the process, they cease to have ciphers under them : 3d, that 
 the quotient is only a number ^vxitten at length, instead of the usual 
 way. For example, the first quotient is 200+20+6, or 226; the second 
 is 20000+7000+60+9, or 27069. Strike out, therefore, all the ciphers 
 and the numbers which come above them, excej)t those in the first 
 line, and put the quotient in one line ; and the two examples of the 
 last article will stand thus : 
 
 i8)ao68(226 
 
 1342)36326599(27069 
 
 36 
 
 26S4 
 
 46 
 
 9486 
 
 36 
 
 9394 
 
 108 
 
 9259 
 
 ic8 
 
 8052 
 
 
 
 1 2079 
 
 
 12078 
 
 80. Hence the following rule is deduced : 
 
 I. Write the divisor and dividend in one line, and place parentheses 
 en each side of the dividend. 
 
 II. Take of}' from the left hand of the dividend the least number of 
 figures which make a number greater than the divisor ; find what num- 
 ber 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 PRINCIPLES OP ARITHMETIC. § 8o-8l, 
 
 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 them- 
 selves. 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. hun- 
 dreds, &c. we may expect that there will not be much difference be- 
 tween 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 1, 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 12, if you have sufficient knowledge of the multiplication 
 table (50). For example, I want to divide 132976 by 4. At full length 
 th J process stands thus ? 
 
§ 8l. DIVISION. 41 
 
 4)132976(33244 But you will recollect, without the necessity of 
 
 ^ writing it down, that 13 contains 4 three times with 
 
 '^ a remainder i ; this 1 you will place before 2, the 
 
 next figure of the dividend, and you know that 12 
 
 9 contains 4 3 times exactly, and so on. It will be more 
 
 8 
 
 convenient to write down the quotient thus : 
 
 ^7 4)132976 
 
 ^ 33244 
 
 16 While on this part of the subject, we may men- 
 
 ^" tion, 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 10 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. To multiply by 25, annex two ciphers and 
 divide by 4. 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 remainfler, 
 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 12 hundreds and 36, the first part of which, 
 being hundreds, is divisible by 4, and gives 12 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 1000 is divisible by 8 ; whether therefore the Avhole shall be divi- 
 sible by 8 or not depends on the last three figures: thus, 127946 is not 
 divisible by 8, since 946 is not so. A number is divisible by 3 or 9 
 only when the sum of its digits is divisible by 3 or 9. Take for example 
 1234 ; this is 
 
 • Among the even figures we include 0. 
 e2 
 
42 PRINCIPLES OF ARITHMETIC. § 81-X3. 
 
 I thousand, or 999 and 1 
 
 z hundred, or twice 99 and 2 
 
 3 tens, or three times 9 and 3 
 
 and 4 or 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. It therefore depends on 1+2+3+4, or the sura of the 
 digits, whether 1234 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 : 
 
 100)33429(334 This is, then, the rule: Cut off as many 
 
 3°o figures from the right hand of the dividend 
 
 342 as there are ciphers. These figures -will be 
 
 3°° the remainder, and the rest of the dividend 
 
 429 will be the quotient. 
 
 ^° Or we may prove these results thus : from 
 
 29 (20), 2717316 is 271731 tens and 6; of which 
 10)2717316 
 the first contains 10 271731 times, and the 
 
 271731 and rem. 6. , , ,, , . . , „ 
 
 second not at all ; the quotient is therefore 
 
 271731, and the remainder 6 (72). Again (20), 33429 is 334 hundreds 
 
 and 29 ; of which the first contains 100 334 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 short- 
 ened 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. 
 
-84. DIVISI02J. 
 
 
 I. 1782000)6424700000(3605 
 
 1782)6424700(3605 
 
 5346000 
 
 5346 
 
 10787000 
 
 10787 
 
 10692000 
 
 10692 
 
 9500000 
 
 9500 
 
 8910000 
 
 8910 
 
 590000 
 
 590000 
 
 II. 12500000)42176189300(3428 
 
 123)421761(3428 
 
 36900000 
 
 369 
 
 52761893 
 
 527 
 
 49200COO 
 
 492 
 
 35618930 
 
 356 
 
 24600000 
 
 ^46 
 
 I10IS9300 
 
 1101 
 
 984CCOOO 
 
 984 
 
 43 
 
 117893C0 
 
 1789300 
 
 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 ; hut at the 
 end of the process place on the right of the remainder all those figures 
 which were struck out of the dividend. 
 
 54. 
 
 EXERCISES. 
 
 
 Dividend. 
 
 Divisor. 
 
 Quotient. 
 
 Remainder. 
 
 9694 
 
 47 
 
 206 
 
 12 
 
 175618 
 
 3136 
 
 56 
 
 2 
 
 23796484 
 
 13C000 
 
 183 
 
 6484 
 
 14C02564 
 
 1871 
 
 7484 
 
 
 
 3 103 14420 
 
 7878 
 
 39390 
 
 
 
 3939040647 
 
 6889 
 
 571787 
 
 4 
 
 22876792454961 
 
 43046721 
 
 531441 
 
 
 
 Including both ciphers and others. 
 
44 PRINCIPLES OF ARITHMETIC. § 84-87. 
 
 Shew that 
 
 ^ looxiooxioo— 4':x4.3x4i 
 
 I. = 100x100+100x43+43x43. 
 
 100—43 
 
 ^, 100x100x100+43x43x43 
 
 II. = 100x100—100x43+43x43. 
 
 100+43 
 
 jjj 76x76+2x76x52+52x52 ^ ^^_^ 
 76+52 ' ^"* 
 
 ,„ I2XI2XI2XI2— I 
 
 IV. 1 + 12+12X12+12X12X12 = 
 
 12— 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 mul- 
 tiply 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. Then, 
 200 is 5 times 40, and 50 is 5 times 10. But by (85), 40 divided by 10 
 gives the same quotient as 5 times 40 divided by 5 times 10, and there- 
 fore the quotient of two numbers is not altered by dividing both the diin- 
 dend 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 15. Also, if a number be divided by any number, and the quotient 
 
§ 87-9I. i>ivisiON. 45 
 
 be divided ly anothei, it is the same as if the first number had been 
 divided by the product of the other two. For example, divide 60 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 60 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 la. 
 
 88. The folloAving rules will be better understood by stating them 
 in an example. If 32 be multiplied by 24 and divided by 6, the result 
 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 
 24, 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 express > I 
 thus : 
 
 If n divide a and b without remainder, 
 
 a a 
 
 -Ji-=« (86) -^ = iL (87) 
 
 ^ b c bo 
 
 ab b etc a 
 
 - = «x- (88) T ==~T 
 
 (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 re- 
 mainder, 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 PRINCIPTES 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. If that street should 
 happen to be 136 feet in length, you will be able to measicre it with 
 the rod, because, since 136 contains 4 34 times, you will find that the 
 street is exactly 34 times the lengtli 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. Hence 4 
 is said to measure 136, but not to measure 137. 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. Thus, 15 is a common measure of 180 
 and 75. Two numbers may have several common measures. For 
 example, 360 and 168 have the common measures 2, 3, 4, 6, 24, and 
 several others. Now, this question may be asked : Of all the common 
 measures of 360 and 168, 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 procccd to consider. 
 
 93. If one quantity measure two others, it measures their sum and 
 difference. Thus, 7 measures 21 and 56. It therefore measures 56+21 
 and 56—21, or 77 and 35. This is only another way of saying what 
 was said in (74). 
 
 94. If one number measure a second, it measures every number 
 which the second measures. Thus, 5 measures 15, and 15 measures 30, 
 45, 60, 75, &c. ; all which numbers are measured by 5. It is plain 
 that if 
 
 15 contains 5 3 times, 
 30, or 15+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 112. The 
 quotient is 3, and the remainder 24, that is (72) 360 is three times 112 
 
§ 95-98. DIVISION. 47 
 
 and 24, or 360 = 112x3+24. From this it follows, that 24 is the differ- 
 ence between 360 and 3 times 112, or 24 = 360—112x3. Take any num- 
 ber which measures both 360 and 112 ; for example, 4. Then 
 
 4 measures 360, 
 
 4 measures 112, and therefore (94) measures 112x3, 
 or 112+112+112. 
 
 Therefore (93) it measures 360—112x3, which is the remainder 24. The 
 same reasoning may be applied to all other measures of 360 and 112 ; 
 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 cammon measure of the divisor and 
 remainder. 
 
 dQ. Every common measure of the divisor and remainder is also a 
 common measure of the dividend and divisor. Take the same example, 
 and recollect that 360 = 112x3+24. Take any common measure of the 
 remainder 24 and the divisor 1 12 ; for example, 8. Then 
 
 8 measures 24 ; 
 and 8 measures 112, and therefore (94) measures 112x3. 
 Therefore (93) 8 measures 112x3+24, 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 mea- 
 sure 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. 0. m. of 360 and 112, and observe that 
 
 ♦ For shortness, I abbreviate the words greatest common measure into their initial 
 letters, g. c. m. 
 
48 PRINCIPLES OF ARITHMBTIC § 98-99. 
 
 360 divided by 112 gives the remainder 24, 
 112 divided by 24 gives the remainder 16, 
 
 24 divided by 16 gives the remainder 8, 
 
 16 divided by 8 gives no remainder. 
 
 Now, since 8 divides 16 without remainder, and since it also divides 
 itself without remainder, 8 is the g. c. m. of 8 and 16, because it is im- 
 possible to divide 8 by any number greater than 8 ; so that, even if 
 16 had a greater measure than 8, it could not be common to 16 and 8, 
 
 Therefore 8 is g. c. m. of 16 and 8, 
 
 (97) g. c. m. of 16 and 8 is g. c. m. of 24 and 16, 
 
 g. c. m. of 24 and 16 is g. c. m. of 112 and 24, 
 
 g. c. m. of 112 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 follow- 
 ing ways : 
 
 1 12)360(3 The rule ftr finding the greatest common mea- 
 
 ^^ sure of two numbers is, 
 
 24)1 12(4 I. Divide the greater of the two by the less. 
 
 " II. Make the remainder a divisor, and the 
 
 16)24(1 divisor a dividend, and find another remainder. 
 
 III. Proceed in this way until there is no 
 
 8)16(2 remainder, and the last divisor is the greatest 
 
 common measure required. 
 
 ° 99, You may perhaps ask how the rule is to 
 
 shew when the two numbers have no common 
 measure. The fact is, that there are, strictly 
 speaking, no such numbers, because all numbers 
 tire measured by i ; that is, contain an exact 
 number of units, and therefore i is a common 
 measure of every two nimibers. If they have no other common mea- 
 sure, the last divisor will be i, as in the following example, where the 
 greatest common measure of 87 and 25 is found. 
 
 112 
 
 360 
 
 3 
 
 96 
 
 336 
 
 4 
 
 16 
 
 24 
 
 1 
 
 16 
 
 i6 
 
 2 
 
 
 
 8 
 
 
DIVISION. 
 
 EXERCISES. 
 
 * 
 
 Numbers. 
 
 g. c. ra. 
 
 6197 
 
 9521 
 
 I 
 
 58363 
 
 26C2 
 
 I 
 
 5547 
 
 147008443 
 
 1849 
 
 6281 
 
 326041 
 
 571 
 
 28915 
 
 31495 
 
 5 
 
 1509 
 
 300309 
 
 3 
 
 § 99-102. . DIVISION. 49 
 
 45)87(3 
 
 75 
 
 12)25(2 
 
 24 
 
 1)12(12 
 12 
 
 o 
 
 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 mea- 
 sure. For example, 360 and 504 are both divisible by 4. The quotients 
 are 90 and 126. Now 90 and 126 are both divisible by 9, the quotients 
 of which division are 10 and 14. By (87), dividing a number by 4, and 
 then dividing the quotient by 9, is the same thing as dividing the num- 
 ber 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 10 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 com- 
 mon measure except i {^0). Otherwise, the number w^hich 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. Th© 
 words mult'ple and measure are thus connected: Since 4 is a n:easi!re 
 
 F 
 
50 PRINCIPLES OF AHITHMETIC. § 102-103. 
 
 of 24, 24 is a multiple of 4. The number 96 is a multiple of 8, 12, 24, 
 48, and several others. It is therefore called a common multiple of 8, 
 12, 24, 48, &c. The product of any two numbers is evidently a common 
 multiple of both. Thus, 36x8, or 288, is a common multiple of 36 and 
 8. But there are common multiples of 36 and 8 less than 288 ; and 
 lecause 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 2x4. 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 not even necessary that you should know 
 it is the least. 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 i, the least common multiple 
 of the two numbers is their product. 
 
 The rule then is : To find the least common multiple of two num- 
 bers, find their greatest common measure, and multiply one of the num- 
 l)ers by the quotient which the other gives when divided by the gieatest 
 common measure. To find the least common multiple of three num- 
 bers, find the least common multiple of the firet two, and find the least 
 common multiple of that multiple and the third, and so on. 
 
X03-104- 
 
 FRACTIONS. 
 EXERCISES. 
 
 51 
 
 Numbers proposed. 
 
 14, 21 
 16, 5, 24 
 
 I, 2, 3» 4, 5^ 6, 7, 8, 9, 10 
 
 6, 8, II, 16, 20 
 
 876, 864 
 
 868, 854 
 
 Least common multiple. 
 
 42 
 
 240 
 
 2520 
 
 2640 
 
 63072 
 
 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 
 themselves no divisors greater than unity. "Write them as diviscrr, and 
 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 
 by iill the numbers last brought down. For instance, let it be required 
 to find the least common multiple of all the numbers from 11 to 21. 
 
 2» ^» 3? 5» 7)11 12 13 14 15 16 17 18 19 20 21 
 II I 13 1 I 4 17 3 19 I I 
 
 There are now no common measures left in the row, and the least com- 
 mon multiple required is the product of 2, 2, 3, 5, 7, 11, 13, 4, 17, 3, and 
 19 ; or 232792560. 
 
 SECTION V. 
 
 FRACTIONS. 
 
 104. Suppose it requiied to divide 49 yards into five equal parts, or, 
 r.8 it is called, to find the fifth part of 49 yards. If we divide 45 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 : 
 
52 PRINCIPLES OF ARITHMETIC. § 104- IC5. 
 
 A_ B 
 
 C J _ 
 
 D ^K — 
 
 E L — 
 
 F M — 
 
 G N — 
 
 I K L ]M N 
 H f I I I I I 
 
 Take 5 liiiep, C, D, e, f, and G, each 9 yards in length, and the line 11, 
 
 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, G, 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 
 
 ya:\ls. 
 
 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. It is the fifth part of 4 yards, and is called 
 
 4 
 a FRACTION of a yard. It is written thus, - (23), and is what we must 
 
 add to 9 yards in order to make up the fiftli part of 49 yards. 
 
 The same reasoning would apply to dividing 49 bushels of com, 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. 
 
 "NVe say, then, once for all, that the fifth part of 49 is 9 and -, or 
 
 9+^ ; which is usually written 9-, or if we use signs, — = 9-. 
 5 '5 5 5 
 
 EXERCISES. 
 
 What is the seventeenth part of 1237 ? — Answer, 7a—. 
 
§ 105-107. FRACTIONS. 53 
 
 What are ^221i, 552!l2, and iiZIMi? , 
 
 1974 13710 2424. 
 
 162 23649 2343 
 
 Answer, c , 27 -, 9394 -, 
 
 1974 23710' ^^^^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 terra fraction even includes whole num- 
 557 17 34 51 
 
 bers :* for example, 17 is — , — , — , &c. 
 123 
 
 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 
 
 6 
 
 less than a unit: thus, — is less than a unit; for 6 divided into 6 parts 
 
 17 . 
 
 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 de- 
 nominator are equal, and greater than a unit when the numerator is 
 
 greater than the denominator. 
 2 
 
 107. By - is meant the third part of 2. This is the same as twice 
 
 the third part of i. 
 
 To prove this, let a n be two yards, and divide each of the yards a c 
 and c B into three equal parts. 
 
 I ^ 1 ^ -r 1 1 
 
 A D E C F G B 
 
 Then, because a e, e f, and f b, are all equal to one another, a e is 
 
 2 
 the third part of 2. It is therefore -. But a e is twice a d, and a d 
 
 I 3 2 1 
 
 is the third part of one y.ard, or - ; therefore - is twice - ; that is, in 
 
 2 3 3 3 
 
 order to get the length -, it makes no difference whether we divide tuo 
 
 yards at once into threv^ parts, and take one of them, or whether we 
 
 divide one yard into three parts, and take two of them. By the same 
 
 reasoning, - 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., dtq 
 called whole numbers or integers, when we -wish to distinguish them from fractions. 
 
 f2 
 
54 PRINCIPLES OF ARITHMETIC. § 107-109. 
 
 ciple is the same as tlie 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 2, or of two units, is made by taking 
 one-third out of each of the units, that is, 
 
 2 I 
 
 - = -X2. 
 
 3 3 
 
 This meaning appears ambiguous when the numerator is greater than 
 
 the denominator : thus, — would mean that i is to be divided into 7 
 
 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 
 
 parts, and take 1 5 of them. 
 
 1 08. The value of a fraction is not altered by multiplying the nume- 
 
 rator and denominator bv the same quantity. Take the fraction -, mul- 
 
 15 4- 
 tiply its numerator and denominator by 5, and it becomes — , which is the 
 
 same thing as - ; that is, one-twentieth 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 20 
 
 parts and taking 15 of them, as you get by dividing it into 4 parts and 
 
 taking 3 of them. To prove this. 
 
 I I I M I I I * I I I I * I I I I 
 
 A C D E B 
 
 let A B represent a yard ; divide it into 4 equal paits, a c, c d, d e, and 
 
 E B, and divide each of these parts into 5 equal parts. Then a e is -. 
 
 4 
 But the second division cuts the line into 20 equal parts, of which a b 
 
 15 15 3 
 
 contains 15. It is therefore — . Therefore, — and - are the same thin^r. 
 
 3 20 20 4 * 
 Again, since - is made from — by dividing both the numerator and 
 
 4 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. 
 
 't If 
 
 109. Though the two fractions - and — are the same in value an I 
 4 20 
 
§ 109-IIO. FRACTIONS. 65 
 
 either of them maybe 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 con- 
 venient 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 i. 
 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. 
 
 22x127 
 
 ^% 
 
 2921 
 
 23x127 
 
 23 
 
 2788 
 
 17x164 
 
 11 
 
 4920 
 
 30x164 
 
 30 
 
 93208 
 
 764x122 
 
 764 
 
 13786 
 
 113x122 
 
 113 
 
 888800 
 
 22x40400 
 
 22 
 
 40359600 
 
 999x40400 
 
 999 
 
 95469 
 359784 
 
 121x789 
 
 456x789 
 
 121 
 456 
 
 110. "When the terms of the fraction given are already in factors,* 
 nny one factor in the numerator may be divided by a number, provided 
 some one factor in the denominator is divided by the same. This fol- 
 lows from (88) and (108), 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, 
 4, 6, 8, are factors of 2^, and 6x4, 8x3, 2x2x2x3, are several ways of decom losing 
 24 inro factors. 
 
56 
 
 PRINCIPLES OF ARITHMETIC. 
 
 
 I2XIIXIO 3'xiixio i'xiix5' 
 
 55- 
 
 2X 3X 4 ~ 2 X 3X i' I'xi' xi' 
 
 18x15x13 2'x3'xi' I'xi'xi' 
 
 1 
 
 20x54x52 4'x6'x4' z'xz' x^' 
 
 i6* 
 
 27x28 3'x4' 3^x2' 
 9x70 I'xio' i'x5' 
 
 6 
 5* 
 
 111. As we can, by (108), multiply the numerator and denominator 
 
 of a fraction by any number, without altering its value, we can now 
 
 readily reduce two fractions to two others, which shall have the same 
 
 value as the first two, and which shall have the same denominator. 
 
 24 2 
 
 Take, for example, - and - ; multiply both terms of - by 7, and both 
 
 4 3 7 
 
 terms of- by 3. It then appears that 
 
 3 
 
 2 . 2x7 14 
 
 - IS or -^ 
 
 3 3x7 21 
 
 4 . 4x3 12 
 
 - IS -^-^ or — . 
 7 7x3 21 
 
 14 12 24 
 
 Here are then two fractions — and — , equal to - and -, and 
 21 ^i 2 4 3 7 
 
 having the same denominator, 2 1 ; in this case, - and - are said to be 
 
 3 7 
 reduced to a common denominator. 
 
 157 
 It is required to reduce — , ~, and - to a common denominator. 
 10 6 9 
 
 Multiply both terms of the first by the product of 6 and 9 ; of the se- 
 cond by the product of 10 and 9 ; and of the third by the product of 
 10 and 6. Then it appears (108) that 
 
 I . 1x6x9 54 
 
 — IS ;r^ or -^^— 
 
 10 10x6x9 540 
 
 6 6x10x9 540 
 
 7 . 7xrox6 420 
 
 - IS or . 
 
 9 9x10x6 540 
 
 On looking at these last fractions, we see that all the numeratois 
 and the common denominator are divisible by 6, and (108) this division 
 
 will not alter their values. On dividing the numerators and deno- 
 
 54 450 420 9 75 
 
 minators of^^-^, ^^^-, and ~ — by 6, the resulting fractions are, -^, — , 
 
 70 540 540 540 90 90 
 
 and — . These are fractions with a common denominator, and which 
 
 90 
 
g III-II2. 
 
 FRACTIONS. 
 
 57 
 
 are the same as — . 7, and - ; and therefore these are a more simple 
 10 6 9 
 
 answer to the question than the first fractions. Observe also that 540 
 
 is one common multiple of 10, 6, and 9, namely, 10x6x9, but that 90 is 
 
 the least common multiple of 10, 6, and 9 (103). The following pro- 
 
 cess, therefore, is better. To reduce the fractions — , -, and -, to others 
 
 10 6 9 
 
 having the same value and a common denominator, begin by finding the 
 
 least common multiple of 10, 6, and 9, by the rule in (103), which is 
 
 90. Observe that 10, 6, and 9 are contained in 90 9, 15, and 10 times. 
 
 Multiply both terms of the first by 9, of the second by 15, and of the 
 
 third by 10, and the fractions thus produced are — , — , and — , the same 
 
 90 90 90 
 
 as before. 
 
 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 
 3 
 
 I 
 5 
 
 I 
 6 
 
 
 
 20 
 30 
 
 30 30 
 
 I 2 
 
 3 7 
 
 _3_ 
 14 
 
 12 
 21 
 
 3 
 4 
 
 
 28 24 
 84 84 
 
 18 48 63 
 
 84 84 84 
 
 3 A 
 
 10 
 
 5 
 100 
 
 6 
 
 ICOO 
 
 
 
 30CO 400 
 
 ICCO ICCO 
 
 50 6 
 
 1000 ICCO 
 
 33 
 
 
 281 
 
 
 
 22341 
 
 106499 
 
 379 
 
 
 677 
 
 
 
 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 - and — . These fractions 
 
 reduced, without alteration of their value, to a common denominator, 
 
 15 14 
 
 are — and — . Of these the first must be the greater, because (107) it 
 
 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 PniNCIPLES OF ARITHMETIC. §112-114, 
 
 8 . 8 . 
 
 Thus, -is greater than -, since the first is a 7th, and the last onlv a 
 
 7 9 
 
 9th part of 8. Also, any numerator may be made to belong to as small 
 
 a fraction as we please, by sufficiently increasing the denominator. 
 
 Thus, is — , is , and is (108). 
 
 100 10 1000 100 lOOOOOO lOOOOO 
 
 We can now also increase and diminish the first fraction by the 
 
 second. For the first fraction is made up of 15 of the 30 equal parts 
 
 into which i is divided. The second fraction is 14 of those parts. The 
 
 8um of the two, therefore, must be 15+14, or 29 of those parts; that 
 
 is, -H — is — . The difference of the two must be 15—14, or i of those 
 
 parts ; that is, = -, 
 
 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 nume- 
 rator, and the common denominator for its denominator. 
 
 EXERCISES. 
 
 
 2345 60 
 
 44 
 3 
 
 153 18329 
 427 128 1 
 
 8 1 3 , 4 _. 1834 
 
 10 ICO ICCO lOCO 
 
 
 2 V^-^^l 
 
 7 ^3 91 
 
 1 8 ^ 94 _ 3 
 
 2 16 188 2 
 
 163 
 521 
 
 97 93066 
 881 459C01 
 
 114." Suppose it required to add a whole number to a fraction, lOr 
 
 example, 6 to -. By (106) 6 is — , and — +- is — ; that is, 6+-, or as 
 
 9 4 . 58 9 9 9^ . 9 . 9 
 
 it is usually written, 6-, is — . The rule in this case is: Multiply the 
 
 whole number by the denominator of the fraction, and to the product 
 
 add the numerator of the fraction ; the sum will be the nimicrator of 
 
 the result, and the denominator of the fraction will be its denominator. 
 
 Thus, 3- = — , 22- = — -, 74 — = . This rule is the opposite of 
 
 ' '*4 4 9 9 55 55 
 that in (105). 
 
§ 115-118. FRACTIOXS. 59 
 
 907 . 17230907 
 
 115. From the last nile it appears that 1723 is , 
 
 ^^ 225 . 667225 ^ 99 . 230C099 ^ ^°°^° '^^°°. 
 
 667 — IS — ■ — -, and 23 — ^^ — IS — ^. Hence, when a whole 
 
 1000 1000 I 00000 iCocco 
 
 number is to be added to a fraction whose denominator is i followed by 
 ciphers, the number of wiiich is not less than the number oi 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. 
 
 Reduce the following mixed quantities to fractions : i , 2457 — , 
 
 299 , 2210 i^ccoo ^^^o' 
 
 1207 , and 233 . 
 
 lOOOOCOO lOOCO J 
 
 116. Suppose it required to multiply - by 4, This by (48) is taking 
 
 2 , . 2 2 2 2 3 g 
 
 - four times ; that is, finding -+-H — h-. This by (112) is -; so that to 
 
 3 . 3 3 3 3 3. 
 multiply a fraction by a whole number the rule is : Multiply the nu- 
 merator 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 
 
 7 4^ 
 
 example; multiply — by 6. This (11(5) is — , which, since the numerator 
 
 and denominator are now divisible bv 6, is (lOH) the same as -. It is 
 
 7 . 7 . ' . 
 
 plam that - is made from —7 in the manner stated in the rule. 
 6 36 
 
 118. Multiplication has been defined to be the taking as many of 
 one number as there are units in another. Thus, to multiply 12 by 7 
 b to take as many twelves as there are units in 7, or to take 12 as many 
 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. For example, 
 
 7 13 1+141-1+1 + 1 + 1. 
 
 7 times 12 is 12+12+12+12+12+12+12. 
 When the same thing is done with two fractions, the result is still 
 
60 PRINCIPLES OF ARITHMETIC. §ll8-I20. 
 
 called their product, and the process is still called multiplication. There 
 is this difference, that whereas a whole number is made by adding i to 
 itself a number of times, a fraction is made by dividing i into a number 
 of equal i)arts, and adding one of these parts to itself a number of times. 
 
 This being the meaning of the word multiplication, as applied to frac- 
 
 3 7 
 
 tions, what is - multiplied by - ? Whatever is done with i in order to 
 
 74 8 - ^ 
 
 make - must now be done with - ; but to make -, i is divided into 8 ^ 
 
 4 3 7 3 
 
 parts, and 7 of them are taken. Therefore, to make -x-, - must be di- 
 
 484, 
 vided into 8 parts, and 7 of them must be taken. Now -is,byCI08), 
 
 24 c:„„„ =^4 
 
 4 
 
 the same thing as - . Since — is made by dividing 1 into 32 parts, and 
 
 3* 3^ 
 
 tfiking 24 of them, or, which is the same thing, taking 3 of them 8 times, 
 . 24 , .3 
 
 if — be divided into 8 equal parts, each of them is — ; and if 7 of these 
 
 3^ 21 3 32' 7 21 
 
 parts be taken, the result is — (116j : therefore - multiplied by - is — ; 
 32 4 8 32 
 
 and the same reasoning may be applied to any other fractions. But 
 
 21 3 7 
 
 — is made from - and - by multiplying the two numerators together 
 
 32 4 8 
 
 for the numerator, and the two denominators for the denominator ; 
 
 which furnishes a rule for the multiplication of fractions. 
 
 21 
 
 119. If this product — is to be multiplied by a third fraction, for 
 
 5 ^^ 105 
 
 example, by -, the result is, by the same rule, -—- ; and so on. The 
 5 280 
 
 general rule for multiplying any number of fractions together is therefore : 
 Multiply all the numerators together for the numerator of the pro- 
 duct, and all the denominators together for its denominator. 
 
 jC g 
 
 120. Suppose it required to multiply together -7 and — . The pro- 
 
 iqxS 120 ^ ^° 
 
 duct may be written thus : -7 , and is — r-, which reduced to its lowest 
 
 ^ 16x10 160 
 
 terms (109) is-. This result might have been obtained directly, by 
 
 4 
 observing that 15 and 10 are both measured by 5, and 8 and 16 are both 
 
 measured bv 8, and that the fraction may be written thus: ---^ . 
 
 2x8x2x5 
 
 Divide both its numerator and denominator by 5x8 (108) and (87), and 
 
 3 
 the result is at once - ; therefore, before proceeding to multiply any 
 
 4 
 number of fiactions together, if there be any numerator and any deno- 
 minator, whether belonging to the same fraction or not, which have a 
 common measure, divide them both by that common measure* and uso 
 tbo quotients instead of the dividends. 
 
§ I20-I2I. 
 
 FRACTIONS. 61 
 
 A whole number may be considered as a fraction whose denominator 
 is I ; thus, i6 is — (106) ; and the same rule will apply when one or 
 more of the quantities are whole numbers. 
 
 7470 
 
 268 36448 
 '919" 6864930 
 
 18224 
 3432465 
 
 I 2 3 
 
 a 3 4 
 
 ^ I 
 
 "5-5' 
 
 Ax^7^-^ 
 17 45 45 
 
 2 13 241 
 59 7 19 
 
 6266 
 
 7H7' 
 
 13 601 7813 
 461 II 5071 
 
 Fraction proposed. - 
 
 Square. 
 
 Cube. 
 
 701 
 
 49 140 1 
 
 344472101 
 
 158 
 
 24964 
 
 39443 la 
 
 140 
 
 196CO 
 
 2 744c 00 
 
 141 
 
 19881 
 
 2803221 
 
 355 
 115 
 
 126025 
 12769 
 
 44738875 
 1442897 
 
 From 100 acres of ground, two-thirds of them are taken away ; 50 
 
 acres are then added to the result, and - of the whole is taken ; what 
 
 7 II 
 
 number of acres does this produce ? — Answer, 59 — . 
 
 121. In dividing one whole number by another, for example, ic8 
 
 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 4 
 
 Suppose we take two fractions, for example, - and -, and ask. Can 
 
 4 ^ ^ . 
 
 we, by dividing - into some number of equal parts, and adding a num- 
 
 5 2 
 
 ber of these parts together, produce - ? if so, into how many parts must 
 
 we divide -, and how many of them must we add together ? The 
 
 5 .24, 
 
 solution of this question is still called the division of - by - ; and the 
 
 3 5 ^. 
 
 fraction whose denominator is the number of parts into which - is 
 
 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 
 
 10 12 
 
 does not alter their value (108) ; they then become — and — . The 
 
62 PRINCIPLES OF ARITHMETIC. § 121-123. 
 
 question now is, to divide — into a number of parts, and to produce — 
 
 by taking a number of these parts. Since — is made by dividing i 
 
 ^5 12 
 into 15 parts and taking 12 of them, if we divide — into 12 equal 
 
 1 '5 
 
 parts, each of these parts is — ; if we take 10 of these parts, the result 
 
 10 ^5 10 2 
 
 is — . Therefore, in order to produce — or - (108), we must divide 
 12 '5 4 , 15 3 10 
 
 — or - into 12 parts, and take 10 of them; that is, the quotient is — . 
 15 5 2 4 ' » 1 j^ 
 
 If we call - the dividend, and - the divisor, as before, the quotient in 
 
 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 divideml 
 
 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 
 
 seen by comparing what is required in both cases. In multiplying - 
 
 10 4 5 
 
 by — , I ask, if out of- be taken 10 parts out of 12, how much of a unit 
 
 ^?' 5 40 2 24 
 
 IS taken, and the answer is 7-, or -. Again, in dividing - by -, I ask 
 
 42 °° 3 10 3 5 
 
 what part of - is -, the answer to which is — . 
 5 3 la 
 
 122. By taking the following instance, we shall see that this rule 
 
 can be sometimes simplified. Divide — by — . Observe that 16 is 
 
 33 15 
 4x4, and 28 is 4x7 ; 33 is 3x11, and 15 is 3x5 ; therefore the two frac- 
 tions are and , and their quotient, according to the rule, is 
 
 4x4x3x5 3^^^ , 3^5, 
 
 , in which 4x3 is found both in the numerator and denominator. 
 
 3x11x4x7 ^ 2Q 
 
 The fraction is therefore (108) the same as --^— ^, or — . The rule of 
 
 11x7 77 
 
 the last article, therefore, admits of this modification : If the two nume- 
 rators 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 
 
 15 .2 3 
 
 15, consider is ss the fraction — . The rule gives — as the quotient. 
 
 Therefore, to divide a fraction by a whole number, multiply the deno- 
 minator by that whole number. 
 
23-124. 
 
 
 
 FRACTIONS. 
 EXERCISES. 
 
 
 
 
 Dividend. 
 
 
 Divisor. 
 
 
 
 Quotient. 
 
 33 
 
 
 
 II 
 
 
 
 189 
 
 467 
 151 
 
 
 
 937 
 
 ICI 
 
 
 
 47157 
 136957 
 
 7813 
 5071 
 
 
 
 601 
 
 II 
 
 
 
 13 
 461 
 
 I I I 
 
 -x-x 
 
 What are ^ ^ ^ 
 I 
 
 5 
 
 2 
 
 X 
 
 17 
 
 2 
 
 "17 
 
 2 2 8 
 
 ''^'\ and ^^' 
 
 8 
 II 
 £ 
 II 
 
 II 
 
 63 
 
 559 
 Answer^ — =^^, and i. 
 
 7225 
 
 A can reap a field in 12 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 sepa- 
 rately fill it in 12, II, 10, and 9 hours .' — Answer^ 2-^- hours. 
 
 763 
 
 124. The principal results of this section may be exhibited algebrai- 
 cally as follows ; let a, 6, c, &c. stand for any whole numbers. Then 
 
 (107) - = --.„ (108) - = - 
 
 f^^^\ ^ :i ^ xu ad bo 
 
 (111) - and - are the same as -— and — - 
 ha bd Id 
 
 ,,,«, a b a-hb a b a—b 
 
 (112) - + - = = 
 
 c c c c c c 
 
 (113) 
 
 a c ad+bc a c ad— be 
 
 b d bd b d bd 
 
 ("«) ^3=^ (-i)3<'^v.M.«o.±.0 
 
 * The method of solving this and the folloMdng 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 re- 
 
 a 
 
 present fractions. For example, take the fraction — , whose numerator 
 
 1 
 and denominator are fractional, and multiply its numerator and deno- 
 
 ae 
 
 minator by the fraction -, which gives , which (121) is • , ^. ' > 
 
 ''Z ad 
 
 which, dividing the numerator and denominator by ef (108), is — -. 
 
 a a e be 
 
 But the original fraction itself is -r- ; hence — = — which corre- 
 
 '^ be lit 
 
 d d^f 
 spends 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, w, and a be respectively 
 
 ^1 r i.- P ^ J ^ rr.i_ , p r ps+qr . , . . 
 
 the fractions -, -, and -. Then m+n is - +-, or —, and (m+n)a is 
 
 g s c q s g» 
 
 ps+qr b (ps+qr)b psb+qrb ^ ,. ,,,^. . psb qrb ,., 
 
 ^-—i- X -, or ^^ ^ ' or -^ '—, But this (112) is ''— + i — , which 
 
 qs c qsc qsc qsc qsc 
 
 . pb rb . psb pb , (/'>b rb ,,^^^ ^ pb p h , rb 
 
 18 — + —, since ^—- = —, and- — = —(108). Buf^ =-x.-, and — 
 
 qc sc qsc qc qsc sc qc q c sc 
 
 = -X-. Therefore (m+w)a, or (-+ )- = -x +-x . In a similar man- 
 s c \q f-/c q c s c 
 
 ner the same may be proved of any other formula. 
 The following examples may be useful : 
 
 a e c g 
 
 acfh+bdeg 
 aedh+bqfy 
 
 1 
 
 b 
 
 ob+1 
 
 I 
 
 I _ bc+r 
 
 b+- 
 
 <" abc+a+c 
 bc+l 
 
 • A formula is a name given to any algebraical expression which is commonly used. 
 
§ 125-128. DECIMAL FRACTIONS, 65 
 
 Thus, _L_ = _L^ = JL 
 
 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 FUACTIONS. 
 
 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 10, 100, 1000, &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, 
 . 040 9400 94000 „„^. 3 . 30 300 3000 ,,„ , 
 
 94 is ^ or 22_ or^^^^^ (106) ; -^ is -^—, or-^ — , or-^ (108). 
 
 ^ 10 100 1000 10 100 1000 lOOOO 
 
 The placing of a cipher on the right hand of any number is the same 
 thing as multiplying that number by 10 (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- 
 10 
 
 minator successively by 10, 100, 1000, &c., which will give a series of 
 
 fractions, each of which is equal to — r (108"), viz. -V-» -^ — •, -^ — •> 
 
 16 ' 160 1600 160C0 
 
 o2 
 
66 PRINCIPLES OF ARITHMETIC. § 128. 
 
 70000 
 
 -7 , &c. The denominator of each of these fractions can be divided 
 
 160000 
 
 without remainder by 16, the quotients of which divisions form the series 
 of decimal numbers 10, 100, 1000, 10000, &c. If, therefore, one of the 
 numerators, be divisible by 16, the fraction to which that numerator be- 
 longs has a numerator and denominator both divisible by 16. 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 10, 100, 1000, &c., and which is equal in value to -7. The ques- 
 
 16 
 
 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 16, as follows : 
 
 16)70(4 
 
 16)700(43 
 
 16)7000(437 
 
 16)70000(4375 
 
 64 
 
 64 
 
 64 
 
 64 
 
 6 
 
 60 
 
 60 
 
 60 
 
 
 48 
 
 48 
 
 48 
 
 
 12 
 
 120 
 
 120 
 
 
 
 112 
 
 112 
 
 80 
 80 
 
 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. In this case, since 70000 is 
 
 - 70000 , . , , 16x4375 4375 . ,•. « X. • , 
 
 io'<437S» -h ♦ which is . ^^^-' or •" , gives the fraction required. 
 
 160000 16x10000 10000 
 
 Therefore, to reduce a fraction to a decimal fraction, annex ciphers 
 to the numerator, and divide by the denominator until there is no re- 
 mainder. 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. DECIMAL FRACTIONS. 67 
 
 EXERCISES. 
 
 Reduce to decimal fractions 
 
 I I a I 39^-7 ^^^^453 
 
 2 
 
 4' ^5' 50' 1250' 625* 
 
 5 25 8 a 31416 7^48 
 
 Answer. — , , , , , and . 
 
 10 ICO 100 100 lOCOO lOCOO 
 
 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)1000000000000000000, &c. 
 142857142857142857, «Scc. 
 
 The quotient here is a continual repetition of the figures i, 4, 2, 
 
 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 nimiber 
 
 of figures from the quotient 142857142857, &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 fit)m -, and 
 
 7 
 which will difier still less from it if we put more figures in the numerator 
 
 and more ciphers in the denominator. 
 
 Thus, ±P«Wl by ll'^'^^<^^^^^^otsolj_ 
 
 10 ( than ) 7 70 ( much as j 10 
 
 14 I 2 1 
 
 100 * ' ' 7 * • 700 100 
 
 142 16 I 
 
 1000 • ' * 7 ' * 7000 1000 
 
 1428 14 I 
 
 loooo * ' 7 * 70000 10000 
 
 14285 I 5 I 
 
 icooco ' * * 7 ' 7C0000 icoooo 
 
 142857 J I I 
 
 lOOOOCO * * 7 70C0000 * * • • lOOOOOO 
 &c. &c. &c. &c. 
 
 In the first column is a series of decimal fractions, which come nearer 
 
 and nearer to -, as the third column shews. Therefore, though we can- 
 
 7 1 
 
 not find a decimal fraction which is exactly -, we can find one which 
 
 7 
 differs from it as little as we please. 
 
68 PRINCIPLES OF ARITHMETIC. § 129-131. 
 
 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 - by a million, and then divide both 
 
 7 
 by 7 ; we have then 
 
 I loooooo 1428574- 
 7 ~" 7C00000 "" loooooo 
 
 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 
 
 142857 
 part of a unit. Therefore — is the fraction required. 
 
 EXEUCISES. 
 
 Make similar tables with ) 3 17 .1 
 these fractions (91' ^43' H7* 
 
 The recurring) 3 . , , „ 
 
 quotient of I -^s 3^9670,329670, &c. 
 
 17 
 
 143' 
 I 
 
 247' 
 
 118881,118881, &c. 
 404858299595141700,4048582 &c. 
 
 130. The reason for the recurrence of the figures of the quotient 
 in the same order is as follows : If 1000, &c. be divi'^ed 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 diiferent, that is, let them be i, 2, 3, &c., up to 246, variously dis- 
 tributed. 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 frac- 
 tions, 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 
 
§ 131-132. DECIMAL FRACTIONS. 69 
 
 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. Now, by carrying on the table in 
 
 (129), we shall see that — does not differ from - by ; 
 
 ^ ' locccoco 7 lOOOOOOO 
 
 and if these fractions represented parts of an inch, the first might be 
 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 frac- 
 tions by 
 
 icococcco 
 
 3 locooccoo 
 
 4 57142857 
 
 J13 ^ 31830985 
 
 355 ICOOOOOOO 
 
 355 31415929a 
 
 113 * ' * lOOOGOOOO* 
 
 7 ICOCCCCGO 
 
 132. Every decimal may be immediately reduced to a quantity con- 
 sisting either of a whole number and more simple decimals, or of more 
 
 simple decimals alone, having one figure only in each of the numerators. 
 
 I47'J26 147326 326 
 
 Take, for example, -^^-^^ — . By (115) -^-^ — is 147^^ — ; and since 
 ^ loco ^ ^ icoo 1000 
 
 326 is made up of 300, and 20, and 6 ; bv (112) ~ — = -^ — -1- + 
 
 ' ' TCCO 1000 lOCO 
 
 6 -.^ . X '?co .3 , 20 . 2 ^, „ 147326 
 
 . But ( 1 08) - — IS -i-, and is . Therefore, -^^ — is 
 
 1000 1000 10 1000 100 1000 
 
 32 6 
 made up of 147+ — + 1- . Now, take any number, for example, 
 
 ^ 10 ICO ICOO * ^ 
 
 147326, and form a number effractions having for their numerators this 
 number, and for their denominators i, 10, ico, 1000, looco, &c., and 
 
70 
 
 PRINCIPLES OF ARITHMETIC. 
 
 § 132-133- 
 
 reduce these fractions into numbers and more simple decimuls, in the 
 foregoing manner, which Avill give the table below. 
 
 DECOMPOSITION OF A DECIMAL FRACTION. 
 
 147326 
 
 147326 
 
 147326 6 
 
 ~^'-^—= 14732+ — 
 
 10 10 
 
 147326 
 
 100 
 
 '473^6 
 1000 
 
 147326 
 1 0000 
 
 147326 
 I 00000 
 
 H7326 
 
 I 000000 
 
 '47326 
 
 I 0000000 
 
 2 6 
 
 1473 +— + 
 
 10 100 
 
 326 
 
 147 + -^+ + 
 
 10 100 1000 
 
 732 
 14+—+-^+ 
 
 10 ICO lOGO lOOOO 
 
 4 7 3 * 
 
 10 100 1000 lOOOO lOOOOO 
 
 .i-+^+^+- 
 
 5 .+. ^ 
 
 10 100 loco loooo loocoo 1000000 
 
 147 
 
 ICO 1000 ICCCO lOOOOO lOOOOOO lOOOOOOO 
 
 N.B. The student should write this table himself, and then proceed 
 to make similar tables from the following exercises. 
 
 EXERCtSES. 
 
 Reduce the following fractions into a series of numbers and more 
 simple fractions : 
 
 31415926 
 10 ' 
 
 2700031 
 10 ' 
 
 20730C0 
 10 ' 
 
 3331303 
 1000 ' 
 
 3I4I5926 
 
 &c. 
 
 100 ' 
 
 2700031 
 
 100 ■* 
 
 &c. 
 
 2073000 
 
 100 ' 
 
 &c. 
 
 3331303 
 
 &c. 
 
 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 
 
§ I33-I34* DECIMAL FRACTION'S. 71 
 
 may be made thus : Mark ofif, from the right hand of the numerator, 
 as xnaxiy figures as there are ciphers in the denominator by a point, or 
 any other convenient mark. 
 
 147326 
 
 This will give i4732'6 when the fraction is 
 
 . . . . i473'26 
 
 . . . . i47'326 
 
 10 
 
 100 
 147326 
 
 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 10, the second of 
 that whose denominator is ico, and so on. We now come to those 
 fractions which do not contain a whole number. 
 
 134. The first of these is , in which the number ot ciphers in 
 
 lOOOOOO 
 
 the denominator is the same as the number oi figures in the numerator. 
 If we still folloAv the same rule, and mark off all the figures, by placing 
 the point before them all, thus, '147326, the observation in (133) sti'l 
 holds good ; for, on looking at — in the table, we find it is 
 
 -L+-^+ 
 
 10 ICO ICOO ICCOO ICCCOO lOCCCOO 
 
 The next fraction is , which we find by the table to be 
 
 1 4 7 
 
 100 loco ICCOO loocoo 1000000 lOOOOOOO 
 
 Tn this, I is not divided by 10, but by 100 ; 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. § I34-135, 
 
 point, thus, "0147326. Here the rule holds good, for by that rule this 
 fraction is 
 
 01473 ^ ^ 
 
 10 100 1000 10000 lOCOOO lOOOOOO lOOOOOOO 
 
 o 
 which is the same as the preceding line, since — is o, and need not be 
 
 10 
 
 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 Avhich the fraction contains. 
 The first figure after the point with the denominator 10, the second with 
 the denominator 100, 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 7 
 
 — , "07 for , and so on. The following tables will give the whole of 
 
 10 ICO ** ^ 
 
 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 lo, 
 100, 1000, &c. while those on the left are units multiplied by 10, 100, 
 1000, &c. 
 
DECIMAL FRACTIONS. 
 
 73 
 
 §135. 
 
 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 
 never at the bottom. The reason is, that it is usual in the higher 
 branches of mathematics to use a point placed between two numbers 
 or letters which are multiplied together; thus, 15.16, a.6, a+b.c+d stand 
 for the products of those numbers or letters. 
 
 O OJ 
 O 14^ 
 
 8U 
 
 oh 
 
 51* 
 
 
 
 
 
 
 »* 
 
 
 
 
 
 
 00 
 
 
 
 
 
 
 00 
 
 ^ 
 
 II 
 
 II 
 
 II 
 
 " 
 
 II 
 
 § 
 
 M 
 
 i_' 
 
 ^ 
 
 ,j 
 
 1" 
 
 
 
 
 tj 
 
 
 
 
 M 
 
 
 
 00 
 
 
 
 
 
 
 UJ 
 
 
 
 
 
 
 
 + 
 
 ■»- 
 
 + 
 
 + 
 
 + 
 
 8 
 
 d 
 
 
 
 s 
 
 d 
 
 8 '^ 
 
 w 
 
 00 
 
 
 
 00 
 
 
 
 
 
 00 
 
 00 
 
 
 
 
 
 II 
 
 II 
 
 II 
 
 + 
 
 + 
 
 8 
 
 
 
 
 d 
 
 ^ 
 
 
 N 
 
 
 
 ■^ 
 
 
 
 00 
 
 
 
 
 CX) 
 
 u 
 
 00 
 
 
 
 8 
 
 + 
 
 + 
 
 + 
 
 + 
 
 4 
 
 d 
 
 d 
 
 d 
 
 
 
 ^ 
 
 
 a 
 
 
 
 
 00 
 
 
 
 g 
 
 - 
 
 ° 
 
 + 
 
 O i 
 
 + 
 
 + 
 
74 
 
 PRIXCIPLES OF ARITHMETIC. 
 
 § 135-13^. 
 
 
 I 
 
 is 
 
 1000 
 
 inches 
 
 
 2 
 
 is 
 
 200 
 
 . 
 
 . 
 
 
 3 
 
 is 
 
 30 
 
 . 
 
 . 
 
 
 4 
 
 is 
 
 4 
 
 • 
 
 • 
 
 IV. In I234.-56789 
 
 5 
 
 is 
 
 10 
 
 of 
 
 an inch 
 
 inches the 
 
 6 
 
 is 
 
 6 
 
 100 
 
 • 
 
 • • 
 
 
 7 
 
 is 
 
 7 
 1000 
 
 • 
 
 . . 
 
 
 8 
 
 is 
 
 8 
 
 • 
 
 
 
 lOCOO 
 
 • 
 
 
 n 
 
 la . 
 
 9 
 
 
 
 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 signi- 
 ficant 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 
 
 of ciphers on its right. Take, for example, '3 and '300. The first (135) 
 
 3 300 
 is — , and the second , which is made from the first by multiplying 
 
 both its numerator and denominator by 100, 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, 
 
 54 43^97 
 
 for example, '54 and 4'3297. The first is -^^—^ and the second ^. 
 
 r > JT T J y/ jpq' iocco 
 
 Multiply the numerator and denominator of the first by 100 (108), 
 
 ^400 43297 
 
 which reduces it to — , which has the same denominator as ^. 
 
 But 
 
 5400 
 
 is '5400 (135). In whole numbers, the decimal point should 
 
§ 138-140. DECIMAL FRACTIONS. 7.5 
 
 be placed at the end : thus, 129 should be written 129*. It is, however, 
 
 usual to omit the point ; but you must recollect that 129 and 129*000 
 
 129000 
 are of the same value, since the first is 129 and the second — . 
 
 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 Avriting 
 decimal fractions, which is explained in this chapter, makes the appli- 
 cation of these rules more simple. We proceed to the different cases. 
 
 Suppose it required to add 42*634, 45*2806, a'ooi, 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, 2*0010, and 54*0000. 
 These are decimal fractions, whose numerators are 426340, 452806, 
 
 20010, and 540000, and whose common denominator is loooo. By 
 
 i-i^c^\ xi- • • 426340+452806+20010+540000 ,,, . 1439156 
 
 (112) their sum is - — — — — ^— , which is ^^^ ^ 
 
 I 0000 I 0000 
 
 or 143*9156. The simplest way of doing this is as follows: write the 
 
 decimals down under one another, so that the decimal points may full 
 under one another, thus : 
 
 42*634 
 
 45*2806 
 
 2*OOI 
 
 5+ 
 
 143-9156 
 
 Add the diiferent columns together as in common addition, and place 
 the decimal point under the other decimal points. 
 
 EXERCISES. 
 
 What are i527+64*732094+2-ooi3+*ooooi974; 
 2276*3+* io7+*9+26*3 172+56732*001 ; 
 and i*ii+7-7+*oo39+-ooi42+*8838 ? 
 
 Answer^ 1593*73341374, 59035*6252, 9*69912. 
 
 140. Suppose it required to subtract 91*07324 from 137*321. These 
 
 fractions when reduced to a common denominator are 91*07324 and 
 
 137*32100 (138). Their difference is therefore ^373^100-9^07324^ 
 
 4624776 lOOOOO 
 
 which is or 46*24776, This may be most simplv done as fol- 
 
76 PRINCIPLES OF ARITHMETIC. § 140-142 
 
 lows : write the less number under the greater, so that its decimal point 
 may fall under that of the greater, thus : 
 
 137-321 
 91-07324 
 
 4624776 
 
 Subtract the lower from the upper line, and wherever tliere 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 12362— 274"22io7+*5 ; 
 
 9976-2073942— "00143976728 ; 
 and i'2+'o3+-oo4— '0005 ? 
 
 Answer^ 12088*27893, 9976-20595443272; and i'2335. 
 
 141. The multiplication of a decimal by 10, 100, 1000, &c., is per- 
 formed by merely moving the decimal point to the right. Suppose, for 
 
 132079 
 
 ex-'.mple, 13*2079 is to be muUiplied bv 100. T,he decimal is , 
 
 132c ^9 ^°°°° 
 
 which multiplied by ico is (117) '—, or 1320*79. Again, i*309x 
 
 . 13*^9 /,,/^\ 130900000 ^ , 
 
 icoooo IS ^xiooooo, or (116) or 130900. From these 
 
 1000 lOCO 
 
 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 17*036 by 4*27. The first of 
 
 17036 427 
 these decimals is —^ and the second . By (118) the product of 
 
 1000 100 
 
 these fractions has for its numerator the product of 17036 and 427, and 
 
 for its denominator the product of 1000 and 100 ; therefore this product 
 . 7274372 
 
 IS , or 72*74372. This may be done more shortly by multiply- 
 ing 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. 
 
§ 143-145' DECIMAL FRACTIONS. 77 
 
 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, 
 
 172 lOI 
 
 multiply '172 by 'lo I, or by , The product of these two is 
 
 17372 JOOO "^ ICOO 
 
 " , or '017372 (135). Therefore, when the number of places in 
 
 looooco /J/ \ / > r 
 
 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'oj is 'OOOOI 
 
 56x*oooi is '0056. 
 
 EXERCISES. 
 
 Shew that 
 
 3'002X3*OOZ = 3X3+2X3X"002+'0O2X*002 
 
 n"56o9X5'3i9i = 8*44x8"44— 3* 1209x3* 1209 
 S'aiyxio'ooi = 8xio+8x"ooi+iox*2i7+*ooix*2i7 
 
 Cube. 
 
 570i35"a33o88 
 
 •000005 1777 17 
 
 2-924207 
 
 •000000729 
 
 •I5625X '64 = •! 
 
 i562*5x'o64 = 100 
 •oi5625x*oo64 = 'oooi i5625ooox'o64 = loooooo 
 
 144. The division of a decimal by a decimal number, such as 10, 
 
 100, 1000, &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 
 
 1734229 
 by 1000: the decimal fraction is -, which divided by 1000 (123) 
 
 1734229 ^^°° 
 
 is — — — ^, or 1-734229. If, in the same way, 1*2106 be divided bv 
 
 1000000 
 
 locoo, the result is -00012106. 
 
 145. Before proceeding to shorten the rule for the division of one 
 
 11 2 
 
 Fraction. 
 
 Square. 
 
 82-92 
 
 6875-7264 
 
 •0173 
 
 •00029929 
 
 »*43 
 
 2-0449 
 
 •009 
 
 •ooooS I 
 
 IS-625X 64 = 
 
 1000 
 
 i*5625x -64 = 
 
 I 
 
78 PfilNCIPLES 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 —z is the same fraction as or '4375. 
 
 16 , lOOOO 
 
 As another example, convert -^ into a decimal fraction. Follow the 
 same process as in (128), thus : a 
 
 128)300000000000(234375 480 
 
 256 384 
 
 440 960 
 
 384 896 
 
 560 , 640 
 
 512 640 
 
 480 o 
 
 Since 7 ciphers are used, it appears that 30000000 is the first of the 
 
 series 30, 300, &c,, which is divisible by 128 ; and therefore — -^ 
 
 30000000 23437c ^ 
 
 or, which is the same thing (108), — is equal to ^-^— or 
 
 1280000000 lOOOOOOO 
 
 •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 num- 
 ber 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 
 
 , . , - . , rr,, I 14 14* 14^8 
 decimal fraction as near as we please. Thus, — , , , , 
 
 j.^gr 10 100 1000 10000 
 
 , &c., or •!, -14, '142, '1428, '14285, were shewn to be fractions 
 
 lOOOOO •> -n T •> -r •) -r Ji 
 
 which approach nearer and nearer to -. To find either of these frac- 
 
 7 
 tions, the rule is the same as that in the last article, with this exception. 
 
§ 146-148. 
 
 DECIMAL FRACTIONS. 79 
 
 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 can- 
 not 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. Thus, '1428 is very near to -, but not so near as 
 •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 1000 is divisible by 125, and that the 
 
 quotient is 8. One cipher only has been annexed to the numerator, and 
 
 therefore 100 divided by 125 is '8. Had the fraction been , since 
 
 •' ^ 125 
 
 1000 divided by 125 gives 8, and three ciphers would have been annexed 
 to the numerator, the fraction would have been "ooS. 
 
 148. Suppose that the given fraction has ciphers at the right of its 
 
 denominator; for example, . The annexing a cipher to the nu- 
 
 2500 
 
 merator is the same thing as taking one away from the denominator ; 
 
 310 31 310 31 
 for, (108) is the same thing as , and as — . The rule, 
 
 ^ 2500 ° 250 250 25 
 
 therefore, is in this case : Take away the ciphers from the denominator ; 
 
 EXERCISES. 
 
 Keduce the following fractions to decimal fractions : 
 
 1 36 297 I 
 
 :; — » > -7~» and — -. 
 
 8co 1250 64 128 
 
 Answer, "00125, '0288, 4*640625, and '0078125. 
 Find decimals of 6 places very near to the following fractions : 
 
80 PRINCIPLES OF ARITHMETIC. § 148- 1 50. 
 
 27 156 22 194 2617 I I , _3_ 
 
 49' 33' 37000' 13' 9907* 29C8* 466* 277 
 
 Answer, -551020, 4727272, -000594, 14923076, -266175, '000343, 
 •002145, ^^^ •010S30. 
 
 149. From (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 17-762 by 6-25. These fractions (138), when reduced 
 
 17762 6250 
 to a common denommator, are 17 762 and 6 250, or and . 
 
 17762 ^°°° ^°°° 
 
 Their quotient is therefore -^ — , which must now be reduced to a 
 
 6250 
 
 decimal fraction by the last rule. The process at full length is as 
 
 follows : 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 Here four ciphers have been annexed to the 
 
 '^^° numerator, and one has been taken from the 
 
 5262 denominator. Make five decimal places in the 
 
 5000 
 
 quotient, which then becomes 2-84192, and this 
 
 is the quotient of 17-762 divided by 6-25. 
 
 150. The rule for division of one decimal by 
 
 1200 
 6-jc another is as follows: Equalise the number of 
 
 ._„ decimal places in the dividend and divisor, by 
 
 5625 annexing ciphers to that which has fewest places, 
 
 1250 Then, further, annex as many ciphers to the 
 
 '^3° dividend * as it is required to have decimal places, 
 
 ° throw away the decimal point, and operate as in 
 
 common division. Make the required number of decimal places in 
 
 the quotient. 
 
 Thus, to divide 67173 by "014 to three decimal places, I first write 
 
 6 7173 and -0140, 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 from the divisor; or make up the number of ciphers partly 
 by removing from the divisor and annexing to the dividend, if there be not a iulB- 
 clent number in the divisor. 
 
§ 130. ^ DECIMAL FRACTIONS. 81 
 
 that the divisor '0140 has one cipher, I strike that one out and annex 
 two ciphers to 67 17 3. Throwing away the decimal points, then divide 
 6717300 by 014 or 14 in the usual way, which gives the quotient 479807 
 and the remainder 2. Hence 479*807 is the answer. 
 
 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 
 the 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. Thus, it should be visible, 
 that 26*1 1 9-^7*243 6 has one figure before the decimal point, and that 
 26*ii9-T-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 I7'3i4 
 divided by 61-2 is 173*14 divided by 612, and the decimal point must 
 precede the first figure of the quotient. But 17*314 divide.! by 6617-5 
 is I73'i4 by 66175 5 ^^^d 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 "002 
 
 EXAMPLES. 
 
 3 I -00062 
 
 — 1240, — -- — = -00096875 
 
 -0025 
 
 EXERCISES, 
 e, ^i_ ^ 15-006x15-006— •004X -OCA 
 
 bhew that ; = 1 5-002, and that 
 
 •OIXOI X-OT+2-9X2-9X2 9 
 
 -; = 2 9x2-9 — 2 9X-OI+-OIX-01 
 
 2 91 
 
82 PBINCIPLES OF ARITHMETIC. § 150- 151. 
 
 What are , ^ ^ ■■, and — ; , as far as 6 places 
 
 3-14159 27182818 '1^349 
 
 of decimals? — Answer, '318310, '367879, and i989'20922i. 
 
 Calculate 10 terms of each of the following series, as far as 5 places 
 of decimals. 
 
 I 
 
 I 
 
 I 
 
 
 = 
 
 1-71824. 
 
 2 2x3 
 
 2x3x4 
 
 2x3x4x5 
 
 
 
 I 
 
 I+- + 
 
 2 
 
 I I I 
 
 - + - + - + &C. 
 
 3 4 5 
 
 = 
 
 2-92895. 
 
 
 80 81 82 83 84 ^ 
 
 81 82 83 84 85 
 
 = 
 
 988286. 
 
 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 
 17-8; for though this is too little by -046217, yet it is not so much too 
 little as 17-9 is too great; and the error is less than half a tenth, or 
 — , Again, the same distance, within a hundredth of a mile, is more 
 correctly I7'85 than 17-84, since the last is too little by '006217, which 
 is greater than the half of 'oi; and therefore i7'84+*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 i)lace at 
 a time. 
 
§ I51-T52. DECIMAL FRACTIONS. 83 
 
 ''■•^159' 3'i4i6, 3-142, 3-14, 3'T, 3-0 
 
 2*7182818. 2718282, 2*71828, 2*7183, 2718, 2*72, 2*7, 3*o 
 
 1*9919, 1*992, 1*99, 2*00, 2*o 
 
 15*2. 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 ma> be any where between 8*955 and 8*965. 
 The product, therefore, of the numbers which represent the correct dis- 
 tances will lie between 9'975x8*955 and 9'985x8*965, that is, taking 
 three decimal places in the products, between 89*326 and 89*516. The 
 product of the actual numbers given is 89*4208. 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 
 10 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 
 
 to the first decimal place, the product is within of the truth ; if to 
 
 two decimal places, within ; if to three, within ; and so on. 
 
 200 2COO 
 
 Thus, in the above example, we have 9*98 and 8*96, which are true to 
 two decimal places: their sum divided by 200 is *0947, and their product 
 is 89*4208, which is therefore within '0947 of the truth. If, in fact, we 
 increase and diminish 89*4208 by '0947, 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 suiRciently so for every practical purpose. 
 
84 PRINCIPLES OF ARITHMETIC. § 152-153. 
 
 is exact also, and that this article applies where the numbers given are 
 correct only to a certain number of decimal places. The rule is : Take 
 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 ai;d divisor may be depended upon. Thus, if 
 17*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 53'8o9, or, which will do as well for 
 our purpose, the square of 50, or 2500. The result is something less 
 than '00002, 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, 50 
 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 line one place to the right instead of to the left, as in the following 
 example : 
 
 2221 2221 
 
 1234 4321 
 
 8884 2221 
 
 6663 4442 
 
 444.2 6663 
 
 2221 8884 
 
 2740714 2740714 
 
 Suppose now we wish to multiply 348*8414 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 : 
 
3488414 
 
 
 2470315 
 
 
 17442070 
 
 
 3488414 
 
 
 1046524 
 
 2 
 
 24418 
 
 89S 
 
 1395 
 
 3656 
 
 69 
 
 76828 
 
 i7898'i522 
 
 23188 
 
 § 153. « DECIMAL FRACTIONS. 85 
 
 Cut off, by a vertical line, the first four places 
 of decrmals, and the columns which produced 
 them. It is plain that in forming our abbre- 
 viated rule, we have to consider only, I. all that 
 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 
 23188 jeft of the line, we see 4, 4, 8, 5, 9, of which the 
 first 4 comes from 4x1',* the second 4 from 1x3', 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 
 51-30742, 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 diflferent 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 r here means that the 1 is in the multiplier. 
 
S(i 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 prmciple 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 be avoided 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 ry^ 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 multi- 
 plier 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 136-4072 by 1-30609, retaining 7 decimal 
 jlaces. 
 
 1364072000 
 906031 
 
 1364072000 
 
 409221600 
 
 81844.32 
 
 122766 
 
 I7}?*t6oo798 
 
§ 154-155. 
 
 DECIMAL FRACTIONS. 
 
 87 
 
 In the following examples the first two lines are the multiplicand 
 and multiplier; and the number of decimals to be retained wUl be 
 seen from the results. 
 
 •447I6I8 
 
 33-166248 
 
 3-4641016 
 
 377I92I4. 
 
 1-4142136 
 
 1732-508 
 
 377192 14 
 
 033166248 
 
 346410 I 60 
 
 8161744 
 
 63I2414I 
 3316625 
 
 8052371 
 
 15087686 
 
 346410160 
 
 1508768 
 
 1326650 
 
 242487112 
 
 264034 
 
 33166 
 
 10392305 
 
 3772 
 
 13266 
 
 692820 
 
 2263 
 
 ' 663 
 
 173205 
 
 38 
 
 33 
 
 2771 
 
 30 
 
 10 
 
 z 
 
 6001-58373 
 
 1-6866591 
 
 
 46-90415 
 
 Exercises may be got from article (143). 
 
 165. With regard to division, take any two numbers, for example, 
 16-80437921 and 3'i42, and divide the first by the second, as far as 
 any required number of decimal places, for example, five. This gives 
 the following : 
 
 
 3-142)16-80437 
 
 921(5-34830 
 
 
 15710 
 
 
 
 10943 
 
 
 
 9426 
 
 
 
 15177 
 
 
 (A) 
 
 12568 
 
 
 2609 
 
 2609 
 
 9 
 
 2514 
 
 ^5136 
 
 95 
 
 9632 
 
 94 
 
 9A6 
 
 — 
 
 — - 
 
 1 
 
 2 
 
 061 
 
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 2061. 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, nnd also liOAV many decimals .are required, 
 
 we can tell from the beginning how many figures there will be iu 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, '003178, since this is — , divide by 
 
 100 
 
 •3178 in the usual way, and afterwards multiply the quotient by 100, 
 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. 
 
 EXTRACTION OF THE SQUARE ROOT. 
 
 89 
 
 Places required. 
 Divisor, 
 
 Dividend, 
 
 iiuotieut. 
 
 z 
 •41432 
 
 673-1489 
 41432 
 
 258828 
 248 592 
 
 10237* 
 8286 
 
 1 951 
 1657 
 
 294 
 
 290 
 
 4 
 4 
 
 162471 
 
 3-1415927 
 
 271828180 
 2-51327416 
 
 20500764 
 18849556 
 
 1651208 
 
 1570796 
 
 80412 
 62832 
 
 17580 
 
 15708 
 
 1872 
 ^571 
 
 301 
 
 283 
 
 18 
 
 19 
 
 •86525596 
 
 Examples may be obtained from (143) and (150). 
 
 SECTION VII. 
 
 ON THE EXTRACTION OF THE SQUARE ROOT. 
 
 156. We have already remarked {66)y that a number multiplied by 
 itself produces vrhat is called the square of tliat number. Thus, 169, 
 or 13x13, is the square of 13. 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 V or v' ; thus, V25 means 
 the square root of 25, or 5; V16+9 means the square root of 16+9, 
 and is 5, and must not be confounded with v' 16+^9, which 134+3, ^^ 7- 
 
 • This is written 7 instead of 6, becaue the figure which is abaDdoned in the divi- 
 dend is 9 (151). 
 
 I 2 
 
90 PRIKCIPLES OF ARITHMETIC. § 157-I59. 
 
 157. The following equations are evident from the definition : 
 
 Va-uVa = a 
 w aa = a 
 waby. w ab = ah 
 (v'axv'6)x( Vax /v/6) = -v/ax -/ax V'ixV'i = ah 
 whence ^/ay-Vb = ^/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 nc 
 
 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 
 
 S(iuare root at all. Nevertheless, there are methods of finding fractions 
 
 whose squares shall be as near to 5 as we please, though not exactly 
 
 15 127 
 equal to it. One of these methods gives , whose square, viz. 
 
 15127 15127 228826120 ,.^ „ , "705 4 , . , . 
 
 -r-T— X-— — - or — — -, differs from 5 by only :;: , which is 
 
 6765 6765 45765225 45765225 
 
 less than '0000001: hence we are enabled to use V^ in arithmetical 
 
 and algebraical reasoning : but when we come to the practice of any 
 
 p.roblem, we must substitute for Vs one of the fractions whose square 
 
 is nearly 5, and on the degree of accuracy we want, depends what 
 
 fraction is to be used. For some purposes, may be sufficient, as its 
 
 4 ^^ 
 
 square only differs from 5 by -— =>- ; for others, the fraction first given 
 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 suflficiently 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 0? be a number consisting of any number of parts, for 
 example, four, viz. a, i, c, and d; that is, let 
 
 7 15 
 
 » Meaning, of course, a really fractional number, such as - or — , not one which, 
 
 ihough fractional in form, is whole in reality, such as — or — ., 
 
 5 d 
 
§ 159-160. EXTRACTION OF THK SQUARE ROOT. 91 
 
 The square of this number, found as in (68), will be 
 
 aa+2a(6+j+rf) 
 +bb+zb{c+d) 
 •¥cc-k zed 
 -t-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 2a 
 by each of the succeeding parts, 6, c, and d, and adding the results, we 
 multiplied za 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 
 sc^uaring 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 aris- 
 ing from any new part, even though that part were the least number, 
 which is 1. 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 20, sub- 
 tracted from 2025, gives 1625, the first remainder. I again choose 20, 
 whose square, together with twice itself, multiplied by the preceding 
 part, is 20x20+2x20x20, or 1200 ; which subtracted from 1625, the 
 
gi PBINCIPLKS OF ARITHMETIC. § ibo-lbl. 
 
 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 20+20, gives 609, which 
 is more than 425. I therefore choose 5, which closes the process, since 
 5x5, together with 2x5 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 = 2025. Again, I ask if 13340 
 has, or has not, a square root. Let 100 be the first part, whose square 
 is loooo, and the first remainder is 3340. Let 10 be the second part. 
 Here 10x10+2x10x100 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 115. There is, then, no 
 square root ; for a single additional unit will give a subtraction of 
 ixi+2xix(ioo+io-f5), or 231, which is greater than 115. But if the 
 number proposed had been less by 115, each of the remainders would 
 have been 115 less, and the last remainder would have been nothing. 
 Therefore 13340— 115, or 13225, has the square root 100+10+5, or 
 115; and the answer is, that 13340 hsis no square root, and that 13225 
 is the next number below it which has one, namely, 115. 
 
 16L 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. It is evident 
 (57) that any number which terminates with ciphers, as 4000, has 
 double the number of ciphers in its square. Thus, 4000x4000 = 
 16000000 ; therefore, any square number,* as 49, with an even number 
 of ciphers annexed, as 490000, is a square number. The rootf of 
 490000 is 700. This being premised, take any number, for exanijilc, 
 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. 
 
 f The term ' root' is frequently used as an abbreviation of square root. 
 
§ l6i. EXTRACTION OF THE SQUARE ROOT. 93 
 
 lias four ciphers, and its square root is 200. Let this be the first part 
 chosen: its square subtracted from 76176 leaves 36176, the first re- 
 mainder ; 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 
 100, 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 loN, 
 whose square is loNxioN, or looNN, and whose double multiplied by 
 the former part is 2oNx2oo, or 4000N ; the two together are 4000N+ 
 icoNN. Now, N must be so taken that this may not be greater than 
 36176 : still more 40C0N 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 (80) applies here. Let us try 
 9 tens or 90. Then, 2x90x200+90x90, or 44100, is to be subtracted, 
 which is too great, since the whole remainder is 36176. We then try 
 8 tens or 80, which gives 2x80x200+80x80, or 38400, which is likewise 
 too great. On trying 7 tens, or 70, we find 2x70x200+70x70, or 32900, 
 which subtracted from 36176 gives 3276, the second remainder. The 
 rest of the square root can only be units. As before, let N be this 
 number of units. Then, the sum of the preceding parts being 200+70, 
 or 270, the number to be subtracted is 270X2N+NN, or 540N+NN. 
 Hence, as before, 540N miist be less than 3276, or N must not be greater 
 than the number of times which 3276 contains 540, or (80) which 327 
 contains 54. We therefore try if 6 will do, which gives 2x6x270+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 subtnicted may be 
 ■hortened thus. Let A be the sum of the parts already found, and N 
 a new part: there must then be subtracted 2AN+NN, or (54) 2A+N 
 
94 PRINCIPLES OF ARITHMETIC. § 161-163. 
 
 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 : 
 
 7,6i,76(: 
 40000 
 
 iOO 
 
 70 
 6 
 
 7,6i,7' 
 4 
 
 40o\3,6i,76 
 70J3 29 00 
 
 47)361 
 329 
 
 400 32 76 
 
 140 32 76 
 
 6/ 
 
 
 
 546)3276 
 3276 
 
 
 
 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 61, 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 34,86,78,44,01(59049 
 
 25 00 00 00 00 9000 25 
 
 40 
 
 looooo] 986784401 109) 986 
 
 9000^ 9 8 1 00 00 00 
 
 loooocX 57844.01 11804) 57844 
 
 iSoool 4721600 47216 
 
 47 
 
 iooooc\ 1 06 28 01 118089)1062801 
 
 I8000I 1 06 28 01 1062801 
 
 80 
 
 9/ ° ° 
 
 163. The rule is as follows: To extract the square root of a 
 Dumber ; — 
 
 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 requiretl 
 
§ lb3. EXTRACTION OF THE SQUARE ROOT. 95 
 
 root ; subtract its square from the first period, which gives the first re- 
 mainder. 
 
 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 con- 
 tained 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, i 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 simply, add the second figure of the root to the first divisor. 
 
PRINCIPLES OF ARITHMETIC, § 163-165. 
 
 EXERCISES. 
 
 Numbers proposed. 
 
 73441 
 
 4991900 
 
 6414247921 
 
 903687890625 
 
 42420747482776576 
 
 13422659310152401 
 
 Square roots. 
 
 271 
 
 1730 
 
 80089 
 
 950625 
 
 205962976 
 
 II5856201 
 
 164. Since the square of a fraction is obtained by squaring the 
 
 numerator and the denominator, the square root of a fraction is found 
 
 2 c ? 
 
 bjr taking the square root of both. Thus, the square root of — is -, 
 
 64 8 
 
 since 5 x 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 by the same number may convert both the numerator and 
 
 27 
 denominator into square numbers (108). Thus, — , which appears at 
 
 first to have no square root, has one in reality, since it is the same as 
 
 -7, whose square root is -. 
 10 4 
 
 165. We now proceed from (158), where it was stated that any num- 
 ber 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 fol- 
 lowing, " Find a fraction whose square shall not differ from 2 by so 
 much as 'oooocooi." 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 of 2 
 
 within — of the truth ; by which I mean, to find a fraction - Avhose 
 
 57 a I * 
 
 square is less than 2, but such that the square of -+ — is greater than 
 
 2. Multiply the numerator and denominator of- by the square of 57, 
 
 1-. ^ . 6498 ^ . I 
 
 or 3249, which gives . On attempting to extract the square root 
 
 of the numerator, I find (163) that th re is a remainder 98, and that the 
 
 square number next below 6498 is 6400, whose root is 80. Hence, 
 
 tlie square of 80 is less than 6498, while that of 81 is greater. The 
 
§ 165-168. EXTRACTION OF THE SQUARE ROOT. 97 
 
 square root of the denominator is of course 57. Hence, the square of 
 
 — is less than , or 2, while that of — is greater, and these two 
 
 57 3249 1 57 
 
 fractions only differ by — ; 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 i"4i42, or i'99996i64, is 
 
 less than 2, while an increase of only i in the fourth decimal place, 
 
 giving 1*4143, gives the square 2*00024449, which is greater than 2. 
 
 To take a more general case : Suppose it required to find the square 
 
 1637 
 root of 1*637 true to four places of decimals. The fraction is , 
 
 J lOCO 
 
 whose square root is to be found ^vithin *oooi, or . Annex ciphers 
 
 1 0000 
 
 to the numerator and denominator, until the denominator becomes the 
 
 I , . , . 1637C0000 , „ , 
 
 square of , which gives . extract the square root of the 
 
 ICCOO ICOOCOOOO 
 
 numerator, as in (1G3), which shews that the square number nearest to 
 it is 163700000 — 13564, whose root is 12794. Hence, —^ or 1*2794, 
 
 lOOCO 
 
 gives a square less than 1*637, while 1*2795 gives a square greater. In 
 fact, these two squares are 1*63686436 and 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 .ight, 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 i~ to five places of 
 
 o 
 
 decimals? This is (145) 1*375, and the process is the first example 
 over leaf. The second example is the extraction of the root of *o8i to 
 seven places, the first period being 08, from which the cipher is omitted 
 as useless. 
 
98 
 
 PRINCIPLES OF ARITHMETIC. § r)3-l69. 
 
 > 4 
 
 21) 37 48)410 
 
 ai 384 
 
 227)1650 564) 2600 
 
 1589 2256 
 
 2342) 6100 5686) 34400 
 
 4684 341 16 
 
 23446)141600 569204) 2840000 
 
 140676 2276816 
 
 23452) 92400 569208) 56318400 
 
 •00000241 367222 i(*oo 1 55 3599 
 
 ^5) 
 
 141 
 
 
 125 
 
 305) 
 
 1636 
 
 
 1525 
 
 3103) III72 
 9309 
 
 ^1065) 186322 
 
 155325 
 
 310709) 30997 If 
 2796381 
 
 30332900 
 
 169. When more than half the decimals required have been found, 
 the others maybe simply found by dividing the dividend by the di- 
 visor, as in (155). The extraction of the square root of 12 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 de- 
 pended upon I on which account it is advisable to carry the process so 
 far, that one or even two more decimals shall be obtained than arc 
 absolutely required to be correct. 
 
§ 169. 
 
 EXTRACTION OF THB SQUARE ROOT. 
 
 vd 
 
 12(3*46410161513 
 9 
 
 64) 300 
 
 256 
 
 6S6) 4400 
 4116 
 
 6524) 284CO 
 27696 
 
 B 
 
 «92?20?23026)53725355o83i (77545870549 
 4849742261 18 
 
 52279324713 
 4849742261 1 
 
 3781902102 
 3464101615 
 
 6928 1 ; 
 
 70400 
 69281 
 
 6928201) 111900C0 
 6928201 
 
 69282026) 4261799 
 4156921 
 
 692820321) 104877 
 69282 
 
 6928203225) 3555; 
 3464 J 
 
 69282032301) 95439177500 
 
 692 
 
 317800487 
 277128129 
 
 40672358 
 34641016 
 
 6031342 
 5542562 
 
 4400 
 0321 
 
 4079C0 
 016125 
 
 8203230] 
 
 488780 
 484974 
 
 3806 
 3464 
 
 34* 
 Z77 
 
 692820323023) 261 
 
 207 
 
 53 
 
 57145 1990c 
 8460969069 
 
 6S 
 62 
 
 7253550831 
 
 If from any remainder we cut off the ciphers, and all figures which 
 would come under or on the right of these ciphers, by a vertical line, 
 we find on the left of that line a contracted division, such as those in 
 ^155). Thus, after having found the root as far as 3 "4641 01, we have 
 the remainder 4261799, and th*> divisor 6928202. The figures on the 
 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 oif the first figure. By this alone we 
 might have doubled our number of decimal places, and got the addi- 
 
100 PRINCIPLES OF ARITHMETIC. § 169-170. 
 
 tional figures 615 137, the last 7 being obtained by carrying the con- 
 tracted division one step further with the remainder 53. We have, 
 then, this rule : 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*46410161513. The remainder is 
 537253550831, the divisor 692820323026, and the process is as in (B). 
 Hence the square root of 12 is, 
 
 3-464ioi6i5i377545870549; 
 
 which is true to the last figure, and a little too great ; but the sub- 
 stitution of 8 instead of 9 on the right hand would make it too small. 
 
 
 EXERCISRS. 
 
 
 Numbers. 
 
 
 Square roots. 
 
 •001728 
 
 
 •04 1 56^2 194 
 
 64' 34 
 
 
 8-02122185 
 
 .8074 
 
 
 89-8554394 
 
 10 
 
 
 3-16227766 
 
 J'57 
 
 1-2529964086141667788495 
 
 SECTION VIII. 
 
 ON THE PROPOUTION 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 con- 
 sidering the two together, to esta1)lish 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 11, and ico differs from m by the 
 
§ 170-172. ON PROPORTION OP NUMBERS. 101 
 
 same number. In this point of view, 8 stands to 19 in the same 
 situation in which 100 stands to 1 1 1, the first of both couples differing 
 in the same degree from the second. The four numbers thus noticed, 
 viz.: 
 
 8, 19, 100, III, 
 
 are said to be in arithmetical* proportion. When four numbers are 
 thus placed, the first and last are called the ecrtremes^ and the second 
 and third the means. It is obvious that 111+8 = 100+19, ^^^^ ^s» ^^^ 
 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 11 1+8, by 
 (35), any diminution of 111 will not affect the sum, provided a cor- 
 responding increase be given to 8 ; and, by the definition just given, 
 one mean is as much less than 1 1 1 as the other is greater than 8. 
 
 171. A set or series of numbers is said to be in continued arith- 
 metical 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 : 
 
 I» 
 
 a. 
 
 3, 
 
 4, 
 
 5» 
 
 &c. 
 
 3. 
 
 6, 
 
 9. 
 
 12, 
 
 15, 
 
 &c. 
 
 I 
 I-, 
 
 2 
 
 2, 
 
 I 
 
 2-, 
 
 2 
 
 3» 
 
 I 
 3? 
 
 &c. 
 
 The difference between two succeeding terms is called the common 
 
 difference. In the three series just given, the common differences 
 
 are, i, 3, and -. 
 2 
 
 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 g. 
 
 * This is a very incorrect name, since the term « arithmetical' applies equally to 
 every notion in this book. It is necessary, hovrever. that the pupil should use words 
 in the sense in which they will be used in his succeeding studies. 
 
 k2 
 
102 PRINCIPLES OK ARITHMETIC. § 172-I73. 
 
 Then, since, by the ir.iture of the series, b is as much above a as /is 
 below 1^ (170), a+g = b+f. Again, since c is as much above b as e is 
 below/ (170), b+f = c+e. But a+^ = b+f; therefore a+^ = c+e, and 
 80 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 
 P.S much l)elow d as e is above it, we have c+e = d+d = 2d. But c+e = 
 a+g; therefore, a-i-g = zd. This will give a short rule for finding the 
 sum of any number of terms of an arithmetical series. Let there be 
 7, viz. those just given. Since a+g, b+f, and c+e, are the same, their 
 sum is three times («+//), which with rf, the middle term, or half a+^, 
 is three times and a half a+g, or the sum of the first and last terms 
 
 I n 
 
 multiplied by 3-, or -, or half the number of teims. If there had been 
 
 z 2 
 an even number of terms, for example, six, viz. a, 6, c, rf, e, and/, we 
 
 know now that a+/, b+e, and c+d, are the same, whence the sum is three 
 
 times a+/, or the sum of the first and last terms multiplied by half tne 
 
 number of terms, as before. The rule, then, is : To sum any number of 
 
 terms of an arithmetical progression, multiply the sum of the first and 
 
 hist terms by half the number of terms. For example, what are 99 
 
 terms of the series i, 2, 3, &c. ? The 99th term is 99, and the sum 
 
 is (99+1) — , or , or 4950. The sum of 50 terms of the series 
 
 2 2 
 
 ' ^ 4 5 n . /I 5o\ 50 
 
 -, -, T, -, -, 2, &c. IS I -+^^ ^ or 17x25, or 425. 
 
 3 3 3 3 \3 3/ 2* 
 
 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 multij)lied 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 ijv n— i steps, at each of which the comroon 
 
 dih'erence is add(?d. 
 
§173-175. 
 
 ON- PROPORTION OF NUMBERS. 
 
 103 
 
 
 
 EXERCISES. 
 
 
 
 Given. 
 
 To find. 
 
 
 Series. 
 
 No. of terms. 
 
 Last term. 
 
 Sura. 
 
 4. 
 
 6^, 9, &c. 
 
 33 
 
 84 
 
 1452 
 
 I, 
 
 3, 5, &c. 
 
 28 
 
 55 784 
 
 2, 
 
 20, 38, &c. 
 
 IOO,OCO 
 
 1799984 
 
 8999930C000 
 
 174. The Slim 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 100, and the sum 
 ic,cco. Since io,cco was made b}' multiplying the sum of the first and 
 
 ICO 
 
 last terms by , if we divide by this, we shall recover the sum of the 
 
 ^ IC,COO ICO 
 
 first and last terms. Now, divided by is (122) 200, and the 
 
 first term being i, the last term is 199. We have then to pass from 
 
 I to 199, or through 198, by 99 equal steps. Each step is, therefore, 
 
 198 
 
 -^-, or 2, which is the common difference; or the series is i, 3, 5, &c., 
 99 
 .ip to 199. 
 
 
 Given. 
 
 
 To find. 
 
 Sum. 
 
 No. of terms. 
 
 First term. 
 
 Last term. 
 
 Common diif. 
 
 1809025 
 
 1345 
 
 I 
 
 2689 
 
 2 
 
 44 
 7075600 
 
 10 
 1330 
 
 3 
 
 4 
 
 29 
 
 5 
 1C&36 
 
 45 
 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 10 thousand 
 pounds, that he is much richer than B, who has only 3 thousand ; but 
 we do not say that C, who has 107 thousand pounds, is much richer 
 than D, who has 100 thousand, though tlie difference of fortune is the 
 same in both cases, viz. 7 thoii«ind 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 100 pounds which he 
 had before, while D for every ico 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 10 as 100 is to 
 107, yet, in the light in which we now regard them, 3 is not so near to 
 10 as 100 is to 107, for 3 differs from 10 by more than twice itself, 
 while 100 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 10 to 3 is greater than the ratio or proportion of 107 to 
 ICO. 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, 1, 2, 3, 4, and 6, are parts of 12 ; - is also a 
 
 part of 12, being contained in it 24 times ; 12, 24, 36, &c., are multiples 
 
 of 12 ; and 8, 9, -, &c. are multiple parts of 12, being multiples of 
 
 2 
 some of its parts. And when multiple-parts generally are spoken of, 
 
 the parts themselves are supposed to be included, on the sfime principle 
 
 that 12 is counted among the multiples of 12, 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 12. 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 12 is of 7, we 
 gee that on dividing 7 into 7 parts, and repeating one of these parts 12 
 times, we obtain 12 ; or, on dividing 7 into 14 parts, each of whicli 
 is one-half, and repeating one of these parts 24 times, we obtain 24 
 
 1 2 2 A *X 6 
 
 halves, or 12. Hence, 12 is — , or — , or — of 7 ; and so on. Generally, 
 
 7 ' 14 „ 21 ' 
 when a and b are two whole numbers, 7 expresses the multiple -part 
 
 b * 
 
 which a is of b, and - that which b is of a. Again, suppose it required 
 
 * I . I 15 16 
 
 to determine what multiple-part 2- is of 3-, or — of — . These 
 ^ ^ 7 5 7 5 
 
 fractions, reduced to a common denominator, are — and , of which 
 
 1 35 35 
 
 the second, divided into 112 parts, gives — , which repeated 75 tunes 
 
§ I'JJ-l-j'i. ON PROPORTION OF NUMBERS. 105 
 
 gives — , the first. Hence, the multiple-part which the first is of the 
 
 35 75 . . V , 
 
 second is — —, which being obtained by the rule given in (121), anews 
 
 a '^^ 
 that -, or a divided by i, according to the notion of division there given, 
 
 expresses the multiple-part which a is of h 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 defini- 
 tion 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, 
 
 that of 2 to I-. Since, on dividing two inches into 4 parts, and taking 
 
 "^ 1 
 3 of them, we get i-, the same must be done with all the lines in the 
 
 2 
 
 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 £100, 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 £100. 
 
 a 
 Since, then, the part which a is of 6 is expressed by the fraction -, 
 
 or any other fraction which is equivalent to it, and that which c is of rf 
 
 by -, it follows, that when cr, 5, c, and c?, are proportional, 7 = -:. This 
 a a 
 
 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, 6, c, and rf, being proportionals, that is, a being the 
 same multiple-part of 6 which c is of </, it does not follow that a, rf, b, 
 and c are proportionals, that is, that a is the same multiple-part of of 
 
 * 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 teen called arithmetical. 
 
106 PRINCIPLES OF ARITHMETIC. § 178-182. 
 
 which b is of 0. 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, 6, c, and rf, being proportional in the order 
 written, a and d are called the extremes, and b and c 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 axe similar, and so are 6. and c; while a and 6, a and i/, 
 d and 6, d and c, are dissimilar. It is customary to express the pro- 
 portion by placing dots between the numbers, thus : 
 
 a : b 11 I 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 = g, a+p = 6+7, Or-p — b—q, ap — 
 a b , ap a 
 
 bg, and - = -. It is also evident, that a+p—p, a—p+p, — , and -xja, are 
 p q P P 
 
 all equal to a. 
 
 181. The product of the extremes is equal to the product of the 
 
 means. Let 7 = -}? and multiply these equal numbers by the product 
 
 bd. Then, jxbd = ^ ( 1 16) = arf, and -xbd =^ = cb: hence (180), 
 b b d d 63 3x7 
 
 ad = be. Thus, 6, 8, ai, and 28, are proportional, since -= = 
 
 21 04 4x7 
 
 = -— (180) ; and it appears that 6x28 = 8x21, since both products are 
 
 25 
 
 168. 
 
 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 p I a :: b : q 
 
 a : g :: p : b p : b :: a : q 
 
 b : J) :: q : a q i a :: h l 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 ofa'.q'.'.plby 
 
 divide both ab and pq by bq. Then, — = -, and 7^ = 7 ; hence (180), 
 
 bq q bq 
 
§ 182-184. ON PROPORTION OF NUMBERS. 107 
 
 - = -^ or a '. g '. '. p '. b. The pupil should not fail to prove every one 
 
 g d 
 
 of the eight cases, and to verify them by some simple examples, such 
 
 as 1x6 = 2x3, which gives i : 2 : : 3 : 6, 3 : i : : 6 : 2, &c 
 
 183. Hence, if four numbers be proportional, they are also propor- 
 tional in any other order, provided it be such that similar terms still 
 
 remain similar. For since, when -= , it follows (181) that ad = be, 
 
 b a 
 all the proportions which follow from ad = be, by the last article, follow 
 
 also from 7 = ;}• 
 
 184. From (114) it follows that 1+- = -7—, and if 7 be less than i, 
 
 Ob b 
 
 a b — a a a a — b 
 
 1—7 = — — , whileif7 be greater than i, 7 — 1 = -r-. Also (122), if 
 b b b b b 
 
 -7— be divided by ■—— the result is — -. Hence, a, b, c, and d, being 
 b h a—b 
 
 proportionals, we may obtain other proportions, thus : 
 a c 
 
 Then (114) 1+2=. t+3 
 b d 
 
 a+b c+d 
 or a+b lb'.', c+d '. d 
 
 That is, the sum of the iirst 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 c 
 
 a c .„a , , 
 
 i_ = i__, if- be less than i, 
 b d o 
 
 b—a d—c 
 
 •"— = -r 
 
 tliat is, b—a '. b :'. d—c '. d 
 
 a 
 or, a—b '. b ','. c—d '. d, if - be greater than i. 
 
108 TRINCIPLES OF ARITHMETIC. § 184-187 
 
 An.o« • ^+* c+d , a—b c—d /a , . , , 
 
 Ago. n, since — - = __ and — - = — -- I - being gi eater than i) 
 o a a \o 
 
 dividing the first by the second we have = , 
 
 a— i c—d 
 
 or a+b : a—b '. '. c+d '. c—d 
 and also a+b : b-a '. : c+d .* rf— c, if- be less than i. 
 
 b 
 
 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. 
 
 a+b la : : c+d : c 
 
 a '. a—h '.'. c '. c—d 
 a+c : a—c '.'. l+d '. b—d. 
 
 In these and all others it must be observed, that when such expressions 
 as a—b and c—d occur, it is supposed that a is greater than 6, 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 aie 
 proportional. Thus, if o : 6 : : c I </, and m and n be any two num- 
 bers, we have also the following : 
 
 ma '. b '.'. mc '. d 
 
 a '. mb '.', c '. md 
 
 — \ mb : : - : md 
 n n 
 
 and various others. To prove any one of these, recollect that nothing 
 more is necessary to make four 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 
 
 Mad ,..,,,„, . , c wAc T> ^ . ,^ 
 
 , while that of the means is mbx -, or . But since a : A : : c : rf, 
 
 bv (181) ad — hCy whence, by (180), mad = wjic, and = . Hence, 
 
 a c n n 
 
 - , wii, - , and mrf, are proportionals. 
 n n 
 
 187. If the terms of one proportion be multiplied by the terms of a 
 
 lecond, the products are proportional ; that is, if a : & : : c : </, and 
 
 ma '. 
 
 nb : 
 
 : mc : 
 
 : nd 
 
 a 
 
 b 
 
 c 
 
 d 
 
 — 1 
 
 — * 
 
 * — ] 
 
 \ — 
 
 m 
 
 m 
 
 m 
 
 m 
 
 a 
 
 b . 
 
 , c ^ d 
 
 m 
 
 m 
 
 n 
 
 n 
 
§ 187- 19°' °^ PROPORTJON OF NUMBERS. 109 
 
 p : q ','. r I s, it follows that ap I bq 11 cr '. ds. For, since ad ■» 
 be, and ps = jr, by (180) adps = bcqTf or apxds = bq^cr, whence (182) 
 ap : bq :: cr : ds. 
 
 188. If four mirabere be proportional, any similar powers of these 
 numbers are also proportional ; that is, if 
 
 a '. b :: c : d 
 
 Then aa '. bb '.'. cc '. dd 
 
 aaa '. bbb ', \ ccc '. ddd 
 &c. &c. 
 
 For, if we write the proportion twice, thus. 
 
 
 a : 
 
 b :: 
 
 c : 
 
 d 
 
 
 a '. 
 
 b :: 
 
 c : 
 
 d 
 
 by (187) 
 
 aa : 
 
 bb :: 
 
 cc : 
 
 dd 
 
 But 
 
 a : 
 
 b :: 
 
 c : 
 
 d 
 
 lence (187) 
 
 aaa '. 
 
 bbb :: 
 
 : ccc '. 
 
 ddd 
 
 189. An expression is said to be homogeneous with respect to any 
 two or more letters, for instance, a, 6, and c, when every term of it 
 contains the same number of letters, counting a, 6, and c only. Thus, 
 maab+nabc-irccc is homogeneous with respect to a, i, and c ; and of the 
 third degree, since in each term there is either a, i, and c, or one of 
 these repeated alone, or with another, so as to make three in all. Thus, 
 Suaabc, izabccc, maaaaa, naabbc, are all homogeneous, and of the fifth 
 degree, with respect to a, 6, 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 ». 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 extra.no is equal to the product of the means, is another. 
 
 I. 
 
110 PRINCIPLES OF ARITHMETIC. § 190. 
 
 a and 5, 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 Avill be proportional. For example, if 
 a I b '.: c : d, and if zaaa+^aab and bbb+abb be chosen, which are both 
 homogeneous with respect to a and i, and both of the third degree ; and 
 if the corresponding expressions zccc+lccd and ddd+cdd be formed, which 
 are made from e and d precisely in the same manner as the two former 
 ones from a and 6, then will 
 
 zaaa-Viaab '. bbb+abb 1 1 zccc+'^ccd I ddd+cdd 
 
 To prove this, let 7 be called a*. Then, since - = jr, and 7 = -, it 
 ^ b b b d 
 
 follows that -^ — ^- But since a divided by h gives x, x multiplied 
 by b will give a, or a = bx. For a similar reason, c = dx. Put bx 
 and dx instead of a and c 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, 
 Ixhxbx is the same as bbbxxx. 
 
 Hence, zaaa+iaab = 7,hxbxbx\-ibxbxh 
 
 = •T.bbbxxx+'^bbbxx 
 which is Ibb multiplied by zxxx+^xx 
 
 or bbb (jzxxx+ixx)* 
 
 Similarly, zccc+iccd = ddd {zxxx+'r^xx) 
 
 Also. bbb-k- abb = bbb+bxbb 
 
 = bbb multiplied by 1+* 
 or bbb (i+x) 
 Similarly, ddd+cdd = ddd {j+x) 
 
 Now, bbb : bhb :: ddd: ddd 
 
 Whence (186), bbb(zxxx+2xx) : bbh{i+x) '.'. ddd(zxxx+^xx) I ddd 
 ( i+x), which, when instead of these expressions their equals just found 
 are substituted, becomes zaaa+'^aab '. bbb+abb 11 zccc+iccd '. ddd+cdd. 
 
 » If bx be substituted for a In any expression which is homogeneous with re- 
 spect 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+-ab becomes bxhx+bxb 
 ot bb{xx+-x)', aaa-+bbb hecom^i hxbxbx+-bbb ox bbb{xxx+\)', and so on. 
 
§ 190-I92. ON PROPORTION OF NUMBERS. Ill 
 
 The same reasoning may be applied to any other case, and the pupil 
 may in this way prove the following theorems : 
 
 If a \ b :: c : d 
 
 za-rzb lb:: zc+id : d 
 
 aa+bb I aa—bb : : cc+dd I cc—dd 
 
 mab : zaa+bb : : mcd : 2cc+dd 
 
 191. If the two means of a proportion be the same, that is, if 
 a : b :: b : c, the three numbers, a, i, 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 
 
 1 2 4. 8 16 3a 64 &c. 
 
 2222 2 2 
 
 2 _ ^ _ -^ Sec. 
 
 3 9 27 81 243 729 
 
 Which are in continued proportion, since 
 
 1 : 2 :: 2 : 4 
 
 2 : 4 :: 4 : 8 
 
 2 2 2 
 
 2 : - ;; - : - 
 
 3 3 9 
 222^ 2 
 
 3 • 9 •• 9 • 27 
 &c. &c. 
 
 192. Let a, i, c, </, &c. be in continued proportion ; we have then 
 
 a b 
 a : b :: b : or t = - or ac = bb 
 b o 
 
 b 
 
 b : :: c : d ... - = - ... bd = co 
 
 d 
 
 c d 
 c : d :: d : e ... — = - ... ce = dd 
 
 d e 
 
 Each term is formed from the preceding, by multiplying it by the same 
 
 _, , b /,„»v c , , . a b b 
 
 number. Thus, b = -xa (180); c = 7x6; and smce - = -, - = - 
 
 b °' d d " . b f^ c a b 
 
 or c = -x5. Again, d = -xc, but - = -, which is = - ; therefore, 
 
 f^ a c ^ c b a 
 
 d = -xc, and so on. If, then, - (which is called the common ratio of 
 
 a a 
 
 the series) be denoted by r, we have 
 
 b — ar c = br — arr d == cr = arrr 
 
 and so on ; Avhence the series 
 
112 PRINCIPLES OF ARITHMETIC. § 192-19}. 
 
 a 
 
 b ^ e d 
 
 &c. 
 
 is a 
 
 ar art arrr 
 
 &c. 
 
 Hence 
 
 a '. '.'. a ', arr 
 
 
 (186) 
 
 '.I aa '. aarr 
 
 
 : : aa : Ib 
 because, b being ar, bb is arar or aarr. Again, 
 aid','. a ', arrr 
 (186) r: aaa '. aaarrr 
 
 :: aaa '. hbb 
 Also ale'.', aaaa ', bbbb, and so on ; 
 
 that is, the first bears to the n*** term from the first the same proportion 
 as the n'** 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, 
 provided we afterwards subtract or add the same. For example, 
 
 p = p—q+q—r+r—s-i^s 
 Let us take four terms of the series, i, r, rr, &c. or, 
 
 i+r+ri'+rrr 
 It is plain that 
 
 rrrr—i = rrrr—rrr+rrr—rr-i-r7-—r+r—i 
 Now (54), ?-r— r = r(r— i), rrr—rr = rr{r—i), rrrr—rrr = rrr (r—i\ 
 and the above equation becomes rrrr—i = rrr (r— i) + rr (r— i) + r (r— i) 
 +r-i ; which is (54) rrr+rr-^r+i taken r— i times. Hence, rrrr—i 
 divided by r—i will give i+r+rr-frrr, the sum of the terms required. 
 In this way may be proved the following series of equations : 
 
 rr—i 
 i+r = 
 
 i+r+rr 
 
 i-\^r-\rrr-irrrT 
 
 r—i 
 •1 
 
 i-TT-^TT-^rrr+rrrr = 
 
 r-i 
 
§ I93-J94* O^ PROPORTION OP NUMBERS. 113 
 
 If r be less than unity, in order to find J+r+rr+rrr, observe that 
 
 i—rrrr = j—r+r—rr+rr—rrr+rrr—rrrr 
 
 = I— r-fr(i— r)+j'r(i— r)+rrr(i— r) ; 
 
 W'hence, by similar reasoning, j+r+rr+rrr is found by dividing i—rrrr 
 by I— r ; and equations similar to these just given may be found, 
 which are. 
 
 i+r 
 
 = 
 
 i—rr 
 i—r 
 
 i+r+rr 
 
 = 
 
 i-rrr 
 
 i—r 
 
 i+r+rr+rrr 
 
 = 
 
 I—rrrr 
 
 I—r 
 
 i+r+rr-^rrr+rrrr 
 
 __ 
 
 i—rrrrr 
 
 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+i)*'' term by the diff*erence 
 between i and r. 
 
 194. This may be applied to finding the sum of any number of 
 terms of a continued proportion. Let a, 6, c, &c. be the terms of which 
 it is required to sum four, that is, to find a+bi-c+d, or (192) a+ar 
 +arr+arrr, or (54) a{i +r+rr+rrr), which (193) is xa, or 
 
 r— I i-r 
 
 xa, according as r is greater or less than unity. The first fraction is 
 
 , or (192) . Similarlv, the second is . The rule, 
 
 r— I ^ ' r—i ' I—r 
 
 therefore, is : To sum n terms of a continued proportion, divide the 
 
 difference of the 7i+i^^ and first terms by the difference between unity 
 
 and the common measure. For example, the sum of lo terms of the 
 
 series 1+3+9+27+&C. is required. The eleventh term is 59049, and 
 59049 — I 
 
 is 29524. Again, the sum of 18 tenns of the series 2+1+ 
 
 3-1 
 
 II ,1 131072 
 
 -+ — h&c. of which the nineteenth terra is , is 
 
 2 4 131072 ^_i 
 
 1 3 1070 * 
 
 31072 
 
 EXAMPL"?. 
 
 9 terms of 1+4+ 16 + &c. aie 87381 
 l2 
 
114 PRINCIPLES OF ARITHMETIC. § I94-J97. 
 
 6 12 „ 847422675 
 
 lo terms of ;: + -+ — +&c. are 73 — - 
 
 •' 7 49 201768035 
 
 1 I I ^ 1048575 
 
 ^° rrs^'^" - ^^J8i76 
 
 ly5. The powers of a number or fraction greater than unity increase; 
 
 for since 2- is gi-eater than 1, 2- x 2- is 2- taken more than once, that 
 
 is, is greater than a-, and so on. This increase goes on without limit ; 
 
 2 1 
 
 that is. there is no quantity so great but that some power of 2- is greater. 
 
 I ^. 
 
 To prove this, observe that every power of 2- is made by multiplying the 
 
 1 I ^ 
 
 preceding power by 2-, or bv 1+1-, that is, by adding to the former 
 
 2 ' 2 
 
 power that power itself and its half. There will, therefore, be more 
 
 added to the loth power to form the nth, than was added to the 9th 
 
 power to form the loth. But it is evident that if any given quantity, 
 
 however small, be continually added to 2 , the result vnll come in time 
 
 2 
 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 
 
 ^ 1 
 each step, which is the case when the successive powers of 2- are formed. 
 
 It is evident, also, that the powers of i never increase, being always i ; 
 
 thus, ixi = I, &c. Also, If a be greater than m times 6, the square of a 
 
 is greater than mm times the square of b. Thus, if a = 26+c, where 
 
 a is greater than 26, the square of cr, or aa, which is (68) ^b+^-bc+cc 
 
 is greater than 4ii, 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. 
 
 This decrease continues without limit ; that is, there is no quantity so 
 
 2 521 
 
 small but that some power of - is less. For if - = ;p, - = -, and the 
 
 211 5 2 ' 5 * 
 
 powers of - are — , , and so on. Since or is greater than i (195), 
 
 some power of d? may be found which shall be greater than a given 
 
 I 2 
 
 quantity. Let this be called m ; then — is the corresponding power of- ; 
 
 m 5 
 
 and a fraction whose denominator can be made as great as we please, can 
 
 itself be made as small as wo please (112). 
 
 197. We have, then, in the series 
 
 1 r rr rrr rrrr &c. 
 
 I. A series of increasing terms, if r be greater than i. II. Of terms 
 
g loy. ON PBOPORTION OF NUMBERS. 115 
 
 having the same value, if r be equal to i. III. A series of decreasing 
 terms, if r be less than i. In the first two cases, the sum 
 
 1+r+rr+rrr+ik.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 augment- 
 ation diminishes at every step, we may not certainly say that we can, 
 by any numbjr of such augmentations, make the result as great as we 
 please. To shew the contrary in a simple instance, consider the series, 
 
 I I I I n 
 
 1+- +-+- + — + &c. 
 2 4 8 i6 
 
 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 a. Thus, 
 
 I I 
 
 I + -+- =1+1 — 2, 
 
 / I I\ I II 
 
 ( 1+- + -)+-= I + -+- 
 
 \ a 4/ 4 2 2 
 
 / I 1 i\ 1 
 
 I 1+-+- + )+t; = ^• 
 \ 2 4^8 
 
 H—f - + -+—) + — = 2, &C. 
 
 2 4 8 i6/ i6 
 
 But in the series, every term is only the half of the preceding ; con- 
 sequently no number of terms, however great, can be made as great 
 
 as 2 bv adding one more. The sum, therefore, of i, -, -, -, &c. con- 
 
 ' 24^ 
 
 tinually approaches to 2, diminishing its distance from 2 at every step, 
 
 but never reaching it. Hence, 2 is called the limit of 1+- •r- + &c. 
 
 2 4 
 
 We are not, therefore, to conclude that everp series of decreasing terms 
 has a limit. The contrary may be shewn in tlie very simple series, 
 
 n — +- + — h&c. which may be written thus: 
 
 234 
 
 [-r-+( -+-)+(-+•• -up to- )+( +...up to — T j+l — I-...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 deno- 
 
 minator of its last term. Thus, the fourth lot contains 16 or -^ terms. 
 
 I ^ 
 
 Eacli of these lots may be shewn to be greater than -. Take the third, 
 
 2 
 
ilG PRINCIPLES OP ARITHMETIC. §'97-199. 
 
 for example, consisting of -, — , - , — , — , — , — , and — . All except 
 ^ ** 9 lo II 12 13 14 15 16 
 
 — , the last, are greater than -7; consequently, by substituting — ^ for 
 10 ID 10 
 
 each of them, the amoimt of the whole lot would be lessened ; and aa 
 
 81 I 
 
 it would then become — r, or -, the lot itself is greater than -. Now, 
 
 II 16 2 a 
 
 if to 1+-,- be continually added, the result will in time exceed any 
 
 2 2 
 
 given number. Still more will this be the case if, instead of -, the 
 
 2 
 
 several lots written above be added one after the other. But it is thus 
 
 that the series 1 +-+-, &c. is composed, which proves what was said, 
 
 2 3 
 that this series has no limit. 
 
 198. The series i+r+rr+rrr+&c. always has a limit when r is less 
 
 than I. To prove this, let the term succeeding that at which we stop 
 
 be a, whence (194) the sum is , or (112) . The terms 
 
 ' 1— r I— r I— r 
 
 decrease without limit (196), whence we may take a term so far distant 
 
 from tlie beginning, that a, and therefore , shall be as small as we 
 
 I— r 
 
 please. But it is evident that in this case , though always 
 
 I — r I — r 
 
 less than , may be brought as near to as we please ; that is, the 
 
 I — r I— r J 
 series i+r+7T+&c. continually approaches to the limit . Thus 
 
 1+ — h-+-+&c. where r = -, continuallv approaches to — - or 2, as 
 248 2 . rr ,_^ 
 
 was shewn in the last article. 
 
 + &C. 
 
 + ^ +&C.) is 3 
 + &c. ... 10 
 
 EXERCISES. 
 
 of 2 + 
 
 2 
 
 - + 
 
 3 
 
 2 
 9 
 
 2(14 
 
 I 
 
 - + 
 3 
 
 I 
 9 
 
 . 1 + 
 
 10 
 
 81 
 
 100 
 
 5+^H ^+&c. ... 8^ 
 
 7 49 
 
 199. "When the fraction - is not equal to -,but greater, a is said to 
 o " a c 
 
 have to J a greater ratio than has to d ; and when - is less than -, a 
 
 It 
 
 is said to have to i a less ratio than c has to J. We propose the fol- 
 
§199-202. ON PROPORTION OF NUMBERS. 117 
 
 lowing questions as exercises, since they follow very simply from this 
 definition, 
 
 I. If a be greater than A, and c less than or equal to rf, a Avill 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 6 as c is to rf, and if a have a greater ratio to b than c 
 has to a?, d is less than x ; and if a have a less ratio to b than c to .r, d is 
 greater than of. 
 
 IV. a has to 6 a greater ratio than ax to hx-Vy^ and a less ratio than 
 ax to bx—y. 
 
 200. If a have to 6 a greater ratio than c has to rf, a+c has to b+d a 
 
 less ratio than a has to 6, but a greater ratio than c has to d; or, in 
 
 other words, if - be the greater of the two fractions - and -, will be 
 
 b ° b d b+d 
 
 greater than -, but less than -. To shew this, observe that — must 
 
 d b m+n 
 
 lie between x and y, if j? and y be unequal : for if j? be the less of the 
 
 two, it is certainly greater than or than x ; and ify be the greater 
 
 m+n 
 
 7W V+7iV 
 
 of the two, it is certainly less than , or than ;/. It therefore lies 
 
 a "'-^"' c 
 
 between x and y. Now let - be x, and let - be y : then a = bx^ c = dy. 
 
 bx+dy " "■ 
 
 Now is something between x and y, as was just proved ; therefore 
 
 u+c , ^+" ,, . , , a n , a , c 
 
 r— - IS somethmg between - and -. Again, smce 7 and - are respectively 
 b+d ap cq " ^ b d ap+co 
 
 equal to -— and — -, and since, as has just been proved, -r- — r lies be- 
 ^ bp dfj ' •' r > bp^dq 
 
 tween the two last, it also lies between the two first; that is, if/) and 
 
 Q be any numbers or fractions whatsoever, -■ lies between - and -. 
 
 ^ * bp+dq b d 
 
 201. By the last article we may often form some notion of the value 
 
 of an expression too complicated to be easily calculated. Thus, 
 
 1- , . I J -^ J ' ax+by .. . , ax .by 
 
 lies between - and — , or i and - ; — — lies between and — - — , 
 
 I XX X axx+bbyy axx bbyy 
 
 1 I a+b 
 
 that is, between - and --. And it has been shewTi that lies between 
 
 X by 2 
 
 a and J, the denominator being considered as i+i. 
 
 202. It may also be proved that a fraction such as always 
 
 a b c d . . p+q+r+s 
 
 lies among -, -, -, and -, that is, is less than the greatest of them, and 
 
 p q r s 
 
 greater than the least. Let these fractions be arranged in order of 
 
I|8 PfilNCIPLES OP ARITHMETIC. § 202-204. 
 
 magnitude ; that is, let - be greater than -, - be greater than -, and - 
 d P 9 9 r r 
 
 greater than -. Then by (200) 
 
 a+b 
 
 P+9 c 
 a+b+o 6 
 
 a 
 
 
 b , C 
 
 
 j3 
 
 - and - 
 
 P 
 
 1 
 
 q r 
 
 a+b , a 
 
 ht 
 
 c , d 
 
 and - 
 
 ■M 
 
 - and - 
 
 p+q p 
 
 S) 
 
 r s 
 
 a+b+c a 
 
 •a 
 
 d 
 
 and - 
 
 rt 
 
 
 p+q+r p 
 
 
 s 
 
 p+q+r S 
 
 01 
 
 a+b+c+d 
 p+q+r+s 
 
 whence the proposition is evident. 
 
 203. It is usual to signify " a is greater than 6" by a > 5, and " a is 
 less than 6" by a<b; the opening of A being turned towards the greater 
 quantity. The pupil is recommended to make himself familiar vnih 
 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 qi;e8tion 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, 6, c, rf, e, and/; the combinations of three which can be made out 
 of them are twenty in number, as follow : 
 
 abo 
 
 ace 
 
 bed 
 
 bef 
 
 abd 
 
 acf 
 
 bee 
 
 cde 
 
 abe 
 
 ade 
 
 bef 
 
 edf 
 
 abf 
 
 adf 
 
 bde 
 
 cef 
 
 acd 
 
 aef 
 
 bdf 
 
 def 
 
 The combinations of four are fifteen in number, namely. 
 
§ 2C4-2C6. PERXIUr^TIONS AND COMBINATIONS. 119 
 
 abed 
 abce 
 abcf 
 
 rnd so on. 
 
 205. Each of these combiuations may be written in several different 
 orders ; thus, abed may be disposed in any of the following ways : 
 
 abde 
 
 acde 
 
 adef 
 
 beef 
 
 abdf 
 
 acdf 
 
 bade 
 
 bdef 
 
 ahef 
 
 acef 
 
 bcdf 
 
 cd^ 
 
 abed 
 
 acbd 
 
 aedb 
 
 abde 
 
 adbe 
 
 adeb 
 
 bacd 
 
 cabd 
 
 cadb 
 
 bade 
 
 dabc 
 
 dacb 
 
 bead 
 
 cbad 
 
 cdab 
 
 bdac 
 
 dbac 
 
 dcab 
 
 bcda 
 
 cbda 
 
 cdba 
 
 bdea 
 
 dbca 
 
 dcba 
 
 of which no two are entirely in the same order. Each of these is said 
 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 con- 
 taining one given number, can be made from the counters in another 
 given number, six, for example. If we knew how to find all the per- 
 mutations containing four counters, we might make those which contain 
 five thus : Take any one which contains four, for example, abcf, in which 
 d and e are omitted ; write d and e successively at the end, which gives 
 abvfd, abcfe, and repeat the same process with every other permutation 
 of four; thus, dabc gives dabee 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 db/e, 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. If wo 
 begin in this way to find the permutations of two out of the six, 
 
 a b c d e f 
 
 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. 
 
120 PRINCIPLES OF ARITHMETIC. § 206-207. 
 
 Again, ab gives abc abd abe ahf 
 ac ach acd ace acf 
 
 and here are 30, or 6x5 permutations of 2, each of which gives 4 
 permutations of 3 ; the whole number of the last is therefore 6x5x4, 
 or 120. 
 
 Again abc gives abed abce abcf 
 
 abd abdc abde abdf 
 
 and here are 120, 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 6x5x4x3x2, 
 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, j?, the number of permutations of 
 two will be x{x—\\ that of three will be x{a!—i){x—i.\ that of four 
 x{jx—\){x—%\x—i)^ 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 12x11x10x9, or 
 1 1880. 
 
 EXERCISES. 
 
 207. In how many different ways can eight persons be arranged on 
 eight scats ? 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 diffbrence between th*s problem and the last is left to the ingenuity of 
 the pupil. 
 
§207-209. PERMUTATIONS AND COMBINATIONS. 121 
 
 arrangement which can be made of fifteen persons, to how much wiU 
 
 the whole amount? Answer, £13621608. 
 
 Out of seventeen consonants and five vowels, how many words can 
 
 be made, having two consonants and one vowel in each ? Answer, 4080. 
 
 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, ahca'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 3x2x1 distinct per- 
 mutations, by filling up the vacant places in the above, all Avhich six 
 are the same when the as are not distinguished. Hence, to deduce 
 the number of permutations of a, a, a, b, c, d, from that of aa'a"bcd, 
 
 OX CX^X'^X2X I 
 
 we must divide the latter by 3x2x1, or 6, which gives — ^—^^ 
 
 3x2x1 
 
 or 120. Similarly, the number of permutations of aaaalbbcc is 
 
 9x8x7x6x5x4x3x2x1 
 
 4x3x2x1x3x2x1x2x1* 
 
 EXERCISE. 
 
 How many variations can be made of the order of the letters in the 
 word antitrinitarian ? Answer, 1261260CO. 
 
 209. From the number of permutations we can easily deduce the 
 number of combinations. But, in order to form these combinations 
 independently, we Avill 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 ac, which, according 
 to our rule, furnishes acd. Neither will any combination be repeated 
 
122 PRINCIPLES OF ARITHMETIC. §209-210. 
 
 twice, for acd^ 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 b c d e 
 
 a gives ab ac ad ae 
 
 b be bd be 
 
 cd ce 
 
 d de 
 
 Of these, ab gives abc aid abe 
 
 ac acd ace 
 
 ad ade 
 
 be bed bee 
 
 bd bde 
 
 cd cde 
 
 ae be ce and de give none. 
 
 Of these, ahe gives abed abee 
 
 abd abde 
 
 acd aede 
 
 bed bcde 
 
 Those which contain e give none, as before. 
 
 Of the last, abed gives abede, 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 de- 
 rived directly from that for the number of permutations. Take 7 
 ounters ; then, since the number of permutations of two is 7x6, and 
 
 since two permutations, ba and ai, are in any combination ai, the 
 
 7x6 
 number of combinations is half that of the permutations, or - — , 
 
 2 
 
 Since the number of permutations of three is 7x6x5, and as each 
 combination abc has 3x2x1 permutations, the number of combina- 
 tions of three is . Also, since any combination of four, abcd^ 
 
 1x2x3 
 
 contains 4x3x2x1 peimutations, the number of combinations of four 
 
 is — , and so on. The rule is : To find the number of com- 
 
 1x2x3x4 
 
 binations, each containing n counters, divide the corresponding number 
 
§210-211. PERMUTATIONS AND COMBINATIONS. 123 
 
 of permutations by the product of i, 2, 3, &c. up to n. If x be the 
 
 whole number, the number of combinations of two is : that of 
 
 1x2 
 
 three is -^^ ; that of four is -^^ — — ; and so on. 
 
 1x2x3 1x2x3x4 
 
 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 
 , 10x9x8x7x6x5x4 . , . , , 
 
 out of ten IS :; — , m which the product 7x6x5x4 occurs 
 
 1x2x3x4x5x6x7 
 
 in both terms, and therefore may be removed from both (108), leaving 
 
 — , which is the number of combinations of three out of ten. The 
 
 1x2x3 
 
 same may be shewn in other cases. 
 
 EXERCISES. 
 
 How many combinations of four can be made out of twelve things ? 
 
 Answer^ 495. 
 What number f ^ 1 f ^ 1 { ^^ 
 
 of combinations { ^6 T °^* °^ 1 28 i Answer^ I ^^ ^ 
 
 can be made of |_ 6 J I 15 J L 5005 
 
 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. 
 
121 COMMERCIAL ARITHMETIC. 
 
 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 which have been 
 explained in the preceding book. But there is still one thing wanted 
 — not 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 
 siigle unit (15), and have troated 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 (i 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 con- 
 tinued ever so far, one of the points of division must at last fall exactly on A : neither 
 would the same appear necessarOy true if the division were made mto 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 
 
§ 212. WEIGHTS AND MEASURES. 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 -77- of a mile, and of another of a 
 
 " 180 174 
 
 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, 
 
 7 
 we shall find that the length of the first room is 9 yards and - of a yard, 
 
 10 9 
 
 and that of the second 10 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 accoimt of the fractions - and --. To get a clearer 
 
 , . 9 , ^7 
 idea of these, suppose the yard to be divided into three equal parts, and 
 
 7 I 
 
 each of these parts to be called a foot ; then - of a yard contains 2- 
 
 10 30 9 3 
 
 feet, and —- of a yard contains -- of a foot, or a little more than 
 
 1 87 87 
 
 - of a foot. Therefore the length of the first room is now 9 yards, 
 
 2 feet, and - of a foot ; that of the second is 10 yards and a little more 
 
 I 3 
 than - of a foot. We see, then, the convenience of having large mea- 
 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. Wliat is meant in the 
 words on which this note is written, is, that any part of a foot can be represented as 
 nearly as we please by a numerical fraction of it ; and this is sutficient f^r practical 
 purposes. 
 
 m2 
 
126 COMMEECIAL ARITHMETIC. § 212-213. 
 
 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 
 
 365-24224 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, 
 
 which gives what we call leap-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 
 
 above 365 days is not '25 but '24224 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 128 years, or to about 3 days in 4 
 
 centuries. 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. 1800 and 
 
 1 900 are not leap-years, but 2000 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 60 parts or minutes, 
 
 and each of these again into 60 parts or seconds. One second, marked 
 
 thus, i»,t is therefore the 86400*'' 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. 
 
2 T 3-2 14. 
 
 WEIGHTS AND MEASURES. 
 
 127 
 
 MEASURE OF TIME.* 
 
 
 are 
 
 I minute . 
 
 , . im. 
 
 . . 
 
 I hour , . . 
 
 . ih. 
 
 . . 
 
 1 day . , 
 
 . . Id. 
 
 . . 
 
 1 week . . 
 
 . I wk 
 
 . . 
 
 I year . . 
 
 . . I yr. 
 
 6o seconds 
 6o minutes 
 24 hours . 
 7 days 
 365 days 
 
 214. The second having been obtained, a pendulum can be con- 
 structed which shall, when put in motion, perform one vilration in 
 exactly one second, in the latitude of Greenwich. + 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. 
 The following is the division of the yard. 
 
 MEASURES OF LENGTH. 
 
 The lowest measure is a barleycorn.^ 
 
 I inch 
 
 3 barleycorns are 
 
 12 inches . . . . . 
 
 3 feci , 
 
 5- yards 
 2 
 
 40 poles or 220 yards . . 
 8 furlongs or 1760 yards . 
 
 \ foot . 
 I yard . 
 I •pole . 
 I furlong 
 I mile . 
 
 1 in. 
 1 ft. 
 1 yd. 
 I po. 
 I fur. 
 I mi. 
 
 * ITie measures in italics are those which it is most necessarj- that the student 
 should learn by heart. 
 
 + The lengths of the pendulums which will vibrate in one second are slightly dif- 
 ferent in different latitudes. Greenwich is chosen as the station of the Royal Ob- 
 servatory. We may add, that much doubt ia now entertained as to the system of 
 standards derived from nature being capable of that extreme accuracy which was 
 once attributed to it. 
 
 X The inch is said to have been originally obtained by putting together three grains 
 of barley. 
 
128 
 
 COMMERCIAL ARITHMETIC. 
 
 § 214-Z16. 
 
 I nail 
 
 . I nl. 
 
 I quarter (of a yard) 
 
 . iqr. 
 
 I Flemish ell . . 
 
 I Fl. e. 
 
 1 English ell . . 
 
 I E. e. 
 
 I French ell . . 
 
 I Fr. e. 
 
 Also 6 feet i fathom . i fth. 
 
 69- miles I degree . i deg. or 1®. 
 
 A geographical mile is ^^-th of a degree, and three such miles are one 
 
 60 
 
 nautical league. 
 
 In the measurement of cloth or linen the following are also used : 
 
 2— inches arc 
 4 
 
 4 nails . . . 
 
 3 quarters . . 
 
 5 quarters . . 
 
 6 quarters . . 
 
 215. 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. 
 
 I square foot . 1 sq. ft. 
 
 I square yard . i sq. yd. 
 
 I square pole . i sq. p. 
 
 I rood ... I rd, 
 
 4 roods I acre ... i ac. 
 
 Thus, the acre contains 4840 square yards, which is ten times a 
 square of 22 yards in length and breadth. This 22 yards is the length 
 which land-surveyors' chains are made to have, and the chain is divided 
 into 100 links, each '22 of a yard or 7*92 inches. An acre is then 10 
 
 square chains. It may also be noticed that a square whose side is 
 
 4 1 
 
 69- yards is nearly an acre, not exceeding it by - of a square foot. 
 
 144 square inches are 
 
 9 square feet . . . 
 
 30- square yards . . 
 
 40 square poles . . 
 
 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. 
 
§216-217. WEIGHTS AND MEASURES. 129 
 
 1728 cubic inches are 1 cubic foot . . i c. ft. 
 27 cubic feet . . . i cubic 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 are 
 
 I pint . . 
 
 . ipt. 
 
 2 pints . . 
 
 1 quart . . 
 
 . iqt. 
 
 4 quarts . 
 
 I gallon 
 
 . I gall 
 
 a gallons . . 
 
 I peck* 
 
 . ipk. 
 
 4 pecks . 
 
 . I bushel 
 
 . I bu. 
 
 8 bushels . 
 
 I quarter . 
 
 . iqr. 
 
 5 quarters 
 
 1 load . . 
 
 . lid. 
 
 The gallon in this measure is about 277*274 cubic inches; that is, 
 
 very nearly 277- cubic inches.+ 
 4 
 217. The smallest weight in use is the grain, which is thus deter- 
 mined. A vessel whose interior is a cubic inch, when filled vnth 
 water,t has its weight increased by 252*458 grains. Of the grains so 
 determined, 7000 are a pound averdupois, and 5760 a pound troi/. The 
 
 * This measiure, 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 1 peck 
 
 4 pecks 1 bushel 
 
 3 busliels 1 sack 
 
 12 sacks 1 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." 
 
 t Pure water, cleared from foreign substances by distillation, at a temperature of 
 62» Fahr. 
 
130 COMMERCIAL ARITHMETIC. §217-218. 
 
 first pound is always used, except in weighing precious metals and 
 stones, and also medicines. It is divided as follows : 
 
 AVERDUPOIS WEIGHT. 
 
 27 — grains are i dram 1 dr. 
 
 16 drams, or drachms . . 1 ounce* . . . . i oz, 
 
 16 ounces i pound .... i lb. 
 
 28 pounds 1 quarter . . ♦ . i qr. 
 
 4 quarters i hundred-weight , i cwt, 
 
 20 hundred-weight . . i ton i ton. 
 
 The pound averdupois contains 7000 grains. A cubic foot of watei 
 weighs 62*3210606 pounds averdupois, or ^^j'l-^SgB^i ounces. 
 
 For the precious metals and for medicines, the pound troy, con- 
 taining 5760 grains, is used, but is differently divided in the two cases. 
 The measures are as follow : 
 
 TROY WEIGHT. 
 
 24 grains are i pennyweight . . . i dwt. 
 
 XQ pennyweights . . . i ounce i oz. 
 
 12 ounces i pound i lb. 
 
 The pound troy contains 5760 grains. A cubic foot of water weighs 
 75' 7 374 pounds troy, or 908*8488 ounces. 
 
 apothecaries' weight. 
 
 20 grains are i scruple 3 
 
 3 scruples . . . . i dram 5 
 
 8 drams .... i ounce 5 
 
 12 ounces i pound ft 
 
 218. The standard coins of copper, silver, and gold, are, — the penny, 
 
 which is 10- drams of copper ; the shilling, which weighs 3 penny- 
 
 3 
 weights 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, 
 
 4 
 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 
 drams. 
 
§ 2J8-2I9. WEIGHTS AND MEASURES. 131 
 
 MEASURES OF SIONEY. 
 
 The lowest coin is a farthing, which is marked thus, -, being one 
 
 4 
 fourth of a penny. 
 
 2 farthings are i halfpenny ~d. 
 
 2 
 
 2 halfpence i penny id. 
 
 12 pence i shilling is. 
 
 20 shillings i pound* or sovereign , £ i 
 
 21 shillings i guinea.f 
 
 219. When any quantity is made up of several others, expressed in 
 
 different units, such as £i . 14. 6, or 2cwt. iqr. 3 lbs., 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 104 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, 
 
 I. How many farthings are there in £18 . 12 . 6- ? J 
 
 4 
 Since there are 20 shillings in a pound, there are, in £18, 18x20, or 
 
 360 shillings; therefore, £18 . 12 is 360+12, or 372 shillings. Since 
 
 there are 12 pence in a shilling, in 372 shillings there are 372x12, 
 
 or 4464 pence ; and, therefore, in £18 . 12 . 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 powwd 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, hut the name is still given, 
 
 from custom, to 21 shillings. The pound, which was not a coin, hut 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. 
 
 J Farthings are never written but as parts of a penny. Thus, thr:e 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. §219. 
 
 4470x4, or 17880 farthings; and, therefore, in £18 . 12 , 6- there 
 
 4 
 are 17880+3, or 17883 farthings. The whole of this process may be 
 
 written as follows : 
 
 £iS . 12 . 6^ 
 
 360+12 = 372 
 12 
 
 44.64+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 12, gives the quotient 372, and the remainder 
 6, 4470 pence is 372 shillings and 6 pence. 
 
 Since 372, divided by 20, gives the quotient 18, and the remainder 
 
 12, 372 shillings is 18 pounds and 12 shillings. 
 
 3 9 
 
 Therefore, 17883 farthings is 4470-</., which is 3725. 6-rf., which is 
 -I 4 4 
 
 £18 . 12 . 6^. 
 4 
 The process may be written as follows : 
 
 4)17883 
 
 12)4470 ... 3 
 
 20)372 ... 6 
 
 £18 . 12 . 6- 
 4 
 
 EXERCISES. 
 
 A has £100 . 4 . 11-, and B has 6439a farthings. If A receive 1492 
 2 1 
 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. 
 
§ 219 222. aEDUCTION. 133 
 
 £iS .18.9- 
 
 2 
 1 1 q"* I"^ gtlwt 
 
 2 bs j^oj gdr 
 
 3™ 149>''* 2^' 9'" 
 j^bii jPits igall ^qt* 
 
 i6'' zs"" 47* 
 
 15302 fhrthings. 
 663072 grains. 
 I CO I drams. 
 195477 inches. 
 
 1260 pints. 
 59027 seconds. 
 
 220. The same may be done where the number first expressed is 
 
 fractional. For example, how many shillings and pence are there in 
 
 4 4 4 4 
 
 — of a pound ? Now, — of a pound is — of 20 shillings ; — of 20 
 
 15 *^ 15 ^ 15 " ' 15 
 
 . 4x20 4x4 16 I I 
 
 18 , OT ^^—^ (HO), or — , or (105) 5- of a shilling. Again, - of 
 
 a shilling is - 0T12 pence, or 4 pence. Therefore, £ — = 5*. 4^. 
 
 Also, "23 of a day is '23x24 in hours, or 5'"52 ; and '52 of an hour 
 is '52x60 in minutes, or 3i"''2; and '2 of a minute is '2x60 in seconds, 
 or 12* ; whence '23 of a day is 5'* 31™ 12*. 
 
 Again, suppose it required to find what part of a pound 6s. Sd. is. 
 
 Since 6s. Sd. is 80 pence, and since the whole pound contains 20x12 
 
 or 240 pence, 6s. M. is made by dividing the pound into 240 parts, and 
 
 taking 80 of them. It is therefore £ (107), but = - (108): 
 
 J 240 ^ ' 240 3 ' ^ 
 
 therefore, 6s. 8c?. = £-. 
 
 EXERCISES. 
 
 - of a day is 9'' 36"* 
 •i284iofaday . . . 3*' 4'" 54* '624* 
 '257ofacwt. . . . 28"'« 12°^ S"''- 704 
 
 £•14936 2« I iJ 3^-3856 
 
 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, 5''5 is five seconds and five-tenths o{ a, second. Thus, 
 0*'5 means five-tenths of a second ; 0^-3, three-tenths of an hour. 
 
134 COMMERCIAL ARITHMETIC, §222-224. 
 
 that Appendix. To prevent the subsequent sections from being altered 
 
 in their numbering, I have numbered this paragraph as above. 
 
 223. The rule of addition* of two compound quantities of the same 
 
 sort will be evident from the following example. Suppose it required 
 
 to add £192 . 14 . 2- to £64 . 13 . II-. The sum of these two is the 
 
 ^ 4 
 
 whole of that which arises from adding their several parts. Now 
 
 h.+ ^. = ^d. = £0.0. I- (219) 
 424 4 
 
 Ilrf.+ 2rf. = 13^. = O.T.I 
 
 13s. + 14s. = 27s. = 1.7.0 
 £64 +£192 = 256 .0.0 
 
 The sum of all of which is £257 . 8 . 2- 
 
 4 
 
 This may be done at once, and written as follows : 
 £192 . 14 . 2 
 
 '^ 64 . 13 . II- 
 
 4 
 
 £257, 8. 2- 
 4 
 
 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 me- 
 mory, the process will be easy . 
 
 224. SuBTRAcnoN 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. Suppose it required to subtract £19 . 13 . i<>5 
 from £24 .5.7-. Write these qu.intities under one another thus : 
 
 • Before redding this article and the next, articles (29) and (42) should be read 
 again carefully. 
 
§224-225. SUBTRACTION. 135 
 
 £24. 5. 7; 
 2 
 
 3 
 
 19 . 13 . 10^ 
 
 4 
 
 3 12 
 
 Since - cannot be taken from - or -, add id. to both quantities, 
 
 .4 =^4 4 , 
 
 which will not alter their difference ; or, which is the same thing, add 
 
 4 farthings to the first, and id. to the second. The pence and farthings 
 
 in the two lines then stand thus : j-d. and 1 1^. Now subtract - 
 
 6 .^44 4 
 
 from -, and the difference is -, which must be written under the 
 
 .4 . . 4- 
 
 fai-things. Again, since i id. cannot be subtracted from ^d., add is. to 
 
 both quantities by adding 12c?. to the first, and is. to the second. The 
 
 pence in the first line are then 19, and in the second 11, and the 
 
 difference is 8, which write under the pence. Since the shillings in the 
 
 lower line were increased by i, there are now 145. in the lower, and 55. 
 
 in the upper one. Add 2cs. to the upper and £1 to the lower line, and 
 
 the subtraction of the shillings in the second from those in the first 
 
 leaves 115. Again, there are now ^"'20 in the lower, and ^^24 in the 
 
 upper line, the difference of which is £^ ; therefore the whole difTerence 
 
 of the two sums is £4. . 11 . 8-. If we write do\vn the two 8uu:s with 
 
 4 
 all the additions which have been made, the process will stand thus : 
 
 6 
 
 £24 . 25 . 19- 
 
 4 
 
 20 . 14 . II- 
 4 
 
 Difference £4. . i: 
 
 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. 2 1 lbs. 14 oz. 
 
 Subtract a cwt. 3 qrs. 27 lbs. 12 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. 12 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. §125-227. 
 
 shortened in the following way. Here we do not subtract 27 lbs. from 
 21 lbs., which is impossible, but we increase 21 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 1 qr. or 28 lbs. and add 
 the difference to 21 lbs. 
 
 226. EXERCISES. 
 
 A man has the following sums to receive: £193 . 14 . 11-, 
 
 3 I 4 
 £22 . o . 6-, £6473 .0.0, and £49 . 14 . 4 ; and the following debts 
 
 4 I 2 I 
 
 to pay: £200 . 19 . 6-, £305 . 16 . 11. £22, and £19 .6.0-. How 
 
 4 ^ 5 
 
 much will remain after paying the debts ? Answer, 166190 . 7 . 4-. 
 
 4 
 There are four to^vns, in the order A, B, C, and D. If a man can 
 
 go from A to B in 5'' 20'" 33*, from B to C in 6i> 49™ 2% and from A 
 
 to D in jg^ o™ 17% how long will he be in going from B to D, and from 
 
 C to D ? Answer, 13'' 39"" 44% and 6'» 5o«n 42". 
 
 227. In order to perform the process of Multiplication, it must 
 be recollected that, as in (52), if a quantity be divided into several 
 puts, and each of these parts be multiplied by a number, and the 
 products be added, the result is the same as would arise from multi- 
 plying the whole quantity by that number. 
 
 It is required to multiply £7 . 13 . 6- by 13. The first quantity ia 
 
 4 
 made up of 7 pounds, 13 shillings, 6 pence, and i farthing. And 
 
 I farth. X 13 is 13 farth. or £0. o . 3- (219) 
 
 4 
 
 6 pence x 13 is 78 pence, or 0.6.6 
 
 13 shill. X13 is 169 shill. or 8.9.0 
 
 7 pounds X 13 is 91 pounds, or 91 . o . o 
 
 
 The sum of all these is 
 
 £99. 
 
 15 
 
 X 
 
 which is 
 
 therefore £7 . 13 
 
 ^-3 
 
 • 
 
 
 
 
 This 
 
 process is usually 
 
 written 
 £7 . 
 
 £99 • 
 
 as follows 
 
 ,5.9; 
 
 
 
 
§ 228. DIVISION. 137 
 
 228. Division is performed upon the same principle as in (74), 
 
 viz. that if a quantity be divided into any number of parts, and each 
 
 part be divided by any number, the diiFerent 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. Since 99 divided by 
 
 4 
 13 gives the quotient 7, and the remainder 8, the quantity is made up 
 
 of £13x7, or £91, and £8 . 15 . 9- The quotient of the first, 13 being 
 
 4 
 the divisor, is £7 : it remains to find that of the second. Since ^8 is 
 
 1605., £S . 15 . 9- is 1755. g-d.^ and 175 divided by 13 gives the quotient 
 
 4 4 . I 
 
 13, and the remainder 6; that is, 175s. g-d, is made up of 1695. and 
 
 1 4 
 
 6s. 9-rf., the quotient of the first of which is 13s., and it remains to find 
 
 4 * I I , 
 
 that of the second. Since 65. is jzd.^ 6s. g-d. is 81-J., and 81 divided 
 
 4- 4 I . 
 
 bv 13 gives the quotient 6 and remainder 3; that is, %i-d. is "jM. and 
 
 I 4 12 
 
 3 -</., of the first of which the quotient is 6c?. Again, since 3c?. is — , 
 
 4 I . , . 4 
 
 or 12 farthings, 3-</. is 13 farthings, the quotient of which is 1 farthing, 
 
 or -, without remainder. We have then divided £gg • 15 . 9- into 
 
 4 . 4 
 
 four parts, each of which is divisible by 13, viz. jfc'91, 1695., 78(/., 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. d. £ s. d. 
 
 13)99 15 97(7 13 6- . 
 
 91 ^ 4- 
 
 ~8 
 
 20 
 
 160+15 = ^75 
 
 IL 
 
 45 
 39 
 
 6 
 
 12 
 
 72+9 = 81 
 78 
 
 3 
 
 _4 
 
 124-1 = 13 
 
 o 
 
 n2 
 
138 COMMERCIAL ARITHMETIC. § 228-230. 
 
 Here, each of the numbers 99, 175, 8i, 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. 21 lbs. 7 oz. x 53 = 129 cwt. i qr. 16 lbs. 3 oz. 
 
 2d ^h ^m 27»x 109 = 236^ lO** l6'» 3« 
 
 £27 . 10 . 8 X 569 = £15666 .9.4 
 
 £7.4. 8 X 123 = £889. 14 
 
 8 6 
 
 £166 X — = £40 . 4 . 10 — 
 
 33 33 
 
 £187. 6. 7 X -1- = £5. 12.4I — 
 100 -^ ^425 
 
 4s. 6-d.x 1 121 = £254 .11.2- 
 2 2 
 
 4s. 4rf. X 4260 = 65. 6d, X 2840 
 
 229. Suppose it required to find how many times i*. ^.-d. is contained 
 
 3 ^ 
 
 m £3.19. 10-. 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. 4-rf. is 
 
 3 4 
 
 £•067, and £3 . 19 . 10- is £3*994, and 3*994 divided by '067 is 3994 
 
 by 67, or 597-. This is an extreme case, for the smaller the divisor, 
 
 67 
 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. x oz, ? and 
 jd jh Qva ^jt contain 3°' 46*? Answer^ 17*30758 and 4i4'367257, 
 
 If 2 cwt. 3 qrs. I lb. cost £150 . 13 . 10, how much does i lb. cost ? 
 
 . 13 
 Answer ^ gs. 90. — —, 
 309 
 
 A grocer mixes 2 cwt. 15 lbs. of sugar at iirf. per pound with 14 cwt. 
 
 3 lbs. at $d. per pound. At how much per pound must he sell the 
 
 3 ^53 
 mixture so as not to lose by mixing them P Answer, Kd. . 
 
 4905 
 
 230. There is a convenient method of multiplication called Prac- 
 tice. Suppose I ask. How much do 153 tons cost if each ton cost 
 
 £2.15.7 ? It is plain that if this sum be multiplied by 153, the 
 2 
 
§ 230. 
 
 PRACTICE. 
 
 139 
 
 product is the price of the whole. But this is also evident, that, if I 
 
 buy 153 tons at £2 . 15 . 7- each ton, payment may be made by first 
 
 2 
 putting down £1 for each ton, then 105. for each, then 55., then 6</., 
 
 [5 . 7-, and the 
 
 and then \-d. These sums together make up £2 
 2 J 
 
 reason for this separation of £2 . 15 . 7- into different parts will be soon 
 
 2 
 
 apparent. The process may be carried on as follows : 
 
 1. 153 tons, at £2 each ton, will cost £306 o o 
 
 2. Since \os. is £-, 153 tons, at \os. each, will cost 
 
 £-^^, which is 76 10 c 
 
 2 1 
 
 3. Since 5s. is - of 10*., 153 tons, at 55., will cost half 
 
 2 
 
 as much as the same number at 10s. each, that 
 
 is, - of £-](> . 10, which is 
 
 2 I 
 
 4. Since 6d. is — of 5s., 153 tons, at 6d. each, will 
 
 I '° 
 cost — of what the same number costs at 5s. 
 10 J 
 
 each, that is, — of £38 . 5, which is .... 
 
 1 ^° I 
 
 5. Since i- or 3 halfpence is - of 6d. or 12 halfpence, 
 
 ^ I 4 I 
 
 is3 tons, at i-rf. each, will cost - of what the 
 
 2 4.1 
 
 same number costs at 6d. each, that is, - of 
 
 4 
 £3.16. 6, which is 
 
 38 5 
 
 3 i*"^ 6 
 
 o 19 
 
 The sum of all these quantities is . . 
 which is, therefore, JS2.15.7-X153. 
 
 425 10 7- 
 
 The whole process may be written down as 
 
 follows : 
 
 
 
 iei53 
 
 
 
 
 
 
 £i 
 
 per ton. 
 
 £2 is 2 X £1 
 
 306 
 
 
 
 
 
 2 
 
 
 
 los. is - of £1 
 2 
 
 76 
 
 10 
 
 
 
 1 
 
 3 
 
 
 
 10 
 
 $s. is - of 105. 
 
 38 
 
 5 
 
 
 
 
 
 
 
 5 
 
 6rf. is — of 5s. 
 
 10 
 
 i-d. is - of Sd. 
 
 2 4 
 
 3 
 
 
 16 
 19 
 
 6 
 
 1 
 2 
 
 2 
 
 
 
 
 
 6 
 
 1 
 
 i- 
 
 2 
 
 Sum , . . 
 
 £V^5 
 
 10 
 
 ^\ 
 
 £2 
 
 ■5 7j 
 
140 
 
 COMMERCIAL ARITHMETIC. 
 
 §230. 
 
 ANOTHER EXAMPLE. 
 
 What do 1735 lbs. cost at 9*. lo^rf. per lb. ? The price q*. 10^. 
 
 I 4 I I 4 
 
 is made up of 5s., 4*., lorf., -</., and -d.; of which 55. is - of £1^ 
 
 24 4 
 
 4s. is - of jg*!, loJ. is -: of 5s., -rf. is — of 10c?., and -d. is - of -</. 
 ^5 ' 6-^2 20 4 22 
 
 Follow the same method as in the last example, which gives the fol- 
 
 lowing : 
 
 £1735 
 
 £1 per lb. 
 
 Ss. is - of £1 
 
 433 
 
 15 
 
 
 
 
 
 
 5 
 
 
 
 4 
 
 
 
 
 ^ 
 
 
 
 
 45. is - of£i 
 
 347 
 
 
 
 
 
 1 
 
 
 
 4 
 
 
 
 lod. is - of 55. 
 6 
 
 72 
 
 5 
 
 10 
 
 T 
 
 
 
 
 
 
 10 
 
 I 
 
 -d. is — of lod. 
 
 3 
 
 12 
 
 3-- 
 
 i-» 
 
 
 
 
 
 0- 
 
 2 20 
 
 
 
 
 3 
 
 
 
 2 
 
 -d. is - of -d. 
 
 I 
 
 16 
 
 il 
 
 ■5 
 
 
 
 
 
 0- 
 
 422 
 
 
 
 4 
 
 
 
 
 
 4 
 
 by addition ... 
 
 £2sS 
 
 9 
 
 i 
 
 £0 
 
 9 
 
 id 
 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 togetlier. 
 
 EXERCISES. 
 
 What is the cost of 
 
 243 cwt. at ^■'14 . 18 . 8- per cwt. ?— Answer, ;f 3629 . 1 . c»-. 
 4 4 
 
 169 bushels at £z .1.3- per bushel ? — Jtuwer, ^34^ . 14 . 9--. 
 4 4 
 
 273 qrs. at 19*. 2c?. per quarter ? — Answer, £7.61 . 12 . 6. 
 
 2627 sacks at 7*. %-d. per sack ?— Answer, £ioiz .9.9-. 
 
 • Any fraction of a unit, whose numerator Is unity, 1b generally called an aliquot 
 
 part of that unit. Thus, 2s. and lOf. are both aliquot parts of a pound, being £ 
 
 andjej-. 
 2 
 
 JO 
 
§ 231-233- PRACTICE. 141 
 
 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 . i . 3- per cwt. ? 
 
 Answer^ ^^562*2849, or ^^562 . 5 . 8- . 
 
 II I + 
 
 66-lbs. at IS. 4.-d. per lb. cost £^ . 1 1 . 5-. 
 22 4 
 
 How many pounds, shillings, and pence, will 279*301 acres let for 
 
 if each acre lets for ^3-1076 ? — Ansioer, £867-9558, or £^67 . 19 . 1-. 
 13 12 4- 
 
 "What does - of -^ of 17 bush, cost at - of - of ii'17 . 14 per bushel ? 
 4 13 5 3 I 
 
 Answer, £z'ii/^6^ or £z . 6 . 3-. 
 
 What is the cost of 19 lbs. 8oz. i2dwt. 8gr. ati^4 .4.6 per ounce? — 
 
 Answer, j^999 . 14 . i — . 
 4 5 
 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. For example, 
 
 3 • 3 
 
 what does 125. 3-</. amount to in a year.? This is 147-d, and its half 
 
 7.4 3.5 ^ 
 
 IS TZji., which added to 147-c?. gives 221-rf., which turned into pounds 
 
 '^ 3 3 
 
 is £^^l . 12 . 6. Also, 125. 3-rf.x5 is ^£"3 . 1 . 6^, which added to the 
 
 34 4 
 
 former sum gives £224 . 14 . o- for the yearly amount. In the same 
 
 4 
 
 :3. 
 
 >vay the yearly amount of zs. 3 d, is £j^i . 16 . 5-; that of 6^d. ia 
 
 3 ^ ^4 
 jgio .5.3-; and that of iirf. is £"16 . 14 . 7. 
 
 4 . 
 
 233. An inverse rule may be formed, sufficiently correct for every 
 
 purpose, in the following way : If the year consisted of 360 days, or 
 
 3 
 
 - 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 360/5 such parts, will give 360 parts for each 
 
142 
 
 COMMERCIAL ARITHMETIC. 
 
 § 233-234« 
 
 of the 5 days which we neglected at first. 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 73d 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 i8(/., or -d. for 35., or lod. for £3. The rule, then, is as 
 
 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. For 
 
 3 
 example, how much per day will give £224 . 14 . o- per year ? This 
 
 4 
 is 224*703, and its third is 74*901, which subtracted from 224703, gives 
 
 149*802, which, if they be pence, amounts to 12s. 5-8o2c?., in which 
 
 IS. 6d. is contained 8 times. Subtract 8 farthings, or 2rf., and we have 
 
 12s. 3'8o2c?., which differs from the truth only about — of a farthing. In 
 
 the same way, £100 per year is 5s. s~^' Pcr day. 
 
 4 
 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 
 
 
 
 
 
 
 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 inches. Divide a b and c d (which are equal) each into 6 inches by 
 the points a, 6, c, /, «i, &c. ; and a c and b d (which are also equal) 
 into 4 inches by the points /, g. A, a?, y, and z. Join a and /, b and m. 
 
§234-i36. MEASURES OF LENGTH AND SURFACE. 143 
 
 &.C., and /and a?, &c. Then, the figure a b c d is divided into a number 
 of squares ; for a square is a rectangle whose sides are equal, and 
 therefore a a/E is square, since a a is of the same length as a/, both 
 being i inch. There are also four rows of these squares, with six 
 sfiuares 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 oiijg 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 
 a whole number of inches, contain some inches and a 
 
 fraction. For example, let a b be 3- inches, or (1 14) 
 
 7 . ^ I . 9 
 
 - of an inch, and let a c contain 2- inches, or - 
 2 ' 44 
 
 of an inch. Draw a e twice as long as a b, and a y 
 
 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 - inches, it is 7 
 
 2 . 9 
 
 inches. And since a f is four times a C, or four times - 
 
 4 
 " inches, it is 9 inches. Therefore, the whole rectangle 
 
 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 — square 
 
 inches. But -r- is made by multiplying - and - together (118). From 
 
 5 42 
 
 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. For example, a square whose 
 
 side is 13 inches in length contains 13x13 or 169 square inches. 
 
 236. exercises. 
 
 What is the content, in square feet and inches, of a room whose 
 sides are 42 ft. 5 inch, and 3 1 ft. 9 inch. ? and supposing the piece from 
 
 
 
 c 
 
 D 
 
 1 
 
 
 
 
 
1*^4 COMMERCIAL ARITHMETIC. § 236-238. 
 
 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- 
 inches. 
 
 The sides of a rectangular field are 253 yards and a quarter of a 
 mile ; liow many acres does it contain ? — Answer, 23. 
 
 What te the difference between 18 square miles^ and a square of 18 
 miles long, or 18 miles square 1 — 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 144 square inches, and so on. In a simihar way 
 it may be shewn that the content in cubic inches of a cube, or paral- 
 lelepiped,* 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 <; 
 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 156 yards will cost, if aa 
 
 yards cost 175. ^d. This quantity, reduced to pence, is 2c8d. ; and if 
 
 22 yards cost 2c8rf., each yard costs d. But 156 yards cost 156 
 
 ^^ 208 
 times the price of one yard, and therefore cost x 156 pence, or 
 
 '^^ ^^^ pence (117). Again, if 25- French francs be 20 shillings 
 
 22 '^ I 
 
 sterling, how many francs are in £20. 15? Since 25- francs are 20 
 
 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. A cube is a parallelepiped with equal sides, such as 
 U a die. 
 
§233-239' RULE OF THREE 145 
 
 fortieth part of 51 francs, or — francs. But £20 15s. contain 415 
 
 40 CI 51 
 
 shillings (219); and since i shilling is — francs, 415 shillings is — 
 
 X415 francs, or (117) francs. 
 
 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 
 
 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 2Z and 156 
 
 yards and 208 pence, and the thing required to be found is a number of 
 
 pence. In the second question there are zo and 415 shillings and 25 
 
 2 
 francs, and what is to be found is a number of francs. Write the three 
 
 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 : 
 
 22 yds. 156 yds. 175. 4^.' 
 
 In the second question they will be placed thus : 
 
 20«. £20 155. 25- francs. 
 
 2 
 
 Reduce the first and second quantities, if necessary, to quantities of 
 the same denomination. Thus, in the second question, £zo 15s. must 
 be reduced to shillings (219). 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 
 
 * This generally comes in the same member of the sentence. In some cases the 
 ingenuity of the student must be employed in detecting it. The reasoning of (238) 
 is the best guide. The following may be very often applied. 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. Tims, in the first 
 question, 156 yards must cost more than 22 ; multiply, therefore, by 156. 
 
 O 
 
146 COMMERCIAL ARITHMETIC. § 239-240. 
 
 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 the answer is (238) 
 208x156 17s. 4d.xis6 
 
 pence, or, which is the same thmg, . 
 
 22 '^ ' ' ° 22 
 
 240. The whole process in the first question is as follows :* 
 
 yds. 
 
 yds. 
 
 s. d. 
 
 22 : 
 
 156 
 
 : •• 17.4 
 12 
 
 208 pence. 
 156 
 
 1248 
 
 
 " 
 
 1040 
 208 
 
 a2)32448(i474-rf. and — , or — of a farthing, 
 
 " or (2 19) £6. 2. 10^-^. 
 ' 411 
 
 104 
 
 88 
 
 '54 
 
 108 
 88 
 
 oo 
 
 (228) 4 
 
 80 
 66 
 
 The question might have been solved without reducing 17*. ^d, 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. 
 
240-241. RULE OF THREE. H7 
 
 yds. 
 
 yds. «. d. 
 
 
 22 : 
 
 ; 156 :: 17.4 
 156 
 
 (2: 
 
 
 a2)£i35.4.o(£6.2. lolX 
 ,3. ^" 
 
 (2 
 
 
 3x20+4 = 64 
 
 
 
 44 
 
 
 
 20x12 = 240 
 
 
 
 220 
 
 
 
 20x4 = 
 
 80 
 66 
 
 *4 
 
 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 aie 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 towrads an income-tax of 45. 6d. in the pound ? 
 Here the three given quantities are, £400, 45. 6rf., and £1, 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 : 
 
 £1 : £400 :: 4s. 6rf. 
 
 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 ^r. 22 lbs. cost? 
 
 3 185 
 Answer, £5 . 14 . 5 ~. 
 
 4217 
 
 If a horse go 14 m. 3 fur. 27 yds. in 3'' 26" i2«, how long will he be 
 
 2462 
 in going 23 miles ? Answer ^ 5" 29™ 34* . 
 
 Two persons, A and B, are bankrupts, and owe exactly the same 
 
148 COMMERCIAL ARITHMETIC. §242-244, 
 
 sum ; A can pay 15s. 4-rf. in the pound, and B only 75. 6-rf. At the 
 
 2 4 
 
 same time A has in his possession £1304. 17 more than B; what do 
 
 the debts of each amount to ? Answer, £3340 .8.3-—. 
 
 I 4^5 
 
 For every 12-- acres which one country contains, a second contains 
 
 1 2 
 
 56-. The second country contains 17,300 square miles. How much 
 
 4 
 does the first contain? 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 
 
 4 
 of square miles in the first is 3844-, and its population 3,321,600; and 
 
 the population of the second is 5,536,000. 
 
 If 42- yds. of cloth, 18 in. wide, cost £59 . 14 . 2, how much will 
 
 1 * 4 
 
 118- yds. cost, if the width be 1 yd. ? Answer, £332 . 5 . 2—. 
 
 If £9 . 3 . 6 last six weeks, how long will £100 last .J* 
 
 Ansioer^ 65-— weeks. 
 
 3 367 
 
 How much sugar, worth 9-rf. a pound, must be given for 2 cwt. of 
 
 4 «r 
 tea, worth lod an ounce ? Answer, 32 cwt. 3 qrs. 7 lbs. — . 
 
 39 
 243. Suppose the following question asked : How long will it take 
 
 15 men to do that which 45 liien can finish in 10 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 
 
 450 
 employs one man, that is, in , or 30 days. By this and similar 
 
 reasoning the following questions can be solved. 
 
 EXERCISES. 
 
 If 15 oxen eat an acre of grass in 12 days, how long will it take 26 
 
 oxen to eat 14 acres ? Answer, 96— davs. 
 
 If 22 masons build a wall 5 feet high in 6 days, how long will it 
 
 take 43 masons to build 10 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 
 
§ 244- 
 
 RULE OF THREE. 
 
 149 
 
 thing to be done is to find out, from the first part of the question, t?ie 
 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 
 30 3 3x5 
 
 make ^ or 6 yards. He will, therefore, make one yard iri - or m 
 
 5 - "^ 6 30 
 
 of a day. From this we are to find how long it will take 4 men to make 
 
 68 yards. Since one man makes a yard in — ^ of a day, he will make 68 
 3^5 .0 ^ Mi^x :_ 3x5x68 30 
 
 yards in x68 days, or (116) in 
 
 days ; and 4 men will do this 
 
 3x5x68 . I 
 
 — days, or m 8 
 
 30x4 2 
 
 Again, suppose the question to be : If 5 men can make 30 yards in 
 
 3 days, how much can 6 men do in 12 days ? Here we must first find 
 
 the quantity one man can do in one day, which appears, on reasoning 
 
 similar to that in the last example, to be ■^— yards. Hence, 6 men, 
 J -11 1 6x30 , , , "^^S 12x6x30 
 
 m one day, will make yards, and m 12 days will make =- 
 
 5x3 ^ ' ^ 5X3 
 
 or 144 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 Avith each other, in the 
 
 same line. In the two examples above given, the quantities must be 
 
 written thus : 
 
 5 men. 30 yds. 3 days. 
 
 5 men. 
 
 68 yds. 
 
 SECOND EXAMPLE. 
 30 yds. 
 
 3 days. 
 
 6 men. 12 days. 
 
 Draw a curve through the middle of each line, and the extremities 
 of the other. There will be three quantities on one curve and two on 
 the other. Divide the product of the three by the product of the two, 
 and the quotient is the answer to the question. 
 
 o2 
 
150 COMMERCIAL ARITHMETIC. § 244-245. 
 
 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). 
 
 EXERCISE'S. 
 
 If 6 horses can, in 2 days, plough 17 acres, how many acres will 
 
 I 7 
 
 93 horses plough in 4- days ? Answer^ 59^o* 
 
 J 2 o 
 
 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 125- yards ? 
 
 ^ A ^451 , 
 
 Answer, 75 davs. 
 
 20720 
 
 If the carriage of 6ocwt. through 20 miles cost 3614 los., what weight 
 ought to be carried 30 miles for 1^5 . 8 . 9 ? Answer, 15 cwt. 
 
 If jgioo gain £^ in a year, how much will £850 gain in 3 years and 
 8 months ? Answer, ^i?*; . 16 . 8. 
 
 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 £16 into 
 
 40 equal parts, and take 7 of them. Each of these parts is £ — , and 7 of 
 
 them make — X7, or pounds (116). The process may be written 
 
 40 40 
 
 as below : 
 
 £16 
 
 7_ 
 
 40) II 2(^2 . 164. 
 80 
 
 32 
 20 
 
 640 
 
 4<|_ 
 
 240 
 %40 
 
 c 
 
§245-246. INTEREST. 151 
 
 Suppose it required to take 13 parts out of a hundred from 
 
 ^-56. 
 
 .13.7^. 
 56 . 13 . 
 
 I 
 
 
 
 
 
 
 13 
 
 . 7 ' 
 
 >5o 
 
 
 100)736 . 17 , 
 
 i^(£7 
 
 
 
 700 
 
 
 
 
 
 
 36A20+17 
 
 = 737 
 700 
 
 
 
 
 
 
 37>«i 
 
 t2+l 
 
 -445 
 400 
 
 45x4+2 = 
 
 1S2 
 100 
 
 82 
 
 Let it be required to take 2- parts out of a hundred from £3 125. 
 The result, by the same rule is — '- — -^ or (123) — ; so 
 
 / 100 '^ ' 2CO 
 
 that taking 2- out of a hundred is the same as taking 5 parts out of 200. 
 
 EXERCISES. 
 
 1 12.9 
 
 Take 7- parts out of 53 from £1 10s. Answer^ as. i d. 
 
 '3 3 159 
 
 Take 5 parts out of 100 from £107 13s. ^-d. 
 
 4 3 
 
 Answer, £5.7.8 and — of a farthmg. 
 
 1^56 3s. zd. is equally divided among 32 persons. How much does 
 
 the share of 23 of them exceed that of the rest ? 
 
 Answer y £24. .11.4 — . 
 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 hun- 
 dred must be taken from the second in order to make the first. Thus, 
 instead of saying that £16 ixs. is the half of ^^33 4s., it is said that the 
 first is 50 per cent of the second. Thus, £$ is 2- per cent of £200 ; 
 
 because, if £200 be divided into 100 parts, 2- of those parts are £$. 
 
 2 
 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. § 246. 
 
 parts out of 56 of any sum ? The question amounts to this: If he who 
 
 has £56 gets £100 for them, how much will he who haa 23 receive ? 
 
 This, by (238), is -^^ — , or -^, or 41 — . Hence, 23 out of 56 is 
 
 41 — per cent. 
 
 Similarly 16 parts out of 18 is — , or 88- per cent, and 2 parts 
 
 „ . 2x100 " 9 
 
 out of 5 IS , or 40 per cent. 
 
 From which the method of reducing other fractions to the rate per 
 
 cent is evident. 
 
 Suppose it asked, How much per cent is £6 . 12 . 2 of £12 . 3 ? 
 
 Since the first contains 1586^, and the second i^iSd., the first is 1586 
 
 . 158600 
 out of 2916 parts of the second; that is, by the last rule, it is — — —, 
 
 11^6 I ^910 
 
 or 54 — ^, or £54 .7.9- 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 shil- 
 ling, which is near enough for all practical purposes. For instance, in 
 the last example, which is to find how much £6*608 is of £12*15, 
 6'6o8xioo is 66o'8, which divided by I2'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?— Jn*. £85 .1.8-. 
 4 4 
 
 Goods which are bought for £193 . 12, are sold for £216 . 13 .4; 
 
 how much per cent has been gained by them ? 
 
 Answer, A little less than £11 . 18 . 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 . 18 4 — -. 
 
 1 4 25 
 
 A stockbroker buys £1700 stock, brokerage being at £- per cent; 
 
 o 
 
 what does he receive ? — Answer. £2.2.6. 
 
 • Commission Is what is allowed by one merchant to another for buying or sell- 
 ing goods for him, and is usually a per-centage on the whole sum employed. Broker- 
 age is an allowance similar to commission, under a diflerent 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 payers 
 
§ 246-249« INTEREST. 153 
 
 2 
 
 A ship whose value is ^615^23 is insured at 19- per cent; what 
 
 3 12 
 does the insurance amount to? — Answer, JS3033 . 3.9 — . 
 
 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 jgioo can 
 only pay JS50, he is said to pay los. in the pound. The rule is easily 
 
 derived from the same reasoning as in (246). For example, £50 out 
 50 50x20 I 15 
 
 of J882 is £:r- out of £1. or — - — shillings, or 125. 2 in the 
 
 82 8a *' 441 
 
 pound. 
 
 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 £100 is lent at 4 
 per cent, it must be understood to mean 4 per cent per annum j that is, 
 that 4 pounds are paid every year for the use of £100 
 
 The sum lent is called the principal^ and the interest upon it is 01 
 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 
 ia called simple, and in the second compound. The interest and principal 
 together are called the amount. 
 
 248. What is the simple interest of ^£1049 .16.6 for 6 years and 
 
 one-third, at 4- per cent ? This interest must be 6- times the interest 
 * 3 
 
 any loss they may sustain by accidents from fire, or storms, according to the agree- 
 ment, 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. 
 
 § 249-150. 
 
 of the same sum for one year, which (245) is found by multiplying the 
 sum by 4-, and dividing by 100. The process is as follows : 
 
 (230) 
 
 (82) ., . ^^ 
 
 ICO 
 
 20 
 (228) 
 
 (a) 
 
 £1049 . 
 
 16.6 
 
 ax4 
 
 4199. 
 
 6.0 
 
 1 
 
 ax- 
 
 2 
 
 524. 
 
 18.3 
 
 100)47,24 , 4 . 3(^47 . 4 . 10 
 20 
 
 4^* 
 
 6x6 
 
 5x1 
 
 3 
 
 io,ijt 
 
 (b) £47 . 4.10 — - Int. for one yr. 
 100 
 
 66 
 
 283 . 9.0 
 
 15 . 14. II -^ 
 100 
 
 €299 .4.0 — Int. for 6^ yrs. 
 100 3 ■ 
 
 EXERCISES. 
 
 What is the interest of £105 . 6 . 2 for 19 years and 7 weeks at 3 
 
 fter cent? Answer, £60 . 9, very nearly. 
 
 What is the difference between the interest of J650 . 19 for 7 years 
 
 at 3 per cent, and for 8 years at 2- per cent ? 
 
 Answer, los. z-d. 
 2 
 
 What is the interest of £157 . 17 . 6 for one year at 5 per cent ? 
 
 Answer, £7 . 17 . 10-. 
 2 
 
 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 bo .h 
 
 • Here the 4«. from the dividend is taken in. 
 t Here the Zd. from the dividend is taken in. 
 
250-251. INTEREST. 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 £100, at 5 per cent, compound 
 interest. The following is the process : 
 
 £100 First principaL 
 
 5 First year's interest. 
 105 Amount at the end of the first year. 
 (249) 5 . 5 Interest for the second year on £105. 
 1 10 . 5 Amount at the end of two years. 
 
 5 . 10 . 3 Interest due for the third year. 
 115 • 15 • 3 Amount at the end of three years. 
 100 . 0.0 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 veiy 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 
 i'i57625 ; meaning that £1 will become £1*157625 in 3 years. Now, 
 €100 must become 100 times as great; and 1*157625x100 is 115*7625 
 (141) ; but (221) £'7625 is 155. 3d. J therefore the whole amount of 
 £100 is £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 ; it is 
 required to find what the sum was at first. Whatever the sum was, if 
 we suppose it divided into 100 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 
 120 parts, 100 of those parts are the principal which we want to find, 
 
 * Sufficient tables for all common purposes are contained in the article on Interest 
 in the Penny Cyclopaedia; and ample ones in the Treatise on Annuities and Re- 
 versions, in the Library of Useful Know ledge. 
 
156 COMMERCIAL ARITHMETIC. §251-253. 
 
 and 20 parts are interest upon it ; that is, the principal is £11^11122. 
 
 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 
 the money, and B would gain it. It is fair, therefore, that lie should 
 only pay to B as much as will, with interest, amount in four years to 
 £350, that is (251), ^^291 .13.4. Therefore, £58 .6.8 must be struck 
 oft 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 
 100, and divide the product by 100 increased by the product of the 
 rate per cent and number of years. 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. 
 
 aXERCISES. 
 
 What is the discount on a bill of £138 . 14 . 4) due 2 years hence, 
 
 discount being at 4- per cent ? Answer, £11.9.1. 
 
 2 
 What is the present value of £1031 . 17, due 6 months hence, 
 
 interest being at 3 per cent ? Answer, £1016 , 12. 
 
 253. If we multiply by a+b, or by a—b, when we should multiply 
 
 by a, the result is wrong by the fraction — -, or — 7-, of itself: being 
 
 a+o a—o 
 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 - 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 17, the result is — of itself too small ; and if 
 
 * 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« INTEREST. 157 
 
 we divide by 560 instead of 365, the result is too great by -^, or — - 
 
 365 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 con- 
 venient to suppose the year to contain only 360 days, in which case its 
 73d part (the jzd 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 £18 . 9 . 10, or 18*491. 
 274 18-491 
 
 tSo is - of 360 
 
 9-246 
 
 94 
 
 
 90 is - of 180 
 
 4-623 
 
 4 is — of 360 
 90 
 
 •205 
 
 
 9)14-074 
 
 
 8)1-564 
 
 -196 
 
 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 divii-or; 
 
 2 
 since m-f-365 is 2m-^73o, or — 771-1.73. Thus, in the preceding instance, 
 
 we multiply by 54*8 and divide by 73 ; and 54-8x18-491 = 1013*3068, 
 
 which divided by 73 gives 13-881, very nearly agreeing with the fonner, 
 
 and giving £13 . 17 . 7-, which is certainly within a farthing of the 
 
 truth. 
 
 254. Suppose it required to divide jgioo 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 first has, the second may have £5, and the 
 
 third £^. It is plain that if we divide the £100 into 6+5+9, or 20 
 
 parts, the first must have 6 of those parts, the second 5, and the third 9. 
 
 Therefore (245) their shares are respectively, £ , £ and 
 
 ^icoxo ^ ^ , ^ 20 20 
 
 £ -, or £30, £25, and £45. 
 
 20 
 
158 COMMERCIAL ARITHMETIC. § 254-256. 
 
 EXERCISES. 
 
 Divide JS394 . l^ among four persons, so that their siiares mav be 
 
 as I, 6, 7, and 18. — Answer, £12 .6.7-; £73 . 19 . 9 ; £86 . 6 . 4-; 
 
 2 2 
 
 £221 . 19 . 3. 
 
 Divide £20 among 6 persons, so that the share of each may be as 
 much as those of all who come before pat together. — Answer, The first 
 two have 12s. Sd.; the third £1.5; the fourth £2 . 10 ; the fifth £5 ; 
 and the sixth £10, 
 
 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 jgio 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 £1000. How much of it ought each to have? Each 
 one ought to gain as much for jgi as the others. Now, since there are 
 
 250+130+45, or 425 pounds embarked, which gain £1000, for each 
 
 , - . . « ^1000 m, « .11-. . 1000x250 
 
 pound there is a gam of £ . Therefore A should gam 
 
 425 4^5 
 
 1000x130 1000x45 
 
 pounds, B should gain pounds, and C pounds. 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 . 10 . 2. How much ought 
 
 each to pay of it ? Answer, A must pay £132 . 15 . 2-. 
 
 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. How much of the loss must each 
 
 bear? Answer, A pays £59 . 12, B £14 . 18, and C £74 . 10. 
 
 266. 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 
 
§ 256-257« INTEREST. 159 
 
 is employed, the more profit should be made upon it. If, for example, 
 
 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 1^3 for 6 months, B £j^ for 7 months, and C j6i2 for 2 months, 
 
 and the gain is £100 ; how much ought each to have of it ? Now, 
 
 since A employs £1 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 
 
 ig6x3, or jgi8, for one month; also, B gains as much as if he had 
 
 employed 1^4x7 for one month; and C as if he had employed £12x2 
 
 for one month. If, then, we divide £100 into 6x3+4x7+12x2, or 70 
 
 parts, A must have 6x3, or 18, B must have 4x7, or 28, and C 
 
 12x2, or 24 of those parts. The shares of the three are, therefore, 
 
 6x3x100 4x7x100 12x2x100 
 
 a* "7 » ^~c. > and i,- 
 
 6x3+4x7+12x2 6x3+4x7+12x2 6x3+4x7+12x2 
 
 EXEUCISES. 
 
 A, B, and C embark in an undertaking ; A placing £1 . 6 for 
 a years, B £100 for i year, and C £12 for 1- years. They gain 
 ^£4276 . 7 How much must each receive of the gain ? 
 
 Answer^ A £226 . 10 . 4 ; B £3432 .1.3; 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 . 12 . 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 con- 
 tained in those which have been given. Such is what is commonly 
 called the Rule of Exchange, for such questions as the following ; If 
 
 * This question does not at first appear to fall under the rule. A little thought 
 will serve to shew that what probauly will be the first idea ct the proper method of 
 solution is erroneous. 
 
160 COMMERCIAL ARITHMETIC. §257- 
 
 20 shillings be worth 25- francs, in France, what is £160 worth ? This 
 
 2 
 may evidently be done by the Rule of Three. 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 
 
 to 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. 
 
appe:n dix 
 
 FIFTH EDITION 
 
 CF 
 
 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, 
 lo, loo, looo, &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 100,000 is a ^v^-cipher number. Learn to 
 connect tens, hundreds, thousands, tens of thousands, hundreds of thou- 
 sands, millions, &c. with i, 2, 3, 4, 5, 6, &c. in the mind. "What is a 
 seventeen-cipher number ? 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, 
 &t> in I, 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 from 
 any number. Thus, beginning from four and proceeding by sevens, we 
 
 P 3 
 
162 APPENDIX. 
 
 have 4, ii, i8, 25, 32, &c., along which series you must leam 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 pennit yourself to say, 4 and 7 are 
 
 11, II and 7 are 18, &c. ; but only 4, 11, 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, leam 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, leam also by practice how to pass up from 8 to 
 the next number which ends with 3 (or 13), and to catch the necessary 
 augmentation. Jive, without the necessity of formally undertaking in 
 vtords to subtract 8 from 13. Take rows of numbers, such as 
 
 42605C1864 
 
 and practise this rule 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 
 
 12, leaves 8), 2 and 4, 6 and 4, o and 5, 5 and 5, o and 1, 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." Thus, the row of figures 
 17729638109 will give the following practice: 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 
 5 are 68 ; 38 and 3 are 41 ; 81 and 9 are 90 ; 10 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. Leam the multiplication-table so well as to name the product 
 
ON THE MODE OF COMPUTING. 163 
 
 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 pro- 
 ducts of every successive pair of digits as fast as you can repeat them, 
 namely, 27, 63, o, o, 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 28, without the necessity of 
 saying " 3 times 8 are 24, and 4 is 28." Thus, 179236408 will suggest 
 the following practice, 16, 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. Thus, 2, 7, 6, 9, must immediately suggest " 20 and 9 are 
 29." 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 ; 81 and 
 
 7 are 88 ; 79 and 4 are 33. 
 
 IX. Having four numbers, as 2, 4, 7, 9, vary the last exercise as fol- 
 lows : 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. Thus, 2, 4, 7, 9, are to suggest "15 
 and 4 are 19." 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. Thus, seeing 8 and 53, 
 arrive at and repeat " 6 and 5 over." Common short division is the 
 best practice. Thus, in dividing 236410792 by 7, 
 
 7)236410792 
 
 33772970, remainder 2. 
 
 All that is repeated should be 3 and 2 ; 3 and 5 ; 7 and 5 ; 7 and 2 
 2 and 6 ; 9 and 4 ; 7 and o •, o and 2. 
 
1 64 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). 
 
 47963 6, 15, 17, 23, 31, 3'V; ", i2> 2i» 22, 31, 3V; 9» 
 
 ]}^\ 17, 24, 27, 32, 4''i'; 10, 14, 20, 2i» 2"8'; 7, 9r i'3'- 
 
 26316 
 . In verifying additions, instead of the usual way of 
 
 819 omitting one line, adding without it, and then adding 
 
 66^6 
 ^ the line omitted, verify each column by adding it both 
 
 1 38 1 74 upwards and downwards. 
 
 SuBTBACTiON. The following process is enough. The carriages, being 
 always of one^ need not be mentioned. 
 
 From 79436258190 8 and 2', 4 and 5', 7 and 4', 3 and 5', 6 and 
 
 Take 58645962738 ^'^ 10 and 2', 6 and o', 4 and 9', 7 and 7', 
 
 20790295452 p and o', 5 and 2'. It is useless to stop and 
 
 say, 8 and 2 make 10 ; 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 18', 49', 22', 2', 42', 4'o', 
 
 4692681 21', 58', 26', 2', 49', 4'6\ 
 
 5363064 24', 66', 30', 3', 56', 5'3', 
 
 6033447 27', 74', 3/, 3', 63', 6V. 
 
 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 6i'; 27 and 2 are 29 ; 2 
 
 587255508 and 2 are 4' ; 49 and o are 49'; 
 
 6620702508 46 and 4 are 5V. 
 
 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 fol-ows: 
 
 a7693)44i9728o9662( 15959730 
 165042 
 265778 
 165410 
 269459 
 202226 
 83756 
 6772 
 
 The number of words by which 26577 is obtained from 16540a (the 
 multiplier being 5) is as follows : 15 and 7' are 2^2 ; 47 and 7' are 5"4 j 
 35 and 5' are 4''o ; 39 and 6' are 4''5 ; 14 and 1' 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 toe Supplement to thePenny Cycloptediat article Computation, 
 
166 APPENDIX. 
 
 APPENDIX II. 
 
 Olf VERIFICATION BY CASTING OUT NINES AND ELEVENS, 
 
 The process of casting out the ninesy as it is called, is one which the 
 young computer should learn and practise, as a check upon his com- 
 putations. 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, i, c, d be four numbers, such that 
 
 a = bc+d, 
 and if m be any other number whatsoever, and if a, 6, c, rf, severally 
 divided by wi, give the remainders jo, y, r, s, then 
 
 p and qr+s 
 give the same remainder when divided by m (and perhaps aie themselves 
 equal). 
 
 For instance, 334= 17x19+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 11, 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 2+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 a, 
 
 6 (2+4), 12 (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 246120377 by 9. A proof may be given thus: Ik 
 
ON VERIFICATION, &C. 167 
 
 is obvious that each of the numbers, i, lo, loo, looo, &c. divided by 9, 
 leaves a remainder i, since they are i, 9+1, 99+1, &-c. Consequently, 
 2, 20, 200, &c. leave the remainder 2 ; 3, 30, 300, the remainder 3 ; 
 and so on. 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 
 60 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, 
 that 10004569x3163 = 31644451747. 
 
 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, 
 three, ten, one. The remainders then are, 7, 4, i. Now, 7x4 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 
 a, 4, 7, 14, 5» ii»2, 6, 14, 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, find some additional check be desirable, the method of casting 
 out elevens may be followed after that of casting out the nines. Ob- 
 serve that lo+i, 100— I, looo+i, 10000— I, &c. are all divisible by 
 eleven. From this the following rule for the remainder of division by 
 
168 
 
 APPENDIX. 
 
 II 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 II. 
 
 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 
 1 1 if the figure be negative, is the remainder required. Thus, to divide 
 1642915 by II, and find the remainder, we have i from 6, 5 ; 5 from 
 4, —I ; — I from 2, 3 ; 3 from 9, 6 ; 6 from i, —5 ; —5 from 5, 10; and 10 
 is the remainder. But 164 gives— -i, and 10 is the remainder; 164291 
 gives— 5, and 6 is the remainder. 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, o, 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 num- 
 ber of times 99 exactly. 
 
 APPENDIX III. 
 
 ON SCALES OF NOTATION. 
 
 We are so well accustomed to 10, 100, &c., as standing for ten, ten 
 tens, &c., that we are not apt to remember that there is no reason why 
 10 might not stand for five, 100 for five fives, &c., or for twelve, twelve 
 twelves, &c. Because we invent different columns of numbers, and let 
 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 10 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 i, 2, 3, 4, 
 5, 6, &c., would be represented by i, 10, 11, 100, 10 1, no, in, 1000, 
 looi, loio, ion. iioo, &c. This is the binary scale. If we take the 
 ternary scale, t which 10 stands for 3, we have i, 2, 10, 11, 12, 20, 
 
ON SCALES OF NOTATION. I6r« 
 
 21, 22, lOO, loi, X02, no, &c. In the quinary scale^ in which lo n 
 five, 234 stands for 2 twenty-fives, 3 fives, and 4, or sixty-nine. If we 
 take the duodenary scale, in which 10 is twelve, we must invent new 
 symbols for ten and eleven, because 10 and 1 1 now stand for twelve and 
 thirteen ; use the letters t and e. 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 10 stands for is called the radix of the scale 
 oj 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 Rem'. I Answer . . 1021 121 
 
 5)681 2 Quinary. Decimal. 
 
 Verification^ loooooo means 15625 
 
 5/^3 I 20000 1250 
 
 5)27 1 1000 125 
 
 — 100 25 
 
 5)5 2 
 
 ^ _ 20 10 
 
 5)^ o I I 
 
 O I I02II2I 17036 
 
 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 681 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. Thus, in the quinary system we 
 carry fives instead of tens. I now give an example of multiplication 
 and division : 
 
170 
 
 
 APPENDIX. 
 
 Quinary. 
 
 DeciniaL 
 
 42143 
 
 means 2798 
 
 1^34 
 
 194 
 
 324232 
 
 11192 
 
 232034 
 
 25184 
 
 I 34341 
 
 2798 
 
 42143 
 
 
 I 143 32222 
 
 , 542812 
 
 Duodecimal. 
 
 Decimal. 
 
 4^9)76^4^08(16687 
 
 705)22610744(32071 
 
 4^9 
 
 1460 
 
 2814 
 
 5074 
 
 2546 
 
 1394 
 
 ^?,te 
 
 689 
 
 2546 
 
 
 3650 
 
 
 3320 
 
 
 3308 
 
 
 2^33 
 
 
 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 
 now 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 1643 a (septenary) into the quaternary scale : 
 
 16687 
 
 
 16432 
 
 Duodecimals into Decimals. 
 
 Septenarics into Quaternaries. 
 
 IXI2+6 = 18 
 
 
 1x7+6 = 31 
 
 XI2+6 
 
 
 X7+4 
 
 222 
 
 
 "33 
 
 X12+8 
 
 
 X7+3 
 
 1672 
 
 
 22130 
 
 XI2-7 
 
 
 X7+2 
 
 \nswer .... 32071 
 
 102101a 
 
ON THE DEFINITION OF FRACTIONS. 171 
 
 Owing to our division of a foot into iz equal parts, the duodecimal 
 scale often becomes very convenient. Let the square foot be also divided 
 into 12 parts, each part is la square inches, and the 12th of the 12th 
 is one square inch. Suppose, now, that the two sides of an oblong 
 piece of ground are 176 feet 9 inches 7-i2ths of an inch, and 65 feet 
 II inches 5-i2ths of an inch. Using the duodecimal scale, and duo- 
 decimal fr actions ^ these numbers are i28'97 and SS'^S- Their product, 
 the number of square feet required, is thus found : 
 
 128*97 Answer, 68^8*1446 (duod.) square feet, or 
 
 55'^5 1 1660 square feet 16 square inches — and 
 
 ^12 144 
 
 617^5 of a square inch. 
 
 116055 It would, however, be exact enough to allow 
 
 6175^ 2-hundredth8 of a foot for every quarter of an 
 
 ^'7^^ inch, an additional hundredth for every 3 
 
 68e8i44e inches,* and i-hundredth more if there be a 
 
 12th or 2-i2ths above the quarter of an inch. Thus, 9 — inches 
 
 should be •76+-o3+'oi, or '80, and 11— would be '95 ; and the preceding 
 
 might then be found decimally aa I76'8x65'95 as 11659-96 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 sevenrninths 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. 
 
 7 
 
 1. In - we have the 9th part of 7. 
 
 9 
 
 2. 7-9ths of a unit. 
 
 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 ot 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 i)art8, each is i, and 7 of the parts are 7 ; 
 
 7 
 consequently the fraction which 7 is of 9 is -. The fourth view follows 
 
 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 
 
 more than a change of words : A number reduced to - of its amount 
 
 has every 9 converted into a 7, and any fraction of a 9 which may 
 
 remain over into the corresponding fraction of 7. This is completely 
 
 7 . a . 
 
 proved when we prove the equation - of a = 7 times -. The sixth, 
 
 9 . 9 , 
 
 seventh, and eighth views are illustrated in tlio chapter on proportion. 
 
 When the student comes to algebra, he will find that, in all the 
 
 applications of that science, fractions such as 7 most frequently require 
 
 o 
 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 — . We shall find that we have, in this case, a 
 4l 
 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 
 
 -. We have no notion of the f 4- j th part of 2-, 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 4- ; that the first is some part of a time the second ; that 
 
 5 3 . . I 
 
 there must be swne multiplier which turns every 4- m a number mto 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. 
 
ON THE DEFINITION OF FRACTIONS. 1 73 
 
 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 existence a length of which four lengths and two-fifths 
 
 of a length make up the original length in question. For instance, we 
 
 might say that 6, 6, 2 is a division of 14 into 2- equal parts — 2 equal 
 
 3 
 parts, 6, 6, and a third of a part, 2. So we might agree to say, that 
 
 the (2-jth, or f 2-|rd, or (2-jst (the reader may coin the adjective 
 
 as he pleases) part of 14 is 6. If we divide the line a b into eleven 
 
 equal parts in c, d, e, «&;c., we must then say that a c is the nth part, 
 
 1 i i i I i i i r ~, i I 
 
 ACDEFGH IKLMB 
 
 A D the ( 5- )th, A E the { 3- )th, a f the { 2- Jth, a g the ( 2- Jth, a h the 
 
 / .\ Vw /,\ V^3/ /,\ V4//2X ^ sy / i\ 
 
 ( i^ )th, A I thef I- 1th, AK thef i| 1th, al the ( i- 1th, am thef i— Itli, 
 
 and AB itself the ist part of ab. The reader may refuse the language 
 
 if he likes (though it is not so much in defiance of etymology as talking 
 
 of multiplying by -) ; but when a b is called i, he must either call 
 
 1 ^ 
 
 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 — , the simplest way is to divide the whole unit into 46 
 4f 3 
 
 parts ; 10 of these parts, repeated 4- times, give the whole. The 
 
 |liilliill[illiillil|llilill!l|llilllill|ilill|e 
 A _i J. ^ J. Jl it 
 
 4f 4i 4f 4l 4| 
 
 ( 4- )th is then — , and 2- such parts is -7, or a c. The student should 
 V 5/ 46 2 '^ 46 
 
 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 "Y ? Are we to have a (7)^^ part of a imit P and what 
 is it ? Had there been a 5 in the denominator, we should have taken 
 
 Q.2 
 
*-'"* APPENDIX. 
 
 the part of which 5 will make a unit. As there is - in the denominator, 
 
 we must take the part of which - will be a unit. That part is larger 
 
 I I 5 2 
 
 than a unit ; it is 2- units : 2- is that of which - is 1. The above 
 
 * ^ I I 5 
 
 fraction then directs us to repeat 2- unite 3- times. By extending our 
 
 * 4 
 
 word ' multiplication' to the taking of a part of a time, all multiplica- 
 tions 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 3- shillings, how much does one yard cost ? In 
 such a case as this, the student looks at a more simple question. If 
 5 yards cost 10 shillings, he sees that each yard costs — , or 2 shillings, 
 and, concluding that the same process will give the true result when 
 
 ,1 7 1 
 
 the data are fractional, he forms ^, reduces it by rules to - or i-, and 
 
 2^ •'22 
 
 concludes that i yard costs 18 pence. 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 demon- 
 stration. In the above question we want money which, repeated 2- 
 1 ... 3 
 
 times, shall give 3- shillings. If we divide the shilling into 14 equal 
 
 parts, 6 of these parts repeated 2- times give the shilling. To get 3- 
 
 3 I 2 
 
 times as much by the same repetition, we must take 3- of these 6 parts 
 
 21 I 
 
 2 
 
 at each step, or 21 parts. Hence, — , or 1-, is the number of shillings 
 in the price. 
 
 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 ot 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 addi- 
 tions of its own species, and diminished by subtractions ; thus, 7 and 
 2 give 9, and let 7 with 2 subtracted give 5. But let the addition of a 
 
ON CHARACTERISTICS. 175 
 
 number without the bar diminish the negative number, and the sub- 
 traction increase it. Thus, 7 and 4 are 3, 7 and 12 make 5, 7 with 8 
 subtracted is 15. 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 re- 
 moval 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 1 1 gives 
 7, we say that a loss of 4 incuned at the moment when a loss of 1 1 is 
 removed, is, on the whole, equivalent to a gain of 7 ; and saying that 4 
 diminished by 2 is 6, we say that a loss of 4, accompanied by the removal 
 of a gain of 2, 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, 3'2i4, 1*0083, 8 (which is 8'oo...) 9*999, all have 
 o for their characteristics. But 17*32, 48, 93*116, all have i; 126*03 
 and 126 have 2; 11937264-666 has 7. But when there are no places 
 before the decimal point, look at the first decimal place which is sig- 
 nificant, and make the characteristic negative accordingly. Thus, '612, 
 •121, "9004, in all of which significance begins in the firsft decimal place, 
 have the characteristic 1; but 'oiS and '099 have a; '00017 has 4-, 
 •00000000 1 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 
 dividend. For instance, in dividing 146*08 by •00279. The character- 
 istics are 2 and 3 ; and 2 with 3 removed would be 5. But on looking, 
 we see that the first significant figiires of the divisor, 27, taken by them- 
 selves, 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 abcdefhe the 
 first figures of the quotient, the decimal point must be thus placed, 
 abcde'f. But if it had been to divide '00279 ^Y 146*08, no carriage 
 would have been required ; and 3 diminished by 2 is 5 ; thai is, the first; 
 
■ 176 APPENDIX. 
 
 significant figure of the quotient is in the 5th place. The quotient, then, 
 has "oooo before any significant figure. 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 whicli 
 can be made to give the result at once. 
 
 Two shillings is jg'ioc 
 
 One shilling is £'050 
 
 Sixpence is £'025 
 
 One farthing is £'ooi 04- 
 6 
 
 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 far- 
 things 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 
 •025 or 25 thousandths. But 25 farthings is "026, 26 farthings is '027^ 
 /v.c. Hence the rule for the ^rst three places is 
 
 One in the fir &i for every pair oj shillings ; 50 in the second and third 
 for the odd shilling, if any ; and i for every farthing additional v'ith 
 I ewtra for sixpence. 
 
ON DECIMAL MONEY. 
 
 177 
 
 Thu.^. 
 
 ^, OS. 
 
 3^. = £-oi4 
 
 OS. 
 
 6rf. = £'izs 
 
 O.S\ 
 
 7^. = £-03z 
 
 zs. 
 
 ^-d. = £-139 
 
 1*. 
 
 2-rf. = £-o6o 
 
 2 
 
 3.*. 
 
 2-rf. = £-i6i 
 4 
 
 IS. 
 
 4 
 
 Z3.. 
 
 lo^d. = £-694 
 
 In the fourth and fifth places, and those which follow, it is obvious 
 that we have no produce from any farthings except those above six- 
 pence. For at every sixpence, '00004- ^s converted into "ooi, and this 
 
 o 
 
 has been already accounted for. Consequently, to fill up the/oMWA and 
 
 fifth places. 
 
 Take \for every farthing* above the last sixpence^ and an additional 
 
 1 for every six farthings^ or three halfpence. 
 
 The remaining places arise altogether from 'ooooo- for every farthing 
 
 above the last three halfpence ; for at every three halfpence complete, 
 
 •ocooo- is converted into 'ooooi, and has been already accounted for. 
 6 
 
 Consequently, to fill up all the places after the fifths 
 
 Let the number of farthings above the last three halfpence be a nu- 
 merator, 6 a denominator, and annex the figures of the correspondirig 
 decimal fraction. 
 It may be easily remembered that 
 
 The figures of 7 are 166666... 
 6 
 
 333333.. 
 
 The figures of 2 are 666666. 
 o 
 
 .... |... 833333. 
 
 OS. 3-rf. =^ '01458 3333,.. zs. 6d. = '12500 0000... 
 
 OS. -J-d. = 'O32I29I1666... 25. 9-rf. = •1395813333..- 
 
 The studeut should remember all the multiples of 4 up to 4x25, or 100. 
 
178 
 
 APPENDIX. 
 
 z-d. 
 
 2 
 
 15. ii-d. 
 4 
 
 060416666... 3«. 
 
 z^d. = •i6i;45|83333... 
 
 4 I 
 
 •096J87I5 
 
 13s. 10-d. 
 4- 
 
 69J79 1666 ... 
 
 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, 
 
 or by 1000—40, or by 1000(1- 
 
 100/ • 
 
 that is, from 1000 times the pounds 
 
 subtract 4 per cent of itself. Thus, required the number of farthings in 
 
 ^1663 
 
 i663"59o625xiooo 
 4 per cent of this. 
 
 1663590*625 
 66543-625 
 
 No. of farthings required, 1597047 
 
 What is 47- per cent of jgi66 
 
 [3 . 10 and •6148 of £2971 .16.9? 
 
 i66'69i 
 
 £79 . 3 . 6 
 
 2971-837 
 
 40 P.O. 
 
 66-6764 
 
 •6 
 
 1783*1022 
 
 5 p. c. 
 
 8-3346 
 
 •01 
 
 29*7184 
 
 H p. c. 
 
 4-1673 
 
 •004 
 
 11*8873 
 
 
 
 •0008 
 
 2*3775 
 
 
 79-1783 
 
 
 
 1827-0854 
 
 £1827 
 
 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 alating 1 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. 
 
 Thus, £*i83 is rather less than 35. 8rf., and 6s. 4-rf. is rather greater than 
 
 4 
 £•319; or when the two do not exactly agree, the common money ts the 
 
 greatest. But £'125 and £'35 are exactly 2*. 6d. and 7*. 
 
ON DECIxMAL MONEY. 179 
 
 I 3 
 
 Required the price of ijcwt. 8ilb. 13-oz. at£;3 .11 9- per cwt. 
 true to the hundredth of a farthing. 
 
 3-590625 
 17 
 
 
 
 61-040625 
 
 lb. 56 
 
 I 
 
 2 
 
 1795313 
 
 16 
 
 I 
 
 7 
 
 •512946 
 
 7 
 
 I 
 
 8 
 
 •224414 
 
 2 
 
 I 
 8 
 
 •C64118 
 
 oz. 8 
 
 I 
 
 4 
 
 •016029 
 
 4 
 
 2 
 
 •C08015 
 
 1 
 
 1 
 4 
 
 •C02004 
 
 I 
 2 
 
 2 
 
 •001002 
 
 if^63'664466 
 
 £63.13.3- 
 
 2 
 
 Three men. A, B, C, severally invest £191 . 12 . 7-, £61 . 14 , 8, 
 1 . 4 1 
 
 and £122 . 1.9- in an adventure which yields £511 . 12 . 6-. How 
 a 2 
 
 ought the proceeds to be divided among them? 
 
 A, I9I-63229 
 
 B, 6173333 
 
 C, 122- 08958 Produce of £1. 
 375-4552o)5ir627o8( 1-362686 
 
 136 17188 
 i^362686 1-362686 23 53532 1-362686 
 92236191 3333716 locHoi 85980221 
 
 1 362686 
 
 1 226417 
 
 13627 
 
 S176 
 
 409 
 
 27 
 
 3 
 
 I 
 
 z 611346 
 
 8I76I2 
 
 25710 
 
 1 362686 
 
 13627 
 
 3J83 
 180 
 
 272537 
 
 9538 
 
 27254 
 
 409 
 
 
 1090 
 
 41 
 
 
 122 
 
 4 
 
 
 7 
 
 84^^31 
 
 
 I 
 
 663697 
 
180 
 
 
 APPENDIX. 
 
 
 
 
 26i'i346 . 
 
 . . A's share 
 
 i.^^^. 
 
 2 . 
 
 si 
 
 4 
 
 84*1231 . 
 
 . . B'8 „ . . 
 
 . 84. 
 
 2 , 
 
 -: 
 
 166-3697 . 
 
 . . C's „ . . 
 
 . 166. 
 
 7 . 
 12 
 
 .4^ 
 4 
 
 511-6274 
 
 .61 
 4 
 
 If ever the fraction of a farthing be wanted, remember that the 
 coinage-result is larger than the decimal of a pound, when wo use 
 only three places. From 1000 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. Thus, in 
 
 134*6 123*1 369*7 
 
 5*38 4*92 14*79 . 
 
 •22 *i8 91 
 
 we see that (if we use four decimals only) the pence of the above results 
 
 are nearly 8</. *22 of a farthing, 5-rf. *i8, and /\.--d. *9i. 
 
 2 J 2 m 
 
 A man can pay £2376 .4.4-, his debts being £3293 . 11 . o-. 
 2 ^ 4 
 
 How much per cent can he pay, and how much in the pound ? 
 
 3293*553)2376-2i8o(-72i4756 
 70 7309 
 48598 
 
 2488 Answer^ £72 . 2 . 11- per cent. 
 
 183 I 
 
 1% o . 14 . 5- per pound. 
 
 4 
 
 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 accu- 
 rately, whenever an investigation of the state of his aftairs is made. 
 
ON BOOK-KEEPING. 181 
 
 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. From the first two of these things he obviously knows 
 the third. 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 difierent 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 book- 
 
 R 
 
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 y 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. At full 
 length : " 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 ; bxit 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 
 «vho receive must be debited ; all who give must be credited. 
 
 No cancel is ever made. If 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. 
 
ON BOOK-KIEPING. 183 
 
 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. The debtor-side is 
 always the left. Other books are used, but they are only to help in 
 keeping the ledger correct. Thns 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 are 
 entered at stated periods, in the language of the ledger. 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 tney 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 for their shares 
 of the whole. The only bock 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 axe entered at once 
 in the ledger as they arise. It has appeared that every item is entered 
 twice. 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, tliat 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. 
 
 2 S S S S 
 'S 'S "S '2 'S 
 Q Q Q Q Q 
 
 A, Creditor 
 
 B, Creditor 
 
 C, Creditor 
 
 D, Creditor 
 
 E, Creditor 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 
 23 
 
 19 
 
 32 
 
 ! 
 4 
 
 17 
 
 
 6 
 
 II 
 
 25 
 
 9 
 
 41 
 
 
 10 
 
 2 
 
 14 
 
 28 
 
 16 
 
 
 3 
 
 15 
 
 4 
 
 60 
 
 I 
 
 
 Here the 16 is supposed to appear in D's account as D creditor 
 by C, and in C's account as C debtor to D. And to say that the sum 
 of debtor items is the same as tl.r.t 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 fol- 
 lowing is the way the accounts will stand : the lines in italics 'vvill pre- 
 sently be explained. 
 
 A, Debtor. 
 
 A, 
 
 Creditor. 
 
 B, 
 
 Debtor. 
 
 B, Creditor. 
 
 To B . . 17 
 
 ByB 
 
 . . 23 
 
 To A 
 
 . . 23 
 
 By A . . 17 
 
 To C . . 9 
 
 ByC 
 
 . . 19 
 
 ToC 
 
 . . 41 
 
 ByC . . 6 
 
 To D . . 14 
 
 ByD 
 
 . . 32 
 
 ToD 
 
 . . 28 
 
 ByD . . II 
 
 To E . . 15 
 
 ByE 
 
 . . 4 
 
 ToE 
 
 . . 4 
 
 ByE . . 25 
 
 To Balance 23 
 
 
 
 
 
 Bjf BaJance 37 
 
 78 
 
 
 78 
 
 
 96 
 
 96 
 
ON BOOK-KEEPING, 
 
 185 
 
 C, Debtor. 
 
 C, Creditor. 
 
 D, Debtor. 
 
 D, Creditor. 
 
 To A . . 19 
 
 By A . . 9 ] To A . . 32 
 
 By A . . 14 
 
 ToB . . 6 
 
 By B . . 41 1 ToB . . 11 
 
 By B . . 28 
 
 ToD . . 16 
 
 ByD . . 10 
 
 To C . . 10 
 
 ByC . . 16 
 
 To.E . . 60 
 
 By E . . 2 
 
 ToE . . I 
 
 ByE . . 3 
 
 
 By Balance 39 
 
 To Balance 7 
 
 
 lOI 
 
 lOI f 61 
 
 6^ 
 
 E, Debtor. 
 
 E, Creditor. 
 
 Balance, Debtor. 
 
 Balance, Cred. 
 
 ToA . . 4 
 
 ByA . . 15 
 
 ToB . . 37 
 
 ByA . . 23 
 
 To B . . 25 
 
 ByB . . 4 
 
 ToC . . 39 
 
 ByD . . 7 
 
 ToC . . 2 
 
 ByC . . 60 
 
 
 ByE . . 46 
 
 ToD . . 3 
 
 ByD . . I 
 
 76 
 
 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 
 
 b2 
 
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 dis- 
 charges 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 balan- 
 cing 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 wliich 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 open- 
 ing 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 pro- 
 perty,,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 ;6'5oo 
 in cash at the commencement of the ledger. There will then appear 
 that the merchant has handed over to the cash-box :f 500, and in the 
 stock account will appear, " Stock creditor by cash, iffoo ;*' while in 
 
ON BOOK-KEEPING. 187 
 
 the Gash 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. 5^50," and in Smith and Co.'s account will appear 
 " Smith and Co. creditors by stock, £50." 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 £%o and sell for £()0 
 cash, it is clear that there is an entry " Cash debtor to goods j6'6o,'* 
 and " Goods creditor by cash i^6o." Now, there is an entry of ^80 
 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 £zo less than he might have 
 reckoned upon, without any explanation of the reason ; and there 
 would be a failure of the principle of double -entry. Since it is 
 convenient that the balance-account of the goods should merely re- 
 present the stock in hand at the close, the account of goods there- 
 fore lays the responsibility of £zo upon the profit-and-loss account, 
 or there is the entry " Goods creditor by profit-and-loss, ^£'"20," and 
 also "Profit-and-loss debtor to goods, £'2.0.'"' 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 ad- 
 just 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 ^^200, 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, jSzoo," " Profit- 
 and-loss debtor to house expenses, jf 200." 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 
 ^onsbt of nothing but the necessary matters of account for the next year'^s 
 ledger. 
 
 This transference of accounts^ or transfusion of one account into 
 another, requires attentive coiuideration. The receiving account be- 
 comes creditor for the credits, and debtor for the debits, of the trans- 
 mitting 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-accouut. If the two stand as in Roman 
 letters, the processes in Italic letters will occur before the final close. 
 
 A, Debtor. 
 
 To sundries £100 
 ToB .... 400 
 
 £500 
 
 A, Creditor. 
 By sundries £500 
 
 £500 
 
 B, Debtor. 
 
 To sundries £600 
 To Balance 200 
 
 £800 
 
 B, Creditor. 
 
 By sundries £400 
 By A ... . 400 
 
 £800 
 
 And the entry in the balance account will be, " Creditor by B, £200," 
 shewing that, on these two accounts, the credits exceed the debits by 
 £200. 
 
 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 
 
 • The treatises on book-keeping have described this difference In as peculiar a 
 
ON BOOK-KEEPING. 189 
 
 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 incor- 
 poration 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 liabi- 
 lities. 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 suppoeed to keep 
 them. 
 
190 
 
 4.PPBNDIX. 
 
 APPENDIX VIII. 
 
 ON THE REDUCTION OF FRACTIONS TO OTHERS OP 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, und bring all the quotients 
 into a line. Then write down, 
 
 I 2d Quot. 
 
 ist Quot. 1st Quot. X 2d Quot.+ 1 
 
 Then take the third quotient, multiply the n iraerator 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 -^—^ — . 
 
 13128 
 
 9131)13128^1,2 This is the process for finding the 
 
 1137 3997(3, I , , f , 
 
 5CI 586(1 ic greatest common measure of 9131 and 
 
 201 35(i»2, 13128 in its most compact form, and 
 
 20 9(^1, 5 
 
 8 I the quotients and fractions are : 
 
 12311 15 I 2 I 8 
 
 127 9 16 249 265 779 1044 913 1 
 I 3 10 13 23 358 381 1120 1501 13128 
 
 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 : 
 
 ist Fraction = 
 
 I I 
 
 
 
 ist Quot. I 
 
 
 2d Fraction = 
 
 2d Quot. 2 
 
 
 
 1st Quot. X 2d Quot. + 1 ~ 3 
 
 
 3d Fraction « 
 
 ad Num'.x 3d Quot. + ist Num'. 
 2d Den'. X3d Quot. + ist Den^ 
 
 2x3+1 
 3x3+1 
 
 r 2- 
 10 
 
 4th Fraction = 
 
 3d Num'.x4th Quot. + 2d Num'. 
 3d Den'. X4th Quot. + zd Den'. 
 
 7x1+2 
 10x1+3 
 
 3 -1 
 
ON REDUCTION OF FRACTIONS. 191 
 
 and so on. But we have done something more than merely reascend to 
 
 the original fraction by means of the quotients. The set of fractions, 
 
 -, -, — , — , &c. are continually approaching in value to the original 
 
 I 3 lo 13 
 
 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 
 
 fraction than any of those before it. Thus, - is too great, and - is 
 
 2 , ^ I 3 
 
 too small; but - is not so much too small as - is too great. And 
 
 7 3 _ I 2 
 
 again, — , though too great, is not so much too great as - is too small. 
 10 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 
 
 I 12 
 
 denominator. Thus, - does not err by so much as -, nor - by so much 
 17 ' 19 3i3 
 
 as — , nor — by so much as , nor — by so much as , &c. 
 
 30 10 130 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 nximerator than 249, and a less denominator 
 
 9i'?i 249 
 
 than 358, can come so near to -—— as —~. 
 
 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, 
 
 1 6 249 
 in our instance, we have — and — ^, which, with a common denominator, 
 
 23 358 
 
 23x358, have 5728 and 5727 for their numerators. 
 
 As another example, let us examine this question : The length of 
 
 the year is 365-24224 days, which is called in common life 365- days. 
 
 24224 4 
 
 Take the fraction — -^-— ^, and proceed as in the rule. 
 
 100000 
 
 24224)100000(4, 7, I, 4, 9, 2 
 2496 3 104 
 64 608 
 o 32 
 
 I 2. A _i2_ ill 757 
 4 29 33 161 1482 3125 
 
 and is '24224 in its lowest terms. Hence, it appears that the 
 
 3^2.5 
 
 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 ii6 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 80 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 : 
 
 V43 = 6 + .... S®* down the number whose 
 
 I 545 545 I 66 
 7639293671 
 
 I I 3 I 5 I 3 
 
 154, &c. square root is wanted, say 43. 
 7 " 3i *^c. rpjjjg square root is 6 and a frao- 
 
 I 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 6', a', c', formed in this order, 
 being thus, 
 
 a a' = excess of iV, already formed, over a. 
 
 b b' = quotient of 43— a'* divided by b. 
 
 e c' = integer in the quotient of 6+a divided by b'. 
 
 Thus the second row is formed from the first, as under : 
 
 excess of 7x1 (both just found) over 6. 
 43—6x6 divided by 1. 
 1 1 1 = integer of 6+6 divided by 7 (just found). 
 
 The third row is formed from the second, thus : 
 15= excess of 1x6 over i. 
 76= 43— IX I divided by 7. 
 11= 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, i, 3, i, 5, i, 3, &c. and use them by the rule given at the 
 beginning of this article, as follows : 
 
 1131513 1 I, &c. 
 
 I I 4 5 29 34 131 165 296 
 I 2 7 9 52 61 235 296 531 
 
GENERAL PROPERTIED OF NUMBERS. 193 
 
 Hence, 6 — - is very near the square root of 43, not erring by so much as 
 I ^96 
 
 I I 
 z z 
 
 ^96x531 i6c 1941 
 
 If we try it, we shall find 6 ■; to be -^-r, the square of which h 
 3767481 ■ 7 ^96 296 
 
 ^7616 '""""^^'ITeTe- 
 
 This rule is of use when it is frequently wanted to use one square 
 
 root, and therefore desirable to ascertain whether any easy ajiproximation 
 
 exists by means of a common fraction. For example, a/z is often used. 
 
 29 
 ^2 = i-i- Here it appears that i — does not 
 
 *"* I 70 99 
 
 err bv r- ; consequently, — or 
 
 100- i 70x169. . 70 
 
 is, considering the ease of the 
 
 70 
 122222 2 operation, a fair approximation. In 
 
 99 
 
 1 2 5 12 29 70 fact, — is 1-4142857... the truth being 
 
 2 5 12 29 70 169 1.4142135... 
 
 The following is an additional example : 
 
 V19 = 4+... 
 4123 3 2 442 
 
 I I 3 5 2 5 313 
 
 41213 I 2821312, Sec. 
 
 1*4 5 14- „ 
 - - — — —,&(;. 
 2 3 II 14 39 
 
 APPENDIX IX. 
 
 ON SOME GENERAL PROPERTIES OF NUMBERS. 
 
 Prop. 1. If a fraction be reduced to its lowest terms, so called,* that 
 is, if neither numerator nor denominator be divisible by any integer 
 greater than unity, then no fraction of a smaller numerator and deno- 
 minator can have the same value. 
 
 Let - be a fraction in which a and b have no common measure 
 
 greater than unity : and, if possible, let ;; he a fraction of the same value, 
 
 ** a c a b 
 
 c being less than a, and d less than b. Now, since 7 = -, we have - = - ; 
 
 o a c a 
 
 • This theorem shews that what is called reducing a fraction to its lowest terms 
 
 (namely, dividing numerator and denominator by their jfreatest common measure), 
 
 is correctly so called. 
 
 B 
 
194 APPENDIX. 
 
 let m be the integer quotient of these last fractions (which must exist, 
 since a > c, 6 > rf), and let e and / be the remainders. Then 
 
 a mc+e c me 
 b md+f d md 
 
 Hence, - and — - must be equal, for if not, , , would lie between 
 / md md +j 
 
 — — and -, instead of being equal to the former. Hence, t = ":.» 
 md J o 1 of 
 
 so that if a fraction whose numerator and denominator have no com- 
 mon measure greater than unity, be equal to a fraction of lower 
 
 numerator and denominator, it is equal to another in which the nume- 
 
 a e , 
 rator and denominator are still lower. If we proceed with - = - m a 
 
 d (J "J 
 
 similar manner, we find 7 = 7 where g<e^ h<f, and so on. Now, if 
 
 o h 
 there be any process which perpetually diminishes the terms of a frac- 
 tion by one or more units at every step, it must at last bring either the 
 
 numerator or denominator, or both, to o. Let -7 = — be one of the 
 
 h w 
 
 steps, and let a = kv+x, b = kw+v ; so that = — . Now, if a? = o 
 
 kw+y w 
 
 but not v, this is absurd, for it gives -: = •;— . A similar absurdity 
 
 kw+y kw 
 follows if y be o, but not x ; and if both x and y be = o,then a = kv^ b = 
 
 kw^ or a and b have a common measure, k. Now k must be greater 
 than I, for v and w are less than c and cf, which by hypothesis are less 
 than a and b. Consequently a and b have a common measure k greater 
 than I, which by hypothesis they have not. If, then, a and b be in- 
 tegers not divisible by any integer greater than i, the fraction - is really 
 
 b 
 in its lowest terms. Also a and b are said to be prime to one another. 
 
 Prop. 2. If the product ab be divisible by e, and if c be prime to i, 
 
 it must divide a. Let — = rf, then- = -. Now - is in its lowest terms ; 
 
 c c a c 
 
 therefore, by the last proposition, d and a must have a common mea- 
 sure. Let the greatest common measure be A:, and let a = kl, d = km. 
 
 -y, b km m , wi . , . ., , , , , . b 
 
 1 nen - = — — = -— , and — is also in its lowest terms; but so is -; 
 
 C nt It C 
 
 therefore we must have m = b, I = c, for otherwise a fraction in its 
 lowest terms would be equal to another of lower terms. Therefore 
 a = kc^ or a is divisible by c. And from this it follows, that if a num- 
 ber be prime to two others, it is prime to their product. Let a be prime 
 
GENERAL PROPERTIES OF NUMBERS. 195 
 
 to b and c, then no measure of a can measure either b or c, and no 
 such measure can measure the product be ; for any measure of be which 
 is prime to one must measure the other. 
 
 Prop. 3. If a be prime to b, it is prime to all the powers of A. 
 Every measure* of a is prime to J, and therefore does not divide L 
 Hence, by the last, no measure of a divides 6^ ; hence, a is prime to 
 b'^, and so is every measure of it ; therefore, no measure of a divides bb'K 
 consequently a is prime to b^, and so on. 
 
 Hence, if a be prime to b, a cannot divide without remainder any 
 
 power of b. This is the reason why no fraction can be made into a 
 
 decimal unless its denominator be measured by no primef numbers ex- 
 
 a c 
 cept 2 and 5. For if -7 = , which last is the general form of a deci- 
 
 a " ^°" io"a ' 
 
 mal fraction, let - be in its lowest terms ; then — - — is an integer, whence 
 
 b b 
 
 (Prop. 2) b must divide lo", and so must all the divisors of 6. If, then, 
 
 among the divisors of b there be any prime numbers except 2 and 5, 
 
 we have a prime number (which is of course a number prime to 10) not 
 
 dividing 10, but dividing one of its powers, which is absurd. 
 
 Prop. 4. If 6 be prime to a, all the multiples of 6, as b, 26, .. . up to 
 
 (a— 1)6 must leave different remainders when divided by a. For if, m 
 
 being greater than n, and both less than a, we have mb and nb giving the 
 
 same remainder, it follows that mb—nb^ or (jn—n)b, is divisible by a; 
 
 whence (Prop. 2), a divides m— n, a number less than itself, which is 
 
 absurd. 
 
 If a number be divided into its prime factors, or reduced to a pro- 
 duct of prime numbers only (as in 360 = 2x2x2x3x3x5), and if a, 6, c, 
 &c. be the prime factors, and a, ^8, 7, &c. the number of times they seve- 
 rally enter, so that the number is a*xi'^ xc''x&c., then this can be done 
 in only one way : For any prime number «, not included in the above 
 
 • For that which measures a measure is itself a measure; so that if a measure of 
 a could have a measure in common with b, a itself would have a common measure 
 with 6. 
 
 t A prime number is one which is prime to all numbers except its own multiples, 
 or has no divisors except 1 and itself. 
 
196 APPENDIX. 
 
 list, is prime to a, and therefore to a*, to b and therefore to 5^, and there- 
 fore to a'^'y.l^ . Proceeding in this way, we prove that v is prime to the 
 complete product above, or to the given number itself. 
 
 The number of divisors which the preceding number a'^'b^c^ .... can 
 have, o and itself included, is (a+i)(i8+i)(7+i).... For o* has the 
 divisors i, a^a^ ...a* and no others, a+i in all. Similarly, b^ has $+1 
 divisors, and so on. Now as all the divisors are made by multiplying 
 together one out of each set, their number (page 202) is (a+i)(j8+i) 
 
 (7+1).... 
 
 If a number, w, be divisible by certain prime numbers, say 3, 5, 7, 1 1, 
 
 then the third part of all the numbers up to n is divisible by 3, the fifth 
 
 part by 5, and so on. But more than this: when the multiples of 3 
 
 are omitted, exactly the fifth part of those which remain are divisible 
 
 by 5 ; for the fifth part of the whole are divisible by 5, and the fifth 
 
 part of those which are removed are divisible by 5, therefore the fifth 
 
 part of those which are left are divisible by 5. Again, because the 
 
 seventh part of the whole are divisible by 7, and the seventh part of 
 
 those which are divisible by 3, or by 5, or by 15, it follows that when 
 
 all those which are multiples of 3 or 5, or both, are removed, the seventh 
 
 part of those which remain are divisible by 7 ; and so on. Hence, the 
 
 number of numbers not exceeding w, which are not divisible by 3, 5, 7, 
 
 10 6 4 2 
 or 1 1, is — of - of - of - of n. Proceeding in this way, we find that the 
 
 number of numbers which are prime to w, that is, which are not divisible 
 
 by any one of its prime factors, a, d, c,. . . is 
 
 n^Zi izi £Z! or a'^-^b^^c'^'^ ...{ar-i)(b-i){c-i).... 
 
 a b c \ f\ j\ J 
 
 Thus, 360 being a'3'5, its number of divisors is 4x3x2, or 24, and there 
 are 2^3. 1.2.4 or 9^ numbers less than 360 which are prime to it. 
 
 Prop. 5. If a be prime to 6, then the terms of the series, o, a'*, a',... 
 severally divided by J, must all leave different remainders, until i 
 occurs as a remainder, after which the cycle of remainders will be again 
 repeated. 
 
 Let a-hb give the remainder r (not unity) ; then a^-i-b gives the same 
 remainder as ra-i-b, which (Prop. 4) cannot be r : let it be s. The » 
 
GENERAL PROPERTIES OF NUMBERS. 197 
 
 a^-i-b gives the same remainder as «a-^i, which (Prop. 4) cannot be 
 either r or s, unless « be i : let it be t. Then a*-i-b gives the same 
 remainder as ta-i-b ; if < be not i, this cannot be either r, s, or ^ : let 
 it be u. So we go on getting different remainders, until i occurs as 
 a remainder ; after which, at the next step, the remainder of a-v-b is 
 repeated. Now, i must come at last; for division by b cannot give 
 any remainders but o, i, 2,.... h—i ; and o never arrives (Prop. 3), 
 so that as soon as 6—2 different remainders have occurred, no one of 
 which is unity, the next, which must be different from all that precede, 
 must be i. If not before, then at ah'^ we must have a remainder i ; 
 after which the cycle will obviously be repeated. 
 
 Thus, 7, 7"^, 7^, 7*, &c. will, when divided by 5, be found to give 
 the remainders 2, 4, 3, i, &c. 
 
 Prop. 6. The difference of two wth powers is always divisible with- 
 out remainder by the difference of the roots ; or a"*— 6'" is divisible by 
 a—b ; for 
 
 a"»-i"' = ar'-ar-^b+a'^-^b-b'" = ar-\a-b)+b{ar-'^-h^-^) 
 
 From which, if a"'— i— 6"*-* is divisible by a— i, so is a*"— i*". But a—b 
 is divisible by a—b ; so therefore is a^—b^ ; so therefore is a^—b^ ; and 
 so on. 
 
 Therefore, if a and 5, divided by c, leave the same remainder, a^ 
 and 6^, a^ and i^, &c. severally divided by c, leave the same remainders; 
 for this means that a—b is divisible by c. But o*"— 6"* is divisible by 
 a— J, and therefore by every measure of a— 6, or by c ; but a^—b"* cannot 
 be divisible by c, unless a*" and J"*, severally divided by c, give the 
 same remainder. 
 
 Prop. 7. If 6 be a prime number, and a be not divisible by 6, then 
 a' and (a— i)*+i leave the same remainder when divided by b. This 
 proposition cannot be proved here, as it requires a little more of algebra 
 than the reader of this work possesses.* 
 
 Prop. 8. In the last case, a'^^ divided by b leaves a remainder i. 
 
 * Expand (a — l)*by the binomial theorem; shew that when b is a prime number 
 every coeflBcient which is not unity is divisible by b ; and the proposition follows. 
 
 82 
 
198 APPENDIX. 
 
 From the last, a^—a leaves the same remainder as (a— i)*+i— a or 
 (a— i)*— (a— i) ; that is, the remainder of af'—a is not altered if a be 
 reduced by a unit. By the same rule, it may be reduced another unit, 
 and so on, still without any alteration of the remainder. At last it 
 becomes i*— i, or o, the remainder of which is o. Accordingly, a^—a^ 
 which is a(a*~^— i), is divisible by b\ and since b is prime to a, it 
 must (Prop. 2) divide o*-*— i ; that is, o*-^, divided by 6, leaves a 
 remainder i, if i be a prime number and a be not divisible by b. 
 
 From the above it appears (Prop. 5 and 7), that if a be prime to 
 6, the set i, a, a^, a\ &c. successively divided by 6, give a set of 
 remainders beginning with i, and in which i occurs again at a*~^, if 
 not before, and at a!'~^ certainly (whether before or not), if 6 be a 
 prime number. From the point at which i occurs, the cycle of re- 
 mainders recommences, and i is always the beginning of a cycle. If, 
 then, a"* be the first power which gives i for remainder, m must either 
 be 5— I, or a measure of it, when b is a prime number. 
 
 But if we divide the terms of the series m, »»a, ma^, ma^^ &c. by 
 A, m being less than J, we have cycles of remainders beginning with m. 
 If I, r, *, ?, &c. be the first set of remainders, then the second set is 
 the set of remainders arising from »n, mr, ms, mt, &c. If i never 
 occur in the first set before a*~* (except at the beginning), then all 
 the numbers under b—z inclusive are found among the set i, r, *, /, 
 &c. ; and if m be prime to b (Prop. 4), all the same numbers are found, 
 in a different order, among the remainders of wi, wir, &c But should 
 it happen that the set i, r, s, <, &c. is not complete, then wi, wir, nw, 
 &c. may give a different set of remainders. 
 
 All these last theorems are constantly verified in the process for 
 
 reducing a fraction to a decimal fraction. If m be prime to 6, or the 
 
 fraction -r- in its lowest terms, the process involves the successive 
 b 
 
 division of m, mxio, toxic*, &c. by b. This process can never come 
 to an end unless some power of lo, say lo", is divisible by b ; which 
 cannot be, if b contain any prime factors except 2 and 5. In every 
 other case the quotient repeats itself, the repeating part sometimes 
 commencing from the first figure, sometimes from a later figure. Thus. 
 
GENERAL PROPKRTIES OF NUMBERS. 199 
 
 - yields '142857 142857, &c., but — gives •07(142857X142857), &c., 
 7 I 14 
 
 snd — gives •03(57i428)(57i428), &c. 
 
 In -— , the quotient always repeats from the very beginning whenever 
 
 6 is a prime number and m is less than b ; and the number of iigures 
 
 in the repeating part is then always 6—1, or a measure of it. That it 
 
 must be so, appears from the above propositions. 
 
 Before proceeding farther, we write down the repeating part of a 
 
 quotient, with the remainders which are left after the several figures 
 
 are formed. Let the fraction be — , we have 
 
 Oio5i 581484^639552169742^31137116841271 
 This may be read thus : lo by 17, quotient o, remainder 10 ; 10* by 
 17, quotient 05, remainder 15 ; lo^ by 17, quotient 058, remainder 14; 
 and so on. It thus appears that lo'^ by 17 leaves a remainder i, which 
 is according to the theorem. 
 
 If we multiply 0588, &c. by ant/ number under 17, the same cycle 
 is obtained with a different beginning. Thus, if we multiply by 13, we 
 have 7647058823529411 
 
 beginning with what comes after remainder 13 in the first number. If 
 
 we multiply by 7, we have 4117, &c. The reason is obvious: — X13, 
 13 '7 
 
 or — , when turned into a decimal fraction, starts with the divisor 130, 
 ^7 I . . 
 
 and we proceed just as we do in forming — , when within four figures of 
 
 17 
 
 the close of the cycle. 
 
 It will also be seen, that in the last half of the cycle the quotient figures 
 are complements to 9 of those in the first half, and that the remainders 
 are complements to 17. Thus, in 0^05^58^484, &c. and 974213113, 
 &c. we see 0+9 = 9, 5+4 = 9, 8+i = 9, &c., and 10+7 = 17, 15+2 = 17, 
 14+3 = 17, &c. "We may shew the necessity of this as follows: If 
 the remainder i never occur till we come to use a*~^, then, b being 
 prime, b—i is even; let it be rk. Accordingly, a'''*— i is divisible by 
 b\ but this is the product of a*— i and a*+i, one of which must be 
 divisible by b. It cannot be a*— i, for then a power of a preceding the 
 {h—i)\h would leave remainder i, which is not the case in our instance; 
 it must then be a*+i, so that a^ divided by b leaves a remainder 6—1; 
 
200 APPENDIX. 
 
 and the kth step concludes the first half of the process. Accordingly, 
 
 in our instance, we see, b being 17 and a being 10, that remainder i6 
 
 occurs at the 8th step of the process. At the next step, the remainder 
 
 is that yielded by 10(6—1), or 96+6—10, which gives the remainder 6— 10. 
 
 But the first remainder of all was 10, and io+(6— 10) = b. If ever this 
 
 complemental character occur in any step, it must continue, which we 
 
 shew as follows : Let r be a remainder, and b—r a subsequent remainder, 
 
 the sum being b. At the next step after the first remainder, we divide 
 
 lor by 6, and, at the next step after the second remainder, we divide 
 
 106— lor by b. Now, since the sum of lor and 106— icr is divisible 
 
 by b, the two remainders from these new steps must be such as added 
 
 together will give 6, and so on ; and the quotients added together must 
 
 give 9, for the sum of the remainders lor and 106— lor yields a quotient 
 
 10, of which the two remainders give i. 
 
 If — and 7- be taken, the repeating parts will be found to contain 
 59 61 
 
 58 and 60 figures. Of these we write down only the first halves, as 
 the reader may supply the rest by the complemental property jxist 
 given. 
 
 01694915254237288135593220338, &c. 
 0163934426229508 1967213 I 147540, &c. 
 
 Here, then, are two numbers, the first of which multiplied by any 
 number under 59, and the second by any number under 61, can have 
 the products formed by carrying certain of the figures from one end to 
 the other. 
 
 But, b being still prime, it may happen that remainder i may occur 
 before 6—1 figures are obtained ; in which case, as shewn, the number 
 of figures must be a measure of 6—1. For example, take — . Tlie 
 repeating quotient, written as above, has only 5 figures, and 5 measures 
 41-1. 
 
 Oio*i84ie3379i 
 
 Now, this period, it will be found, has its figures merely transposed, if 
 we multiply by 10, 18, 16, or 37. But if we multiply by any other 
 number under 41, we convert this period into the period of another 
 
ON COMBINATIONS. 201 
 
 fraction whose denominator is 41. The following are 8 periods which 
 may be found. 
 
 Ol0^184l63379l 
 030436^32733^2 
 "-"so/iaS? ^973 
 04093172352564 
 
 ^9'^8'39921 Ss 
 119426^1431746 
 2286348122-38911 
 327624535^22515 
 
 To find — , look out for m among the remainders, and take the period 
 41 34 
 
 in which it is, beginning after the remainder. Thus, — is '8292682926, 
 
 ic 41 
 
 &c., and — is '3658536585, &c. These periods are complemental, 
 
 41 
 four and four, as 02439 ^^^ 975^0, 07317 and 92682, &c. And if 
 
 the first number, 02439, ^^ multiplied by any number under 41, look 
 
 for that number among the remainders, and the product is found in the 
 
 period of that remainder by beginning after the remainder. Thus, 
 
 02439 multiplied by 23 gives 56097, and by 6 gives 14634. 
 
 The reader may try to decipher for himself how it is that, with no 
 
 more figures than the following, we can extend the result of our division. 
 
 The fraction of which the period is to be found is — . 
 
 87 
 
 87)100(01149425 
 
 
 
 130 
 
 
 
 430 
 
 
 
 820 - 
 
 
 01149425x25 
 
 370 
 
 
 28735625x25 
 
 220 
 
 
 718390625x25 
 
 460 
 
 
 17959765625x25 
 
 25 
 
 
 448994140625 
 
 OII49425287356 
 
 ^5 
 
 
 
 7 
 
 [8390625 
 
 1795976 5625 
 
 
 
 448994 
 
 01149425287356321839080459771011494 
 
 APPENDIX X. 
 
 ON COMBINATIONS. 
 
 There are some things connected with combinations which I place in 
 an appendix, because I intend to demonstrate them more briefly than 
 the matters in the text. 
 
 Suppose a number of boxes, say 4, in each of which there are 
 
202 APPENDIX. 
 
 counters, say 5, 7, 3, and 11 severally. In how many ways can one 
 counter be taken out of each box, the order of going to the boxes not 
 being regarded. Answer^ in 5x7x3x11 ways. For out of the first box 
 we may draw a counter in 5 different ways, and to each such drawing 
 we may annex a drawing from the second in 7 different ways — giving 
 5x7 ways of making a drawing from the first two. To each of these 
 we may annex a drawing from the third box in 3 ways — giving 5x7x3 
 drawings from the first three ; and so on. The following statements may 
 now be easily demonstrated, and similar ones made as to other cases. 
 
 If the order of going to the boxes make a diflference, and if a, 6, c, d 
 be the numbers of counters in the several boxes, there are 4X2X3XIX 
 ax6xcx(/ distinct ways. If we want to draw, say a out of the first box, 
 3 out of the second, i out of the third, and 3 out of the fourth, and if 
 the order of the boxes be not considered, the number of ways is 
 
 a— 1 ,6— li— 2 ,rf— irf— 2 
 
 a X X c X a 
 
 223 i 3 
 
 If the order of going to the boxes be considered, we must multiply the 
 preceding by 4x3x2x1. If the order of the drawings out of the boxes 
 makes a difference, but not the order of the boxes, then the number 
 of ways is 
 
 a{ar-i)b{b-i)[h-z)cd{d-x){d-z) 
 
 The nth power of a, or a", represents the number of ways in which 
 a counters differently marked can be distributed in n boxes, order of 
 placing them in each box not being considered. Suppose we want to 
 distribute 4 differently-marked counters among 7 boxes. The first 
 counter may go into either box, which gives 7 ways ; the second counter 
 may go into either ; and any of the first 7 allotments may be combined 
 with any one of the second 7, giving 7x7 distinct ways; the third 
 counter varies each of these in 7 different ways, giving 7x7x7 in all; 
 and so on. But if the counters be un distinguishable, the problem is a 
 very different thing. 
 
 Required the number of ways in which a number can be compounded 
 of other numbers, different orders counting as different ways. Thus, 
 
ON COMBINATIONS. 203 
 
 1+3+1 and 1+1+3 are to be considered as distinct ways of making 5. 
 It will be obvious, on a little examination, that each number can be 
 composed in exactly twice as many ways as the preceding number. 
 Take 8 for instance. If every possible way of making 7 be written 
 down, 8 may be made either by increasing the last component by a 
 unit, or by annexing a unit at the end. Thus, 1+3+2+1 may yield 
 1+3+2+2, or 1+3-12+1+1: and all the ways of making 8 will thus be 
 obtained ; for any way of making 8, say a+b+c+d, must proceed from 
 the following mode of making 7, a+b+c+(d—i). Now, (d—i) is either 
 o — that is, d is unity and is struck out — or (d—i) remains, a number 
 I less than d. Hence it follows that the number of ways of making 
 n is 2"~'. For there is obviously i way of making i, 2 of making 
 2 ; then there must be, by our rule, z^ ways of making 3, 2^ ways of 
 making 4 ; and so on. 
 
 { 
 
 \ 1+2+1 
 I1+3 
 
 2+1+1 
 
 /2+2 
 
 \r 
 
 This table exhibits the ways of making i, 2, 3, and 4. Hence it 
 follows (which I leave the reader to investigate) that there are twice 
 as many ways of forming a+b as there are of forming a and then 
 annexing to it a formation of b ; four times as many ways of forming 
 a-i-b+c as there are of annexing to a formation of a formations of b and 
 of c ; and so on. Also, in summing numbers which make up a+i, there 
 are ways in which a is a rest, and ways in which it is not, and as many of 
 one as of the other. 
 
 Required the number of ways in which a number can be compounded 
 of odd numbers, different orders counting as different ways. If a be 
 the number of ways in which n can be so made, and b the number of 
 ways in which w+i can be made, then a+b must be the number of 
 ways in which n+z can be made; for every way of making 12 out of 
 odd numbers is either a way of making 10 with the last number 
 
•20 1 APPENDIX. 
 
 increased by 2, or a way of making 11 with a i annexed. Thus, 
 1+5+3+3 gives 12, formed from 1+5+3+1 giving 10. But 1+9+1+1 is 
 formed from 1+9+1 giving 11. Consequently, the number of ways 
 of forming 12 is the sum of the number of ways of forming 10 and of 
 forming 11. Now, i can only be formed in i way, and 2 can only 
 be formed in 1 way; hence 3 can only be formed in i+i or 2 ways, 4 
 in only 1+2 or 3 ways. If we take the series 1, 1, 2, 3, 5, 8, 13, 21, 
 34> 55» ^9* &c. in which each number is the sum of the two preceding, 
 then the wth number of this set is the number of ways (orders counting) 
 in which n can be formed of odd numbers. Thus, 10 can be formed 
 in 55 ways, i.i in 89 ways, &c. 
 
 Shew that the number of ways in which mk can be made of numbers 
 divisible by m (orders counting) is 2*-^ 
 
 In the two series, 11 12 3469 13 19 28, &c. 
 01011122 3 4 5, &c., 
 
 the first has each new term after the third equal to the sum of the 
 last and last but two ; the second has each new term after the third 
 equal to the sum of the last but one and last but two. Shew that 
 the nth number in the first is the number of ways in which n can be 
 made up of numbers which, divided by 3, leave a remainder i ; and 
 that the nth number in the second is the number of ways in which 
 n can be made up of numbers which, divided by 3, leave a remainder 2. 
 
 It is very easy to shew in how many ways a number can be made 
 up of a given number of numbers, if difi*erent orders count as different 
 ways. Suppose, for instance, we would know in how many ways la 
 can be thus made of 7 numbers. If we write down 12 units, there 
 are 1 1 intervals between unit and unit. There is no way of making 
 12 out of 7 numbers which does not answer to distributing 6 partition- 
 marks in the intervals, i in each of 6, and collecting all the units 
 which are not separated by partition-marks. Thus, 1+ 1+3+2+ 1+2+2, 
 which is one way of making 12 out of 7 numbers, answers to 
 
ON COMBINATIONS. 205 
 
 in which the partition-marks come in the ist, 2d, 5th, 7th, 8th, and 
 loth of the II intervals. Consequently, to ask in how many ways iz 
 can be made of 7 numbers, is to ask in how many ways 6 partition- 
 marks can be placed in 11 intervals; or, how many combinations or 
 selections can be made of 6 out of 1 1. The answer is, 
 
 11x10x9x8x7x6 
 
 —, or 462. 
 
 1x2x3x4x5x6 
 
 Let us denote by wi„ the number of ways in which m things can 
 be taken out of n things, so that m„ is the abbreviation for 
 
 n— 1 W — 2 . 71— 7«+l 
 
 nx X ....as far as 
 
 23 m 
 
 Then »n„ also represents the number of ways in which m+i numbers 
 can be put together to make 71+ 1. What we proved above is, that 6j j 
 is the number of ways in which we can put together 7 numbers to make 
 12. There will now be no difficulty in proving the following : 
 
 2"= n-i„+2„+3„, 
 
 In the preceding question, o did not enter into the list of numbers 
 used. Thus, 3+1+0+0 was not considered as one of the ways of putting 
 together four numbers to make 5. But let us now ask, what is the number 
 of ways of putting together 7 numbers to make 12, allowing o to be 
 in the list of numbers. There can be no more (nor fewer) ways of doing 
 this than of putting 7 numbers together, among which o is not included, 
 to make 19. Take every way of making 12 (o included), and put on 
 I to each number, and we get a way of making 19 (o not included). 
 Take any way of making 19 (o not included), and strike off i from 
 each number, and we have one of the ways of making 12 (o included). 
 Accordingly, 6^3 is the number of ways of putting together 7 numbers 
 (o being allowed) to make 12. And (tti— i)„+m-i is the number of 
 ways of putting together m numbers to make n, o being included. 
 
 This last amounts to the solution of the following: In how many 
 ways can n counters (undistinguishable from each other) be distributed 
 into m boxes ? And the following will now be easily proved : The 
 number of ways of distributing undistinguishable counters into b boxes 
 
205 APPENDIX. 
 
 is I — i)6+c_i, if any box or boxes may be left empty. But if there 
 
 must be i at least in each box, the number of ways is (6— i)e_i; if 
 
 there must be 2 at least in each box, it is (6— I)e_^_l ; if there must 
 
 be 3 at least in each box, it is (6— i)c_2fr-i ; and so on. 
 
 The number of ways in which m odd numbers can be put together 
 
 to make n, is tlie same as the number of ways in which m even numbers 
 
 (o included) can be put together to make n—m ; and this is the number 
 
 of ways in which m numbers (odd or even, o included) can be put 
 
 together to make -(n—m). Accordingly, the number of ways in which 
 
 2 
 m odd numbers can be put together to make n is the same as the 
 
 number of combinations of m—i things out of -(w— »») + »»— i, or 
 
 -(w+m)— I. Unless n and m be both even or both odd, the problem 
 
 is evidently impossible. 
 
 There are curious and useful relations existing between nimibers ol 
 
 combinations, some of which may readily be exhibited, under the simple 
 
 expression of m„ to stand for the number of ways in which m things 
 
 may be taken out of n. Suppose we have to take 5 out of 12 : Lot 
 
 the 12 things be marked a, b, c, &c. and set apart one of them, a. 
 
 Every collection of 5 out of the 12 either does or does not include a. 
 
 The number of the latter sort must be 5^; the number of the former 
 
 sort must be 4^, since it is the number of ways in which the other four 
 
 can be chosen out of all but a. Consequently, S12 ™ust be 5ii+4n» 
 
 and thus we prove in every case, 
 
 mn — mn-l-r{m— 1 )n-l 
 
 o„ and n„ both are i ; for there is but one way of taking none^ and 
 but one way of taking all. And again «i„ and (n—m)n are the same 
 things. And if m be greater than n, »w« is o ; tor there are no ways of 
 doing it. We make one of our precedmg results more symmetrical if 
 we write it thus, 
 
 2'' = o„+i„+2„+....+n„ 
 
 If we now write down the table of symbols in which the m+ith 
 
ON COMBINATIONS. 
 
 207 
 
 &c. 
 
 ^1 3i. &c. 
 
 ^3 33» &C. 
 
 number of the nth row represents m„^ the 
 number of combinations of m out of w, we 
 see it proved above that the law of for- 
 mation of this table is as follows : Each 
 number is to be the sum of the number 
 
 &c. I&c. &c, &c. &c. 
 above it and the number preceding the number above it. Now, the 
 first row must be i, i, o, o, o, &c. and the first column must te i, i, i, i, 
 &c. 80 that we have a table of the following kind, which may be carried 
 as far as we please : 
 
 
 o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 I 
 
 
 I 
 
 O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 2 
 
 I 
 
 
 
 
 
 
 
 
 
 .0 
 
 
 
 
 
 3 
 
 
 3 
 
 3 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 4 
 
 6 
 
 4 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 5 
 
 lO 
 
 10 
 
 5 
 
 1 
 
 
 
 
 
 
 
 
 
 6 
 
 
 6 
 
 15 
 
 20 
 
 15 
 
 6 
 
 I 
 
 
 
 
 
 
 
 7 
 
 
 7 
 
 21 
 
 35 
 
 35 
 
 21 
 
 7 
 
 I 
 
 
 
 
 
 8 
 
 
 8 
 
 28 
 
 56 
 
 70 
 
 56 
 
 28 
 
 8 
 
 1 
 
 
 
 9 
 
 
 9 
 
 36 
 
 84 
 
 126 
 
 126 
 
 84 
 
 36 
 
 9 
 
 I 
 
 lO 
 
 
 lO 
 
 45 
 
 120 
 
 210 
 
 252 
 
 210 
 
 120 
 
 45 
 
 10 
 
 Thus, in the row 9, imder the column headed 4, we see 126, which 
 is 9x8x7x6-^(1x2x3x4), the number of ways in which 4 can be chosen 
 out of 9, which we represent by 49. 
 
 If we add the several rows, we have i+i or 2, 1+2+ 1 or 2^, next 
 1+3+3+1 or 2^, &c. which verify a theorem already announced; and the 
 law of formation shews us that the several columns axe formed thus : 
 
 I I 
 I I 
 
 121 
 I 2 
 
 I 3 3 I 
 
 I 3 3 
 
 121 1331 1464 I, &c. 
 
 so that the sum in each row must be double of the sum in the precedinij;. 
 But we can carry the consequences of this mode of formation further. 
 If we make the powers of i+a- by actual algebraical multiplication, we 
 
208 APPENDIX. 
 
 see that the process makes the same oblique addition in the formation 
 of the numerical multipliers of the powers ot r. 
 
 
 Here are the second and third powers of i-vx : the fourth, we can tell 
 beforehand from the table, must be i+^je+Sai^+^j^+a* ; and so on. 
 Hence we have 
 
 (l+.2')" = 0„+l^+2^+2nX^+..'-+n„X'* 
 
 which is usually written with the s3Tiibol8 o„, i„, &c. at length, thus, 
 
 (i+xY = i+nx+n x^+n x^+Scc. 
 
 2 23 
 
 This is the simplest case of what in algebra is called the binomial 
 theorem. If instead of i+x we use x+a, we get 
 
 (^f«)" = x"+i„ax''-^+z„a^x"-^+ina^x"-^+....+nna" 
 We can make the same table in another form. If we take a row of 
 ciphers beginning with unity, and setting down the first, add the next, 
 and then the next, and so on, and then repeat the process with one 
 step less, and then again with one step less, we have the following : 
 
 I 
 
 n 
 
 
 
 
 
 
 
 
 
 I I 
 
 I 
 
 I 
 
 I 
 
 I 
 
 1 
 
 I 2 
 
 3 
 
 4 
 
 J 
 
 6 
 
 
 ^ 3 
 
 6 
 
 10 
 
 T5 
 
 
 
 I A 
 
 10 
 
 20 
 
 
 
 
 I 5 
 
 15 
 
 
 
 
 
 I 6 
 
 
 
 
 
 
 In the oblique columns we see i i, i 2 i, i 3 3 i, &c. the same as in 
 the original table, and formed by the same additions. If, before making 
 
ON COMBINATIONS. 209 
 
 the additions, we had always multiplied by a, we should have got the 
 several components of the powers of i+a, thus. 
 
 
 o 
 
 o 
 
 o 
 
 o 
 
 
 a 
 
 d^ 
 
 a^ 
 
 a' 
 
 
 za 
 
 3«2 
 
 ^^ 
 
 
 
 Za 
 
 ea^ 
 
 
 
 
 .4« 
 
 
 
 
 
 
 
 
 
 where the oblique columns i+a, i+za+a\ i+^a+^a^+a^ &c., give the 
 several powers of if a. If instead of beginning with i, o, o, &c. we 
 had begun with p, o, o, &c. we should have got p, ^0x4^, px6a^, &c. 
 at the bottom of the several columns ; and if we had written at the 
 top .r^, a;\ j?^, j?, i, we should have had all the materials for forming 
 p{je+ay by multiplying the terms at the top and bottom of each column 
 together, and adding the results. 
 
 Suppose we follow this mode of forming j9(a?+a)^+j(a?+a)-+r(j?+a)+*. 
 
 ar^ a; I x i 1 
 
 5^ o o r o 3 
 
 q qa qa^ r ra 
 
 q zqa r 
 9 
 
 pa^+ T^paa^-k- zpara^+pe^+qa^+zqax+qa^+ra^+ra+s 
 — pa^+('ipa+q) r^+( ■zpa^+2qa+r)a;-i^pa^+qa"+ra'rs 
 
 Now, observe that all this might be done in one process, by entering 
 q, r, and * under their proper powers of a; in the first process, as follows • 
 
 x^ or X I 
 
 p q r 8 
 
 p pa+q pa"+qa+r pa^+qar+ra+s 
 
 p 2pa+q 3^a-+2ya^ r 
 
 p 3pa+q 
 
 f 
 P 
 
 t2 
 
 *3 
 
 a=- 
 
 X 
 
 I 
 
 p 
 
 
 
 
 
 
 
 p 
 
 pa 
 
 pd^ 
 
 pa^ 
 
 p 
 
 zpa 
 
 Zpa' 
 
 
 p 
 
 Zpa 
 
 
 
 p 
 
 
 
 
210 APPENDIX. 
 
 This process* is the one used in Appendix XI., with the slight altera- 
 tion of varying the sign of the last letter, and making subtractions ' 
 instead of additions in the last column. As it stands, it is the most 
 convenient mode of writing w+a instead of j? in a large class of alge- 
 braical expressions. For instance, what does zx^+x^+^^'^+ja^+g become 
 when ir+5 is written instead of d? ? The expression, made complete, is, 
 zx^ + la* + oa^ + 3^- + 70? + 9 
 10375 
 278 1397 6994 
 1078 6787 
 
 z 
 
 II 
 
 55 
 
 z 
 
 21 
 
 160 
 
 z 
 
 31 
 
 315 
 
 z 
 
 41 
 
 520 
 
 ^ 51 
 
 Answer^ zx^-\-^ia^+^203^+z6^ia^+6'j2>'jx-{-6g<)^ 
 
 APPENDIX XL 
 
 ON HORNER'S METHOD OF SOLVING EQUATIONS. 
 
 TiiK rule given in this chapter is inserted on account of its excellence 
 as an exercise in computation. The examples chosen will require but 
 little use of algebraical signs, that they may be understood by those who 
 know no more of algebra than is contained in the present work. 
 To solve an equation such as 
 
 zx*+x^—ix = 416793, 
 or, as it is usually written, 
 
 2<r*+a?^—3J?— 416793 = o, 
 we must first ascertain by trial not only the first figure of the root, but 
 also the denomination of it : if it be a 2, for instance, we must know 
 whether it be 2, or 20, or 200, &c., or '2, or "02, or "002, &c. This must 
 
 • The principle of this mode of demonstration of Horner's method was stated in 
 Young's Algebra (1823), being the earliest elementary work In which that method 
 was given. 
 
HORNER S METHOD OF SOLVING EQUATIONS. 211 
 
 be found by trial ; and the shortest way of making the trial is as follows: 
 Write the expression in its complete form. In the preceding case the 
 form is not complete, and the complete form is 
 
 aj?*+c.t'3+ lar^— 3 J?— 41 6793. 
 
 To find what this is when x is any number, for instance, 3000, the best 
 way is to take the first multiplier (2), multiply it by 3000, and take in 
 the next multiplier (o), multiply the result by 3000, and take in the 
 next multiplier (i), and so on to the end, as follows : 
 
 2x3000+0 = 6000 ; 6000x3000+1 = 1 800000 1 
 18000001x3000—3 = 54000002997 
 54000002997x3000—416793 = 162000008574207 
 
 Now try the value of the above when x — 30. We have then, for the 
 steps, 60 (2x30+0), 1801, 54027, and lastly, 
 1620810-416793, 
 
 or J? = 30 makes the first terms greater than 416793. Now try jr = ao 
 which gives 40, 801, 160 17, and lastly, 
 
 320340-416793, 
 
 or X — 20 makes the first terms less than 416793. Between 20 and 30, 
 then, must be a value of of which makes 2x*+a^—^x equal to 416793. 
 And this is the preliminary step of the process. 
 
 Having got thus far, write down the coefiicients J-s, o, -ri, —3, and 
 —416793, each with its proper algebraical sign, except the last, in which 
 let the sign be changed. This is the most convenient way when the 
 last sign is — . But if the last sign be +, it may b more con venient 
 to let it stand, and change all which come before. Thus, in solving 
 a?^— i2j?+i = o, we might write 
 
 —I o +12 I 
 whereas in the instance before us, we write 
 
 +2 o +1 —3 416793 
 
 Having done this, take the highest figure of the root, properly named, 
 which is 2 tens, or 20. Begin with the first column, multiply by 20, 
 
212 APPENDIX. 
 
 and join it to the number in the next column ; multiply that by 20, 
 and join it to the number in the next column ; and so on. But when you 
 come to the last column, subtract the product which comes out of the 
 preceding column, or join it to the last column after changing its sign. 
 When this has been done, repeat the process with the numbers which 
 now stand in the columns, omitting the last, that is, tlie subtracting 
 step ; then repeat it again, going only as far as the last column but two, 
 and so on, until the columns present a set of rows of the following ap- 
 pearance : 
 
 a b c d e 
 f g h i 
 k I m 
 n 
 P 
 to the formation of which the following is the key : 
 
 /= zoa+b, g = 20/+ c, A = 20^+rf, t = e-2oA, 
 A: = 2oa+/, l^'i.ok-kg^ m = 2o/+7i, 
 n = 2oa+Ar, = 20W+ /, 
 p = 2oa+n. 
 We call this Horner''s Process^ from the name of its inventor. The 
 result is as follows : 
 
 201-3 416793 (20 
 40 801 16017 96453 
 
 80 2401 64037 
 120 4801 
 160 
 We have now before us the row 
 
 2 160 4801 64037 96453 
 which fiimishes our means of guessing at the next, or units' Hgure of the 
 root. 
 
 Call the last column the dividend^ the last but one the divisor^ and 
 all that come before antecedents. See how often the dividend contains 
 the divisor; this gives the guess at the next figure. The guess is a true 
 
hormeb's method op solving equations. 213 
 
 one,* if, on applying Homer's process, the divisor result, augmented as 
 it is by the antecedeirt processes, still go as many times in the dividend. 
 For example, in the case before us, 96453 contains 64037 once ; let i 
 be put on its trial. Horner's process is found to succeed, and we have 
 for the second process, 
 
 a 160 4801 64037 96453 
 
 162 4963 69000 27453 
 
 164 5127 74127 
 
 166 5293 
 168 
 
 As soon as we come to the fractional portion of the root, the process 
 assumes a more+ methodical form. 
 
 The equation being of the fourth degree, annex four ciphers to the 
 dividend, three to the divisor, two to the antecedent, and one to the 
 previous antecedent, leaving the first column as it is ; then find the new 
 figure by the dividend and divisor, as before,J and apply Homer's pro- 
 cess. Annex ciphers to the results, as before, and proceed in the same 
 way. The annexing of the ciphers prevents our having any thing to do 
 with decimal points, and enables us to use the quotient-figures without 
 paying any attention to their local values. The following exhibits the 
 whole process from the beginning, carried as far as it is here intended 
 to go before beginning the contraction, which will give more figures, as 
 in the rule for the square root. The following, then, is the process as 
 far as one decimal place : 
 
 * Various exceptions may arise when an equation has two nearly eqxial roots. 
 But I do not here introduce algebraical diflSculties ; and a student might give himself 
 a hundred examples, taken at hazard, without much chance of lighting upon one 
 ■which gives any difficulty. 
 
 + This form might be also applied to the integer portions ; but it is hardly needed 
 in such instances as usually occur. See the article Involution and Evolution in the 
 Supplement to the Penny Cyclopcedia. 
 
 X After the second step, the trial will rarely fail to give the true figure. 
 
214 
 
 
 APPENDIX. 
 
 
 o 
 
 I 
 
 -3 
 
 416793(21 3 
 
 40 
 
 801 
 
 16017 
 
 964S3 
 
 Ho 
 120 
 160 
 
 2401 
 4801 
 
 64037 
 690CO 
 
 274530000 
 47339778 
 
 
 4963 
 
 74127C00 
 
 
 162 
 164 
 166 
 
 51^7 
 529300 
 
 75730074 
 
 77348376 
 
 
 1680 
 
 534358 
 539434 
 
 
 
 r686 
 
 544528 
 
 
 
 1692 
 
 
 
 
 1698 
 
 
 
 
 1704 
 
 
 
 
 If we now begin the contraction, it is good to know beforehand on 
 what number of additional root-figures we may reckon. We may be 
 pretty certain of having nearly as many as there are figures in the divisor 
 when we begin to contract — one less, or at least two less. Thus, there 
 being now eight figures in the divisor, we may conclude that the con- 
 traction ^vill give us at least six more figures. To begin the contraction, 
 let the dividend stand, cut off one figure from the divisor, two from the 
 column before that, three from the one before that, and so on. Thus, 
 our contraction begins with 
 
 |ooo2 i[704 5445]28 7734837[6 47339778 
 
 The first column is rendered quite useless here. Conduct the process 
 as before, using only the figures which are not cut oflF. But it will be 
 better to go as far as the first figure cut oif, carrying from the second 
 figure cut off. We shall then have as follows : 
 
 I 704 5445p8 7734837 
 
 • c^cc ^ 7767570 
 
 5455 
 5465 
 
 5475 
 
 7 7800364 
 
 6 47339778(6 
 
 6 734354 
 
 At the next contraction the column 1I704 becomes I00X704, and ia quite 
 useless. The next step, separately written (which is not, however, neces- 
 sary in working), is 
 
HORNER S METHOD OF SOLVING EQUATIONS. 
 
 215 
 
 541759 78003648 734354(0 
 
 Here the dividend 734354 does not contain the divisor 780036, and we, 
 therefore, write o as a root figure and make another contraction, or begin 
 with 
 
 I, 
 
 54759 
 
 78003 
 7800S 
 7S013 
 
 648 
 5 
 
 4 
 
 734354(9 
 32277 
 
 At the next contraction the first column becomes 1 00547 59, and is quite 
 useless, so that the remainder of the process is the contracted division. 
 
 7801 34)32277(4137 
 
 1072 
 292 
 58 
 
 3 
 and the root required is 2i'36o94i37. 
 
 I now write down the complete process for another equation, one 
 root of which lies between 3 and 4 : it is 
 
 •loai'+i 
 
 30 
 31 
 
 32 
 33 o 
 33 I 
 33 2 
 33 300 
 33 30 3 
 9 33 30 6 
 
 9 33 30 90 
 
 9 33 
 
 9 33 
 
 09I33 
 
 17 
 
 — 10 
 
 — I 
 1700 
 1791 
 1S8300 
 189231 
 190163C0 
 19025631 
 19034963CO o o 
 1903524299 o 9 
 1903552298 2700 
 1903560698 o 5 
 1903569097 8 5 
 1903569144 5 2 
 1903569191 I 
 1903569193 o 
 1903569194I9 
 
 — 1( 3" 1 1 10390520730990796 
 2000 
 209000 
 19769000 
 743369000000 
 172311710273000 
 991247447681 
 39462875420 
 
 1391491559 
 
 58993123 
 
 1886047 
 
 172835 
 
 1515 
 
 183 
 
 1% 
 
 I 
 
 The student need not repeat the rows of figures so far as they come 
 under one another : thus, it is not necessary to repeat 190356. But he 
 must use his own discretion as to how much it would be safe for him 
 to omit. I have set down the whole process here as a guide. 
 
'216 APPENDIX. 
 
 The following examples will serve for exercise: 
 
 1. 2.t^— locr— 7 = o a?= 7"io58ii33. 
 
 2. a^*-ta!^+a;^+x = 6ooo j? = 8-531437726. 
 
 3. «^+3<r'— 4<r— 10 = o a? = 1-895694916504. 
 
 4. a?^+iood;-2— 5jp— 2173 = o .T = 4-582246071058464, 
 6. v2 = 1*259921049894873164767210607278.* 
 
 6. J?''— 6a?= 100 d? = 5-071351748731. 
 
 7. <2'^+2a?'^+3,r = 300 0? = 5-95525967122398. 
 
 8. ar^+x = 1000 X ■=■ 9-96666679. 
 
 9. 27oooa?^+270ooa^ = 26999999 ^ ~ 9*9666666 
 
 10. tT^— 6a? = 100 x= 5-0713517487. 
 
 11. ar*— 4<r'*+7J?^— 863 = a? = 4-5195507. 
 
 12. x^—%ox-V% = 0? = 4-66003769300087278, 
 
 13. .2?^+a''+d?— 10 = o 07=1-737370233. 
 
 14. a^— 46^?^^— 36d?+i8 = X = 46*7616301847, or a- = -3471623191. 
 
 15. jr'+^6j;'2— 360?— 18 = d? = 1*1087925037. 
 
 16. 899:0?^— i62838A'^+74627i.r— 81000 = d?=*iii222333444555.,. .., 
 
 17. 729<r3— 486d:'^+99J?— 6 =0 a? = •iiii..., or •2222.,.., or '3333 ,., . 
 
 18. 2d?^+3<2'2— 40? = 500 X = 5-93481796231515279. 
 
 19. a'^+2a?''+a?— 150 = o .t- = 4-668409014554198325374299120170589^. 
 
 20. o^-^rx = a?^+5oo x = 8-240963558144858526963. 
 
 21. d?^+2af'+3ar— loooo = x = 20*852905526009. 
 
 22. 4?*— 4J*— 2000 = X = 4-581400362, 
 
 23. lOiF^— 330?^— iij-— 100 = X = 4*146797808584278785. 
 
 24. x*+x^+x'^+x = 127694 X = 18*64482373095. 
 
 25. ioa?3+iia?2+i2a' = looooo a? = 21*1655995554508805. 
 
 26. x^-i-x =13 X = 2*209753301208849. 
 
 27. x^+x^—/^—i6oo = x= 11*482837157. 
 
 28. x^-zx = 5 
 
 X = 2*094551481542326591482386540579302963857306105628239. 
 
 • The solution of jfl+Ox^+Ox—i — 0. 
 
THE APPLICATION OF ARITHMETIC TO GEOMETRY. 217 
 
 29. a?*— 8o^3+24u?^—6j7— 80379639 = X = 123,* 
 
 30. 0?^— 242a?''^— 63i5a?+2577096 = a' = 123.* 
 
 31. 20?*— 3a?'+6a'— 8 = o J? = i'4i42i35623730950488o3.* 
 
 32. a?^— 19^^+1 32a?*'*— 302a'+2oo = o x= 1*02804, 014, or 6-57653, ar 
 
 7*39543t. 
 
 33. 'jx*—iix^+6x'^+sx = 21$ 4: = 27o648o4938579i.t 
 
 34. 7jrV6^*+5<r'+4a?'^+3<r = 11 j; =«7707688i9622658522379296505.t 
 
 35. 4<*^+7J?*+9d?*+6j?34.5^2^3^ = 792 
 j?=2'052042i768796o53652i40434oi28i2oi97346o27559954554i7242i4.+ 
 
 36. 2187^?*— 24303^+9454?^— 150^+8 = o a? = *iiii..., or "2222 , or 
 
 •3333...., or -4444.... 
 
 APPENDIX XII. 
 
 RULEi FOR THE APPLICATION OF ARITHMETIC TO GEOMETRY. 
 
 The student should make himself familiar with the most common terms 
 of geometry, after which the following rules will present no difficulty. 
 In them all, it must be understood, that when we talk of multiplying 
 one line by another, we mean the repetition of one line as often as 
 there are units of a given kind, as feet or inches, in another. In any 
 other sense, it is absurd to talk of multiplying a quantity by another 
 quantity. All quantities of the same kind should be represented in 
 numbers of the same unit ; thus, all the lines should be either feet 
 and decimals of a foot, or inches and decimals of an inch, &c. And 
 in whatever unit a length is represented, a surface is expressed in the 
 corresponding square units, and a solid in the corresponding cubic units. 
 This being understood, the rules anply to all sorts of units. 
 
 To find the area of a rectangle. Multiply together the units in 
 
 • These examples are taken from a paper on the subject, b)- Mr. Peter Gray, in 
 the Mechanics' Magazine. 
 
 t These examples are taken from the late Mr. Peter Nicholson's Essay on Invo- 
 lution and Evolution. 
 
218 APPENDIX. 
 
 two sides which meet, or multiply together two sides which meet ; the 
 product is the number of square units in the area. Thus, if 6 feet and 
 5 feet be the sides, the area is 6. 5, or 30 square feet. Similarly, the 
 area of a square of 6 feet long is 6x6, or 36 square feet (234). 
 
 To find the area of a parallelogram. Multiply one side by the per- 
 pendicular distance between it and the opposite side ; the product is the 
 area required in square units. 
 
 To find the area of a trapezium.* Multiply either of the two sides 
 which are not parallel by the perpendicular let fall upon it from the 
 middle point of the other. 
 
 To find the area of a triangle. Multiply any side by the perpen- 
 dicular let fall upon it from the opposite vertex, and take half the 
 product. Or, halve the sum of the three sides, subtract the three sides 
 severally from this half sum, multiply the four results together, and find 
 the square root of the product. The result is the number of square 
 units in the area ; and twice this, divided by either side, is the perpen- 
 dicular distance of that side froni its opposite vertex. 
 
 To find the radius of the internal circle which touches the three sides 
 of a triangle. Divide the area, found in the last paragraph, by half the 
 sum of the sides. 
 
 Given the two sides of a right-angled triangle, to find the hypothenuse. 
 Add the squares of the sides, and extract the square root of the sum. 
 
 Given the hypothenuse and one of the sides, to find the other side. 
 Multiply the sum of the given lines by their difference, and extract the 
 square root of the product. 
 
 To find the circumference of a circle from its radius, very nearly. 
 Multiply twice the radius, or the diametor, by 3* 141 5927, taking as 
 many decimal places as may be thought necessary. For a rough com- 
 putation, multiply by 22 and divide by 7. For a very exact computation, 
 in which decimals shall be avoided, multiply by 355 and divide by 113. 
 See (131), last example. 
 
 To find the arc of a circular sector, very nearly, knowing the radius 
 
 • A four- sided figure, which has two sides parallel, and two sides not paraileL 
 
THE APPLICATION OF ARITHMETIC TO GEOMETRY. 219 
 
 and the angle. Turn the angle into seconds,* multiply by the radius, 
 and divide the product by 206265. The result will be the number of 
 units in the arc. 
 
 To find the area of a circle from its radius^ very nearly. Multiply 
 the square of the radius by 3-1415927. 
 
 To find the area of a sector, very nearly, knowing the radius and the 
 angle. Turn the angle into seconds, multiply by the square of the 
 radius, and divide by 206265x2, or 412530. 
 
 To find the solid content of a rectangular parallelepiped. Multiply 
 together three sides which meet : the result is the number of cubic miits 
 required. If the figure be not rectangular, multiply the area of one 
 of its planes by the perpendicular distance between it and its opposite 
 plane. 
 
 To find the solid content of a pyramid. Multiply the area of the 
 base by the perpendicular let fall from the vertex upon the base, and 
 divide by 3. 
 
 To find the solid content of a prism. B-'ultiply the area of the base 
 by the perpendicular distance between the oijposite bases. 
 
 To find the surface of a sphere. Multiply 4 times the square of the 
 radius by 3*1415927. 
 
 To find the solid content of a sphere. Multiply the cube of the radius 
 
 by 3-I4I5927X-, or 4*18879. 
 3 
 To find the surface of a right cone. Take half the product of the 
 
 circumference of the base and slanting side. To find the solid content^ 
 
 take one-third of the product of the base and the altitude. 
 
 To find the surface of a right cylinder. Multiply the circumference 
 of the base by the altitude. To find the solid content, multiply the area 
 of the base by the altitude. 
 
 The weight of a body may be found, when its solid content is known, 
 if the weight of one cubic inch or foot of the body be known. But it 
 
 * The right angle is divided into 90 equal parts called degrees, each degree into 
 60 equal parts called minutes, and each minute into 60 equal parts called tecortd*. 
 Thus, 2° 1^' 40" means 2 degrees, 15 minutes, and 40 seconds. 
 
220 APPENDIX. 
 
 is usual to form tables, not of the weights of a cubic unit of different 
 
 bodies, but of the proportion which these weights bear to some one 
 
 amongst them. The one chosen is usually distilled water, and the 
 
 proportion just mentioned is called the specific gravity. Thus, the 
 
 specific gravity of gold is 19*362, or a cubic; foot of gold is 19*362 times 
 
 as heavy as a cubic foot of distilled water. Suppose now the weight of 
 
 a sphere of gold is required, whose radius is 4 inches. The content of 
 
 this sphere is 4x4x4x4.' 1888, or 268'o832 cubic inches; and since, by 
 
 (217), each cubic inch of water weighs 252*458 grains, each cubic inch 
 
 of gold weighs 252*458xi9'362, or 4888091 grains; so that 268*0832 
 
 cubic inches of gold weigh 268*08 32x4888*091 grains, or 227- pounds 
 
 2 
 
 troy nearly. Tables of specific gravities may be found in most works 
 of chemistry and practical mechanics. 
 
 The cubic foot of water is 908*8488 troy ounces, 75*7374 troy pounds, 
 997*1369691 averdupois ounces, and 62*3210606 averdupois pounds. 
 For all rough purposes it will do to consider the cubic foot of water as 
 being 1000 common ounces, which reduces tables of specific gravities to 
 common terms in an obvious way. Thus, when we read of a substance 
 which has the specific gravity 4*1172, we may take it that a cubic foot 
 of the substance weighs 4117 ounces. For greater correctness, diminish 
 this result by 3 parts out of a thousand. 
 
 THE END. 
 
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 Single Volumes, 18s., ornamental boards ; or 6 Double Ones, £\ Is., cl, lettered. 
 *#* Also, handsomely hidf-hound morocco, 6 volumes, £1 lis. 6d. 
 Contents :— The Planets ; are they inhabited Worlds? Weather Prognostics. Po- 
 pular Fallacies in Questions of Physical Science. Latitudes and Longitudes. Lunar 
 Influences. Meteoric Stones and Shooting Stars. Railway Accidents. Light. Com- 
 mon Things.— Air. Locomotion in the United States. Comeiary Influences. Common 
 Things.— Water. The Potter's Art. Common Things.— Fire. Locomotion and Trans- 
 port, their Influence and Progress. Tlie Moon. Common Things.— The Earth. The 
 Electric Telegraph. Terrestrial Heat. The Sun. Earthquakes and Volcanoes. Baro- 
 meter, Safety Lamp, and Whitworth's Micrometric Apparatus. Steam. The Steam 
 Engine. The Eye. The Atmosphere. Time. Common Things.— Pumps. Common 
 Things.— Spectacles— The Kaleidoscope. Clocks and Watches. Microscopic Dra^v^ng 
 and Engraving. The Locomotive. Thermometer. New Planets.— Leverrier and Adams's 
 Planet. Magnitude and Mhmteness. Common Things. — The Almanack. Optical 
 Images. How to Observe the Heavens. Common Things.— The Looking Glass. Stellar 
 Universe. The Tides. Colour. Common Things.— Man. Magnifying Glasses. In- 
 stinct and Intelligence. The Solar Microscope. The Camera Lucida. The Magic 
 Lantern. The Camera Obscura. The Microscope. The White Ants ; their Manners 
 and Habits. The Surface of the Earth, or First Notions of Geography. Science and 
 Poetry. The Bee. Steam Navigation. Electro-Motive Power. Thunder, Lightning, 
 and the Aurora Borealis. The Printing-Press. The Crust of ihe Earth. Comets. 
 The Stereoscope, The Pre- Adamite Earth. Eclipses. Sound. 
 
 Lardners Animal Physics, or the Body and its Functions 
 
 familiarly Explained. 520 Illustrations. 1 vol., small 8vo. 12s. 6d. cloth. 
 
 Lardners Animal Physiology for Schools {chiefly taken 
 
 from the " Animal Physics"). 190 Illustrations. 12mo. 3s. 6d. cloth. 
 
 Lardners Hand- Book of Mechanics. 
 
 357 niustrations. 1 vol., small Svo., 5s. 
 
 Lardners Hand-Book of Hydrostatics, Pneumatics, and 
 
 Heat. 292 Illustrations. 1 vol., small 8vo,, Ss. 
 
 Lardner's Hand-Book of Optics. 
 
 290 Illustrations. 1 vol., small 8vo., 5s. 
 
 Lardners Hand-Book of Electricity, Magnetism, and 
 
 Acoustics. 395 Illustrations. 1 vol., small Svo. 5s. 
 
 Lardner's Hand-Book of Astronomy and Meteorology, 
 
 forming a companion work to the " Hand-Book of Natural Philosophy." 37 Plates, 
 and upwards of 200 Illustrations on Wood. 2 vols., each 5s., cloth lettered. 
 
 Lardner^s Natural Philosophy for Schools. 
 
 328 Illustrations. 1 vol., large 12mo., 3s. fid. cloth. 
 
 Lardner's Chemistry for Schools. 
 
 170 Illustrations. 1 vol., large I2mo. 33. 6d. cloth. 
 
WORK8 PUBLISHED BY 
 
 Pictorial Illustrations of Science and Art. Large Printed 
 
 Sheets, each containing from Sffto 100 EnRi-aved Figures 
 
 Part I. Is. 6(1 
 
 1. Mechanic Powers. 
 
 2. Machinery. 
 
 3. Watch ana Clock Work 
 
 Part in. Is. 6d. 
 
 7. Hydrostatics. 
 
 8. Hydraulics. 
 
 9. Pneumatics. 
 
 Parr II. Is. 6d. 
 
 4. Elements of Machinery. 
 
 5. Motion and Force. 
 
 6. Steam Engine. 
 
 Lardner's Popular Geology. {From " The Museum of 
 
 Science and Art.") 201 Illustrations. 2s. 6d. 
 
 Lardners Common Things Explained. Containing : 
 
 Air-Earth— Fire— Water— Time— The Almanack— Clocks and Watches— Spec- 
 tacles — Colour — Kaleidoscope— Pumps — Man— The Eye — The Printing Press — 
 The I'otter's Art — Locomotion and Transport — The Surface of the Earth, or First 
 Notions of Geography. (From "The Museum of Science and Art.") With 233 
 Illustrations. Complete, 5s., clotli lettered. 
 
 *»* Sold also in Two Series, 2s. 6d. each. 
 
 Lardner''s Popular Physics. Containing: Magnitude and 
 
 Minuteness— Atmosphere— Thunder and Lightning — Terrestrial Heat— M-teoric 
 Stones — Popular Fallacies— Weather Prognostics — Thermometer — Barometer — 
 Safety Lamp — Whitwortli's Micrometric Apparatus — Electro-Motive Power — 
 Sound — Magic Lantern — Camera Obscura— Camera Lucida — Looking Glass — Ste- 
 reoscope -Science and Poetry. (Fi-ora " The Museum of Science and Art.") With 
 85 Illustrations. 2s, 6d. cloth lettered. 
 
 Lardners Popular Astronomy. Containing : How to 
 
 Observe the Heavens— Latitudes and Longitudes — The Earth— The Sun— The 
 Moon — The Planets: are tlicy Inhabited?— The New Planets — Leverrier and 
 Adams's Planet— The Tides— Lunar Influences— and tlie Stellar Universe— Light 
 —Comets— Cometary Influences— Eclipses— Terresti'ial Rotation — Lunar Rota- 
 tion—Astronomical Instruments. (From "The Museum of Science and Art.") 
 182 Illustrations. Complete, 4s. 6d. clotli lettered. 
 
 *** Sold also in Two Series, 2s. Gd. and 2s. each. 
 
 Lardner on the Microscope. {From *' The Museum of 
 
 Science and Art.") I vol. 147 Engravings. 23. 
 
 Lardner on the Bee and White Ants; their Manners 
 
 and Habits; with Illustrations of Animal Instinct and Intelligence. (From " The 
 Museum of Science and Art.") I vol. 135 Illustrations. 2s., cloth lettered. 
 
 Lardner on Steam and its Uses ; including the Steam 
 
 Engine and Locomotive, and Steam Navigation. (From " The Museum of Science 
 and Art.") I vol., with 89 Illustrations. 2s. 
 
 Lardner on the Electric Telegraph, Popularised. fVith 
 
 100 lUnstrations. (From "The Museum of Science and Art.") 12mo., 250 pages. 
 28.. cloth lettered. 
 *^* The following Works from " Lardner's Museum of Science and Art," may 
 also be had arranged as descnbed, handsomely half bound morocco, cloth sides. 
 
 Common Things. Two series in one vol 7s. 6d. 
 
 Popular Astronomy. Two series in one vol 7s. Od. 
 
 Electric Telegra])h, with Steam and its Uses. In one vol. . 7s. Od. 
 
 Microscope and Popular Physics. In one vol 7s. Od. 
 
 Popular Geology, and Bee and White Ants. In one vol. . 7s. 6d. 
 
 Lardner on the Steam Engine, Steam Navigation, Roads, 
 
 and Railways. Explained and Illustrated. Eighth Edition. With numerous Illus- 
 trations. 1 vol. large 1 2mo. 8s. 6d. 
 
 A Guide to the Stars for every Night in the Year. In 
 
 Eight Planispheres. With an Introduction. 8vo. 6s., cloth. 
 
 Minasi's Mechanical Diagrams, For the Use of Lee- 
 
 turers and Schools. 15 Sheets of Diagrams, coloured, 15s., illustrating the follow- 
 ing subjects: 1 and 2. Composition of Forces.— 3. Equilibrium.— 4 and 5. Levers. 
 —6. Steelyard, Brady Balance, and Danish Balance. — 7. Wheel and Axle.— 8. 
 Inclined Plane.— 9, 10, 11. Pulleys.— 12. Hunter's Screw.— 13 and 14. Toothed 
 Wheels.— 15. Combination of the Mechanical Powers. 
 
WALTON AND MABERLY. 
 
 LOGIC. 
 
 De Morgaris Formal Logic; or. The Calculus of Inference^ 
 
 Necessary and Probable. 8vo. 6s. 6d. 
 
 NeiVs Art of Reasoning: a Popular Exposition of the 
 
 Principles of Logic, Inductive and Deductive; with an Introductory Outline of 
 the History of Logic, and an Appendix on recent Logical Developments, with 
 Notes. Crown 8vo. 4s. 6d., cloth. 
 
 ENGLISH COMPOSITION. 
 
 NeiVs Elements of Rhetoric ; a Manual of the Laius of 
 
 Taste, including the Theory and Practice of Composition. Crown 8vo. 4s. 6d., cl. 
 
 DRAWING. 
 
 Lineal Drawing Copies for the earliest Instruction. Com- 
 
 prising upwards of 200 sutjects on 24 sheets, mounted on 12 pieces of thick paste- 
 board, in a Portfolio. By the Author of " Drawing for Young Children." 6s. 6d. 
 
 Easy Drawing Copies for Elementary Instruction. Simple 
 
 Outlines without Perspective. 67 subjects, in a Portfolio. By the Author of 
 " Drawing for Young Children." 6s. 6d. 
 
 Sold also in Two Sets. 
 Set I. Twenty-six Subjects mounted on thick pasteboard, in a Portfblio. 3s. 6d. 
 Set II. Forty-one Subjects mounted on tliick pasteboard, m a Portfolio. 3s. 6d. 
 The copies are suflficiently large and bold to be drawn from by forty or fifty children 
 at the same time. 
 
 SINGING. 
 
 A Musical Gift from an Old Friend, containing Twenty- 
 four New Songs for the Young. By W. E. Hickson, author of the Moral Songs of 
 " The Singing Master." Svo. 2s. 6d. 
 
 The Singing Master. Containing First Lessons in Singing, 
 
 and the Notation of Music ; Kudiments of the Science of Harmony ; The First 
 Class Tune Book; The Second Class Tune Book; and the Hymn Tune Book. 
 Sixth Edition. Svo. 6s., cloth lettered. 
 
 Sold also in Five Parts, any of which may be had separately, 
 
 I. — First Lessons in Singing and the Notation of Music. 
 
 Containing Nineteen Lessons in the Notation and Art of Reading Music, as adapted 
 for the Instruction of Children, and especially for Class Teaching, with Sixteen 
 Vocal Exercises, arranged as simple two-part harmonies. Svo. Is., sewed. 
 
 II. — Rudiments of the Science of Harmony or Thorough 
 
 Bass. Containing a general view of the principles of Musical Composition, the 
 Nature of Chords and Discords, mode of applying them, and an Explanation of 
 Musical Terms connected with this branch of Science. Svo. Is., sewed. 
 
 III. — The First Class Tune Book. A Selection of Thirty 
 
 Single and Pleasing Airs, arranged with suitable words for yoimg children. Svo. 
 Is., sewed. 
 
 I y. — The Second Class Tune Book. A Selection of Vocal 
 
 Music adapted for youth of different ages, and arranged (with suitable words) as 
 two or three-part harmonies. Svo, Is. 6d. 
 
 V. — The Hymn Tune Book. A Selection of Seventy 
 
 popular Hymn and Psalm Tunes, arranged -with a view of facilitating the progress 
 of Children learning to sing in parts. Svo. Is. 6d. 
 *»* The Vocal Exercises, Moral Songs, and Hymns, with the Music, may also be had, 
 printed on Cards, price Twopence each Card, or Twenty-five for Three Shillings. 
 
8 
 
 WORKS PUBLISHED BY WALTON AND MABERLY. 
 
 CHEMISTRY. 
 Gregori/s Hand-Book of Chemistry. For the use of 
 
 Students. By William Geegory, M.D., late Professor of Chemistry in the 
 University of Edinburgh. Fourth Edition, revised and enlarged. lllu.strated by 
 Engravings on Wood. Complete in One Volume. Large 12mo. I8s. cloth. 
 *»* The Work may also be hud in two Volumes, as under. 
 * Inorganic Chemistry. Fourth Edition, revised and enlarged. 6s. 6d. cloth, 
 
 OaoANic Chemistry. Fourth Edition, very carefully revised, and gieatly 
 enlarged. 12s., cloth. 
 
 (Sold separately.) 
 
 Chemistry for Schools. By Dr. Lardner. 190 AliLStra- 
 
 tions. Large 12mo. 3s. 6d. cloth. 
 
 Liebig's Familiar Letters on Chemistry, in its Relations 
 
 to Physiology, Dietetics, Agi-iculture, Commerce, and Political Economy. Fourth 
 Edition, revised and enlarged, with additional Letters. Edited by Dr. Blytu. 
 Small 8vo. 7s. 6d. cloth. 
 
 Liehig^s Letters on Modern Agriculture, Small Svo. 6s. 
 Liehig's Principles of Agricultural Chemistry ; with Special 
 
 Reference to the late Researches made in England. Small 8vo. 38. Cd., cloth. 
 
 Liebig's Chemistry in its Applications to Agriculture and 
 
 Physiology. Fourth Edition, revised. Svo. 6s. 6d., cloth. 
 
 Liebig^s Animal Chemistry ; or, Chemistry in its Appli- 
 cation to Physiology and Pathology. Third Edition. Part I. (the first half of the 
 work). Svo. 6s. 6d., clotii. 
 
 Liebig's Hand-Book of Organic Analysis ; containing a 
 
 detailed Account of the various Methods used in determining the Elementary 
 Composition of Organic Substances. Illustrated by 85 Woodcuts. 12mo. 5s., cloth. 
 
 Bunsen's Gasometry ; comprising the Leading Physical and 
 
 Chemical Properties of Gases, together with the Methods of Gas Analysis. Fifty- 
 nine Illustrations. Svo. 8s. 6d., cloth. 
 
 W'ohlers Hand-Book of Inorganic Analysis ; One Hundred 
 
 and Twenty-two Examples, illustrating the most important processes for deter- 
 mining the Elementary composition of Mineral substances. Edited by Dr. A. W. 
 HoFMANN, Professor in the Royal College of Chemistry. Large 12rao. 
 
 Parnell on Dyeing and Calico Printing. (Reprinted from 
 
 Parnell's " Applied Chemistry in Manufactures, Arts, and Domestic Economy, 
 1844.") With Illustrations. Svo. 78., cloth. 
 
 / GENERAL LITERATURE. 
 
 De 'Morgans Book of Almanacs. With an Index of 
 
 Reference by which the Almanac may be found for every Year, whether in Old 
 Style or New, fVom any Epoch, Ancient or Modem, up to a.d. 2000. With means 
 of finding the Day of New or Full Moon, from b.c. 2000 to a.d. 2000. 6s., cloth 
 lettered. 
 
 Guesses at Truth. By Two Brothers. New Edition. 
 
 With an Index. Complete 1 vol. Small Svo. Handsomely bound in cloth with 
 red edges. lOs. 6d. 
 
 RudalVs Memoir of the Rev. James Crabb ; late of South- 
 
 ampton. With Portrait. Large 12mo., 6s., cloth. 
 
 Herschell (R.H). The Jews; a brief Sketch of their 
 
 Present State and Future Expectations. Fcap. Svo. Is. 6d., cloth. 
 
 
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