Elements of arithmetic E-Book

Augustus de Morgan

• Herausgeber: Librorium Editions
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2022

Beschreibung

Imagine a multitude of objects of the same kind assembled together; for example, a company of horsemen. One of the first things that must strike a spectator, although unused to counting, is, that to each man there is a horse. Now, though men and horses are things perfectly unlike, yet, because there is one of the first kind to every one of the second, one man to every horse, a new notion will 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 horsemen, 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.

Leseprobe

ELEMENTS OF ARITHMETIC.

BY AUGUSTUS DE MORGAN,

OF TRINITY COLLEGE, CAMBRIDGE;

FELLOW OF THE ROYAL ASTRONOMICAL SOCIETY, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY; PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.

“Hominis studiosi est intelligere, quas utilitates proprie afferat arithmetica his, qui solidam et perfectam doctrinam in cæteris philosophiæ partibus explicant. Quod enim vulgo dicunt, principium esse dimidium totius, id vel maxime in philosophiæ partibus conspicitur.”—Melancthon.

“Ce n’est point par la routine qu’on e’instruit, c’est par sa propre réflexion; et il est essentiel de contracter l’habitude de se rendre raison de ce qu’on fait: cette habitude s’acquiert plus facilement qu’on ne pense; et une fois acquise, elle ne se perd plus.”—Condillac.

1858

© 2022 Librorium Editions

ISBN : 9782383834786

ELEMENTS OF ARITHMETIC.

PREFACE.

TABLE OF CONTENTS.

BOOK I.

SECTION I. NUMERATION.

SECTION II. ADDITION AND SUBTRACTION.

SECTION III. MULTIPLICATION.

SECTION IV. DIVISION.

SECTION V. FRACTIONS.

SECTION VI. DECIMAL FRACTIONS.

SECTION VII. ON THE EXTRACTION OF THE SQUARE ROOT.

SECTION VIII. ON THE PROPORTION OF NUMBERS.

SECTION IX. ON PERMUTATIONS AND COMBINATIONS.

BOOK II.

SECTION I. WEIGHTS, MEASURES, &C.

SECTION II. RULE OF THREE.

SECTION III. INTEREST, ETC.

APPENDIX TO

I. ON THE MODE OF COMPUTING.

APPENDIX II. ON VERIFICATION BY CASTING OUT NINES AND ELEVENS.

APPENDIX III. ON SCALES OF NOTATION.

APPENDIX IV. ON THE DEFINITION OF FRACTIONS.

APPENDIX V. ON CHARACTERISTICS.

APPENDIX VI. ON DECIMAL MONEY.

APPENDIX VII. ON THE MAIN PRINCIPLE OF BOOK-KEEPING.

APPENDIX VIII. ON THE REDUCTION OF FRACTIONS TO OTHERS OF NEARLY EQUAL VALUE.

APPENDIX IX. ON SOME GENERAL PROPERTIES OF NUMBERS.

APPENDIX X. ON COMBINATIONS.

APPENDIX XI. ON HORNER’S METHOD OF SOLVING EQUATIONS.

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 augmented by the addition of eleven new Appendixes,[1] 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 instantaneous conversion of the usual fractions of a pound sterling into decimal fractions, gives the possessor the greater part of the advantage which he would derive from the introduction of a decimal coinage.

At the time when this work was first published, the importance of establishing arithmetic in the young mind upon reason and demonstration, was not admitted by many. The case is now altered: schools exist in which rational arithmetic is taught, and mere rules are made to do no more than their proper duty. There is no necessity to advocate a change which is actually in progress, as the works which are published every day sufficiently shew. And my principal reason for alluding to the subject here, is merely to warn those who want nothing but routine, that this is not the book for their purpose.

A. De Morgan.

London, May 1, 1846.

ELEMENTS OF ARITHMETIC.

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, yet, 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 horsemen, 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 there is a pebble; that is, in common language, the number of pebbles and of horsemen is the same. Suppose that while the second company passes, he drops a pebble for each man into a second basket: he will then have two baskets of pebbles, by which he will be able to convey to any other person a notion of how many horsemen there were in each company. When he wishes to know which company was the larger, or contained most horsemen, he will take a pebble out of each basket, and put them aside. He will go on doing this as often as he can, that is, until one of the baskets is emptied. Then, if he also find the other basket empty, he says that both companies contained the same number of horsemen; if the second basket still contain some pebbles, he can tell by them how many more were in the second than in the first.

3. In this way a savage could keep an account of any numbers 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 prætor, one of the magistrates, used to go every year in great pomp, and drive a nail into the door of the temple of Jupiter; a way of remembering the number of years which the city had been built, which probably took its rise before the introduction of writing.

4. In process of time, names would be given to those collections of pebbles which are met with most frequently. But as long as small numbers only were required, the most convenient way of reckoning them would be by means of the fingers. Any person could make with his two hands the little calculations which would be necessary for his purposes, and would name all the different collections of the fingers. He would thus get words in his own language answering to one, two, three, four, five, six, seven, eight, nine, and ten. As his wants increased, he would find it necessary to give names to larger numbers; but here he would be stopped by the immense quantity of words which 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 preceded it.

One

eleven

ten and one[2]

two

twelve

ten and two

three

thirteen

ten and three

four

fourteen

ten and four

five

fifteen

ten and five

six

sixteen

ten and six

seven

seventeen

ten and seven

eight

eighteen

ten and eight

nine

nineteen

ten and nine

ten

twenty

two tens

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.

a hundred and one

ten tens and one

&c. &c.

a thousand

ten hundreds

ten thousand

a hundred thousand

a million

ten hundred thousand

or one thousand thousand

ten millions

a hundred millions

&c.

6. Words, written down in ordinary language, would very soon be too long for such continual repetition as takes place in calculation. Short signs would then be substituted for words; but it would be impossible to have a distinct sign for every number: so that when some few signs had been chosen, it would be convenient to invent others for the rest out of those already made. The signs which we use areas follow:

0

1

2

3

4

5

6

7

8

9

nought

one

two

three

four

five

six

seven

eight

nine

I now proceed to explain the way in which these signs are made to represent other numbers.

7. Suppose a man first to hold up one finger, then two, and so on, until he has held up every finger, and suppose a number of men to do the same thing. It is plain that we may thus distinguish one number from another, by causing two different sets of persons to hold up each a certain number of fingers, and that we may do this in many different ways. For example, the number fifteen might be indicated either by fifteen men each holding up one finger, or by four men each holding up two fingers and a fifth holding up seven, and so on. The question is, of all these contrivances for expressing the number, which is the most convenient? In the choice which is made for this purpose consists what is called the method of numeration.

8. I have used the foregoing explanation because it is very probable that our system of numeration, and almost every other which is used in the world, sprung from the practice of reckoning on the fingers, which children usually follow when first they begin to count. The method which I have described is the rudest possible; but, by a little alteration, a system may be formed which will enable us to express enormous numbers with great ease.

9. Suppose that you are going to count some large number, for example, to measure a number of yards of cloth. Opposite to yourself suppose a man to be placed, who keeps his eye upon you, and holds up a finger for every yard which he sees you measure. When ten yards have been measured he will have held up ten fingers, and will not be able to count any further unless he begin again, holding up one finger at the eleventh yard, two at the twelfth, and so on. But to know how many have been counted, you must know, not only how many fingers he holds up, but also how many times he has begun again. You may keep this in view by placing another man on the right of the former, who directs his eye towards his companion, and holds up one finger the moment he perceives him ready to begin again, that is, as soon as ten yards have been measured. Each finger of the first man stands only for one yard, but each finger of the second stands for as many as all the fingers of the first together, that is, for ten. In this way a hundred may be counted, because the first may now reckon his ten fingers once for each finger of the second man, that is, ten times in all, and ten tens is one hundred (5).[3] Now place a third man at the right of the second, who shall hold up a finger whenever he perceives the second ready to begin again. One finger of the third man counts as many as all the ten fingers of the second, that is, counts one hundred. In this way we may proceed until the third has all his fingers extended, which will signify that ten hundred or one thousand have been counted (5). A fourth man would enable us to count as far as ten thousand, a fifth as far as one hundred thousand, a sixth as far as a million, and so on.

10. Each new person placed himself towards your left in the rank opposite to you. Now rule columns as in the next page, and to the right of them all place in words the number which you wish to represent; in the first column on the right, place the number of fingers 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.

7th.

6th.

5th.

4th.

3rd.

2nd.

1st.

I.

5

7

fifty-seven

II.

1

0

4

one hundred and four.

III.

1

1

0

one hundred and ten.

IV.

2

3

4

8

two thousand three hundred

and forty-eight.

V.

1

5

9

0

6

fifteen thousand nine

hundred and six.

VI.

1

8

7

0

0

4

one hundred and eighty-seven

thousand and four.

VII.

3

6

9

7

2

8

5

three million, six hundred and

ninety-seven thousand,

two hundred and eighty-five.

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 shewn 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 as many thousands of yards; and so on: that is, if we suppose a figure to move from any column to the one on its left, it stands for ten times as many yards as before. Recollect this, and you may cease to draw the lines between the columns, because each figure will be sufficiently well known by the place in which it is; that is, by the number of figures which come upon the right hand of it.

13. It is important to recollect that this way of writing numbers, which has become so familiar as to seem the natural method, is not more natural than any other. For example, we might agree to signify one ten by the figure of one with an accent, thus, 1′; twenty or two tens by 2′; and so on: one hundred or ten tens by 1″; two hundred by 2″; one thousand by 1‴; and so on: putting Roman figures for accents when they become too many to write with convenience. The fourth number in the table would then be written 2‴ 3′ 4′ 8, which might also be expressed by 8 4′ 3″ 2‴, 4′ 8 3″ 2‴; or the order of the figures might be changed in any way, because their meaning depends upon the accents which are attached to them, and not upon the place in which they stand. Hence, a cipher would never be necessary; for 104 would be distinguished from 14 by writing for the first 1″ 4, and for the second 1′ 4. The common method is preferred, not because it is more exact than this, but because it is more simple.

14. The distinction between our method of numeration and that of the ancients, is in the meaning of each figure depending partly upon the place in which it stands. Thus, in 44444 each four stands for four of something; but in the first column on the right it signifies only four of the pebbles which are counted; in the second, it means four collections of ten pebbles each; in the third, four of one hundred each; and so on.

15. The things measured in (11) were yards of cloth. In this case one yard of cloth is called the unit. The first figure on the right is said to be in the units’ place, because it only stands for so many units as are in the number that is written under it in (6). The second figure is said to be in the tens’ place, because it stands for a number of tens of units. The third, fourth, and fifth figures are in the places of the hundreds, thousands, and tens of thousands, for a similar reason.

16. 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 corn there are in a barrel of wine.

18. All things which are true of some numbers of one unit are true of the same numbers of any other unit. Thus, 15 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 mentioned in this chapter.

I. Ten signs are used, one to stand for nothing, the rest for the first nine numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these is called a cipher.

II. Higher numbers have not signs for themselves, but are signified by placing the signs already mentioned by the side of each other, and agreeing that the first figure on the right hand shall keep the value 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 hundreds. For the 6 is 6 thousands, or 6 ten hundreds, or sixty hundreds; which, with the 7, or 7 hundreds, is 67 hundreds: similarly, the 678 is 678 tens. This number may then be expressed either as

1 ten thousand 6 thousands 7 hundreds 8 tens and 5;

or 16 thousands 78 tens and 5; or 1 ten thousand 678 tens and 5;

or 167 hundreds 8 tens and 5; or 1678 tens and 5, and so on.

21. EXERCISES.

I. Write down the signs for—four hundred and seventy-six; two thousand and ninety-seven; sixty-four thousand three hundred 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 10111 is greater than 9879.

22. You now see that the convenience of our method of numeration arises from a few simple signs being made to change their value as they change the column in which they are placed. The same advantage arises from counting in a similar way all the articles which are used in every-day life. For example, we count money by dividing it into pounds, shillings, and pence, of which a shilling is 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. 1. 11, where £ shews that the 1 in the first column is a pound. Here is a system of numeration in which a number in the second column on the right means 12 times as much as the same number in the first; and one in the third column is twenty times as great as the same in the second, or 240 times as great as the same in the first. In each of the tables of measures which you will hereafter meet with, you will see a separate system of numeration, but the methods of calculation for all will be the same.

23. In order to make the language of arithmetic shorter, some other signs are used. They are as follow:

I. 15 + 38 means that 38 is to be added to 15, and is the same thing as 53. This is the sum of 15 and 38, and is read fifteen plus thirty-eight (plus is the Latin for more).

II. 64-12 means that 12 is to be taken away from 64, and is the 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. 9 × 8 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.

IV. 108/6 means that 108 is to be divided by 6, or that you must 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.

and so on.

24. It often becomes necessary to speak of something which is true not of any one number only, but of all numbers. For example, take 10 and 7; their sum[4] is 17, their difference is 3. If this sum and difference be added together, we get 20, which is twice the greater of the two numbers first chosen. If from 17 we take 3, we get 14, which is twice the less of the two numbers. The same thing will be found to hold good of any two numbers, which gives this general proposition,—If the sum and difference of two numbers be added together, the result is twice the greater of the two; if the difference be taken from the sum, the result is twice the lesser of the two. If, then, we take any numbers, and call them the first number and the second number, and let the first number be the greater; we have

The brackets here enclose the things which must be first done, before the signs which join the brackets are made use of. Thus, 8-(2 + 1) × (1 + 1) signifies that 2 + 1 must be taken 1 + 1 times, and the product must be subtracted from 8. In the same manner, any result 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 2b. The equations then become

This may be explained still further, as follows:

25. Suppose a number of sealed packets, marked a, b, c, d, &c., on the outside, each of which contains a distinct but unknown number of counters. As long as we do not know how many counters each contains, we can make the letter which belongs to each stand for its number, so as to talk of the number a, instead of the number in the packet marked a. And because we do not know the numbers, it does not therefore follow that we know nothing whatever about them; for there are some connexions which exist between all numbers, which we call general properties of numbers. For example, take any number, multiply it by itself, and subtract one from the result; and then subtract one from the number itself. The first of these will always contain the second exactly as many times as the original number increased by one. Take the number 6; this multiplied by itself is 36, which diminished by one is 35; again, 6 diminished by 1 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 contained in the packet marked a, or of the number a. If we represent a multiplied by itself by aa,[5] we have, by (23)

aa - 1

———

a + 1.

a - 1

26. When, therefore, we wish to talk of a number without specifying any one in particular, we use a letter to represent it. Thus: Suppose we wish to reason upon what will follow from dividing a number into three parts, without considering what the number is, or what are the parts into which it is divided. Let a stand for the number, and b, c, and d, for the parts into which it is divided. Then, by our supposition,

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

If each part be doubled, the whole number will be doubled, or

If we diminish one of the parts, as d, by a number x, we diminish the whole number just as much, or

27. EXERCISES.

What is

aa - bb

———

a - b

What is the difference between (a + b)(c + d) and a + bc + d, for the following values of a, b, c, and d?

a

b

c

d

Ans.

1

2

3

4

10

2

12

7

1

25

1

1

1

1

1

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 1 thous. 8 hund. 3 tens and 4; 2799 is 2 thous. 7 hund. 9 tens and 9.

Each number is thus broken up into four parts. If to each part of the first you add the part of the second which is under it, and then put together what you get from these additions, you will have added 1834 and 2799. In the first number are 4 units, and in the second 9: these will, when the numbers are added together, contribute 13 units to the sum. Again, the 3 tens in the first and the 9 tens in the second will contribute 12 tens to the sum. The 8 hundreds in the first and the 7 hundreds in the second will add 15 hundreds to the sum; and the thousand in 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

1 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 hundreds, 3 tens, and 3 units,

which (19) is written 4633.

30. The former process, written with the signs of (23) is as follows:

Therefore,

But

1 × 10 + 3

1 × 100 +

2 × 10

1 × 1000 +

5 × 100

3 × 1000

Therefore,

4 × 1000 +

6 × 100 +

3 × 10 + 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:

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

10

2

3

4

5

6

7

8

9

10

11

3

4

5

6

7

8

9

10

11

12

4

5

6

7

8

9

10

11

12

13

5

6

7

8

9

10

11

12

13

14

6

7

8

9

10

11

12

13

14

15

7

8

9

10

11

12

13

14

15

16

8

9

10

11

12

13

14

15

16

17

9

10

11

12

13

14

15

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).

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,[6] which is the same as that in (29).

Rule I. Place the numbers under one another, units under units, tens under tens, and so on.

II. Add together the units of all, and part the whole number thus obtained into units and tens. Thus, if 85 be the number, part it into 8 tens and 5 units; if 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.

IV. Add together all the numbers in the column of tens, remembering to take in (or carry, as it is called) the tens which you were told to recollect in III., and divide this number of tens into tens and hundreds. Thus, if 335 tens be the number obtained, part this into 33 hundreds and 5 tens.

V. Place the number of tens under the tens, and remember the number of hundreds.

VI. Proceed in this way through every column, and at the last column, instead of separating the number you obtain into two parts, write it all down before the rest.

Example.—What is

1805 + 36 + 19727 + 3 + 1474 + 2008

1805

36

19727

3

1474

2008

——-

25053

The addition of the units’ line, or 8 + 4 + 3 + 7 + 6 + 5, gives 33, that is, 3 tens and 3 units. Put 3 in the units’ place, and add together the line of tens, taking in at the beginning the 3 tens which were created by the addition of the units’ line. That is, find 3 + 0 + 7 + 2 + 3 + 0, which gives 15 for the number of tens; that is, 1 hundred and 5 tens. Add the line of hundreds together, taking care to add the 1 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 thousands, or 1 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 1 + 1, 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 corner to corner, when added together give the number 24156.

2016

4212

1656

3852

1296

3492

936

3132

576

2772

216

252

2052

4248

1692

3888

1332

3528

972

3168

612

2412

2448

288

2088

4284

1728

3924

1368

3564

1008

2808

648

684

2484

324

2124

4320

1764

3960

1404

3204

1044

2844

2880

720

2520

360

2160

4356

1800

3600

1440

3240

1080

1116

2916

756

2556

396

2196

3996

1836

3636

1476

3276

3312

1152

2952

792

2592

36

2232

4032

1872

3672

1512

1548

3348

1188

2988

432

2628

72

2268

4068

1908

3708

3744

1584

3384

828

3024

468

2664

108

2304

4104

1944

1980

3780

1224

3420

864

3060

504

2700

144

2340

4140

4176

1620

3816

1260

3456

900

3096

540

2736

180

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 b stand for two numbers, as in (24). It is impossible to tell what their sum will be until the numbers themselves are known. In the mean while a + b stands for this sum. To say, in algebraical language, that the sum of a and b is not altered by adding c to a, provided we take away c from b, we have the following equation:

which may be written without brackets, thus,

For the meaning of these two equations will appear to be the same, on consideration.

38. The use of the brackets must here be noticed. They mean, that the expression contained inside them must be used exactly as a single letter would be used in the same place. Thus, pa signifies that a is taken p times, and (m + n)a, that a is taken m + n times. It is, therefore, a different thing from m + na, which means that a, after being taken n times, is added to m. Thus (3 + 4) × 2 is 7 × 2 or 14; while 3 + 4 × 2 is 3 + 8, or 11.

39. When one number is taken away from another, the number which is left is called the difference or remainder. The process of finding the difference is called subtraction. The number which is to be taken away must be of course the lesser of the two.

40. The process of subtraction depends upon these two principles.

I. The difference of two numbers is not altered by adding a number to the first, if you add the same number to the second; or by subtracting a number from the first, if you subtract the same number from the second. Conceive two baskets with pebbles in them, in the first of which are 100 pebbles more than in the second. If I put 50 more 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 difference.

II. Since 6 exceeds 4 by 2,

and 3 exceeds 2 by 1,

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 c be three numbers, of which a is greater than b (40), I. leads to the following,

Again, if c be less than a and 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 q and r is subtracted from p; but in the second, first q and then r are subtracted from p, which is the same as subtracting as much as q and r together, or q + r. Therefore p-q-r is p-(q + r). In order to shew how to remove the brackets from p -(q-r) without altering the value of the result, let us take the simple instance 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,

By the same kind of reasoning,

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 1 unit.

Now 2 units exceed

1 unit

by 1 unit.

6 tens

3 tens

3 tens.

7 hundreds

6 hundreds

1 hundred.

7 thousands

4 thousands

3 thousands.

5 ten-thousands

3 ten-thous.

2 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. 1 hund. 3 tens and 1 unit,

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 difficulty immediately, since 8, being greater than 4, cannot be taken from it. But from (40) it appears that we shall not alter the difference of two numbers if we add the same number to both of them. Add ten to the first number, that is, let there be 14 units instead of four, and add ten also to the second number, but instead of adding ten to the number of units, add one to the number of tens, which is the same thing. The numbers will then stand thus,

6 ten-thous. 1 thous. 2 hund. 7 tens and 14 units.[7] 3 ten-thous. 9 thous. 6 hund. 3 tens and 8 units.

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 1, which are the same things. And since the thousands in the lower cannot be subtracted from the thousands in the upper line, add 1 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 1, which are the same things; and at the close the numbers which we get will be,

6 ten-thous. 11 thous. 12 hund. 7 tens and 14 units. 4 ten-thous. 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,

2 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 1, 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 beginning 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

0 ten-thous. 0 thous. 8 hunds. 1 ten and 8 units;

the first two signs are nothing, and the rest is

8 hundreds, 1 ten, and 8 units, or 818.

The second does not differ from the first, except in its being said that there are no thousands and no tens of thousands in the number, which may be known without their being mentioned at all. You may ask, perhaps, why this does not apply to a cipher placed in the middle of a number, or at the right of it, as, for example, in 28007 and 39700? But you must recollect, that if it were not for the two ciphers in the first, the 8 would be taken for 8 tens, instead of 8 thousands; and if it were not for the ciphers in the second, the 7 would be taken for 7 units, instead of 7 hundreds.

46. EXAMPLE.

What is the difference between

3708291640030174

and

30813649276188

Difference

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

10000000 - 2731634.

IV. Find 18362 + 2469 and 18362-2469, add the second result to the first, and then subtract 18362; subtract the second from the first, and then subtract 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.

VI. In the following table subtract b from a, and b from the remainder, and so on until 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.

Remainder.

23604

9999

2

3606

209961

37173

5

24096

74712

6792

11

0

4802469

654321

7

222222

18849747

3141592

6

195

987654321

123456789

8

9

SECTION III.MULTIPLICATION.

47. I have said that all questions in arithmetic require nothing but addition and subtraction. I do not mean by this that no rule should ever be used except those given in the last section, but that all other rules only shew 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 sum of five seventeens, or I ask the following question: There are five heaps of pebbles, and seventeen pebbles in each heap; how many are there in all? Write five seventeens in a column, and make the addition, which gives 85. In this case 85 is called the product of 5 and 17, and the process of finding the product is called multiplication, which 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.

17

17

17

17

17

85

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 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 enormous 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 products of all numbers as far as 10 times 10 by means of the following table,[8] which must be committed to memory.

1

2

3

4

5

6

7

8

9

10

11

12

2

4

6

8

10

12

14

16

18

20

22

24

3

6

9

12

15

18

21

24

27

30

33

36

4

8

12

16

20

24

28

32

36

40

44

48

5

10

15

20

25

30

35

40

45

50

55

60

6

12

18

24

30

36

42

48

54

60

66

72

7

14

21

28

35

42

49

56

63

70

77

84

8

16

24

32

40

48

56

64

72

80

88

96

10

20

30

40

50

60

70

80

90

100

110

120

11

22

33

44

55

66

77

88

99

110

121

132

12

24

36

48

60

72

84

96

108

120

132

144

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.

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