Chapter II: 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.

[2] 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 _decem_.

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.

[3] The references are to the preceding articles.

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.

V. When two numbers, or collections of numbers, with the foregoing signs, are the same, the sign = is put between them. Thus, that 7 and 5 make 12, is written in this way, 7 + 5 = 12. This is called an _equation_, and is read, seven _plus_ five _equals_ twelve. It is plain that we may construct as many equations as we please. Thus:

12
7 + 9 - 3 = 12 + 1; --- - 1 + 3 × 2 = 11,
2

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

[4] Any little computations which occur in the rest of this section may be made on the fingers, or with counters.

(1st No. + 2d No.) + (1st No. - 2d No.) = twice 1st No.

(1st No. + 2d No.) - (1st No. - 2d No.) = twice 2d No.

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 2_a_, and twice _b_ by 2_b_. The equations then become

(_a_ + _b_) + (_a_ - _b_) = 2_a_,

and (_a_ + _b_) - (_a_ - _b_) = 2_b_.

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

[5] This should be (23) _a_ × _a_, but the sign × is unnecessary here. It is used with numbers, as in 2 × 7, to prevent confounding this, which is 14, with 27.

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,

_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

2_a_ = 2_b_ + 2_c_ + 2_d_.

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

_a_ - _x_ = _b_ + _c_ + (_d_ - _x_).

27. EXERCISES.

What is _a_ + 2_b_ - _c_,
where _a_ = 12,
_b_ = 18,
_c_ = 7?--_Answer_, 41.

_aa_ - _bb_
What is ----------- ,
_a_ - _b_

where _a_ = 6 and _b_ = 2?--_Ans._ 8.

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