14. When the numbers are large, the subtraction is performed, part at a time, by taking successively from the units of each order in the greater number, the corresponding units in the least. That this may be done conveniently, the numbers are placed as 9587 and 345 in the following example;

and under each column is placed the excess of the upper number, in that column, over the lower, thus ;

5, taken from 7, leaves 2,

4, taken from 8, leaves 4,

3, taken from 5, leaves 2,

and writing afterwards the figure 9, from which there is nothing to be taken; the remainder, 9242, shows how much 9587 is greater than 345.

That the process here pursued gives a true result is indisputa ble, because in taking from the greater of the two numbers all the parts of the least, we evidently take from it the whole of the

least.

15. The application of this process requires particular attention, when some of the orders of units in the upper number are greater than the corresponding orders in the lower.

If, for instance, 397 is to be taken from 524.

In performing this question we cannot at first take the units in the lower number from those in the upper; but the number 524, here represented by 4 units, 2 tens, and 5 hundreds, can be expresssed in a different manner by decomposing some of its collections of units, and uniting a part with the units of a lower order. Instead of the 2 tens and 4 units which terminate it, we can substitute in our minds 1 ten and 14 units; then taking from

these units the 7 of the lower number, we get the remainder 7. By this decomposition, the upper number now has but one ten, from which we cannot take the 9 of the lower number, but from the 5 hundred of the upper number we can take 1, to join with the ten that is left, and we shall then have 4 hundreds and 11 tens; taking from these tens the tens of the lower number, 2 will remain. Lastly, taking from the 4 hundreds, that are left in the upper number, the three hundreds of the lower, we obtain the remainder 1, and thus get 127 as the result of the operation. This manner of working consists, as we see, in borrowing, from the next higher order, an unit, and joining it according to its value to those of the order, on which we are employed, observing to count the upper figure of the order from which it was borrowed one unit less, when we shall have come to it.

16. When any orders of units are wanting in the upper number, that is, when there are ciphers between its figures, it is necessary to go to the first figure on the left, to borrow the 10 that is wanted. See an example.

As we cannot take the 5 units of the lower number from the 2 of the upper, we borrow 10 units from the 7000, denoted by the figure 7, which leaves 6990; joining the 10 we borrowed to the figure 2, the upper number is now decomposed into 6990 and 12; taking from 12 the 5 units of the lower number, we obtain 7 for the units of the remainder.

This first operation has left in the upper number 6990 units or 699 tens instead of the 700, expressed by the three last figures on the left; thus the places of the two ciphers are occupied by 9s, and the significant figure on the left is diminished by unity. Continuing the subtraction in the other columns in the same manner, no difficulty occurs, and we find the remainder, as put down in the example.

17. Recapitulating the remarks made in the two preceding arti cles, the rule to be observed in performing subtraction may be

given thus. Place the less number under the greater, so that their units of the same order may be in the same column, and draw a line under them; beginning at the right, take successively each figure of the lower number from the one in the same column of the upper; if this cannot be done, increase the upper figure by ten units, counting the next significant figure, in the upper number, less by unity, and if ciphers come between, regard them as 95.

18. For greater convenience, when it is necessary to decrease the upper figure by unity, we can suffer it to retain its value, and add this unit to the corresponding lower figure, which, thus increased, gives, as is wanted, a result one less than would arise from the written figures. In the first of the following examples, after having taken 6 units from 14, we count the next figure of the lower number 8, as 9, and so in the others.

Method of proving Addition and Subtraction.

19. In performing an operation, according to a process, the correctness of which is established upon fixed principles, we may nevertheless sometimes commit errors in the partial additions and subtractions, the results of which we seek in the memory. To prevent any mistake of this kind, we have recourse to a method, the reverse of the first operation, by which we ascertain whether the results are right; this is called proving the operation.

The proof of addition consists in subtracting successively from the sum of the numbers added, all the parts of these numbers, and if the work has been correctly performed, there will be no remainder. We will now show by the example given in article. 11, how to perform all these subtractions at once.

527

2519

9812

73

8

Sum 12939

1120

We first add the numbers in the left hand column, which here contains thousands, and subtract the sum 11 from 12, which begins the preceding result, and write underneath the difference 1, produced by what was reserved from the column of hundreds, in performing the addition. The sum of the column of hundreds, taken by itself, amounts to but 18; if we take this from the 9 of the first result, increased by borrowing the one thousand, considered as ten hundred, that remains from the column preceding it on the left, the remainder 1, written beneath, will show what was reserved from the column of tens. The sum of the last, 11, taken from 13, leaves for its remainder 2 tens, the number reserved from the column of units. Joining these 2 tens with the 9 units of the answer, we form the number 29, which ought to be exactly the sum of the column of units, as this column is not affected by any of the others; adding again the numbers in this column, we ought to come to the same result, and consequently to have no remainder. This is actually the case, as is denoted by the 0 written under the column. The process, just explained, may be given thus; To prove addition, beginning on the left, add again each of the several columns, subtract the sums respectively from the sums written above them and write down the remainders which must be joined, each as so many tens to the sum of the next column on the right; if the work be correct there will be no remainder under the last column.

20. The proof of subtraction is, that the remainder, added to the less number, exactly gives the greater. Thus to ascertain the exactness of the following subtraction,

324

297

227

524

we add the remainder to the smaller number, and find the sum, in reality, equal to the greater.

Multiplication.

21. WHEN the numbers to be added are equal to each other, addition takes the name of multiplication, because in this case the sum is composed of one of the numbers repeated as many times as there are numbers to be added. Reciprocally, if we wish to repeat a number several times, we may do it, by adding the number to itself as many times, wanting one, as it is to be repeated. For instance, by the following addition,

16

16

16

16

64

the number 16 is repeated four times, and added to itself three times.

To repeat a number twice is to double it; 3 times, to triple it; 4 times, to quadruple it; and so on.

22. Multiplication implies three numbers, namely, that which is to be repeated, and which is called the multiplicand; the number which shows how many times it is to be repeated, which is called the multiplier; and, lastly, the result of the operation, which is called the product. The multiplicand and multiplier, considered as concurring to form the product, are called factors of the product. In the example given above, 16 is the multiplicand, 4 the multiplier, and 64 the product; and we see that 4 and 16 are the factors of 64.