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.

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 ake 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 least number, exactly gives the greatest. Thus to ascertain

the exactness of the following subtraction,

524

297

227

524

we add the remainder to the smallest number, and find the sum, in reality, equal to the greatest.

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.

23. When the multiplicand and multiplier are large numbers, the formation of the product, by the repeated addition of the multiplicand, would be very tedious. In consequence of this, means have been sought of abridging it, by separating it into a certain number of partial operations, easily performed by memory. For instance, the number 16 would be repeated 4 times, by taking separately, the same number of times, the six units and the ten, that compose it. It is sufficient then to know the products arising from the multiplication of the units of each order in the multiplicand by the multiplier, when the multiplier consists of a single figure, and this amounts, for all cases that can occur, to finding the products of each one of the 9 first numbers by every other of these numbers.

24. These products are contained in the following table, attributed to Pythagoras.

25. To form this table, the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, are written first on the same line. Each one of these numbers is then added to itself and the sum written in the second line, which thus contains each number of the first doubled, or the product of each number by 2. Each number of the second line is then added to the number over it in the first, and their sums are written in the third line, which thus contains the triple of each number in the first, or their products by 3. By adding the numbers of the third line to those of the first, a fourth is formed, containing the quadruple of each number of the first, or their products by 4; and so on, to the ninth line, which contains the products of each number of the first line by 9.

It may not be amiss to remark, that the different products of any number whatever by the numbers 2, 3, 4, 5, &c. are called multiples of that number; thus 6, 9, 12, 15, &c. are multiples of 3.

26. When the formation of this table is well understood, the mode of using it may be easily conceived. If, for instance, the product of 7 by 5 were required; looking to the fifth line, which contains the different products of the 9 first numbers by 5, we should take the one directly under the 7, which is 35; the same

method should be pursued in every other instance, and the product will always be found in the line of the multiplier and under the multiplicand.

27. If we seek in the table of Pythagoras the product of 5 by 7, we shall find, as before, 35, although in this case 5 is the multiplicand, and 7 the multiplier. This remark is applicable to each product in the table, and it is possible, in any multiplication, to reverse the order of the factors; that is, to make the multiplicand the multiplier, and the multiplier the multiplicand.

As the table of Pythagoras contains but a limited number of products, it would not be sufficient to verify the above conclusion by this table; for a doubt might arise respecting it in the case of greater products, the number of which is unlimited; there is but one method independent of the particular value of the multiplicand and multiplier of showing that there is no exception to this remark. This is one well calculated for the purpose, as it gives a good illustration of the manner, in which the product of two numbers is formed. To make it more easily understood, we will apply it first to the factors 5 and 3.

If we write the figure 1, 5 times on one line, and place two similar lines underneath the first, in this manner,

the whole number of 1s will consist of as many times 5 as there are lines, that is, 3 times 5; but, by the disposition of these lines, the figures are ranged in columns, containing 3 each. Counting them in this manner, we find as many times 3 units as there are columns, or 5 times 3 units, and as the product does not depend on the manner of counting, it follows that 3 times 5 and 5 times 3 give the same product. It is easy to extend this reasoning to any numbers, if we conceive each line to contain as many units. as there are in the multiplicand, and the number of lines, placed one under the other, to be equal to the multiplier In count

ing the product by lines, it arises from the multiplicand repeated as many times as there are units in the multiplier; but the assemblage of figures written presents as many columns as there Arith.