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

A

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 five 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 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 counting 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 are units in a line, and each column contains as many units as as there are lines; if, then, we choose to count by columns, the number of lines, or the multiplier, will be repeated as many times as there are units in a line, that is, in the multiplicand. We may therefore, in finding the product of any two numbers, take either of them at pleasure, for the multiplier.

28. The reasoning, just given to prove the truth of the preceding proposition, is the demonstration of it, and it may be remarked, that the essential distinction of pure mathematics is, that no proposition, or process, is admitted, which is not the necessary consequence of the primary notions on which it is founded, or the truth of which is not generally established by reasoning independent of particular examples, which can never constitute a proof, but serve only to facilitate the reader's understanding the reasoning, or the practice of the rules.

29. Knowing all the products given by the nine first numbers, combined with each other, we can, according to the remark in article 23, multiply any number by a number consisting of a single figure, by forming successively the product of each order of units in the multiplicand, by the multiplier; the work is as follows;

526

7

3682

The product of the units of the multiplicand, 6, by the multiplier, 7, being 42, we write down only the 2 units, reserving the 4 tens to be joined with those that will be found in the next higher place.

The product of the tens of the multiplicand, 2, by the multiplier, 7, is 14, and adding the 4 tens we reserved, we make them 18, of which number we write only the units, and reserve the ten for the next operation.

The product of the hundreds of the multiplicand, 5, by the multiplier, 7, is 35; when increased by the 1 we reserved, it becomes 36, the whole of which is written, because there are no more figures in the multiplicand.

30. This process may be given thus; To multiply a number of several figures by a single figure, place the multiplier under the units of the multiplicand, and draw a line beneath, to separate them from the product. Beginning at the right, multiply successively, by the multiplier, the units of each order in the multiplicand, and write the whole product of each, when it does not exceed 9; but, if it contains tens, reserve them to be added to the next product. Continue thus to the last figure of the multiplicand, on the left, the whole result of which must be written down.

Examples. 213 by 6. Ans. 1458. 8943 by 9. Ans. 80487. It is evident that, when the multiplicand is terminated by Os, the operation can commence only with its first significant figure; but to give the product its proper value, it is necessary to put, on the right of it, as many Os as there are in the multiplicand. As for the Os which may occur between the figures of the mul tiplicand, they give no product, and a 0 must be written down when no number has been reserved from the preceding product, is shown by the following examples :

as

Multiply

730 by 3. Ans. 2190.· 20508 by 5. Ans. 102540.

297000 by 7. Ans. 2079000.

8104 by 4. Ans. 32416. 360500 by 9. Ans. 2163000. 9097030 by 9. Ans. 81873270.

31. The most simple number, expressed by several figures, being 10, 100, 1000, &c., it seems necessary to inquire how we can multiply any number by one of these. Now if we recollect the principle mentioned in article 6, by which the same figure is increased in value 10 times, by every remove towards the left, we shall soon perceive, that to multiply any number by 10, we must make each of its orders of units ten times greater; that is, we must change its units into tens, its tens into hundreds, and so on, and that this is effected by placing a 0 on the right of the number proposed, because then all its significant figures will be advanced one place towards the left.

For the same reason, to multiply any number by 100, we should place two ciphers on the right; for, since it becomes ten times greater by the first cipher, the second will make it ten times greater still, and consequently it will be 10 times 10, or 100 times, greater than it was at first.

Continuing this reasoning, it will be perceived that, according to our system of numeration, a number is multiplied by 10, 100, 1000, &c. by writing on the right of the multiplicand as many ciphers as there are on the right of the unit in the multiplier.

32. When the significant figure of the multiplier differs from unity, as, for instance, when it is required to multiply by 30, or 300, or 3000, which are only 10 times 3, or 100 times 3, or 1000 times 3, &c. the operation is made to consist of two parts; we at first multiply by the significant figure, 3, according to the rule in article 30, and then multiply the product by 10, 100, or 1000, &c. (as was stated in the preceding article) by writing one, two, three, &c., ciphers on the right of this product.

Let it be required, for instance, to multiply 764 by 300.

764 300