30. When the multiplier has one or more ciphers subjoined. RULE. Write down the cipher or ciphers for the right hand part of the product, then multiply every figure of the multiplicand by the significant figure or figures of the multiplier, (art. 29.) and set the products in their order to the left of the ciphers k.

31. When the multiplier consists of two or more significant figures,

and is greater than 12.

RULE I. Multiply every figure of the multiplicand by the right hand figure of the multiplier, and set down the product (as in art. 29).

II. In like manner multiply by the next figure (or the right hand but one), set the product in a line below the former, proceed thus with every figure, setting down each succeeding product below the preceding, observing to place the right hand figure of each under the figure you multiply by.

k The truth of the rule may be shewn from art. 29: thus, let 479 be multiplied by 40; now if the multiplier had been only 4, the product (by art. 29.) would have been 1916; but the multiplier 40 is ten times 4, therefore the product must be ten times 1916, or 19160, by art. 19. In like manner 87 multiplied by 500 is 43500; for 87 multiplied by 5 is 435; but 500 is 100 times 5, whence the product by 500 must be 100 times the product by 5, or 43500; and the like is true in every case.

3. Add all the products together, and their sum will be the

product required'.

36. Multiply 850467 By 234

3401868 2551401 1700934

Product 199009278

37.

Explanation.

First, I multiply by 4, and place the product (3401868) so that the right hand figure 8 may stand under the multiplier 4. Next I multiply by 3, and place the product (2551401) below the last, so that its right hand figure 1 may stand under the multiplier 3. I next multiply by 2, and place the product (1700934) below the last, so that its right hand figure 4 may stand under the multiplier 2. Lastly, I add the three products together, and their sum is the last line or product required.

39.

38. 827135

67

5789945

4962810 55418045

312758

789

2814822

2502064

2189306 246766062

Product 82622438.
Product S8246944.
Product 138933048.

1 It has been shewn, that multiplying by a single figure, or units, produces figures of the same denomination with those multiplied; whence it follows, that multiplying by tens will produce figures, the denominations of which will be tenfold what they would have been had the multiplier been units; multiplying by hundreds will produce figures one hundred-fold greater in denomination, &c. Now removing the second line of the product one place to the left, is equivalent to increasing its simple value ten-fold; removing the third line two places is the same as increasing it to one hundred-fold its simple value, &c. (for these removals answer the same purpose as subjoining a cipher or ciphers, which would actually increase the products as above.) Whence it follows, that the first figure of every product ought to stand under the multiplying figure which produces it: also it is plain, that the sum of all the products will be the product of the whole multiplicand into the whole multiplier, since it is made up of all the products of the parts. Thus, in ex. 36. the first line or product is 4 times the multiplicand; the second product is 30 times the multiplicand; the third product 200 times the multiplicand; and therefore the sum of the three products (or the last line in the example) will be 234 times the multiplicand: whence the rule is obvious.

32. To prove the truth of these operations by casting out the

nines.

RULE I. Having made a cross like the sign for multiplication, add the figures in the multiplicand together, rejecting every nine as it arises, and reserving only the remainder to carry set the last remainder on the left of the cross.

11. Add the figures in the multiplier together, casting out the nines as before, and set the last remainder on the right of the

cross.

III. Multiply the two remainders together, and, having cast out the nines from the product, set the remainder at the top of the cross; if there are no nines, the product itself must be set.

IV. Lastly, cast the nines out of the product, and set the remainder at the bottom of the cross: if the bottom and top figures are alike, the operation is right, except you have made a mistake of nine exactly; but if they are not alike, the work is certainly wrong.

Thus to prove Ex. 43. I say, 2 and 7 are 9; this I omit; and go on, 3 and 8 are 11; (omitting the 9) I carry 2 to the 1, which make 3, and 4 are 7; this I put down on the left of the cross.

X

Next, beginning with the multiplier, I say, 4 and 5 are 9, which I omit ; and go on, 6 and 7 are 13, which is 4 above 9; omitting the 9, I put down 4 on the right side of the cross.

Thirdly I multiply 7 by 4, and the product is 28; in this number I find there are 3 nines (which I omit) and 1 over; therefore I put I at the top.

Lastly, I cast the nines out of the product; thus, 1 and 2 are 3 and 5 are 8 and 5 are 13, which is 9 and 4 over; (omitting the 9) I carry 4 to the 8 are 12, which is 9 and 3 over; (omitting the 9) I carry 3 to 5 are 8 and 3 are 11, which is 2 above 9 (omitted); carry 2 to 8 are 10, which is 1 above 9; this 1 I set down at the bottom of the cross, and finding that it agrees with the figure at the top, I conclude that the operation is right, except I have made a mistake of 9.

33. When ciphers are intermixed with the other figures in the multiplier, they need not be regarded, provided the first figure of each line in the multiplying, be put under the figure you multiply by m.

Ex. 45. Although here are 5 figures in the multiplier, yet one being a cipher, there are only 4 figures to multiply by: the difference made by the cipher is, that it occasions the third and fourth lines to stand one place more to the left, than they would if the cipher was not there.

Ex. 46. Here are 6 figures in the multiplier, but only 4 multiplying figures; the two ciphers remove the third and fourth lines two places to the left.

In examples of this kind, when ciphers are subjoined to the right of both factors, they are to be omitted in the multiplying, and all of them are to be written to the right hand of the adding.

In Ex. 47, I multiply by 34, and take no notice of the two ciphers in the multiplying part; but I bring them both down to the right hand of the adding. Ex. 48. Here are 5 ciphers subjoined to both factors; they are omitted in the work, and set down at the end of the adding, as before.

34. When the figures 11 or 12 stand together in a multiplier, the multiplication by both figures may be performed in one. line, (see Ex. 24. 25. Art. 29.) provided the right hand figure of every product stand under the units place of its respective multiplier, as before directed ".

- This rule is sufficiently evident from the note on Art. 31.

This rule depends on the reasons given in the note on Art. 31,

Ex. 49. In this example the multiplier consists of 11, 4, and 12. I multiply first by 12, putting the right hand figure 8 of the product under the units 2. I next multiply by 4, putting the right hand figure 6 of the product under the said multiplier 4. Lastly, I multiply by 11, putting the right hand figure 4 of the product under the right hand I of the 11. And note, that always in multiplying by a double figure, the units place of the product must stand under the right hand figure of such multiplier.

Ex. 50. Here the multipliers are 12, 12, and 11. The truth of these operations may be proved by multiplying by every figure singly (Art. 31.), or by changing the places of the factors, viz. multiplying the multiplier by the multiplicand.

35. When the multiplier is a composite number, that is, the product of two or more numbers in the table.

RULE. Multiply by one of the component parts, and multiply the product by another, and this last product by another, and so on, when there are several component parts; but this rule is seldom applied when the multiplier is found to consist of more than two parts, or three at most ".

The operations may be proved by Art. 31.

=

• A number which is the product of two or more numbers (each greater than unity) is called a composite number; and the numbers of which it is the product are called its component parts. Thus 45 is a composite number, the component parts of which are 9 and 5; for 9 × 5 45. The terms composite and component are derived from the Latin con, with, and pono, to place. P Suppose it were required to multiply 234 by 14: by Art. 28, if I add 234 taken successively 7 times, the sum will be the same as 234 multiplied by 7. Now if I double this sum, or multiply it by 2, the result will be the same as though I had taken down 234 14 times, and added the whole together; that is, 234 multiplied by 14 is the same as if it were multiplied by 7 and the product multiplied by 2. Again, if 234 be multiplied by 2, and the product be taken 7 times, the result will evidently be the same as it would had I taken the given number 14 times; and the same thing may be shewn in every similar case. The truth of the rule may likewise be proved by Art. 81.