If the multiplier be a composite number, we may multiply by its component parts, instead of the number itself.

Example-4276 by 42, or by 7 times 6. 4276

No. 1. Multiply 5492 by 72 No. 4. Multiply 378914 by 54

520813 by 63 56417 by 144

If the multiplier be 5, which is the half of 10, we may annex a cipher, and then divide by 2. If it be 25, which is the fourth part of 100, we may annex two ciphers, and then divide by 4: if 125, which is the eighth part of 1000, three ciphers, and then divide by S. Other contractions may be found in a similar

manner.

RULE III.

1. To multiply any number by 9, which is one less than 10, annex a cipher to the number and then subtract the multiplicand from that number, increased by the cipher, and the remainder will be the product.

2. To multiply by 99, or any number of nines, annex as many ciphers as there are nines in the multiplier, and then subtract the multiplicand. A similar rule may be found for other figures as well as for 9.

No. 1. Multiply 4875967 by 99994 No. 3. 4759378 by 7997

2.

4169875 5995

4.7093856 399958

It often happens that a line of the product is more easily found from a former line than from the multiplicand itself,

In the first example, instead of multiplying by 8, we may multiply 1894104 by 2, and in the second, instead of multiplying by 3, we may divide 2235408 by 2.

The product of two or more figures may sometimes be obtained at once, from the product of a figure already found.

In the first example, we first multiply by 6; then, because 6 times 9 is 54, we multiply that product by 9, instead of multiplying successively by 4 and 5.

In the second example, we first multiply by 9; then, because 3 times 9 is 27, we multiply the first line of the product by 3, instead of multiplying separately by 7 and 2; lastly, because twice 27 is 54, we multiply the second line of the product by 2, instead of multiplying separately by 4 and 5.

When this method is followed, care must be taken to place the right hand figure of each product under the right hand figure of that part of the multiplier from which it is derived.

Contractions may sometimes be obtained by beginning the work at the highest place of the multiplier, instead of the lowest.

METHODS OF PROOF.

1. Repeat the operation, using the multiplier as the multiplicand, and the multiplicand as the multiplier.

2. Cast the nines out of the multiplicand and also out of the multiplier, if above 9. Multiply the excesses together, and cast the 9's, if necessary, out of their product. Then cast the 9's out of the product of the factors, required to be multiplied together, and observe if this excess correspond with the former. The results are generally placed round a cross, thus:

C

1. There are twelve signs in the Ecliptic, and every sign contains 30 degrees; how many degrees does the Ecliptic contain ?

2. How many balls in 18 boxes, each containing 365?

3. How much hay will grow on 425 acres, supposing two crops, the first 212 loads, and the second 70 loads per acre, distinguishing the crops?

4. The sum of two numbers is 4584, and the less is 1876: what is their product?

5. What is the difference between twelve times sixty-seven, and twelve times, seven and sixty?

6. How many miles will a person walk, in five years, supposing he travels twenty-three miles each day?

7. Suppose 200 men take a prize, and each man's share is £160, what is the value of the prize?

8. How many letters in a book of 7 vols., each vol. containing 328 pages, each page 34 lines, and each line 36 letters?

DIVISION.

In DIVISION two numbers are given, and it is required to find how often the one is contained in the other.

Thus it may be required to find how many times 9 is contained in 58, and the answer is 6 times, with a remainder of 4. Here the 9, or number divided by, is called the Divisor; the (58) or number to be divided, the Dividend; and 6, or the number of times the divisor (9) is contained in the dividend, (58) is called the Quotient.

As the operation of Division would be tedious, when the divisor is contained a great many times in the dividend, it is proper to shorten the labour, as much as possible, by every con venient method that can be discovered.

When the divisor is less than 12, place the divisor on the left-hand of the dividend, separating them by a short line, then consider how often the divisor is contained in the first or lefthand figure of the dividend; if this figure shall be less than the divisor, take the first two, or even three, if necessary, and set down the quotient under the right-hand figure employed; if there shall be any remainder, carry it as so many tens to the next figure of the dividend, and continue this operation till all the dividend is exhausted.