4. From 2637804 take 2376982.

5. From 3762162 take 826541.

6. From 78213606 take 27821890.

Ans. 260822.

Ans. 293 621.

Ans. 50391716.

7. The Arabian method of notation was first known in England about the year 1150; how long was it thence to the year 1776? Ans. 626 years.

8. Sir Isaac Newton was born in the year 1642, and aied in 1727; how old was he at the time of his decease? Ans. 85 years,

SIMPLE MULTIPLICATION.

Simple Multiplication is a compendious method of addition, and teaches to find the amount of any given number of one denomination, by repeating it any proposed number of times.

The number, to be multiplied, is called the multiplicand. The number, to multiply, is called the multiplier. The number, found from the operation, is called the pro'duct.

Both the multiplier and multiplicand are, in general, called terms or factors.

difference of the corresponding figures, taken together, will evidently make up the difference of the given numbers, Q. E. D. The truth of the method of proof is evident; for the difference of two numbers, added to the less, is manifestly equal to the greater.

Use of the table in Multiplication.

Find the multiplier in the first column on the left, and the multiplicand in the first line; and the product is in the common angle of meeting, or against the multiplier, and under the multiplicand.

Use of the table in Division.

Find the divisor in the first column on the left, and the dividend in the same line; then the quotient will be, over the dividend, the first number of the column.

RULE.*

1. Place the multiplier under the multiplicand, so that units may stand under units, tens under tens, &c. and draw a line under them.

* DEMON. 1. When the multiplier is a single digit, it is plain, that we find the product; for by multiplying every figure, that is, every part of the multiplicand, we multiply the whole; and writing down the products, that are less than ten, and the excesses above tens respectively in the places of the figures

2. Begin at the right, and multiply the whole multiplicand severally by each figure in the multiplier, setting the first figure of every line produced directly under the figure you are multiplying by, and carrying for the tens, as in addition. 3. Add all the lines together, and their sum is the product,

multiplied, and carrying the number of tens in each product to the product of the next place is only gathering together the similar parts of the respective products, and is therefore the same thing, in effect, as writing the multiplicand under itself so often as the multiplier expresses, and adding the several repetitions together; for the sum of each column is the product of the figures in the place of that column; and these products, collected together, are evidently equal to the whole required product.

2. If the multiplier consists of more than one digit; having then found the product of the multiplicand by the first figure of the multiplier, as above, we suppose the multiplier divided into parts, and find, after the same manner, the product of the multiplicand by the second figure of the multiplier; but as the figure we are multiplying by stands in the place of tens; the product must be ten times its simple value; and therefore the first figure of this product must be placed in the place of tens; or, which is the same thing, directly under the figure we are multiplying by. And proceeding in this manner separately with all the figures of the multiplier, it is evident, that we shall multiply all the parts of the multiplicand by all the parts of the multiplier; or the whole of the multiplicand by the whole of the multiplier; therefore the sum of these several products will be equal to the whole required product. Q. E. D.

The reason of the method of proof depends on this proposition, namely," that two numbers being multiplied together, either of them may be made the multiplier, or the multiplicand, and the product will be the same." A small attention to the nature of the numbers will make this truth evident; for 3×7=

Method of Proof.

Make the former multiplicand the multiplier, and the multiplier the multiplicand, and proceed as before; and if the product be equal to the former, the product is right.

21=7x3; and, in general, 3×4X5X6, &c.=4×3×6×5, &c. without any regard to the order of the terms; and this is true of any number of factors whatever.

The following examples are subjoined to make the reason of the rule appear as plain as possible.

Beside the preceding method of proof, there is another very convenient and easy one by the help of that peculiar property of the number 9, mentioned in addition; which is performed thus.

RULE 1. Cast the nines out of the two factors, as in addition, and write the remainder.

2. Multiply the two remainders together, and, if the excess of nines in their product be equal to the excess of nines in the total product, the answer is.'ght.

4215 3 excess of 9s in the multiplicand.

878 5 ditto in the multiplier.

33720

29505

33720

3700770 6 ditto in the product excess of 9s in 3x5.

DEMONSTRATION OF THE RULE. Let M and V be the number of 9s in the factors to be multiplied, and a and b what remains; then M+a and N+6 will be the numbers themselves, and their product is Mx N+ Mxb+Ñxa+axb; but the first three of these products are each a precise number of 9s, because one of their factors is so; therefore, these being cast away, there remains only axb; and if the 9s be also cast out of this, the excess is the excess of the 9s in the total product; but a and b are the excesses in the factors themselves, and a×b their product; therefore the rule is true. Q. E. D.

This method is liable to the same inconvenience with that in addition.

Multiplication may also very naturally be proved by division; for the product being divided by either of the factors, the quotient will evidently be the other; but it would have been contrary to good method to give this rule in the text, because the pupil is supposed as yet to be unacquainted with division.