truth, that, if two unequal quantities be equally increased, their difference is not thereby altered.

RULE FOR SUBTRACTION.

Write the smaller number under the greater, placing units under units, &c. Begin with the units, and subtract each figure in the lower number from the figure over it. When a figure in the upper number is smaller than the figure under it, consider the upper figure to be 10 more than it is, and the next upper figure on the left hand, to be 1 less than it is.

PROOF. Add together the remainder and the smaller number: their sum will be equal to the greater number, if the work be right.

1. What is the difference between 70240 and 69418? 2. How much is the excess of the number 482724 above the number 194750?

3. Suppose 479021 to be a minuend, and 38456 the subtrahend; how much is the remainder?

5. Subtract fifty-one thousand from one hundred bil lion, eighteen thousand, five hundred and one.

V.

MULTIPLICATION.

MULTIPLICATION is the operation by which a number is produced, equal to as many times one given number, as there are units in another given number. It is an abridged method of finding the sum of several equal quantities, by repeating one of those quantities.

The number to be multiplied or repeated is called the multiplicand; it may be viewed as one of several equal quantities, whose sum is to be produced by the operation. The number to multiply by is called the multiplier; it indicates how many such quantities as the multiplicand are to be united, or, how many times the multiplicand is to be repeated. The number resulting from the operation is called the product.

The multiplicand and multiplier, considered as concurring to form the product, are called factors of the product. Either factor may be used as the multiplier of the other; that is, the multiplicand and multiplier may change places, and the product will be still the same. For example, 4×3=12. 3 X 4=12.

When a product arises from more than two factors, the numbers may be denoted thus, 6 X3 X5=90; but, in forming the product, a distinct operation is necessary to oring in each factor, after the first two. The numbers, 6, 3, 5, would, therefore, be multiplied into each other thus, 6X3=18; 18×5=90.

Factors may be arranged in any succession whatever, since the mere order in which they are brought into the operation cannot affect their final product. For example, 5 X3 X4 60. 4X3X5-60. 3X5X4=60.

The products of small numbers may be committed to memory; but when the product of factors consisting of several figures is required, it is necessary to multiply each figure in the multiplicand by each figure in the multiplier, and denote the several products in such order that they shall represent their respective values. When simple units are employed as the multiplier, the product of each figure in the multiplicand is of the same degree as the figure multiplied; that is, units multiplying units give units, units multiplying tens give tens, units multi plying hundreds give hundreds, &c. When tens are employed as the multiplier, the product of each figure in the multiplicand is one degree higher than the figure multiplied; that is, tens multiplying units give tens, tens multiplying tens give hundreds, tens multiplying hundreds give thousands, &c. When hundreds are employed as the multiplier, the product of each figure in the multiplicand is two degrees higher than the figure multiplied; and so on.

RULE FOR MULTIPLICATION.

Write the multiplier

under the multiplicand, placing units under units, &c.

When there is but one figure in the multiplier, begin with the units, multiply each figure in the multiplicand separately, and place each product under the figure in

the multiplicand from which it arose; observing to carry the tens to the left as in addition.

When there is more than one figure in the multiplier, multiply by each figure separately, and write its product in a separate line, placing the right hand figure of each line under the figure by which you multiply; and finally, add together the several products. The sum will be the whole product.

Abbreviations of the above rule may frequently be adopted, as follows.

When there are ciphers standing between other figures, in the multiplier, they may be disregarded.

When ciphers stand on the right of either factor, or both, they may be disregarded till the multiplication is performed, and then annexed to the product.

When either factor is 10, 100, 1000, &c., merely place the ciphers in this factor on the right hand of the other factor, and it becomes the product.

When the multiplier is a number that can be produced by multiplying two smaller numbers together, multiply the multiplicand first by one of the smaller numbers, and the product thence arising by the other.

1. Suppose 479265 to be a multiplicand, and 9236 the multiplier; how much is the product?

2. Suppose 26537 to be one factor, and 873643 another; how much is their product?

3. Suppose the numbers 725, 38046 and 91, to be factors; how much is the product?

4. What is the product of 62392 × 4003 ?

5. What is the product of 248000 × 9400 ?

6 What is the product of 24 X 300 X13X10002? 7. Multiply one hundred five million, by one thousand.

For the purpose of determining whether any error has happened in the process of multiplication, the following method of trial, which depends on the peculiar property of the number 9, and which is called casting out the nines, may be practised.

Add together the figures of the product, horizontally,

rejecting or dropping the number 9 as often as the sum amounts to that nuniber, and proceeding with the excess, and finally denote the last excess. Perform the same operation upon each of the factors; then multiply together the excesses of the factors, and cast out the nines from their product. If the excess of this smaller product be equal to the excess of the larger product first found, the work may be supposed to be right. It is, however, to be observed, that, although this test furnishes satisfactory evidence of the correctness of an operation, it is not an infallible proof; for, if a product chance to contain an error of just 9 units of any degree, the excess of its horizontal sum is not thereby altered.

In order to perceive why the excess above nines found in the horizontal sum of a product, must be equal to the excess found in the product of the excesses of the factors, observe that, by the law of notation, a figure is increased nine times its value by its removal one place to the left; and hence, however far a figure is removed from the place of units, when its nines are excluded, its remainder can be only itself. Therefore, any number, and the horizontal sum of its figures, must have equal remainders when their nines are excluded. This being understood, observe that, since factors composed of entire nines will give a product consisting of entire nines, it follows, that any excess above nines in a product, must arise from an excess above nines in the factors. Therefore, the product of the excesses of the factors, must contain the same excess that is contained in the product of the whole factors.

VI.

DIVISION.

DIVISION is the operation by which we find how many times one number is contained in another. It is the converse of multiplication; the product and one factor being given, and the other factor resulting from the operation.

The number which corresponds to the product in multiplication, is the number to be divided, and is called the dividend. The given factor is the number to divide by, and is called the divisor. The factor to be found, that is, the number which shows how many times the dividend contains the divisor, is called the quotient.

As multiplication has been shown to proceed from addition, so division may be shown to proceed from subtraction. If we repeatedly subtract the divisor from the dividend till the latter is exhausted, the number of subtractions performed will answer to the number of units in the quotent. For example, if the dividend be 24, and the divisor 6, the quotient may be found by subtraction thus, 24-6=18, 18—6=12, 12—6—6, 6—6=0. Here 6 is subtracted four times from 24, and there is nothing remains; therefore,4 is the number of times that 6 is contained in 24. In division, this operation is denoted thus, 24÷6=4; or thus, 244.

Division not only investigates the number of times the dividend contains the divisor, but it also serves to divide the dividend into as many equal parts as the divisor contains units; the quotient being one of these parts. This effect of the operation may be understood by considering, that, since the divisor and quotient are factors of the dividend, they must each indicate how many of the other the dividend contains.

It may be observed, that all the preceding operations begin at the place of simple units; division, however, must begin at the highest degree of units; for, the number of times that the divisor is contained in the higher units of the dividend must be taken out first, in order that any remainder, or excess above an exact number of times, may be carried down to the lower degrees of units, and divided therewith.

When the divisor is not contained an exact number of times in the dividend, there will be a remainder at the end of the operation. This remainder, being a part of the dividend, is to be divided; but its quotient will be smaller than a unit, since a quantity in the dividend just equal to the divisor, gives only a unit in the quotient.