responding figure in the minuend, add ten to the latter, and then take the figure in the subtrahend from the sum, putting down the remainder, as before; and in this case add 1 to the next figure to the left in the subtrahend, to compensate for the ten borrowed in the preceding place.

Thus proceed till all the figures are subtracted.

Note. It is customary to place the minuend above the subtrahend; but this is not absolutely necessary. Indeed, it is often convenient in computation to find the difference between a number and a greater that naturally stands beneath it; it is therefore, expedient to practise the operation in both ways, so that it may, however it occur, be performed without hesi

tation.

Here the five figures on the right of the subtrahend are each less than the corresponding figures in the minuend, and may therefore be taken from them, one by one. But the sixth

figure, viz. 8, cannot be taken from the 5 above it. Yet, as a unit in the seventh place is equivalent to 10 in the sixth, this unit borrowed (for such is the technical word here employed) makes the 5 become 15. Then 8 taken from 15 leaves 7, which is put down; and 1 is added to the 9 in the 7th place of the subtrahend, to compensate or balance the 1 which was borrowed from the 7th place in the minuend. Recourse must be had to a like process whenever a figure in the subtrahend exceeds the corresponding one in the minuend.

Other Examples.

From 8217 From 44444
Take 3456 Take 3456

Take 45624

Take 21498
From 76262 From 80200

Remains 4761 Remains 40988 Remains 54764 Remains 34576

SECTION IV.-Multiplication of Whole Numbers.

MULTIPLICATION of whole numbers is a rule by which we find what a given number will amount to when it is repeated as many times as are represented by another number.*

* This definition, though not the most scientific that might be given, is placed here, because others depend implicitly, if not explicitly, on proportion, and therefore cannot logically be introduced thus early in the course.

The number to be multiplied, or repeated, is called the multiplicand, and may be either an abstract or a concrete number.

The number to be multiplied by is called the multiplier, and must be an abstract number, because it simply denotes the number of times the multiplicand is to be repeated.

Both multiplicand and multiplier are called factors.

The number that results from the multiplication is called the product.

Before any operation can be performed in multiplication, the learner must commit to memory the following table of products, from 2 times 2 to 12 times 12.

10

288

63 72 81 90 99 108

20 30 40 50 60 70 80 90 100 110 120

11 22 33 44 55

66

77 88 99 110 121 132

12 24 36 48 60 72 84 96 108 120 132 | 144

It is very advantageous in practice to have this table carried on, at least intellectually, to 20 times 20. All the products to this extent are easily remembered.

X 3

=

The learner will perceive that in this table 7 times 5 is equal to 5 times 7, or 7 × 5 = 35 5 x 7. In like manner that 8 24 3 x 8, 4 × 11 = 44 = 11 x 4, and so of other products. This is often made a subject of formal proof, as well as that 3 x 5 x 8: 3 x 8 x 5 = 5 x 3 x 8 = 5 x 8 x 3, &c. But to attempt the demonstration of things so nearly axiomatical as these is quite unnecessary.

4827

56

Previously to exhibiting the rules, let us take a simple example, and multiply 4827 by 8. Here placing 8 the numbers as in the margin, and multiplying in their order 7 units by 8, 2 tens by 8, 8 hundreds by 8, 4 thousands by 8, the several products are 56 units, 16 tens, 64 hundreds, 32 thousands: these placed in their several ranks, according to the rules of notation, and then added up, give for the sum of the whole, or for the product of 4827 multiplied by 8 the number 38616.

16

64

32

38616

CASE I. To multiply a number, consisting of several figures, by a number not exceeding 12.

Multiply each figure of the multiplicand by the multiplier, beginning at the units; write under each figure the units of the product, and carry on the tens to be added as units to the product following.

CASE II. To perform multiplication when each factor exceeds 12.

Place the factors under each other (usually the smallest at bottom), and so that units stand under units, tens under tens, and so on.

Multiply the multiplicand by the figure which stands in the unit's place of the multiplier, and dispose the product so that its unit's place shall stand under the unit of the multiplicand; then multiply successively by the figure in the place of tens, hundreds, &c. of the multiplier, and place the first figure of each product under that figure of the multiplier which gave the said product.

The sum of these products will be the product required.

Note.-Multiplication may frequently be shortened by separating the multiplier into its component parts or factors, and multiplying by them in succession. Thus, since 132 times any number are equal to 12 times 11 times that number, the first example may be performed in this manner :

So, again, the multiplier of the second example, viz. 672576, divides into three numbers, 600000, 72000, and 576; where,

=

omitting the cipher, we have 72 12 x 6, and 576 8 x 72. Hence the operation may be performed thus:

Multiplicand 821436
Multiply by 6

in the 6th place

59143392 for 72 thousands. 473147136 for 576 units.

552478139136: three lines saved.

Other modes of contraction will appear as we proceed.

SECTION V.-Division of Whole Numbers.

DIVISION is a rule by which we determine how often one number is contained in another. Or, it is a rule by which, when we know a product of one of the faetors which produced it, we can find the other.

The number to be divided is called the dividend.
That by which it is divided, the divisor.

That which results from the division, the quotient, when division and multiplication are regarded as reciprocal ope

rations.

The dividend is equivalent to the product.

The divisor

The quotient

RULE.

multiplier
multiplicand.

Draw a curved line both on the right and left of the dividend, and place the divisor on the left.

Find the number of times the divisor is contained in as many of the left hand figures of the dividend as are just necessary, and place that number on the right.

Multiply the divisor by that number, and place the product under the above mentioned figures of the dividend.

Subtract the said product from that part of the dividend under which it stands, and bring down the next figure of the dividend to the right of the remainder.

Divide the remainder thus increased, as before; and if at any time it be found less than the divisor, put a cipher in the quotient, bring down the next figure of the dividend, and continue the process till the whole is finished: the quotient figures thus arranged will be that required.