To this Rule belong three principal Numbers, viz.

1. The Multiplicand, or Number to be increased or multi

plied.

2. The Multiplier, or Number by which the Multiplicand is increased or multiplied.

3. The Product, or Number produced in multiplying. Note.-Before any operation can be performed in this Rule, it is absolutely necessary that the following Table be got by heart; as the ready performance of this and all the following Rules entirely depends upon having a perfect knowledge of it.

12 3

4

TABLE.

5 6 7 8 9 10 11 121 2468|10|12|14|16|18| 20| 22| 24 3|9|12|15| 18 | 21 | 24 | 27 30 33 36

Seek the greater of the two Digits in the upper line, and underneath it, against the lesser, taken in the left-hand column, is the Product sought. Thus, to multiply 9 by 6, seek 9 in the upper line, and under it, against 6 on the left, is 54 the Product: and so of any other.

Note. For the conveniency of dividing by 11 or 12, I have continued the Table to 12 Times, or else in Multiplication it is only required to 9 Times.

Case 1. To multiply by a single Figure.

RULE.

1. Place the Multiplier underneath the Unit's Place of the Multiplicand. 2. Multiply the Unit Figure of the Multiplicand by the Multiplier; if their Product be less than ten, set it down un

der its own Place of Units; but if their Product exceed ten or tens, then set down the Excess only (as in Addition), and bear or carry the said ten or tens in mind, until you have multiplied the next Figure of the Multiplicand by the same Figure of the Multiplier, and to their Product add one for each ten borne in mind, setting down the excess of their Sum above ten, or tens, as before; and so proceed until all the Figures of the Multiplicand are multiplied by the Multiplier.

PROOF.

The most sure and unerring way is by Division. But as the learner is supposed not yet to know that Rule, he cannot prove by it; let him therefore make the Multiplicand the Multiplier, and if the Product come out the same as before, the Work is right.

Some Masters that teach, and several Authors that write of, Arithmetic, prove Multiplication by the Cross. But this method is not to be depended upon, as it will prove a Sum to be right, when at the same time the work is utterly false. But it will not prove a Sum false that is right.

Case 2. When the Multiplier consists of several Figures.

RULE.

1. Place each Figure in the Multiplier respectively under its own kind in the Multiplicand.

2. Multiply the Multiplicand by each Figure of the Multiplier, as before; observing to place the first Figure of each

respective Product underneath that Figure of the Multiplier by which you multiply.

3. Add the several Products together, and the Sum will be the desired, or total, Product.

Case 3. When Ciphers are intermixed with the Figures in

the Multiplier.

RULE.

Omit them, and place the first Figure of each particular

Product under its respective Multiplier.

EXAMPLES.

(21) 10746047

(22) 804700625

40500108

207008009

Case 4. When there are Ciphers at the right-hand of either, or both the Multiplier and the Multiplicand.

RULE.

Proceed as before, neglecting the Ciphers until the particular Products are added together, and to that Sum place the Number of Ciphers that are at the end of both Factors, on the right-hand.

EXAMPLES.

(23) 1460900

(24) 2768000

8700

24600

If it be required to multiply any Number by 10, 100, 1000, &c. it is only annexing the Ciphers of the Multiplier to the right-hand of the Multiplicand, and the work is done.

Case 5. When the Multiplier is such a Number that any two Figures in the Table, being multiplied together, will produce it.

RULE.

Multiply the given Number by one of those Figures, and that Product by the other; which will give the desired Product.

Case 6. When the Multiplier is any Number between 10

and 20.

RULE.

Multiply by the Figure in the Unit's Place, and, as you multiply, add to the Product of each single Figure that of the Multiplicand, which stands next on the right-hand.

V. DIVISION

TEACHETH us to find how often one Number is contained in another, or to divide any Number or Quantity given into any Parts assigned, and serves instead of many Subtractions. In this Rule there are three Numbers real, and a fourth accidental; viz.

1. The Dividend, or Number to be divided.

2. The Divisor, or Number by which you divide.

3. The Quotient, or Number that shows how often the Divisor is contained in the Dividend.

4. The Remainder, which is always less than what you divide by.

Case 1. When the Divisor is not greater than 12.

RULE.

First seek how often the Divisor is contained in the first Figure of the Dividend, or, in case the first Figure of the Dividend be less than the Divisor, then in the first two Figures of the Dividend, and set the quotient Figure down accordingly; and, if any thing remain, carry it to the next Figure in the Dividend, where it must be reckoned as so many Tens; that is, if one remain, you call it 10; if two, 20; if five, 50, and so on; bearing in mind the Remainder of each Figure, and adding it to the next, until you have made use of all the Figures in the Dividend. This is called Short Division.

PROOF.

Multiply the Quotient by the Divisor, and, as you multiply, add the Remainder, if any, or add the whole Remainder to the Product at last, and if it come the same as the Dividend, the Work is right.