dred and fifty-two; how many pounds had he remaining? Ans. 148 pounds.

15. Borrowed one thousand two hundred and thirty-four guineas, and paid in part nine hundred and eighty-seven; what sum is there still remaining due? Ans. 247 guineas.

16. There is a person, who if he lives until the year 1820 will be seventy-five years old; in what year was he born? Ans. in 1745.

17. King George the Second came to the throne in 1727, and died in 1760 ; how long did he reign? Ans. 33 years.

18. The western empire was destroyed by Odoacer, king of the Heruli, in the year 476, and the eastern empire submitted to Mahomet the Second, emperor of the Turks in 1453; how many years did the latter exist after the former? Ans. 977 years.

MULTIPLICATION.

28 Multiplication' is a short method of addition; it teaches how to find the sum that arises from repeating one number, called the multiplicand, as often as there are units in another number, called the multiplier.

The number sought, or that which arises from the operation, is called the product.

The multiplicand and multiplier are frequently called Terms or Factors, and the product is sometimes called the Factum.

The mark denoting Multiplication is × ; it is named into, and shews that the number standing before the sign is to be multiplied into (or by) the number which follows it; thus 3 x 4 denotes that 3 is to be multiplied by 4, or taken 4 times .

To perform the operations in this rule with ease, a table of the products of every two numbers, each not exceeding 12, has been contrived; it is called the Multiplication Table, and must be well understood, learned by heart, and remembered.

f The name Multiplication comes from the Latin multus many, and plico to fold; product from produco to produce; factor a maker or doer, and factum a thing done or made.

8 When two or more numbers are to be multiplied by any number, the vinculum is placed over all the former, and a point is frequently interposed between the factors instead of the sign ; thus 3+4.5 denotes that the sum of 3 and 4 (or 7) is to be multiplied by 5; also 3+7+2.6-4 shews that the sum of 3, 7, and 2, (viz. 12,) is to be multiplied by the difference of 6 and 4, (viz. 2;) and the like in other cases.

To find the product of two numbers, look for one of them in the top line of the Table, and for the other in the left hand column; then the number which stands directly under the first, and level with the second, is the product required: thus to find 3 times 5, look for 3 at top, and 5 on the left, then under 3 and level with 5 stands 15, which is the product of 3 and 5. To find 10 times 9, under 10 and level with 9 stands 90, the product. To find 6 times 11, under 6 and level with 11 stands 66, the product, &c.

To shew that Multiplication is derived immediately from Notation, let it be required to find 3 times 4; put down the units in 4 three times, and then count the whole, (Art. 12.) thus, 1, 1, 1, 1 1, 1, 1, 1 these counted amount to 12, therefore 3 times 4 are 12; and the like may be shewn in all cases.

1,1,1,1;

But the rule is commonly derived from Addition thus; to find the product of 3 times 4, put 4 down three times, and find the sum. So if I want to find 9 times 8, I must put nine eights under each other, add them together, and the sum will be 72, or the product of 9 times 8; by this method the Multiplication Table was first formed.

4 12

Simple Multiplication, or Multiplication of whole numbers, is performed by the following rules.

29. When the multiplier does not exceed 12.

RULE I. Under the right hand figure of the multiplicand write the multiplier.

II. Multiply every figure in the multiplicand by the multiplier, and set the product, if it be less than 10, under the figure multiplied.

III. But if the product be 10, or more, set down the units only, and carry 1 for every 10 to the next; multiply the next figure, and after you have multiplied it, (not before,) carry the ones to the product.

IV. Proceed in this manner till all the figures are multiplied; at the last (or left hand) figure, the whole product must be set downi.

i The sum of numbers is evidently of the same denomination with the numbers added; it cannot be of any other; consequently the product of any number multiplied by another must be of the same denomination with the number mul tiplied. Hence if units be multiplied by any whole number, the product is units; if tens be multiplied, the product will be tens; if hundreds be multiplied, the product will be hundreds, &c. Let it be required to multiply 574 by 4; now the 4 units, multiplied by 4 produces 16 units; the 7 tens multiplied by 4 produces 28 tens, or 280; and the 5 hundreds multiplied by 4 produces 20 hundreds, or 2000. Wherefore these several products when added together will give the product of the two given numbers.

From this example, the truth and reason of the Rule will be manifest. With respect to the method of carrying, it is plain from Notation that 10 units are 1 ten, 20 units 2 tens, &c. 10 tens are 1 hundred, 20 tens 2 hundreds, &c. And in general, any number of tens of an inferior denomination will be so many units of the next superior; and therefore I is always carried for every ten, from the inferior to the next superior denomination, as directed in the Rule.

1. Multiply 375294 by 2. OPERATION. Multiplicand 375294 Multiplier 2 Product 750588

EXAMPLES.

Explanation.

Having written the multiplier 2 under the right hand figure of the multiplicand, namely, under the 4, I begin by multiplying that figure; thus I say twice 4 are 8, and put it down; next I say twice 9 are 18, put down 8 and carry 1; then twice 2 are 4 and 1 I carried 5, put down 5; then twice 5 are 10, put down o and carry 1; then twice 7 are 14 and 1 carried 15, put down 5 and carry 1; lastly, twice 3 are 6 and 1 carried 7, I put it down, and the work is finished.

2. Multiply 968754 by 3.

OPERATION.

Multiplicand 968754

Multiplier

3

Product 2906262

Explanation.

Here I say 3 times 4 are 12, put down 2 and carry 1; 3 times 5 are 15 and 1 carried make 16, put down 6 and carry 1; 3 times 7 are 21 and 1 carried 22, put down 2 and carry 2; 3 times 8 are 24 and 2 carried 26, put down 6 and carry 2; 3 times 6 are 18 and 2 carried 20, put down 0 and carry 2; lastly, 3 times 9 are 27 and 2 carried make 29, I put down the whole 29, and the work is finished.

30. When the multiplier has one or more ciphers subjoined. RULE. Write down the cipher or ciphers for the right hand part of the product, then multiply every figure of the multiplicand by the significant figure or figures of the multiplier, (art. 29.) and set the products in their order to the left of the ciphers.

31. When the multiplier consists of two or more significant figures,

and is greater than 12.

RULE I. Multiply every figure of the multiplicand by the right hand figure of the multiplier, and set down the product (as in art. 29).

II. In like manner multiply by the next figure (or the right hand but one), set the product in a line below the former, proceed thus with every figure, setting down each succeeding product below the preceding, observing to place the right hand figure of each under the figure you multiply by.

* The truth of the rule may be shewn from art. 29: thus, let 479 be multiplied by 40; now if the multiplier had been only 4, the product (by art. 29.) would have been 1916; but the multiplier 40 is ten times 4, therefore the product must be ten times 1916, or 19160, by art. 19. In like manner 87 multiplied by 500 is 43500; for 87 multiplied by 5 is 435; but 500 is 100 times 5, whence the product by 500 must be 100 times the product by 5, or 43500; and the like is true in every case.