row, it is evident that in this way seven rows, of four trees each, may be made of them. But the number of trees remains the same, which way soever they are counted.

Now whatever be the number of trees in each row, if they are all alike, it is plain that as many rows, of four each, can be made, as there are trees in a row. Or whatever be the number of rows of seven each, it is evident that seven rows can be made of them, each row consisting of a number equal to the number of rows. In fine, whatever be the number of rows, and whatever be the number in each row, it is plain that by taking one from each row a new row may be made, containing a number of trees equal to the number of rows, and that there will be as many rows of the latter kind, as there were trees in a row of the former kind.

The same thing may be demonstrated abstractly as follows: 6 times 5 means 6 times each of the units in 5; but 6 times 1 is 6, and 6 times 5 will be 5 times as much, that is, 5 times 6.

Generally, to multiply one number by another, is to repeat the first number as many times as there are units in the second number. To do this, each unit in the first must be repeated as many times as there are units in the second. But each unit of the first repeated so many times, makes a number equal to the second; therefore the second number will be repeated as many times as there are units in the first. Hence the product of two numbers will always be the same, whichsoever be made multiplier.

What will 254 pounds of meat cost, at 7 cents per pound? This question will show the use of the above proposition; for 254 pounds will cost 254 times as much as 1 pound; but 1 pound costs 7 cents, therefore it will cost 254 times 7. But since we know that 254 times 7 is the same as 7 times 254, it will be much more convenient to multiply 254 by 7. It is easy to show here that the result must be the same; for 254 pounds at 1 cent a pound would come to 254 cents;, at 7 cents a pound therefore it must come to 7 times as much.

Operation.
254
7

Ans. 1778 cents.
(tens); write the 7

Here say 7 times 4 are 28; reserving the 2 (tens) write the 8 (units); then 7 times 5 (tens) are 35 (tens) and 2 (tens) which were reserved are 37 (tens) and reserve the 3 (hundreds);

then 7 times 2 (hundreds) are 14 (hundreds) and 3 which were reserved are 17 (hundreds). The answer is 1778 cents; and since 100 cents make a dollar, we may say 17 dollars and 78 cents.

The process of multiplication, by a single figure, may be expressed thus: Multiply each figure of the multiplicand by the multiplier, beginning at the right hand, and carry as in addition.

IV. What will 24 oxen come to, at 47 dollars apiece?

It does not appear so easy to multiply by 24 as by a num ber consisting of only one figure; but we may first find the price of 6 oxen, and then 4 times as much will be the price of 24 oxen.

A number which is a product of two or more numbers is called a composite or compound number. The numbers, which, being multiplied together, produce the number, are called factors of that number. 4 is a composite number, its factors are 2 and 2, because 2 times 2 are 4. 6 is also a composite number, its factors are 2 and 3. T'he numbers 8,

9, 10, 12, 14, 15, &c. are composite numbers; some of them have only two factors, and some have several. The sign X, a cross, in which neither of the marks is either horizontal or perpendicular, is used to express multiplication. Thus 3 X 2 = 6, signifies 2 times 3 are equal to 6. 2 × 3 X 530, signifies 3 times 2 are 6, and 5 times 6 are 30.

Numbers which have several factors, may be divided into a number of factors, less than the whole number of factors, in several ways. 24, for example, has 4 factors, thus, 2 × 2 × 2 × 3 = 24. This may be divided into 2 factors and into 3 factors in several different ways. Thus 4 X 6=24;~ ́

2 × 2 × 6=24; 3 x 824; 2 × 1224; 2 × 6 × 2 = 24.

When several numbers are to be multiplied together, it will make no difference in what order they are multiplied, the result will always be the same.

What will be the price of 5 loads of cider, each load containing 7 barrels, at 4 dollars a barrel?

Now 5 loads each containing 7 barrels, are 35 barrels. 35 barrels at 4 dollars a barrel, amount to 140 dollars. Or we may say one load comes to 28 dollars, and 5 loads will come to 140 dollars. Or lastly, 1 barrel from each load will come to 20 dollars, and 7 times 20 are 140.

Thus 7

Or

Or

54

7

What is the price of 23 loads of hay, at 34 dolls. a load?

It is easy to see that we may multiply by any other number in the same manner.

This operation may be expressed as follows. To multiply by a composite number: Find two or more numbers, which being multiplied together will produce the multiplier; multiply the multiplicand by one of these numbers, and then that product by another, and so on, until you have multiplied by all the factors, into which you had divided the multiplier, and the last product will be the product required.

If the multiplier be not a composite number, or if it cannot be divided into convenient factors: Find a composite number as near as possible to the multiplier, but smaller, and multiply by it according to the above rule, and then add as many times the multiplicand, as this number falls short of the multiplier.

V. I have shown how to multiply any number by a single figure; and when the multiplier consists of several figures, how to decompose it into such numbers as shall contain but one figure. It remains to show how to multiply by any number of figures; for the above processes will not always be found convenient.

The most simple numbers consisting of more than one figure are 10, 100, 1000, &c. It will be very easy to multiply by these numbers, if we recollect that any figure written in the second place from the right signifies ten times as many as it does when it stands alone, and in the third place, one hundred times as many, and so on. If a zero be annexed at the right of a figure or any number of figures, it is evident that they will all be removed one place towards the left, and consequently become ten times as great; if two zeros be annexed they will be removed two places, and will be one hundred times as great, &c. Hence, to multiply by

any number consisting of 1, with any number of zeros at the right of it, it is sufficient to annex the zeros to the multiplicand.

VI. When the multiplier is 20, 30, 40, 200, 300, 2000, 4000, &c. These are composite numbers, of which 10, or 100, or 1000, &c. is one of the factors. Thus 20 = 2 × 10; 30= 3 x 10; 300 = 3 × 100 ; &c. In the same

manner 387000387 X 1000.

How much will 30 hogsheads of wine come to, at 87 dollars per hogshead?

It appears that it is sufficient in this example to multiply by 3 and then annex a zero to the product. If the number of hogsheads had been 300, or 3000, two or three zeros must have been annexed. It is plain also that, if there are zeros on the right of the multiplicand, they may be omitted until the multiplication has been performed, and then annexed to the product.