Elementary arithmetic, with brief notices of its history... by Robert Potts.

16

MULTIPLICATION AND DIVISION.
ART. 1. The multiplication of one number by another is the process
of finding the number produced by the addition of one of them as
many times as there are units in the other, as 5 multiplied by 3 or
3 times 5 is 15; and is the same as 5 added 3 times.
The number produced is called the product, the number to be multiplied the multiplicand, and the number indicating the number of
times the multiplier. These are also called the factors of the product.
Division is the process of finding how many times a less number is
contained in a greater, and is equal to the number of times the less
number can be subtracted from the greater, as the number 5 is contained 3 times in 15, or 5 can be subtracted exactly three times from 15.
The greater number is named the dividend, the less number the
divisor, and the number of times the divisor is contained in the dividend, the quotient.
The operations of multiplication and division are, one the reverse
of the other; for since the product of 5 multiplied by 3 is 15, it
follows that 15 divided by 5 gives 3 for the quotient.
The divisor and quotient are convertible, as for instance, if 15 be
the dividend, and 3 the divisor, the quotient is 5; but if 5 be the
divisor, the quotient is 3.
The names of dividend, divisor, and quotient in the operation of
division correspond to those of product, multiplier, and multiplicand
in the process of multiplication.
The processes of multiplication and division being one the reverse of
the other, as the multiplicand multiplied by the multiplier produces
the product, so the quotient multiplied by the divisor will reproduce
the dividend when there is no remainder after the division. If, however,
there be a remainder, it must be added to the product of the divisor
and quotient to reproduce the complete dividend.
Any product is said to be a multiple of each factor, and each
factor is called a submultiple of the product. Thus 15 is a multiple
of each of the factors 5 and 3; and 5 and 3 are each submultiples of 15.
2. PeOB. To find theyproduct of any tzwo single figures.
Let it be required to find the product of 5 and 3. The product of
5 multiplied by 3 means that 5 is to be repeated three times; that is,
the sum of 5 + 5 + 5 is 15. All the products of two single figures can
be thus found by the simple process of addition.
The product of any two numbers is the same, whichever vmay be amcde
the multiplier.
For instance, 5 multiplied by 3 gives the same product as 3 multiplied by 5.