figures may fall in a column straight below each other; but observe to increase the first figure of every line with what would arise from the figures omitted, in this manner namely 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c. ; and the sum of all the lines will be the product as required, commonly to the nearest unit in the last figure.

EXAMPLES.

1. To multiply 27.14986 by 92-41035, so as to retain only four places of decimals in the product.

2. Multiply 480.14936 by 2.72416, retaining only four decimals in the product.

3. Multiply 2490-3048 by 573286, retaining only five decimals in the product.

4. Multiply 325-701428 by 7218393, retaining only three decimals in the product.

DIVISION OF DECIMALS.

DIVIDE as in whole numbers; and point off in the quotient as many places for decimals, as the decimal places in the dividend exceed those in the divisor*.

* The reason of this Rule is evident; for, since the divisor multiplied by the quotient gives the dividend, therefore the number of decimal places in the dividend, is equal to those in the divisor and quotient, taken together, by the nature of Multiplication; and consequently the quotient itself must contain as many as the dividend exceeds the divisor.

Another way to know the place for the decimal point is this: The first figure of the quotient must be made to occupy the same place, of integers or decimals, as that figure of the dividend which stands over the unit's figure of the first product.

When the places of the quotient are not so many as the Rule requires, the defect is to be supplied by prefixing ciphers.

1

When there happens to be a remainder after the division; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required.

WHEN the divisor is an integer, with any number of ciphers annexed cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers, if need be; then proceed as before.

CONTRACTION II.

HENCE, if the divisor be 1 with ciphers, as 10, 100, or 1000, &c.; then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left, as the divisor hath ciphers; prefixing ciphers if need be.

WHEN there are many figures in the divisor; or when only a certain number of decimals are necessary to be retained in the quotient; then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual.

Let each remainder be a new dividend; and for every such dividend, leave out one figure more on the right-hand side of the divisor; remembering to carry for the incrcase of the figures cut off, as in the 2d contraction in Multiplication.

Note. When there are not so many figures in the divisor as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient; after which begin the contraction.

EXAMPLES.

1. Divide 2508-92806 by 92-41035, so as to have only four decimals in the quotient, in which case the quotient will contain six figures.

92-4103,5) 2508-928,06 (27-1498 92.4103,5) 2508-928,06 (27-1498

2. Divide 4109.2351 by 230-409, so that the quotient may

contain only four decimals.

Ans. 17.8345.

3. Divide 37-10438 by 5713.96, that the quotient may contain only five decimals. Ans. 00649. 4. Divide 913-08 by 2137-2, that the quotient may contain only three decimals.

REDUCTION OF DECIMALS.

CASE I.

To reduce a Vulgar Fraction to its equivalent Decimal.

DIVIDE the numerator by the denominator, as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required*.

*The following method of throwing a vulgar fraction, whose denominator is a prime number, into a decimal consisting of a great number of figures, is given by Mr. Colson in page 162 of Sir Isaac Newton's Fluxions.

EXAMPLE.

Let be the fraction which is to be converted into an equivalent decimal.

=

=

Then, by dividing in the common way till the remainder becomes a single figure, we shall have 03448, for the complete quotient, and this equation being multiplied by the numerator 8, will give 275844, or rather 27586, and if this be substituted instead of the fraction in the first equation, it will make 0344827586. Again, let this equation be multiplied by 6, and it will give •2068965517; and then by substituting as before

03448275862068965517,7, ;

=

and so on, as far as may be thought proper; each fresh multiplication doubling the number of figures in the decimal value of the fraction.

In the present instance the decimal circulates in a complete period of 28 figures, i. e. one less than the denominator of the fraction. This, again, may be divided into equal periods, each of 14 figures, as below:

·03448275862068
96551724137931

in which it will be found that each figure with the figure vertically below it makes 9; 0+9=9; 3+6=9; and so on. This circulate also comprehends all the separate values of,,, &c. in corresponding circulates of 28 figures, only each beginning in a distinct place, easi- › ly ascertainable. Thus, 06896, &c. beginning at the 12th place of the primitive circulate. ·103448, &c. beginning at the 28th place. So that, in fact, this circle includes 28 complete circles.

=

=

See, on this curious subject, Mr. Goodwyn's Tables of Decimal Circles, and the Ladies' Diary for 1824.

To find the Value of a Decimal in terms of the Inferior

Denominations.

MULTIPLY the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder to the right-hand, as there are places in the given decimal.

Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before.

Proceed in the same manner through all the parts of the integer; then the several denominations separated on the left-hand will make up the answer.

Note, This operation is the same as Reduction Descending in whole numbers.

EXAMPLES.

1. Required to find the value of 775 pounds sterling.