Sect. I.

Division by the Lines.

Problem III.

One Number being given to be divided by another, to find the Quotient.

In Multiplication both of whole Numbers and mixt, the Proportion is:

    Which Quotient shall ever consist of so many Figures as the Dividend hath more than the Divisor, except when the Divisor does not exceed so many of the first Figures of the Dividend; then it shall have one place more.

    1. Example: Let it be required to divide 24 by 4; The Proportion is, 4 : 1 :: 24 : 6.

    Therefore, Set 4 upon B, to 1 upon A, and then against 24 upon B, is 6 upon A, which is the Quotient sought.

    2. Example: Let it be required to divide 1768 by 26: The proportion is, 26 : 1 :: 1768 : 68.

    Therefore, Set 26 upon B, to 1 upon A, and then against 1768 upon B, is 68 (the Quotient) upon A.

    3. Example: Suppose 952 were to be divided by 14: The proportion is, 14 : 1 :: 952 : 68.

    Therefore, Set 14 upon A, to 1 upon B, then against 952 upon A, you have 68 upon B, which is the Quotinet requir’d.

    Observe here, the Dividend hath but one Figure more than the Divisor, yet the Quotient consists of two Figures, because the Divisor doth not exceed so many of the first Figures of the Dividend; but in the First and Second Examples the Quotients have but so many Figures as the Dividend hath more than the Divisor, because the Divisor doth not exceed so many of the first Figures of the Dividend, according to the general Rule above given.

    This shews (in all Cases) how many Figures must be in the Quotient, the Value of the first of which may be found by the Rule given in the Introduction, Pag. 8.

    Or it will thus appear, that the Quotient in the last Example is 68, and not 6.8, nor .68: Set 14 upon A, to 1 upon B,

    This is but the Converse to the Third Example of Multiplication; and as by that it is manifest, that if 2 were multiplied by 14, the Product would be 28; if 3, it would be 42; if 4, 56, &c. So here it is evident, that if 28 be divided by 14, the Quotient will be 2; if 42, it will be 3; if 56, it is 4, &c. thus by reading on the Proportion from the Divisor 14 : 28, 56, 70, &c. I find at last, that if the Dividend were but 95.2, the Quotient would be 6.8, but the Dividend being 952, the Quotient must be 68; for by taking away the prick, the Fractions in each are made whole Numbers.

    By these Examples it is also evident that at once setting of the Rule we both Multiply and Divide.

    For if 14 be a Multiplicator, set 1 on B, against 14 on A; then against any Multiplicand upon B, you have the Product upon A; as appear’d by the third Example of Multiplication.

    And without moving the Rule, if you suppose 14 to be a Divisor, then against any Dividend upon A, you have the Quotient upon B; as in the last Example of Division.

    How by any Divisor to find a Multiplicator, was shown in pag. 13 of the Introduction; it may also be perform’d by the Lines more readily.

    Thus having a Divisor, to find a Multiplicator:

    Set the Divisor given upon A, to 1 upon B, and then against 1 (towards the left hand) upon A, is the Multiplicator upon B. Example; Suppose 25 were a Divisor given to find a Multiplicator.

    Set 25 upon A, to 1 upon B, and then against 1 (towards the left hand) upon A, is .04 the Multiplicator sought.

    By a Multiplicator to find a Divisor: This is but the Converse of the former; for having .04 a Multiplicator upon B, set it against 1 upon A, and then against 1 upon B, is 25 the Divisor, as before.

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This page was last updated on 29 March 2000.