Sect. I.

Multiplication by the Lines.

Problem II.

One Number being given to be multiplied by another, to find the Product.

In Multiplication either of whole Numbers, mixt or decimal Fractions; the Proportion is:

    And the Product of any two Numbers shall have so many places as there be in both the Numbers given, except where the lesser of them do not exceed so many of the first Figures of the Product; then it will have one less.

    1. Example: Let it be required to multiply 6 by 4.

    Say, [1 : 4 :: 6 : 24] which Analogy or Proportion may be read thus; As 1 is to 4 So is 6 to 24.

    Therefore, Set 1 upon the Line B, to 4 on the Line A, and then against 6 upon B, is 24 upon A; which is the Product sought.

    Note, The Unit or first Term may be taken upon either of the Lines A or B; but the first and third terms must be counted upon one and the same Line, and the Second on the other Line, where the fourth will also be found.

    The Letters A and B may serve to distinguish the Lines.

    2. Example: Let the two Numbers given be 68 and 26, to find the Product. The Proportion is [1 : 26 :: 68 : 1768.]

    Therefore, Set 1 upon B, to 26 upon A, then against 68 upon B, is 1768 on A, which is the Product sought; Or,

    Set 1 upon B, to 68 upon A, then against 26 upon B, is 1768 (on A) the Product. Therefore it matters not which of the Numbers given be made the Multiplicator; and note also, that the Product hath as many places as are in both the Numbers given, because the least of them (viz. 26) doth exceed so many of the first Figures of the Product, according to the Rule before given.

    3. Example: Let 68 be multiply’d by 14. The Proportion is, 1 : 14 :: 68 : 952.

    Therefore, Set 1 upon B, to 14 upon A, then against 68 upon B, is 952 the Product upon A, and here the Product consists of one place less than thre be in the two Numbers given, because the lesser of them (viz. 14) doth not exceed so many of the first Figures of the Product.

    Now that the Product last found is 952, and not 95.2, nor 9.52 will thus appear: Set 1 upon B, against 14 on A.

    By this ’tis evident, that the Lines are in effect a Table of Multiplication, for having set 1 to the Multiplicator, against any Multiplicand, you have the Product; so if 2 be multiply’d by 14, the Product is 28; if 3, the Product will be 42; if 4, the Product will be 56; if 5, 70 &c. Hence I conclude that if the Multiplicand had been but 6.8, the Product would have been 95.2, but the Multiplicand being 68, the Product must be 952; for by taking away the prick, the Fractions in each are made whole Numbers.

    When of two Numbers given to be multiply’d, the one consists of whole Numbers or mixt, and the other of Fractions only, make the whole or mixt number the Multiplicator, and having set 1 against it, seek the Fraction towards the left hand, and against it you have the Product.

    4. Example: Let the two Numbers be 27.5, and .8.

    Set 1 on A, to 27.5 on B, and then against .8 (which being less than 1, I seek towards the left hand) on A, is 22, the Product on B.

    And notwithstanding a Number of more than four places cannot be exactly expressed on a Line of this length, yet the Product of any Multiplication may be discover’d to six or seven places at least.

    5. Example: Suppose I were to multiply 2482 by 54, The proportion is,

1 : 54 :: 2482 : 134028

    Therefore set 1 against A to 54 upon B, and then against 2, upon A, is 108 upon B: Now suppose,

    So the Product will have six places, and against 2842 the Multiplicand, you may discern (upon the Line) the four first of them (viz.) 1340; the two last may be found by multiplying (in ones mind) the two last Figures of the Multiplicand, by the two last of the Multiplicator; for so shall you discover the two last figures of the Product, which in this Example will be 28, which plac’d behind the four first (viz.) 1340, makes the Product compleat 134028.

    I might here add more Examples, but these already given may serve, they containing (I think) sufficient Directions for all the Variety that can happen in Multiplication.

Next page
Return to index

This page was last updated on 29 March 2000.