feveral external Denominations into one Chap. V. external Denomination: Of which there fore we shall speak, Chap. 10, treats of Reduction. which Way. But there is also another Way which 14. may be used in this Cafe, and is much Another fhorter than that by Reduction: Namely, by refolving the Multiplier given into fuch Digits, as are, when multiply'd one into the other, equivalent to the Multiplier, and fo multiplying by each of the faid Digits fingly. Thus, fuppofing twenty-feven Perfons were each to be paid 245 1. 6 s. 10 d. and I would know, what Sum is requifite to pay the fame. Because 9×3=27, therefore I multiply 2451. 6 s. 1od, first by 9, and then the Product thereof by 3; and the last Product will be the Sum fought, and the fame as will arise by multiplying 245 7. 6. 10 d, (reduc'd into one Denomination) by 27. As will appear by comparing the following Example, with the Example contain'd in Sect. 7, of Chap. 10, concerning Reduction. Chap. V. feveral ex nominati-. 15. As for Multiplicators of several exterof Multi- nal Denominations, they never occur, plicators of but when the external Denominations are ternal De- of various Measures, as Yards, Feet, Inches, &c. And in this Cafe, the Operation may be perform'd, by reducing both the Multiplicand and Multiplicator into one Denomination, as is fhewn, § 8, Chap. 10, of Reduction. ons. concerning the fame. 16. There is also another Method in this Further, laft Cafe, which is called Cross Multiplication. But this, as well as Reduction, depending on Divifion, it will be improper to speak of it here, before we have 1poken of Divifion. And therefore it fhall be referr'd likewife to Chap. 10; Where it shall be added by Way of Annotation, to the Reduction belonging to this Cafe. The Chap. V The best or moft certain Way of proving, whether Multiplication be perform'd 17. aright, is (according to the fourth gene- of proral Rule) by Divifion, as fhall be fhewn ving Mulin the following Chapter of Divifion. It tiplication by cafting remains only here to fhew, how the Mul- away 9. tiplication of Numbers of one external Denomination, may with certainty enough for common Matters be prov'd by cafting away 9. Namely, 9 being caft away, as often as may be, out of the Multiplicand, and likewife distinctly out of the Multiplicator, multiply the two Remainders one into the other, cafting away also 9 out of what arifes, if it be large enough. The Remainder thence arifing, will be equal to the Remainder of the Product, after 9 is caft away therefrom, as often as may be; if the Multiplication prov'd hereby be rightly perform'd. For Inftance, in the firft Example of this Chapter, the Remainders will be 8, and in the fifth Example will be 6, as appears by the faid Examples here repeated. C H A P. VI Of the Divifion of Integers. Du Chap. VI I. Divifion,' Ivifion fhews, how often the Divi. for is contain'd in the Dividend; what, As whence the Number found by this Ope- also the ration, is called the Quotient. The Dividend, feveral Numbers are plac'd, either and Quo Dividend thus, Divifor 3)6(2 Quotient, or thus, Dividend 6 (2 Quotient. And the O Divifor 3 Divifor, tient. 2. quot Part, What is properly Divifion, is perform'd by the Quotient's fhewing, how An Alioften the Divifor is contain'd in, or may what. As be taken out of the Dividend; as 3)6(2, alfo Multiand when the Divifor does thus exactly and Subdivide the Dividend, it is call'd an ali- ftraction, quot Part of the Dividend. Thus 3 and 2 are aliquot Parts of 6. But because it frequently happens, that the Divifor is not exactly contain❜d in the Dividend, but after the Divifion there remains fomewhat of the Dividend; therefore in order to know what the faid Remainder is, the Quotient must be multiplied plication, why re quifite to Divifion. |