But if ddd be made the firft Term of the Dividend, then the Divifion will stand thus, If the Divifor be not an Aliquot Part of the Dividend, the Quotient may, in fome Cafes, be continued to an Infinite Series: But if, after you have plac'd as many Terms in the Quotient as you think proper, you have a Mind to have the exact Quotient, place the laft Remainder over your Divifor, with a Line between them, which Fraction with its proper Sign + or annexed to the before found Quotient, will exhibit the exact Quotient required. PART II. Of Frattional or Broken Duantities. CHAP, I. Notation of Fractionri Quantities. FRational Quantities are exprefs'd, or fet down, like Vulgar in common Arithmetick, How they came to be fo, may be seen by the General Rule in the Beginning of Division. Thele Fractional Quantities are manag'd in Algebra, as broken Numbers in Arithmetick. CHA P. II. Keduction of Fractional Duantities; Sec. 1. To reduce Frations having different Demo minato2s, to Fractions of the fame Value, that shall have a common Denominatoz. M Kule. Ultiply all the Denominators together, and referve their Product for a new and common Denominator; then Multiply any of the Numerators, and all the Denominators, but its own, together, and their Product put for a Numerator over the common Denominator; fo this Fraction is equal to that whofe Numerator you Multiplyed. Do fo with the rest of the Nume rators, and you'll have the Fractions required. Examples Examples. a 1. Let it be required to bring — and to -to one Denomination, First, bxcbc, is the common Denominator. Secondly, a x cac; therefore xc Thirdly, bxdbd; therefore ac a is = bc bd Confequently and are the two Fractions required. bc ac bc ab 2. Let cdf, and ad be, all of them, brought to e fd a common Denominator. First, ex fdx:cd: Denominator. efde-efdd is the common Secondly, rdfx fd x :cd:= ccddff-cdddff; abxex:cd:= abec abed; and:a+d:xexfd == aefdddef, are the Numerators; Sec. 2. To reduce a whole Duantity into an Equivalent Fraction of a given Denomination. Kule. Multiply the whole Quantity by the given Denominator, and under the Product place the faid Denominator with a Line between them; and you will have the Fraction required. Exam Examples. 1. Let it be required to bring a+b into a Fraction, whose Denominator fhall be d-a. First: ab:x:d-a:=da + d b -a a ba, 2. b being reduc'd to a Fraction, whofe Denominator will be bg-br Note, When whole Quantities are to be fet down Fraction. wife; fubfcribe an Unit for the Denominator; thus ab is ab and baa b + aa I Lemma to Sect. 3. How to find the greatest common Divifo (or greateft common pealure) of two propos'd whole Duantities. Kule. Divide the greater propos'd Quantity by the leffer, and, if any thing remains, Divide your Divifor thereby, and, if any thing yet remains, Divide your laft Divifor thereby; and thus proceed 'till nothing remains, provided the firft Term of each Remainder measures the firft Term of the next foregoing Divifor; and the laft Divifor is the greatest common one required. But here, Note, that when the firft Term of any Divifor (or Remainder) does not measure the firft Term of its refpective Dividend (or next foregoing Divifor), Then find a Meafure of fuch Divifor, which Meafure you are to confider three ways; viz. 1. As being not Prime to the Dividend; or {3} Prime to the Dividend, and that, upon Dividing the Divifor by it, the first Term of the Quotient fhall not be a Mea{not be a bea fure of the first Term of the Dividend. In Cafe 1. First find (by Inspection, or by what is faid here) the greatest common Measure of the Divifor's faid Measure and of the Dividend Then Divide the Dividend by the faid greatest common Measure, and the Divifor by its faid Measure, and the |