+4caa + 4 cda +41c+dd + daa + dda + ddd (aa+4ac +dd aaa + daa 10 +4caa +4cda +dda + ddd N. B. The foregoing Dividend may be writ in the following In which the Members of the fecond Term, as alfo they of the third are United, by adding in each Term the Factors of the Letter (a), in refpect of which the Terms of the Dividend were plac'd. But if ddd be made the firft Term of the Dividend, then the Divifion will stand thus, * That is, g+hxa - g+hx b for the Di= 18: aa+ba+ca - bc for the Divi= visor, an 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 as a Numerator over your Divifor, which Fraction annex to the before found Quotient, with its proper Sign + or Examples. +x+xx+xxx+ &c. Sine Fine. Or I - X 1- $ ) 1 - 5 + 1 + 1 + 1 + B = 2 114,54,25 +1125 1-,5=15/1.0/2 + 5-25° 4,25 8 16 +x xx + xx 64 eee aa ce ade e + 3 4 by halving 3 1 8 like manner 1+ 2+ 37 +00 +4 10-1 of eeeee thy multiplying. be: 4 y numer, you'll find so much value of the frail as multiphed accordingly thrid 2 +26+ eeeeeee aaaa eeeee aa 2 + + + 2 256 2 ~ ~38 128 3 می 2 ,24998 n.5 By reducing the tracions 16 5 23. 5 D 2 125 1499935 3125 195 1.2 3 쫌.. +125 100 PART PART II. Of Fractional or Bioken Quantities. CHA P.-I. Rotation of Frational Quantities. FRational Quantities are exprefs'd or fet down like Vulgar Fractions, in common Arithmetick. Thus { 5b-4a 2bc Numerators. 4d-76' 'd Denominators. How they came to be fo, may be seen by the general Rule in the beginning of Divifion. Thefe Fractional Quantities are manag'd in Algebra, as Broken Numbers in Arithmetick. CHA P. II. Reduction of Fracional Quantities. Sect. 1. To reduce Fractions having different Denominato2s, to Fraaiens of the fame Value, that fhall have a common Denominatoj, Kule. Multiply all the Denominators continually by each other, (i. e. Multiply the firft Denominator by the fecond, and that Product by the third, and fo on) and referve their Product for a new and common Denominator; then Multiply any of the Numerators by all the Denominators but its own, continually, and the laft Product put for a Numerator over the faid common Denominator; fo this Fraction is equal to that, whose Numerator yoy Multiplied into all the Denominators, but its own continually. Do fo with the reft of the Numerators, and you'll have your defire, Examples 1 Examples. a d C 1. Let it be required to bring and to one Denomination. First, bx c be, is the common Denominator. Thirdly, bx dbd; therefore ac bc a bd d is bc Secondly, edf x fdc-dccddff-cdddff; ab xexc-d abec-abed; and adxex fdaefd + ddef, are the Numerators, Therefore ccddffcdddff efdc efdd And aefdddef btc are the Fractions required. efdc efdd' 3. If and b be brought to one Denomination, d they will be bbbc bd bd babb da bd pectively. Sect. 2. To reduce a whole Quantity into an Equivalent Fraction of a given Denomination. Multiply the whole Quantity by the given Denominator, under which Product place the faid Denominator with a Line between them, and you will have the Fraction required. i. Let it be required to bring ab into a Fraction, whose Denominator fhall be da. First, abxa— da db — aa —ba, 2. b being reduc'd to a Fraction, whofe Denominator will bq-br Note, when whole Quantities are to be fet down Fraction-wife; ab fubfcribe an Unit for the Denominator; thus ab is = and aa aa+b +b= Lemma to Sect. 3. How to find the greatest common Divifoz (or common Meature) of two given whole Quantities. Kule. Divide the Greater given Quantity by the Lefs, and if any thing remains, Divide your Divifor thereby, and if any thing yet remains, Divide your last Divifor thereby; and thus proceed 'till nothing remains (as in Vulgar Fractions), if the firft Term of each Remainder be a Sub-Multiple of the firft Term of the next foregoing Divifor: But here Note, that every Remainder whose firft Term is not a Sub-Multiple of the firft Term of the next foregoing Divifor, must be Divided by fuch of its Sub-Multiples as is Prime to that Divifor, and will give the firft Term of the Quotient a Sub-Multiple of the firft Term of that Divifor by which Quotient that Divifor muft be Divided, and if there be a Remainder, the faid Quotient will be your next Dividend to be Divided by this Remainder, or &c. And the laft Divifor is the greatest common one required. *Note, this Divifion is not neceffary nor proper, if the firft Term of be a Sub-Multiple of the firft Term of b. Suppofe the Greater of the two given Quantities to beb, and the Lefsc; and fuppofe c to be* Divided by a SubMultiple thereoff Prime to b, and giving the firft Term of the Quotient a Sub Multiple of the firft Term of b 1, if the first Term of c be a Subthen fuppofe b to be Divided by ƒ (where Note, that fwill be Aultiple of the first Term of b); and the Remainderr. Again, fuppofer to be Divided by a Sub Multiple thereof. =g Prime to, and giving the firft |