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 84 A L B R A. G E mon meafure, and place the quotients’ in their room, V R O B LEM VII. and you ftiall have a fraflion equivalent to the given 7"9 find the great eft common meafure of two numbers ; that is, the greateft number that can divide them ffattion exprefled in the lead: terms. . both without a remainder. 2 5 M Vjabc = 5 x 7_ Sl2aa—5l2ab 156 a a + 156 a b_ Rule. Firfi divide the greater number by the lefler, Thus 2slc)i25bcx acd if ,there is ne remainder, the lefTer number is the T -M-t-S* greateu common divifor required. If there is a re- IX a — lib' mainder, divide your laft divifdr by it; and thus pro? When unit is the greateft common meafure of the ceed continually, dividing the latt divifor by its re-' numbers and quantities, then the fraction is already in its Bi/uider, till there is no remainder left, and then the Lift divilbr is the greateft common meafure required. . loweft terms. Thus e, da cannot be reduced lower. Thus, the-greateft common meafure of 4; and 63 whofe greateft common meafure is is 9 ; and the greateft common meafure of 256 unit,And,are numbers faid to be prime to one another. and 48 is 16. • If a vulgar fraflion is to be reduced to a decimal (that 45) 63 (1 48) 2256 (5 is, a fraftion whofe denomination is 10, or any of its 45 4° powers,) “ annex as many cyphers as you pleafe to the “ numerator, then divide it by the denominator, the T8) 4r(2 16) 48 (3' “ quotient ftiallandgive a decimal equal to the vulgar frac36 48 “ tion propofed.” Thus, - 9)1 18 (2 o —= .66666, tec. -2-5 = .6 ; 3 18 -y = .2857142, £}C. o Much after the fame manner the greateft common Thefe fractions are' added arid fubtrafled like whole meafure of algebraic quantities is difcovered; only the numbers; only care muft be taken to fet Jimilar places remainders that arife in the operation are to be divione another, as units above units, and tenths aded by their ftmple divifors, and the quantities are al- above tenths, be. They are multiplied and divided as ways to be ranged according to' the dimenftons of the boveinteger numbers; only there vtuft be as tnany decimal fame letter. in the produfi as in both the multiplicand and Thus to find the greateft common meafure of a% — bx places multiplier; and in the quotient as many as there are in the and a* — 2 a b fi- xb* • dividend more than in the divifor. And in divifion the a* — £*) a7 — 2 a b fi-b* (1 quotient may be continued to any degree of exaflnels a‘— you pleafe, by adding cyphers to the dividend. Theground of thefe operations is eafily underftood from the — 2 a b 2 V Remainder, general rules for adding, multiplying, and dividing which divided by —2 b is reduced to fraftions. a — V) a*—b*1 (a fir b a* — b C h a p. VI. Of the Involution of 0*0. Qu ANTITIES. Therefore a — b is the greateft common meafure required. produfls arifing from the continual multiplicaThe ground of this operation is, That any quantity tionThe of the fame quantity were palled (in zChap.z III.) that meafures the divifor and the remainder (if there is the powers Thus, a, a, a , be. any) muft alfo meafure the dividend; becaufe the dividend are the powersof ofthata;quantity. b, a% bx, a3 b>, be. are is equal to the fum .of the divifor multiplied into the quo- the powers of a b. Inandthea fame chapter, the rule for tient, and of the remainder added together. Thus, in the multiplication of powers of the fame quantity is, “ To the laft example, a—b meafures the divifor a* —b*, “ add the exponents, and make their fum the exponent 4 s therefore likev and the remainder — 2 a h fi-2. b it muft produft.” Thus a X « = a5, ; and a3 hl X 9 wife meafure their fum a? — 2 abfi-b1. You muft ob- a“6 ofb1 the a b*. In Chap. IV. you have the rule for diferve in this operation to make that the dividend which viding—powers of the fame quantity, which is “ To fubhas the higheft powers of the letter, according to which “ tradl the exponents, and make the difference the expothe quantities are ranged. “ nent of the quotient.” PROBLEM VIII. fi 4 1 4 4 3=a5—^ i3”1 Thus, a~=a - =a ;7 and a^-fb To reduce any fraflion to its loweft terms. Rule. Find the greateft comrtion meafufe of the nu- Ifyou divide a lejfer power by a greater, the expomerator and denominator; divide them by that com- nent of the quotient muft, by this rule, be negative. Thus,