Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/127

 A L G E B R 95 quantities a and b have any common meaIt is obvious, that the pofitive value 7 gives the folu- fureIfx,anythistwoquantity alfo meafure their fum or tion of the queftion; the negative value —5 being, in difference a=+=.b. Letx xfliall be found in a as many times as the prefent eafe, ufelefs. Any; equation of this form where the unit is found in m, fo that a — m x, and in b as many greatefl index of the unknown quantity^, is double to the times as unit is found in «, fo that b-=nx-, then lhall index ofj in the other term, may be reduced to a qua- a-z+zb—mx-=+inx—inz^=MY.x; fo that x fhall be found in dratic zl -i-az — i, by putting j/m = z, and confequently q—yrb, as often as unit is found in m—yzn : now lince m n are integer numbers, niz+zn mull be an integer y2,ntz^zt. And this quadratic refolved as above gives and number or unit, and therefore x mull meafure a=fzb. b It is alfo evident, that if x meafure any number as a, 2 s/ 4 ' it mult meafure any multiple of that number. If it be found in a as many times as unit is found in m fo that And feeing yw—2= — rAr b + —, y — a—mx, then it will be found in any multiple of a, as na, as many times as unit is found in mn; for na—tnnx. If two quantities a and b are propofed, and b meafure a by the units that are in m (that is, be found in a as times as unit is found in ni) and there be a remain- ■ Examp. “ The produdt of two quantities is a, and many c, and if x be fuppofed to be a common meafure of “ the Turn of their fquares b. Shi. The quantities?” der a and b, it (hall be a'lfo a meafure of c. For by the fuppofition a—?nb--c, fince it contains b as many times as Supp. ’ y y* there are units in w, and there is c befides of remainder. . . x*—b—y* Therefore a—mb—c. Now x is fuppofed to meafure a and b, and therefore it meafures mb, and conlequently whence b—y*=jpr„ a—mb, which is equal to c . If c meafure b by the units in r., and there be a remult, byy*4 . . by1—-y^zza1 mainder d, fo that b—nc--d, and b—nc—d, then lhall tranfp. y *—^y*=—•z*x alfo meafure d; becaufe it is fuppofed to meafure b, Put nWy'* =*.*•. . and confeq. . .y4=2i, and it is and it has been proved that it meafures c, and confequently nc, and b — nc which is equal to d. Whence, —iz -j 4 = a x add —-4 as, after fubtra&ing b as often as poffible from a, the remainder c is meafured by x; and, after fubtradling c as often as pollible from b, the remainder d is alfo meafured ext. y'~ 2 V 44 by x; fo, for the fame reafon, if you fubtrad; d as ofas poffible from c, the remainder (if there be any) —- =±= — n and, feeingy=-y/'A, ten muft ftill be meafured by x: and if you proceed. Hill fubtrading every remainder from the preceding remainder, till you find fome remainder, which, lubtraded from the preceding, leaves no further remainder, but exadly meafures it, this laft remainder will ftill be meafnred by x, any common meafure of a and b. Chap. XIII. 0/ Surds. The laft of thefe remainders, viz. that which exadly meafures the preceding remainder, muft be a common If a iefler quantity meafures a greater fo as to leave meafure of a and b: fuppofe that d was this laft remainno remainder, as la mcafures 10.?, being found in it five der, and that it meafured c by the units in r, then ftiall times, it is faid to be an aliquot part of it, and the greater is faid to be a multiple of the Idler. The lefier c—rd, and we lhall have thefe equations,^ quantity in this cafe is the -greatejl common meafure of a — mh-f-c the two quantities : for as it meafures the greateft, fo b=nc + d it alfo meafures itfelf, and no quantity can meafure it c — rd. that is greater than itfelf. When a third quantity meafures any two propofed Nowit is plain that fince d meafures>, it muft alfo- ‘ quantities, as 2a meafures ba and 10a, it is faid to be a meafurc nc, and therefore muft meafure nc--d, or b. ' con.mon meafure of thefe quantities; and if no greater And fince it meafures b and c, it muft meafure ??ib--cy or a fo that it muft be a common meafure of a and b. quantity meafure them both, it is called their greatejl But further, it muft be tiaPir greatefl common meafure ; common meafure. every common meafure of a and b muft meafure d, Thofe quantities are faid to be commsnfurable which for by the laft article; and the greateft number that meahave any common meafure ; but if there can be no quan- fures d, is itfelf, which therefore is the greateft comtity found that meafures them both, they are faid to be mon meafure of a and b. incommenfurable and if ahy one quantity be called raby continually fubtrading every remainder tional, all others that have any common meafure with fromButtheif,preceding remainder, you can never find one that it, are alfo called rational: But thofe that have no Common meafure with it, are called irrational quantities. meafures that which precedes it exadly, no quantity can be