A treatise of algebra, in three parts. Containing

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surescio a Ming Mund in it sive times, it is sal to e ah aliquot parco it, and the greateris sal to e a multiple of the est. The essquantit in his case is the greates common mea- Iure of the wo quantities fortas it meas es thagreater, socii esse me ures itself, an no quantit can me ore it that is reverissim iuris When a thir quantit measure any propost quantities, is et measures, and

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io a it is nid in besa emison me in os sese quantities an is no greater quantit measum them both, tris their greatast common

can eis quantity Ound that measures themhoth, the are sal to e iseommensurabis andi any one quantityie calle rational, MLothemthat have an common measere missi is are esse calle rationes: ut tho that have no common meaiure illi si are calle irratio I quantities.

93. Is any two quantities iandis have any common measere his quantit x mal esse . measere thei sum and dis renc ais Let

times as uni is Bund in is se that a x, then it illis Aund in an multiple of a. as

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hys. 1 two quantitie a and Lare propos , andis measure a by the uniis that are in m thalis, e Bund in aras an times asinit is seundinim and there M a rei nainde e and i suppost to eis common measure os candis, it halli alio a mea ire of e. o by the supposition armis AE e sincerit containsis asmana times a there are unit in and thereis etesides remaining ther ore a --mbNo x is suppostario meas ire a an and

sequently - r. a. Whicli is qual

lce me ures bi the unita in , and there be

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No it is lain that me, measores e, it must also measere ne and theresere must me sure ne . cor b. And sincerit me uresin and coit must measure L is, o a so that it mustia a common measere of iandis. But urther,it must be thei ruates common measore, sor e very comnis measere os a and bisust meastred by the laxarticle and the greate numberctat molares cisalself, hicli heresere is thegrotest common me ure M a an b.

8 97. ut is, by continuassy subtractim eve

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rem aliiGer more than iis half, and procee in this manner, o wili come at a remainde lesi

Euclid Prop. 1. Book io than an assignatae quantiis It appears there re that i thecis mainder c S c ne ver end the willaecomeles than an assignable quantity, as whichthere re anno postibi molare them, and theres e cannot be a common measure os aand 948. In the fame way the greate commonmeasure of two number is discovered. Unitis a cismmqn measure of an integer numbers, and w numbere are sal to e prime to ach

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are me ast numbers that can e assume in an give proportion; sor i these ad an common meastire, then the quotienis that would isti dividin them by that omino me lare,ould e in the fame proportion, an be- in tes than the numbers thenascives thesenumbers mouid not e the ea in the fame proportionis against the supposition. 6 99. The least number in an proportionalwavs meatur an Other Umber that a re in

the fame proportion suppos a an boo est least os ali integer umber in the fame proportion, and that e and Lare other numbers in that proportion, then illis measure si and

For is cand b are not aliquot paris of e and then the must contain the same number of the same hin sos paris os e and , and there redividinius into paris of c, an b into an equalnumber of like paris of d, and callinione of the firmni, an one of the atteris then a mi is, se Will the sum Dal theis e to thesum os ali the su stat is, b; there re aland b, ill nolle the least in the fame proportion againit the supposition. Thereserea andi inuit be aliquot paris of and Z ence me se that numbers hicli are prime to achother are the least in the fame proportioni soris there,ere other in the fame proportio testtha them, these ould meas ure them by the H amo'

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a common mealare against the supposition. acii. I two number a and re prime toe, then stin thei product-be also prime to GF n is D suppost them is have an commonmeasure as an suppo statu measeres ab by the unit in e so that dei ad then statId: a ba But sinces measures and His suppolia tole prime to a it sellam byaoo. that danda are prime minach other, and theresere by Art. 99. d must meas e L andret sine cis suppost to me ure si hic is prime to it soliuWs that cis esse prime tostat rudis prime to a number hic it me fures, whic is absurd. 1oa It talows fio the las articie, thati a and e re prime to ach other, then a Milibe prime to e Fori supposing that iis equat in b, then ab illie eques o a' and conisquenti mill e prime to in the sime manne e illie prime to a.

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suppi sed tot the re est common measure ofa an it sellorus that, and mare the leasi os

allis umber in the fame proportion, and there- fore, mea res h andis measu res I. ut a cis supposie to e les thamna, that is, a lessthan a theresere es is des inanis, o that agreater ould meassere a less, hicli is absurd. There restandis cannot measure an number

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must measu reuior c., expres an integer number, and et an stactio reduce torat lo est ternas, sothat, and mma be prime to rach ther, antconsequently an in tu allo prime to it,ill

is a

consequently willieri si action in iis least

Urms, and an neve be qua to an integeroumber Theresere the square of the mixtium-ber is stili a narit number, an never

an integer. In lae fame manne the cube, hi-quactra te, or an poWc os a mix number, is stili a mixtisumber an neve an integer. Itiollom se ciliis that ne quare o V an integer anus te an integer or an comm ηψG .lle. Suppos that the in eger proposita is , and that the quare roo oscit scies than a F i, butyeare than then it must be an incommen