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 96 A L G E B R A. be found that •will meafure both a and b; and therefore each other, they would have a common mcafure, which, becaufe it would jneafure c, would aifo meafure a, which they will be incommenfurable to each other. For if there was any common meafure of thefe quan- is meafured by c, therefore a and b would have a comtities, as Xj it would neceffarily meafure all the remain- mon meafure, againft the fuppofition. ders c, d. See. For it would meafure a—mb, or c, and If two numbers a and b are prime to c, then lhall their confequently b—nc, or d, and fo on; now thefe remain- produdf ab be alfo prime to c: For if you fuppofe them ders decreafe in fuch a manner, that they will neceflarily to have any common meafure as d, and fuppofe that d become at length lefs than x, or any afltgnable quantity. meafures .<i£ by the units in e, fo that de—ab, then fliall For c muft be lefs than ; becaufe c is lefs than b, and d •. a w b e. But fince d meafures c, and c is fuppofed therefore lefs than mhi and confequently lefs than 4-r + to be prime to a, it follows that d and a are prime to i^mb, or 4^* In like manner d muft be lefs than b; for each other; and therefore d muft meafure b; and yet, d is lefs than c, and confequently lefs than d-^uc, or fince d is fuppofed to meafure c which is prime to b, it 44. The third remainder, in the fame manner, muft be follows that d is aifo prime to b; that is, d is prime to lefs than 4c, which is itfelf lefs than 4« • Thus thefe a number which it meafures, which is abfurd. remainders decreafe, fo that every one is lefs than the It follows from the laft article, that if a and c are half of that which preceded it next but one. Now if prime to each other, then a' will be prime to c : For to b, then ab will be equal to from any quantity you take away more than its half, by1 fuppofing that a is equal and from the remainder more than its half, and proceed a ; and confequently a1 will be prime to c. In the in this manner, yon will come at a remainder .lefs than fame manner r* will be prime to a. any aftignable quantity. It appears therefore, that if If two numbers a and £ are both prime to other two the remainders c, d, Sec. never end, they will become lefs c, d, then (hall the product ab be prime to the product than any affignable quantity, as x, which therefore can- cd; for ab will be prime to c and aifo to d, and therenot poffibly meafure them, and therefore cannot be a fore, by the fame article, cd will be prime to ab. From this it follows, that if a and c are prime to each common ineafure of a and b. In the fame way the greateft common meafure of two other, then ftiall a1 be prime to cl, by fuppofing, in numbers is difeovered. Unit is a common meafure of all the laft, that a—b,3 and c=.d. It is aifo evident that integer numbers, and two numbers are faid to be prime will be prime to c, and in general any power of a to to each other when they have no greater common mea- any power of c whatfoever. fure than unit; fuch as 9 and 25. Such always are Any two numbers, a and b, ,being given, to find the the leaft numbers that can be affumed in any given pro- leaft numbers that are in the fame proportion with them, portion ; for if thefe had any common meafure, then the divide them by their greateji common meafure x, and the quotients that would arife by dividing them by that quotients c end d Jhall be the leaf numbers in the fatne common meafure would be in the fame proportion, and, proportion nuith a and b. being lefs than the numbers themfelves, thefe numbers For if there could be any other numbers in that prowould not be the leaft in the fame proportion; againft portion lefs than c and d, fuppofe them to be e and f, and thefe being in the fame proportion as a and b would the fuppofition. The leaft numbers in any proportion always meafure meafure them: And the number by which they would any other numbers that are in theAfame proportion. meafure them, would be greater than x, becaufe e and f Suppofe a and b to be the leaft of all integer numbers in are fuppofed lefs than c and d,, fo that x would not be the fame proportion, and that c and d are other num- the greateft common meafure of a and b-, againft the bers in that proportion, then will a meafure c, and b fuppofition. Let it be required to find the leaft number that any meafure d. For if a and b are not aliquot parts of c and d, then two given numbers, as a and b, can meafure. Firft, “ If they muft contain the fame number of the fame kind of “ they are prime to each other, then their produdt ab is parts of c and d; and therefore dividing a into parts of c, “ the leaft number which they can both meafure.” and b into an equal number of like parts of d, and call- For if they could meafure a lefs number than ab as c, ing one of the firft nt, and one of, the latter n; then as fuppofe that c is equal 10 ma, and to nb; and fince c is m is to n, fo will the fum of all the m'% be to the fum lefs than ab, therefore tna will be lefs than ab, and m of all the »’s ; that is, m : n :: a •. b, therefore a and b lefs than b; and nb being lefs than ab, it follows that n will not be the leaft in the fame proportion; againft muft be lefs than a; but fince ma—nb, and confequently the fuppofition. Therefore a and b muft be aliquot parts a '. b •. n •. m, and * and b are prime to each other, it of c and d. Hence we fee that numbers which are prime would follow that a would meafure n, and b meafure in, to each other are thedeaft in the fame proportion; for if that is, a greater number would meafure a lefs, which is there were others in the fame proportion lefs than them, abfurd. thefe would meafure them by the fame number, which But if the numbers a and h are not prime to each otherefore would be their common meafure againft the ther, and their greateft common meafure is x, which fuppofition, for wre fuppofed them to be prime to each meafures a by the units in in, and meafures b by the uother. nits in n, fo that a—mx, and b—nx, then lhall an (which If two numbers a and b are prime to one another, and is equal to bm, becaufe a:b:: rnx t ax ; tu : n, and therea third number c meafures one of them <7, it will be fore an—bni) be the leaft number that a and b can both prime to the.other b. For if c and b were not prime tp meafure. For if they could meafure any number c lefs than