| XXX | (97) | XXX |
than na fo that crr-Ia—kh, then a l : m : n :: k : l •, and bee^fe x is fuppofed to be the greateft common iheafure of a and b, it follows that m and n are the lealt pf all numbers iii the fame proportion, and therefore in meafares k, arid n meafures /. But as c is fiippofed to be lefs than fta, that is, Ifefs than ha, therefore l is lefs than n, fo that a greater would naeafure a leffer, which is abfurd, Therefore^* and b cannot meafure any number lefs than an; v/hicll they both meafure, becaufe na—mb. It follows from this reafpning, that if a and b meafure any quantity c, the leaft quantity na, which is meafltred by « and b, will alfo meafure c. For if you fuppofe, as before, that c—la, you will find, that n muft meafure /, and na muft meafure la or c. Let a exprefs any integer number, and — any fraction reduced to its loweft terms, fo that m and n maybe prime to each other, and confequently an--m alfo prime to «, it will follow that an-^-m1 will be prime to. n2, and a J confequently — ^ will be a fraction in its leaft terms, and can never be equal to an integer number. Therefore the fquare of the mixt number «*f"~ is ftill a mixt number, and never an integer. In the fame manner, the cube, biquadrate, or apy power of a mixt number, is ftill a mixt number, and never an integer. It follows from this, that the fquare root of an integer mujl be an integer or an incommenfurable. Suppofe that the integer propofed is B, and that the fquare root of it is lefs than tf-i-i, but greater than a, then it muft be an incommenfurable ; for if it is a commenfurable, let it be where — reprcfents any fraction reduced to its leaft terms; it would follow, that fquared would give an integer number B, the contrary of which we have demonftratjed. It follows from the laft article, that the fquare roots of all numbers but of 1, 4, 9, 16, 25, 36, 49, 64, 8r, 100, 121, 144, &c. (which are the fquares of the integer numbers r, 2, 3, 4, 5, 6, 7, 8, 9, xo, 11, 12, &c.) are inconmenfurables : after, the fame manner, the cube roots of all numbers but of the cubes of 1, 2, 3, 4, 5, 7, 8, 9, &c. are incommenfurables; and quantities that are to one another in the proportion of fuch numbers muft allb have their fquare roots or cube roots incommenfurable. The t oots of fuch numbers being incommenfurable are exprefled therefore'by placing the proper radical fign over them; thus, 4/ 2, 4/ 3, 4/5, 4/ C, 4/ 7, 4/S, 4/ 10, &c. exprefs numbers incommeniurahle with unit. Thefe numbers, though they are incommenfurable theihfelves with unit, are commenfuraobe in power with it, becaufe their powers are integers, that is, multiples of unit. . They may alfo be commenfurable fometnnes with one another, as the 4/ 8, and the 4/ 2, becaufe they are to one another as 2 to x : And when they have Von. 1. No. $. 3
B R A. 97 a common meafure, as v/ 2 is the common meafure of both, then their ratio is reduced to an expreflion in the leaft terms, as that of commenfurable quantities, by dividing them by their greateft common; sneafure. . This common meafure is found as in commenfurable quantities, only the root of the common meafure is to be made their common divifor. Thus, V 3 ; zy'4 =2, and /iSa . 2 A rational quantity may be reduced to th© form of any given furd, by raifing the quantity to the power that is denominated by the name of the furd,2 and then fetting the radical fign over it thus, a—*/a —./a! a* 1 » n and 4=v'l6= . :y'64==y'256=^ j 4 s 1024 — —A/a'—s/a i n V'4 As furds may be confidered as powers with fraflional exponents, “ they are reduced to others of the famevatc“ lue that Ihall have the fame radical fign, by reducing thefe fractional exponents to fractions having the fame “ value and a common denominator.” Thus, A/a=za”r, and ■“=“» ”= and therefore a/a and a/a, reduced to the fame radical fign, become and y’tf’V If y°u are to reduce 4/3 and 4/2 to the fame denominator, confider, 4/3 as equal to 3"*', the 4/2 as equal to 2T, whofe indices reduced to a common denominator, you have 3^ and 2T /—2s, and confequently V1—Vand 4/2=v 2* =4/4 ; fo that the propofed furds 4/3 and 4/2 are reduced to other equal firds 4/27 and 4/4, having a common radical fign. Surds of the fame rational quantity are multiplied by adding their exponents, and divided by fit bt railing them;
- X4/rf=<f
3 *rXT* 1±2 thus 4/tf 6 =asr ~/l/a:‘ and s a/a a j , y__ T x =v* ; 4/« aT m n m--n 4^X4/*=*— ; Afa
- =
4/2X4/2=4^2J= « 7— 4/2 6 4/32; 4/2 =4/2. If the furds are of different rational quantities, as and 4/^3, and -have the fame fign, “ multiply “ thefe rational quantities into one another, or divide Bb “ them
