ALGEBRA. 



tor and denominator by the greatest 

 quantity that measures them both. 



The greatest common measure of two 

 quantities is found by arranging them ac- 

 cording' to the powers of some letter, and 

 then dividing the greater by the less, and 

 tiie preceding divisor always by the last re- 

 mainder, till the remainder is notliine- ; the 

 last divisor is the greatest common measure 

 required. 



Let a and b be the two 6) a (/> 

 quantities, and let b be 

 contained in a, p times, with c) b {q 

 a remainder c ; again, let c 

 be contained in b, q times, </)c(r 

 with a remainer d, and so 



on, till nothing remains ; let 



d be the last divisor, and it 

 will be the greatest common 

 measure of a and b. 



The truth of this rule depends upon 

 these two principles : 



1. If one quantity measure another, it 

 will also measure any multiple of that 

 quantity Let x measure y by the units 

 in M, then it will measure c y by the units 

 in n c. 



2. If a quantity measure two others, it 

 will measure their sum or difference. 

 Let a be contained in x, m times, and in 

 y, n times ; then m a = x and n a = y ,- 

 therefore xy=m ana=mn .a; i. e. 

 a is contained in x^-y, in times, or it 

 measures xy by the units in mn. 



Now it appears from what has been 

 said, that a p b = c, and b q c = d ; 

 every quantity therefore, which measures 

 a and 6, measures p b, and a p b, or c,- 

 hence also it measures q c, and 6 q c, 

 or d ; that is, every common measure of a 

 and 6 measures d. 



Ex. To find the greatest common mea- 

 sure of a* x* and a) a 1 x ax*-\-x~i, 



4 #4 



and to reduce to 



a3 l x t ' 



its lowest terms. 



a* x 1 a3 



a* a>x 



alx a 1 x 1 ar3 -f- x* 



leaving out 2 x 1 , which is found in each 

 term of the remainder, the next divisor is 

 a 1 x 1 . 



a 1 x- is therefore the greatest common 

 measure of the two quantities, and if they 

 be respectively divided by it, the frac- 



tion is reduced to 



its lowest 



terms. 



The quantity 2 x-, found in every term 

 of one of the divisors, 2 a 1 x 1 2 x*, but 

 not in every term of the dividend, a? a 3 

 x ax*-^-xi, must be left out; other- 

 wise the quotient will be fractional, 

 which is contrary to the supposition made 

 in the proof of the rule ; and by omitting 

 this part, 2x l , no common measure of the 

 divisor and dividend is left out ; because, 

 by the supposition, no part of 2 x 1 is 

 found in all the terms of the dividend. 



To find the greatest common measure 

 of three quantities, a b c ; take d the great- 

 est common measure of a and b, and the 

 greatest measure of d and c is the great- 

 est common measure, required. In the 

 same manner, the greatest common mea- 

 sure of four or more quantities may be 

 found. 



If one number be divided by another, 

 and the preceding divisor by the remain- 

 der, according to what has been said, the 

 remainder will at length be less than any 

 quantity that can be assigned. 



Fractions are changed to others of equal 

 value with a common denominator, by multi- 

 plying each numerator by every denominator 

 except its own, for the new numerator , and 

 all the denominators togetlier for the common 

 denominator. 



ace 

 Let T, 3,7 be the proposed fractions ; 



adf cbf edb 



then '* are fractlons of 



the same value with the former, having 

 the common denominator b d f. For 

 adf a cbf c *edb e 



numerator and denominator of each frac- 

 tion having been multiplied by the same 

 quantity, viz. the product of the denomi- 

 nators of all the other fractions. 



When the denominators of the propo- 

 sed fractions are not prime to each other, 

 find their greatest common measure -, 

 multiply both the numerator and deno- 



