ALGEBRA, 



tor and denominator by the greatest a- #*)a3 a- x a x3-}-x*(a z 

 quantity thst measures them both. a> a -r 1 



The greatest common measure of tioo 

 quantities is found by arranging them ac- 

 cording to the powers of some letter, and 

 then di-j'ding the greater t>y the less, and 

 the preceding divisor ahvays by the last re- 

 mainder, till the remainder is nothing , the 

 last divisor is the greatest common measure 

 required. 



Let a and b be the two b} a (p 

 quantities, and let b be 

 contained in a, p times,\vith fc) b (q 

 a remainder c; again, let c 

 be contained in #, q times, d) c ( ? 

 with a remainder d, and so 



on, till nothing remains; let 



d be the last divisor, and it 

 will be the greatest com- 

 mon measure of a and b. 



TheUruth 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 n, then will it 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 t n times ; then m a = x and n a = y ; 

 therefore xy=m an =ra;i . a ; i. e. 

 a is contained in x#, mn times, or it 

 measures x:y by the units of mn. 



Now it appears from what has been 

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

 even quantity therefore, which measures 

 a and b_ measures/* b, and a p 6, or c ,- 

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

 or df that is, every common measure of 

 a and b measures d. 



Ex. To find the greatest common mea- 

 sure of a 4 a;* and a3 a 1 x a a^-f-x?, 



a* x* 

 and to reduce --- to 



a z x-J-.r3 



its lowest terms. 



a3 a 1 a: ax 1 -{-x3 



'x a- 



leaving out 2 x*, which is found in each 

 term of the remainder, the next divisor 



a 1 x- is therefore the greatest common 

 measure of the two quantities, and if : hey 

 be respectively divided by it, the frac- 

 tion is reduced to - , its lowest 



x 

 terms. 



The quantity 2 x 1 , found in every term 



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



not in every term of the dividend, 3 a 1 

 x a x 1 -f ^ ; , 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, 2 x% no common measure of the 

 divisor and dividend is left out; because, 

 by the supposition, no part of 2 x* 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 6, 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 togetJierfor the common 

 denominator. 



Let -,-be the proposed fractions; 



adf cbf edb 



iT^~f> 7T777' 7"T7> are tractions of 



s a 1 x 1 . 



the same value with the former, having 

 the common denominator b d f. For 



a df_q c bf c edb e 



TdJ~b' b~Tf~ d'* Tdf~~f 

 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- 



