AND OTHER SURD EQUATIONS. 231 



we immediately infer, from the fact of the last quantity 

 imder each of the radicals being positive, that the equations 

 (A), (B), (C), (D), have each one, and only owe, root. (See 

 the latter portion of art. 7.) 



Now, in each of the foregoing cases, the advantage of 

 the ancient over the modem method, if we may so distin- 

 guish them, is too obvious to require argument. That 

 method, as I have before intimated, is easily adapted so as 

 to comprise within it every class of surd equations, and will 

 always be found of important service when the sign of the 

 radical is to be taken strictly as indicated. The considera^ 

 tion of a few particular examples, will sufficiently show the 

 general applicability of the ancient system of solution to 

 irrational equations, and will tend to illustrate more clearly 

 than any number of general observations, the peculiar value 

 of that system. 



K the equation 



4 + Va: -^ 3 + Va: + 21 = 



(see art 5), be proposed for solution, the ancient method 

 at once suggests the following assumptions : — put 



a? rz V ^ — 3, and £c zz 's/ a: -{- 21, 



I s 



then we shall have 



4 -|- as 4- iK =i: 0, 



1 a 



24-4-a:* — a^ — 0; 



1 s 



whence we readily find x m 1^ and a: :=z — 5. Now the 



1 3 



latter of these values, being negative, is rejective ; and, since 

 a has only one root, we immediately infer that the equation 



is impossible. We likewise learn that the only possible 

 equation analogous to the proposed one, is 



4: -\' j^x — 3 — a/x -\- 21 zz 0, 

 the root of which is [a; = ^ + 3 = (aP) — 21 =] 5. The 



