AJtD OTHER SURD EQUATIONS, 22f 



would probably have noticed it as such, as he has done 

 other 'incongruities' further on (pp. 217 — 18); but perhaps 

 he felt that he could not call that an * incongruity,' which 

 was in fact a solution of the congeneric surd equation, cor- 

 responding to the given one. Lastly, let me observe, that 

 we can hardly suppose that enquiry was directed to the two 

 examples given above, and only to them. And the dis- 

 covery of one surd equation without any root, would ac- 

 count for the studious avoidance of surd formulae, which we 

 see in the above portion of the Vija-ganita — a magnificent 

 work, which would be more generally studied, did the his- 

 tory of science hold the position which it deserves in the 

 estimation of the learned." 



In the passage above cited, Mr. Cockle has, I think, 

 shown good reason for suspecting that the ancient Oriental 

 algebraists were not altogether ignorant of the existence of 

 impossible equations ^ or at least of the fact, that in the 

 solution of surd equations foreign roots are frequently 

 evolved. The system of solution employed is admirably 

 adapted for the exclusive determination of the possible 

 root; and it is difficult to conceive why that system was so 

 uniformly adopted, if it were not to avoid the contradictory 

 results above referred to. A very slight extension of the 

 ancient system will enable us to compnehend within it 

 every irrational equation; for by making suitable assump- 

 tions, as we shall see, every such equation may be resolved 

 into as many simultaneous rational ones as there are 

 radicals and rational terms of a; in the proposed ; and by 

 rejecting all negative roots, the true and only value, or 

 values, of the unknown will be determined by the common 

 process. This system of solution, as we shall also see, has 

 the subsidiary advantage of preventing the necessity for 

 adopting the method of experimental verification, in order to 

 ascertain to what equations the foreign roots introduced by 

 the operation for eliminating radicals severally belong. 



