228 MB. E. HAELEY ON IMFOSSIBUB 



avoidance of surds, which, as we have seen, takes place in 

 the above examples, became a rule of proceeding in conse- 

 quence of the contradictory results to which surd equations 

 sometimes lead us ? Bhascara was aware of the double 

 sign which attaches to a square root (p. 135), and has 

 used that double sign to obtain two positive roots of a 

 quadratic (p. 216), and I believe that he also admitted, 

 in all cases, two roots of a quadratic ; for we see him (p. 135), 

 squaring a negative quantity, considered by itself, without 

 reference to other quantities; and further, when wo see 

 him rejecting the root 5 because it is 'incongruous ' (p. 217), 

 he qualifies the rejection by, as I presume, assigning as a 

 ground, that 'people do not approve negative absolute 

 numbers,' and negative quantities are by means of this 

 root 5, introduced into the conditions of the question. Now 

 suppose for a moment, that, in the first attempts at the 

 solution of the two problems given above, the qucesitum, had 

 been taken as ya, then the algebraist would have had in 

 the first example, 72 and |, and in the second 100 and 4, 

 as the values of ya. It would have been seen (for, as in 

 other cases, both values would certainly have been tried), 

 that the second value would in neither case satisfy the 

 required conditions. I think it also highly probable, that 

 the reason of the failure would have been seen, as the 

 double value of the radical in the enunciation would natu- 

 rally offer itself as a mode of explanation. The question 

 then is, whether ya was originally taken as the qucesitum in 

 those examples; and I confess that I cannot help seeing, 

 in the introduction of square roots into the enunciation of 

 the first two examples of quadratic equations, and in the 

 assumption that ya is something other than the qucesitxmij 

 a marked desire to overcome a natural tendency to make 

 such assumption — a tendency which the writer had found 

 to lead to error. Add to this, that if the writer had con- 

 sidered the error in question as a mere * incongruity,' he 



