AND OTHER SURD EQUATIONa 225 



root of (1). This numerical value, it may be interesting to 

 observe, is otherwise obtainable, thus : — Multiplying both 

 members of (1) by V ^ — V .J? -f* !> we have 



X — X — 1 — ; .*. ;p rz 1 ; .'.x-zz^ 

 as before. This is evidently the root, however, of the equa- 

 tion 



V* — Va: + 1 =: 0. 



9. In a series of original essays, entitled, " Horas Alge- 

 braicae," published in the Mechanics^ Magazine, Mr. 

 Cockle has given a very interesting and general discussion 

 of the theory of surd equations. Ho7'cb VIII., IX., and X., 

 contain valuable disquisitions on the algebra of impossibles; 

 tlie history of which is given in the last-mentioned Horce, 

 To show that there is good ground for supposing that the 

 existence of impossible equations was known, or at least 

 suspected, by certain ancient philosophers, Mr. Cockle 

 cites two very curious arid interesting solutions from the 

 Vija-ganita. My own remarks on those solutions I re- 

 serve until I have given Mr. Cockle's discussion, which 

 is so interesting and instructive in all its parts, that I feel 

 sure no apology will be deemed necessary for introducing 

 it here entire. 



" I am inclined to think,"* says Mr. Cockle, " but I oflPer 

 the opinion with great hesitation, that the existence of im- 

 possible equations has been known for many ages — or, if 

 known should seem too strong a word, I will state some 

 circumstances which tend to indicate that the existence of 

 such equations was at least suspected by those philosophers 

 —whether Caucasian or Mongolian, Indo- German or 

 Indian, or other, is not now the question — by those philo- 

 sophers whose labours are preserved to us in the lAlavatif 



* See Mechanics' Magazine, Volume xlix., pp. 655 — 7. Some of tiie 

 foot-notes we omit as comparatively imimportant ; and the rest, for obTi- 

 0U8 reasons, are transferred to the text, and bracketed. 



2g 



