"with Mr. T. S. Davies's Notes on some of the Topics. 355 



adapting it to the higher orders. This course I shall adopt 

 in pointing out the paths which discovery has taken; but it is 

 not perhaps the most strictly correct one ; for in solving a 

 cubic by the method handed to us by Cardan, we have to 

 satisfy two simultaneous equations of the second order. Equa- 

 tions of this order (and with them the process of elimination) 

 are, in fact, in the ordinary course of study, brought under our 

 notice before we come to the subject of cubics of the first 

 order ; and perhaps the first example that we see of the trails- 

 formation of an equation is presented in the solution of two 

 quadratic equations of the second order each of which has on 

 one side a number and on the other a homogeneous function 

 of the two unknowns. 



Engaged, then, on a subject matter that seems to set pre- 

 cise classification at defiance, unless we take a view other than 

 the historical one of the solution of the cubic, I shall for the 

 present confine myself to the subject of equations which in- 

 volve only one unknown quantity : and, first, to their solutions. 



1. The solution of linear and quadratic equations was un- 

 derstood by the Hindoo algebraists, and the processes em- 

 ployed did not differ in any essential respect from those now 

 in use {c). But something more than reduction and evolution 



(c). But the Hindu method of "completing the square" is cer- 

 tainly more convenient in practice than the Italian one which is in 

 common use. It is surprising, indeed, that the Hindu process is 

 not more generally insisted on in our elementary books, than we 

 find it to he. Even where it is noticed at all, it is never so much 

 spoken of in respect to its utility as it is under the character of an 

 exotic curiosity, the chief merit of which arises from its being " old " 

 and being " Indian." 



was required for the purpose of solving a cubic. The prin- 

 ciple employed by Tarlalea (and Ferrei?) was that of render- 

 ing the equation indeterminate by introducing two unknowns 

 instead of one. The introduction of tisoo unknowns suffices 

 not only for the solution of a cubic, but also for that of a bi- 

 quadratic, as has been shown by Pilatte {Annales de Mathe- 

 matiqnes, tome ii. pp. 152-154'). But Euier had previously 

 solved the biquadratic by substituting for x the sum of three 

 unknowns. 



The solution of Ferrari, the first in order of time, was 

 effected, not by substituting for x, but by giving the biqua- 

 dratic a new form by means of a subsidiary quantity intro- 

 duced for that purpose into the equation. I have shown (Phil. 

 Mag. S. 'A. vol. xxii. p|). 502-503) that a cubic can be solved 

 by the same means. 



All these solutions are direct : they do not depend upon any 

 2 A 2 



