118 Mr. Cockle on the Method of Vanishing Groups. 



It was the opinion of an illustrious analyst, now no morcj 

 that the actual solution of the quadratic equation in general 

 algebraical symbols contributed, in a greater degree than the 

 solution of the cubic and of the biquadratic, to the advance- 

 ment of algebra as a science (Murphy, Theory of Equations, 

 pagel). And we may regard such solution under an aspect which 

 did not meet the view of that great writer, — we may regard it 

 as containing the germ of a process which enables us to 

 exhibit far more general functions than those involved in the 

 quadratic in the form of sums of powers, and which, combined 

 with the indeterminate method when occasion requires, is ca- 

 pable of most extensive application in analysis. But it must 

 never be forgotten that it was in the solution of a cubic that 

 this indeterminate method — this great device of making a de- 

 terminate problem algebraically indeterminate, and so solving 

 it — was probably first elicited : and this circumstance ought 

 to influence us materially in estimating the rank which the first 

 discoverer of the solution of cubic equations is entitled to hold 

 in scientific history. 1 consider the method of vanishing 

 groups, when applied to the theory of equations, as the com- 

 bination (accompanied by full developments) of the principle 

 involved in the solution of the cubic with the process employed 

 in that of the quadratic. Suggested, as it were, by a question 

 in analytical geometry of two dimensions, that method admits 

 of application in the analytical geometry of three dimensions; 

 and 1 have in fact so applied it in my Chapters on Analytical 

 Geometry, now in course of publication in the Mechanics' 

 Magazine. Its processes would evidently be capable of a similar 

 extension to an analytical geometry of n dimensions, had we 

 any knowledge of an entity possessed of more than three dimen- 

 sions. Were such things within the bounds of our faculties, 

 there would be functions of their dimensions, linear ones for 

 instance, which would bear an analogy to the corresponding 

 ones of the three-dimensioned entity space (and the one- 

 dimensioned entity time), and which would by this method be 

 discussed in a manner similar, or at all event^but slightly dif- 

 ferent from, that of space (or time). 



In conclusion, I may mention a point in which our results 

 so closely resemble those deduced in another department of 

 science, as to have suggested to me the name " Diophantic '* 

 as one which would be appropriate to our processes. 1 mean, 

 that as in the Diophantine algebra we seek to form an equa- 

 tion, of which, if one side be zero the other shall consist of 

 powers (squares, cubes, &c.) of numerical quantities connected 

 by the signs plus and minus, so in our analysis the object is 

 in general to obtain an equation, of which, one side being 

 zero, the other shall consist of powers of algebraical quantities 



