[ 37 ] 



III. On a new Imaginary in Algebra, By James Cockle, 

 Esq.^ M.A., of Trinity College^ Cambridge, and Barrister- 

 at-Latio of the Middle Temple*. 



ALGEBRA may be regarded under the triple aspect pre- 

 sented by the words Identity, Equivalence, and Impossi- 

 bility f; but the latter view will fall more particularly within 

 the scope of the present observations. The ordinary algebra, 

 it is true, takes cognizance, not only of negative and unreal 

 quantities, but sometimes of questions involving impossibility. 

 This impossibility is indicated either by contradictory arith- 

 metical results or, occasionally, by the symbol infinity %. But, 

 neither in the one case nor the other, does the indication of 

 impossibility furnish us with the elements of a calculus. Un- 

 real results, on the contrary, although not subjects of concep- 

 tion, like number, nor directly interprelable, like negative 

 quantities, are yet not only indirectly interpretable, but also 

 important instruments of investigation. Why is this? My 

 answer is, — because impossibility has never yet been symbol- 

 ized. And I would add that, before this is done, we ought to 



* Communicated by T. S. Davies, Esq., F.R.S.L. & Ed., &c., who requests 

 us to annex the following note. 



[It will of course be understood that I do not pledge myself to an agree- 

 ment with Mr. Cockle's views on the geo/nefnca/ signification of his i,^', i. 

 With respect to them as algebraical symbols, I would not here offer an 

 opinion : but with respect to the geometrical interpretation, I take a totally 

 different view, as will be inferred from a short paper of mine printed in 

 vol. xxix. (pp. 1 71-175) of the Philosophical Magazine, under the signature 

 " Shadow." 



Mr. Cockle has undoubtedly the weight of cotemporary scientific autho- 

 rity on his side of the question; and, indeed, I believe I stand nearly alone 

 in the view I take of these questions. It would, however, be as unphilo- 

 sophical a mode of searching for the truth as it would be disingenuous in 

 the discussion, to suppress the expression of all views which differ from 

 those which 1 may happen to entertain. It is always a matter of far less 

 moment who is right than what is true. 



Mr. Cockle very kindly put his paper into my hands for perusal before 

 he printed it; and J have much pleasure in fulfilling his request by forward- 

 ing it to you. 



I may add, that it is much to be desired that the history of the attempts 

 that have been made to give an explanation of the symbols of incongruity 

 should be published. A strict discrimination between the views of differ- 

 ent algebraists and geometers might prevent the waste of much valuable 

 time and power; for there can be no doubt that a large portion of the spe- 

 culations which have been put forward in recent times are essentially iden- 

 tical with much earlier ones. 



Little Heath, Charlton, T. S. D.] 



Nov. 25, 1848. 



t Phil. Mag. S. 3, vol. xxxii. p. 352. 



X Peacock, Report on Analysis (Third Report of British Association), 

 pp. 237-238. 



