Page:Mind (New Series) Volume 9.djvu/363

V. -LOGICAL THEORY OF THE IMAGINARY. BY PROF. G. J. STOKKS. TN his address before the meeting of the British Association at Southport Prof. Cayley, having referred to the amount of discussion which the notion of negative magnitudes has occasioned in philosophy, said : " But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasise the assertion) underlying and per- vading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a loot* in quo of imaginary points and figures) in geometry : I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry." Prof. Cayley then pro- ceeded to say " considering the prominent position which the notion occupies say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored ; it should at least be shown that there is a right to ignore it." It is evident from the tone of the passage I have quoted that Prof. Cayley was not satisfied with the attitude adopted towards this notion by the majority of those who have treated the subject. Most writers seem to have adopted the view that however useful such an expression as v/ 1 may be in technical mathematics, still^even in pure algebra, the expres- sion is essentially devoid of meaning. The dissatisfaction implied in the above quotation must be my excuse for venturing to dissent from the eminent writers who have held this view and for attempting to evolve from the logi- cal standpoint an interpretation for the imaginary of pure algebra. Prof. A. Macfarlane in his pamphlet on The Imayinary of Alijebra has divided analysts into three classes with re- spect to the theory and use of V 1 : " fit, those who have considered it as undefined and uninterpreted, and consequently