Page:Mind (New Series) Volume 9.djvu/366

 352 G. ,i. STOKES : We might therefore infer from this, that, inasmuch as ^1 has two roots, one positive and the other negative, which are disjunctively related to each other as alternatives, so the J 1 will involve the same roots, no longer disjunctively hut conjunctively related. Or, the expression Jl will mean, simply, that 1 is to he multiplied with another 1 similar in sign to itself ; whereas, v ' 1 will mean a 1 which is to be multi- plied by a 1 dissimilar in sign to itself. The whole mystery, therefore, underlying this symbol is that the identity or equivalence in the factors which is quantitatively implied, does not extend to the qualitative relation represented by the signs + and. This is already recognised in all inter- pretations involving the concrete application of the imaginary symbol. If we substitute for J 1 a symbol, say, + (-), expressive of this logical analysis, we shall find it acquires different meanings in different systems of mathematical analysis. It is identical with the " law of duality " of Boole. If V be an independently interpretable logical function, V (1 V) = O. Boole terms this equation the condition of the interpretability of logical functions. It is quite clear that Boole regarded V (1 V) as discharging an analogous func- tion in logic to that performed by J - 1 in algebra ; though the work contains no hint of the substantial identity of the two, which is here maintained. In point of fact, the diverse application which the function receives in logic and in mathematics establishes points of contrast sufficient to obscure the identity. In logic, precisely that element is excluded which characterises the imaginary in its application to mathematics. I shall refer again to this point when con- cluding, here merely remarking that the only writer who has attempted to make a mathematical use of the purely logical form is Hegel, V (1 - V) is the Notion of Hegel. If we turn from the logical calculus to trigonometry and substitute in De Moivre's theorem the symbol I have pro- posed for the usual J-l, we shall find that the results work out identically with the ordinary form of the theorem. We can apply also to this symbol the interpretation of rota- tion through a right angle, and as an immediate consequence we might arrive at that double interpretation of a line pro- posed many years ago by the late Prof. Sylvester in the Messenger of Mathematics and which subsequently formed the subject of controversy between Cayley and Sylvester in con- nexion with the Carnot-D'Alembert problem. The issue between these eminent mathematicians depended as I con- ceive on this, that Cayley and Mrs. Ladd-Franklin (who also