Page:Mind (New Series) Volume 9.djvu/367

 LOGICAL THEORY OK THE IMAGINARY. MM took part in the controversy) regarded the signs + and as admitting of alternative interpretation in time or space, whereas Sylvester held the necessity of maintaining Doth interpretations at once. Before leaving this subject of the application of the imagi- nary to geometry, I cannot forbear from touching on a question which has been agitated in the pages of Nature and elsewhere, viz., the principle that the square of a vector should be negative. It has been claimed that to omit the ( - ) is not only essential to the physicist but is more con- sistent with ordinary algebra. Here again, the principle of the interpretation of the imaginary, advocated above, gives the clue. The imaginary, as a conjunctive relation of + and -, is on the one side identically related to a given direction, and in this relation would answer to the ordinary operations of algebra ; but as non-identically or dissimilarly related (and in a directional calculus such as Hamilton's this is the dominant point of view) the sign must be retained. Hamilton was therefore justified in saying " every line in tridimensional space has its square equal to a negative number, which is one of the most novel but essential ele- ments of the whole quaternion theory " (Lectures, p. 53). The analysis we have given admits of other illustrations as in Determinants. I now pass on to the more general question of its effect on our conception of the relation of the logical calculus to other branches of analysis. The idea of a symbolical calculus which should be perfectly general and applicable to all kinds of investigations is one which has frequently presented itself to both logicians and mathema- ticians. The idea occurs in the Discours de la Methode of Descartes. Such a calculus, Leibnitz seems to have had before his mind under the name of Characteristica universalis and Cornte also in a passage in the Synthese Subjective seems to have contemplated the same idea. It might also be said that Newton's definition of algebra as Arithmetica universalis implies the conception as an ultimate consequence. Boole maintained such a view in an article in the Philosophical Magazine. The idea of the isolation of the specious, universal or formal element of arithmetic or any other science, seems to lead to the conception of a theory of forms which should be perfectly pure and admit of general application, varied only by the conditions of the peculiar matter to which such a calculus is applied. Thus, the " principle of the perma- nence of equivalent forms " is regarded by Peacock as ex- pressing the law of transition from an algebra arithmetically conditioned to a more universal, a symbolical algebra. The 23