from attributing to this scale any one direction rather than 

 another in tridimensional space, as having such or such a 

 zenith distance, or such or such an azimuth, rather than such 

 or such another. And the progression on this scale from ne- 

 gative to positive infinity, obtained by combining a quanti- 

 tative element with the contrast between two opposite direc- 

 tions, corresponds less to the conception of space itself (though 

 we have seen that considerations of space might have sug- 

 gested it) than to the conception of time ; the variety which 

 it admits is not tri- but uni- dimensional ; and it would, 

 in the language of some philosophical systems, be said to 

 appertain rather to the notion of intensive than of extensive 

 magnitude. Though answering precisely to the progression 

 of the quantities called real in algebra, it has, when viewed 

 from the geometrical side, somewhat the same sort of imagi- 

 nariness, and yet (it is believed) of utility, as compared with 

 lines in space, which the square root of an ordinary negative 

 has, when compared with positive and negative quantities. 

 This analogy becomes still more complete when we observe 

 that (in this theory) the fourth proportional to any direction 

 X in space, and either of the two directions A or B upon the 

 scale, is the direction opposite to X ; so that, if a vector-unit 

 in any determined direction X had been taken for positive 

 unity, then each of the two scalar units in the directions A 

 and B (in common, it is true, with every vector-unit perpen- 

 dicular to X) might have been called, by the general nomen- 

 clature of multiplication, a square root of negative one. 



It is, however, a peculiarity of the calculus of quaternions, 

 at least as lately modified by the author, and one which seems 

 to him important, that it selects no one direction in space as 

 eminent above another, but treats them as all equally related 

 to that extra-spatial, or simply scalar direction, which has 

 been recently called " Forward." In this respect it differs in 

 its processes from the Cartesian method of coordinates, and seems 

 often to admit of being more simply and directly applied to 



