8 
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 ¢ime; the variety which 
it admits is not ¢rz- but wni- 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, ifa 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 fecently called ‘* Forward.” In this respect it differs in 
its processes from the Cartesian method of coordinates, andseems 
often to admit of being more simply and directly applied to 
