[951 
951] 
NON-EUCLIDIAN GEOMETRY. 
481 
distances of the same kind, having a common unit, the quadrant, represented by \nr; 
and in fact, any distance may be considered indifferently as a linear, an angular, or 
a dihedral distance: the word, perpendicular, usually represented by x, refers of course 
to a distance = \tc. We have moreover the distance of a point from a plane, that 
of a point from a line, and that of a plane from a line. Two lines which do not 
meet may be x, and in particular they may be reciprocal: in general, they have 
two distances; and they have also a “ moment ” and “ comoment,” the values of 
which serve to express those of the two distances. Lines may be, in several distinct 
senses, as will be explained, parallel; and for this reason the word parallel is never 
used simpliciter; the notion of parallelism does not apply to planes, nor to points. 
Elliptic space has been considered and the theory developed in connexion with 
the imaginaries called by Clifford biquaternions, and as applied to Mechanics: I refer 
to the names, Ball, Buchheim, Clifford, Cox, Gravelius, Heath, Klein, and Lindemann: 
in particular, much of the purely geometrical theory is due to Clifford. Memoirs by 
Buchheim and Heath are referred to further on. 
1894), 
Geometrical Notions. Art. Nos. 1 to 16. 
ine, plane, 
iz. instead 
a quadric 
ry surface 
3 so-called 
r hat more 
particular, 
rs of two 
leorems of 
case also 
elism and 
7, infinity 
' instance) 
ew theory, 
lere is no 
}his plane, 
nes, there 
:h of the 
is nothing 
nsequently 
two such 
yen lines. 
1. The Absolute is a general quadric surface: it has therefore lines of two 
kinds, which it is convenient to distinguish as directrices and generatrices: through 
each point of the surface there is a directrix and a generatrix, and the plane 
through these two lines is the tangent plane at the point. A line meets the surface 
in two points, say A, C; the generatrix at A meets the directrix at C; and the 
Fig. 1. 
A 
v \ 
/ 
D\ / 
¿kN. \ U' 
%\ \ ns 
G 
directrix at A meets the generatrix at 6; and we have thus on the surface two new 
points B, D; joining these we have a line BJD, which is the reciprocal of AC; viz. 
BD is the intersection of the planes BAD, BCD which are the tangent planes at 
A, C respectively, and similarly AC is the intersection of the planes ABC, ADC 
which are the tangent planes at B, D respectively. 
ation: we 
tween two 
these are 
According to what follows, reciprocal lines are x, but x lines are not in general 
reciprocal; thus the two epithets are not convertible, and there will be occasion 
throughout to speak of reciprocal lines. 
C. XIII. 61 
61