Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

57

8. THE AJXILIARY SPACE E.
The discussion of the following paragraphs will be clarified by
making use of familiar propositions of the projective geometry.
In so doing, we avail ourselves of theorems which are in essence
analytic. Their geometric clothing lends vividness to their
meaning and helps greatly in their application. We consider
then a projective geometry in which a point is fixed by the homogeneous coordinates
~0: 1:L:L.
A linear homogeneous equation defines a plane, two such equations
a straight line. In this geometry we define a system of projective
measurement, based upon the fundamental quadric
t + k(1 + 2 + + ) = 0.
The distance A between two points is by definition given by the
relation
cos klA + =22 + 33)
V ~' + k 2( + 12 + )) 12 + 2 k2('2 + + 2 + 2)
Any collineation which leaves the fundamental quadric invariant
we shall call a movement of the projective space. Such a movement leaves distance and angle unaltered. The space in which
this geometry prevails we shall call the auxiliary space E.
The points of I may be made to correspond to the points of S
by placing
x.= (_ = 1 2, 3)
' t V 2E + k2(+ 2 + E + ) '
where the sign of the radical is the same for all values of i. It is
clear that geodesic lines and surfaces in S correspond to straight
lines and planes in E and conversely. Geodesic distances and
angles in S correspond to projective lengths and angles in 2 and
a displacement in S corresponds to a movement in I and conversely.
Now if k is zero or pure imaginary, x0 is always positive, since