657 
of Edinburgh, Session 1879 - 80 . 
If, therefore, we define A° + B° + C 0 - 180° as the excess of the 
triangle, we have the proposition that — 
The excess of every triangle is positive , and is proportional to its 
area. 
The conclusions drawn above (p. 650) for hyperbolic space 
follow here, mutatis mutandis . In particular, we see that we may 
apply Euclidean planimetry to infinitely small figures. On this 
remark we can, as will be done later, found a system of planimetry 
4L 2 
for elliptic space, and determine P. The result is P = . Hence, 
writing p for i , and e for the radian measure of the excess, we have 
A — p 2 e 
where p is a linear constant characteristic of the elliptic space. 
It is easy after what has now been established to work out the 
propositions corresponding to Euclid’s first book. The conclusions 
will, of course, be subject to certain modifications, but these are easily 
found. I may mention in particular that the propositions concerning 
the curves of equidistance already given for hyperbolic space, hold 
with very slight modification for elliptic space, the main difference 
being that the equidistants are convex instead of concave to the given 
straight line. 
Theory of Parallels. 
In elliptic space there is, of course, no such thing as a parallel, 
because there are no infinitely distant points on a straight line.* 
If 0 (fig. 16) be a point outside the line ID I'; then it is easy to 
see that the two segments of the perpendicular from 0 are respec- 
tively the least and greatest distances from the given line. If OD 
be the least distance, then, as OP, starting from OD, revolves about 
0, OP continually increases, until it has rotated through 180°, and 
then it is at its maximum, after which it decreases again. 
It can easily be shown that, as OP revolves from OD, the angle 
OPD decreases, until OP is perpendicular to OD, and then OPD 
is at its minimum value. After that, as may be easily shown by 
producing the line backwards through O, the angle again increases. 
* In the language of modern geometry the points at infinity on a straight 
line in elliptic space are imaginary. 
