653 
of Edinburgh, Session 1879-80. 
In figure 1 1 KOL' and K'OL are the two parallels ; all lines lying 
in the angles KOL, K'OL', are non-intersectors, all those lying in 
KOK', LOL' are intersectors. The fact that in hyperbolic space 
there are two parallels through a given point to a given straight line 
is expressed in modern geometry by saying that in hyperbolic space 
a straight line has two distinct real points at infinity. 
After what has been laid down, the following propositions either 
are immediately evident, or can be proved with very little trouble. 
If a line is parallel to another at any point , it is so at every point 
of itself . 
Parallelism is mutual. 
Lines which are parallel to the same line are parallel to one 
another. 
Lines that are parallel continually approach one another on the 
side towards which they are parallel. 
Non-inter sectors in the same plane have a minimum distance, which 
is the common perpendicular. 
The angle which a parallel through 0 to L makes with the per- 
pendicular on L is called the parallel angle. 
The parallel angle is a function of the length of the perpendicular, 
increasing tvhen the perpendicular diminishes. 
If 6 be the angle, p the length of the perpendicular, then it may 
be shown by methods which I shall presently explain that 
_ 
tan l$ = e p , 
When p = 0, 6 = E ; when p = oo , 6 = 0. 
Geometry of Elliptic Space. 
For simplicity I take single elliptic space, but there will be no 
difficulty in modifying what follows so as to make it apply to double 
elliptic space. 
In single elliptic space every straight line returns into itself ; and 
two straight lines intersect in only one point. Thus, starting from 
any point P, and proceeding in any direction continuously, we at last 
return to the point P ; the length L travelled over in this process is 
called the length of the complete straight line. 
It is obvious that in single (as well as in double) elliptic space 
