652 
Proceedings of the Royal Society 
Theory of Parallels. 
If 0 (figs. 9 and 10) be any point outside a line, P any point in it 
to the right of the foot of the perpendicular, then the limiting position 
of OP, when P is moved in the direction DI to the right, without 
limit, is called the parallel through 0 to DI. The corresponding 
limiting line on the other side of OD is called the parallel through 
0 to DI'. 
Thus 
OK //DI 
OK' // DI' . 
It is obvious, from the uniformity of space, that OK and OK' 
make equal angles with OD. Whether they are parts of the same 
line or not, remains to be seen. 
As P moves off along DI the angle at P diminishes without limit. 
This is easily shown (fig. 10) by taking PP X = OP, P X P 2 = OP 2 and 
so on ad. inf. 
In homaloidal space the parallel to DI through 0 is the perpen- 
dicular to DO at the point 0 : for the sum of the three angles of the 
triangle DPO is always 2R, and P diminishes without limit, hence the 
angle at 0 approaches nearer to R than by any assignable quantity. 
Thus in homaloidal space the two parallels OK, OK' are parts of 
the same straight line, and all the lines through 0 cut IDI', except 
the parallel, which may be said to cut it at an infinite distance. In 
the language of modern geometry there is but one point at infinity 
on the line IDI'. 
In hyperbolic space there are two parallels through a given point 
to a given straight line. 
For as we move P away from D the area of ODP, and consequently 
its defect, constantly increases, but the angle OPD constantly dimi- 
nishes, hence the angle at 0 can never exceed a certain angle which 
is less than a right angle. 
It follows, therefore, that if we take any line IDI' and any external 
point 0, we must classify the lines through 0 as follows : — (1) inter- 
sectors, (2) non-intersectors, (3) two parallels. 
ber of solutions, the angles of the squares becoming less and their area greater 
as p increases. The area of the'greatest possible square tile that we could use 
would be 27 rp 2 , but the lengths of the sides would be infinite. 
