658 Proceedings of the Boyal Society 
The line 01, perpendicular to OD, is all that there is in elliptic 
space to represent a parallel through 0 to the line I'DI. 
! 
General Conclusions. 
If I have succeeded in my attempt to explain the results of modern 
research concerning the axioms of geometry, it will he apparent that, 
even if we overlook the possibility of space being non-uniform, in 
the sense of having properties depending on position and direction, 
it is still possible to develop three self-consistent kinds of geometry 
— the hyperbolic, the homaloidal, and the elliptic. It is impossible, 
it appears to me, to say on a priori grounds that any one of these is 
more reasonable than the others. If, therefore, a priori ground is to 
be sought for the axioms of geometry, such tests of its firmness “ as 
the inconceivability of the opposite ” and others like it are not to be 
relied upon. They are merely an appeal to ignorance. 
If, on the other hand, we view the question from the side of 
experience, three alternatives are open to us. We may hold that 
space is homaloidal and therefore infinite. In this case we extend 
to the infinite part of space which we do not know the results of our 
experience of the finite part of it that we do know. 
Again, we may hold that space is hyperbolic and therefore infinite. 
In this case experience teaches us that the radius of the sphere of oar 
experience is infinitely small compared with the linear constant of 
space ; for Lobatschewsky calculated from astronomical observations 
the sum of the three angles of triangles whose smallest sides were 
about double the distance of the earth from the sun, and found that 
the difference from two right angles was not greater than the probable 
error of observation. 
Lastly, we may suppose that space is elliptic and therefore finite, 
in this case we must admit that our experience extends to but an 
infinitely small fraction of its whole extent, since no sensible excess 
can be found in the largest triangles with which we are acquainted. 
Before leaving this subject, it may be well to illustrate with some 
care what is meant by the words finite and infinite as I have used 
them. They have, of course, a purely relative meaning. In the 
geometry of homaloidal space no distinction can be bailt on the 
relative dimensions of figures apart from their form. Owing to the 
