644 
Proceedings of the Royal Society 
acute angled triangle, but reject in the meantime, at all events, all 
that follow in the first book. 
Next we adopt Euclid’s propositions concerning angles at a point, 
viz., I. 13, 14, 15; also the propositions as to congruency I. 4, 5, 6, 
8, and the first part of 26, with a protest to the effect that in many 
cases his demonstrations are needlessly circuitous and difficult. All 
that is wanted for the demonstration of these propositions is the 
defining property of the straight line and the ordinary axioms and 
definitions as to equality. 
Different Kinds of Space. 
Before going farther, we must distinguish the different cases that 
may arise when we consider two intersecting straight lines. 
1. They may never intersect again and be of infinite length (i.e., 
each is non-re-entrant). Space which has this characteristic is called, 
for the present, hyperbolic space. We shall see, however, by and by 
that another case must be distinguished under this head, that, viz., 
of homaloidal or Euclidean space. 
2. They may intersect again. Space having this characteristic is 
called elliptic space. 
The simplest space of this kind is that in which a straight line 
returns into itself, so that the next point in which two straight lines 
intersect is the point in which they first intersected. In this kind 
of space, which I shall call single elliptic space, two straight lines 
intersect in only one point ; and there is no exception to the state- 
ment that two points determine a straight line. 
The next simplest case would be that in which two straight lines 
intersect a second time in a distinct point, and then re-enter at the 
next point of intersection which coincides with the original one. 
This might be called double elliptical space. I am not yet certain* 
whether the symmetry of space will allow us to carry this multiplicity 
* I have not been able to find a definite settlement of this question by any 
of the great authorities on hyper space. Frischauf takes double elliptic space 
as the representative of elliptic space, and seems to hold that this is the only 
possible kind. Klein (“ Mathematische Annalen,” vi. 125) takes single elliptic 
space, and criticises Frischauf ’s view (“ Fortschritte der Mathematik,” viii. 
313, 1876). Newcomb (Borchardt’s Journ., lxxxiii. p. 293) professes himself 
unable to settle the question. If the notion of double elliptic space cannot be 
shown to be self-contradictory, then it would appear that the question becomes 
simply one of the choice of axioms. See note below, p. 661. 
