645 
of Edinburgh, Session 1879-80. 
of elliptical space farther. In the meantime, I may remark that in 
a space of this second kind we must, as already explained, admit 
exceptions to the statement that two points determine a straight line. 
In what follows I take single elliptical space as the representative 
of elliptical space generally, although on account of the non-existence 
of a closed surface of uniform positive curvature, on which a pair of 
geodetics intersect only once, the conclusions of the geometry of single 
elliptical space appear in some respects more bizarre than those of 
double elliptical space, whose planimetry is mirrored by the geodesy 
of a sphere. 
It is obvious that Euclidean, or homaloidal, space is included in 
hyperbolic space as above defined. We shall afterwards show, 
however, that it may be regarded as a limiting case of elliptic space. 
It is therefore the transition case lying between the other two. 
Sketch of the Geometry of Hyperbolic, (Infinite) Space. 
From the definition of this kind of space it is clearly infinite. 
Here I must insist on the distinction between infinite and unbounded, 
a distinction first brought into notice by Riemann. The uniformity 
of space necessarily involves the notion that it is unbounded, but 
by no means necessitates that it shall be infinite in extent ; in fact, 
I shall point out directly that a single elliptical space is necessarily of 
finite extent.* 
After the propositions relating to congruency already proved, the 
next fundamental proposition to be established is the following : — 
In hyperbolic space the sum of the three angles of a rectilineal 
triangle cannot exceed two right angles. 
The following proof of this proposition is due in substance to 
Bolyai. Legendre had given another, but he failed to see exactly 
the nature of the assumptions on which he founded. 
ABC (fig. 1) is any triangle, 0 the middle point of BC, OD = OA ; so 
that CD falls within the angle BCL. (Here we assume that a straight 
line is non-re-entrant, and that a pair of straight lines never intersect 
twice.) Then DOC — f AOB ; and ADC is equal in area to ABC, and 
* An ellipse and a eircle are unbounded but finite lines ; a hyperbola is both 
unbounded and infinite. 
t I adopt the sign - used by continental writers for congruent to, or equal 
in every respect to. 
