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of Edinburgh, Session 1879-80. 
If we take a region in this space whose greatest linear dimension is 
an infinitely small fraction of p, then the defect of every triangle 
within that region will he infinitely small, and its geometry will not 
differ sensibly from that of a homaloidal space. This is often 
expressed by saying that hyperbolic space is homaloidal in its 
smallest parts. 
It appears, therefore, that, even in hyperbolic space, Euclid’s 
planimetry will apply to infinitely small figures. For instance, the 
ratio of the circumference of a circle to its diameter will be 
7r = 3T4159 . . , . (the ordinary transcendental constant), when 
the diameter is made infinitely small. We may, therefore, if we 
please, measure our angles in radians (circular measure), and in 
fact use all the formulae of homaloidal plane trigonometry, if proper 
restrictions be observed. 
It should also be noticed that the existence of this length p related 
to the space, but not directionally related, suggests the possibility of 
explaining the properties of tridimensional space by subsuming it 
in a space of four or more dimensions. I have not chosen to enter 
into speculations of this nature, partly because their development 
has been entirely analytical hitherto ; and partly because, so far as I 
can see at present, it may be justly contended that the conceivability 
of hyperspace of three dimensions rests on different grounds from 
that which we must necessarily assume when we attempt to add 
another dimension. In this, however, I may be but one of those 
whom Gauss playfully called Boeotians.* 
* Before leaving this part of the subject, I may mention the curious solution 
of the problem of dividing a plane in hyperbolic space into a network of 
regular polygons. 
If n be the number of sides of each polygon, p the number of polygons round 
a point of the network, A the area of each of the w-gons, then 
A—mrp^f 1 — — — — \ , 
V n jo J 
with the condition - + — ~ ■ 
n p 2 
Suppose, for instance, we wish to divide a plane into squares, i. e. , regular 
four-sided figures. Then n = 4. If jp = 4, i. e. , if the angles of the square be 
right angles, A = 0, which does not, strictly speaking, give a solution. The 
2 
next case is p = 5, so that A = ^ 7rp 2 is the area of the smallest finite square 
with which we could pave a plane floor. Of course there are an infinite num- 
