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of Edinburgh , Session 1879 - 80 . 
existence of similar figures, the geometrical experience of a cheese 
mite in homaloidal space would not be different from that of a being 
one of whose habitual walking steps was from the sun to the dog star. 
In hyperbolic or elliptic space the case is otherwise. In either of 
these two kinds of space we might divide intelligent beings into two 
classes according to their bodily dimensions. We might have a race 
of micranthropes, whose bodily dimensions and the radius of whose 
sphere of experience were infinitely small compared with the linear 
constant of space. For instance, if the space were elliptic, the world 
of the micranthropes would be but an infinitely small fraction of the 
elliptic universe. It must be noticed, however, that from the point 
of view of a micranthrope, his world need not be a prison-house by 
any means, for he would compare it not w r ith the linear constant of 
universal space, of whose magnitude he must necessarily be ignorant, 
but with some arbitrary standard such as the length of his own arm, 
and so considered his world would to him be infinite, if we only suppose 
him small enough. Again, we might have a race of macranthropes, 
whose bodily dimensions were comparable with the linear constant 
of space. In the case of an elliptic and finite space, we could, of 
course, conceive one of these himself so great that there would not be 
room enough in the universe for another as great. 
The geometry of the micranthropes would, of course, be homaloidal. 
The axioms of Euclid would appear to them strictly in accordance 
with experience, and, although they lived in part of an elliptic or 
hyperbolic space, their prejudices would render the conceptions of 
the general properties of such a space as difficult to them as they are 
to us. On the other hand, the geometry of the macranthropes would 
be elliptic or hyperbolic, as the case might be. A hyperbolic 
macranthrope would, of course, be familiar with the fact thatthe defect 
of a triangle diminishes as its area diminishes. If he were a 
mathematician he would be aware of the relation of proportionality, 
and might speculate concerning triangles of zero defect, much as we 
do about absolute zero of temperature. If Euclid’s geometry were to 
fall into the hands of an instructed macranthrope, he would very 
likely regard it as the production of some macranthropic lunatic, who 
had meditated on the fact that the defect of a triangle diminishes 
with its area, until he had so far lost his wits as to commit the 
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