978 Transactions. —Miscellaneous. 
will c cut the circle in two points, and each of Ton points will be at a dis- 
, tance from O equal to t —m, or ; vm, according as the 
distance is measured in T" one eniin or the other: 
And their distance from each other will be equal to 2m, 
that is to say, it will be comparatively small. But each 
point on the polar of O will be at a distance from O equal 
Fig. 2. to ri Henee each point on the circle will be at a distance 
from this polar ander tom. Moreover, every point at a distance of m from 
the ien will be a point on the circle, because it will be at a distance 
of i-m from O. But the locus of points at a dis- 
baci of m from the ents line 4, B, will consist 
of two branches, C, D, and E, F, one on either side 
of A, B, and at the same distance from it, along 
their whole length. It is true that these branches 
form in soli a single continuous line. A point travelling along from C 
to D, and further in the same direction, would ultimately appear at FK, 
travel along to F, and then, after a further journey, reappear at the point C. 
But this does not alter the fact that when a small portion only of this line 
is contemplated, it presents the appearance of two straight lines, each of 
them parallel to, and equi-distant from 4, B. 
In the limiting case, where the radius becomes equal to ! 3: €, D, and 
E, F, both of them coincide with 4, B. The circle merges into a straight 
b and becomes, in fact, the polar of its own centre. It is not, indeed, 
quite accurate to say that it merges into a straight line, for it reduces itself, 
rather to two coincident straight lines, and its equation in co-ordinate 
geometry would be one of the second degree. 
In regard to the surface here treated of, it is easy to see that, as with 
the sphere, the smaller the portion of it we bring under our consideration, 
the more nearly its properties approach to those of the plane. Indeed, if 
we consider an area that is very small as compared with the total area of 
the surface, its properties will not differ sensibly from those of the plane. 
And on this ground it has been argued that the universe may in reality be 
of finite extent, and that each of its geodesie lines may return into itself, 
provided only that its total magnitude be very great as compared with any 
magnitude which we can bring under our observation. 
In conclusion, I cannot do better than quote to you the passage in which 
Professor Clifford explains what must be the constitution of space, if this 
hypothesis should be true:—* In this ease," he says, “ the universe, as 
own, is again a valid conception; for the extent of space is a finite 
number of eubie miles. And this comes about in a curious way. If you 
