FaaxkLAND.—Sümplest Continuous Manifoldness of Two Dimensions. 277 
it is also of the same shape all along, it satisfies Leibnitz’s definition of a 
straight line; and it is, in fact, a geodesic line of the surface. 
Hence we have this second proposition, that all points in the surface 
opposite to a given point lie in a straight line. 
From the method of its construction, this straight line is further from 
the given point than any other line in the surface. Travelling from the 
given point as a centre, in whatever direction we might set out, we should, 
after completing half our journey, arrive at this furthest straight line, we 
should eross it at right angles, and we should then keep getting nearer 
and nearer to our starting point, until we finally reached it from the oppo- 
site side. 
Each separate point in the surface, moreover, has a separate furthest 
line. For if any two points be taken, the points opposite to them on the 
straight line which joins them will be distinct. Hence their furthest lines 
will cut this joining line in two separate points. They must, therefore, be 
two separate lines, for the same straight line cannot cut another straight line 
in two separate points. In a similar manner it may be shown that each 
straight line in the surface has a separate furthest point. Hence there exists 
a reciprocal relation between the points and straight lines of the surface,— 
a relation which we may express by saying that every point in the surface 
has a polar, and that every straight line in the surface has a pole. It is 
then easy to show that when a point is made to move along a straight line, 
its polar will turn about a point, and that when a straight line is made to 
turn about a point, its pole will move along a straight line. 
It is interesting to compare these propositions with the corresponding 
ones in spherical geometry. There, too, each point has a furthest geodesic 
line, that is to say, a geodesic line which is further from it than any other 
geodesic line on the sphere, but each geodesic line has two furthest points, 
or poles, instead of having only one. Hence there is not that perfect 
reciprocity of relationship between points and geodesic lines which exists 
in the surface we have been examining; and this is one of the many ways 
in which the sphere shows itself to be inferior to that surface in simplicity. 
The most astounding fact I have elicited in connection with this surface 
is one which comes out in the theory of the circle. Defining a circle as the 
locus of points equi-distant from a given point, we shall find that it assumes 
a very extraordinary shape when its radius is at all nearly equal to half 
the entire length of a straight line. For, let us again figure to ourselves a 
number of straight lines radiating from a point. Let / be the total length 
of each straight line. Then, the supposition that we have to make is, that 
the radius of our circle shall be nearly equal to $ . Let us suppose it equal 
to + — n, where m is small as compared with l. Each of the radiating lines 
