Franxtanp.—Simplest Continuous Manifoldness of Two Dimensions. 275 
of a spherical surface, they will converge towards a point which is separated 
from ihe starting-point by half the length of a geodesic line. And this is 
the only case we are able to conceive. The surface of a sphere is the only 
doubly-extended ifoldness of uniform positive curvature which geometry 
recognizes, and it is the only one which we can figure to ourselves in 
thought. It is not, however, the simplest of such manifoldnesses. To obtain 
the simplest case, we must suppose that the point towards which two geodesic 
lines converge is separated from their starting-point, not by half, but by the 
entire length of a geodesic line; or, what amounts to the same thing, that 
it coincides with the starting-point. It is true that we are utterly unable to 
figure to ourselves a surface in which two geodesic lines shall have only one 
point of intersection, and shall yet enclose a space. But we are perfectly 
at liberty to reason about such a surface, because there is nothing self- 
contradictory in the definition of it, and because therefore the analytical 
conception of it is perfectly valid. It is the simplest continuous manifold- 
ness of two dimensions, and of finite extent, and those few properties of it, 
which I have worked out, appear to me to be very beautiful. In order to 
make my observations more intelligible, I shall for the future speak of it as 
a surface, and its geodesic lines I shall speak of as straight lines. I have 
the highest authority for using this nomenclature ; and, though it will 
impart to my theorems a very paradoxical sound, it is calculated, I think, 
to give a juster idea of their meaning than if I were to use the more 
accurate, but less familiar terms. 
Assuming, then, as the fundamental properties of our surface, that every 
straight line is of finite extent (in other words, that a point moving along 
it, will arrive at the position from which it started after travelling a finite 
distance), and that two straight lines cannot have two points in common, 
the first corollary I propose to establish is, that all straight lines in the 
surface are of equal extent. 
Let A B be two straight lines in the given surface. If possible, let 4 
be greater than B. From 4 cut off a portion equal to B. Let P Q be 
the extreme points of this segment, and let R be any point in B. Apply 
the line 4 to the line B in such a manner that the point P falls on the 
point H. Then, since, in a surface of uniform eurvature, equal lengths of 
geodesie lines may be made to coincide, the segment P Q will coincide with 
the entire straight line B. Hence Q will fall upon R. But P coincides 
with R, and P and Q do not coincide with one another, since P Q is less 
than the entire straight line 4; therefore, Q cannot coincide with R. 
Hence A cannot be greater than D. 
The straight lines here spoken of are of course not terminated straight 
lines. What the proposition asserts is, that the entire length of all straight 
