274 — Transactions. — Miscellaneous. 
line there are only two directions in which we can set out. — It is therefore 
a line-aggregate of one dimension. Considered as a point-aggregate it has 
two dimensions, and accordingly it is a manifoldness of two dimensions. 
In the same way it will be seen that solid space is a manifoldness of three 
dimensions. 
I have endeavoured by these preliminary remarks to explain what is 
meant when we speak of a continuous manifoldness of two dimensions. It 
is the object of this paper to communicate some results I have arrived at 
respecting the properties of the simplest of such manifoldnesses which has a 
finite extent. The existence of the particular manifoldness I shall endeavour 
to describe has been referred to in a remarkable lecture by Professor Clifford 
on the Postulates of the Science of Space, but I am not aware that its pro- 
perties have not hitherto been worked out in detail. 
The simplest of all doubly extended continuous manifoldnesses is the 
plane, but it is not a manifoldness of finite extent. It reaches to infinity 
in every direction, and its area is greater than any assignable area. It is 
therefore not the manifoldness of which we are in search. Now the cir- 
cumstance in which the plane differs from those doubly extended manifold- 
nesses, which are next to it in order of simplicity, is the possibility that 
figures constructed in it may be magnified or diminished to any extent 
without alteration of shape; in other words, that figures which can be 
constructed in it at all can be constructed to any scale. That this property 
is not possessed by curved surfaces may be seen by considering the case of 
a spherical triangle. If the sides of a triangle constructed on a given sphere 
be all of them increased or diminished in the same proportion, the shape of 
the triangle will not remain the same. Now it has been found by Professor 
Riemann that this property of the plane is equivalent to the following two 
axioms :—(1.) That two geodesic lines which diverge from a point will 
never intersect again ; or, as Euclid puts it, that two straight lines cannot 
enclose a space; and (2.) that two geodesic lines which do not intersect will 
make equal angles with every other geodesic line. This second is precisely 
equivalent to Euclid’s twelfth axiom. Deny the first of these axioms and 
you have a manifoldness of uniform positive curvature ; deny the second, 
and you have one of uniform negative curvature. The plane lies midway 
between the two, and its curvature is zero at every point. 
Let us consider, then, the case of a doubly-extended manifoldness, of 
which the curvature is uniform and positive. The first of the before- 
mentioned two axioms is no longer true. Geodesic lines, diverging from a 
point, do not continue to diverge for ever. They meet again, and enclose a 
space. The first question which presents itself, is with reference to the 
