FaaNKLAND.—Sünplest Continuous Manifoldness of Two Dimensions. 978 
to construct a perfectly self-consistent scheme of propositions, all of them 
valid as analytical conceptions, but all of them perfectly incapable of being 
realized in thought. 
Many of the results he arrived at were very curious ; such as for 
instance that the three angles of a triangle would not be together equal to 
two right angles, but would be together less than two right angles by a 
quantity proportional to the area of the triangle. If we were to increase the 
sides of such a triangle, keeping them always in the same proportion, the 
angles would become continually smaller and smaller, until at last the three 
sides would cease to form a triangle, because they would never meet at all. 
There are many other assumptions, at variance with the axioms of 
Euclid, which may be made respecting distance-relations, and which yield 
self-consistent schemes of propositions differing widely from the propositions 
of geometry. We see therefore, that geometry is only a particular branch 
of a more general science, and that the conception of space is a particular 
variety of a wider and more general conception. This wider conception, of 
which time and space are particular varicties, it has been proposed to denote 
by the term manifoldness. Whenever a general notion is susceptible of a 
variety of specializations, the aggregate of all such specializations is called 
amanifoldness. Thus space is the aggregate of all points, and each point 
is a specialization of the general notion of position. In the same way, time 
is the aggregate of all instants, and each instant is a specialization of the 
general notion of position in time. Space and time are, in fact, of all 
manifoldnesses the ones with which we are by far the most frequently con- 
cerned. Now there is an important feature in which these two manifold- 
nesses agree. ‘They are both of them of such a nature that no limit can be 
conceived to their divisibility. However near together two points in space 
may be, we can always conceive the existence of intermediate points, and 
the same thing holds in regard to time. Mathematicians express this fact by 
saying that space and time are continuous manifoldnesses. But there is 
another feature, equally important with the foregoing, in regard to which 
space and time are strikingly contrasted. If we wish to travel away from 
any particular instant in time, there are only two directions in which we 
can set out. We must either ascend or descend the stream. But from a 
point in space we can set out in an infinite number of directions. This diffe- 
rence is expressed by saying that time is a manifoldness of one dimension, and 
that space is a manifoldness of more than one dimension. An aggregate of 
points in which we could only travel backwards or forwards, would be, not 
solid space, but a line. A line therefore is a manifoldness of one dimension. 
A surface, again, may be regarded as an aggregate of lines; and it is an ag- 
gregate of such a nature that if we wish to travel away from a particular 
Il 
