STRUCTURE OF THE UNIVERSE—HEAPS 175 
a good way, but like other escapist philosophies it must be consid- 
ered and estimated for what it is worth. It involves spatial curvature. 
The idea of curved space is now quite a familiar idea to most 
people. Eddington, Jeans, Einstein, and others have written books 
for popular consumption and the sales have been very gratifying. 
Even the pulp magazines.do not hesitate to invoke the fourth dimen- 
sion as a mode of escape for the hero or the villian. A simple way 
of approaching the concept of spatial curvature is as follows. Think 
of a straight line along one dimension. Given a second dimension 
at right angles to the first, then we have the possibility of curving the 
line into the second dimension. Think of a plane surface, like a sheet 
of paper flat on a desk. Given a third dimension, at right angles 
to the desk, we have the possibility of curving the paper sheet into 
this third dimension. Think of a solid filling three dimensions. Give 
a fourth dimension at right angles to the other three, we then have 
a possibility of curving the solid into the fourth dimension. It is 
only because we have three-dimensional minds that we cannot see 
this fourth dimension. 
A mathematician may speak of space itself as being curved without 
reference to any solid matter in it. For example, consider the earth 
to be perfectly smooth. If we were two-dimensional creatures instead 
of being three-dimensional, we might draw a big circle on the earth’s 
surface, measure its diameter and its circumference, and then find 
that the circumference was not equal to z times the diameter. We 
would not know that the circle was not flat (since we are assumed 
to be two-dimensional), but we could certainly infer a curvature 
of our flat space and even determine its radius if we knew enough 
about ordinary Euclidean geometry, which would work pretty well 
for small circles on the earth’s surface. 
The mathematical description of the universe to which allusion 
was made at the beginning of the lecture involved curving of three- 
dimensional space in somewhat the same fashion as described above 
for the two-dimensional space. If space actually is curved in this 
way our ordinary solid geometry, Euclidean geometry, would not 
be quite correct. In order to find out whether it is correct, measure- 
ments of certain kinds must be made. For example, if a negative 
parallax could ever be observed for a single star, a spherically curved 
space would be implied. The mathematician Schwarzschild, a good 
many years ago, attempted to find what curvature of space would 
be possible according to certain types of non-Euclidean geometry. 
In dealing with these geometries he said, “One there finds oneself, 
if one but will, in a geometrical fairyland, but the beauty of this 
fairy tale is that one does not know but that it may come true.” 
