176 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1944 
Schwarzschild’s results need not be considered here because his data 
were limited and because we have at present more detailed modes of 
procedure than he used. There are at least two mathematicians who 
have achieved the unique distinction of having a universe named 
after them. They are Einstein, and a Dutchman named de Sitter. 
Both universes are non-Euclidean and the Einstein universe appears 
to be the more popular. The curvature of the Einstein universe is 
determined by the amount of matter in it, and if it is not a static 
universe, by certain other factors. A chunk of matter produces quite 
a large local curvature, which is evidenced to us by what we call 
gravitational attraction. 
This universe is not infinite in extent. It is a closed universe with 
a finite volume but having no boundaries, just as the surface of a 
sphere is a closed surface of finite area yet has no bounding edges. 
In this universe one might expect to see a star in two directions, first 
by looking directly at it, second, by looking in the exactly opposite 
direction at light rays which have gone completely around the circuit 
of the universe in the opposite direction. Star images have not been 
seen in this way, possibly because their light is too faint after the 
long trip around the universe. There is also the possibility that the 
theory is wrong. It has, however, been seriously suggested that two 
very faint nebulae, observed in a certain direction, may actually be 
the backs of two of our nearest neighbors, as seen the long way 
around. 
The theory of a finite, closed universe is very attractive in many 
respects. We may again use the term “intellectually satisfactory” 
in this connection, largely because this universe can be given a concise 
mathematical description and in terms that explain the gravitational 
effects of matter. There is also, in many individuals, a. definite 
repugnance to the idea of infinite space. In discussing the stars 
Kant, in 1755, says, “There is here no end, but an abyss of real 
immensity in presence of which all the capability of human concep- 
tion sinks exhausted.” The finite mind likes to set up a blank wall 
somewhere, in order to end it all. It is probably intellectually satis- 
factory to know that one can start out in imagination and not have 
to get farther away forever and ever, but will eventually get back to 
the good, old, familiar region of the starting point. 
With this picture of a finite, closed universe in mind we may now 
return to the question regarding the nebulae. Why should they 
appear to be crowded together at great distances from us? The 
answer might be that the curvature of space appears to make them 
crowd into smaller and smaller volumes as their distance increases. 
If this is true it is possible to calculate what radius of curvature of 
the universe would give the observed apparent crowding of the 
