SPIRAL NEBULA WIRTZ 141 



reason. The understanding can Avell conceive what our imagina- 

 tion can not picture. That is the case here. Man, bodily con- 

 structed in three dimensions, can not picture things in four or more 

 dimensions. Nevertheless, the following conceit may help us better 

 to understand the space of de Sitter. 



If we eliminate two dimensions of a four-dimensional space, the 

 latter reduces to a surface. But what is the character of this surface? 



We can take in at a glance the phenomena which could be under- 

 stood by a being of two dimensions living on the surface of a sphere. 

 From these phenomena he could infer the mathematical characteris- 

 tics of his cosmical space. This being would recognize the extended 

 but not infinite character of his sphere of existence. Further, he 

 would recognize something which is really the radius of the three- 

 dimensional sphere upon whose surface he is constrained to exist 

 and which might become an object for research. It is the radius 

 of curvature of his w^orld space. He can talk of it, he can determine 

 a linear measure of it. A spatial representation in three dimen- 

 sions will never be possible to him who can know only two dimen- 

 sions. The two-dimensional observer, carrying out certain direct 

 measures upon this surface-of-a-sj)here limited universe, will find 

 that the value of his determinations of the number tt will not be 

 constant. It will be found that it is dependent upon the greatness of 

 the circle, that it is always less than 3.14159, and that the value of tt 

 which he finally decides upon will be the maximum limit toward 

 which all his values tend to approach. 



The strange cosmos of de Sitter, reduced to tw^o dimensions, 

 does not become such a simple surface-of-a-sphere world but one 

 lying on a hyperbola of revolution. Our physicist, confined upon 

 the surface of this hyperboloid, can make certain observations com- 

 parable to those of our real world. For instance, he will find that 

 the number of Archimedes is always less than our value of ir. It may 

 be mentioned as remarkable that all the phenomena of probability 

 from which we may determine a value of tt will lead to a value which 

 is greater than the value found from Euclidean geometry. Taken by 

 itself this is possibly only a curiosity and might signify little. 

 Nevertheless certain modern astronomical phenomena can be given 

 an interpretation which points the same way. 



This phenomenon occurs in de Sitter's world, although its exist- 

 ence was not known at the time he constructed his cosmology (1917). 

 With increasing distance from the center of his coordinate system, 

 natural phenomena occur slower and slower, and of course all natural 

 clocks go likewise slower and slower. Especially is this true with 

 atomic vibrations. This decrease in vibration time can be measured 

 for it will be impressed upon the light coming to us. It means that 



