236 THE MECHANISM OF LIFE 



continued somehow in a straight line, T. Now this does not 

 describe the universe when we take account of relative positions 

 and motions. Somehow or other the co-ordinates must be 

 curved^ and to allow for this we have to introduce a fifth dimen- 

 sion. This is " quite all right," and the mathematics of a five- 

 dimensional geometry are as straightforward (though immensely 

 more difficult) as are those of three dimensions. The trouble is- 

 that we cannot visualise five dimensions. Still, the mathe- 

 matical results are there — a fact of very great significance, for an 

 abstract result of this kind always means possible action. 

 Qiven any abstruse mathematical result, and we may be pretty 

 certain that by-and-by it will mean something " real " — ^that is^ 

 something that practical science, and by-and-by even industry,, 

 will do. 



S3 we live in a non- Euclidean universe of four dimensions 

 which is "in" a continuance of a fifth dimension. Leave it at- 

 that and see what are the consequences. 



The consequences (for our speculations of Chapter XI.) are that 

 space — cosmic space, that is — is " finite but unbounded." Think 

 about this by analogy: the surface of the earth to us is finite, 

 but unbounded. One can go on travelling over it in " straight "" 

 lines (geodesies) without being brought up anywhere ; there 

 would be no end to our journeying if we could live for ever. We 

 might come back to the same place from which we started, though 

 we should be travelling in a " straight " line, but even then we 

 could still go on in the same direction. 



Now extend that to cosmic space, and suppose that it, too, is 

 finite and unbounded. We are stepping out of non-Euclidean, 

 two-dimensional, curved space into non- Euclidean, four-dimen- 

 sional, curved space. This means that the universe is finite but 

 unboimded. We can go on through it with the velocity of light 

 for ever, and always in the same direction. There is no end to 

 our journey. Perhaps we might come back to the starting-place, 

 perhaps not ; if we did, we should still be facing the same way. 

 It means that the galactic universe is finite (and Einstein even 

 tells us how big it is). Outside it there is nothing, not even 

 space. Light travels in what we call curved lines, and never 

 goes outside the universe from which it originated. 



The notion is of immense speculative importance. 



Suppose, though, that the time dimension is also curved I 

 Einstein does not seem to have worked out that. Perhaps the 

 reader may see to what extraordinary speculations this may 

 lead. 



