TENDENCIES IN MATHEMATICAL SCIENCE 339 



would be profoundly modified by any considerable inaccuracy in the 

 hypothesis that our actual space is Euclidean. The observed truth 

 of physical science, therefore, constitutes overwhelming empirical 

 evidence that this hypothesis is very approximately correct, even 

 if not rigidly true. Russell. 



The most suggestive and notable achievement of the last century 

 is the discovery of Non-Euclidean geometry. Hilbert. 



What Vesalius was to Galen, what Copernicus was to Ptolemy, 

 that was Lobatchewski to Euclid. There is, indeed, a somewhat 

 instructive parallel between the last two cases. Copernicus and 

 Lobatchewski were both of Slavic origin. Each of them has brought 

 about a revolution in scientific ideas so great that it can only be com- 

 pared with that wrought by the other. And the reason of the trans- 

 cendent importance of these two changes is that they are changes 

 in the conception of the Cosmos. . . . And in virtue of these two 

 revolutions the idea of the Universe, the Macrocosm, the All, as 

 subject of human knowledge, and therefore of human interest, has 

 fallen to pieces. Clifford. 



Geometrical axioms are neither synthetic a priori conclusions 

 nor experimental facts. They are conventions : our choice, amongst 

 all possible conventions, is guided by experimental facts ; but it 

 remains free, and is only limited by the necessity of avoiding all 

 contradiction. ... In other words, axioms of geometry are only 

 definitions in disguise. That being so what ought one to think of 

 this question : Is the Euclidean Geometry true ? The question is 

 nonsense. One might as well ask whether the metric system is 

 true and the old measures false ; whether Cartesian co-ordinates are 

 true and polar co-ordinates false. Poincart. 



To make non-Euclidean geometry intelligible to laymen the 

 following illustration has been given by Helmholtz : 



Think of the image of the world in a convex mirror. ... A 

 well-made convex mirror of moderate aperture represents the objects 

 in front of it as apparently solid and in fixed positions behind its 

 surface. But the images of the distant horizon and of the sun in the 

 sky lie behind the mirror at a limited distance, equal to its focal 

 length. Between these and the surface of the mirror are found the 

 images of all the other objects before it, but the images are diminished 



