SPACE 669 



metry). They therefore suffice to prove that these bodies move ac- 

 cording to the Euclidean group; or at least that they do not move 

 according to the Lobatschewskian group. That they may be com- 

 patible with the Euclidean group is easily seen; for we might make 

 them so if the body a(3y were an invariable solid of our ordinary geo- 

 metry in the shape of a right-angled triangle and if the points 

 abcdefgh were the vertices of the polyhedron formed of two regular 

 hexagonal pyramids of our ordinary geometry having abcdef 

 as their common base, and having the one g and the other h as their 

 vertices. Suppose now, instead of the previous observations, we note 

 that we can as before apply a/3y successively to ago, bgo, ego, dgo, ego, 

 igo, oho, bho, clw, dho, eho, fho, and then that we can apply a/3 (and 

 no longer ay) successively to ab, be, cd, de, cf, and fa. These are 

 observations that could be made if non-Euclidean geometry were true, 

 if the bodies a(3y, oabcdefgh were invariable solids, if the former 

 were a right-angled triangle, and the latter a double regular hexagonal 

 pyramid of suitable dimensions. These new verifications are there- 

 fore impossible if the bodies move according to the Euclidean group ; 

 but they become possible if we suppose the bodies to move according 

 to the Lobatschewskian group. They would therefore suffice to show, 

 if we carried them out, that the bodies in question do not move ac- 

 cording to the Euclidean group. And so, without making any hypo- 

 thesis on the form and the nature of space, on the relations of the 

 bodies and space, and without attributing to bodies any geometrical 

 property, I have made observations which have enabled me to show 

 in one case that the bodies experimented upon move according to 

 a group, the structure of which is Euclidean, and in the other case that 

 they move in a group, the structure of which is Lobatschewskian. It 

 cannot be said that all the first observations would constitute an exper- 

 iment proving that space is Euclidean, and the second an experiment 

 proving space is non-Euclidean: in fact, it might be imagined (note 

 that I use the word imagined) that there are bodies moving in such a 

 manner as to render possible the second series of observations: and 

 the proof is that the first mechanic who came our way could con- 

 struct it if he would only take the trouble. But you must not con- 

 clude, however, that space is non-Euclidean. In the same way, just 

 as ordinary solid bodies would continue to exist when the mechanic 

 had constnicted the strange bodies I have just mentioned, he would 

 have to conclude that space is both Euclidean and non-Euclidean. 

 Suppose, for instance, that we have a large sphere of radius E, and 

 that its temperature decreases from the centre to the surface of the 

 sphere according to the law of which I spoke when I was describing 

 the non-Euclidean world. We might have bodies whose dilatation is 

 negligible, and which would behave as ordinary invariable solids; 

 and, on the other hand, we might have very dilatable bodies, which 



