These are determinations we may make without having in advance any notion about form or about the metric properties of space. They in no wise bear on the 'geometric properties of bodies.' And these determinations will not be possible if the bodies experimented upon move in accordance with a group having the same structure as the Lobachevskian group (I mean according to the same laws as solid bodies in Lobachevski's geometry). They suffice therefore to prove that these bodies move in accordance with the Euclidean group, or at least that they do not move according to the Lobachevskian group. That they are compatible with the Euclidean group is easy to see. For they could be made if the body [alpha][beta][gamma] was a rigid solid of our ordinary geometry presenting the form of a right-angled triangle, and if the points _ABCDEFGH_ were the summits of a polyhedron formed of two regular hexagonal pyramids of our ordinary geometry, having for common base _ABCDEF_ and for apices the one _G_ and the other _H_. Suppose now that in place of the preceding determination it is observed that as above [alpha][beta][gamma] can be successively applied to _AGO_, _BGO_, _CGO_, _DGO_, _EGO_, _AHO_, _BHO_, _CHO_, _DHO_, _EHO_, _FHO_, then that [alpha][beta] (and no longer [alpha][gamma]) can be successively applied to _AB_, _BC_, _CD_, _DE_, _EF_ and _FA_. These are determinations which could be made if non-Euclidean geometry were true, if the bodies [alpha][beta][gamma] and _OABCDEFGH_ were rigid solids, and if the first were a right-angled triangle and the second a double regular hexagonal pyramid of suitable dimensions. Therefore these new determinations are not possible if the bodies move according to the Euclidean group; but they become so if it be supposed that the bodies move according to the Lobachevskian group. They would suffice, therefore (if one made them), to prove that the bodies in question do not move according to the Euclidean group. Thus, without making any hypothesis about form, about the nature of space, about the relations of bodies to space, and without attributing to bodies any geometric property, I have made observations which have enabled me to show in one case that the bodies experimented upon move according to a group whose structure is Euclidean, in the other case that they move according to a group whose structure is Lobachevskian. And one may not say that the first aggregate of determinations would constitute an experiment proving that space is Euclidean, and the second an experiment proving that space is non-Euclidean. In fact one could imagine (I say imagine) bodies moving so as to render possible the second series of determinations. And the proof is that the first mechanician met could construct such bodies if he cared to take the pains and make the outlay. You will not conclude from that, however, that space is non-Euclidean.