to the Lobatschewskian group. That they may be compatible with the Euclidean group is easily seen; for we might make them so if the body α β γ were an invariable solid of our ordinary geometry in the shape of a right-angled triangle, and if the points abcdefgh were the vertices of a 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 α β γ successively to ago, bgo, cgo, dgo, ego, fgo, aho, bho, cho, dho, eho, fho, and then that we can apply αβ (and no longer α γ) successively to ab, bc, cd, de, ef, and fa. These are observations that could be made if non-Euclidean geometry were true. If the bodies α β γ, 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 therefore 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 according to the Euclidean group. And so, without making any hypothesis 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
