To know the height of the mainmast does not suffice for calculating the age of the captain. When you have measured every bit of wood in the ship you will have many equations, but you will know his age no better. All your measurements bearing only on your bits of wood can reveal to you nothing except concerning these bits of wood. Just so your experiments, however numerous they may be, bearing only on the relations of bodies to one another, will reveal to us nothing about the mutual relations of the various parts of space.
7. Will you say that if the experiments bear on the bodies, they bear at least upon the geometric properties of the bodies? But, first, what do you understand by geometric properties of the bodies? I assume that it is a question of the relations of the bodies with space; these properties are therefore inaccessible to experiments which bear only on the relations of the bodies to one another. This alone would suffice to show that there can be no question of these properties.
Still let us begin by coming to an understanding about the sense of the phrase: geometric properties of bodies. When I say a body is composed of several parts, I assume that I do not enunciate therein a geometric property, and this would remain true even if I agreed to give the improper name of points to the smallest parts I consider.
When I say that such a part of such a body is in contact with such a part of such another body, I enunciate a proposition which concerns the mutual relations of these two bodies and not their relations with space.
I suppose you will grant me these are not geometric properties; at least I am sure you will grant me these properties are independent of all knowledge of metric geometry.
This presupposed, I imagine that we have a solid body formed of eight slender iron rods, OA, OB, OC, OD, OE, OF, OG, OH, united at one of their extremities O. Let us besides have a second solid body, for example a bit of wood, to be marked with three little flecks of ink which I shall call α, β, γ. I further suppose it ascertained that αβγ may be brought into contact with AGO (I mean α with A, and at the same time β with G and γ with O), then that we may bring successively into contact αβγ with BGO, CGO, DGO, EGO, FGO, then with AHO, BHO, CHO, DHO, EHO, FHO, then αγ successively with AB, BC, CD, DE, EF, FA.
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 αβγ 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 αβγ can be successively applied to AGO, BGO, CGO, DGO, EGO, AHO, BHO, CHO, DHO, EHO, FHO, then that αβ (and no longer αγ) 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 αβγ 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.