Experiments only teach us the relations of bodies

6. Experiments only teach us the relations of bodies to one another.

They do not and cannot give us the relations of bodies and space, nor the mutual relations of the different parts of space.

“Yes!” you reply, “a single experiment is not enough, because it only gives us one equation with several unknowns; but when I have made enough experiments I shall have enough equations to calculate all my unknowns.” If I know the height of the main-mast, that is not sufficient to enable me to calculate the age of the captain.

When you have measured every fragment of wood in a ship you will have many equations, but you will be no nearer knowing the captain’s age.

All your measurements bearing on your fragments of wood can tell you only what concerns those fragments;

Similarly, your experiments, however numerous they may be, referring only to the relations of bodies with one another, will tell you nothing about the mutual relations of the different parts of space.

7. Will you say that if the experiments have reference to the bodies, they at least have reference to the geometrical properties of the bodies.

First, what do you understand by the geometrical properties of bodies?

I assume that it is a question of the relations of the bodies to space.

These properties therefore are not reached by experiments which only have reference to the relations of bodies to one another, and that is enough to show that it is not of those properties that there can be a question.

Let us therefore begin by making ourselves clear as to the sense of the phrase: geometrical properties of bodies.

When I say that a body is composed of several parts, I presume that I am thus enunciating a geometrical property, and that will be true even if I agree to give the improper name of points to the very small parts I am considering.

When I say that this or that part of a certain body is in contact with this or that part of another body, I am enunciating a proposition which concerns the mutual relations of the two bodies, and not their rela-experiment and geometry.

Admitting this, I suppose that we have a solid body formed of 8 thin iron rods, oa, ob, oc, od, oe, of , og, oh, connected at one of their extremities, o. And let us take a second solid body—for example, a piece of wood, on which are marked three little spots of ink which I shall call α β γ.

I now suppose that we find that we can bring into contact αβγ with ago; by that I mean α with a, and at the same time β with g, and γ with o. Then we can successively bring into contact αβγ with bgo, cgo, dgo, ego, fgo, then with aho, bho, cho, dho, eho, f ho;

And then αγ successively with ab, bc, cd, de, ef , f a.

These are observations that can be made without having any idea beforehand as to the form or the metrical properties of space. They have no reference whatever to the “geometrical properties of bodies.”

These observations will not be possible if the bodies on which we experiment move in a group having the same structure as the Lobatschewskian group (I mean according to the same laws as solid bodies in Lobatschewsky’s geometry).

They therefore suffice to prove that these bodies move according to the Euclidean group; or at least that they do not move according to the Lobatschewskian group. That they may be compati- ble with the Euclidean group is easily seen; for we might make them so if the body αβγ were an invariable solid of our ordinary geometry in the shape of a right-angled triangle, and if the points abcdef gh 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 αβγ successively to ago, bgo, cgo, dgo, ego, f go, aho, bho, cho, dho, eho, f ho, and then that we can apply αβ (and no longer αγ) successively to ab, bc, cd, de, ef , and f a. These are observations that could be made if non-Euclidean geometry were true. If the bodies αβγ, oabcdef gh 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 theexperiment and geometry.

It cannot be said that all the first observations would constitute an experiment proving that space is Euclidean, and the second an experiment proving that 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 construct it if he would only take the trouble.

But you must not conclude, however, that space is non-Euclidean. In the same way, just as ordinary solid bodies would continue to exist when the mechanic had constructed the strange bodies I have just mentioned, he would have to conclude that space is both Euclidean and non-Euclidean. Suppose, for in- stance, that we have a large sphere of radius R, 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 would behave as non-Euclidean solids.

We might have two double pyramids oabcdef gh and o 0 a 0 b 0 c 0 d 0 e 0 f 0 g 0 h 0 , and two tri- angles αβγ and α 0 β 0 γ 0 .

The first double pyramid would be rectilinear, and the second curvilinear. The trian- gle αβγ would consist of undilatable matter, and the other of very dilatable matter. We might therefore make our first observations with the double pyramid o 0 a 0 h 0 and the triangle α 0 β 0 γ 0 .

And then the experiment would seem to show—first, that Euclidean geometry is true, and then that it is false. Hence, experiments have reference not to space but to bodies.