mind already familiar with geometry would reason as follows: Evidently, if there is to be compensation, the various parts of the external object, on the one hand, and the various sense organs, on the other hand, must be in the same relative position after the double change. And, for that to be the case, the various parts of the external object must likewise have retained in reference to each other the same relative position, and the same must be true of the various parts of our body in regard to each other.
In other words, the external object, in the first change, must be displaced as is a rigid solid, and so must it be with the whole of our body in the second change which corrects the first.
Under these conditions, compensation may take place.
But we who as yet know nothing of geometry, since for us the notion of space is not yet formed, we can not reason thus, we can not foresee a priori whether compensation is possible. But experience teaches us that it sometimes happens, and it is from this experimental fact that we start to distinguish changes of state from changes of position.
Solid Bodies and Geometry.—Among surrounding objects there are some which frequently undergo displacements susceptible of being thus corrected by a correlative movement of our own body; these are the solid bodies. The other objects, whose form is variable, only exceptionally undergo like displacements (change of position without change of form). When a body changes its place and its shape, we can no longer, by appropriate movements, bring back our sense-organs into the same relative situation with regard to this body; consequently we can no longer reestablish the primitive totality of impressions.
It is only later, and as a consequence of new experiences, that we learn how to decompose the bodies of variable form into smaller elements, such that each is displaced almost in accordance with the same laws as solid bodies. Thus we distinguish 'deformations' from other changes of state; in these deformations, each element undergoes a mere change of position, which can be corrected, but the modification undergone by the aggregate is more profound and is no longer susceptible of correction by a correlative movement.
Such a notion is already very complex and must have been relatively late in appearing; moreover it could not have arisen if the observation of solid bodies had not already taught us to distinguish changes of position.
Therefore, if there were no solid bodies in nature, there would be no geometry.
Another remark also deserves a moment's attention. Suppose a solid body to occupy successively the positions α and β; in its first position, it will produce on us the totality of impressions A, and in its second position the totality of impressions B. Let there be now a second solid body, having qualities entirely different from the first, for example, a different color. Suppose it to pass from the position α, where it gives us the totality of impressions A´, to the position β, where it gives the totality of impressions B´.
In general, the totality A will have nothing in common with the totality A´, nor the totality B with the totality B´. The transition from the totality A to the totality B and that from the totality A´ to the totality B´ are therefore two changes which in themselves have in general nothing in common.
And yet we regard these two changes both as displacements and, furthermore, we consider them as the same displacement. How can that be?
It is simply because they can both be corrected by the same correlative movement of our body.
'Correlative movement' therefore constitutes the sole connection between two phenomena which otherwise we never should have dreamt of likening.
On the other hand, our body, thanks to the number of its articulations and muscles, may make a multitude of different movements; but all are not capable of 'correcting' a modification of external objects; only those will be capable of it in which our whole body, or at least all those of our sense-organs which come into play, are displaced as a whole, that is, without their relative positions varying, or in the fashion of a solid body.
To summarize:
1º We are led at first to distinguish two categories of phenomena:
Some, involuntary, unaccompanied by muscular sensations, are attributed by us to external objects; these are external changes;
Others, opposite in character and attributed by us to the movements of our own body, are internal changes;
2º We notice that certain changes of each of these categories may be corrected by a correlative change of the other category;
3º We distinguish among external changes those which have thus a correlative in the other category; these we call displacements; and just so among the internal changes, we distinguish those which have a correlative in the first category.
Thus are defined, thanks to this reciprocity, a particular class of phenomena which we call displacements.
The laws of these phenomena constitute the object of geometry.
Law of Homogeneity.—The first of these laws is the law of homogeneity.
Suppose that, by an external change α, we pass from the totality of impressions A to the totality B, then that this change α is corrected by a correlative voluntary movement β, so that we are brought back to the totality A.
Suppose now that another external change α´ makes us pass anew from the totality A to the totality B.
Experience teaches us that this change α´ is, like α, susceptible of being corrected by a correlative voluntary movement β´ and that this movement β´ corresponds to the same muscular sensations as the movement β which corrected α.
This fact is usually enunciated by saying that space is homogeneous and isotropic.
It may also be said that a movement which has once been produced may be repeated a second and a third time, and so on, without its properties varying.
In the first chapter, where we discussed the nature of mathematical reasoning, we saw the importance which must be attributed to the possibility of repeating indefinitely the same operation.
It is from this repetition that mathematical reasoning gets its power; it is, therefore, thanks to the law of homogeneity, that it has a hold on the geometric facts.
For completeness, to the law of homogeneity should be added a multitude of other analogous laws, into the details of which I do not wish to enter, but which mathematicians sum up in a word by saying that displacements form 'a group.'
The Non-Euclidean World.—If geometric space were a frame imposed on each of our representations, considered individually, it would be impossible to represent to ourselves an image stripped of this frame, and we could change nothing of our geometry.
But this is not the case; geometry is only the résumé of the laws according to which these images succeed each other. Nothing then prevents us from imagining a series of representations, similar in all points to our ordinary representations, but succeeding one another according to laws different from those to which we are accustomed.
We can conceive then that beings who received their education in an environment where these laws were thus upset might have a geometry very different from ours.
Suppose,
