'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.