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 [alpha] and [beta]; 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 [alpha], where it gives us the totality of impressions _A'_, to the position [beta], 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.