A 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.