Changes of State and Changes of Position

If the concept of geometrical space is not imposed upon our minds, and if, on the other hand, none of our sensations can furnish us with that concept, how then did it ever come into existence?

None of our sensations, if isolated, could have brought us to the concept of space; we are brought to it solely by studying the laws by which those sensations succeed one another.

We see at first that our impressions are subject to change.

But among the changes that we ascertain, we are very soon led to make a distinction. Sometimes we say that the objects, the causes of these impressions, have changed their state, sometimes that they have changed their position, that they have only been displaced.

Whether an object changes its state or only its position, this is always translated for us in the same manner, by a modification in an aggregate of impressions. How then have we been enabled to distinguish them?

If there were only change of position, we could restore the primitive aggregate of impressions by making movements which would confront us with the movable object in the same relative situation. We thus correct the modification which was produced, and we re-establish the initial state by an inverse modification.

If, for example, it were a question of the sight, and if an object be displaced before our eyes, we can “follow it with the eye,” and retain its image on the same point of the retina by appropriate movements of the eyeball.

These movements we are conscious of because they are voluntary, and because they are accompanied by muscular sensations.

But that does not mean that we represent them to ourselves in geometrical space. So what characterises change of position, what distinguishes it from change of state, is that it can always be corrected by this means. It may therefore happen that we passspace and geometry.

1. Involuntarily and without experiencing muscular sensations—which happens when it is the object that is displaced
2. Voluntarily, and with muscular sensation—which happens when the object is motionless, but when we displace ourselves in such a way that the object has relative motion with respect to us.

If this be so, the translation of the aggregate A to the aggregate B is only a change of position. It follows that sight and touch could not have given us the idea of space without the help of the “muscular sense.”

Not only could this concept not be derived from a single sensation, or even from a series of sensations; but a motionless being could never have acquired it, because, not being able to correct by his movements the effects of the change of position of external objects, he would have had no reason to distinguish them from changes of state.

Nor would he have been able to acquire it if his movements had not been voluntary, or if they were unaccompanied by any sensations whatever.

Conditions of Compensation

How is such a compensation possible in such a way that two changes, otherwise mutually independent, may be reciprocally corrected?

A mind already familiar with geometry would reason as follows:—If there is to be compensation, the different parts of the external object on the one hand, and the different organs of our senses on the other, must be in the same relative position after the double change.

For that to be the case, the different parts of the external body on the one hand, and the different organs of our senses on the other, must have the same relative position to each other after the double change; and so with the different parts of our body with respect to each other.

In other words, the external object in the first change must be displaced as an invariable solid would be displaced, and it must also be so with the whole of our body in the second change, which is to correct the first. Under these conditions compensation may be produced.

But we who as yet know nothing of geometry, whose ideas of space are not yet formed, we cannot reason in this way—we cannot predict à priori if compensation is possible. But experiment shows us that it sometimes does take place, and we start from this experimental fact in order to distinguish changes of state from changes of position.

Solid Bodies and Geometry.—Among surrounding objects there are some which frequently experience displacements that may be thus corrected by a correlative movement of our own body—namely, solid bodies.

The other objects, whose form is variable, only in exceptional circumstances undergo similar displacement (change of position without change of form). When the displacement of a body takes place with deformation, we can no longer by appropriate movements place the organs of our body in the same relative situation with respect to this body; we can no longer, therefore, reconstruct the primitive aggregate of impressions.

It is only later, and after a series of new experiments, that we learn how to decompose a body of variable form into smaller elements such that each is displaced approximately according to the same laws as solid bodies.

We thus distinguish “deformations” from other changes of state.

In these deformations each element undergoes a simple change of position which may be corrected; but the modification of the aggregate is more profound, and can no longer be corrected by a correlative movement.

Such a concept is very complex even at this stage, and has been relatively slow in its appearance. It would not have been conceived at all had not the observation of solid bodies shown us beforehand how to distinguish changes of position.

If, then, there were no solid bodies in nature there would be no geometry.

Suppose a solid body to occupy successively the positions α and β. In the first position, it will give us an aggregate of impressions A, and in the second position the aggregate of impressions B. Now let there be a second solid body, of qualities entirely different from the first—of different colour, for instance.

Assume it to pass from the position α, where it gives us the aggregate of impressions A 0 to the position β, where it gives the aggregate of impressions B 0.

In general, the aggregate A will have nothing in common with the aggregate A 0 , nor will the aggregate B have anything in common with the aggregate B 0.

The transition from the aggregate A to the aggregate B, and that of the aggregate A 0 to the aggregate B 0 , are therefore two changes which in themselves have in general nothing in common. Yet we consider both these changes as displacements; and, further, we consider them the same displacement.

How can this be? It is simply because they may be both corrected by the same correlative movement of our body. “Correlative movement,” therefore, constitutes the sole connection between two phenomena which otherwise we should never have dreamed of connecting.

On the other hand, our body, thanks to the number of its articulations and muscles, may have a multitude of different movements, but all are not capable of “correcting” a modification of external objects.

Those alone are capable of it in which our whole body, or at least all those in which the organs of our senses enter into play are displaced en bloc—i.e., without any variation of their relative positions, as in the case of a solid body.

To sum up:

1. We distinguish 2 categories of phenomena:

• involuntary: these are unaccompanied by muscular sensations, and attributed to external objects—they are external changes;
• voluntary the second, of opposite character and attributed to the movements of our own body, are internal changes.

2. We notice that certain changes of each in these categories may be corrected by a correlative change of the other category.

3. We distinguish among external changes those that have a correlative in the other category—which we call displacements; and in the same way we distinguish among the internal changes those which have a correlative in the first category.

Thus by means of this reciprocity is defined a particular class of phenomena called displacements. The laws of these phenomena are the object of geometry.