Why now have all these spaces three dimensions? Go back to the "table of distribution" of which we have spoken. We have on the one side the list of the different possible dangers; designate them by _A1_, _A2_, etc.; and, on the other side, the list of the different remedies which I shall call in the same way _B1_, _B2_, etc. We have then connections between the contact studs or push buttons of the first list and those of the second, so that when, for instance, the announcer of danger _A3_ functions, it will put or may put in action the relay corresponding to the parry _B4_.
As I have spoken above of centripetal or centrifugal wires, I fear lest one see in all this, not a simple comparison, but a description of the nervous system. Such is not my thought, and that for several reasons: first I should not permit myself to put forth an opinion on the structure of the nervous system which I do not know, while those who have studied it speak only circumspectly; again because, despite my incompetence, I well know this scheme would be too simplistic; and finally because on my list of parries, some would figure very complex, which might even, in the case of extended space, as we have seen above, consist of many steps followed by a movement of the arm. It is not a question then of physical connection between two real conductors but of psychologic association between two series of sensations.
If _A1_ and _A2_ for instance are both associated with the parry _B1_, and if _A1_ is likewise associated with the parry _B2_, it will generally happen that _A2_ and _B2_ will also themselves be associated. If this fundamental law were not generally true, there would exist only an immense confusion and there would be nothing resembling a conception of space or a geometry. How in fact have we defined a point of space. We have done it in two ways: it is on the one hand the aggregate of announcers _A_ in connection with the same parry _B_; it is on the other hand the aggregate of parries _B_ in connection with the same announcer _A_. If our law was not true, we should say _A1_ and _A2_ correspond to the same point since they are both in connection with _B1_; but we should likewise say they do not correspond to the same point, since _A1_ would be in connection with _B2_ and the same would not be true of _A2_. This would be a contradiction.
But, from another side, if the law were rigorously and always true, space would be very different from what it is. We should have categories strongly contrasted between which would be portioned out on the one hand the announcers _A_, on the other hand the parries _B_; these categories would be excessively numerous, but they would be entirely separated one from another. Space would be composed of points very numerous, but discrete; it would be _discontinuous_. There would be no reason for ranging these points in one order rather than another, nor consequently for attributing to space three dimensions.
But it is not so; permit me to resume for a moment the language of those who already know geometry; this is quite proper since this is the language best understood by those I wish to make understand me.
When I desire to parry the stroke, I seek to attain the point whence comes this blow, but it suffices that I approach quite near. Then the parry _B1_ may answer for _A1_ and for _A2_, if the point which corresponds to _B1_ is sufficiently near both to that corresponding to _A1_ and to that corresponding to _A2_. But it may happen that the point corresponding to another parry _B2_ may be sufficiently near to the point corresponding to A1 and not sufficiently near the point corresponding to _A2_; so that the parry _B2_ may answer for _A1_ without answering for _A2_. For one who does not yet know geometry, this translates itself simply by a derogation of the law stated above. And then things will happen thus:
Two parries _B1_ and _B2_ will be associated with the same warning _A1_ and with a large number of warnings which we shall range in the same category as _A1_ and which we shall make correspond to the same point of space. But we may find warnings _A2_ which will be associated with _B2_ without being associated with _B1_, and which in compensation will be associated with _B3_, which _B3_ was not associated with _A1_, and so forth, so that we may write the series
_B1_, _A1_, _B2_, _A2_, _B3_, _A3_, _B4_, _A4_,
where each term is associated with the following and the preceding, but not with the terms several places away.
Needless to add that each of the terms of these series is not isolated, but forms part of a very numerous category of other warnings or of other parries which have the same connections as it, and which may be regarded as belonging to the same point of space.
The fundamental law, though admitting of exceptions, remains therefore almost always true. Only, in consequence of these exceptions, these categories, in place of being entirely separated, encroach partially one upon another and mutually penetrate in a certain measure, so that space becomes continuous.
On the other hand, the order in which these categories are to be ranged is no longer arbitrary, and if we refer to the preceding series, we see it is necessary to put _B2_ between _A1_ and _A2_ and consequently between _B1_ and _B3_ and that we could not for instance put it between _B3_ and _B4_.
There is therefore an order in which are naturally arranged our categories which correspond to the points of space, and experience teaches us that this order presents itself under the form of a table of triple entry, and this is why space has three dimensions.
