“If a child contemplates his hand, he is conscious of its existence in a double manner—­in the first place by its tangibility, the second by its image on the retina of his eye.  By repeated groping about and touching, the child knows by experience that his hand retains the same form and extension through all the variations of distance and position under which it is observed, notwithstanding that the form and extension of the image on the retina constantly change with the different position and distance of his hand in respect to his eye.  The problem is thus set to the child’s understanding:  how to reconcile to his comprehension the apparently contradictory facts of the invariableness of the object together with the variableness of its appearance.  This is only possible within a space of three dimensions, in which, owing to perspective distortions and changes, these variations of projection can be reconciled with the constancy of the form of a body.”

Thus we have come to the idea of a three-dimensional space in order to overcome the apparent contradictoriness of facts of sensible experience.  Should we observe in three-dimensional space contradictory facts our reason would be forced to reconcile these contradictions, also, and if they could be reconciled by the idea of a four-dimensional space our reason would accept this idea without cavil.  Furthermore, if from our childhood, phenomena had been of daily occurrence requiring a space of four or more dimensions for an explanation conformable to reason, we should feel ourselves native to a space of four or more dimensions.

Poincare, the great French mathematician and physicist, arrived at these same conclusions by another route.  By a process of mathematical reasoning of a sort too technical to be appropriately given here, he discovers an order in which our categories range themselves naturally, and which corresponds with the points of space; and that this order presents itself in the form of what he calls a “three circuit distribution board.”  “Thus the characteristic property of space,” he says, “that of having three dimensions, is only a property of our distribution board, a property residing, so to speak, in human intelligence.”  He concludes that a different association of ideas would result in a different distribution board, and that might be sufficient to endow space with a fourth dimension.  He concedes that there may be thinking beings, living in our world, whose distribution board has four dimensions, and who do consequently think in hyperspace.

THE NEED OF AN ENLARGED SPACE-CONCEPT