"If it be conceded that lines have breadth, then all we have to do is to assign some definite breadth to each line—say the one-thousandth of an inch—and allow for it. But the lines of the geometer have no breadth. All the micrometers of which Mr. Buckle speaks depend, either directly or indirectly, upon lines for their graduations, and the positions of these lines are indicated by rulings or scratches. Now, in even the finest of these rulings, as, for example, those of Nobert or Fasoldt, where the ruling or scratching, together with its accompanying space, amounts to no more than the one hundred and fifty thousandth part of an inch, the scratch has a perceptible breadth. But this broad scratch is not the line recognized by the microscopist, to say nothing of the geometer. The true line is a line which lies in the very center of this scratch and it is certain that this central line has absolutely no breadth at all."[6]
It must be very evident that if Mr. Buckle's contention that geometrical lines have breadth were true, then some of the fundamental axioms of geometry must be false. It could no longer hold true that "the whole is equal to all its parts taken together," for if we divide a square or a circle into two parts by means of a line which has breadth, the two parts cannot be equal to the whole as it formerly was. As a matter of fact, Mr. Buckle's lines are saw-cuts, not geometrical lines. Geometrical points, lines, and surfaces, have no material existence and can have none. An ideal conception and a material existence are two very different things.
A very interesting book[7] has been written on the movements and feelings of the inhabitants of a world of two dimensions. Nevertheless, if we know anything at all, we know that such a world could not have any actual existence and when we attempt to form any mental conception of it and its inhabitants, we are compelled to adopt, to a certain extent, the idea of the third dimension.
But at the same time we must remember that since the ordinary mechanic and the school-boy who has studied geometry, find no difficulty in conceiving of points without magnitude, lines without breadth, and surfaces without thickness—conceptions which seem to have been impossible to Buckle, a man of acknowledged ability—it may be possible that minds constituted slightly differently from that of myself and some others, might, perhaps, be able to form a conception of a fourth dimension.
Leaving out of consideration the speculations of those who have woven this idea into romances and day-dreams we find that the hypothesis of a fourth dimension has been presented by two very different classes of thinkers, and the discussion has been carried on from two very different standpoints.
The first suggestion of this hypothesis seems to have come from Kant and Gauss and to have had a purely metaphysical origin, for, although attempts have been made to trace the idea back to the famous phantoms of Plato, it is evident that the ideas then advanced had nothing in common with the modern theory of the existence of a fourth dimension. The first hint seems to have been a purely mathematical one and did not attract any very general attention. It was, however, seized upon by a certain branch of the transcendentalists, closely allied to the spiritualists, and was exploited by them as a possible explanation of some curious and mysterious phenomena and feats exhibited by certain Indian and European devotees. This may have been done merely for the purpose of mystifying and confounding their adversaries by bringing forward a striking illustration of Hamlet's famous dictum—
"There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy."
A very fair statement of this view is thus given by Edward Carpenter:[8]
"There is another idea which modern science has been familiarizing us with, and which is bringing us towards the same conception—that, namely, of the fourth dimension. The supposition that the actual world has four space-dimensions instead of three makes many things conceivable which otherwise would be incredible. It makes it conceivable that apparently separate objects, e. g., distinct people, are really physically united; that things apparently sundered by enormous distances of space are really quite together; that a person or other object might pass in and out of a closed room without disturbance of walls, doors or windows, etc., and if this fourth dimension were to become a factor of our consciousness it is obvious that we should have means of knowledge which, to the ordinary sense, would appear simply miraculous. There is much, apparently, to suggest that the consciousness attained to by the Indian gñanis in their degree, and by hypnotic subjects in theirs, is of this fourth dimensional order.
"As a solid is related to its own surface, so, it would appear, is the cosmic consciousness related to the ordinary consciousness. The phases of the personal consciousness are but different facets of the other consciousness; and experiences which seem remote from each other in the individual are perhaps all equally near in the universal. Space itself, as we know it, may be practically annihilated in the consciousness of a larger space, of which it is but the superficies; and a person living in London may not unlikely find that he has a back door opening quite simply and unceremoniously out in Bombay."