“It is an anciently established doctrine that Form or Figure, which we may call the most complex mode of extension, is resolvable into relative magnitude of parts. An equilateral triangle is one of which the three sides are alike in magnitude. An ellipse is a symmetrical closed curve, of which the transverse and conjugate diameters are one greater than the other. A cube is a solid, having all its surfaces of the same magnitude, and all its angles of the same magnitude. A cone is a solid, successive sections of which, made at right angles to the axis, are circles regularly decreasing in 153 magnitude as we progress from base to apex. Any object described as narrow is one whoso breadth is of small magnitude when compared with its length. A symmetrical figure is a figure in which the homologous parts on opposite sides are equal in magnitude. Figures which we class as similar to each other, are such that the relation of magnitude between any two parts of the one, is equal to the relation of magnitude between the corresponding parts of the other. Add to which, that an alteration in the form of anything, is an alteration in the comparative sizes of some of its parts—a change in the relations of magnitude subsisting between them and the other parts, and that by continuously altering the relative magnitudes of its parts, any figure may be changed indefinitely. Hence, figure being wholly resolvable into relations of magnitude we may go on to analyze that out of which these relations are formed—magnitude itself.” (pp. 224, 225.)
Next therefore, of Magnitude:
“What is a magnitude, considered analytically? The reply is, It consists of one or more relations of position. When we conceive anything as having a certain bulk, we conceive its opposite limiting surfaces as more or less removed from each other; that is, as related in position. When we imagine a line of definite length, we imagine its termini as occupying points in space having some positive distance from each other; that is, as related in position. As a solid is decomposable into planes; a plane into lines; lines into points; and as adjacent points can neither be known nor conceived as distinct from each other, except as occupying different places in space—that is, as occupying not the same position, but relative positions—it follows that every cognition of magnitude, is a cognition of one or more relations of position, which are presented to consciousness as like or unlike one or more other relations of position.” (p. 226.)
And finally, of Position:
“This analysis of itself brings us to the remaining space-attribute of body—Position. Like magnitude, Position cannot be known absolutely; but can be known only relatively. 154 The notion of position is, in itself, the notion of relative position. The position of a thing is inconceivable, save by thinking of that thing as at some distance from one or more other things. The essential element of the idea will be best seen, on observing under what conditions only, it can come into existence. Imagine a solitary point A, in infinite space; and suppose it possible for that point to be known by a being having no locality, what now can be predicated respecting its place? Absolutely nothing. Imagine another point B to be added. What can now be predicated respecting the two? Still nothing. The points having no attributes save position, are not comparable in themselves; and nothing can be said of their relative position, from lack of anything with which to compare it. The distance between them may be either infinite or infinitesimal, according to the measure used; and as, by the hypothesis, there exists no measure—as space contains nothing save these two points; the distance between them is unthinkable. But now imagine that a third point C is added. Immediately it becomes possible to frame a proposition respecting their positions. The two distances, A to B, and A to C, serve as measures to each other. The space between A and B may be compared with the space between A and C; and the relation of position in which A stands to B becomes thinkable, as like or unlike the relation in which A stands to C. Thus, then, it is manifest that position is not an attribute of body in itself, but only in its connection with the other contents of the universe.
“It remains to add, that relations of position are of two kinds: those which subsist between subject and object; and those which subsist between either different objects, or different parts of the same object. Of these the last are resolvable into the first. It needs but to remember, on the one hand, that in the dark a man can discover the relative positions of two objects only by touching first one and then the other, and so inferring their relative positions from his own position towards each; and on the other hand, that by vision no knowledge of their relative positions can be 155 reached save through a perception of the distance of each from the eye; to see that ultimately all relative positions may be decomposed into relative positions of subject and object.
“These conclusions—that Figure is resolvable into relative magnitudes; that magnitude is resolvable into relative positions; and that all relative positions may finally be reduced to positions of subject and object—will be fully confirmed on considering the process by which the space-attributes of body become known to a blind man. He puts out his hand and touching something, thereby becomes cognizant of its position with respect to himself. He puts out his other hand, and meeting no resistance above or on one side of the position already found, gains some negative knowledge of the thing’s magnitude—a knowledge which three or four touches on different sides of it serve to render positive. And then, by continuing to move his hands over its surface, he acquires a notion of its figure. What, then, are the elements out of which, by synthesis, his perceptions of magnitude and figure are framed? He has received nothing but simultaneous and successive touches. Each touch established a relation of position between his centre of consciousness and the point touched. And all he can know respecting magnitude and figure—that is, respecting the relative position of these points to each other—is necessarily known through the relative positions in which they severally stand to himself.
“Our perceptions of all the space-attributes of body being thus decomposable into perceptions of position like that gained by a single act of touch; we have next to inquire what is contained in a perception of this kind. A little thought will make it clear that to perceive the position of anything touched, is really to perceive the position of that part of the body in which the sensation of touch is located. Whence it follows that our knowledge of the positions of objects, is built upon our knowledge of the positions of our members towards each other—knowledge both of their fixed relations, and of those temporary relations they are placed in by every change of muscular adjustment. That 156 this knowledge is gained by a mutual exploration of the parts—by a bringing of each in contact with the others—by a moving over each other in all possible ways; and that the motions involved in these explorations, are known by their reactions upon consciousness; are propositions that scarcely need stating. But it is manifestly impossible to carry the analysis further without analysing our perception of motion. Relative position and motion are two ideas of the same experience. We can neither conceive motion without conceiving relative position, nor discover relative position without motion. In the present, therefore, we must be content with the conclusion that, whether visual or tactual, the perception of every statical attribute of body is resolvable into perceptions of relative position which are gained through motion.” (pp. 226—229.)
In further prosecution of the analysis:
“How do we become cognizant of the relative positions of two points on the surface of the body? Such two points, considered as coexistent, involve the germinal idea of Space. Such two points disclosed to consciousness by two successive tactual sensations proceeding from them, involve the germinal idea of Time. And the series of muscular sensations by which, when self-produced, these two tactual sensations are separated, involve the germinal idea of Motion. The questions to be considered then are—In what order do these germinal ideas arise? and—How are they developed?