On the other hand, take geometrical ideas and remove them from the sensible sphere; and all that you base upon them will be only unmeaning words. The ideas of substance, cause, and relation do not flow from geometrical ideas; if we regard them alone, we see an immense field extending into regions of unbounded space; but the coldness and silence of death reign there. If we would introduce beings, life, and motion into this field we must seek them elsewhere; we must use other ideas, and combine them, so that life, activity, and motion may result from their combination, in order that geometrical ideas may contain something besides this inert, immovable, and vacant mass, such as we imagine the regions of space to be beyond the confines of the world.
35. Geometrical ideas, properly so called, as distinguished from sensible representations, are not simple ideas, since they necessarily involve the ideas of relation and number. Geometry cannot advance one step without comparing them; and this comparison almost always takes place by the intervention of the idea of number. Hence it is that geometrical ideas, apparently so unlike purely arithmetical ideas, are really identical with them so far as their form or purely ideal character is concerned; and are only distinguishable from them when they refer to a determinate matter, such as extension as presented in its sensible representation. The inferiority therefore of geometrical ideas already mentioned, only refers to their matter, or to their sensible representations, which are presupposed to be an indispensable element.
36. Another consequence of this doctrine, is the unity of the pure understanding, and its distinction from the sensitive faculties. For, the very fact that the same ideas apply alike to sensible and to insensible objects, with no other difference than that arising from the diversity of the matter perceived, proves that above the sensitive faculties there is another faculty with an activity of its own, and elements distinct from sensible representations. This is the centre where all intellectual perceptions unite, and where that intrinsic force resides, which, although excited by sensible representations, develops itself by its own power, makes itself master of these impressions, and converts them, so to speak, by a mysterious assimilation, into its own substance.
37. Here we repeat what we have already remarked, concerning the profound ideological meaning involved in the acting intellect of the Aristotelians, so ridiculed because not understood. But we leave this point and proceed to the careful analysis of geometrical ideas, to discover, if possible, a glimpse of some ray of light amid the profound darkness which envelops the nature and origin of our ideas.
CHAPTER VI.
IN WHAT THE GEOMETRICAL IDEA CONSISTS; AND WHAT ARE ITS RELATIONS WITH SENSIBLE INTUITION
38. In the preceding chapters we have distinguished between pure ideas and sensible representations, and we seem to have sufficiently demonstrated the difference between them, although we limited ourselves to the geometrical order. But we have not explained the idea in itself; we have said what it is not, but not what it is; and although we have shown the impossibility of explaining simple ideas, and the necessity of our being satisfied with indicating them, we do not wish to be confined to this observation, which may seem to elude the difficulty rather than to solve it. Only after due investigations, by which we shall be better able to understand what is meant by designate, will it be allowable to confine ourselves to their designation, for it will then be seen that we have not eluded the difficulty. Let us begin with geometrical ideas.
39. Is a geometrical idea, without any accompanying or preceding sensible representation, possible? It would seem that we can have none. What meaning has the idea of the triangle if not referred to lines forming angles and enclosing a space? And what do lines, angles, and space mean, without sensible intuition? A line is a series of points, but it represents nothing determinate, nothing susceptible of geometrical combinations, except it be referred to that sensible intuition in which the point appears to us as an element generating by its movement that continuity which we call a line. What would become of angles without the real or possible representation of these lines? What would become of the area of the triangle were we to abstract a space, a surface which is or may be represented? We might challenge all the ideologists in the world to assign any sense to the words used in geometry if absolute abstraction be made all sensible representation.
40. Geometrical ideas, such as we conceive them, have a necessary relation to sensible intuition. In order the better to understand this relation, let us define the triangle to be the figure enclosed by three right lines. This definition involves the following ideas: space, enclosed, three, lines. With a space and three lines which do not enclose the figure, we have no triangle; the word enclosed cannot therefore be omitted. If you enclose a space, but with more than three lines, the result will not be a triangle; and if you take less than three lines you can have no enclosure. The idea of three is therefore necessary to the idea of the triangle. It is useless to add that the idea of line is as necessary as the others, since without it no triangle can be conceived. Different and distinct ideas, it is true, are here combined, but they are all referred to one sensible intuition, although in an indeterminate manner. We here abstract the longness or shortness of the lines and their forming larger or smaller angles. But we cannot thus abstract in the case of determinate intuitions; for every determinate intuition has its own peculiar qualities; otherwise it would not be a determinate representation, and consequently not sensible as it is supposed to be. But although the reference be to an indeterminate intuition, it always supposes some intuition either actual or possible, since otherwise the material of combination would be wanting to the understanding; and the four ideas involved in the triangle would be empty and unmeaning forms, and their combination extravagant if not absurd.
41. The idea then of the triangle seems to be simply the intellectual perception of the relation between the lines presented to the sensible intuition, considered in all its generality, without any determining circumstance limiting it to particular cases or species. This explanation admits nothing intermediate between the sensible representation and the intellectual act, which, exercising its activity upon the materials presented by sensible intuition, perceives their relations, and this pure and simple perception constitutes the idea.
42. We shall understand this better if, instead of the triangle, we take a many-sided figure, such as a polygon of a million sides, which cannot be clearly presented to the sensible intuition. The idea of this figure is as simple as that of the triangle; we perceive it by an intellectual act, express it by a single word, and can calculate its properties and relations with the same exactness and certainty as we can those of the triangle, although it is absolutely impossible to represent it distinctly to our imagination. When we reflect upon what it offers to the intellectual act, we notice the same elements as in the idea of the triangle, with this single difference that the number three is changed into million. We can have no sensible representation of all these lines; but the understanding has sufficiently combined the idea of line with that of number to perceive its object, a million. Here, then, we perceive the same elements as in the triangle; but it is upon these elements, considered in general without any other determination than results from the fixed number, that the perceptive act operates.
43. The idea of a polygon in general, abstracting the number of its sides, offers in its sensible representation, nothing determinate to the mind, nothing but the abstract idea of a right line, the general idea of an enclosed space. The relation which these objects of the intellectual, act even in the midst of their indeterminateness, have amongst themselves, is perceived by the intellectual act. This perceptive act is the idea. Every thing beyond this is useless, and not only useless but affirmed without reason.
44. It will perhaps be asked how the understanding can perceive what passes without it, since sensible intuition is a function of a faculty distinct from the understanding? In reply, we shall abstract the questions discussed in the schools concerning the powers of the mind, and be content to remark that whether these be really distinct among themselves, or only one power