30. A classification of our ideas into geometrical and non-geometrical naturally occurs when we fix our attention upon the difference of objects to which our ideas may refer. The former embrace the whole sensible world so far as it can be perceived in the representation of space; the latter include every kind of being, whether sensible or not, and suppose a primitive element which is the representation of extension. In their divisions and subdivisions the latter present simply the idea of extension, limited and combined in different ways; but they offer nothing in relation to the representation of space, and even when they refer to it, they only consider it inasmuch as numbered by the various parts into which it may be divided. Hence the line which in mathematics separates geometry from universal arithmetic; the former is founded upon the idea of extension, whereas the latter considers only numbers, whether determinate, as in arithmetic properly so called, or indeterminate, as in algebra.
31. Here we have to note the superiority of non-geometrical to geometrical ideas, a superiority plainly visible in the two branches of mathematics, universal arithmetic and geometry. Arithmetic never requires the aid of geometry, but geometry at every step needs that of arithmetic. Arithmetic and algebra may both be studied from their simplest elementary notions to their highest complications without ever once involving the idea of extension, and consequently without making use of one single geometrical idea. Even infinitesimal calculus, in a manner originating in geometrical considerations, has been emancipated from them and formed into a science perfectly independent of the idea of extension. On the contrary, geometry cannot take a single step without the aid of arithmetic. The comparison of angles is a fundamental point in the science of geometry, but it cannot be made except by measuring them; and their measure is an arc of the circumference divided into a certain number of degrees, which must be counted; and thus we come to the idea of number, the operation of counting, that is, into the field of arithmetic.
The very proof by superposition, notwithstanding its eminently geometrical character, stands in need of numeration, inasmuch as the superposition is repeated. We do not require the idea of number to demonstrate by means of superposition the equality of two arcs perfectly equal; but in order to appreciate the relation of their quantity we compare two unequal arcs and follow the method of placing the less upon the greater several times, we count, we make use of the idea of number, and find we have entered upon the ground of arithmetic. We discover the equality of two radii of a circle, when we compare them by superposition, abstracting the idea of number; but if we would know the relation of the diameter to the radii, we employ the idea of two; we say the diameter is twice the radius, and again enter the domains of arithmetic. As we proceed in the combination of geometrical ideas, we make use of more and more arithmetical ideas. Thus the idea of the number three necessarily enters into the triangle; and the sum of three and the sum of two both enter into one of its most essential properties; the sum of the three angles of a triangle is equal to two right angles.
32. The idea of number cannot be replaced by the sensible intuition of the figure whose properties and relations are under discussion. In many cases this intuition is impossible, as, for example, in many-sided figures. We have little difficulty in representing to our imagination a triangle, or even a quadrilateral figure, but the difficulty is greater in the case of the pentagon, and greater still in the hexagon and heptagon; and when the figure attains a great number of sides, one after another escapes the sensible intuition, until it becomes utterly impossible to appreciate it by mere intuition. Who can distinctly imagine a thousand-sided figure?
33. This superiority of non-geometrical over geometrical ideas is very remarkable, since it shows that the sphere of intellectual activity expands in proportion as it rises above sensible intuition. Extension, as we have before seen,3 serves as the basis not only of geometry, but also of the natural sciences, inasmuch as it represents in a sensible manner the intensity of certain phenomena; but it can by no means enable us to penetrate their inmost nature, and guide us from that which appears to that which is. This and other subordinate ideas are, so to speak, inert, and from them springs no vital principle to fecundate our understanding, and still less the reality; they are an unfathomable depth in which our intellectual activity may toil, perfectly certain of never finding any thing in it which we ourselves have not placed there; they are a lifeless object which lends itself to all imaginable combinations without ever being capable of producing any thing, or of containing any thing not given to it. The naturalists in considering inertness as a property of matter, have perhaps regarded more than they are aware the idea of extension, which presents the inertness most completely.
34. The ideas of number, cause, and substance abound in results, and are applicable to all branches of science. We can scarcely speak without expressing them; it might almost be said that they are constituent elements of intelligence, since without them it vanishes like a passing illusion. They extend to every thing, apply to every thing, and are necessary, whenever objects are offered to the intellectual activity, in order that the intellect can perceive and combine them. It makes no difference whether the objects be sensible or insensible, whether there be question of our intelligence or of others subject to different laws; whenever we conceive the act of understanding we conceive also these primitive ideas as elements indispensable to the realization of the intellectual act. They exist and are combined independently of the existence, and even of the possibility, of the sensible world; and they would also exist in a world of pure intelligences, even if the sensible universe were nothing but an illusion or an absurd chimera.
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 exercising its activity upon different objects and in different manners, it will be alike necessary to admit a consciousness common to all the faculties. The soul which feels, thinks, recollects, desires, is one and the same, and is alike conscious of all these acts. Whatever be the nature of the faculties by which she performs these acts, she it is that performs them and knows that she performs them. There is then in the soul a single consciousness, the common centre where dwells the inward sense of every activity exercised, and of every affection received, to whatever order they may belong. However, supposing the case the most unfavorable to our theory, that the faculty to which sensible intuition corresponds, is really distinct from the faculty which perceives the relations of the objects offered by sensible intuition; does it therefore follow that the understanding cannot without something intermediate exercise its activity upon objects presented by this intuition? Certainly not. The act of pure understanding and that of sensible intuition, are indeed different, but they meet in consciousness, as in a common field; and there they come in contact, the one exercising its perceptive activity upon the material supplied by the other.