say how the intellect perceives without the idea, as it is to say how the supposed representation refers to its object. How does our idea refer to an object? If by itself, then by itself alone, since it is purely internal, it refers to the external, and requires no intermediary to place the subject in relation with external objects. What it does, the intellectual act of itself alone can also do. If we perceive the relation of the idea with the object by means of another idea, this intermediate idea presents the same difficulty as the preceding idea; and so at last we must come to a case in which there is a transition from the intellect to the object without any intermediary.
If we see an object which is the image of another not known, we shall see the object in itself, but we shall not know that it has the relation of image, unless informed that it has: we shall know its reality, but not its representation. The same will happen in ideas which are images; these, therefore, do not at all explain how the transition from the internal act to the object is made; for this would require them to do for the understanding that which we find them unable to do for themselves.
27. There is something mysterious in the intellectual act, which men seek to explain in a thousand different ways, by rendering sensible what they inwardly experience. Hence so many metaphorical expressions, useful only so long as they serve merely to call and fix the attention, and give an account of the phenomenon, but hurtful to science if they go beyond these limits, if it be forgotten that they are metaphors, and are never to be confounded with the reality.
By intelligence we see what there is in things, we experience the act of perception; but when we reflect upon it we grope in the dark, as if there were a dense cloud about the very source of light, preventing us from seeing it with clearness. Thus the firmament is at times flooded with the light of the sun, although the sun is encircled with clouds and hidden from our view, so that we cannot even determine its position upon the horizon.
28. One cause of obscurity in this matter is the very effort to clear it up. The act of the understanding is, in its objective part, exceedingly luminous, since by it we see what there is in objects; but in its subjective nature, or in itself, it is an internal fact, simple indeed, but incapable of being explained by words. This is not a peculiarity of the intellectual act, it is common to all internal phenomena. What is it to see, to taste, to hear? What is a sensation, or feeling of any kind whatsoever? It is an inward phenomenon, of which we are conscious, but which we cannot decompose into parts; nor can we explain with words the combination of these parts. A word is enough to indicate the phenomenon, but this word has no meaning for him who does not now experience this phenomenon, or has not oat some former time experienced it. No possible explanations would ever enable a man born blind to understand color, or a deaf man sound.
The act of understanding belongs to this class; it is a simple fact which we can point out, but not explain. An explanation supposes various notions, the combination of which may be expressed by language; in the intellectual act there are none of these. When we have said, I think, or, I understand, we have said all. This simplicity is not destroyed by objective multiplicity; the act by which we compare two or more objects is just as simple as the act by which we perceive a single object. If one act be not enough, more will follow; and finally one act will unite or sum them all up; but it will not be a composite act.
CHAPTER V.
COMPARISON OF GEOMETRICAL WITH NON-GEOMETRICAL IDEAS
29. The idea is a very different thing from the sensible representation, but it has certain necessary relations with it which it will be well to examine. When we say necessary, we speak only of the manner in which our mind, in its actual state, understands, abstracting the intelligence of other spirits, and even that of the human mind when subject to other conditions than those imposed by its present union with the body. So soon as we quit the sphere in which our experience operates, we must be very cautious how we lay down general propositions, and take care not to extend to all intelligences qualities which are possibly peculiar to our own, and which, even with respect to it, will perhaps be entirely changed in another life. Having made these previous observations, which will be found of great utility to mark the limits of things there is danger of confounding, we now proceed to examine the relations of our ideas with sensible representations.
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,
Book III.
