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.