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,
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.
[3] Book III.
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.