hematical points, in
order to give a termination to bodies; and others eluded the force of
this reasoning by a heap of unintelligible cavils and distinctions. Both
these adversaries equally yield the victory. A man who hides himself,
confesses as evidently the superiority of his enemy, as another, who
fairly delivers his arms.

Thus it appears, that the definitions of mathematics destroy the
pretended demonstrations; and that if we have the idea of indivisible
points, lines and surfaces conformable to the definition, their
existence is certainly possible: but if we have no such idea, it is
impossible we can ever conceive the termination of any figure; without
which conception there can be no geometrical demonstration.

But I go farther, and maintain, that none of these demonstrations
can have sufficient weight to establish such a principle, as this of
infinite divisibility; and that because with regard to such minute
objects, they are not properly demonstrations, being built on ideas,
which are not exact, and maxims, which are not precisely true. When
geometry decides anything concerning the proportions of quantity, we
ought not to look for the utmost precision and exactness. None of its
proofs extend so far. It takes the dimensions and proportions of
figures justly; but roughly, and with some liberty. Its errors are never
considerable; nor would it err at all, did it not aspire to such an
absolute perfection.

I first ask mathematicians, what they mean when they say one line or
surface is EQUAL to, or GREATER or LESS than another? Let any of them
give an answer, to whatever sect he belongs, and whether he maintains
the composition of extension by indivisible points, or by quantities
divisible in infinitum. This question will embarrass both of them.

There are few or no mathematicians, who defend the hypothesis of
indivisible points; and yet these have the readiest and justest answer
to the present question. They need only reply, that lines or surfaces
are equal, when the numbers of points in each are equal; and that as
the proportion of the numbers varies, the proportion of the lines and
surfaces is also varyed. But though this answer be just, as well as
obvious; yet I may affirm, that this standard of equality is entirely
useless, and that it never is from such a comparison we determine
objects to be equal or unequal with respect to each other. For as the
points, which enter into the composition of any line or sur