numbers; and according as they correspond or not to that standard, we
determine their relations, without any possibility of error. When two
numbers are so combined, as that the one has always an unite answering to
every unite of the other, we pronounce them equal; and it is for want of
such a standard of equality in extension, that geometry can scarce be
esteemed a perfect and infallible science.

But here it may not be amiss to obviate a difficulty, which may arise
from my asserting, that though geometry falls short of that perfect
precision and certainty, which are peculiar to arithmetic and algebra,
yet it excels the imperfect judgments of our senses and imagination. The
reason why I impute any defect to geometry, is, because its original and
fundamental principles are derived merely from appearances; and it may
perhaps be imagined, that this defect must always attend it, and keep it
from ever reaching a greater exactness in the comparison of objects or
ideas, than what our eye or imagination alone is able to attain. I own
that this defect so far attends it, as to keep it from ever aspiring to a
full certainty: But since these fundamental principles depend on the
easiest and least deceitful appearances, they bestow on their
consequences a degree of exactness, of which these consequences are
singly incapable. It is impossible for the eye to determine the angles of
a chiliagon to be equal to 1996 right angles, or make any conjecture,
that approaches this proportion; but when it determines, that right lines
cannot concur; that we cannot draw more than one right line between two
given points; it's mistakes can never be of any consequence. And this is
the nature and use of geometry, to run us up to such appearances, as, by
reason of their simplicity, cannot lead us into any considerable error.

I shall here take occasion to propose a second observation concerning our
demonstrative reasonings, which is suggested by the same subject of the