We might proceed, after the same manner, in fixing the proportions of
quantity or number, and might at one view observe a superiority or
inferiority betwixt any numbers, or figures; especially where the
difference is very great and remarkable. As to equality or any exact
proportion, we can only guess at it from a single consideration; except
in very short numbers, or very limited portions of extension; which are
comprehended in an instant, and where we perceive an impossibility of
falling into any considerable error. In all other cases we must settle
the proportions with some liberty, or proceed in a more artificial
manner.

I have already I observed, that geometry, or the art, by which we fix
the proportions of figures; though it much excels both in universality and
exactness, the loose judgments of the senses and imagination; yet never
attains a perfect precision and exactness. It's first principles are
still drawn from the general appearance of the objects; and that
appearance can never afford us any security, when we examine, the
prodigious minuteness of which nature is susceptible. Our ideas seem to
give a perfect assurance, that no two right lines can have a common
segment; but if we consider these ideas, we shall find, that they always
suppose a sensible inclination of the two lines, and that where the angle
they form is extremely small, we have no standard of a I @ right line so
precise as to assure us of the truth of this proposition. It is the same
case with most of the primary decisions of the mathematics.

There remain, therefore, algebra and arithmetic as the only sciences, in
which we can carry on a chain of reasoning to any degree of intricacy,
and yet preserve a perfect exactness and certainty. We are possest of a
precise standard, by which we can judge of the equality and proportion of