face, whether
perceived by the sight or touch, are so minute and so confounded with
each other, that it is utterly impossible for the mind to compute their
number, such a computation will Never afford us a standard by which we
may judge of proportions. No one will ever be able to determine by an
exact numeration, that an inch has fewer points than a foot, or a foot
fewer than an ell or any greater measure: for which reason we seldom or
never consider this as the standard of equality or inequality.

As to those, who imagine, that extension is divisible in infinitum, it
is impossible they can make use of this answer, or fix the equality of
any line or surface by a numeration of its component parts. For since,
according to their hypothesis, the least as well as greatest figures
contain an infinite number of parts; and since infinite numbers,
properly speaking, can neither be equal nor unequal with respect to each
other; the equality or inequality of any portions of space can never
depend on any proportion in the number of their parts. It is true, it
may be said, that the inequality of an ell and a yard consists in the
different numbers of the feet, of which they are composed; and that of
a foot and a yard in the number of the inches. But as that quantity we
call an inch in the one is supposed equal to what we call an inch in
the other, and as it is impossible for the mind to find this equality by
proceeding in infinitum with these references to inferior quantities: it
is evident, that at last we must fix some standard of equality different
from an enumeration of the parts.

There are some [See Dr. Barrow's mathematical lectures.], who pretend,
that equality is best defined by congruity, and that any two figures
are equal, when upon the placing of one upon the other, all their parts
correspond to and touch each other. In order to judge of this definition
let us consider, that since equality is a relation, it is not, strictly
speaking, a property in the figures themselves, but arises merely from
the comparison, which the mind makes betwixt them. If it consists,
therefore, in this imaginary application and mutual contact of parts, we
must at least have a distinct notion of these parts, and must conceive
their contact. Now it is plain, that in this conception we would run up
these parts to the greatest minuteness, which can possibly be conceived;
since the contact of large parts would never render the figures equal.
But the m