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
minutest parts we can conceive are mathematical points; and consequently
this standard of equality is the same with that derived from the equality
of the number of points; which we have already determined to be a just but
an useless standard. We must therefore look to some other quarter for a
solution of the present difficulty.

There are many philosophers, who refuse to assign any standard of
equality, but assert, that it is sufficient to present two objects, that
are equal, in order to give us a just notion of this proportion. All
definitions, say they, are fruitless, without the perception of such
objects; and where we perceive such objects, we no longer stand in need
of any definition. To this reasoning, I entirely agree; and assert, that
the only useful notion of equality, or inequality, is derived from the