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 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.