Note 38 — Chapter XXIV · OF THE REMAINING LAWS OF NATURE.

The axiom, "Equals subtracted from equals leave equal differences," may be demonstrated from the two axioms in the text. If A = a and B = b, A - B = a - b. For if not, let A - B = a - b + c. Then since B = b, adding equals to equals, A = a + c. But A = a. Therefore a = a + c, which is impossible.

This proposition having been demonstrated, we may, by means of it, demonstrate the following: "If equals be added to unequals, the sums are unequal." If A = a and B not = b, A + B is not = a + b. For suppose it be so. Then, since A = a and A + B = a + b, subtracting equals from equals, B = b; which is contrary to the hypothesis.

So again, it may be proved that two things, one of which is equal and the other unequal to a third thing, are unequal to one another. If A = a and A not = B, neither is a = B. For suppose it to be equal. Then since A = a and a = B, and since things equal to the same thing are equal to one another, A = B: which is contrary to the hypothesis.