[37] Supra, book i. ch. v. § 1, and book ii. ch. v. § 5. [38] 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. [39] Geometers have usually preferred to define parallel lines by the property of being in the same plane and never meeting. This, however, has rendered it necessary for them to assume, as an additional axiom, some other property of parallel lines; and the unsatisfactory manner in which properties for that purpose have been selected by Euclid and others has always been deemed the opprobrium of elementary geometry. Even as a verbal definition, equidistance is a fitter property to characterize parallels by, since it is the attribute really involved in the signification of the name. If to be in the same plane and never to meet were all that is meant by being parallel, we should feel no incongruity in speaking of a curve as parallel to its asymptote. The meaning of parallel lines is, lines which pursue exactly the same direction, and which, therefore, neither draw nearer nor go farther from one another; a conception suggested at once by the contemplation of nature. That the lines will never meet is of course included in the more comprehensive proposition that they are everywhere equally distant. And that any straight lines which are in the same plane and not equidistant will certainly meet, may be demonstrated in the most rigorous manner from the fundamental property of straight lines assumed in the text, viz. that if they set out from the same point, they diverge more and more without limit. [40] _Philosophie Positive_, iii. 414-416. [41] See the two remarkable notes (A) and (F), appended to his _Inquiry into the Relation of Cause and Effect_. [42] Supra, pp. 119, 120.