If the postulate is assumed that a straight line is the shortest path between two points, then the first part of this theorem requires no further proof, and the second part follows at once from the axiom of inequalities. This seems the better plan for beginners, and the proposition may be considered as semiobvious. Euclid proved the first part, not having assumed the postulate. Proclus tells us that the Epicureans (the followers of Epicurus, the Greek philosopher, 342-270 B.C.) used to ridicule this theorem, saying that even an ass knew it, for if he wished to get food, he walked in a straight line and not along two sides of a triangle. Proclus replied that it was one thing to know the truth and another thing to prove it, meaning that the value of geometry lay in the proof rather than in the mere facts, a thing that all who seek to reform the teaching of geometry would do well to keep in mind. The theorem might simply appear as a corollary under the postulate if it were of any importance to reduce the number of propositions one more. If the proposition is postponed until after those concerning the inequalities of angles and sides of a triangle, there are several good proofs. [Illustration] For example, produce _AC_ to _X_, making _CX_ = _CB_. Then [L]_X_ = [L]_XBC_. [therefore] [L]_XBA_ > [L]_X_. [therefore] _AX_ > _AB_. [therefore] _AC_ + _CB_ > _AB_. The above proof is due to Euclid. Heron of Alexandria (first century A.D.) is said by Proclus to have given the following: [Illustration] Let _CX_ bisect [L]_C_. Then [L]_BXC_ > [L]_ACX_. [therefore] [L]_BXC_ > [L]_XCB_. [therefore] _CB_ > _XB_. Similarly, _AC_ > _AX_. Adding, _AC_ + _CB_ > _AB_. THEOREM. _If two sides of a triangle are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater._ Euclid stated this more briefly by saying, "In any triangle the greater side subtends the greater angle." This is not so satisfactory, for there may be no greater side. THEOREM. _If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater._ Euclid also stated this more briefly, but less satisfactorily, thus, "In any triangle the greater angle is subtended by the greater side." Students should have their attention called to the fact that these two theorems are reciprocal or dual theorems, the words "sides" and "angles" of the one corresponding to the words "angles" and "sides" respectively of the other. It may also be noticed that the proof of this proposition involves what is known as the Law of Converse; for (1) if _b_ = _c_, then [L]_B_ = [L]_C_; (2) if _b_ > _c_, then [L]_B_ > [L]_C_; (3) if _b_ < _c_, then [L]_B_ < [L]_C_; therefore the converses must necessarily be true as a matter of logic; for