PROBLEMS for Great Theorems SOME MISSING EUCLIDEAN PROPOSITIONS One thing that some people find surprising about...
PROBLEMS for Great Theorems SOME MISSING EUCLIDEAN PROPOSITIONS One thing that some people find surprising about Book 1 of the Elements is not the familiar propositions that appear but the familiar ones that are omitted. Everyone may have his or her own "favorite" that appears nowhere in Euclid, but the following are mine: 16. Prove a result sometimes called "Playfair's Postulate": Through a point not on a line there can be drawn one and only one parallel to the given line. Assume that you are inserting this in the Elements as Proposition 1.31. 17. Prove the following result as if it were to be included as Proposition 1.16%: From a point not on a line there can be drawn one and only one perpendicular to the given line. Note that proving the uniqueness of parallels (Problem #16) requires the Parallel Postulate, but proving the uniqueness of perpendiculars (Problem #17) does not. tanning i annidistant from the trianale!