Mathematical recreations and essays . r Playfairsassertion as axiomatic, that is, as being a part of our conceptionof plane space as derived from experience. This is not in-consistent with admitting that mathematicians can conceivea more general view of space { non-Euclidean space) andthat to them the space of our experience is only, to the highestdegree of approximation, Euclidean space*. But many writersdo not accept Playfairs assertion as axiomatic. On such anissue no argument is possible. Attempted Direct Demonstrations of the Proposition. Thedifficulties connected with the subject of
Mathematical recreations and essays . r Playfairsassertion as axiomatic, that is, as being a part of our conceptionof plane space as derived from experience. This is not in-consistent with admitting that mathematicians can conceivea more general view of space { non-Euclidean space) andthat to them the space of our experience is only, to the highestdegree of approximation, Euclidean space*. But many writersdo not accept Playfairs assertion as axiomatic. On such anissue no argument is possible. Attempted Direct Demonstrations of the Proposition. Thedifficulties connected with the subject of parallelism led tovarious attempts to prove the proposition directly and thenceto deduce some property of parallelism equivalent to Euclidspostulate. Substantially this was the method used by Thalesand Pascal. I will mention one or two of these attempts. Play fairs Rotational Proof of the Proposition. First I willdescribe an attempt, given by Playfair f in 1813. His argumentis as follows. An angle is measured by the amount of turning. H of a vector. Let ABC be any triangle. Suppose we havea rod AL placed along AB with one end at ^. If we rotateit clockwise round J. as a pivot through the angle BAG itwill move from AL to AK. It will make no difference if wenow slide the rod along AK so that the end moves from A * See A. Cayley, British Association Report, London, 1883, p. 9. + See the notes appended to J. Playfairs Elements of Geometry^ p. 432 inthe fifth edition. Playfair finds the sum of the exterior angles and thencededuces the sum of the interior angles, but the method is the same as thatgiven above. 318 THE PARALLEL POSTULATE [CH. XV to (7. If we now turn the rod, in the same direction as before,round 0 as a pivot through the angle AGB it will move fromCK to OH. It will make no difference if we now slide it backalong GB so that the end moves from G to B. If we now turnthe rod, again in the same direction, round 5 as a pivot throughthe angle GBA it will move from BH to BA. W
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