HPS391H1 Chapter Notes - Chapter 5: Euclidean Space, Reductio Ad Absurdum, Parallel Postulate
29 Apr 2016
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Discovered certain gaps in euclid"s definitions and postulates for plane geometry. This postulate is more complicated than the others, to deny it would go against common sense. Even euclid himself did not quite trust it, so he postponed using it a proof until his twenty- ninth proposition. The postulate was doubted by the greeks of euclid"s time and in the centuries following him. He criticized the parallel postulate as follows: this ought even to be struck out of the postulate altogether; for it is a theorem involving many difficulties, which ptolemy, in a certain book, set himself to solve . The statement that since [the two lines] converge more and more as they are produced, they will sometimes meet is plausible but not necessary example of hyperbola. The fifth axiom should be able to proven from other axioms: ptolemy. He assumed hilbert"s parallel postulate without realizing it. Hilbert"s parallel postulate: logically equivalent to euclid"s fifth axiom.
