HPS391H1 Chapter Notes - Chapter 6: Hyperbola, Parallel Postulate, Reductio Ad Absurdum
29 Apr 2016
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There was quite a lot of opposition to this postulate because it did not hold true for everything. For instance, proctus takes the example of a hyperbola to prove this. The hyperbola approaches its asymptotes closer and closer but the lines never meets the axis. Yet, this idea has perplexed mathematicians through the ages. It isn"t enough to just assume the postulate as an axiom. It must be proved using the other axioms that have definite proofs. The first person to attempt a proof was ptolemy. But he assumes hilbert"s postulate (which is logically euclid"s postulate v) to be true. So in a way he assumed what he was trying to prove. Proculus makes the next proof (hard to type out, best would be to read the two paragraphs at the bottom of page 82 in the reader, proculus attempted to prove ). Proculus" argument is correct until the last step. He assumes that xy increases indefinitely, i. e. without bounds.
