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Published online by Cambridge University Press: 05 June 2012
This ought even to be struck out of the Postulates altogether; for it is a theorem … the converse of it is actually proved by Euclid himself as a theorem. … It is clear then from this that we should seek a proof of the present theorem, and that it is alien to the special character of postulates.
Proclus (410–85)The most reliable information about Euclid and early Greek geometry is based on the commentaries of Proclus whose objections to Postulate V are stated in the quote. To its author and early readers the Elements provided an idealized description of physical space. From this viewpoint it is natural to understand the objections to Postulate V. The phrase “if produced indefinitely” strains the intuition based on constructions with compass and straight edge. Furthermore, Euclid studiously avoided using Postulate V for the first 28 propositions of Book I. The first application was to prove 1.29, the converse of 1.27 and 1.28. Several of the previous propositions are related to their neighbors as converses with proofs that simply observe the contradiction to the earlier statement were the converse false. Proposition 1.29 does not yield to this logic. Why introduce such an unnatural statement to prove it?
To eliminate this “blemish” on Euclid's great work, subsequent generations heeded Proclus's call and either sought a proof of Postulate V from the other assumptions, or they tried to replace it with a more self-evident assumption.
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