The Problem With Postulate 5


For 2100 years after the appearance of the Elements, a steady trickle of subtle thinkers were disturbed by Postulate 5. It wasn’t as simple as the other axioms. No one doubted it was true, but it seemed out of place as a basic assumption.


Early 19th Century Euclidean Geometry Parallel Straight Line Common Notion Equal Side 
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  1. 3.
    Neutral geometry. The term seems to have originated with Prenowitz and Jordan in Basic Concepts of Geometry (Blaisdell, 1965). Most books use the term “Absolute geometry” introduced in 1832 by János Bolyai, one of the founders of non-Euclidean geometry.Google Scholar
  2. 4.
    Wallis’ Postulate. Wallis actually made the stronger assumption that “to every figure there exists a similar figure of arbitrary magnitude.” (Bonola, Non-Euclidean Geometry (1906; Dover reprint, 1955), pp. 15–17.)Google Scholar
  3. 5.
    letter. From Bonola, op. cit., pp. 65–66.Google Scholar
  4. 7.
    Metatheorem. The distinction between “theorem” and “metatheorem” is largely in the eye of the beholder. So far in this article I’ve called a result a “theorem” (Theorem A, p. 133 and Theorem B, p. 134) when its hypothesis was known to be possible in Neutral geometry, and a “metatheorem” when the possibility of its hypothesis had not been established. Since a conditional statement does not assert the hypothesis, but merely that it entails the conclusion, this pattern is not binding. Nonetheless I will follow it; thus this is a “metatheorem,” even though it has the ring of a “theorem.”Google Scholar

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© Birkhäuser Boston 2008

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