Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- PART A Prelude and themes: Synthetic methods and results
- 1 Spherical geometry
- 2 Euclid
- 3 The theory of parallels
- 4 Non-Euclidean geometry
- PART B Development: Differential geometry
- PART C Recapitulation and coda
- Riemann's Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry
- Solutions to selected exercises
- Bibliography
- Symbol index
- Name index
- Subject index
3 - The theory of parallels
Published online by Cambridge University Press: 05 November 2012
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- PART A Prelude and themes: Synthetic methods and results
- 1 Spherical geometry
- 2 Euclid
- 3 The theory of parallels
- 4 Non-Euclidean geometry
- PART B Development: Differential geometry
- PART C Recapitulation and coda
- Riemann's Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry
- Solutions to selected exercises
- Bibliography
- Symbol index
- Name index
- Subject index
Summary
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 C.E.)To be sure, it might be possible that non-intersecting lines diverge from each other. We know that such a thing is absurd, not by virtue of rigorous inferences or clear concepts of straight and crooked lines, but rather throught experience and the judgement of our eyes.
G. S. KLÜGEL (1763)Some of the most reliable information about Euclid and early Greek geometry is based on the commentaries of Proclus, the leader of the Academy in Athens in the fifth century of the common era, whose objections to Postulate V are stated in the epigram. 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 avoided using Postulate V in the proofs of the first twenty-eight propositions of Book I. It is first called upon in the proof of Proposition I.29, which is the converse of Propositions I.27 and I.28.
- Type
- Chapter
- Information
- Geometry from a Differentiable Viewpoint , pp. 27 - 42Publisher: Cambridge University PressPrint publication year: 2012