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
4 - Non-Euclidean geometry
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
I have created a new universe from nothing. All that I have hitherto sent you compares to this only as a house of cards to a castle.
J. BOLYAI (1823)The non-Euclidean geometry throughout holds nothing contradictory.
C. F. GAUSS (1831)In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it has come to us from Euclid.
N. I. LOBACHEVSKIĬ (1840)Between Euclid's time and 1829, the year Lobachevskiĭ's On the Principles of Geometry appeared, most of the critics of The Elements were concerned with the “purification” of Euclid's work from its perceived imperfections. So strong was the conviction that Postulate V depended on Postulates I through IV that some of these critics did not see in their work the basis for a new geometry. An important contribution appeared in 1763 with the Göttingen doctoral thesis of G. S. KLÜGEL (1739–1812), written under the direction of Abraham Kaestner (1719–1800), in which Klügel examined 30 attempts to prove Postulate V, finding all of them guilty of some form of petitio principii. In his introduction he states:
If all the attempts are given thorough consideration, it becomes clear that Euclid correctly counted among the axioms a proposition that cannot be proven in a proper manner with any others.
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- Information
- Geometry from a Differentiable Viewpoint , pp. 43 - 74Publisher: Cambridge University PressPrint publication year: 2012