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Post-Euclidean Geometry General Focus on results after Euclid closely connected to Elements Hellenic Geometry Why was there a post-Euclidean change? Roman dominance and Roman’s distaste for the impractical. Heron Apparently lived in 1st century AD. His book Metrica was concerned with numerical calculations related to geometry, some- thing unemphasized by his notable predecessors. Unconcerned with Euclidean niceties related to working with natural units. (E.g., he had no qualms about multiplying 4 lengths together.) Heron’s formula for the area of a triangle in terms of its side lengths. Possibly known earlier by Archimedes. His derivation of this formula. Ptolemy ( 150 AD) Ptolemy’s Theorem Possibly known earlier by Hipparchus. His derivation Its trigonometric signi±cance. The Almagest A symbol of bad astronomy? Pappus ( 340 AD) Theorem: “The ratio of rotated bodies is the composite of the ratio of the areas rotated
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Unformatted text preview: and the ratio of straight lines drawn similarly from their centers of gravity to the axes of rotation. Non-Euclidean Geometry Attempts to prove the parallel postulate. Why were they undertaken? What was their common pitfall? Saccheri ( 1700) and his quadrilaterals. (Hypotheses of the acute, obtuse, right angle.) Legendre and his proof of the parallel postulate. Questioning the parallel postulate. Philosophical issues Strict adherence to only axioms and postulates Logical necessity versus empirical justication How to give a proof of the (relative) consistency of non-Euclidean geometry. Models Reinterpretation of terms Discoveries of the properties of Non-Euclidean Geometry Apparently discovered independently by Lobachevskii (,1829), Janos Bolyai (1823,1832), and Gauss (1824,)....
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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