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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 significance. 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 justi±cation ⋆ 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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