Unformatted text preview: • Brianchon ( ∼ 1810) ◦ A student of Monge. ◦ His theorem: If 6 tangents to conic form a circumscribing hexagon, the 3 lines connect- ing opposite vertices are concurrent. Dual of Pascal’s Theorem. • Poncelet ◦ A student of Monge. ◦ First to view projective geometry as a new branch of mathematics. ◦ Captured in Russia in Napoleonic wars. Did work in prison. ◦ Principle of Duality ⋆ Proposed and used by (but not really proved by) Desargues. ⋆ Content: The words “line” and “point” can be interchanged in the plane. The words “plane” and “point” can be interchanged in 3-dimensional space. The language needs to be smoothed out. E.g., interchange “lie on” and “pass through”. ◦ Principle of Continuity ⋆ More ambiguous and controversial than the Principle of Duality. ⋆ Poncelet: “Let us consider an arbitrary figure in a general position and indeterminate in some way, taken from all those that one can consider without breaking the laws, the conditions, the relationships which exist between the diverse parts of the sys-...
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- Fall '10
- Geometry, Projective geometry, Monge, Desargues, certain particular modiﬁcations, homogeneous polynomial equations, regular coordinates