history-page18 - Should we be impressed? Eect on...

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Euclidean Geometry Precursors Pythagoras Considered right angles to be morally exemplary. Transformation of areas How to dissect a polygon and reassemble the pieces to form a square. · By Pythagorean Theorem, it’s suFcient to solve the case where the polygon is a triangle. · We can ±rst turn the triangle into a rectangle and then turn the rectangle into a square. The 3 Classical Problems Squaring the Circle Proved impossible (using only Euclidean means) by Lindemann in 1882 when he proved π is transcendental. Doubling a Cube Proved impossible (using only Euclidean means) by Pierre Wantzel in 1837. Trisecting an Angle Proved impossible (using only Euclidean means) by Pierre Wantzel in 1837. All can be done if the rules are loosened. Typical complaint about the loosened rules is that they involve creating intermediate objects more complicated than the ±nal problem to be solved. Zeno’s Paradoxes E.g., motion is impossible.
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Unformatted text preview: Should we be impressed? Eect on Pythagoreans? Sign over the entrance to Platos Academy: Let no ungeometrical person enter. Eudoxus Theory of proportion Method of Exhaustion: Indirectly lets us ascertain volume by approximation. Euclid Elements Miscellaneous Oldest extant manuscript: 4th century AD? With few exceptions, the individual results are not original to Euclid, but the organi-zation probably is. Intended as a textbook, not a research monograph. Not intended to contain all known results. Elements in the sense of fundamental, not in the sense of simple and uncomplicated. Synthetic rather than analytic . Superlatives Eves: No work, except the Bible, has been more widely used, edited, or studied, and probably no work has exercised a greater inuence on scientic thinking. Over 1000 editions irst printed in 1482...
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