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# history-page18 - Should we be impressed ◦ E²ect 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 sufficient to solve the case where the polygon is a triangle. · We can first 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 final problem to be solved. Zeno’s Paradoxes E.g., motion is impossible.
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Unformatted text preview: Should we be impressed? ◦ E²ect on Pythagoreans? • Sign over the entrance to Plato’s 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 in³uence on scienti±c thinking.” ⋆ Over 1000 editions ⋆ ´irst printed in 1482...
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