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Seminaire Bourbaki vol 1976 77 Exposes 489-506 by A. Dold, B. Beckmann

By A. Dold, B. Beckmann

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This segment contains excerpts from the video The Theorem of Pythagoras. It begins with a statement of the Pythagorean Theorem and its interpretation in terms of areas of squares constructed on the sides of a right triangle. An animated proof is shown, suggested by a diagram in a Chinese manuscript. This is followed by a discussion of Pythagorean triples using a moving ruler as the hypotenuse of a variable 24 Tom M. Apostol Fig. 11. Animated segment showing examples of Pythagorean triples right triangle.

Cervone hindsight should have been obvious) is that since the objects displayed by Geomview are in three-dimensional perspective, the ones near the edge of the images have a slightly different viewing angle than the ones near the middle. Usually this is not a problem, but in some images, where the viewing angle was sensitive to small changes, we had to compensate for it. There was also an associated bug in Geomview that required a work-around. One of the most difficult conversion issues involved lighting and coloration of the various surfaces.

The Race for the Calculus. In an intellectual marathon, which can be thought of as a relay race toward the calculus covering several centuries, mathematicians competed to find a general method for calculating areas and volumes of curved figures. The first lap was a victory by Archimedes, who calculated the area of a circular disk and the volume of a sphere. Interest was revived many centuries later with the advent of algebra and analytic geometry, which made it possible to analyze area and volume problems with the help of algebraic equations.

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