All Categories, Antiquity, Essays - Studies, Αγγλικά
Exhaustion: From Eudoxus to Archimedes
May
Franz Lemmermeyer
24grammata.com / free ebook/
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Abstract
Disclaimer: Eventually, I plan to polish this and use my own diagrams; mso far, most of it is lifted from the web.
Exhaustion is a method invented by Eudoxus to prove results about lengths, areas and volumes of geometrical objects. In Euclid’s elements (Book XII on the measurement of figures), the following propositions, among others, are proved by the method of exhaustion:
• XII.1. Similar polygons inscribed in circles are to one another as the squares on their diameters.
• XII.2. Circles are to one another as the squares on their diameters.
• XII.6. Pyramids of the same height with polygonal bases are to one another as their bases.
• XII.7. Corollary: Any pyramid is the third part of the prism with the same base and equal height.
• XII.10. Any cone is the third part of the cylinder with the same base and equal height.
• XII.18. Spheres are to one another in triplicate ratio of their respective diameters.
24grammata.com / free ebook/
[download]