April 2016
Intermediate to advanced
430 pages
8h 58m
English
Content preview from Differential Geometry of Curves and Surfaces, 2nd Edition
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CHAPTER 8
The Gauss-Bonnet Theorem and Geometry of Geodesics
Historians of mathematics often point to Euclid as the inventor, or at least the father in some metaphorical sense, of the synthetic methods of mathematical proofs. Euclid’s Elements, which comprises 13 books, treats a wide variety of topics but focuses heavily on geometry. In his Elements, Euclid presents 23 geometric definitions along with five postulates (or axioms), and in Books I-VI and XI-XIII, he proves around 250 propositions about lines, circles, angles, and ratios of quantities in the plane and in space.
Most popular texts about the nature of mathematics agree that Euclid’s geometric methods held a foundational importance in the development of mathematics (see, e.g., [ ...
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