Titlebook: Euclid—The Creation of Mathematics; Benno Artmann Book 1999 Springer-Verlag New York, Inc. 1999 Euclid.Euclid‘s elements.Geometry.Math.Vol

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The Origin of Mathematics 4: Squaring the Circle,Theorem II.14 solves an important problem: Every rectilinear figure can be squared. As usual in mathematics, a problem is solved only to beget another one. The next most prominent figure is the circle. How to square it? Proclus observes in his comment on Prop. I.45, which is the last step before II.14:

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Euclid Book III: About the Circle,Equal circles are those the diameters of which are equal, or the radii of which are equal.

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The Origin of Mathematics 5: Problems and Theories,In section C of Book III Euclid presents the prototype of a mathematical theory. He has a clear sense of its architecture. Let us recapitulate the main steps:

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The Origin of Mathematics 6: The Birth of Rigor,Our historical reconstructions about the pentagon maybe hypothetical. Nevertheless, we can use them as an example for some remarks on rigor in mathematics. What is meant by saying that an argument is rigorous and not just intuitively right?

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The Origin of Mathematics 7: Polygons After Euclid,In Prop. IV. 16 Euclid constructs a regular 15-gon by superimposing an equilateral triangle on a regular pentagon (Fig. 13.1).

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The Origin of Mathematics 9: Nicomachus and Diophantus,Aside from Euclid, there are two mathematicians from antiquity whose books about arithmetic have survived.

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