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When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the necessary calculus is being described in geometric language. Fortunately for us lesser mortals, the mathematician-astrophysicist Chandrasekhar has written a book that rephrases Newton's arguments in the language of modern-day calculus. However, if we stick to Newton's way of doing things, how can we improve the rigor of his synthetic-geometric arguments?

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Maybe you could give some examples from Newton? Synthetic descriptions of plane geometry after Hilbert have been pretty successful, also for non-Euclidean planar, but I'm not sure how that helps you. –  Will Jagy Oct 21 '12 at 21:59

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