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# sc-gs - THE MATHEMATICS OF NEWTONS PRINCIPIA MATHEMATICA...

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1 THE MATHEMATICS OF NEWTON’S PRINCIPIA MATHEMATICA George E. Smith Philosophy Department Tufts University January 2010 With appreciation to D. T. Whiteside Niccolò Guicciardini Bruce Pourciau Curtis Wilson

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2 A Treatise of the Method of Fluxions and Infinite Series, with its Application to the Geometry of Curve Lines – 1671
3 NEWTON ON CURVATURE – 1671

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4 From Lecture 3, “Optical Lectures” (1670-72) … the generation of colors includes so much geometry, and the understanding of colors is supported by so much evidence, that for their sake I can thus attempt to extend the bounds of mathematics somewhat, just as astronomy geography, navigation, optics, and mechanics are truly considered mathematical sciences even if they deal with physical things: the heavens, earth, seas, light, and local motion. Thus although colors may belong to physics, the science of them must nevertheless be considered mathematical, insofar as they are treated by mathematical reasoning . Indeed, since an exact science of them seems to be one of the most difficult that philosophy is in need of, I hope to show – as it were, by my example – how valuable mathematics is in natural philosophy. I therefore urge geometers to investigate nature more rigorously, and those devoted to natural sciences to learn geometry first. Hence the former shall not entirely spend their time in speculations of no value to human life, nor shall the latter, while working assiduously with an absurd method, perpetually fail to reach their goal. But truly with the help of philosophical geometers and geo-metrical philosophers, instead of conjectures and probabilities that are being blazoned everywhere, we shall finally achieve a natural science supported by the greatest evidence .
5 From Principia , Preface to the First Edition For the whole difficulty of philosophy seems to be to find the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces . It is to these ends that the general propositions in Books 1 and 2 are directed, while in Book 3 our explication of the system of the universe illustrates these propositions. For in Book 3, by means of propositions demonstrated mathemati- cally in Books 1 and 2 we derive from celestial phenomena the forces of gravity by which bodies tend toward the Sun and toward the indivi- dual planets. Then the motions of the planets, the comets, the moon, and the sea are deduced from these forces by propositions that are also mathematical .

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6 From Preface to the First Edition, cont. If only we could derive the other phenomena of nature from mechanical principles by the same kind of reasoning! For many things lead me to have a suspicion that all phenomena may depend on certain forces by which the particles of bodies, by causes yet unknown, either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede. Because these forces are unknown, philosophers have hitherto made trial of nature in vain
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sc-gs - THE MATHEMATICS OF NEWTONS PRINCIPIA MATHEMATICA...

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