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English: Propositions 15 and 16 of Book 2 derive the resistive acceleration, in the opposite direction to motion, required to make a body follow the path of an equiangular spiral under the effect of a centripetal acceleration proportional to a power of the radius. Newton’s method is geometrical with a limiting procedure, involving arcs that are described as ‘minimally small’. In the Scholium to Section 1 of Book 1, Newton explains that when: ‘....I speak in what follows of quantities as minimally small or vanishing or ultimate, take care not to understand quantities that are determinate in magnitude, but always think of quantities that are to be decreased without limit’. By following exactly the same method it is possible to find the corresponding results in the case where a body follows the path of a more general smooth curve, with the central acceleration a more general function of the radius. By using the familiar functions of trigonometry and the notation of calculus, Newton’s method is demonstrated in a simpler manner here. The various assertions in Proposition 15 are either stated without proof or are difficult to justify from dynamical considerations. In spite of the lack of rigor, as with most of Newton’s proofs, the final results are correct.

See also

Principia, A New Translation, by I. Bernard Cohen and Anne Whitman – University of California Press, 1999

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