File:Principia Book 1 Proposition 41 Cor 3.jpg

English: Proposition 41, Book 1 enables the path of a body acted on by a central force as a function of the distance of the body from a fixed centre to be found. The method involves 3 integrations.

The Corollary considers a body moving under the influence of an inverse cube force directed to or from a fixed point and whose initial direction is perpendicular to the radius. The path of the body is obtained directly from an associated ellipse or hyperbola, rather than from the integrals of the Proposition.

This Corollary has caused much confusion, because it gives no indication of how it is follows from Proposition 41. Also its final sentence is obscure: ‘All this follows from the foregoing (prop. 41), by means of the quadrature of a certain curve, the finding of which, as being easy enough, I omit for the sake of brevity.’

In 1694 David Gregory asked Newton for details of the Corollary, and received a letter of reply explaining how the Corollary follows from the integrals of the Proposition. The ‘certain curve’ referred to above being an ellipse or hyperbola.

A copy of the letter can be found in:

The Oxford Handbook of the History of Mathematics, by Eleanor Robson, Jacqueline Stedall, Oxford University Press, 2009, p 722

See also:

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

The Visualization of Quadratures in the Mystery of Corollary 3 to Proposition 41 of Newton’s Principia, by Herman Ehrlichson in Historia Mathematica 21 (1994), pp 148-161

The Mathematical Papers of Isaac Newton’, by D. T. Whiteside, Cambridge University Press, 1967-1981, Vol. 6, pp 435-438