File:Principia Resistance Trajectory Edition 2.jpg

English: Proposition 10 of Book 2 in Edition 1 of Principia is incorrect. It concerns the problem of finding the resistance that would cause a body to follow a specified curve under the action of gravity. Edition 2 contains a very different, complicated proof which does produce the correct result. However, it contains exactly the same error that was present in the 1st edition. Newton made many attempts to find the correct proof (see ‘The Mathematical Papers of Isaac Newton’, by D. T. Whiteside, Cambridge University Press, 1967-1981’ Vol 8). Eventually he found a way to obtain the correct result although the method is erroneous. He probably thought it was correct since it gave the correct result. In any case he was desperate to have a proof since Edition 2 was in the process of being printed when he was made aware of the error. It’s clear that Newton never understood the cause of the error, that the time required to traverse a small arc of the curve contained an additional term which the proof in Edition 2 succeeded in eliminating. If that had been deliberate, it could have been regarded as a highly ingenious solution to a difficult problem. In fact, it was purely a case of trial and error, and many apparently equally valid attempts which gave the wrong result were discarded. However, he did achieve his goal, and although his critics were able to gloat about the error and the need to change Proposition 10 so radically, they were not able to find anything wrong with it. Given the methods he used, and the urgency of the matter, it is highly unlikely Newton would have been able to find a legitimate proof.

See link below for more detailed analysis.

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

File:Principia_Resistance_Trajectory.jpg

The error in the Edition 2 version of Proposition 10 had been previously noted and fully analysed by Prof. Michael Nauenberg. Please see the following link:

http://physics.ucsc.edu/~michael/prop10book2proofs.pdf