In your opinion, what are some of the best books/papers on Newton and his work that accurately cover the connections between his geometric proofs in the Principia and his development of the calculus and his disputes with others on these topics?

So far, some articles in The Cambridge Companion to Newton seem pretty good to me in addressing the interplay between Newton's calculus and his geometric proofs of physics in the Principia as well as the associated disputes on priority and philosophical approach (physical intuition versus pure algebraic manipulation). See also the intro of Needham's Visual Complex Analysis.

Edit March 2020: This question is about Newton's mathematics, but I can't resist quoting Keynes on Newton, which aptly places Newton's diverse interests into an understandable perspective:

Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind that looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.

  • $\begingroup$ See also "Newton: The algebraist and geometer" in The Royal Society's Newton Tercentenary Celebrations, 1947. $\endgroup$ – Tom Copeland Jun 3 '17 at 12:32
  • $\begingroup$ Newton, Methodus fluxionum et serierum infinitarum, 1664-1671. Newton, Philosophiæ Naturalis Principia Mathematica , 1687. $\endgroup$ – Tom Copeland Jun 3 '17 at 19:33
  • $\begingroup$ Borrowing Needham's phrase, I would say the historical debate that focuses on claims of the use of geometry vs. calculus in the proofs for Principia revolves around the imposition of an artificial dichotomy--algebraic calculus vs. geometric calculus--fueled by territorial disputes on priority between two camps. $\endgroup$ – Tom Copeland Jun 5 '17 at 21:07

The standard book about Newton's life is Never at Rest by Richard Westfall. On my opinion it is a very good book, it covers his life in great detail, and gives a general overview of his activities (not only in physics) but in astronomy, history, theology, alchemy, and as the Mint administrator. On physics, the latest English translation of Principia by Cohen (UCB Press, blue cover, not the cheaper editions!), has very comprehensive introduction and commentaries, (Introduction is about 1/3 of the total size of the book!) which covers all aspects of Principia (and Principia contains Newton's most important contributions to physics). I do not know an equally good commentaries on his Optics.

"Newton's Principia for the common reader" by S. Chandrasekhar, written by one of the famous physicists of 20th century. This is a commentary to the first part of Principia.

" Huygens and Barrow, Newton and Hooke" by V. Arnold, is a very interesting (and very small) book written by a prominent mathematician. It explains very much of Newton's physics for a book of such small size.

I listed only my favorite general books. There is an enormous specialized literature (books and articles) on various aspects of Newton's physics, for example, on his derivation of Kepler's laws, on his theory of the Moon, on the theory of motion in resisting medium, on tides, on optics etc. The books listed above give a good general overview.

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  • $\begingroup$ I particularly enjoyed Arnol'd's book with his keen insights and signature provocative asides. Of all the authors mentioned, he perhaps had the deepest appreciation of the analytic geometric approach and the work of Barrow and Hooke. $\endgroup$ – Tom Copeland Oct 30 '17 at 19:23

I agree with the citation of Westfall's biography of Newton, the 'Cambridge Companion' and the 1999 translation and introduction of the 'Principia'.

In addition, for the calculus and an account of the dispute about it I would suggest A R Hall's 'Philosophers at War'.

I would also cite excellent books and papers by: Niccolo Guicciardini (esp. books including 'Reading the Principia', 'Isaac Newton on Mathematical Certainty and Method', & papers); William L Harper (esp. 'Isaac Newton's Scientific Method').

Also valuable is almost everything by I Bernard Cohen, e.g.
his earlier (1972/78) 'Introduction to Newton's Principia'.

The book on the Principia by Chandrasekhar is a strange mixture. Some parts of it can deliver great clarity and insight, even though it tends to project modern ways of thinking backwards on to a time before the concepts had yet developed. Other parts are so wrong on verifiable facts they show Chandrasekhar was telling the truth when he claimed (in the prologue) to have made "no serious attempt to enlarge my knowledge derived from the Principia by any significant collateral reading."

But Newton and his work are subjects on which a number of myths have developed (i.e. not based on evidence of primary documents, especially of Newton & his contemporaries, even contradicting the sources). Some of the myths are longstanding, even some of the better books have not been quite free of them. It's worth looking at a range of authors and at original documents whenever you can, to help tell fact from speculation, to see directly in what sort of terms Newton and his contemporaries held their discussions and argued, and identify some of the uncertainties. The published volumes of the Correspondence of Isaac Newton (7 vols) and the Mathematical Papers of Isaac Newton (8 vols) are bulky but can be found in some libraries and are well worth the discovery.

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D. T. Whiteside's abundant commentary in his Mathematical Papers of Isaac Newton (8 volumes, Cambridge University Press, 1967-1981) is widely regarded as the authoritative work on Newton's mathematics. (See Whiteside's obituaries in The Guardian, The Independent, The Times, Oxford Dictionary of National Biography, The Investigation of Difficult Things.)

Regarding your specific question (“connections between his geometric proofs in the Principia and his development of the calculus”), Whiteside later wrote:

How often am I still asked: ‘Did Newton use calculus to obtain the theorems in his Principia?’ How, without seeming to patronize, do you lay the groundwork on which you can reply that the question is ill-formed and therefore meaningless? I will not here go into the reasons why.$\smash{^4}$

  1. I have done so in ‘The mathematical principles underlying Newton's Principia Mathematica’, J. Hist. Astron. 1, 116-38 (1970); and more generally in pertinent footnotes in vol. 6 of my edition of The mathematical papers of Isaac Newton (Cambridge University Press, 1974).
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  • $\begingroup$ The first Whiteside quote strikes me as gratuitous and rather dismissively pompous. $\endgroup$ – Tom Copeland Jun 3 '17 at 19:36
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    $\begingroup$ I don't really distinguish limiting arguments using tangent line segments from the basic notions of the calculus (having myself a physics background). It's irrelevant to me whether the convenience of the notation and terminolgy of fluxions and more algebraic algorithms were used in the Principia or earlier work of Newton. $\endgroup$ – Tom Copeland Jun 3 '17 at 19:53
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    $\begingroup$ On page 120 of Whiteside's paper: Clifford Truesdal's claim that the Principia is "a book dense with the theory and application of the infinitesimal calculus" is just. $\endgroup$ – Tom Copeland Jun 5 '17 at 1:12
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    $\begingroup$ @Tom Copeland : Tom Copeland's comment seems good to me, and in substance seems also yet closer in agreement with the thought underlying Truesdell's remark, from which a little more may relevantly be quoted even if it puts the matter less tactfully: ". . . a modern mathematician, according little respect to those who confuse notations with notions, finds the Principia a book dense with the theory and application of the infinitesimal calculus." (Essays in the History of Mechanics, 1968, 99 (at n.4)). $\endgroup$ – terry-s Jun 5 '17 at 11:41
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    $\begingroup$ About DT Whiteside as "the authority": true that practically anybody interested in Newton's work owes Whiteside a very considerable debt for his scholarship and the immense work and commitment he gave to produce it. But that doesn't mean his pronouncements deserve to be treated as if on tablets of stone from heaven. Occasionally he was unduly (and aggressively -- see above) dismissive and made careless mistakes, his commentary not even quite free from howlers, unfortunately. His reputation, deserved on many points, can lend undeserved credibility even to some myths about Newton. $\endgroup$ – terry-s Jun 5 '17 at 12:00

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