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Editing the Principia

‘At last, in 1687, Newton resolved to unveil himself and to reveal what he was; the Mathematical Principles of Natural Philosophy appeared. This book, in which the most profound geometry provides a foundation for a wholly novel physics, did not at once receive all the applause that it deserved and which one day it was to receive. As it is a very learned work, very sparing in words, and with propositions arising in it very swiftly from the principles, so that the reader is forced to supply by himself all the connection between the two, time had to pass before the public could understand it. Great geometers could do so only after studying it with care; mediocre geometers only embarked upon it when excited by the testimony of their greater brethren; but in the end, when the book was sufficiently well known, all the applause that it had won so slowly broke out on all sides, constituting a single paean of praise. Everyone was struck by the original intelligence shining through the book; by the creative spirit which has throughout a fortunate century been shared only with three of four men among all the most learned nations.’

Bernard le Bovier de Fontenelle’s Eloge [cited in Rupert Hall, 1999).

Edward Worth collected not one but two copies of Newton’s Philosophiæ naturalis principia mathematica: the second edition printed at Cambridge in 1713 and the third edition printed at London in 1726. Worth was clearly trying to collect copies of the Principia but his pursuit of the first edition of 1687 may have been foiled by its limited print run: Hall (1958 and 1974) suggests that the 1687 edition numbered no more than 300 copies and that it ‘rapidly became an unobtainable book’.

Philosophiæ naturalis principia mathematica (London, 1726), titlepage.

As the title page of the 1713 edition suggests, there were other reasons, beyond increased availability, that might have drawn Worth to it: it had been edited and extended to include, among other things, the famous General Scholium. As the above quotation by de Fontenelle suggests, it also took some time for the Principia to make its mark. The difficulties associated with reading the text are well known and those who did devote their attention to it were anxious that a second edition be forthcoming to clear up some of the mistakes they had been able to find in the text. As Newton’s annotations to his own copy of the first edition attest, he himself was also thinking about another edition but his nervous breakdown delayed the process. Likewise, as Hall (1958) relates, Christiaan Huygens (1629-1695) and Fatio de Duillier (1664-1753) were also preparing lists of mistakes (printer’s or otherwise), which they felt should be corrected in any new edition. Some commentators, such as Gottfried Wilhelm Leibniz (1646-1716) and Johann Bernouilli (1667-1748), felt that more extensive changes were required. In particular, the latter drew attention to a problem with proposition X in Book II: ‘Let a uniform force of gravity tend straight toward the plane of the horizon, and let the resistance be as the density of the medium and the square of the velocity jointly; it is required to find, in each individual place, the density of the medium that makes the body more in any given curved line, and also the velocity of the body and resistance of the medium’.*


Philosophiæ naturalis principia mathematica (Cambridge, 1713), beginning of Proposition 10.

Bernouilli, a staunch defender of Leibnizian claims in the calculus dispute, pointed out that Newton’s solution to proposition X (which he had based on his method of series) had been false. He was quick to use this against Newton, arguing that not only had Newton not invented calculus but that he was incapable of inventing it. When Bernouilli’s criticism of proposition X was made known to Newton by his nephew Nicolas Bernouilli in September 1712, Newton acted on the information (but did not acknowledge the source). As Hall (1958) relates, a memorandum in Cambridge University Library states that:

‘Mr Newton corrected the error himself, shewed him the correction & told him that the Proposition should be reprinted in the new Edition wch was then coming abroad. The Tangents of the Arcs GH and HI are first moments of the Arcs FG and FH & should have been drawn the same way with the motion describing those arcs, whereas through inadvertency one of them had been drawn the contrary way, & this occasioned the error in the conclusion.’

Given the timing of Bernoulli’s announcement, proposition X provided Newton with a challenge: to present the new findings in the space allocated to his initial version, since the text for the Proposition section had already been printed off. As Whiteside (1981) has shown, Newton spent over a hundred manuscript pages working this out. In the end he managed to do it by adding in a new leaf (pp. 231/232). This had the advantage of correcting his mistake but it drew silent attention to his late change to the work (and therefore, indirectly, his debt to Bernouilli), a fact that did not go un-noticed in the Acta eruditorum.

The editor of the second edition, Roger Cotes (1682-1716), likewise had a difficult task: Newton’s inability to accept criticism was well known and the author had already made it clear that he did not expect Cotes to clear up every mistake – something Newton felt should be left to the reader’s own judgment. Cotes, who had been appointed Plumian professor of astronomy at Cambridge, was keen to produce a properly edited text and did not limit himself to Newton’s initial suggestion that he should correct ‘only such faults as occur in reading over the [proof] sheets to correct them as they are printed’. His correspondence with Newton demonstrates that while there was little debate between the two men over Book I and the first part of Book II that after that Cotes became increasingly interventionist, attempting to make Newton’s style easier to understand. In the main, Cotes concentrated on correcting mathematical errors and did not try too forcefully to change Newton’s mind on the addition of the General Scholium, which Newton had sent him late in the process of publication – on 28 March 1713. The delay between 1709 and 1713 may be explained by the long drawn out editing process which Cotes and Newton engaged in. Hall (1974) has noted the contrast between the store of manuscripts Newton left behind and his relatively few published works. Newton found it difficult to commit himself in print and was frequently unhappy with the published version of his great works. This clearly impacted on the publishing process of the second edition.

Cotes’ preface to the second edition concentrated on defending Newtonianism against Cartesianism and Leibnizian claims. Cotes’s attack on Cartesian vortices was clearly in line with Newton’s work since, as Cohen (1999) suggests, the very name of the Principia was a challenge to Descartes’ famous Principia Philosophiae (1644). Newton’s General Scholium which concluded Book 3, not only sought to provide a proper conclusion to the work (something which had been missing in the original edition which had ended with an examination of the orbits of comets), but also to reinforce his condemnation of Cartesianism. This is evident from the very first line which declares that ‘The hypothesis of vortices is beset with many difficulties’.

In the General Scholium Newton went beyond mathematics and physics to argue that the universe could not have come into being ‘without the design and dominion of an intelligent and powerful being’, framing his analysis of the attributes of God by declaring that ‘to treat of God from phenomena is certainly a part of ‘natural’ philosophy’. This section was followed by his famous summary of his conclusions about gravity:

‘Thus far I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity. Indeed, this force arises from some cause that penetrates as far as the centers of the sun and planets without any diminution of its power to act, and that acts not in proportion to the quantity of the surfaces of the particles on which it acts (as mechanical causes are wont to do) but in proportion to the quantity of solid matter, and whose action is extended everywhere to immense distances, always decreasing as the squares of the distances. Gravity toward the sun in compounded of the gravities towards the individual particles of the sun, and at increasing distances from the sun decreases exactly as the squares of the distances as far out as the orbit of Saturn, as is manifest from the fact that the aphelia of the planets are at rest, and even as far as the farthest aphelia of the comets, provided that those aphelia are at rest. I have not as yet been able to deduce from hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies, and the laws of motion and the law of gravity have been found by this method. And it is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.’*

Philosophiæ naturalis principia mathematica (Cambridge, 1713), titlepage

Worth’s copy of the third edition of the Principia contains his brief note of Newton’s death on the frontispiece portrait of the edition. The 1726 edition had been edited by Henry Pemberton (1694–1771), who, like Worth, was ‘a physician by profession, a mathematician by avocation’ (Hall 1974). Why Newton appointed Pemberton and not a better known mathematician is unclear. Perhaps he felt that Pemberton was less likely to argue every point with him and though Pemberton’s editing process added little to the work, his edition allowed Newton to add further material which he had been preparing in the latter part of the 1710s. Newton outlined these changes in his preface to the third edition:

‘In this third edition, supervised by Henry Pemberton, M. D., a man greatly skilled in these matters, some things in the second book concerning the resistance of mediums are explained a little more fully than previously, and new experiments are added concerning the resistance of heavy bodies falling in air. In the third book, the argument proving that the moon is kept in its orbit by gravity is presented a little more fully; and new observations, made by Mr Pound, on the proportion of the diameters of Jupiter to each other have been added. There are also added some observations of the comet that appeared in 1680, which were made in Germany during the month of November by Mr. Kirk, and which recently came into our hands; these observations make it clear how closely parabolic orbits correspond to the motions of comets. The orbit of that comet, by Halley’s computations, is determined a little more accurately than heretofore, and in an ellipse. And it is shown that the comet traversed its course through nine signs of the heavens in this elliptical orbit just as exactly as the planets move in the elliptical orbits given by astronomy. There is also added the orbit of the comet that appeared in the year 1713, which was calculated by Mr Bradley, professor of astronomy at Oxford.’*

Perhaps the most important change in the third edition was the addition of the ‘Leibniz Scholium’ in Book II Section II. Here Newton deliberately claimed the invention of the calculus, seeking to continue the debate even though Leibniz had been dead nearly ten years. Given that the vast majority of the Principia eschewed Newton’s ‘new analysis’ and instead concentrated on geometrical proofs, using it as a text to support his position in the calculus priority dispute with Leibniz was a sometimes fraught affair. At the height of the calculus dispute Newton tried to explain his concentration on geometrical proof in the Principia in a review of John Collins’ Commercium epistolicum, which he printed anonymously in the Philosophical Transactions in 1715: ‘By the help of this new Analysis Mr Newton found out most of the Propositions in his Principia Philosophiae. But because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically that the systeme of the heavens might be founded upon good Geometry. And this makes it now difficult for unskilful men to see the Analysis by wch those Propositions were found out.’ As Guicciardini concludes (1999), ‘external and internal evidence seems to be against Newton’s statement’ that the Principia was initially based on his ‘analytical method of fluxions’.

Much has been written about the Principia (including Newton’s many revisions between his various editions). As Cohen (1999) relates, perhaps the clearest indication of what Newton hoped to achieve may be seen in a draft preface which he wrote prior to the third edition:

‘The aim of the Book of the Principles was not to give detailed explanations of the mathematical methods, nor to provide exhaustive solutions to all the difficulties therein relating to magnitudes, motions, and forces, but to deal only with those things which relate to natural philosophy and especially to the motions of the heavens; and thus what contributed little towards this end I have either entirely omitted or only lightly touched on, omitting the demonstrations.’

Philosophiæ naturalis principia mathematica (London, 1726), Mandate to print

*[Translations from I Bernard Cohen, and Anne Whitman, (1999) Isaac Newton. The Princpia. Mathematical Principles of Natural Philosophy. A New Translation (University of California Press)].


Bernard Cohen, I and Whitman, Anne (1999) Isaac Newton. The Princpia. Mathematical Principles of Natural Philosophy. A New Translation (University of California Press).

Feingold, Mordechai (2004), The Newtonian Moment. Isaac Newton and the Making of Modern Culture (New York and Oxford).

Guicciardini, Niccolò (1999), Reading the Principia. The Debate on Newton’s Mathematical Methods for Natural Philosophy from 1687 to 1736 (Cambridge).

Hall, A. Rupert (1958), ‘Correcting the Principia’, Osiris 13, pp. 291-326.

Hall, A. Rupert (1974), ‘Newton and his Editors: The Wilkins Lecture, 1973’, Notes and Records of the Royal Society of London, 29, no. 1, pp. 29-52.

Hall, A. Rupert (1999), Isaac Newton. Eighteenth-Century Perspectives (Oxford).

Johnson, W. (2004), ‘Pemberton, Henry (1694–1771)’, Oxford Dictionary of National Biography, Oxford University Press.

Westfall, Richard S. (2004), ‘Newton, Sir Isaac (1642–1727)’, Oxford Dictionary of National Biography, Oxford University Press.






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