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Newton and Leibniz

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Newton and Leibniz

‘He laid the foundation of all his discoveries before he was twenty-four years old, and communicated most of them in loose tracts and letters to the Royal Society, of which an ample account is given in the Commercium Epistolicum. And now I am on that subject, give me leave, sir, to observe to you, that since many new lights have appeared, relating to that dispute, it is expected from your candour and justice, that you will, in some measure, recall several passages in your works, printed before these discoveries were made. In your Eloge upon the Marquis de L’Hopital, you say, ‘Le calcul differential inventé par Monsieur Leibnitz et en meme-tems par Monsieur Newton.’ I am confident you are persuaded (as I am credibly informed the Germans now are) not only that Sir Isaac invented the method of fluxions, many years before Mr Leibnitz knew any thing of it, but that Mr. Leibnitz took it from him. If the chain of circumstances, and the clear evidence which has been laid before the world, were not sufficient, Mr. Leibnitz’s manner of defending himself would convince every body of what I have advanced.’

John Conduitt’s Memoir [cited in Hall, 1999].


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Portrait of Gottfried Wilhelm Leibniz: Courtesy of the National Library of Medicine.

Gottfried Wilhelm Leibniz (1646-1716) had studied at the Universities of Leipzig, Jena and Altdorf, gaining his doctorate from the latter university in 1666. While he received some education in mathematics at these universities it was really his sojourn in Paris from 1672 to 1676, where he had the chance to meet Christiaan Huygens (1629-1695) and other members of the Académie Royale des Sciences, which encourage his studies. His discovery of the calculus dates from the latter end of his stay though his rules of the differential calculus would only be published in 1684 in the Acta Eruditorum, a scientific journal which he had helped found. Unlike the more secretive Newton, Leibniz was eager to promote his differential calculus and it soon gained adherents such as the Marquis de l’Hospital (1661-1704) in France and the Bernoulli brothers in Basel. Gucciardini (1999) reminds us that it was followers like Johann Bernoulli (1667-1748 ) who would promote and develop the Leibnizian calculus in the early eighteenth century, by focusing on developing the rules of differentiation and integration of transcendental quantities.Newton and Leibniz2
Collins’ Commercium Epistolicum (London, 1722), titlepage.

Worth purchased two works dealing with different aspects of Newton’s disputes with Leibniz. The first was the second edition of John Collins’ Commercium Epistolicum (London, 1722) which had initially been published at London in 1712 by the Royal Society. This was a report on the priority question and it unsurprisingly found in favour of Newton. Collins (1626-1683), a mathematician, was the author of texts on a wide range of topics: An Introduction to Merchants’ Accompts (London, 1652), The Sector on a Quadrant (London, 1658), Geometrical Dialling (London, 1659) and, in the same year, The Mariners’ Plain Scale (London). He had been elected a Fellow of the Royal Society in 1667 and helped with the publications of other members such as Isaac Barrows’ Lectiones opticae (1669) and Lectiones geometricae (1670) and John Wallis’ Mechanica (1669-71) and Algebra (1685). His zeal in constructing a scientific correspondence with a wide range of mathematicians in Britain and the rest of Europe led Isaac Barrow to call him the ‘English Mersenne’ and though he was dead prior to the publication of the Commercium Epistolicum it was essentially made possible due to his efforts in collecting letters (dating from the 1670s) from Newton on the subject of the calculus.

As Conduitt’s quotation makes clear, Newton’s priority dispute with Gottfried Wilhelm Leibniz was clearly one of the motivating factors behind Conduitt’s account of Newton’s life and works. The calculus dispute generated much publication and was complicated by the fact that both men had produced various versions of their calculi at different times. Newton had started with his analytical method of fluxions in the 1670s but moved away from this to focus on first and ultimate ratios, while Leibniz had developed his theories in a range of letters and papers. As Bertoloni Meli argues (1993), on one level, there was relatively little difference between the Leibnizian calculus and Newton’s analytical calculus, a view with which Guicciardini (1999) concurs, pointing out that both men had similar approaches to the existence of infinitesimals. But whereas Newton sought to eliminate infinitesimals from the Principia, Leibniz utilised them. Likening the difference between a Newtonian and a Leibnizian mathematician to that between two computer programmers using different computer languages, Bertoloni Meli suggests that, though the end results might be the same, the process of getting there was radically different. Difference also lay in the uses made of the methods and in interpretations of results.

Leibniz himself sought to apply the differential calculus to natural philosophy in a number of papers in the Acta Eruditorum in January and February 1689. Though Leibniz stated that he first viewed the Principia following these papers (in April 1689 in Rome), it seems clear that he had access to the Principia prior to this. The two papers Leibniz produced for the Acta Eruditorum demonstrate his initial attempts to use his differential calculus to examine what he would later call ‘dynamics’, a general theory of motion. His papers make it clear that he disagreed with Newton’s idea of a gravitational force, preferring instead the Cartesian concept of a fluid vortex.


Newton and Leibniz3

Recueil de diverses pieces, sur la philosophie, la religion naturelle, l’histoire, les mathematiques, &c. Par Mrs. Leibniz, Clarke, Newton, & autres autheurs célèbres (Amsterdam, 1720), titlepage.

In a sense, the priority dispute was just the tip of the iceberg when it came to differences between Newton and Leibniz, an intellectual iceberg which gradually came into view in publications such as the other book collected by Worth on the subject: Pierre Des Maizeaux’s Recueil de diverses pieces sur la Philosophie, La Religion Naturelle, l’Histoire, les Mathématiques, &c. Par Mrs. Leibniz, Clarke, Newton & autres Auteurs célèbres (Amsterdam, 1720). This two-volume work contains the Leibniz-Clarke correspondence from 1715-6 and a number of letters concerning the calculus dispute and other issues. The Leibniz-Clarke correspondence had been facilitated by Princess Caroline of Brandenburg-Ansbach (1683-1737) who, as Princess of Wales, had been the recipient of a letter from Leibniz in 1714 which had commented on the decline of natural religion in her new home. Princess Caroline had initially been introduced to Leibniz while she was at the Brandenburg court and later, as princess of Hanover had, with the encouragement of Electress Sophia, given him patronage. As Princess of Wales, and later as Queen of England, Caroline would, however, later prove to be a staunch supporter of Newton’s cause – a fact demonstrated both by her support for Samuel Clarke and by the dedication of Newton’s posthumous work, The Chronology of Ancient Kingdoms amended to her. Clarke had originally published the correspondence under the title A collection of papers, which passed between the late learned Mr. Leibnitz, and Dr. Clarke, in the years 1715 and 1716 (London, 1717). Worth’s copy of the correspondence was by Pierre Des Maizeaux, who became a Fellow of the Royal Society in 1720. Des Maizeaux’s book did not earn him the friendship of Newton who opposed his publication of the work.

One of the chief protagonists, Samuel Clarke (1675-1729), had first drawn attention to his credentials as a follower of Newton when he defended a proposition from the Principia as part of his B.A. disputation at Cambridge in 1695. He followed this by including Newtonian notes to his translation of Rohault’s Traité de physique (1697). At the request of the author, Clarke was responsible for translating Newton’s Opticks into Latin in 1706 – and is it clear that he and Newton were firm friends. Clarke’s Boyle Lectures of 1704 and 1705 had already established him as a leading theologian and his credentials as a Newtonian were second to none. It was, therefore, natural that he should be called on to engage with Leibniz’s criticism that Newtonianism had led to a decline in natural religion in England. Leibniz’s comment to Bernoulli in 1716, that he was ‘now grinding a philosophical axe with Newton or, what amounts to the same thing, with his champion, Clarke, a royal almoner’ illustrates that he was fully aware that Clarke and Newton were working together in response to Leibniz’s charges. Indeed, it might be said that at times Clarke was more Newtonian than Newton himself. As Perl (1969) suggests, Clarke ‘characteristically asserts categorically what Newton has said conjecturally or hypothetically’.

At the heart of the Leibniz-Clarke/Newton dispute lay a completely different philosophical world view. For Cassirer (1943), the chief difference lay in their philosophical approaches, Newton emphasising facts and Leibniz emphasising forms. Iltis (1973) draws attention to four major areas of disagreement: in theology, philosophy of matter, causality, and mechanics. Theologically there was an intellectualist-voluntarist clash with Leibnizians arguing that a world which required God’s intervention undermined the concept of His foresight, while Newtonians were equally exercised by the need to stress the capability of God to intervene, should He so will it. Linked to this were their philosophies of matter for Newton argued that the ultimate source of force and motion was God, given that matter was dead and static, whereas followers of Leibniz argued that matter was alive and held within it a force for change. Arguments over rival concepts of force informed a series of experiments undertaken by followers of both men in the period 1718-28, and particularly following ‘s Gravesande’s conversion to a Leibnizian viewpoint in 1722. This vis viva (‘living force’) controversy over the ‘force’ of a moving body predated the epistolary exchanges of Leibniz and Clarke in 1715-16. In 1686 Leibniz had published his own views on vis viva as part of his critique of Descartes’ Principia philosophiae of 1644 which had advocated an absolute quantity of motion. As Iltis (1971) explains, Leibniz in his 1686 ‘Brevis demonstatio’, argued that what was absolute and indestructible was not quantity of motion m | v| but vis viva, mv2.

Clarke’s interest did not stop with Leibniz’s death in 1716. In 1728 he was still writing vociferously in the Philosophical Transactions ‘concerning the proportion of velocity, and force in bodies in motion’ (and using much the same arguments as he had against Leibniz in 1716):

‘It has often been observed in general that Learning does not give men Understanding; and that the absurdist things in the world have been asserted and maintained by persons whose education and studies should seem to have furnished them with the greatest extent of Science.

That knowledge in many languages and Terms of Art and in the History of Opinions and Romantick Hypotheses of Philosophers, should sometimes be of no effect in correcting Man’s Judgment, is not so much to be wondered at. But that in Mathematicks themselves, which are a real Science, and founded in the Necessary Nature of Things; men of very great abilities in abstract computations, when they come to apply those computations to the Nature of Things, should persist in maintaining the most palpable absurdities, and in refusing to see some of the most evident and obvious truths; is very strange.

An extraordinary instance of this, we have had of late years in very eminent Mathematicians, Mr. Leibniz, Mr. Herman, Mr. ‘s Gravesande, and Mr. Bernoulli; (who in order to raise a Dust of Opposition against Sir Isaac Newton’s philosophy, the glory of which is the application of abstract Mathematics to the real phenomena of Nature, ) have for some years insisted with great Eagerness, upon a principle which subverts all Science, and which may easily be made to appear… to be contrary to the necessary and essential Nature of Things.

What they contend for is, That the Force of a Body in Motion is proportional, not to its Velocity, but to the Square of its Velocity.

The Absurdity of which Notion I shall first make appear and then shew what it is that had led these gentlemen into Errour.’


Bertoloni Meli, Domenico (1993), Equivalence and priority: Newton versus Leibniz (Oxford).

Cassirier, Ernst (1943), ‘Newton and Leibniz’, The Philosophical Review, 53, no. 4, pp 366-91.

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

Gascoigne, John (2004), ‘Clarke, Samuel (1675–1729)’, Oxford Dictionary of National Biography, Oxford University Press.


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

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

Ilstis, Caroline (1971), ‘Leibniz and the Vis Viva Controversy’, Isis 62, no. 1, pp. 21-35.

Illtis, Caroline (1973), ‘The Leibnizian-Newtonian Debates: Natural Philosophy and Society Pyschology’, The British Journal for the History of Science, 6, no. 4, pp 343-77.

Perl, Margula R. (1969), ‘Physics and Metaphysics in Newton, Leibniz, and Clarke’, Journal of the History of Ideas 30. no. 4, pp. 507-526.

Scriba, Christoph J. (2004), ‘Collins, John (1626–1683)’, Oxford Dictionary of National Biography, Oxford University Press.

Taylor, Stephen (2004), ‘Caroline [Princess Caroline of Brandenburg-Ansbach] (1683–1737), queen of Great Britain and Ireland, and electress of Hanover’, Oxford Dictionary of National Biography, Oxford University Press.

Westfall, Richard S. (1980), Never at Rest. A Biography of Isaac Newton (Cambridge).

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