Newton and Mathematicians:

Sterling detail.
Lineæ tertii ordinis Neutonianæ, sive illustratio tractatus d.
Neutoni de Enumeratione linearum tertii ordinis. (Oxford, 1717), detail.

Newton’s influence on mathematical physics in the early eighteenth century has often been overshadowed by the Leibnizian calculus. Newton’s geometrical style certainly proved unattractive to later mathematicians who favoured a more abstract approach but this is to ignore the impact made by Newton’s mathematics at the time – and, indeed, to simplify his legacy. For, as Guicciardini rightly argues (2004), ‘the Newtonian mathematical heritage was, in fact, complex and fractured.’ This was true not only because Newton’s mathematical manuscripts and printed books offered a range of sometimes competing approaches, but also because the motivations behind the publication of Newton’s mathematical works were not always straightforward. The priority dispute with Leibniz loomed in the background behind the publication of many of his works.

Newton’s appreciation for a geometrical approach may represent the influence of Isaac Barrow (1630-1677), his predecessor in the Lucasian chair. Newton began developing it in the 1670s where his geometrical approach to the method of fluxions overtook his earlier ‘analytical method of fluxions’ so evident in the new analysis of ‘De methodis serierum et fluxionum’ of 1670-71. His earlier innovative method of series and fluxions were hardly used in his Principia of 1687 which concentrated instead on a geometrical version of the calculus of fluxions.

Lineæ tertii ordinis Neutonianæ, sive illustratio tractatus d.
Neutoni de Enumeratione linearum tertii ordinis. (Oxford, 1717), titlepage.

Worth collected books by two Scottish mathematicians, both dedicated followers of Newton. The first, James Stirling (1692-1770), was the author of Lineae tertii ordinis Neutonianae printed at Oxford in 1717. Stirling had come to Oxford in 1711, matriculating at Balliol College and gaining not one but two scholarships. Though he lost both of these in 1715 due to his Jacobite sympathies he remained in Oxford until 1717, the year in which he travelled to the Continent with Nicholas Tron, the Venetian ambassador to whom he had dedicated his work.

The Lineae tertii ordinis Neutonianae (1717) sought to extend Newton’s work on plane curves while simultaneously examining three problems proposed by Leibniz. Guicciardini (1989) states that it was ‘the first systematic work devoted to Newton’s classification of cubics’. Stirling continued to work on differential method, communicating his findings to the Royal Society and eventually publishing them in his magnum opus, Methodus differentialis, sive, Tractatus de summatione et interpolatione serierum infinitarum (London, 1730), a work which was not collected by Worth.

Geometria Organica (London, 1720), titlepage.

Worth’s second text was by Stirling’s most important mathematical correspondent, Colin MacLaurin (1698-1746), the Scottish mathematician who did more than most to popularize Newtonianism. Worth was dead by the time MacLaurin produced his two seminal Newtonian works: the Treatise of Fluxions (1742), which defended the Newtonian version of the calculus against the criticism of Berkeley, and his An Account of Sir Isaac Newton’s Philosophical Discoveries (1748). It was, however, clear even before the advent of these two works that MacLaurin was a staunch follower of Newton. His MA thesis in 1713 had been on the subject of gravity and within a few years he had become a protégé of Newton’s. Appointed in 1717 as Professor of Mathematics at Marischal College, Aberdeen, MacLaurin had concentrated on ‘organic geometry’ and published some of his findings in the Philosophical Transactions: ‘On the Construction and Measure of Curves’ (1718) and ‘A New Method of Describing all Kinds of Curves’ (1719). These papers attracted the attention of Newton and it was Newton who encouraged him to publish the fruits of his research as Geometria Organica (1720), a text which was collected by Worth.

The book was dedicated to Newton and in his preface MacLaurin stoutly defended the study of pure geometry and asserted the underlying importance of mathematics for natural philosopher. Clearly utilizing Newton’s work in the former’s Enumeratio linearum tertii ordinis, MacLaurin went beyond him to invent pedal curves. The book is divided into two parts: the first part deals with curves of all orders, and opens with a discussion of Newton’s organic description of conics. This theme is also taken up in part II where MacLaurin introduces his theory of pedal curves in Chapter III. His Geometria organica proved to be a very influential work and MacLaurin himself an influential purveyor of Newtonianism. His award of a prize in 1725 by the Paris Académie Royale des Sciences ensured not only his own fame but also aided the spread of Newtonian concepts on the continent.

His appointment in the same year as a deputy for James Gregory (1666-1742), then professor of mathematics at the University of Edinburgh, was due to Newton’s intervention and it is not surprising to learn that, following Newton’s death in 1727, John Conduitt approached MacLaurin to collaborate with him on a biography of Newton (later published posthumously in 1748 as An Account). MacLaurin’s commitment to Newtonianism runs through all his works and may be seen in the notes of his lecture course at Edinburgh, which ranged from Euclid to Newton’s Principia. Tweedie (1915), citing the Scots Magazine for August 1741, gives us the following summary of his mathematical course at Edinburgh:

‘He taught three classes during the same session, and sometimes a fourth upon such of the abstruse parts of the science as are not explained in the former three. In the first, he began with demonstrating the grounds of vulgar and decimal arithmetic; then proceeded to Euclid; and after explaining the six books, with the plain trigonometry, and use of the tables of logarithms, sines, etc., he insisted on surverying, fortification, and other practical parts, and concluded with the elements of algebra. He gave geographical lectures, once in the fortnight, to this class of students.

In the second, he repeated the algebra again from its principles, and advanced farther in it; then proceeded to the theory and mensuration of solids, spherical trigonometry, the doctrine of the sphere, dialling and other practical parts. After this he gave the doctrine of the conic sections, with the theory of gunnery, and concluded with the elements of astronomy and optics.

In the third class, he began with perspective; then treated more fully of astronomy and optics. Afterwards he prelected on Sir Isaac Newton’s Principia and explained the direct and inverse method of fluxions. At a separate hour he began a class of experimental philosophy, about the middle of December, which continued thrice every week till the beginning of April; and at proper hours of the night described the constellations, and shewed the planets by telescopes of various kinds.’

Geometria Organica (London, 1720), Table VII.

Sources

Gillespie, Charles Coulson (ed) (1976), ‘Stirling, James’, Dictionary of Scientific Biography (New York), pp. 67-70.

Grabiner, Judith V. (2004), ‘Newton, MacLaurin, and the Authority of Mathematics’, The American Mathematical Monthly 111, no. 10, pp. 841-852.

Guicciardini, Niccolò (1989), The Development of Newtonian Calculus in Britain 1700-1800 (Cambridge).

Guicciardini, Niccolò (2004), ‘Dot-Age: Newton’s Mathematical Legacy in the Eighteenth Century’ Early Science and Medicine 9, no. 3, Newtonianism: Mathematical and ‘Experimental’, pp 218-256.

Sageng, Erik Lars (2006), ‘MacLaurin, Colin (1698–1746)’, Oxford Dictionary of National Biography, Oxford University Press.

Scott, J. F. (1973), ‘MacLaurin, Colin’, Dictionary of Scientific Biography (New York), pp. 609-612.

Tweedie, Charles (1915), ‘A Study of the Life and Writings of Colin MacLaurin’, The Mathematical Gazette 8, no. 119, pp. 133-151.

Tweedle, Ian (1998), ‘The Prickly Genius: Colin MacLaurin (1698-1746), The Mathematical Gazette 82, no. 495, pp 373-78.

Tweddle, Ian (2004), ‘Stirling, James (1692–1770)’, Oxford Dictionary of National Biography, Oxford University Press.