MacLaurin, Colin

(1698-1746), mathematician and natural philosopher

by Erik Lars Sageng

© Oxford University Press 2004 All rights reserved

MacLaurin, Colin (1698-1746), mathematician and natural philosopher, was born in Kilmoden, Argyll, Scotland, in February of 1698, the third son of John MacLaurin (1658-1698), a minister in the Scottish church, and his wife, Mary, daughter of John Cameron. He lost both his parents in childhood, his father dying only six weeks after Colin's birth. After his mother's death in 1707, his uncle Daniel MacLaurin, also a minister, undertook the care of Colin and his elder brother John MacLaurin, the second brother, Daniel, having died young. John was sent to the University of Glasgow, and became a minister in that city. Colin in 1709 followed his brother to the University of Glasgow, where he continued the classical education he had begun in the parish schools, and began mathematical studies under the direction of Robert Simson (1687-1768). The influences of Simson's interest in the history of mathematics, his enthusiasm for classical Greek methods, and his piety can all be seen in MacLaurin's subsequent career.

In 1713 MacLaurin defended a thesis on the power of gravity, and was awarded the degree of master of arts. In this thesis he refuted a number of theories as to the cause of gravity and argued that it can only be accounted for as the direct result of divine will; he demonstrated in the manner of Isaac Newton's Principia that the observed law of terrestrial gravity suffices to explain lunar and planetary motion; and he argued from the evident grandeur and fitness of the creation to the omnipotence and benevolence of the creator. At about this same time, MacLaurin began a correspondence with Colin Campbell (1644-1726), something of a Mersenne figure for Scottish and English mathematicians and natural philosophers in this period. In 1714 MacLaurin sent Campbell several mathematical papers, in one of which he provided fluxional demonstrations to a number of propositions from the Principia, and another entitled De viribus mentium bonipetis. In this latter paper, MacLaurin applied the method of fluxions to the analysis of the forces with which our minds are attracted to various goods, in a manner analogous to the mathematical analysis of celestial mechanics.

MacLaurin spent another year at the university after earning his master's degree, reading divinity. He left the university in 1714, and continued his studies independently at the home of his uncle in Kilfinan, where he lived until 1717, when, at the age of nineteen, he was appointed to the chair of mathematics at Marischal College in Aberdeen.

Mathematical career
Colin MacLaurin was a younger contemporary, and to some extent a protégé of Isaac Newton, and he wrote the first thorough, systematic, axiomatic development of the method of fluxions, the Newtonian version of the calculus. However, his relative isolation in Scotland--and his heavy teaching duties, about which he persistently complained--denied him the fame commensurate with his stature. He came to the attention of Newton's circle in London early in his career when he published two papers on the construction and mensuration of curves in the Philosophical Transactions for 1718 and 1719. Edmond Halley, then secretary of the Royal Society, invited him to go to London in 1719. He was admitted to membership in the Royal Society (1719), and often visited Newton, who encouraged him to publish the rest of his work on the description of curves. MacLaurin did so in his first major book, the Geometria organica, which was published under Newton's imprimatur in 1720. In its dedication and preface MacLaurin praises his patron Newton for having determined the proper relationship between mathematics and natural philosophy, and he defends and justifies the pursuit of pure geometry. He asserts the dependence of natural philosophy, the ultimate purpose of which is to reveal God's work, on mathematics, and argues that mathematics and mathematical relationships underlie not only attempts to investigate nature, but nature herself.

MacLaurin's Geometria organica is organic in the sense of instrumental; it treats curves from the point of view of their theoretical construction by means of abstract instruments. MacLaurin uses this method to generalize the Newtonian organic generation of conics to all orders of curves, and applies it to the classification of curves on the model of Newton's Enumeratio linearum tertii ordinis. Among other things in this treatise, he invents pedal curves, he states Cramer's paradox, thirty years before Cramer published it and attributed it to MacLaurin, he describes many of the particular curves thought to have been discovered in the nineteenth century, and he treats many applications to mechanics, including problems of centripetal forces and motion in resisting media. The influence of Geometria can be seen in the work of the later geometers Poncelet, Chasles, Steiner, Grassmann, and Salmon. Also in 1720, MacLaurin published De linearum geometricarum proprietatibus, in which he studies the curvature and harmonic properties of curves, and properties of their tangents and secant lines. Poncelet was to make extensive use of this work in his Traité des propriétés projectives des figures (1822).

MacLaurin in 1721 made his second trip to London, where he was engaged by Lord Polworth, the king's ambassador to the Congress of Cambrai, as a tutor and travelling companion for his son, who was to accompany his father to Cambrai and then set off in the following spring on a tour of France. MacLaurin returned to Aberdeen to finish the year's teaching and to obtain a leave of absence, but he left at the beginning of the summer vacation in May without such permission, and, apparently with no further correspondence with the university officials, stayed away for three years.

MacLaurin continued his mathematical work during this time--he wrote an important treatise on the percussion of bodies for a prize-contest on the subject set up by the Paris Académie Royale des Sciences, and he also carried on research which was to figure later in the MacLaurin-Braikenridge controversy. Despite significant competition MacLaurin won the Académie Royale's contest, thereby giving a considerable boost to the forces of Newtonianism on the continent. In 1724 MacLaurin's charge was overcome by a fever and died at Montpellier in southern France. Thus deprived of his patron, MacLaurin returned to his job in Aberdeen. He was there in January 1725, when he expressed regret for his long unexcused absence and was accepted back. In January of 1726, however, the governors of Marischal College learned 'by the publick newsprints' (Tweedie, 135) that he had accepted a position at the University of Edinburgh, and his position in Aberdeen was declared vacant.

MacLaurin had been appointed at Edinburgh the previous November as a deputy for James Gregory (1666-1742), who had become too infirm to carry out his teaching duties. There was some difficulty about funds for a second professor of mathematics, since Gregory was to retain the income from the professorship for the rest of his life, more or less as a pension. The appointment was apparently secured for MacLaurin by the intervention of Newton, who offered to pay £20 a year toward MacLaurin's salary if that would facilitate his appointment. MacLaurin had to pay Gregory a considerable sum for consenting to his appointment, and then Gregory failed to die as quickly as expected, surviving and depriving MacLaurin of the full salary until 1744.

MacLaurin taught a three-year course from elementary to advanced mathematics, beginning with arithmetic and Euclid, and working up to the Principia and the method of fluxions. He also taught experimental philosophy, surveying, fortification, geography, theory of gunnery, astronomy, and optics. He wrote his A Treatise of Algebra at this time for use in his courses, although it did not appear in print until after his death. On its publication the Treatise became a popular textbook, going through many editions and remaining in use through the rest of the century.

In 1726 MacLaurin published 'A letter ... concerning equations with impossible roots' in the Philosophical Transactions. In this paper he extended Newton's rules for determining the number of impossible, that is, complex roots of polynomial equations. George Campbell published a paper in the 1728 Philosophical Transactions carrying the same methods further than MacLaurin had done in 1726; Campbell's results seemed to grow directly out of MacLaurin's 1726 paper, or worse, to have been taken from MacLaurin's teaching at Edinburgh, where Campbell had been a private teacher at least since 1719. MacLaurin hastened to publish the continuation of his 1726 piece, 'A second letter ... concerning the roots of equations', in the 1729 Philosophical Transactions, in which he delivered and proved several propositions that Campbell had given without proof in his paper, and several that went far beyond Campbell.

Defence of Newtonian philosophy
Some time after Newton died in 1727 John Conduitt asked MacLaurin to collaborate in the writing of his biography; the project lost momentum with Conduitt's death in 1737 but MacLaurin continued with his part in it and the work was published posthumously in 1748 as An Account of Sir Isaac Newton's Philosophical Discoveries. MacLaurin discusses Newton's method of investigation and of 'philosophizing', as well as his discoveries, and begins with a long history of natural philosophy since the ancients, in which he again asserts the importance for religion of sound natural philosophy. Though a number of other general expositions of Newton's thought were published during the eighteenth century, MacLaurin's Account has long been recognized as the leading authoritative statement of mainstream Newtonianism. As with his prize-winning Treatise on Percussion of 1724, one of the priorities of his Account was to combat the rival natural philosophy promulgated by the followers of Leibniz.

On 18 July 1733 MacLaurin married Anne Stewart, the daughter of the late solicitor general for Scotland, Walter Stewart. They had seven children, four daughters and three sons; two of them, Barbara and Walter, died in 1739, but John MacLaurin (later Lord Dreghorn), Colin, and three daughters survived their father.

In the same year William Braikenridge published Exercitatio geometrica de descriptione linearum curvarum, and in the January-March 1735 issue of the Philosophical Transactions, a continuation under the title, 'A general method of describing curves, by the intersection of right-lines; moving about points in a given plane'. Exercitatio geometrica contained what has come to be called the Braikenridge-MacLaurin theorem. The Philosophical Transactions article treats the construction of cubics and quartics along the lines pursued by MacLaurin in his Geometria organica. These publications gave rise to a priority dispute and to charges of plagiarism, both apparently originated by Braikenridge. MacLaurin responded by publishing a paper dated 27 November 1722, which he declared he had written while en route for Cambrai. This paper contains the Braikenridge-MacLaurin theorem, and carries its generalization much further than Braikenridge had done. MacLaurin's theorem contains Pascal's theorem on a hexagon inscribed in a conic as a special case.

MacLaurin was one of two co-secretaries of the Edinburgh Philosophical Society on its foundation in 1737. It was originally the Edinburgh Society for Improving Medical Knowledge, but on MacLaurin's urging the plan was expanded to include natural philosophy and antiquities, and it became the Royal Society of Edinburgh in 1783.

In 1734 George Berkeley had published The Analyst, or, A Discourse Addressed to an Infidel Mathematician, wherein it is Examined whether the Object, Principles, and Inferences of the Modern Analysis are More Distinctly Conceived, or More Evidently Deduced, than Religious Mysteries and Points of Faith. The main point of Berkeley's pamphlet was that it is wrong for freethinkers to complain of the incomprehensibility of religion, and to throw up mathematics as the model of right reasoning, since mathematical doctrines are also incomprehensible and, not being susceptible of rigorous demonstration, are accepted on faith. As an example, Berkeley fixed on 'the modern analysis', that is, the Newtonian method of fluxions and the Leibnizian differential and integral calculus. Besides objecting to particular demonstrations and procedures, Berkeley's criticism of the method of fluxions amounted to the well-substantiated assertion that it was founded inescapably either on infinitesimals or on a shifting of hypotheses, both of which were logically indefensible.

MacLaurin's magnum opus, the Treatise of Fluxions, published in 1742, was begun as a response to Berkeley's Analyst. MacLaurin founded the method of fluxions on a limit concept drawn from the method of exhaustions in classical geometry, avoiding the use of infinitesimals, infinite processes, and actually infinite quantities, and avoiding any shifting of the hypothesis. In addition, he went on in this treatise of over 760 pages to demonstrate that the method so founded would support the entire received structure of fluxions and the calculus, and could deal effectively with all of the challenge problems then being exchanged between British and continental mathematicians.

MacLaurin's response to Berkeley was informed by his belief that mathematics, properly understood, is necessarily based on real, actually existent entities, which belief made it impossible for MacLaurin--as for Berkeley--to accept a system based on infinitesimals, and by his ideas about the role of mathematics in religion, both directly, as 'the surest bulwark against the skeptics' (letters, Aberdeen University, MS 206), and by way of natural philosophy, the ultimate purpose of which is to support natural religion. These ideas led MacLaurin both to emphasize the importance of sound foundations in such a vital enterprise, and to be offended by the suggestion that mathematics is dangerous to religion, or that mathematicians are liable to lead men to infidelity. He stresses the value of the methods of the ancients, especially their insistence on deduction from clear and distinct principles. He describes geometry's fall from grace with the introduction of indivisibles and infinitesimals--an introduction of mysteries into a science where there should be none--and its salvation at the hands of Newton, who placed these systems on a sound basis with the method of fluxions.

The Jacobite rising and death
MacLaurin took a leading role in preparing the defence of Edinburgh against the highland army of Prince Charles Edward Stuart in the Jacobite rebellion of 1745. MacLaurin was supervising the loading of cannon on 16 September when he heard the news that 'a packed meeting' (Collected Letters, 126) (the volunteers, organized to defend the city, were all out manning the walls) had voted to capitulate. When Edinburgh was occupied MacLaurin fled south into England where he was invited to stay with Thomas Herring, a zealous whig and the archbishop of York. He returned to Edinburgh on 16 November, after the Jacobites' departure, having travelled for three days from York. It is generally reported that MacLaurin arrived back in Edinburgh mortally ill, after a difficult journey both ways on horseback, including a fall and exposure to unpleasant weather. He was, however, able to return to his duties, although, as he reports in a letter of 9 December, he had caught 'the most dangerous cold' (Collected Letters, 132) he had ever had, from which, he said, he was then recovering. He apparently did not recover entirely, and a dropsy of the belly was diagnosed, but resisted treatment; he died on 14 June 1746. He continued to dictate the concluding chapter of his Account of Sir Isaac Newton's Philosophical Discoveries, 'Of the supreme author and governor of the universe, the true and living God', until a few hours before his death, and he remained the natural philosopher to the end, asking his friend and eulogist, the anatomy professor Alexander Monro, to account for various phenomena he experienced as his body failed.

MacLaurin was buried in Greyfriars churchyard, Edinburgh; his tombstone is set in the exterior south wall of the church. His wife and family were left financially insecure, and put Patrick Murdoch in charge of editing MacLaurin's writings, which resulted in the appearance in 1748 of the Account of Sir Isaac Newton's Philosophical Discoveries and A Treatise of Algebra, the former published 'for the author's children' with a large subscription.

ERIK LARS SAGENG

Sources  
P. Murdoch, 'An account of the life and writings of the author', in C. Maclaurin, An account of Sir Isaac Newton's philosophical discoveries, ed. P. Murdoch (1748); repr. with introduction and index of names by L. L. Laundan (1968) [repr. 1968]
C. Tweedie, 'A study of the life and writings of Colin MacLaurin', Mathematical Gazette, 8 (1915-16), 133-51; 9 (1917-19), 303-6; 10 (1920-21), 209
H. W. Turnbull, Bicentenary of the death of Colin Maclaurin (1951)
C. MacLaurin, 'Journall of what pass'd relating to the defence of Edinburgh from the monday September 2d till monday September 16', NL Scot., MS 1342
C. MacLaurin, letters, 1720-43, and journal, 1722-4, U. Aberdeen L., MS 206
The collected letters of Colin MacLaurin, ed. S. Mills (1982)
'A short account of the University of Edinburgh, the present professors in it, and the several parts of learning taught by them', Scots Magazine, 3 (1741), 371-4
marriage contract, Dalry office, register of deeds, 1 July-31 Dec 1746, vol. 160

Archives  
BL, papers relating to spheroids and Nugae Poeticae, Add. MSS 4437, 52247
NL Scot., journals and letters
RS, papers
U. Aberdeen L., MSS and letters
U. Edin. L., special collections division, lecture notes and papers
U. Glas. L., notebooks and letters |  NA Scot., letters to Sir John Clerk
NRA, priv. coll., letters to James Stirling
U. Glas. L., special collections department, papers relating to projected biography by J. C. Eaton
U. Glas. L., special collections department, letters to James Spreull, etc. [copies]

Likenesses  
C. Metz, oils, 1746?; in possession of Jessie Maclaren, Hartwell House, Moffat, Dumfriesshire in 1954 ; negative, U. Glas. L.
McLaron Young, photograph, 1959 (after C. Metz), U. Glas.
Earl of Buchan, pencil and chalk drawing (after J. Ferguson), Scot. NPG
S. Freeman, stipple (after model by Percey), NPG, RS
Page, stipple, NPG
Trotter, stipple, NPG


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