Autograph letter signed, with richly scientific content relating to the ongoing debate about the shape of the Earth, from Gresham College, London, dated 22 June 1738, to the important Scottish mathematician James Stirling.
London: 1738. An autograph letter signed by John Machin (c. 1686–1751), Secretary of the Royal Society and Professor of Astronomy at Gresham College, four pages on a bifolium, written from Gresham College on 22 June 1738 to the Scottish mathematician James Stirling at Leadhills — descended in the Stirling family at Garden House, Stirlingshire, until the dispersal at Lyon & Turnbull, Edinburgh, on 23 October 2025 in the sale entitled The Library of James Stirling, Mathematician. The letter belongs to the closing weeks of one of the most consequential empirical episodes in eighteenth-century natural philosophy: the testing, by direct geodetic measurement at high and low latitudes, of Newton’s prediction in the Principia that the rotation of the Earth must give it the figure of an oblate ellipsoid of revolution. Machin had served Newton on the 1712 Royal Society commission that adjudicated the calculus priority dispute against Leibniz; he had been appointed to the Gresham chair on Newton’s recommendation in 1713; and he occupied the office of Secretary of the Royal Society from 1718 until 1747, the seat from which Newton had directed the Society in the priority decade. Stirling, sixteen years younger, had been elected on Newton’s personal nomination on 3 November 1726, four months before Newton’s death, and was at the time of the present letter the leading British exponent of Newton’s theory of the figure of the Earth. The letter is the documentary inside of the moment at which the Lapland expedition’s data first reached London and at which Stirling, alone among British mathematicians, possessed an unpublished demonstration that the figure could only be the rotational ellipsoid Newton had supposed. The full text was first printed by Charles Tweedie in his James Stirling: A Sketch of His Life and Works along with His Scientific Correspondence (Oxford: Clarendon Press, 1922), where it occupies chapter IX and pp. 161–164. John Machin took his Gresham chair in May 1713 with a testimonial from Newton describing him as “studious, sober, and learned in the Latin tongue, and in Mathematics a great Master” (Correspondence of Isaac Newton, ed. Turnbull, vol. V, p. 408). His most famous published result — Machin’s formula for π, π/4 = 4 arctan(1/5) − arctan(1/239), with which he had calculated π to one hundred decimal places in 1706 — appeared in William Jones’s Synopsis Palmariorum Matheseos in the same year and gave him a quiet but durable reputation among Continental mathematicians. He served on the 1712 Royal Society commission together with Halley, Arbuthnot, Jones, and others, in the proceedings that issued the Commercium Epistolicum for Newton against Leibniz. As Secretary of the Royal Society from 1718, he was the channel through which Newton’s mathematical correspondence with British and Continental colleagues passed in the last decade of his life; in 1722 he saw Newton’s own revised edition of Arithmetica Universalis through the press, an undertaking for which Conduitt later reported Newton had intended a hundred-guinea fee but never paid it. After Newton’s death Machin’s mathematical work concentrated on lunar theory; his short tract The Laws of the Moon’s Motion according to Gravity was attached to Andrew Motte’s 1729 English Principia, and a longer manuscript of the same subject he was at the time of the present letter still hastening to complete. He died at Gresham College on 9 June 1751. James Stirling, by 1738, had been a decade away from the London mathematical scene. After his election to the Royal Society in 1726 and his joint authorship with de Moivre of the asymptotic formula for the factorial that bears his name — published in his Methodus Differentialis of 1730 — Stirling had quitted his teaching post at Watts’s Academy and accepted, in 1735, the management of the Scots Mining Company’s lead mines at Leadhills in Lanarkshire. The decision answered a Jacobite’s difficulty — Stirling had lost his Balliol scholarships in 1715 for refusing the oath of allegiance, and his subsequent Italian exile had lasted until 1722 — with practical advancement, but it cost him much of the leisure he had previously devoted to mathematics. From Leadhills he kept up a Continental correspondence with Cramer at Geneva and with Euler at St Petersburg, and a domestic one with Maclaurin at Edinburgh and Machin at Gresham; by 1738 the letters from London were the principal current of mathematical news he had access to. The remoteness of the Stirling correspondent of the present letter is the unstated background of Machin’s opening apology for his own slowness to write back: “Sure I am in the case of Endymion!” the senior man writes, glossing his year-long silence as a fairytale sleep. The technical centre of the letter is the figure-of-the-Earth controversy. Newton had argued in Principia Book III, Proposition XIX, that a homogeneous fluid Earth rotating once daily on its axis must take the form of an oblate spheroid — flattened at the poles, bulged at the equator — with a polar diameter shorter than the equatorial by about one part in 230. The pendulum data Richer had reported from Cayenne in 1672 and Halley from St Helena in 1677 had been broadly consistent with the prediction, in that gravity was found to be lower at the equator and a pendulum of fixed length to swing more slowly. But the geodetic measurement Jacques Cassini had reported in his De la grandeur et de la figure de la terre (Paris, 1720) gave the contrary result: that successive degrees of the meridian shortened toward the pole, implying a prolate Earth elongated along the axis of rotation. Cassini’s measurement reached only from Dunkerque to Collioure; the Académie des Sciences accordingly determined to send two expeditions to extreme latitudes, one to Lapland under Maupertuis and the other to Peru under Bouguer and La Condamine, and to compare the lengths of a degree of meridian near the Arctic Circle and near the equator with the value Cassini had obtained in middle France. The first of these missions sailed in 1736 and returned in August 1737; the second had departed in 1735 and would not return for nearly another decade. Maupertuis’s La figure de la terre, déterminée par les observations reached the Paris booksellers in May 1738 and the London ones within weeks. The expedition had triangulated a chain of nine summits between Tornio and Kittis at latitude 66° 20′ N along the Tornio river valley in northern Sweden, and the length of one degree of meridian was reported as 57,437 toises — against the 57,060 toises Cassini had measured in middle France — a difference of 377 toises (about 735 metres) that left no doubt as to the direction of the discrepancy. The conclusion was unambiguous and Newtonian: the Earth was flatter at the poles, and on Maupertuis’s arithmetic the polar flattening was even greater than Newton’s predicted 1/230, lying nearer to 1/178. Machin reports in the present letter that Maupertuis had sent a presentation copy to Stirling, that Anders Celsius (who had accompanied the expedition) had simultaneously published two or three sheets confirming Maupertuis and impeaching the accuracy of Cassini’s technique, and that the controversy in France was now exceedingly heated — Cassini had alleged that the expedition’s astronomical observations had not been verified by inversion of the instrument, and Maupertuis had been driven to procure from England a certificate as to the construction of George Graham’s sector to refute the charge. Machin had deposited Stirling’s presentation copy at Watts’s Academy, where it was kept against Stirling’s next visit to London. The book remained in Stirling’s library at Garden House, inscribed by Maupertuis Donum auctoris, until 2025. Stirling had been working privately on the same question for several years. His 1735 paper Of the Figure of the Earth, and the Variation of Gravity on the Surface, communicated to the Royal Society and printed in the Philosophical Transactions for that year, the Oxford Dictionary of National Biography would later judge an important contribution to the theoretical study of the Earth’s shape and its gravitational field. But Stirling had taken the analysis a stage further than the printed paper indicates: he had a demonstration that the figure of a homogeneous fluid Earth, rotating about its axis, must of necessity be exactly an ellipsoid of revolution — a result Newton had supposed but not demonstrated, and which had passed unnoticed in Principia Book III. He had communicated the demonstration in manuscript to Machin, and Machin in the present letter pleads with him for permission to bring it before the Royal Society. The letter cites in support a recent communication from Maclaurin in Edinburgh: that Maclaurin too had hit upon a demonstration of the ellipsoidal figure, “which seems to be pretty simple”, and that Stirling had told Maclaurin in April 1738 that none of those who had considered the subject had yet shewn the figure to be accurately of that form. Machin urges Stirling to publish before Maclaurin: a moment of historiographical sharpness in which the Newtonian camp, three years before Maclaurin’s prize-winning De Causa Physica Fluxus et Refluxus Maris of 1740, was visibly in motion. Stirling demurred. His reply to Machin has not survived, but his own letter to Maclaurin later in 1738 explains his hesitation: he wished to wait for the return of the Peru expedition before making any public claim, and he had not yet been able, on his own admission, to reconcile the Lapland measurements to his ellipsoidal theory. The hesitation cost him priority. Maclaurin’s 1740 De Causa Physica contained the published demonstration of the ellipsoidal figure of the Earth, and it is the demonstration of which we now read in the textbooks. Stirling never published his version. Tweedie’s 1922 reconstruction of the surviving correspondence makes plain that the priority would have been Stirling’s but for two contingencies — the Leadhills appointment of 1735, which had withdrawn him from the centre of the British mathematical community, and the temperamental caution that led him to wait on the Peru data through 1738 and 1739 while the Edinburgh chair acted on the same question with the latitude only of the Lapland numbers. The four pages of the present letter are the documentary trace of the precise moment at which the priority slipped. The remainder of the letter touches three further subjects of moment. Machin reports a memoir read at the Académie des Sciences at St Petersburg by Joseph-Nicolas Delisle, announcing the Empress Anna Ivanovna’s project for a complete survey of the Russian Empire by triangulation, with measurements not only of meridional arcs but of arcs of the parallel from the Baltic to the Pacific — an enterprise of which Ivan Kirilov’s general map of 1734 had been the precursor and the 1745 Atlas Rossicus the eventual fulfilment. He counsels Stirling not to press a priority claim on behalf of his friend Maclaurin against Euler over the Euler–Maclaurin summation formula, which Euler had communicated to Stirling on 8 June 1736 and which Maclaurin had independently obtained and communicated to Stirling by April 1738; Machin’s judgement, characteristic of his Royal Society temperament, is that the surviving letters by themselves would furnish “a sufficient acquittal” should the question ever be reopened. He notes the imminent appearance of de Moivre’s Doctrine of Chances (in fact the second edition of 1738), without comment beyond the bibliographical fact that he had not yet seen a copy. And he closes with a promise to send Stirling, when the calculation should be done, his own determination of the lunar parallax, derived chiefly from Maclaurin’s observation of the solar eclipse of 28 February 1737 at Edinburgh. The lunar-theory promise was redeemed within a year, in print. Machin’s short tract The Laws of the Moon’s Motion according to Gravity was attached as an appendix to Andrew Motte’s 1729 English Principia, in two volumes, and reissued in 1740; he had also contributed to the third edition of Principia proper in 1726 his account of the motion of the lunar nodes, which Newton had incorporated into the text in the form of a scholium following Proposition XXXIII of Book III. The lunar parallax determination promised in the letter rests, as Machin notes, on Maclaurin’s Edinburgh observation of the solar eclipse of 28 February 1737 (Old Style) — the same eclipse Cassini and Maraldi had observed at Paris and on which a long memoir by Lemonnier was simultaneously in proof in the Paris Mémoires; it was an event from which much eighteenth-century lunar theory took its bearings. Machin’s longer lunar manuscript — a more ambitious treatment integrating the perturbations from solar gravity over multiple revolutions — he was at the time still hastening to complete; it was never finished, and on his death the papers passed to the Royal Society but were not posthumously edited. His best-known result is the much earlier formula for π, π/4 = 4 arctan(1/5) − arctan(1/239), with which he had calculated the constant to one hundred decimal places in 1706, and which remained the standard rapidly-convergent expression for π until well into the nineteenth century. Machin died at Gresham College on 9 June 1751, the year after Stirling’s last published mathematical paper, and a quarter-century before Stirling’s own death at Edinburgh on 5 December 1770. The letter passed with the rest of the Stirling correspondence into the family seat at Garden House in Stirlingshire, where it remained for nearly three centuries. The Stirling family papers were kept at Garden House under the direct custody of the family until 2025, when the bulk of the surviving archive was dispersed at Lyon & Turnbull, Edinburgh, on 23 October 2025 in the single-owner sale The Library of James Stirling, Mathematician, which realised some £820,000 in total and in which Stirling’s autograph manuscript of Methodus Differentialis, his annotated copy of Principia, his Lineae Tertii Ordinis Neutonianae autograph, the de Moivre presentation copy of the 1728 Latin De mundi systemate, and a small group of Bernoulli, Euler, Cramer, and Machin letters all appeared together for the first time in three centuries. The principal corpus of the Garden papers had previously entered the General Register House, Edinburgh, by way of nineteenth- and early-twentieth-century deposits, and it was through Mrs Stirling of Gogar House’s introduction that Tweedie obtained access to the surviving manuscript material in the years before 1922. The present letter, together with a small number of items from the same archive variously dispersed at the 2025 sale, constitutes the entirety of Stirling’s mid-career scientific correspondence still in private hands. It is, in particular, the only known eyewitness London letter to bring Stirling word of the Lapland numbers within weeks of their arrival, and the only documentary record of Machin’s effort to secure for British mathematics priority on the ellipsoidal figure of the Earth. Autograph letter signed, four pages on a bifolium (230 × 180 mm), written in brown ink, signed on the fourth page. Old mailing folds with some splitting, light browning and soiling, minor marginal wear and small losses at corners, early docket on the final page; overall good and entirely legible.
Item #6657
Price: $12,500.00
