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First edition. On the Figure of the Earth: The Lapland and Peru Expeditions. 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Lapland expedition's data first reached London and at which Stirling, alone among British mathematicians, possessed an unpublished demonstration that the figure could only be the rotational ellipsoid Newton had supposed.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 ?, ?/4 = 4 arctan(1/5) ? arctan(1/239), with which he had calculated ? 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. Signed.