Autograph letter signed, in Latin, with richly scientific content, from Padua, dated 29 April 1719, to the Scottish mathematician James Stirling, then in Venice.
[Padua: 1719]. An autograph letter signed by Nikolaus I Bernoulli (1687–1759), in Latin, written from Padua on 29 April 1719 to the Scottish mathematician James Stirling, then in Venice — preserved undisturbed at the Stirling family seat of Garden House in Stirlingshire from the eighteenth century until 2025, the earliest item of Stirling’s scientific correspondence recorded in private hands, and one of the earliest of his mathematical manuscripts known to survive anywhere. Its three principal subjects — the motion of a cycloidal pendulum in a resisting medium (Newton, Principia Book II), Brook Taylor’s 1718 public challenge to the Continental mathematicians on the integration of rational functions, and a method for the determination of areas by means of infinite series proposed by Johann Bernoulli — bear directly on the active mathematical frontier of the spring of 1719, and each leads, by a short path, back to the Newton–Leibniz calculus priority dispute in which the two correspondents were at that moment figures of considerable interest. The letter closes with a Latin appendix of seven theorems by Johann Bernoulli on the integration of rational functions, treating Taylor’s challenge among other cases — mathematical content, not epistolary filler — and thereby takes on the character of a short memoir. Stirling would use the underlying ideas in the paper Methodus Differentialis Newtoniana Illustrata submitted to the Royal Society six weeks later, and would refer explicitly to Bernoulli’s pendulum question in the letter he wrote to Newton from Venice on 17 August 1719 offering to serve as a go-between in the priority dispute. A full transcription and English translation of the present letter is available on request. Nikolaus I Bernoulli (1687–1759) belonged to the second generation of the great Basel mathematical dynasty. Born in Basel on 10 October 1687, the son of a painter and city councillor, and a nephew of Jakob I and Johann I Bernoulli, he was trained in mathematics by Jakob from boyhood, defended his master’s dissertation on infinite series (written by Jakob) under Jakob’s supervision in 1704, and took his law doctorate in 1709 with a thesis on the application of the new probability theory to jurisprudence, De Usu Artis Conjectandi in Jure — an important early essay on quantitative legal reasoning. He travelled widely in the years immediately following: Paris in 1710; London in 1712, where he was received by Newton, Halley, de Moivre, and, as Tweedie conjectured, very possibly made the acquaintance of the young James Stirling during a stop at Oxford; and Paris again later in 1712, where his three-month residence at the country house of Pierre Rémond de Montmort resulted in the group of letters on probability printed in the second edition of Montmort’s Analyse sur les Jeux de Hazard (1713) and, most consequentially, in Nikolaus’s first formulation of what has since been known as the St Petersburg problem. In the same years he saw his uncle Jakob’s posthumous Ars Conjectandi through the Basel press (also 1713). By 1716, on the warm recommendation of Leibniz — to whom Johann Bernoulli had earlier commended his nephew — he had been appointed Professor of Mathematics at Padua, a chair vacated by Jakob Hermann’s removal to Frankfurt an der Oder. He held the Paduan chair until 1722, returning then to Basel, where he would serve as Professor of Logic from 1722 and Professor of Law from 1731 until his death in 1759. In the priority dispute he sided unambiguously with his uncle Johann in the Leibnizian interest, and it was he who first pointed out to the Bernoullis an error in Newton’s handling of higher-order derivatives in Principia Book II — the issue that sits in the background of the present letter’s opening subject. James Stirling (1692–1770) was born on 11 May 1692 at Garden House, Stirlingshire, the third son of the Jacobite laird Archibald Stirling, acquitted of high treason for his part in the Brig o’Turk gathering of 1708. Educated at Balliol College, Oxford, from 1710, Stirling held a Snell exhibition and, later, a Bishop Warner exhibition. The Hanoverian succession of 1714, the Jacobite rebellion of 1715, and the subsequent requirement that scholars swear the oath of allegiance to George I placed him in an impossible position: as a non-juror he lost his scholarships, was accused at assizes of corresponding with exiled Jacobites and of cursing the king (he was acquitted), and was ultimately denied a degree — though he remained in residence at Balliol for some time afterwards. In April 1717 his first book, the Lineae Tertii Ordinis Neutonianae, a mathematical commentary on Newton’s classification of the cubic curves, was printed at the Sheldonian Theatre and added four further species to Newton’s seventy-two; Newton himself was among the subscribers, and the work was dedicated to Nicolas Tron, the Venetian ambassador to London, whose patronage drew Stirling to Italy later the same year. Stirling’s Italian residence — from 1717 until probably 1722, a period during which he became known in his family as the Venetian — is sparsely documented in other respects, but is the context out of which the present letter springs. He attended the University of Padua (certainly in 1721, and very likely earlier), formed a close friendship with Nikolaus Bernoulli there, and submitted to the Royal Society on 18 June 1719 his paper Methodus Differentialis Newtoniana Illustrata, a prospectus of what would become his magnum opus Methodus Differentialis (London, 1730). On returning to Britain by 1722 he passed through a teaching post at Watt’s Academy in Covent Garden, was elected Fellow of the Royal Society on Newton’s nomination on 3 November 1726, and corresponded over the following decade on roughly equal terms with Maclaurin, de Moivre, Euler, and Gabriel Cramer. In 1735 he became manager of the Scots Mining Company’s lead mines at Leadhills, a post he held for the rest of his life and which, as he told Maclaurin, left him too little time for mathematics; the decision cost him the Edinburgh chair that fell vacant on Maclaurin’s death in 1746. The textual context of the letter is the priority dispute. In 1712 the Royal Society, at Newton’s direction, had published the Commercium Epistolicum assembling documents intended to prove Newton’s priority in the invention of the calculus over Leibniz. By 1719 Johann Bernoulli had become, in the memorable phrase, the head of Leibniz’s party, and the Acta Eruditorum of Leipzig was the standing organ of the Continental side. Stirling’s correspondence with Newton in August 1719 — written four months after the present letter, and substantially informed by his conversations with Nikolaus at Padua and Venice — exists precisely in this context: he had promised Newton to test the Bernoullian positions, to report on Johann’s recent Acta articles, and, should Nikolaus wish it, to carry Nikolaus’s own vindication of himself to Newton in London. That it had been Nikolaus, rather than Johann, who had first identified Newton’s error in Principia II Prop. 10 made the situation delicate; Nikolaus had already written privately to Newton in 1719 to clear himself of John Keill’s accusation that he bore ill will. Stirling’s Venice letter to Newton, in which the whole mathematical content of the present Bernoulli letter is echoed and expanded, records that he had found in Nikolaus a young man whose modesty surpassed what could be expected of one so closely tied to the head of the Leibnizian party. The present letter is the documentary inside of that conversation. Bernoulli writes in reply to a letter from Stirling that has not survived, and opens with customary thanks for the invitation to regular correspondence and for Stirling’s willingness to take up where they had left off in their discussions at Venice. He then turns to the first substantive matter, the motion of a cycloidal pendulum in a resisting medium. Galileo had assumed, on the basis of small-amplitude observations, that a simple circular pendulum was isochronous; Huygens, in the Horologium Oscillatorium of 1673, had shown that true isochronism requires the bob to trace not a circle but a cycloid, and had introduced the celebrated cycloidal cheeks by which a pendulum could be constrained to the isochronous path. Newton, in Section VI of Book II of the Principia, had extended the analysis to the motion of a cycloidal pendulum through a resisting medium. His Proposition XXVI, Theorem XXI, established that if the resistance is proportional to the first power of the velocity the motion remains isochronous; his Proposition XXVII, Theorem XXII, addressed the more realistic case in which the resistance is proportional to the square of the velocity, and obtained only an approximate result. Bernoulli, in the present letter, reports that he has by calculation established the impossibility of a heavy body oscillating in a cycloid under the linear-resistance assumption, but confesses that he does not yet know how to demonstrate this impossibility a priori from physical principles. He asks Stirling to seek the construction of a curve in which the abscissae represent the arcs of the cycloid traversed in each oscillation and the ordinates the resistance or velocity of the bob at the ends of those arcs — noting Newton’s remark at Principia p. 282 that this curve is approximately an ellipse. Stirling’s response, several months later in Venice, would take the form of a remark to Newton himself: that in the linear-resistance hypothesis the pendulum has zero velocity at its lowest point and therefore can perform only half an oscillation before coming to rest — a result Jacopo Riccati had independently obtained, and which led Nikolaus, as Stirling remarked to Newton with some edge, to raise the question of how in that hypothesis a pendulum can be said to oscillate at all. The second subject is Brook Taylor’s 1718 challenge to the Continental mathematicians on the integration of rational functions. The problem of integrating rational and algebraic functions had been systematically attacked by Newton as early as 1665–66, shortly after his discovery of the inverse relation between derivative and integral; his tables of such integrals, much developed in the October 1666 tract on fluxions, were extended in his letter of October 1676 to Oldenburg — the second of the two epistolae posteriores intended for Leibniz — to include an infinite family of rational and algebraic antiderivatives of a general form. In 1702, with Newton’s work still unpublished, Johann Bernoulli and Leibniz in separate publications had addressed the integration of rational functions by way of partial-fraction decomposition, and the problem of factorising the denominator had remained a standing challenge. Roger Cotes, writing to William Jones on 5 May 1716, had announced a general method for the questions raised in Leibniz’s 1702 paper; he died the same year, the method was in his posthumous papers, and his Harmonia Mensurarum, published only in 1722, would eventually contain it. In 1718 Taylor — evidently intending to force the Continental hand on a subject in which British mathematicians were, through Cotes, already ahead — sent via Montmort a challenge to the Continental mathematicians to find, by quadrature of circle or hyperbola (that is, in closed form by trigonometric or logarithmic functions), the integral of a rational function of a specific form, subject to the restriction that the imaginary roots of the denominator are not admitted and that a particular exponent λ is restricted to the geometric progression 2, 4, 8, 16, 32. The restriction on λ reflected exactly the limits of Cotes’s own solution. The challenge was intended for Johann Bernoulli, who published his response in the Acta Eruditorum of May 1719; Jacob Hermann and Eustachio Manfredi produced further solutions. Bernoulli, in the present letter, reports having received from his uncle the Leipzig solution, adds that he had himself obtained a solution some time earlier, and transmits to Stirling an appendix of seven of his uncle’s theorems bearing on rational integration, of which the seventh treats Taylor’s problem itself. He forbears to give proofs, remarking in the Latin formula customary for such appendices that their demonstrations will readily be supplied by other geometers. The final substantive passage concerns Johann Bernoulli’s method for the quadrature of areas by means of infinite series, and responds to examples Stirling had sent in his now-lost earlier letter. The use of infinite series in integration was as old as Newton’s own work of the 1660s, and by 1719 the comparison of different series expansions — their rates of convergence, their manipulability under substitution, the relations between them — had become a specialised technical craft. Much of Stirling’s own mathematical career would be devoted precisely to the acceleration of convergence of slowly converging series: the 1730 Methodus Differentialis is the classical monument of that programme. That Stirling was already in 1719 exchanging examples of quadrature by series with a Bernoulli is itself a measure of the sophistication he had reached by his late twenties, and of the authority his Italian residence had given him within the Continental circles. Stirling does not appear to have replied to the present letter, at least not in any document that has survived; the next known item in his correspondence with Nikolaus is ten years later. He did, however, report back to Newton, and the report makes the value of Bernoulli’s questions visible from the British side. Writing from Venice on 17 August 1719, Stirling noted that Nikolaus had proposed to him the pendulum-resistance problem, that he had worked out the simplest cases and found (as Bernoulli had found before him, and as Riccati had independently obtained) the result of the half-oscillation, and — the sharper observation — that he suspected Johann Bernoulli of setting his nephew to comb through Newton’s writings for anything that might be cavilled at. Stirling’s defence of Nikolaus against this reading is revealing: he had never heard Nikolaus speak of Newton without respect; Nikolaus had shown him Johann Bernoulli’s Acta piece against John Keill; on the specific complaint that Newton’s construction of a curve at Principia II Prop. 4 could not be reduced to logarithms, Nikolaus himself had pointed out to Stirling that Corollary 2 of the Proposition does exactly that, and had wondered aloud at his uncle’s oversight. The Stirling–Newton correspondence is preserved; the Bernoulli–Stirling correspondence for this year, with the exception of the present letter, is not. The pairing is what makes this document, on its own, a primary witness of the 1719 state of the priority dispute. The letter was first printed and translated in Charles Tweedie’s James Stirling: A Sketch of His Life and Works along with His Scientific Correspondence (Oxford: Clarendon Press, 1922), Tweedie having obtained access to the Stirling family papers — which he calls throughout the Garden letters — through the generosity of Mrs Stirling of Gogar House. The Garden papers remained 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 sale entitled The Library of James Stirling, Mathematician — a single-owner sale that realised some £820,000 in total and in which Stirling’s autograph manuscript of Methodus Differentialis, his annotated copy of Principia, and Abraham de Moivre’s presentation copy to Stirling of the 1728 Latin De mundi systemate all appeared together for the first time in three centuries. The present letter, together with a small number of other items from the same archive now variously dispersed, constitutes the entirety of Stirling’s early scientific correspondence still in private hands; the bulk of the material presented by Tweedie has long since entered institutional collections, principally at the General Register House, Edinburgh. Nothing of equivalent intimacy with the mathematical frontier of the late 1710s — with Newton in London, Johann Bernoulli in Basel, Taylor in his London challenge, and Nikolaus Bernoulli in Padua reading back through the Principia — appears outside of public institutions. One leaf, 295 × 205 mm, written in brown ink on both sides, the second side used only on the upper half, with mathematical formulae throughout; signed. Old folds, light browning and spotting, slight marginal wear, old docket on the verso, overall very good.
Item #6655
Price: $18,500.00
