BERNOULLI, Nikolaus I.

I. Autograph letter signed to James Stirling, in Latin, with five diagrams, Basel, 22 October 1729; II. Autograph letter signed to James Stirling, in Latin, Basel, 1 April 1733. [With:] STIRLING, James. Contemporary copy of Stirling's reply to Bernoulli's first letter, September 1730. [With:] MACHIN, John. Autograph letter signed to James Stirling, London, n.d. (between October 1729 and September 1730), responding to Bernoulli's first letter.

Basel & London: 1729–1733.

A four-document mathematical dossier, descended in the Stirling family at Garden House, Stirlingshire, from the eighteenth century down to the present generation. It preserves, in a single file, the full three-way exchange between Nikolaus I Bernoulli in Basel, James Stirling in London, and John Machin in London over the eighteen months from October 1729 to April 1733 — the first substantive continental reception of John Machin’s Laws of the Moon’s Motion (appended to Andrew Motte’s 1729 English translation of Newton’s Principia) and the prepublication reception, in advance of its appearance in 1730, of Stirling’s forthcoming Methodus Differentialis. The pair of letters from Bernoulli treats three distinct subjects — the interpolation of series, Machin’s lunar theory, and a catalogue of five mathematical problems — but the second letter carries, in passing, a datum of first-class importance in the history of geometry: the announcement of a new species of plane cubic curve, overlooked in both Newton’s Enumeratio Linearum Tertii Ordinis (1704) and Stirling’s Lineae Tertii Ordinis Newtonianae (1717), and assigning to Bernoulli priority in the classification of the curve over Edmund Stone (1730), Thomas Murdoch (1746), and Jean-Paul de Gua de Malves (1740). Bernoulli’s discovery, taken together with François Nicole’s of 1731 — the two species having been missing from the equation xy² = Ax² + Bx + C in the Newton–Stirling enumeration — closes the classification of plane cubics at seventy-eight species, the result subsequently confirmed and retained in W. W. Rouse Ball’s memoir on Newton’s classification in the Transactions of the London Mathematical Society of 1891. Full transcriptions and translations of the four documents are available on request.

Nikolaus I Bernoulli (1687–1759), born in Basel on 10 October 1687 to a line of painters and city councillors and trained in mathematics from boyhood by his uncle Jakob, had by the date of the first letter completed the central phase of his career. His London journey of 1712 had brought him into the circle of Newton, Halley, and de Moivre, very possibly with a stop at Oxford where he may have first met the young Stirling; his Paris residence the same year, at the country house of Pierre Rémond de Montmort, had produced the letters on probability printed in the second edition of Montmort’s Analyse sur les Jeux de Hazard (1713) and his first formulation of what is now known as the St Petersburg problem; his Padua chair, obtained on Leibniz’s recommendation, he had resigned in 1722 to return permanently to Basel, where he had served as Professor of Logic from 1722 and, from 1731, as Professor of Law — a chair he would hold for the remaining twenty-eight years of his life. The correspondence with Stirling was, by 1729, already ten years old; a letter of April 1719 from Padua survives as the earliest-known item of Stirling’s scientific correspondence in private hands, and in the decade between that letter and the present first letter the two had remained in occasional contact. James Stirling (1692–1770), the Jacobite non-juror Balliol mathematician who had left Oxford without a degree and spent 1717–1722 in Italy as the Venetian of the family’s private nomenclature, had by 1729 settled into a teaching post at William Watt’s Academy in Little Tower Street, Covent Garden, had been elected Fellow of the Royal Society on Newton’s nomination in November 1726, and was completing the text of the Methodus Differentialis for the London press of George Strahan, where it would appear the following year as his magnum opus — the classical treatise on the summation and interpolation of infinite series and the enduring monument of his mathematical career. John Machin (c. 1686–1751), the third correspondent in the file, was Professor of Astronomy at Gresham College from 1713, Secretary of the Royal Society from 1718, and the mathematician who had first computed π to a hundred places in 1706 using what is now called Machin’s formula; his Laws of the Moon’s Motion, appended to Motte’s 1729 translation, was a prospectus for the larger lunar-theory treatise he had begun in 1717 and never brought to publication, its manuscript remains preserved by the Royal Astronomical Society.

The first letter, dated Basel 22 October 1729 and covering four sides with five mathematical diagrams, opens with Machin. A copy of The Laws of the Moon’s Motion, Bernoulli reports, had just been given to him by Pierre Louis Moreau de Maupertuis, who passed through Basel on his return from his London stay (where he had become a Fellow of the Royal Society in May 1728). The object of Bernoulli’s attention is Machin’s handling of the equant, the Ptolemaic construct devised to preserve a semblance of uniform circular motion: in Ptolemy’s original scheme, the moon moved at a constant angular speed not about the centre of its orbit but about a point displaced from that centre, the punctum aequans, and Machin, in adapting the notion to the post-Newtonian dynamics of the Principia, had asserted that a body describing equal areas in equal times about a fixed point must be such that somewhere within the orbit there exists a place about which the body’s motion appears more uniform than about any other. Bernoulli disputes the construction: even if a point F can be found from which the velocities at the two apsides appear equal, it does not follow that the equant-orbit approximates a circle, nor that F is the point from which the motion over the entire revolution is most nearly uniform. Bernoulli supplies a figure in demonstration, and raises a series of further objections to the argument of Machin’s pages 41–42.

Stirling, receiving the letter in London, took the unusual step of passing Bernoulli’s objections directly to Machin. The Machin letter offered here in the dossier is Machin’s response — an autograph reply addressed to Stirling at the Academy in Little Tower Street (the address leaf preserves this direction) — in which Machin defends the construction and promises, in the promised larger work, to supply the solution to a further problem he perceives Bernoulli to have raised and then set aside: the determination of the point in the locus from which the planet appears to move equally swiftly at the apsides as at one of the middle distances. Machin’s letter thereby supplies the direct documentary evidence — otherwise unattested — of an exchange between Machin and Bernoulli brokered entirely through Stirling. Stirling’s own reply to Bernoulli, preserved in the file as a contemporary copy in his own hand and dated September 1730, transmits Machin’s letter with a characteristically generous evaluation: Machin’s Laws, he writes, were composed in haste and are only a prospectus; the larger treatise will clarify the obscurities of Book III of the Principia and will carry the theory of gravity further than Newton himself had taken it. Bernoulli, in his second letter of April 1733, concedes a portion of Machin’s case — the definition of the place of most uniform motion he now acknowledges to be correct — but presses a further ambiguity in Machin’s own words, in which the equality of velocities at the apsides had been asserted to imply, rather than be implied by, a property of the curve as a whole. The promised larger treatise of Machin, as Bernoulli could not then know, would never appear; its manuscripts remain in the Royal Astronomical Society archive.

The second topic of the first letter is the interpolation of infinite series — the central concern of Stirling’s Methodus Differentialis, then in the press, a copy of which Stirling would dispatch to Bernoulli with his reply the following September. Bernoulli observes that one of Stirling’s interpolation theorems — that for the series whose terms are proportioned by the factors (r/p), (r+1)/(p+1), (r+2)/(p+2), and so on — can be derived by quadrature of curves from a theorem he himself had communicated to Montmort nineteen years earlier, in 1710. Where the difference between p and r is large, he reports on the authority of Gabriel Cramer’s recent letters, the series can be made to converge rapidly by a method of Cramer’s whose demonstration Bernoulli would welcome seeing. Stirling, in his September 1730 reply, is polite but firm. He sends the Methodus Differentialis itself rather than rehearsing its content in a letter; he doubts that Bernoulli’s method of interpolation will generalise beyond the single special case in which (p – r) is a moderate multiple of the step b; and he points out that the theorem Cramer had sent Bernoulli can be derived, as easily as from Bernoulli’s Montmort-letter theorem of 1710, from Wallis’s result of 1655 that xⁿ/n is the area of the curve whose ordinate is xⁿ⁻¹ — a result, Stirling concedes, that in 1730 is better known in its fluxional rather than its Arithmetica Infinitorum form. Bernoulli, in his second letter, accepts the limitation on his own method (it applies only when the difference between p and r is divisible by b and not too large), but is sceptical that Stirling’s series can all be derived from Wallis as straightforwardly as Stirling had claimed. The correspondence forms, in effect, a specialist peer-review of the Methodus Differentialis at the moment of its publication, conducted between two of the very few mathematicians in Europe capable of such a review.

The third and most miscellaneous portion of the first letter is a set of five special mathematical problems, with Bernoulli’s own constructions supplied and Stirling’s evaluations coming back in the September 1730 reply. The first problem concerns the resolution of the fraction 1/(1 + qzⁿ + z²ⁿ) into a sum of partial fractions of the form (a + bz)/(1 + cz + z²) — a descendant of the 1702 programme of Johann Bernoulli and Leibniz on the factorisation of polynomials for the purpose of rational integration, and of Roger Cotes’s full solution in the posthumous Harmonia Mensurarum (1722). The second is a problem of Maupertuis’s: two fires A and B, of intensities in the ratio p : q, with the heat at a given object falling off as the inverse square of the distance; along what curve CD must a man placed at C retreat to feel the least heat? Bernoulli supplies a geometrical construction, and Stirling notes in reply that the Swedish mathematician Samuel Klingenstierna (1698–1765) has given a simpler solution. The third is a problem of Daniel Bernoulli — Nikolaus’s younger cousin, then at St Petersburg — on the rotation of a curve ANOD about a fixed point A, locating the curve traced by the furthest point O in each position and determining what curves make the ratio of their areas constant; Bernoulli gives both curves as algebraic, a claim Stirling finds to hold only in special cases. The fourth is a dynamical problem on the motion of a body through a resisting medium in which the resistance is proportional to the square of the velocity: given a curve of descent CBA, find a curve Abc such that the body, ascending along Abc with the velocity acquired in descent, traverses an arc Ab equal to the arc of descent BA. Stirling’s solution, reported in the September 1730 reply, extends to the case in which CBA and Abc are portions of a single curve — a case Bernoulli had described as requiring deeper investigation. Stirling’s evaluation of the problem-set as a whole is frank: since the solutions depend on no new principles, and since he cannot tell for what purpose the problems had been proposed, he has not troubled himself greatly over them, especially as Bernoulli has already found them. Bernoulli’s rejoinder to this, in the second letter, defends the method of mutual problem-exchange as exactly the practice by which the analytical art grows.

The second letter, dated Basel 1 April 1733, continues the Machin and interpolation discussions and closes with the contribution that, more than any other passage in the file, justifies the dossier’s historical interest. Bernoulli reports that he has identified a species of plane cubic curve absent from both Newton’s and Stirling’s enumerations. The context is Newton’s Enumeratio Linearum Tertii Ordinis, appended to the Opticks of 1704, in which Newton had classified the cubic curves into seventy-two species by the properties of their diameters and asymptotes. Stirling, in his first book Lineae Tertii Ordinis Newtonianae of 1717, had added four species overlooked by Newton, proving each of the Newtonian propositions analytically and extending the enumeration to seventy-six. Two species were still missing, both among the cubics represented by the equation xy² = Ax² + Bx + C in one of its orientations. Nicole had located one of them in 1731, corresponding to xy² = p²(x + a²)(x + b²) — an oval and two infinite branches. Bernoulli in the present letter announces the second, corresponding to xy² = ±p²(x ± a²)² — an acnode and two infinite branches. The discovery completes the classification of plane cubics at seventy-eight species, the number that has remained canonical ever since. Several other mathematicians were to rediscover the curve independently in the following decade — Edmund Stone in the Method of Fluxions of 1730, Jean-Paul de Gua de Malves in the Usages de l’Analyse de Descartes of 1740, and Patrick Murdoch in the Genesis Curvarum per Umbras of 1746 — but the priority, as Tweedie established in 1922, belongs to the present letter of April 1733. The second letter also corrects an error of Newton’s retained in the Horsley edition of the Opera Omnia, and closes with Bernoulli’s complaint that his name, through a confusion between him and his cousins Nicolas II (of Padua, d. 1726) and Daniel (of St Petersburg), has been dropped from the catalogue of members of the Royal Society and that he hopes the error will be corrected.

The contextual significance of the file is the documentary record of an early 1730s mathematical exchange conducted at the very front of Continental and British analysis. Nikolaus Bernoulli at Basel, Maupertuis in transit between Paris and Basel with copies of the 1729 English Principia, Daniel Bernoulli and Samuel Klingenstierna at St Petersburg, Gabriel Cramer at Geneva, Stirling and Machin in London, and the shade of Newton (who had died in March 1727) all enter the dossier through its cross-references. The topics — the equant in post-Newtonian lunar theory, the interpolation of slowly converging series, partial-fraction rational integration, variational problems of the two-fires type, the dynamics of resisting media, and the enumeration of plane cubics — constitute in miniature the working agenda of the generation that bridged Newton and Euler. That the three principals in the immediate exchange are Bernoulli, Stirling, and Machin is itself notable: the priority dispute of the previous generation, in which Bernoulli as a young man had sided with Leibniz, here gives way to collaborative scientific exchange conducted in the friendly-problem idiom that Bernoulli defends in the second letter.

The correspondence was first printed in Charles Tweedie’s James Stirling: A Sketch of His Life and Works along with His Scientific Correspondence (Oxford: Clarendon Press, 1922) — the edition of the Garden letters for which Tweedie obtained access to the family papers through Mrs Stirling of Gogar House — and is discussed by Tweedie at pp. 206–210. The cubic-curve discovery is analysed by Tweedie at p. 208 with reference to the 1891 memoir of W. W. Rouse Ball on Newton’s classification. The priority of Bernoulli over Stone, Nicole (for the acnode species, as distinct from Nicole’s own oval species), de Gua, and Murdoch is there established; it has not since been disputed. The four documents have descended in the Stirling family at Garden House, Stirlingshire, from Stirling himself through to the present generation.

References: Tweedie, James Stirling: A Sketch of his Life and Works along with his Scientific Correspondence, Oxford: Clarendon Press, 1922 (the two Bernoulli letters, Stirling’s reply, and the Machin letter printed and partly translated at pp. 204–215). Ball, W. W. R., ‘On Newton’s Classification of Cubic Curves,’ Proceedings of the London Mathematical Society 22 (1890–91), pp. 104–143. Tweddle, James Stirling’s Methodus Differentialis: An Annotated Translation of Stirling’s Text, London: Springer, 2003. Guicciardini, The Development of Newtonian Calculus in Britain, 1700–1800, Cambridge University Press, 1989. Whiteside (ed.), The Mathematical Papers of Isaac Newton, vol. VII (The Principia and related material), Cambridge University Press, 1976. Stewart, The Works of Maclaurin, 1982. On Machin: Tweedie, op. cit. pp. 210–215; Cook, Edmond Halley: Charting the Heavens and the Seas, Oxford, 1998, pp. 391–395.

I. Bernoulli, 22 October 1729: 4 pages, 220 × 158 mm, addressed and signed at head Viro clarissimo Jacobo Stirling S.P.D. Nic. Bernoulli, with five mathematical diagrams in the text (short split to centre of transverse fold). II. Bernoulli, 1 April 1733: 4 pages, 370 × 245 mm, addressed and signed at head Viro clarissimo Jacobo Stirling Nicolas Bernoulli S.P.D. III. Stirling, September 1730: 4 pages, contemporary copy in Stirling’s hand. IV. Machin, n.d. (October 1729–September 1730): 2 pages, 200 × 160 mm, addressed To Mr Stirling at the Academy in little Tower Street on the integral address leaf, a faint pencilled annotation at the end of the letter (Mr Gregory, Mr Klingensterna, Mr Campbell, Mr [?], 2 for myself), possibly by Stirling. All four documents in very good condition, the usual folds and light spotting consistent with their age and their preservation together as a working file.

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Item #6656

Price: $48,500.00