<p style='text-align: justify;'>Add. 3964 contains papers providing information on Newton’s work on algebra, infinite series, fluxions, and methods of interpolation via finite differences. </p><p style='text-align: justify;'><a href='' onclick='store.loadPage(1);return false;'>3964.1</a>, probably written in the late 1670s, concerns Diophantine equations. It is related to a page in the Waste Book [<a href='/view/MS-ADD-04004/184'>Add.4004: fol. 91v</a>], the commonplace book which Newton used to recast in neater form his working notes.</p><p style='text-align: justify;'><a href='' onclick='store.loadPage(9);return false;'>3964.2</a> begins with a very elementary statement, from the viewpoint of a modern reader. Newton shows that the real roots of f(x)=0 are represented in Cartesian coordinates by the intersections of the graph with the base line y=0. This paper is certainly related to Newton’s Lucasian lectures on algebra (Dd. 9. 68).</p><p style='text-align: justify;'><a href='' onclick='store.loadPage(13);return false;'>3964.3</a> consists of parts of a long treatise, first entitled <i>Matheseos Universalis Specimina</i>, later revised as <i>De Computo Serierum</i>, which Newton composed in 1684. Other parts are in the Macclesfield Collection 9597.2.6 and 9597.2.7. The item was probably split up c.1709-1710 when William Jones was contemplating an edition of Newton’s mathematical papers. Newton was prompted to write this treatise after being presented with a copy of David Gregory’s <i>Exercitatio Geometrica de Dimensione Figurarum</i> (Edinburgh, 1684), in which results on infinite series by James Gregory (David’s uncle) were made public. Newton felt that his priority was challenged and, very much as he had done in the case of Leibniz with his two letters of 13 June 1676 and 24 October 1676 addressed to Henry Oldenburg, he set himself the task of systematizing for publication some of his results in the <i>De Analysi per Aequationes Numero Terminorun Infinitas</i> (Royal Society, MS 81/4) and in the so-called <i>De Methodis Serierum et Fluxionum</i> [<a href='/view/MS-ADD-03960/325'>Add. 3960.14</a>]. In the <i>Specimina</i> Newton referred to James Gregory’s work as well as to his own correspondence with Leibniz, whereas in the <i>De Computo Serierum</i> he dropped all references to Gregory and Leibniz. The structure and careful hand suggests that Newton contemplated either print publication or scribal circulation of this summa of his mathematical researches on series, fluxions and finite differences. </p><p style='text-align: justify;'><a href='' onclick='store.loadPage(41);return false;'>3964.4</a> and <a href='' onclick='store.loadPage(43);return false;'>3964.5</a> are rare examples in Newton’s hand of researches on finite differences and interpolations. The extant manuscript evidence suggests that Newton’s work on interpolation dates from the middle of the 1670s to the middle of the 1690s. It was partly published in Lemma 5, Book 3, of the <i>Principia</i> (1687). Newton systematized his researches in a short treatise entitled <i>Methodus Differentialis</i>, which appeared in the collection of mathematical essays edited by William Jones (<i>Analysis per Quantitatum, Series, Fluxiones ac Differentias</i> (1711), pp. 93-101). Jones transcribed Newton’s autograph, which does not survive. That is why these fragments, together with similar texts in the Waste Book [<a href='/view/MS-ADD-04004/167'>Add.4004, fols 82r-84r</a>] are so important. </p><p style='text-align: justify;'>On the 8th<sup>h</sup>of May 1675, Newton sent some of his results on finite differences to John Smith, an accountant and compiler of mathematical tables (<i>Correspondence</i>, 1, pp. 342-5). In January 1711/12, William Jones, the mathematical tutor and librarian of the future Earl of Macclesfield, who had just acquired Collins’s papers, passed a copy of Newton’s letter to Smith to Roger Cotes, the Plumian Professor of Astronomy in Cambridge and editor of the second edition (1713) of Newton’s <i>Principia</i> (Trinity College Library, Cambridge, MS R.16.38, fols 308r-9r). A copy of Newton’s letter to Smith was also made for John Keill, a Scottish mathematician based in Oxford (UL O.XIV.278.9 (iv)).</p><p style='text-align: justify;'>The tract ‘Of quadrature by Ordinates’ [<a href='' onclick='store.loadPage(41);return false;'>3964.4</a>] is dated by Whiteside October 1695 (<i>Mathematical Papers</i>, 7, p. 700 n. 2). It deals with methods of numerical integration via polynomials which approximate a tabulated function. Newton’s researches on finite differences are now remembered by the names attached to methods such as the Newton-Bessel (or Newton-Cotes) and Newton-Stirling formulas.</p><p style='text-align: justify;'><a href='' onclick='store.loadPage(51);return false;'>3964.6</a> is a draft (in English and then in Latin) of Newton’s attempts to reply to a challenge set by Leibniz in a letter to Antonio Schinella Conti dated 25 November 1715. Newton’s solution appeared anonymously in the <i>Philosophical Transactions of the Royal Society</i> for January-March 1716. Leibniz had received suggestions from Johann Bernoulli, a Swiss mathematician based in Basel who actively participated to the controversy over the invention of the calculus, which divided British from Continental mathematicians. Their idea was to ‘feel the pulse of our English analysts’ (Newton <i>Correspondence</i>, 6, p. 253). The problem was to find the plane curves that intersect at a given angle all of the members of family of curves that lie in the same plane. The problem was understood as a way to compare the Newtonian and the Leibnizian versions of the calculus in relation to a particularly difficult problem. These researches, carried out mostly by Continental mathematicians, led to the formulation of partial differential equations. </p><p style='text-align: justify;'><a href='' onclick='store.loadPage(55);return false;'>3964.7</a> is a letter by John Collins addressed to Newton, probably written in July 1675. Collins, Newton’s most important mathematical correspondent in this period, provides information on recent publications, the most exciting being the Latin translation from an Arabic manuscript of books 5, 6, and 7 of Apollonius’s <i>Conics</i> by Christian Ravius (1669). Collins also deals with his exchanges with Michael Dary, a gauger active in Bristol, Newcastle and London, who had asked Newton’s advice on the calculation of the volumes of solids of revolutions (obtained by revolution of circles, ellipses and parabolas around the axis). Collins sends also a list of errata noted by John Wallis concerning Isaac Barrow’s editions (1675) of Archimedes’s <i>Opera</i> and Apollonius’s <i>Conics</i>. We find here Newton immersed in his lively interactions with the London mathematical practitioners and in activities concerning the publication and edition of mathematical books.</p><p style='text-align: justify;'>Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge.</p>
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