Newton Papers : Papers Connected with the Principia on Mathematical Problems

Newton, Isaac, Sir, 1642-1727

Newton Papers

Add. 3967 concerns some of Newton’s mathematically more interesting researches connected with the Principia. In some cases, the mathematical methods deployed in the Principia are just hinted at, so that the reader is left with little indication of how the results have been obtained. Add. 3967.1 is a solution of the so-called Kepler equation. In the Astronomia Nova (1609) Johannes Kepler had provided a method for determining the position in function of time of a planet orbiting in an elliptical trajectory. Newton faced this problem in Section 6, Book 1 of his Principia developing a method that can be understood as the Newton-Raphson iterative procedure of approximation. In Add. 3967.1, Newton seems to be responding to a criticism advanced by Nicolas Fatio de Duillier in his annotated copy of the first edition of the Principia on p. 112, line 8 (Bodleian Library, Oxford, shelfmark 4o Z.22.Art), which made the method used more effective. The folios in Add. 3967.2 are badly damaged by fire. The earliest material that they include is a draft letter to the Oxford professor John Wallis (27 August 1692 <a href='' onclick='store.loadPage(24);return false;'>Add. 3967.2, fol. 12v</a>). The paper of that draft is then reused to present Newton’s method for determining the solid of least resistance; that is, the shape of the solid of revolution which experiences the less resistance in a rare medium, composed of equal particles arranged freely at equal distances from one another. Newton deploys his method of fluxions in order to tackle a problem that belongs to the calculus of variations, a mathematical theory developed in the eighteenth century, most notably by Leonhard Euler in Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (Lausanne, 1744). Newton presented the solution of this very (at that time) difficult problem in the Scholium to Proposition 35 (34 in the second and third editions), book 2, of his Principia. The preparatory manuscripts for this Scholium are in <a href='/view/MS-ADD-03965/218'>Add. 3965.10, fols 107v, 134</a>. In 1694, another Oxford mathematician, David Gregory, asked Newton for a proof (lacking from the printed Principia). <a href='' onclick='store.loadPage(19);return false;'>Add. 3967.2</a> provides a draft of Newton’s reply (14 July 1694) to that request for clarification. Indeed, Gregory included a treatment of the solid of least resistance copied based on Newton’s fluxional approach in his manuscript, dated late autumn 1694, ‘Isaaci Newtoni Methodus Fluxionum’, which circulated widely among Newton’s closest mathematical allies (Gregory’s original is in the library of the University of St. Andrews, MS QA 33G8D12, a fair copy is in Christ Church (Oxford), while transcripts attributed to William Jones and John Keill can be found in the Macclesfield CollectionAdd.9597.9.3 and 9597.9.4 and in the Lucasian Papers in the University Archives, O.XIV.278.13). Newton’s solution for the solid of least resistance was printed in one of the appendices of Andrew Motte’s English translation of the Principia (London, 1729), pp. v-vii. Add. 3967.3 gathers material related to Newton’s study of atmospheric refraction. In autumn 1694, the Astronomer Royal John Flamsteed prompted Newton to provide a table of atmospheric refraction, which was to be used in correcting the displacement in the observed position of the stars due to the bending of light traversing the atmosphere. Newton set himself to solve the problem from first principles. He assumed light to consist of corpuscles accelerated by a force determined by the index of refraction. The latter depended on the variation of density of the atmosphere (studied in the annotations on barometric measurements gathered in Add. 3967.4). Newton dealt with the mathematical theory of optical refraction in Section 14, book 1 of his Principia. In 1721 Newton’s tabula refractionum was published by Edmond Halley in appendix to a paper of his. For Newton’s studies on atmospheric refraction one should see also 9597.2.118 and 3970.3, fols 341r/341v.It is interesting to note that in Add. 3967.4, fol. 38v one finds Newton’s annotations on the problem of orthogonal trajectories that Leibniz and Johann Bernoulli posed as a challenge to English mathematicians in late 1715.Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge.