Newton Papers : Papers on geometry and related subjects.

Newton, Isaac, Sir, 1642-1727

Newton Papers

In Add. 3963 one finds a number of papers on geometry that Newton penned in the 1680s and 1690s. It was in these two decades that he devoted particular attention to Greek geometrical works, particularly by Apollonius and by Pappus. Like many of his contemporaries, Newton searched for a geometric method of discovery alternative to the symbolic, algebraic one, that, it was believed, the Ancients possessed. One of the reasons why Newton distanced himself from algebra, a mathematical tool that he had mastered so well, as a means of discovery was the fact that geometry was aesthetically more pleasing to him. He often affirmed that the ancient geometrical method was more elegant than the Cartesian one. The enthusiastic acknowledgement of the elegance and conciseness of geometry compared to the ‘tediousness’ of the ‘algebraic calculus’ is a topos that recurs frequently in Newton’s mature and late mathematical manuscripts [<a href='/view/MS-ADD-04004/180'>Add. 4004, 89v</a>]. The importance for Newton of such aesthetic evaluations can hardly be overestimated. Newton’s search for a geometric analysis led him to read a compilation by the late antique mathematician Pappus (4th Century AD) known to him from the Mathematicae collectiones, first published in Latin translation in Urbino in 1588 and probably used by Newton in an edition of 1660. Newton’s attention was particularly focused on a branch of the lost Euclidean corpus: the three books on Porisms. ‘Porisms’ are elliptically referred to in the seventh book of the Collectiones, where Pappus tells his readers that the ancients possessed a method of discovery, a ‘method of analysis’, that allowed them to reach their extraordinary results. This method had been illustrated in several works, of which Euclid’s three books on porisms were the most advanced. Early modern mathematicians were tantalized and tried to reconstruct this method from Pappus, who provided some lemmas as an introduction to the reading of Euclid’s work. For Pappus’s fourth-century readers everything was quite clear, since they had Euclid’s work, but for early modern mathematicians the situation was really frustrating, since that work was lost (as it still is). Newton came to the conclusion that porisms consisted in the kind of results that nowadays we would classify as pertaining to projective geometry. In the 1690s he planned a treatise in three volumes on geometry [<a href='' onclick='store.loadPage(225);return false;'>Add. 3963.11</a>]. Newton was also interested in a related area, what was known as ‘organic geometry’, the art of tracing curves via instruments (‘organa’ in Greek). On this topic, he might have found inspiration in Frans van Schooten, Organica Conicarum Sectionum in Plano Descriptio (Leiden, 1657). Newton’s beautiful organic construction of conics can be found in a letter to John Collins of 1672 [<a href='/view/MS-ADD-03977/83'>Add. 3977.10, f.1v</a>], in his Principia (1687) as Lemma 21, Book 1 [<a href='/view/PR-ADV-B-00039-00001/177'>Adv.b.39.1</a>], and in several other of his manuscripts and printed works. Another field of interest was a geometric reformulation of the method of fluxions in terms of limits of ratios and sums of geometrical magnitudes. The Geometria Curvilinea [<a href='' onclick='store.loadPage(95);return false;'>Add. 3963.7</a>], tentatively dated to around 1680, is a treatise devoted to this version of the method of fluxions. It was recast and further developed as Section 1, Book 1 of the Principia (1687). Newton’s purpose was to avoid the use of infinitely small magnitudes (the ‘method of indivisibles’) in order to be on ‘safer grounds’ using these ‘demonstrated principles’ (Principia (1687), p. 35). In the eighteenth century Newton’s approach to the method of fluxions in terms of limits was considered more rigorous compared to the calculus based on infinitely small magnitudes. Colin Maclaurin in his Treatise of Fluxions (1742) as well as Jean Le Rond D’Alembert, in his article ‘Limite’ in the Encyclopédie (1751) endorsed Newton’s theory of limits, a theory which was brought to a new level of perfection by Augustin-Louis Cauchy in his Cours d’Analyse (1821). Newton’s researches on porisms, organic and projective geometry were continued in Britain by Robert Simson and Matthew Stewart, and later on the Continent by Jacob Steiner and Michel Chasles. Add. 3963 includes a number of notes by Newton’s late eighteenth-century editor, Samuel Horsley, in which he expresses his opinion about the suitability of publishing Newton’s thoughts on these matters, in particular approving his remarks on the porisms of Euclid.Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge.