Newton Papers : Enumeration of lines of the third order

Newton, Isaac, Sir, 1642-1727

Newton Papers

The papers in Add. 3961 mostly relate to Newton’s researches on the study and classification of cubic curves. The folios do not always follow the sequence in which they must have been composed by Newton. Since the inception of analytic geometry – most notably with Descartes’s Géométrie (1637), which Newton carefully studied in its Latin translation (1659-61) – European mathematicians became interested in the algebraic representation of plane curves. As Descartes showed, and John Wallis further developed, conic sections can be represented by second-degree polynomial equations in two variables (in Cartesian coordinates, as we would say nowadays), and they can be divided into circle, parabola, ellipse and hyperbola. Newton studied Wallis’s algebraic treatment of conics in Wallis, Operum Mathematicorum, Pars Altera (Oxford, 1656).The question naturally arises of how to move a step further and study the graphs of third-degree polynomials. This is a question Newton asked himself quite early: the ‘Enumeratio Curvarum Trium Dimensionum’, part of <a href='' onclick='store.loadPage(3);return false;'>Add. 3961.1</a>, was probably written in 1667-8. Newton deploys Cartesian axes (something Descartes did not do) and has no qualms in using negative coordinates. In his classification of cubics (in the end he will subdivide them into 72 ‘species’, 6 more were added later by James Stirling, François Nicole, and Nicolaus I Bernoulli), Newton shows a full command of algebra and calculus, but he has also deep geometrical insights into projective geometry. He states that all cubic curves can be obtained by centrally projecting the five ‘divergent parabolas’, very much as all conics can be obtained by projecting the circle. We should add that Newton shows his scribal dexterity in accurately drawing the cubic curves [<a href='' onclick='store.loadPage(45);return false;'>Add. 3961, fols 22r-34r</a>].Newton resumed these researches in the middle of the 1670s, and once again in the middle of the 1690s (the dating is luckily facilitated by Newton’s use of a dated letter [<a href='' onclick='store.loadPage(140);return false;'>Add. 3961.2, fol 16v</a>]). As far as the 1670s writings, mention should be made of the ‘Errores Cartesij Geometriae’ [<a href='' onclick='store.loadPage(157);return false;'>Add. 3961.4, fols 24r-25r</a>] in which Newton advances some very insightful criticisms on Descartes’s study of algebraic curves. In 1777, Samuel Horsley, who studied this manuscript, valued it ‘not worth publishing’ [<a href='' onclick='store.loadPage(153);return false;'>Add. 3961.4, fol. 22r</a>].Newton published the 1690s treatise on cubics as Enumeratio Linearum Tertii Ordinis, added as an Appendix to his Opticks (1704) (pp. 139-62 + 6 Tables). The manuscript of the text prepared for the printer can be found in <a href='' onclick='store.loadPage(109);return false;'>Add. 3961.2, fols 1r-14r</a>. The figures are missing. In the eighteenth century, the Enumeratio elicited considerable attention. Notable works are James Stirling, Lineae Tertii Ordinis Neutonianae (Oxford, 1717), Jean Paul Gua de Malves, Usages de l’Analyse de Descartes (Paris, 1740), and Gabriel Cramer, Introduction à l'Analyse des Lignes Courbes Algébriques (Geneva, 1750). The interest in Newton’s program of curve classification somewhat faded, most likely because of the almost unmanageable intricacy related to attempts to classify quartic curves.Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge.