Historical Summary

Isaac Newton started to work on what is now called the calculus in 1664 under Barrow at Cambridge (Selection IV.14). One of his early sources was the Latin edition by F. van Schooten of the Géométrie of Descartes, which also had contributions to the infinitesimal calculus. Newton’s first manuscript notes date from 1665. Here we see emerge his "pricked" letters, such as x for our dx/dt. Studying Wallis’s Arithmetica infinitorum he also discovered the binomial series. Then, in 1669, having studied Nicolas Mercator’s Logarithmotechnia (London, 1668) and James Gregory’s De vera circuli et hyperbolae quadrotura (Padua, 1667), he composed the manuscript later published as De analysi per aequationes numero terminorum infinitas (ed. W. Jones, London, 1711). Expanding on his fluxional methods, he wrote another text in 1671, entitled Methodus fluxionum et serierum infinitorum, first published, in English translation, as The method of fluxions and infinite series, ed. John Colson (London, 1736); the original was first published by Samuel Horsley in the Opera omnia (London, 1779–1785), under the title Geometria analytica.Then, in 1676, in two letters to Henry Oldenburg, the secretary of the Royal Society and, like Mersenne at an earlier date, a man whose scientific contacts connected him with practicallyall who worked in the exact sciences, Newton presented some of his results, especially on the binomial series and on fluxions. The letters were destined for Leibniz, then in his early struggles for the discovery of his own calculus (see Selection V.1). After some time, Newton’s attention was directed toward mechanics and astronomy; the result was the immortal Philosophiae naturalis principia mathematica (London, 1687; Principia for short), with its exposition of the planetary theory on the basis of the law of universal gravity. Newton did not explain his theory of fluxions in this book, preferring to give his proofs in classical geometric form as Huygens had done. However, some of his lemmas and propositions present, in carefully chosen language, a few products of his meditations on the calculus, and we reprint them here as Selections V.5, 6. Then, finally, in an attempt to collect his thoughts on fluxions, Newton produced in 1693 a manuscript that was eventually published as Tractatus de quadratura curvarum (London, 1704), which we have chosen for Selection V.7, being, as it seems, part of the last formulation that Newton gave to his theory of fluxions.The Analysis per aequationes, the Quadratura curvarum, and the Methodus fluxionum have been republished, in their eighteenth-century English translations, by D. T. Whiteside, The mathematical works of Isaac Newton (Johnson Reprint Co., New York, London, 1964).Our first selection of Newton’s work gives essential parts of his two letters of 1676 to Oldenburg, dealing in the main with the binomial series. By applying Wallis’s methods of interpolation and extrapolation to new problems, Newton had taken the concept of negative and fractional exponents from Wallis, and so had been able to generalize the binomial theorem, already known for a long time for positive integral exponents (see Selection I.5 on the Pascal triangle), to these more generalized exponents, by which a polynomial expression was changed into an infinite series. He then was able to show how a great many series that already existed in the literature could be regarded as special cases, either directly or by differentiation or integration.Here follow the two letters from Newton to Oldenburg; they are taken from The correspondence of Isaac Newton, ed. H. W. Turnbull (Cambridge University Press, New York, 1959), vol. 1.Other examples give the solution of the equations and series for sin x and for sin2 x, the solution of Kepler’s problem (to divide a semicircle by a line through a given point on the diameter into two sections of which the areas are in given proportion) for an ellipse, the rectification of the are of an ellipse and a hyperbola, the area of a hyperbola with the aid of the series for the logarithm, the quadrature of the quadratrix tan x/a, and the volume of a segment of an ellipsoid of rotation. Newton, careful not to give too much away, selected these examples from results that were already known.Leibniz answered in his letter of August 17 with an account of several of his own results in finding quadratures, hinting at his possession of a general method. He also offered several series, among them as the ratio of the area of a circle to the circumscribed square—a series which Leibniz had already mentioned to friends in 1673, but which James Gregory had found before.Newton was interested, and answered as follows:Newton continues by mentioning in a guarded way, by means of an anagram, that he has a general method for finding tangents and quadratures, which is not limited by irrationalities He gives a series representation of the binomial integralbut without explanation (which Leibniz had no trouble in finding, as a marginal note to the letter shows). Among examples Newton gives the rectification of the cissoid a formula in series which we can writeand the solution of x fromin the formA few years after Newton’s discovery of the binomial series James Gregory, the Scottish mathematician, rediscovered it independently. We know it from letters of Gregory to the London mathematician John Collins (1625–1683), a correspondent of Newton’s and of many other mathematicians. We give here a translation from Gregory’s letter of November 20, 1670, taken from James Gregory tercentenary memorial volume, ed. H. W. Turnbull (Bell, London, 1939), 131–133.This statement gives for which we recognize as the binomial expansion of whose logarithm isBy pure equation Gregory means an equation of the form rational. He adds an example which gives the daily rate percent at Compound interest equivalent to 6 percent per annum; he takesThenthe correct value is 100.0159919.On the atmosphere of intellectual tension typical of the period, see for example the accounts in J. E. Hofmann, Die Entwicklungsgeschichte der Leibnizschen Mathematik (Leibnizens Verlag, Munich, 1949), 60–87, 194–205, or J. F. Scott, The mathematical work of John Wallis (Taylor and Francis, London, 1938), chaps. 9, 10. See further H. W. Turnbull, The mathematical discoveries of Newton (Blackie and Son, Glasgow, 1945).