VI.14 Isaac Newton

b. Woolsthorpe, England, 1642; d. London, 1727

Calculus; algebra; geometry; mechanics; optics; mathematical astronomy

Newton entered Trinity College, Cambridge, in 1661, and it was in Cambridge that he spent most of his formative years, first as a student, then as a Fellow, and then, from 1669, as Lucasian Professor of Mathematics. His election to the Lucasian Chair was engineered by his mentor Isaac Barrow, a talented mathematician and theologian who was the first to hold the prestigious chair. In 1696 Newton moved to London to take up the post of Warden of the Mint. He resigned his professorship in 1702.

It appears that Newton’s interest in mathematics began in 1664. In that year he embarked on a course of self-instruction, reading VIÈTE’S [VI.9] works (1646), Oughtred’s Clavis Mathematicae (1631), DESCARTES’S [VI.11] La Géométrie (1637), and Wallis’s Arithmetica Infinitorum (1656). From Descartes, Newton learned how useful it could be to relate algebra to geometry, since plane curves could be represented by algebraic equations in two unknowns. Descartes had, however, imposed strict limitations on the class of curves allowed in La Géométrie: “geometrical” (i.e., algebraic) curves were admitted but “mechanical” (i.e., transcendental) ones were not. In common with many of his contemporaries, Newton felt that such limitations ought to be overcome and that a “new analysis” capable of dealing with mechanical curves ought to be possible. He found the answer in infinite series.

Newton had learned how to deal with infinite series from Wallis’s work, and it was while elaborating one of Wallis’s techniques that, in the winter of 1664, he obtained his first great mathematical discovery: the binomial theorem for fractional powers. This provided him with a method for expanding into power series a large class of “curves,” including transcendental curves, which could now be given an “analytical” representation to which the rules of algebra could be applied. Termwise application of the relation (which he knew from Wallis and which is expressed in familiar Leibnizian notation as ∫ xⁿdx = x^{n + 1} / (n + 1)) allowed him to “square” a variety of curves when they were expanded as a power series. (In the seventeenth century, squaring a curvilinear figure meant finding a square the area of which is equal to that of the curvilinear figure.)

Isaac Newton

A few months later, Newton, with extraordinary insight, realized that most of the problems dealt with by his contemporaries could be reduced to two classes: problems in which one is required to find the tangent to a curve, and problems in which one is required to find the area subtended by a curve. He conceived geometrical magnitudes as being generated by continuous motion. For example, the motion of a point generates a line and the motion of a line generates a surface. These he called “fluents,” while the instantaneous rate of flow he called the “fluxion.” Basing his intuitions on kinematical models, he formulated a version of what is known today as THE FUNDAMENTAL THEOREM OF CALCULUS [I.3 §5.5]. Namely, he proved that tangent and area problems are inverses of each other. In modern terms, Newton was able to reduce quadrature problems (i.e., calculating curvilinear areas) to the search for primitive functions (indefinite integrals). He built “catalogues of curves” (tables of integrals), deploying techniques equivalent to substitution of variables and integration by parts. He developed an efficient algorithm that allowed him to tackle both the direct (differential) and inverse (integral) methods of fluxions. He was able to calculate the tangent to and curvature of any known curve, and perform integrations of many classes of (what we now call) ordinary differential equations. Such mathematical tools allowed him to explore the properties of cubics, and he classified seventy-two different species of them. His results on series and on the direct and inverse methods of fluxions were published in De Quadratura Curvarum and those on cubics in the Enumeratio Linearum Tertii Ordinis, both works appearing in 1704 as appendices to Opticks. His Arithmetica Universalis, the text in which he collected together his lectures on algebra, appeared in 1707.

Before 1704, Newton, displaying his characteristic reluctance to publish, had divulged his discoveries on the fluxional method through letters and manuscripts rather than in printed form. In the meantime LEIBNIZ [VI.15], later than Newton but independently, had also discovered the differential and integral calculus, and had printed it as early as 1684–86. Newton was convinced that Leibniz had stolen the idea from him, and from 1699 onward he engaged Leibniz in a bitter quarrel over priority.

In the early 1670s Newton began distancing himself from the modern symbolic style that had characterized his youthful researches. He turned to geometry in the hope of restoring a hidden geometric method of discovery: the “method of analysis,” known to the ancient Greeks. In fact, geometry dominates Newton’s masterpiece, the Philosophiae Naturalis Principia Mathematica. In this work, which appeared in 1687, Newton presented his theory of gravitation. Newton was convinced that the ancient method was superior to the modern symbolic one that he identified with Cartesian analysis. In his attempts to rediscover the method, he developed elements of projective geometry. (This sprang from the idea that the ancients were able to solve complex problems related to conics by using projective transformations.) An important result is his solution of Pappus’s locus problem, which appears in book I of the Principia (1687). Here he shows that a conic is the locus of points, the product of whose distances from two given lines is proportional to the product of its distances from a third and fourth given line. He then applied projective transformations to determine the conic tangent to m given lines that passes through n given points, when m + n = 5.

The Principia contains a rich array of mathematical results. In book I Newton presents the “method of first and ultimate ratios,” in which he deploys geometric limit procedures in order to determine tangents, curvatures, and curvilinear areas, the latter containing the basic ingredients of what today is known as the RIEMANN INTEGRAL [I.3 §5.5]. He also shows that “ovals” are algebraically nonintegrable. In dealing with the so-called Kepler problem, Newton approximates the roots of x − d sin x = z (d and z given) by a technique equivalent to the NEWTON-RAPHSON METHOD [II.4 §2.3]. In book II he inaugurates VARIATIONAL METHODS [III.94] by tackling the problem of the solid with least resistance. And in book III, in dealing with cometary paths, he presents a method of interpolation which inspired research by mathematicians such as Stirling, Bessel, and GAUSS [VI.26]. In his masterpiece, Newton had shown how productive the application of mathematics to natural philosophy could be: most notably, his studies on the Moon’s motion, the precession of the equinoxes, and the tides were seminal in stimulating eighteenth-century perturbation theory.

Further Reading

Newton, I. 1967–81. The Mathematical Papers of Isaac Newton, edited by D. T. Whiteside et al., eight volumes. Cambridge: Cambridge University Press.

Pepper, J. 1988. Newton’s mathematical work. In Let Newton Be! A New Perspective on His Life and Works, edited by J. Fauvel, R. Flood, M. Shortland, and R. Wilson, pp. 63–79. Oxford: Oxford University Press.

Whiteside, D. T. 1982. Newton the mathematician. In Contemporary Newtonian Research, edited by Z. Bechler, pp. 109–27. Dordrecht: Reidel. (Reprinted, 1996, in Newton. A Critical Norton Edition, edited by I. B. Cohen and R. S. Westfall, pp. 406–13. New York/London: W. W. Norton & Co.)

Niccolò Guicciardini