Newton, Isaac (1642–1727)

6. Mathematics

Newton engaged in extensive mathematical research throughout much of the period from 1664 until he left for London in 1696, and even after that he produced new results on some problems. Besides the many lines along which he developed and applied the calculus, he made substantial discoveries in algebra and in pure as well as analytic geometry. The mathematics of the Principia is itself a new form of synthetic geometry, incorporating limits. (Contrary to the still-persisting myth, there is no evidence that Newton first derived his results on celestial orbits within the symbolic calculus and then recast them in geometric form.)

Newton’s invention of the calculus grew out of his attempts to solve several distinct problems, often employing novel extension of the ideas and methods of others. For example, his initial algorithms for derivatives of algebraic curves combined Cartesian techniques with the idea of an indefinitely small, vanishing increment. He exploited a method of indivisibles that John Wallis had used in obtaining integrals of algebraic curves; but he reconceptualized the method to represent an integral that grows as the curve extends incrementally, and then joined this with the binomial series to yield integrals via infinite series of a much wider range of curves. Geometrical representations of these results revealed the inverse relation between integration and differentiation. Then, adapting his Lucasian predecessor Isaac Barrow’s idea to treat curves as arising from the motion of a point, Newton recast the results on derivatives, replacing indefinitely small increments with his ’fluxions’. The first full tract on fluxions, ‘To Resolve Problems by Motion’, is dated 1666, but the first manuscript to circulate was De analysi per aequationes infinitas of 1669.

Mathematicians in England used Newtonian methods and notation into the nineteenth century. But the Leibnizian tradition had gained so much momentum by the time Newton’s works appeared that the calculus, as we know it, stems far more from that tradition. Ironically, it was such figures as the Bernoullis and Euler (belonging to the Leibnizian tradition) who recast the Principia into the language of the calculus.