---
title: "Isaac Newton"
chunk: 3/17
source: "https://en.wikipedia.org/wiki/Isaac_Newton"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:07:12.165802+00:00"
instance: "kb-cron"
---

=== Mathematics ===
Newton's work has been said "to distinctly advance every branch of mathematics then studied". His work on calculus, usually referred to as fluxions, began in 1664, and by 20 May 1665 as seen in a manuscript, Newton "had already developed the calculus to the point where he could compute the tangent and the curvature at any point of a continuous curve". His work by 1665 amounted to a systematic calculus that unified differentiation and integration, which he applied to the dynamic analysis of algebraic and transcendental curves, an approach described by scholar Tom Whiteside as "radically novel, indeed unprecedented" and which later directly informed the theory of central-force orbits in the Principia. Another manuscript of October 1666, is now published among Newton's mathematical papers. Newton recorded a definitive tract of calculus in what is called his "Waste Book". He was self-taught in mathematics and did his research without help, as according to scholar Richard S. Westfall, "By every indication we have, Newton carried out his education in mathematics and his program of research entirely on his own." His work De analysi per aequationes numero terminorum infinitas, sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins that August as the work "of an extraordinary genius and proficiency in these things". 
Newton later became involved in a dispute with the German polymath Gottfried Wilhelm Leibniz over priority in the development of calculus. Both are now credited with independently developing calculus, though with very different mathematical notations. However, it is established that Newton came to develop calculus much earlier than Leibniz. Despite this, the notation of Leibniz is recognised as the more convenient notation, being adopted by continental European mathematicians, and after 1820, by British mathematicians.

The historian of science A. Rupert Hall notes that while Leibniz deserves credit for his independent formulation of calculus, Newton was undoubtedly the first to develop it, stating:But all these matters are of little weight in comparison with the central truth, which has indeed long been universally recognized, that Newton was master of the essential techniques of the calculus by the end of 1666, almost exactly nine years before Leibniz . . . Newton's claim to have mastered the new infinitesimal calculus long before Leibniz, and even to have written — or at least made a good start upon — a publishable exposition of it as early as 1671, is certainly borne out by copious evidence, and though Leibniz and some of his friends sought to belittle Newton's case, the truth has not been seriously in doubt for the last 250 years. Hall further notes that in Principia, Newton was able to "formulate and resolve problems by the integration of differential equations" and "in fact, he anticipated in his book many results that later exponents of the calculus regarded as their own novel achievements." Hall notes Newton's rapid development of calculus in comparison to his contemporaries, stating that Newton "well before 1690 . . . had reached roughly the point in the development of the calculus that Leibniz, the two Bernoullis, L'Hospital, Hermann and others had by joint efforts reached in print by the early 1700s".
Despite the convenience of Leibniz's notation, it has been noted that Newton's notation could also have developed multivariate techniques, with his dot notation still widely used in physics. Some academics have noted the richness and depth of Newton's work, such as the physicist Roger Penrose, stating "in most cases Newton's geometrical methods are not only more concise and elegant, they reveal deeper principles than would become evident by the use of those formal methods of calculus that nowadays would seem more direct." The mathematician Vladimir Arnold stated that "Comparing the texts of Newton with the comments of his successors, it is striking how Newton's original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibniz."
His work extensively uses calculus in geometric form based on limiting values of the ratios of vanishingly small quantities: in the Principia itself, Newton gave demonstration of this under the name of "the method of first and last ratios" and explained why he put his expositions in this form, remarking also that "hereby the same thing is performed as by the method of indivisibles." Because of this, the Principia has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times and in Newton's time "nearly all of it is of this calculus." His use of methods involving "one or more orders of the infinitesimally small" is present in his De motu corporum in gyrum of 1684 and in his papers on motion "during the two decades preceding 1684".
It has been argued that Newton had an imprecise or limited understanding of limits. However, the mathematician Bruce Pourciau contends that in his Principia, Newton actually demonstrated a more sophisticated understanding of limits than he is generally credited with, including being the first to present an epsilon argument.