---
title: "Isaac Newton"
chunk: 4/17
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"
---

Newton had been reluctant to publish his calculus because he feared controversy and criticism. He was close to the Swiss mathematician Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton's Principia, and corresponded with Leibniz. In 1693, the relationship between Duillier and Newton deteriorated and the book was never completed. Starting in 1699, Duillier accused Leibniz of plagiarism. The mathematician John Keill accused Leibniz of plagiarism in 1708 in the Royal Society journal, thereby deteriorating the situation even more. The dispute then broke out in full force in 1711 when the Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud; it was later found that Newton wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both men until Leibniz's death in 1716.
Newton's first major mathematical discovery was the generalised binomial theorem, valid for any exponent, in 1664–65, which has been called "one of the most powerful and significant in the whole of mathematics." He discovered Newton's identities (probably without knowing of earlier work by Albert Girard in 1629), Newton's method, the Newton polygon, and classified cubic plane curves (polynomials of degree three in two variables). Newton is also a founder of the theory of Cremona transformations, and he made substantial contributions to the theory of finite differences, with Newton regarded as "the single most significant contributor to finite difference interpolation", with many formulas created by Newton. He was the first to state Bézout's theorem, and was also the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula) and was the first to use power series with confidence and to revert power series. He introduced the Puisseux series. He also provided the earliest explicit formulation of the general Taylor series, which appeared in a 1691-1692 draft of his De Quadratura Curvarum. He originated the Newton-Cotes formulas for numerical integration. Newton's work on infinite series was inspired by Simon Stevin's decimals. He also initiated the field of calculus of variations, being the first to formulate and solve a problem in the field, that being Newton's minimal resistance problem, which he posed and solved in 1685, later publishing it in Principia in 1687. It is regarded as one of the most difficult problems tackled by variational methods prior to the twentieth century. He then used calculus of variations in his solving of the brachistochrone curve problem in 1697, which was posed by Johann Bernoulli in 1696, and which he famously solved in a night, thus pioneering the field with his work on the two problems. He was also a pioneer of vector analysis, as he demonstrated how to apply the parallelogram law for adding various physical quantities and realised that these quantities could be broken down into components in any direction. He is credited with introducing the notion of the vector in his Principia, by proposing that physical quantities like velocity, acceleration, momentum, and force be treated as directed quantities, thereby making Newton the "true originator of this mathematical object". 
Newton was probably first to develop a system of polar coordinates in a strictly analytic sense, with his work in relation to the topic being superior, in both generality and flexibility, to any other during his lifetime. His 1671 Method of Fluxions work preceded the earliest publication on the subject by Jacob Bernoulli in 1691. He is also credited as the originator of bipolar coordinates in a strict sense. 
A private manuscript of Newton's which dates to 1664–66 contains what is the earliest known problem in the field of geometric probability. The problem dealt with the likelihood of a negligible ball landing in one of two unequal sectors of a circle. In analysing this problem, he proposed substituting the enumeration of occurrences with their quantitative assessment, and replacing the estimation of an area's proportion with a tally of points, which has led to him being credited as founding stereology. 
Newton was responsible for the modern origin of Gaussian elimination in Europe. In 1669 to 1670, Newton wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which he then supplied. His notes lay unpublished for decades, but once released, his textbook became the most influential of its kind, establishing the method of substitution and the key terminology of 'extermination' (now known as elimination).
In the 1660s and 1670s, Newton found 72 of the 78 "species" of cubic curves and categorised them into four types, systemising his results in later publications. However, a 1690s manuscript later analysed showed that Newton had identified all 78 cubic curves, but chose not to publish the remaining six for unknown reasons. In 1717, and probably with Newton's help, James Stirling proved that every cubic was one of these four types. He claimed that the four types could be obtained by plane projection from one of them, and this was proved in 1731, four years after his death.
Newton briefly dabbled in probability. In letters with Samuel Pepys in 1693, they corresponded over the Newton–Pepys problem, which was a problem about the probability of throwing sixes from a certain number of dice. For it, outcome A was that six dice are tossed with at least one six appearing, outcome B that twelve dice are tossed with at least two sixes appearing, and outcome C in which eighteen dice are tossed with at least three sixes appearing. Newton solved it correctly, choosing outcome A, Pepys incorrectly chose the wrong outcome of C. However, Newton's intuitive explanation for the problem was flawed.

=== Optics ===