Newton and Leibniz Isaac Newton (1642/43 – 1726/27)

• born in rural Lincolnshire, son of a rich farmer

• went to a good school

• student at Cambridge 1661–65, 67–68

• Isaac Barrow resigned his chair 1669 in favor of N, N dodged the requirement to become a priest

• wrote religious tracts in the 1690s

• warden of the Royal Mint 1696, cracked down on counterfeiters

• knighted 1705 by Queen Anne Newton’s key works

• (Philosophiae Naturalis) Principia Mathematica (1687) foundation of Newtonian physics (mechanics, laws of motion, gravitation, planetary motions)

• Opticks (1704) theory of light (as particles) and color

• De analysi (1711), de methodis (1736) Newton’s foundations of calculus, which he already used in Principia

• plus works on theology, occultism, alchemy Characteristics of Newton’s approach to calculus

• Keep in mind that the concept of function did not exist!

• Newton came to consider power series by looking at the binomial theorem and generalizing it for arbitrary rational exponents

• Newton thought of variables x as motions (“fluents”) with respect to time, and would call its derivative ẋ “fluxion”.

• He used infinitesimals “x+ẋo” for infinitesimal movements to derive rules of differentiation and integration, reverted to power series when explicit integration was hard

• Solved extremal problems, computed curvatures and areas Why did Newton not publish his results?

• He did write about his methods in the beginning of Principia.

• He was probably aware that the “infinitesimals” weren’t rigidly defined and was afraid of criticism.

• He generally wasn’t fast to publish.

• He was wary that people might steal his results. Gottfried Wilhelm Leibniz (1646–1716)

• born in Leipzig, son of a philosophy professor

• studied philosophy in Leipzig 1661– 64 (PhD), law 1665–66 (promotion)

• landed some administrative/political jobs due to his connections

• met Huygens in Paris 1672, who embarrassed L by exposing his patchy knowledge in math, commenced self-study

• courtier at duke’s court in Hanover 1676 with vast freedom

• died fallen out of favor, unmarked grave for 50 years. Leibniz’s key works

• Dissertatio de arte combinatorica (1666), a philosophical work seeking to encode all thought symbolically

• Théodicée (1712), central philosophical work

• Several shorter works on philosophy, most important his theory of monads (“Monadologie”, 1714)

• Several papers on calculus in Acta eruditorum, first one “a new method for maxima and minima, and the same for tangents, unhindered by either fractional or irrational powers, and a unique kind of calculus for that, by G. G. L.” 1684 Leibniz’s approach to calculus

• a discrete, arithmetic approach: an-a1 = (an-an-1)+(an-1-an-2)+…+(a2-a1)

• wrote dx for a small difference, ∫ y for a sum of a sequence, so that ∫ dx = x

• found differentiation rules such as d(xy)=xdy+ydx

• fundamental theorem was obvious

• “arithmetic quadrature of the circle” = series expression for π

• like Newton, was aware of the shortcomings of his infinitesimals The controversy

• Newton found the results first, but Leibniz published first

• Completely different approaches

• Newton accused Leibniz of plagiarism, especially after L’s visit to London

• Newton made the Royal Society condemn Leibniz, halting all communication between British and Continental mathematicians

• Leibniz’s notation proved more flexible, resulting in English mathematics lagging behind for a whole century. From the introduction of the English translation of de l’Hôpital’s Analyse des Infiniment Petits by E. Stone