**Leibniz vs. Newton,** Pre-May Seminar April 11, 2011

**Leibniz vs. Newton,or Bernoulli vs. Bernoulli?** Pre-May Seminar April 11, 2011

**Jakob Bernoulli (1654-1705)**

**Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)**

**ActaEruditorum, June 1696** I, Johann Bernoulli, address the most brilliant mathematicians in the world.

**ActaEruditorum, June 1696** I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.

**ActaEruditorum, June 1696** I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect.

**ActaEruditorum, June 1696** I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

**Brachistochrone Problem** Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

**Galileo Galilei** "If one considers motions with the same initial and terminal points then the shortest distance between them being a straight line, one might think that the motion along it needs least time. It turns out that this is not so.” - Discourses on Mechanics (1588)

**Galileo’s curves of quickest descent, 1638**

**Galileo’s curves of quickest descent, 1638**

**Curve of Fastest Descent**

**Solutions and Commentary** • June 1696: Problem proposed in Acta

**Solutions and Commentary** • June 1696: Problem proposed in Acta • Bernoulli: the “lion is known by its claw” when reading anonymous Royal Society paper

**Solutions and Commentary** • June 1696: Problem proposed in Acta • Bernoulli: the “lion is known by its claw” when reading anonymous Royal Society paper • May 1697: solutions in ActaEruditorum from Bernoulli, Bernoulli, Newton, Leibniz, l’Hospital

**Solutions and Commentary** • June 1696: Problem proposed in Acta • Bernoulli: the “lion is known by its claw” when reading anonymous Royal Society paper • May 1697: solutions in ActaEruditorum from Bernoulli, Bernoulli, Newton, Leibniz, l’Hospital • 1699: Leibniz reviews solutions from Acta

**The bait…** ...there are fewer who are likely to solve our excellent problems, aye, fewer even among the very mathematicians who boast that [they]... have wonderfully extended its bounds by means of the golden theorems which (they thought) were known to no one, but which in fact had long previously been published by others.

**The Lion** ... in the midst of the hurry of the great recoinage, did not come home till four (in the afternoon) from the Tower very much tired, but did not sleep till he had solved it, which was by four in the morning.

** I do not love to be dunned [pestered] and teased by** foreigners about mathematical things ...

**Nicolas Fatio de Duillier** “I am now fully convinced by the evidence itself on the subject that Newton is the first inventor of this calculus, and the earliest by many years;

**Nicolas Fatio de Duillier** “I am now fully convinced by the evidence itself on the subject that Newton is the first inventor of this calculus, and the earliest by many years; whether Leibniz, its second inventor, may have borrowed anything from him, I should rather leave to the judgment of those who had seen the letters of Newton, and his original manuscripts.

**Nicolas Fatio de Duillier** “I am now fully convinced by the evidence itself on the subject that Newton is the first inventor of this calculus, and the earliest by many years; whether Leibniz, its second inventor, may have borrowed anything from him, I should rather leave to the judgment of those who had seen the letters of Newton, and his original manuscripts. Neither the more modest silence of Newton, nor the unremitting vanity of Leibniz to claim on every occasion the invention of the calculus for himself, will deceive anyone who will investigate, as I have investigated, those records.”

**Table IV from Acta, 1697**

**Snell’s Law for Light Refraction,Fermat’s Principle of** Least Time

**The math…** • Sin q = Cos a = 1/Sec a • 1/sqrt[1+Tan^2 a] • 1/sqrt[1+(dy/dx)^2] Galileo: v = sqrt[2gy] Sin q / v = constant

**Cycloid**

**Jakob challenges Johann…** “ Given a starting point and a vertical line, of all the cycloids from the starting point with the same horizontal base, which will allow the point subjected only to uniform gravity, to reach the vertical line most quickly.”

**Cycloid: the “Helen of geometers”**

**Cycloid: the “Helen of geometers”** Gilles Personne de Roberval (1602-1675) at the College Royal 1634-1675.

**Cycloid: the “Helen of geometers”** Gilles Personne de Roberval (1602-1675) at the College Royal 1634-1675. Area under One Arch = 3 x Area of Generating Circle

**Cycloid: the “Helen of geometers”** Gilles Personne de Roberval (1602-1675) at the College Royal 1634-1675. Area under One Arch = 3 x Area of Generating Circle Never publishes, but Torricelli does.

**Cycloid and Pascal** • 23 November 1654: Religious Ecstasy

**Cycloid and Pascal** • 23 November 1654: Religious Ecstasy • 1658: Toothache!

**Cycloid and Pascal** • 23 November 1654: Religious Ecstasy • 1658: Toothache! • Pascal proposes a contest

**Cycloid and Pascal** • 23 November 1654: Religious Ecstasy • 1658: Toothache! • Pascal proposes a contest • Controversy!

**Calculus of Variations**

**Calculus of Variations** • Bernoulli & Bernoulli

**Calculus of Variations** • Bernoulli & Bernoulli • Euler

**Calculus of Variations** • Bernoulli & Bernoulli • Euler • Lagrange

**Calculus of Variations** • Bernoulli & Bernoulli • Euler • Lagrange • Gauss

**Calculus of Variations** • Bernoulli & Bernoulli • Euler • Lagrange • Gauss • Poisson

**Calculus of Variations** • Bernoulli & Bernoulli • Euler • Lagrange • Gauss • Poisson • Cauchy

**Calculus of Variations** • Bernoulli & Bernoulli • Euler • Lagrange • Gauss • Poisson • Cauchy • Hilbert

**Sources** • Great Feuds in Mathematics – Hal Hellman • Applied Differential Equations – Murray R. Spiegel • Differential Equations – George F. Simmons • Isaac Newton, A Biography – Louis T. More • A History of Mathematics (2nded) – Carl B. Boyer • http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html