Leibniz vs. Newton,Leibniz vs. Newton,

Pre-May SeminarPre-May Seminar

April 11, 2011April 11, 2011

Leibniz vs. Newton,Leibniz vs. Newton,

or Bernoulli vs. or Bernoulli vs. Bernoulli?Bernoulli?

Pre-May SeminarPre-May Seminar

April 11, 2011April 11, 2011

Jakob Bernoulli (1654-1705)Jakob Bernoulli (1654-1705)

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

1748)1748)

Acta EruditorumActa Eruditorum, June 1696, June 1696I, Johann Bernoulli, address the most brilliant I, Johann Bernoulli, address the most brilliant mathematicians in the world. mathematicians in the world.

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

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

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

Brachistochrone ProblemBrachistochrone Problem

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

Galileo GalileiGalileo Galilei

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

- Discourses on Mechanics (1588)- Discourses on Mechanics (1588)

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

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

Curve of Fastest DescentCurve of Fastest Descent

Solutions and CommentarySolutions and Commentary

June 1696: Problem proposed in June 1696: Problem proposed in ActaActa

Solutions and CommentarySolutions and Commentary

June 1696: Problem proposed in June 1696: Problem proposed in ActaActa Bernoulli: the “lion is known by its Bernoulli: the “lion is known by its

claw” when reading anonymous claw” when reading anonymous Royal Society paperRoyal Society paper

Solutions and CommentarySolutions and Commentary

June 1696: Problem proposed in June 1696: Problem proposed in ActaActa Bernoulli: the “lion is known by its Bernoulli: the “lion is known by its

claw” when reading anonymous claw” when reading anonymous Royal Society paperRoyal Society paper

May 1697: solutions in May 1697: solutions in Acta Acta EruditorumEruditorum from Bernoulli, Bernoulli, from Bernoulli, Bernoulli, Newton, Leibniz, l’HospitalNewton, Leibniz, l’Hospital

Solutions and CommentarySolutions and Commentary

June 1696: Problem proposed in June 1696: Problem proposed in ActaActa Bernoulli: the “lion is known by its Bernoulli: the “lion is known by its

claw” when reading anonymous Royal claw” when reading anonymous Royal Society paperSociety paper

May 1697: solutions in May 1697: solutions in Acta Acta EruditorumEruditorum from Bernoulli, Bernoulli, from Bernoulli, Bernoulli, Newton, Leibniz, l’HospitalNewton, Leibniz, l’Hospital

1699: Leibniz reviews solutions from 1699: Leibniz reviews solutions from ActaActa

The bait…The bait…

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

The LionThe Lion

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

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

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

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

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

Table IV from Table IV from ActaActa, 1697, 1697

Snell’s Law for Light Snell’s Law for Light Refraction,Refraction,

Fermat’s Principle of Least Fermat’s Principle of Least TimeTime

The math…The math…

Sin Sin Cos Cos Sec Sec 1/sqrt[1+Tan^2 1/sqrt[1+Tan^2 ]]1/sqrt[1+(dy/dx)^2]1/sqrt[1+(dy/dx)^2]

Galileo: Galileo: v = sqrt[2gy]v = sqrt[2gy]

Sin Sin v v = constant= constant

CycloidCycloid

Jakob challenges Johann…Jakob challenges Johann…

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

Cycloid: the “Helen of Cycloid: the “Helen of geometers”geometers”

Cycloid: the “Helen of Cycloid: the “Helen of geometers”geometers”

Gilles Personne de Gilles Personne de Roberval (1602-Roberval (1602-1675)1675)

at the College Royal at the College Royal 1634-1675.1634-1675.

Cycloid: the “Helen of Cycloid: the “Helen of geometers”geometers”

Gilles Personne de Gilles Personne de Roberval (1602-Roberval (1602-1675)1675)

at the College Royal at the College Royal 1634-1675.1634-1675.

Area under One Arch Area under One Arch = 3 x Area of = 3 x Area of Generating CircleGenerating Circle

Cycloid: the “Helen of Cycloid: the “Helen of geometers”geometers”

Gilles Personne de Gilles Personne de Roberval (1602-1675)Roberval (1602-1675)

at the College Royal at the College Royal 1634-1675.1634-1675.

Area under One Arch Area under One Arch = 3 x Area of = 3 x Area of Generating CircleGenerating Circle

Never publishes, but Never publishes, but Torricelli does.Torricelli does.

Cycloid and PascalCycloid and Pascal

23 November 1654: Religious 23 November 1654: Religious EcstasyEcstasy

Cycloid and PascalCycloid and Pascal

23 November 1654: Religious 23 November 1654: Religious EcstasyEcstasy

1658: Toothache!1658: Toothache!

Cycloid and PascalCycloid and Pascal

23 November 1654: Religious 23 November 1654: Religious EcstasyEcstasy

1658: Toothache!1658: Toothache! Pascal proposes a contestPascal proposes a contest

Cycloid and PascalCycloid and Pascal

23 November 1654: Religious 23 November 1654: Religious EcstasyEcstasy

1658: Toothache!1658: Toothache! Pascal proposes a contestPascal proposes a contest Controversy!Controversy!

Calculus of VariationsCalculus of Variations

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli EulerEuler

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli EulerEuler LagrangeLagrange

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli EulerEuler LagrangeLagrange GaussGauss

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli EulerEuler LagrangeLagrange GaussGauss PoissonPoisson

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli EulerEuler LagrangeLagrange GaussGauss PoissonPoisson CauchyCauchy

Calculus of VariationsCalculus of Variations

Bernoulli & BernoulliBernoulli & Bernoulli EulerEuler LagrangeLagrange GaussGauss PoissonPoisson CauchyCauchy HilbertHilbert

SourcesSources Great Feuds in Mathematics Great Feuds in Mathematics – Hal Hellman– Hal Hellman Applied Differential Equations Applied Differential Equations – Murray R. – Murray R.

SpiegelSpiegel Differential Equations Differential Equations – George F. – George F.

SimmonsSimmons Isaac Newton, A Biography Isaac Newton, A Biography – Louis T. More– Louis T. More A History of Mathematics A History of Mathematics (2(2ndnd ed) – Carl B. ed) – Carl B.

BoyerBoyer http://www-history.mcs.st-and.ac.uk/http://www-history.mcs.st-and.ac.uk/

HistTopics/Brachistochrone.htmlHistTopics/Brachistochrone.html