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)
Acta Eruditorum, June 1696
I, Johann Bernoulli, address the most brilliant
mathematicians in the world.
Acta Eruditorum, 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.
Acta Eruditorum, 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.
Acta Eruditorum, 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 Acta Eruditorum
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 Acta Eruditorum
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 (2nd ed) – Carl B.
Boyer
 http://www-history.mcs.stand.ac.uk/HistTopics/Brachistochrone.html