Johann Bernoulli

1667 - 1748

Spirit of the AgeDevelopment, improvement or discovery of:

Logarithms

Rings of Saturn

Pendulum clock

Barometer

Air pump

Sextant

Compound microscope

Telescope

Thermometer

Number theory

Analytic geometry

Kinematics of falling bodies

Christiaan Huygens

Galileo

Born in Basel, Switzerland

Father was wealthy spice and drug merchant

10th child of his parents

Younger brother of Jakob Bernoulli (Jakob was 5th child) (12 years difference in age

Father wanted sons to follow him in the family business (Jakob started in theology, Johann started in medicine)

Went to University of Basel in 1683 (age 16)

Was tutored in math by brother Jakob

Leibnitz published his first calculus findings in 1684

Johann’s early yearsAlso called John or Jean

Jakob Bernoulli

Was already a mathematician and physicist when Leibnitz published his article

Published articles on calculus in 1690

Bernoulli equation named for him:nyxQyxPy )()('

,

Held the mathematics chair at University of Basel from 1687 to 1705

Proposed the catenary problem and the isoperimetric problem

Also called James or Jaques (1654-1705)

Newton versus Leibnitz

Newton discovered first: about 1655 to 1666

Didn’t publish until 1704

Leibnitz discovered about 1682-1683

Published in 1684

Johann—professional and personal

Proud Competitive Jealous

Loved a good fight

Increasing rivalry between the brothers

“The Bernoulli’s took their mathematics in deadly earnest. Some of their letters about mathematics bristle with strong language that is usually reserved for horse thieves.” E.T. Bell

Published his first independent mathematical paper in 1691

1691—lectured in Geneva for several months

1691—traveled to Paris and met with many outstanding mathematicians including L’Hopital

Finished his medical degree in 1694 (never practiced medicine)

Married Dorothea Falkner in 1694

Son Nicolaus born in 1695

Took Chair of Mathematics at University of Groningen in 1695

L’Hopital hired Johann to tutor him in the new calculus

Johann and L’Hopital continued to correspond after Johann left Paris

For a considerable monthly fee: “I shall ask you to give me occasionally some hours of your time to work on what I shall ask you—and also to communicate to me your discoveries, with the request not to mention them to others. …for it would not please me if they were made public.”

Johann and L’Hopital

Marquis Guillaume Francois de L’Hopital (1661-1704)

L’Hopital published first textbook on differential calculus in 1696. Johann was barely acknowledged. “And then I am obliged to the gentlemen Bernoulli for their many bright ideas; particularly to the younger Mr. Bernoulli who is now a professor in Groningen. “

L’Hopital’s Rule

A

C

M

Dd

N

P

O

B

g

b

f

0

01lim

2

0

x

e x

x apply L’Hopital’s rule 2

1

2lim

][

]1[lim

2

0

2

0

x

x

x

x

e

xdx

d

edx

d)("

)('lim

0

0

)(

)(lim

xg

xf

xg

xfaxax

In his textbook, L’Hopital uses the same or nearly the same examples given in letters from Johann.

The Brachistochrone Problem

From “brachistos” meaning shortest and “chronikos” meaning time

Proposed by Johann in June 1696

“…If two points A and B are given in a vertical plane, to assign to a mobile particle M the path AMB along which, descending under its own weight, it passes from the point A to the point B in the briefest time.”

Leibnitz asked Johann to extend the contest time from January 1, 1697 to Easter to allow foreign mathematicians more time.

The Brachistochrone Problem“so few have appeared to solve our extraordinary problem, even among those who boast that through … their golden theorems, which they imagine known to no one, have been published by others long before.”

Newton solved the problem in 12 hours.

“I do not love to be …teased by foreigners about mathematical things.”

Sir Isaac Newton I recognize the lion by his paw.

Received 5 correct answers: Johann, Jakob, Leibnitz, L’Hopital, and an unsigned entry from England…..

The Brachistochrone Problem –Johann’s solution

AF

H

EC

G

O

K

LM

nm

c

D

Taken from A Source Book in Mathematics, 1200-1800, p. 394.

B

Snell’s law

a

vK

sin

Velocity of a falling body ghv 2

The Brachistochrone Problem Drawings

#I, II, III—Johann; VI thru VIII—Jakob: IX and X—L’Hopital; XII and XIII-”Anonymous” solution

How to insult like JohannYou have to really enjoy insulting someone

Insult with eloquence and style

“so few have appeared to solve our extraordinary problem, even among those who boast that through … their golden theorems, which they imagine known to no one, have been published by others long before.”

aimed at Sir Isaac Newton

, “…I would not have minded so much if (the student) had not been one of the worst students, an utter ignoramus, not known, respected, or believed by any man of learning, and his is certainly not in a position to blacken an honest man’s name, let alone a professor know throughout the learned world…”

aimed at a student of the University of Groningen

Johann’s later years

1705 Jakob died of tuberculosis. Johann was named to the mathematics chair at Basel, even though he had offers from several other universities.

1712 and 1713 Newton vs. Leibnitz (again) Johann solved a more general version of the ballistics equation and validated Leibnitz’s methods

1714 Johann published a book improving navigational methods and discussing an early understanding of kinetic energy

1720 Johann took a young Leonhard Euler as a student

1727 After the death of Newton, Johann is considered the foremost mathematician in Europe

Johann’s accomplishments

Co-discoverer of calculus of variations

Provided L’Hopital with enough material for a textbook

Teacher of Euler

His sons Nicholas, Daniel, and Johann all became mathematicians and held mathematics chairs at European universities.

Worked on and contributed to:

Differential geometry

Description of exponential calculus

Divergence of harmonic series

Principle of conservation of energy

Transmission of motion

Motion of the planets

Navigation

Johann died January 1, 1748 at the age of 80 years.

His tombstone was inscribed with “Archimedes of his age”

Bibliography

Struik, D.J. “The Origin of L’Hopital’s rule”. Mathematics Teacher. April 1963: pp. 257-260.2) Young, Robyn V., editor. Notable Mathematicians: From Ancient Times to the Present. New York: Gale Research, 1998.Kramer, Edna E. Biographical Dictionary of Mathematicians, Volume 1. New York: Simon & Schuster Macmillan, 1991. Article by E.A. Fellman and J.O. Fleckenstein. Katz, Victor J. A History of Mathematics: An Introduction, 2nd Edition. New York: Addison Wesley Longman, 1998.Boyer, Carl B. A History of Mathematics, 2nd Edition. New York: John Wiley & Sons, 1991.Struik, Dirk J. A Concise History of Mathematics, Fourth Revised Edition. New York: Dover Publications, Inc., 1987.Bell, E.T. Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincare. New York: Simon & Schuster, 1965.Seife, Charles. Zero: The Biography of a Dangerous Idea. New York: Penguin Books, 2000.Dunham, William. Journey through Genius: The Great Theorems of Mathematics. New York: Penguin Books, 1990.Larson, Ron, Robert Hostetler, and Bruce H. Edwards. Calculus: Early Transcendental Functions, Fourth Edition. New York: Houghton Mifflin Company, 2007.Bell, E.T. Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincare. New York: Simon & Schuster, 1937.Gillespie, Charles Coulston, Editor in Chief. Dictionary of Scientific Biography, Volume IV. New York: Charles Scribner’s Sons, 1971.Safra, Jacob E., Chairman of the Board. The New Encyclopaedia Britannica, 15th Edition, Volume 2. Chicago: Encyclopaedia Britannica, Inc., 2005. http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Bernoulli_Johann.html Mac Tutor History of Mathematics/Biographies/Johann Bernoulli. Article by J.J. O’Connor and E.F. Robertson, September 1998. Accessed March 16, 2008.http://www-history.mcs.st-andrews.ac.uk/HistTopics/Brachistochrone.html MacTutor History of Mathematics/History Topics/Brachistochrone. Article by J.J. O’Connor and E.F. Robertson, February 2002. Accessed March 16, 2008.Woolf, Henry Bosley, Editor in Chief. Webster’s New Collegiate Dictionary. Springfield, Massachusetts: G.&C. Merriam Co., 1979.Durant, Will and Ariel. The Age of Louis XIV, The Story of Civilization, Volume VIII. New York: Simon and Schuster, 1963.Struik, D.J., editor. A Source Book In Mathematics, 1200-1800. Cambridge, Massachusetts: Harvard University Press, 1969.