1. The Bernoullis and Euler
100 Years of Mathematics
Claudette Reep and Henry Williams
July 2, 2008

2. And Calculus was Defined
Sir Isaac Newton
Leibniz
John Bernoulli
James Bernoulli
Brooks Taylor
Nicolas Bernoulli
John Bernoulli
Colin Maclaurin
Daniel Bernoulli
Leonhard Euler

4. The Bernoulli Brothers
James & John
Father – Nicolaus ( ) was a spice trader. Grandfather was a Calvinist who left Holland to avoid persecution by Catholic Spain.

5. James Bernoulli December 27, 1654 – August 16, 1705.
Master’s in philosophy, Univ. Basel 1671.
Licentiate in theology 1676.
Studied math/astronomy during this time (against his parent’s wishes).
Probably influenced others in his family to study math/physics.
A.K.A. Jacob; Jacques

6. Travels Geneva – worked as a tutor.
France – studied with followers of Descartes.
Netherlands – met Hudde.
England – met with Boyle and Hooke.

7. James returned to Basel and began teaching at the university there in He would later be appointed mathematics professor in He married the lovely Judith Stupanus in They were blessed with a daughter and a son (Nicolaus), neither of whom became mathematicians or physicists.
Leibniz’s publications on the calculus were very obscure to mathematicians at the time and James Bernoulli was the first to try to understand and apply his theories.

8. Published Works
Important work on infinite series and his law of large numbers in probability theory.
Five treatises containing results such as :
diverges and converges to a finite limit less than 2.
Studies of exponential series from examinations of compound interest.
In a 1690 paper, he used the word integral with its integration meaning for the first time.

9. Other Accomplishments
James solved what is now called “the Bernoulli Equation:”
His most original work, Ars Conjectandi contains “Bernoulli Numbers.”

10. Auf Wiedersehen James died of tuberculosis in 1705.
He asked that a logarithmic spiral be placed on his tombstone.
Epitaph : “I shall arise the same though changed.”
He found the properties of the logarithmic spiral to be almost magical. The Latin inscription is ‘Eadem Mutata Resurgo.’

11. John Bernoulli July 27, 1667 – January 1, 1748
10th child (12 years younger than James)
In 1682, at age 15, worked in father’s spice trade.
Entered University of Basel to study medicine in 1683 (studied mathematics with brother James, especially Leibniz’s papers).
A.K.A. Johann; Jean. He disliked the spice trading business. After two years of studying with James, John is said to have become his equal in mathematical skill.

12. 1691
At age 24, John solved the catenary problem posed by his brother James (as did Leibniz and Huygens). This assured him a place among leading mathematicians.
Moved to Geneva to lecture on differential equations.
Made his way to Paris.
During the early 1690’s the friendly rivalry once enjoyed by the brothers descended into open hostility. Even though they worked on similar topics, they never published joint work.

13. While in Paris John met Guillaume François Antoine de L’Hôpital.
Developed a strong friendship and a long correspondence with Varignon.
Also began a fruitful correspondence with Leibniz.
Leibniz’s most major correspondence was the one he carried out with John. John produced many papers during this time in addition to his letters (all while working on his doctoral dissertation).
In discussing proper notation, Leibniz suggested the elongated s for calculus summatorius. Bernoulli preferred I for claculus integralis. They comprimised by using Leibniz’s symbol and John’s word (which was really James’).
Notre Dame
Image from:

14. The L’Hôpital Controversy
L’Hôpital hired John Bernoulli to tutor him in the new calculus methods.
The lessons continued via letters after John left Paris.
Their financial agreement evidently included L’Hôpital receiving the latest discoveries first to be used as he pleased.
L’Hôpital published the first calculus book in 1696 based on the lessons sent to him by John.
The book said little to acknowledge the fact that it was based on John’s lectures and after L’H’s death in 1704, he protested that he was the author. In 1922, a copy of Bernoulli’s course made by his nephew Nicolaus was found in Basel. It was virtually identical to the L’H calculus text (L’H had corrected some errors). Few had believed Bernoulli until these proofs were discovered. The rule we now call L’Hopital’s is a result of John Bernoulli.

15. 1694 : Math and Marriage Married Drothea Falkner.
Considered the function
Investigated series using integration by parts.
By viewing integration as the inverse of differentiation, he had success in integrating DE’s.
Summed series and discovered addition theorems for trig and hyperbolic functions using DE’s.
Journeyed to Holland with wife and 7-month old baby in Sept
Three of their sons became mathematicians : Nicolaus II, Daniel, and Johann II.

16. His outstanding contributions to mathematics led to John receiving two offers of chairs. On the advice of Huygens, John accepted the position at Groningen (Netherlands).
The Bernoulli’s decided to move back to Basel after receiving a letter from Drothea’s father pleading to see his daughter and grandchildren before he died. Once into the journey, John learned that his brother had died two days before they had left Groningen. John had intended to fill the chair of Greek at B.U. upon his return, but was soon appointed to James’ chair in mathematics.

17. Other Highlights/lowlights
Expelled son Daniel from his house for winning a prize he had expected.
Tutored young Leonhard Euler on Saturday mornings (Euler later became his doctoral student).
Became involved in the Newton/Leibniz controversy in 1713.
Published Hydraulica around (he had backdated the work in an attempt to obtain priority over Daniel).
Of course, Bernoulli supported Leibniz by showing the power of his brand of calculus to solve certain problems Newton had failed to. He also incorrectly supported Descartes over Newton in the argument between vortex theory and the theory of gravitation, which delayed acceptance of Newton’s physics.

18. Good Night, John Boy
John attained great fame in his lifetime, being elected fellow to academies of Paris, Berlin, London, St. Petersburg, and Bologna.
He died at age 80.
On his tombstone : “Archimedes of his age.”

19. Calculus- The Next Generation
Daniel Bernoulli
And
Leonhard Euler

20. Daniel Bernoulli 1700-1782 Born in Groningen, Netherlands
Moved to Basel at the age of 5
Father was to assume chair of mathematics at Basel University
The son of Johann (John) Bernoulli
Brothers were Nicolaus and Johann(II)
A family of talented mathematicians where rivalries and jealousy were very strong

21. Daniel Education Father wanted him to pursue a career in business
Attended Basel University where he studied philosophy and logic
Obtained Masters degree in 1716
Father told him there was no money in mathematics
Yet, he continued to learn calculus from his father and older brother, Nicolaus
Later he studied Medicine at Heidelberg in 1718 and Strasburg in 1719
1720- completed his doctorate in medicine at Basel
Dissertation was on the mechanics of breathing
Based on the conservation of energy from his father’s studies

22. Daniel begins to publish
Mathematical exercises was publish in 1724
Based on four topics
Game of faro (probablility based)
Fluid flow (a predecessor to future studies)
Riccati differential equation
Geometry

23. Daniel’s St. Petersburg Experience
In 1725 he and his brother, Nicolaus, assumed chairs in mathematics positions at St. Petersburg
Less than a year later his brother dies of fever
In 1727 Euler arrived in St. Petersburg to work with Daniel at the request of Johann, Daniel’s father
This was a catalyst for Daniel’s renewed spirit

24. St. Petersburg Discoveries
Defined the simple nodes and the frequencies of oscillation of a system
Probability and political economy
Inverse relationship between the moral value of the increase in a person’s wealth and the amount of that wealth
Applied these studies to insurance
Most importantly was his work with hydrodynamics
Hydrodynamica
Completed in 1734 and finally published in 1738
The first correct explanation of water flowing from a hole in a container
The basis for the kinetic theory of gases

25. Daniel Leaves St. Petersburg
In 1733 Daniel leaves for Basel to assume a position that was not related to math
Had been unhappy at St. Petersburg for a long time
This allowed Euler to assume the position at St. Petersburg
In 1734 He become a joint winner for the Grand Prize of the Paris Academy for his work in astronomy
Shared the prize with his father
Father was very upset that his son had been declared his equal
He was banned from his father’s house
In later years his father attempted to gain credit for his son’s work by backdating the publishing date of his book
Hydraulica – dated 1732, but probably actually published in 1739
Bernoulli Principle equation with explanation
Bernoulli Principle Simulation
In 1750 he was appointed chair of physics and taught there for 26 more years.

26. Notable Awards and Recognitions
10 time winner of the Grand Prize of the Paris Academy
Astronomy and Nautical Topics
Daniel Bernoulli was successful in assisting in the development of mathematical physics by combining the works of Newton and Liebniz
Using Newton’s basic equations
Using the powerful calculus of Leibniz
Fellow of the Royal Society

27. “Bernoulli's active and imaginative mind dealt with the most varied scientific areas. Such wide interests, however, often prevented him from carrying some of his projects to completion”
H Straub, Biography in Dictionary of Scientific Biography (New York ). From website:

28. Nicolaus(II) and Johann(II)
Nicolaus(II) Bernoulli
Older brother to Daniel and his father’s favorite son
Assisted father in defending Liebniz claim to priority in calculus
Had a promising career and shared an appointment at St. Petersburg with Daniel
Less than a year after arriving in St. Petersburg he died of fever
Johann(II) Bernoulli
Younger brother to Daniel
Studied law and in 1727 he obtained the degree of doctor of jurisprudence
Won the Prize of the Paris Academy four times
Assumed his father’s chair after his death

29. Leonhard Euler ( )
Born the son of a Lutheran pastor near Basel, Switzerland
Life can be viewed in four phases
Basel, Switzerland ( )
St. Petersburg Academy ( )
Berlin Academy ( )
St. Petersburg ( )
europe/lgcolor/chcolor.htm

30. Basel, Switzerland 1707-1726 Attended University of Basel
Initially entered in the study of Theology
John Bernoulli convinced Euler’s father to allow him to study mathematics
Received his masters degree in 1723 at the age of 15
(Burton, 2007)
The city of Basel

31. St. Petersburg Academy 1726-1741
St. Petersburg Academy
1727 Euler arrives in Russia to accept a position in the Academy
The same day as Empress Catherine I died
He became a ship’s officer in the Russian Navy and almost accepted a lieutenancy
1730 Euler accepts a position as physics professor at the Academy
1733 Euler becomes the chief mathematician of the Academy, succeeding his friend, Daniel Bernoulli
During this time, following a severe illness, Euler lost the sight in one eye.
(Burton, 2007)

32. 1734 Euler married Katharina Gsell, who was from a Swiss family
He fathered thirteen children
Only five survived past infancy
Sons helped him to publish in his later years,
following the loss of his eyesight
1735 was when he was able to prove that
This solution, to the what had become known as the “Basel problem”, had eluded many top mathematicians of the time including the Bernoullis, Leibniz, Stirling, and de Moivre. He continued to prove many more summations during this time.
Euler won the Grand Prize of the Paris Academy in 1738 and 1740
By 1739 Euler had found the rational coefficients C in ζ(2n) = Cπ^(2n) in terms of the Bernoulli numbers.
1740 he published Methodus inveniendi lineas curvas
Calculus of variations

33. Berlin Academy
“Académie Royale des Sciences et Belles Lettres”
1741 Euler accepted the director of mathematics position at the Berlin Academy
Initially was held in high esteem by Frederick the Great
By the time he left Euler was not on the best of terms with Frederick
Euler was traditional and not much into the spirit of the Enlightenment
Wrote over 380 articles during this time
Euler wrote Letters to a German Princess
Instructions on scientific matters
Based on his religious beliefs
Printed in many languages
O’Conner & Robertson
Frederick the Great
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34. 1748 he wrote his great calculus textbook
Introductio in analysin infinitorum
Bases the calculus on the theory of elementary functions, rather than geometric curves.
Was based on the ideas of Johann Bernoulli
Introduction of Euler’s Formula
In 1751 he published his full theory of logarithms of complex numbers including
1755 he published Institutiones calculi differentialis
Study of the calculus of finite differences
Euler continued to receive a salary from the St. Petersburg Academy and therefore continued to send them papers to publish
Article

35. Return to St Petersburg 1766-1783
Shortly after returning he lost sight in his other eye
He was able to continue publishing with the help of his children and servants
1770 he published Institutiones calculi integralis
A study of integrals
Included beta and gamma functions
Ordinary and partial differential equations
He had a phenomenal memory!
After losing his sight completely he is recorded to have arrived at the correct calculation in his mind when two of his students disagreed in their results at the fifteenth significant figure in a sum of a 17 term series.
(Burton, 2007)
View of the port of St. Petersburg to the West
Image from:

36. A Few Other Euler Notables:
Euler was a Fellow of the Royal Society
Euler contributed to the study of differential geometry which was later rediscovered by Gauss
Euler published works on mechanics, hydrostatics and astronomy
We gained much of our current notation from Euler
Some of these include
f(x) for a function (1734)
e for the base of natural logs (1727)
i for the square root of -1 (1777)
π for pi, ∑ for summation (1755)

37. Euler’s Death
“…the pipe , which he held in his hand, dropped from it, and he ceased to calculate and to breath”
Nicolas Condorcet’s Elogium of Euler
Euler suffered a stroke around noon and passed away later that evening

38. A Final Look at Euler’s Writings
He wrote or dictated over 700 books and papers in his lifetime
Approximately half of these were after he became blind
It is estimated that it will take more than 75 large volumes to publish his collected works
This was begun in 1911 by the Swiss Society of Natural Sciences
(Burton, 2007)

39. References
Burton, D. M. (2007). The history of mathematics: An introduction. New York, NY:
McGraw-Hill.
O' Conner, J. J. & Robertson, E. F. (1998). Daniel Bernoulli. Retrieved June 28, 2008,
from MacTutor Web site:
O' Conner, J. J. & Robertson, E. F. (1998). Jacob (Jacques) Bernoulli. Retrieved June 30,
2008, from MacTutor Web site:
O' Conner, J. J. & Robertson, E. F. (1998). Johann Bernoulli. Retrieved June 30, 2008,
from MacTutor Web site:
O' Conner, J. J. & Robertson, E. F. (1997). Johann (II) Bernoulli. Retrieved June 30,
2008, from MacTutor Web site:
O' Conner, J. J. & Robertson, E. F. (1998). Leonhard Euler. Retrieved June 28, 2008,
from MacTutor Web site:
O' Conner, J. J. & Robertson, E. F. (1997). Nicolaus (II) Bernoulli. Retrieved June 30,
2008, from MacTutor Web site:
Pictures of notable mathamaticians were obtained from the MacTutor Website (
All other pictures were credited beneath them.