The Bernoullis and Euler 100 Years of Mathematics

The Bernoullis and Euler 100 Years of Mathematics 1690 -1790 Claudette Reep and Henry Williams July 2, 2008

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

http: //www. mathsoc. spb. ru/pantheon/bernoull/Bernoulli_family. gif

The Bernoulli Brothers James & John

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.

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

§ James returned to Basel and began teaching at the university there in 1683. He would later be appointed mathematics professor in 1687. He married the lovely Judith Stupanus in 1684. 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 apply his theories.

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.

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

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. ”

John Bernoulli § July 27, 1667 – January 1, 1748 § 10 th 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).

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.

While in Paris § John met Guillaume Notre Dame Image from: http: // en. wikipedia. org/wiki/Image: DSC 00733_Notre_Dame_Paris_from_east. jpg François Antoine de L’Hôpital. § Developed a strong friendship and a long correspondence with Varignon. § Also began a fruitful correspondence with Leibniz.

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. http: //z. about. com/d/math/1/0/G/G/hopital. jpg http: //www. mrtbooksla. com/shop_image/product/14204. jpg

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.

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). http: //www. auwerda. nl/home/uploads/content/Image/martini. jpg 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.

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 1738 -1739 (he had backdated the work in an attempt to obtain priority over Daniel).

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. ”

Calculus- The Next Generation Daniel Bernoulli And Leonhard Euler

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 http: //www. map-of-netherlands. co. uk/map-of-groningen. htm

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 http: //www-groups. dcs. st-and. ac. uk/~history/Biographies/Bernoulli_Daniel. html

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 http: //www-groups. dcs. st-and. ac. uk/~history/Biographies/Bernoulli_Daniel. html

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

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

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.

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

§ “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 1970 -1990). From website: http: //www-groups. dcs. st-and. ac. uk/~history/Biographies/Bernoulli_Daniel. html

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 http: //www. mathsoc. spb. ru/pantheon/bernoull/nic 2. html § 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 http: //www-history. mcs. st-andrews. ac. uk/Printonly/Bernoulli_Johann(II). html

Leonhard Euler (1707 -1783) § Born the son of a Lutheran pastor near Basel, Switzerland § Life can be viewed in four phases § § Basel, Switzerland (1707 -1726) St. Petersburg Academy (1726 -1741) Berlin Academy (1741 -1766) St. Petersburg (1766 -1783) http: //www. worldatlas. com/webimage /countrys/ europe/lgcolor/chcolor. htm

Basel, Switzerland 1707 -1726 § Attended University of Basel http: //www. erscp 2007. net/cms/index. php? id=24 § 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 http: //en. wikipedia. org/wiki/Basel

http: //en. wikipedia. org/wiki/Image: Kunstkamera_%28 Saint-Petersburg%29. jpg St. Petersburg Academy 1726 -1741 § 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) http: //www-history. mcs. st-andrews. ac. uk/Biographies/Euler. html

§ 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(2 n) = Cπ^(2 n) in terms of the Bernoulli numbers. § 1740 he published Methodus inveniendi lineas curvas § Calculus of variations http: //www-history. mcs. st-andrews. ac. uk/Biographies/Euler. html

Berlin Academy 1741 -1766 “Académie Royale des Sciences et Belles Lettres” § 1741 Euler accepted the director of mathematics position at the Berlin Academy http: //www. historycooperative. org/journals/ahr/112. 3/sauter. html § 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 Frederick the Great Picture attained from: http: //images. google. com/imgres? imgurl=http: //www. hillsboro-museums. com/ files/Fredrick-der-Gross 3. jpg&imgrefurl=http: //www. hillsboro-museums. com/ Who-were-the-Mennonites. html&h=350&w=286&sz=17&hl=en&start=5&um=1&tbnid =jf. Cv. Hxn_hva 85 M: &tbnh=120&tbnw=98&prev=/images%3 Fq%3 Dfredrick%2 Bthe%2 Bgreat%26 um%3 D 1%26 hl%3 Den%26 rls%3 Dcom. microsoft: *: IE-Search. Box%26 rlz% 3 D 1 I 7 DKUS%26 sa%3 DN § Instructions on scientific matters § Based on his religious beliefs § Printed in many languages § O’Conner & Robertson

§ 1748 he wrote his great calculus textbook § Introductio in analysin infinitorum § Bases the calculus on 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 http: //www-history. mcs. st-andrews. ac. uk/Biographies/Euler. html § Euler continued to receive a salary from the St. Petersburg Academy and therefore continued to send them papers to publish § Article http: //www-groups. dcs. st-and. ac. uk/~history/Bookpages/Euler 6 a. gif

§ Shortly after returning he lost sight in his other eye § He was able to continue publishing with the help of his children and servants Return to St Petersburg 1766 -1783 § 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. View of the port of St. Petersburg to the West (Burton, 2007) Image from: http: //www. russiablog. org/Portof. St. Petersburg. JPG

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) http: //www-history. mcs. st-andrews. ac. uk/Biographies/Euler. html

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 § http: //books. google. com/books? id=13 d. Kav 77 v. Gs. C&pg=PA 187&lpg=PA 187&dq=euler's+death+he+ceased&source=web&ots=Yl. M_Ny. Ye 2 P&sig=Dt 5 LMn. A 4 C 4 vk. GQSJPSoy 09 ay 7 Eg&hl=en&sa=X&oi=book_result&resnum=8&ct=result#PPA 187, M 1 § Euler suffered a stroke around noon and passed away later that evening

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)

References § § § § § Burton, D. M. (2007). The history of mathematics: An introduction. New York, NY: Mc. Graw-Hill. O' Conner, J. J. & Robertson, E. F. (1998). Daniel Bernoulli. Retrieved June 28, 2008, from Mac. Tutor Web site: http: //www-history. mcs. st-andrews. ac. uk/Biographies/Bernoulli_Daniel. html O' Conner, J. J. & Robertson, E. F. (1998). Jacob (Jacques) Bernoulli. Retrieved June 30, 2008, from Mac. Tutor Web site: http: //www-history. mcs. standrews. ac. uk/Biographies/Bernoulli_Jacob. html O' Conner, J. J. & Robertson, E. F. (1998). Johann Bernoulli. Retrieved June 30, 2008, from Mac. Tutor Web site: http: //www-history. mcs. standrews. ac. uk/Biographies/Bernoulli_Johann. html O' Conner, J. J. & Robertson, E. F. (1997). Johann (II) Bernoulli. Retrieved June 30, 2008, from Mac. Tutor Web site: http: //www-history. mcs. standrews. ac. uk/Biographies/Bernoulli_Johann(II). html O' Conner, J. J. & Robertson, E. F. (1998). Leonhard Euler. Retrieved June 28, 2008, from Mac. Tutor Web site: http: //www-history. mcs. st-andrews. ac. uk/Biographies/Euler. html O' Conner, J. J. & Robertson, E. F. (1997). Nicolaus (II) Bernoulli. Retrieved June 30, 2008, from Mac. Tutor Web site: http: //www-history. mcs. standrews. ac. uk/Biographies/Bernoulli_Nicolaus(II). html Pictures of notable mathamaticians were obtained from the Mac. Tutor Website (http: //wwwhistory. mcs. st-and. ac. uk/) All other pictures were credited beneath them.