Euler At 300: An Appreciation

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Overview

Leonhard Euler (1707–1783) was the most important mathematician of the 18th century. His collected works, which number more than 800 books and articles, fill over 70 large volumes. He revolutionised real analysis and mathematical physics, single-handedly established the field of analytic number theory, and made important contributions to almost every other branch of mathematics. A great pedagogue as well as a great researcher, his textbooks educated the next generation of mathematicians. This book compiles over 20 papers, based on some of the most memorable contributions from mathematicians and historians of mathematics at academic meetings across the USA and Canada, in the years approaching Leonhard Euler's tercentenary. These papers will appeal not only to those who already have an interest in the history of mathematics, but will also serve as a compelling introduction to the subject, focused on the accomplishments of one of the greatest mathematical minds of all time.

Meet the Author

Robert E. Bradley is Professor in the Department of Mathematics and Computer Science at Adelphi University in New York. He is President of the Euler Society.

Lawrence A. D'Antonio is Associate Professor of Mathematics at Ramapo College of New Jersey, Mahwah.

C. Edward Sandifer is Professor of Mathematics at Western Connecticut State University.

Introduction

When an important mathematician celebrates a landmark birthday, other mathematicians sometimes gather together to give papers in appreciation of the life and work of the great person. When a mathematician as influential and productive as Leonhard Euler celebrates an anniversary as important as the 300th, a single meeting isn't sufficient to contain all of the contributions. During the years leading up to Leonhard Euler's tercentenary in 2007, at more than a dozen academic meetings across the USA and Canada, mathematicians and historians of mathematics paid homage to Euler in presentations detailing his many achievements. This book collects together some of the most memorable of these papers.
Leonhard Euler (1707—1783) was the most important mathematician of the 18th century. His collected works, which number more than 800 books and articles, fill over 70 large volumes. Between 1910 and 1913, Swedish Mathematician Gustav Eneström published a comprehensive census of Euler's works. He enumerated 866 works, including books, journal articles, and some published items of correspondence. He assigned each of them a number, which are now referred to as "Eneström numbers." Much like the opus numbers of classical music composers, this gives scholars a quick way to identify Euler's writings. These numbers, in the form E9, E41, and E251, are used freely throughout this collection.
Leonhard Euler was born on April 15, 1707 in Basel, Switzerland, son of Paul Euler, a Protestant minister, and Margaret Brucker. He grew up in the suburban village of Riehen and at age 14 he enrolled at the University of Basel, expecting to follow his father intothe clergy. At the University, though, he began taking Saturday lessons in mathematics from Johann Bernoulli, probably the best mathematical mind in Continental Europe at that time. Euler thrived under Bernoulli's tutelage, and his mentor soon persuaded the Eulers, father and son both, that Leonhard should pursue a career in mathematics.
Though Bernoulli's career advice eventually proved to be wise, in the short term it was not very practical, and in 1727 when Euler finished his studies in Basel, the only position he could find was at the newly-founded Academy of Peter the Great in St. Petersburg.
Czar Peter I died in 1725 and his wife, Empress Catherine I died in 1727, so when Euler arrived in St. Petersburg the country as a whole was in some disarray and the Academy suffered a severe absence of leadership and financial support. Most members of the Academy were almost entirely unproductive. Euler used the opportunity provided by the lack of supervision to pursue his own interests in mathematics and mathematical physics.
Many of Euler's best ideas have their beginnings in his first years in St. Petersburg. He established his fame among his contemporaries in 1735 when he solved the Basel problem, the best-known outstanding problem of the time. Several of our contributors, including Pengelley and Sandifer, describe aspects of this episode.
He also began his lifelong project of making the principles of physics mathematical, especially mechanics, using the new tool of calculus. He made his first large step in this program in 1731 with E33, Tentamen novae theoriae musicae, "Thoughts on a new theory of music," in which, among other things, he establishes the mathematical foundations of the evenly tempered musical scale, the scale to which most modern musical instruments are still tuned. Despite his popular connection with well tempered tuning of the clavier, Johann Sebastian Bach ultimately favored the even tempering that Euler described.
Influential though the Music Theory was, it took until 1739 to see publication, so it was not the first book that Euler published. That distinction goes to his Mechanica, E15 and E16, written in the years 1734 to 1736 and published in two volumes in 1736. This great treatise on the mechanics of point masses was only the first of a long series of important works in mathematical physics. It was followed in 1749 by Scientia navalis, E110 and E111, establishing new principles of fluid mechanics and their applications to shipbuilding. His Theoria motus corporum solidorum seu rigidorum, (Theory of the motion of solid or rigid bodies), E289, used calculus to explain the phenomena of rotating bodies, thus virtually completing the subject that the modern undergraduate physics curriculum calls~"mechanics."
The mid 1730s brought some stability to Russia, and enough additional funding that Euler earned a salary increase. This gave him the resources to marry Katerina Gsell in January 1734, and their son, Johann Albrecht, was born in November of that year, the first of their 13 children.
The 1730s marked Euler's first work in a variety of other subjects as well. He had written a short paper on the calculus of variations, E9, in about 1727, and through the 1730s he systematized his study of the subject. In 1735 he wrote his paper on the Bridges of Königsberg, E53. This paper is often cited as the beginning of the field of topology.
Probably Euler's greatest impact on the day-to-day activities of today's mathematicians and physicists came from this era, the gradual standardization of the symbols p and e for the constants we use today.
In 1741 the death of Empress Elizabeth I again left Russia in political turmoil. Riots and mobs attacking foreigners made Euler fear for the safety of his young family. When the Prussian King Frederick II offered him a position in his new Academy in Berlin, Euler accepted and left Russia for 25 years.
Frederick II later became known as Frederick the Great for his many military conquests. For about half of the time Euler spent in Berlin, Frederick was away at war. As he had done in St. Petersburg, Euler used these relatively unsupervised intervals to great effect. He wrote the pioneering text on the calculus of variations, E65, the Methodus inveniendi, and what is regarded as one of the finest mathematics books ever written, the Introductio in analysin infinitorum, E101 and E102. Euler designed this two-volume work to help students make the difficult step from algebra to calculus.
While in Berlin, Euler also worked on celestial mechanics, number theory, ballistics, a differential calculus textbook, hydraulics, the theory of machines, differential equations and a dozen other topics, as well as discovering his polyhedral formula, V - E + F = 2. More about Euler's activities in Berlin and the intellectual backdrop of Europe in the Enlightenment can be found in Rüdiger Thiele's article on Euler in the 1750s.
When the Seven Years War ended in 1763, Frederick returned to Berlin and gave much of his attention to the Academy. Euler did not appreciate such close supervision from his King, so when he had the opportunity to return to the Academy in St. Petersburg in 1766, he and his family again transplanted themselves.
Euler was treated as a celebrity when he arrived in St. Petersburg, and his son Johann Albrecht was appointed Secretary of the Academy. He finished the three books he had been working on, Letters to a German Princess, E343, E344 and E417 in 1768 and 1772, his integral calculus textbook, Institutionum calculi integralis, E342, E366 and E385 in the years 1768 to 1770, and a textbook on the construction and sailing of ships, Théorie complette de la construction et de la manoeuvre des vaisseaux, E426 in 1773. With his life's plan for publishing books complete, he turned his attention to two final goals, to leave the St. Petersburg Academy a legacy of papers to publish for 20 years after his death and to train a generation of mathematicians to succeed him. Despite unsuccessful cataract surgery in 1771 that left him almost completely blind and a house fire that destroyed his library and many of his personal possessions, he was entirely successful in the first of these goals. The Academy continued publishing his papers regularly until 1830, almost 50 years after his death.
Euler was less successful at training the next generation. Though he worked closely with a group of successors and they showed good promise while they were under his wing, who today remembers the names Johann Albrecht Euler, Anders Lexell, Nicolaus Fuss, Mikhail Golovin, S. K. Kotelnikov, Stepan Rumovsky or Petr Inokhodtsev? Though they helped Euler write over 300 papers between 1766 and his death in 1783, they, unlike Euler, were not able to work effectively without the supervision of their teacher. After Euler's death, the Academy began a long and gradual decline into mediocrity.
In a very real sense, the celebration of Euler's tercentenary in the American mathematical community began with a Contributed Paper Session on January 11, 2001, at the Joint Mathematics Meetings in New Orleans, LA. The session, Mathematics in the Age of Euler," was sponsored by the Mathematical Association of America (MAA) and organized by William Dunham and V. Frederick Rickey. Dunham is the author of the 1999 book {\it Euler: The Master of Us All\/} and Rickey was an organizer of the NSF-funded summer Institutes on the History of Mathematics and Its Use in Teaching. Well-known both for these achievements and for their outstanding abilities as speakers, Rickey and Dunham attracted an impressive array of speakers, who made presentations on the mathematics of Euler and his contemporaries. Four of the papers presented at that meeting are included in this volume: the pieces by Barnett, Heine, McKinzie ("The Formalist Argument") and Sandifer ("Euler's Fourteen Problems").
Less well-known, but just as important for the course of Euler studies, was the conversation that took place that evening among Ron Calinger, John Glaus and Ed Sandifer. Over dinner at Delmonico's, they conceived of an academic society devoted to the study of Euler's life, his achievements in mathematics and physics, and the broader historical issues related to his place in the European Enlightenment. Thus the Euler Society was founded, although it was not officially incorporated as a non-profit organization until 2003. The\break Euler Society has held annual summer meetings since August 2002. The largest group of papers in this volume consists of those given at meetings of the Euler Society: the papers by D'Antonio, Godard, Klyve & Stemkoski, Langton ("Combinatorics"), Lathrop & Stemkoski, McKinzie ("Harmonic Series"), Pengelley, and Thiele ("The Decade 1750—60").
The Canadian Society for History and Philosophy of Mathematics (CSHPM), whose annual summer meetings are a crucial activity for North American historians of mathematics, has also played host to a large number of Euler-related talks in the years leading up to the Euler tercentenary. We have included four papers from CSHPM meetings: the pieces by Baltus, Bradley (both articles), and Sandifer ("Euler Rows the Boat").
Sectional meetings of the American Mathematical Society (AMS) sometimes include sessions on the history of mathematics. Papers in this volume by Jardine and Sandifer ("The Basel Problem") were first presented at AMS meetings in Boston, MA, in 2002 and New York, NY, in 2003, respectively.
To complete this volume, we have included two more papers from MAA-sponsored events. Langton's paper on lunes was given at the MAA Contributed Papers Session on Mathematics in the Second Millennium" at the 2002 Joint Mathematics Meeting in San Diego, CA. Thiele's paper on the function concept was a special lecture sponsored by HOMSIGMAA (the MAA's History of Mathematics Special Interest Group) at the 2004 MathFest in Providence, RI.
Janet Heine Barnett's paper Enter, Stage Center: The Early Drama of the Hyperbolic Functions" has previously appeared in Mathematics Magazine, February 2004, pp. 15—30. Otherwise, the papers in this collection have not appeared elsewhere in their present form. We would like to note that Carolyn Lathrop was an undergraduate and Lee Stemkoski was a graduate student when they co-authored "Parallels in the Work of Leonhard Euler and Thomas Clausen." Both Stemkoski and Dominic Klyve were graduate students when they wrote their paper The Euler Archive: Giving Euler to the World."
This book is arranged thematically. It begins with three articles with a broad historical perspective. The rest of the chapters describe the details of many of Euler's mathematical discoveries. Results in pure mathematics come before those in applied mathematics. The mathematics papers begin with one article each on algebra and geometry. These are followed by eight chapters on analysis, the first four of which cover a variety of topics from the function concept to the theory of elliptic integrals, while the latter four are concerned with infinite series. These are followed by three pieces on combinatorial or probabilistic topics. The collection concludes with five papers on applied topics, from fluid mechanics to shipbuilding and mapmaking.
Many people contribute to the creation of a volume such as this one, and the editors wish to express their appreciation. We begin by thanking the fifteen authors, whose work the editors get three opportunities to enjoy: first of all when they made their presentations at the various meetings over the last several years, second in manuscript form as they worked with the authors to recast their spoken words as written ones, and finally they can enjoy their creative efforts in the form of this book for many years to come. We join our authors in extending t to the organizers of the many sessions and meetings that gave the authors venues to present their papers.
All of the authors—indeed, all Euler scholars—are grateful to Dominic Klyve and Lee Stemkoski for creating and maintaining the Euler Archive (www.eulerarchive.org). They conceived of the archive, which they describe in chapter 3 of this book, at the first meeting of the Euler Society in 2002. With the assistance of Rachel Esselstein, Alison Setyadi, Erik Tou and a group of energetic Dartmouth undergraduates, Lee and Dominic have made more than 95% of Euler's original publications available on the worldwide web. Almost as important to scholars is the searchable data base of bibliographic information on Euler's writings. The archive is an efficient and powerful tool.
We thank the editors and staff at the MAA, especially Jerry Alexanderson, Bev Ruedi, Elaine Pedreira and Don Albers, for their hard work, insight and patience. To these we add special thanks to Theresa Sandifer. When the editorial work got particularly heavy for a while, she pitched in to help proofread and edit manuscripts. That help is greatly appreciated. We are also grateful to our wives and families (and cats) for their patience and support through the years it has taken to bring this book to fruition.

Robert E. Bradley & Lawrence A. D'Antonio
Adelphi University & Ramapo College of New Jersey
Garden City, NY & Mawah, NJ

C. Edward Sandifer
Western Connecticut State University
Danbury, CT

Introduction; Leonhard Euler, the decade 1750–1760 Rüdiger Thiele; Euler's fourteen problems C. Edward Sandifer; The Euler archive: giving Euler to the world Dominic Klyve and Lee Stemkoski; The Euler-Bernoulli proof of the fundamental theorem of algebra Christopher Baltus; The quadrature of Lunes, from Hippocrates to Euler Stacy G. Langton; What is a function? Rüdiger Thiele; Enter, stage center: the early drama of the hyperbolic functions Janet Heine Barnett; Euler's solution of the Basel problem - the longer story C. Edward Sandifer; Euler and elliptic integrals Lawrence D'Antonio; Euler's observations on harmonic progressions Mark McKinzie; Origins of a classic formalist argument: power series expansions of the logarithmic and exponential functions Mark McKinzie; Taylor and Euler: linking the discrete and continuous Dick Jardine; Dances between continuous and discrete: Euler's summation formula David J. Pengelley; Some combinatorics in Jacob Bernoulli's Ars Conjectandi Stacy G. Langton; The Genoese lottery and the partition function Robert E. Bradley; Parallels in the work of Leonhard Euler and Thomas Clausen Carolyn Lathrop and Lee Stemkoski; Three bodies? Why not four? The motion of the Lunar Apsides Robert E. Bradley; 'The fabric of the universe is most perfect': Euler's research on elastic curves Lawrence D'Antonio; The Euler advection equation Roger Godard; Euler rows the boa C. Edward Sandifer; Lambert, Euler, and Lagrange as map makers George W. Heine, III; Index.

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