Leonhard Euler

Swiss mathematician and physicist

• Born: April 15, 1707
• Birthplace: Basel, Switzerland
• Died: September 18, 1783
• Place of death: St. Petersburg, Russia

Euler had a tremendous impact on almost all fields of mathematics, opening new and more fruitful courses of inquiry. One of the most prolific mathematical writers ever, his founding of the field of analysis was particularly important, and his notations remain in common use in mathematics.

Early Life

Leonhard Euler (LAY-awn-hahrt OY-luhr) was born to a Calvinist minister, Paul Euler, and his wife, Marguerite Brucker, the daughter of a minister, in Basel, Switzerland, on April 15, 1707. The family soon moved to the suburb of Riehen. Little is known of Euler’s childhood, but the information that is available indicates that his interest in mathematics was quite logical, because his father was an excellent mathematician in his own right.

Paul Euler had studied under Jakob I Bernoulli, a member of the famous Bernoulli family of mathematicians, while he was studying for his degree in theology. The elder Euler gave Leonhard his first instruction at home. During this period, the younger Euler studied some of the most difficult texts in mathematics available at the time. He later went to live with his grandmother in Basel, where he went to the local school (Gymnasium). Euler was not satisfied with the mathematics instruction offered there and received private tutoring from Johann Burckhardt.

When Euler was almost fourteen, at his father’s wish he entered the University of Basel to study theology. Although Leonhard was quite devout and worked dutifully, he had no desire to become a minister, and he filled his free time with mathematics. In fact, in time he received limited tutoring from Johann I Bernoulli. Bernoulli suggested texts for Euler and agreed to explain any difficulties during his free time on Saturdays. Since Euler did not want to disappoint Bernoulli, he worked very hard to ensure that he did not waste the professor’s time.

Life’s Work

In 1724, at age seventeen, Euler received a master’s degree. His father was concerned about the progress Leonhard was making in theology, but the Bernoullis convinced Paul Euler that the young man was extremely gifted in mathematics and that the gift should not be wasted. Thus, Euler was free at a very young age to pursue a career in mathematics. Euler began working independently and submitted a solution to a problem in navigation proposed by the Academy of Sciences in Paris in 1727. Although he received only an honorable mention from the academy, his name was placed before many of the people who could influence his career.

Unfortunately, there were many mathematicians who were not ready to accept one so young. When Euler applied for a post as professor of mathematics at the University of Basel, his name was not forwarded, probably because of his age. As such positions were rare in his home country, Euler was very discouraged, but the Bernoullis encouraged him with news of the newly formed Academy at St. Petersburg in Russia. This institution had a twofold purpose: Its members received a stipend to continue their own work, and, from time to time, the czar might pose practical problems to be solved by the members. Both Nikolaus and Daniel Bernoulli held positions there, and they wrote to Euler that there would soon be an opening in medicine. Therefore, Euler began to study anatomy so that he would be qualified for the position when it became available. He received the appointment in 1727, and he traveled to Russia, intending to accept this medical position. The reigning monarch, Catherine I, died before he could take up his appointment, however, and he instead joined the mathematical group, unnoticed in the change of regimes.

Although the political situation in Russia was not entirely satisfactory, the Russian academy offered Euler security and a comfortable lifestyle, and he was able to marry Catharina Gsell and begin a family. Except for a two-decade stint in Berlin, he made St. Petersburg his permanent home. His work for the first six years was fairly routine as a member of the physics staff, but in 1733 he became the leading member in mathematics when Daniel Bernoulli left to return home.

Euler threw himself into his work with fervor. (Indeed, when the Swiss prepared to publish Euler’s writings in the twentieth century, the project’s editors were stunned by the amount of material found in St. Petersburg.) His work at the Russian academy was diverse, spanning navigation, cartography, ballistics, mechanics, measurement, and especially mathematics. During this first, fourteen-year period in Russia, Euler wrote nearly one hundred articles and memoirs for publication. He also maintained correspondence with the most widely known European mathematicians, both for himself and in the name of the Academy of Sciences. Indeed, as the result of his strenuous pursuit of a Parisian prize in the field of astronomy, Euler developed an illness that resulted in the loss of sight in one eye.

In 1741, Euler was invited to Berlin by Frederick the Great of Prussia as part of the reorganization and refurbishing of the Berlin Academy. Euler accepted this position, which he filled from 1741 until 1766. He also maintained his membership in the St. Petersburg Academy, as well as in the Royal Society of London, to which he was elected in 1749. He continued to write for the Russian academy during this time, as he was still in their employ. While living abroad, Euler received a stipend from Russia in addition to his recompense for his post in Berlin. While in Russia, he was supported well enough to have several servants. Euler and Frederick got along so poorly that at least once Frederick tried to remove him as the director of the academy but was convinced by others that this would be a mistake. Nevertheless, Euler more than earned his pay. He worked on many applications for Frederick, including coinage, insurance, and pensions, and he held several administrative posts. In addition, he produced almost three hundred mathematical papers and tutored some of Frederick’s relatives.

By 1766, however, the situation with Frederick had become so bad that Euler, then fifty-nine, decided to accept the invitation of Catherine the Great to return to Russia. Because he had regularly sent memoirs back to that country and had enjoyed its financial support while in Berlin, the move seemed logical. He was to live there for the remainder of his life.

Soon after his return, Euler began to develop a cataract in his remaining eye. For a man of lesser gifts, blindness would have been a career-ending disability. Euler, however, began to train himself to solve problems mentally and dictate the results to others, principally his sons Johann Albrecht and Christoph, who were also mathematicians. He succeeded so completely that he was able to work in this fashion for another fifteen years, holding his post at the academy and actually producing more papers than ever before. During this time, Euler produced some of his best work, including his analysis of the effects of the gravitational pull of the Earth and the Sun upon the motion of the Moon. Although he did benefit from discussions with his peers, all the work had to be done without the aid of writing partial results or ideas. He also produced a monograph on integral calculus, work on fluid mechanics, and won a prize for work in astronomy.

Despite a lifetime of work, Euler was most comfortable with his family. A devoted family man, he even held his children while working on mathematics when they were still small. Working with his sons in his later life was also quite fulfilling for him. Although he had chosen not to pursue theology as a youth, Euler never left his church, and he held daily services with his family. Euler’s wife died in 1776, and he soon was married to his first wife’s sister, with whom he lived until his death in St. Petersburg on September 18, 1783.

Significance

The extent of Leonhard Euler’s work was vast. In mathematics, he developed much of the notation in current use, in addition to a considerable amount of theory. Euler was the first to treat trigonometry as a field in itself rather than a branch of geometry, and he developed spherical trigonometry. Thus, he led the way in its development as a discipline. He made great progress in calculus, writing two texts, Institutiones calculi differentialis (1755) and Institutiones calculi integralis (1768-1770), that are still used by mathematicians as reference works. Included in these books are several discoveries Euler made concerning differential equations and partial differential equations. Euler made significant refinements to the fundamental theorem of algebra. He was also extremely interested in summation of infinite series and developed much of the basis upon which convergence theories would later be founded.

Although he produced a great quantity of work in physics, in part in response to requests by monarchs, Euler’s major contribution in this field was his imposing analysis of mechanics. He was far more interested in the mathematical aspects of physical problems and thus was able to systematize his study. Euler published his results in Mechanica sive motus scientia analytice exposita (1736) and Theoria motus corporum solidorum seu rigidorum (1765). The former work was the first attempt to establish clear solutions to mechanical problems. Other sciences in which Euler worked include astronomy, navigation, and optics, yet Euler’s foremost field was mathematical analysis, a field that owes its foundation to Euler’s book Introductio in analysin infinitorum (1748; Introduction to Analysis of the Infinite, 1988-1990). Of particular interest to mathematicians is his development of function theory and notation.

The republication of Euler’s work began in 1911 in Leipzig, Germany, and moved to Lausanne, Switzerland, in 1942. Three series have been produced, Opera mathematica, Opera mechanica et astronomica, and Opera physica, in which each work is reproduced in the original language of publication. Although only those papers that Euler personally prepared for publication are included, it is estimated that to include them all would take more than fifty volumes. Euler ranks as one of the most prolific mathematicians in history.

Bibliography

Bell, Eric T. “Analysis Incarnate.” In Men of Mathematics. New York: Simon & Schuster, 1937. Each chapter in this book deals with a major mathematician, dating from ancient Greece to the early twentieth century. This chapter on Euler includes biographical information and a limited discussion of his work.

Boyer, Carl B. A History of Mathematics. 2d ed., rev. New York: Wiley, 1989. Boyer’s book is a standard though extensive history, and his discussion of Euler and his work is both interesting and clear.

Dunham, William. Euler: The Master of Us All. Washington, D.C.: Mathematical Association of America, 1999. In this book aimed at readers with a knowledge of mathematics, Dunham describes Euler’s many contributions to the field. Topics include number theory, logarithms, infinite series, complex variables, algebra, and geometry.

Eves, Howard. An Introduction to the History of Mathematics. 5th ed. Philadelphia: Saunders College, 1983. Although the treatment of Euler is extremely brief, Eves is excellent in placing Euler within the evolution of mathematics.

Havil, Julian. Gamma: Exploring Euler’s Constant. Princeton, N.J.: Princeton University Press, 2003. Euler first described how gamma was a constant in many areas of mathematics. Almost three hundred years later, however, the nature of this constant remains a mystery. Havil examines Euler’s discovery and subsequent developments in the understanding of gamma.

Struik, Dirk J. A Concise History of Mathematics. 4th rev. ed. New York: Dover, 1987. A standard history of mathematics; the treatment of Euler and his work is concise yet informative.

Youschkevitch, A. P. “Leonhard Euler.” In Dictionary of Scientific Biography. Vol. 4. New York: Charles Scribner’s Sons, 1971. This article is of particular note for at least two reasons. First, Youschkevitch was a fellow of the Soviet Academy of Sciences, an outgrowth of the St. Petersburg Academy. As such, he had easy access to Euler’s work. Second, the article contains an extensive bibliography (seventy entries).