Euler, Leonhard (1707–1783)

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EULER, LEONHARD (1707–1783)

EULER, LEONHARD (1707–1783), the most prominent and productive mathematician of the Enlightenment, who laid the foundations for numerous new fields. Born in Basel to a Protestant minister and the daughter of another, Euler was destined for the clergy. His propensity for mathematics appeared early, however, and when he entered the University of Basel at the age of thirteen, he studied under the noted mathematician Johann I Bernoulli (1667–1748). He received his master's degree in philosophy in 1723 and joined the department of theology.

From the beginning, however, Euler worked hard to secure a position as a professional mathematician. His close association with the Bernoulli clan of mathematicians, which was to last throughout his life, proved invaluable in this. In 1727 he followed Johann I's two sons, Nikolaus II and Daniel, to the newly established St. Petersburg Academy of Sciences. Although initially invited to serve as professor of physiology, Euler ultimately succeeded Daniel Bernoulli as the academy's professor of mathematics in 1733. In the same year he married Katrina Gsell, daughter of a Swiss painter residing in St. Petersburg. During his years in St. Petersburg, Euler set the pattern for his subsequent career with a prodigious output of articles, treatises, and books on all aspects of mathematics. He was also active in various practical duties of the academy, including the mapping of Russian territories and studies of shipbuilding and navigation.

In 1741 Euler accepted the invitation of Frederick II of Prussia to join the newly reorganized Berlin Academy of Sciences. His twenty-five years in Berlinwere marked by his close association with the academy's president, Pierre-Louis Moreau de Maupertuis (1698–1759), as well by his active participation in several controversies that rocked the "Republic of Letters." Most celebrated among these were a dispute on Leibnizian monads, which Euler vehemently opposed; a controversy about Maupertuis's "Principle of Least Action," in which Euler supported his colleague against Johann Samuel König and Voltaire; and a prolonged debate with d'Alembert and Daniel Bernoulli on the equations describing vibrating strings, which drew in all the leading mathematicians in Europe.

After Maupertuis's death in 1759 Euler became the de facto leader and administrator of the Berlin Academy, but without the official title of president. His strained relations with Frederick II led him to accept an invitation from Catherine the Great to rejoin the St. Petersburg Academy. He returned to St. Petersburg in 1766 and remained there until his death in 1783.

Euler's mathematical output was prodigious, and his collected works include no less than 856 separate works, both published and unpublished. His contributions span all mathematical fields known in his time, as well as several that he founded himself. Euler wrote three textbooks on the differential and integral calculus, which included extensive discussions of differential equations and means to their solution: Introductio in Analysin Infinitorum (1748), Institutiones Calculi Differentialis (1755), and Institutiones Calculi Integralis (1768–1770). In these works Euler insists that the calculus is essentially a relationship between algebraic functions and is not based on geometry. He has no place for the traditional interpretation of differentials and integrals as determining the tangent of a curve or the area beneath it, and his calculus textbooks include none of those familiar graphics. The notion of "function" was a novel one at this time, and he defined it as any algebraic expression including variables and constants.

In the Introductio Euler presents the differential calculus as a special case of the calculus of finite differences when the difference reaches zero. At that point, the ratio between the difference in the value of the function f(x) and the difference in the value of the variable x is 0/0. Whereas most mathematicians considered this expression to be more or less meaningless, according to Euler, it is the basis of the calculus and can take on any value whatsoever. The calculus, he argued, was a procedure to determine the specific value taken on by this ratio in each particular case.

Euler, along with Joseph-Louis Lagrange, founded the calculus of variations, which deals with the extremum characteristics of functions as a whole, rather than the point characteristics dealt with by the differential calculus. His work in this field played a crucial role in supporting Maupertuis during the controversy over his principle of least action. Using the calculus of variations, Euler demonstrated that the fundamental laws of motion were those that demonstrated the least amount of "action," as defined by Maupertuis. Maupertuis viewed this result as a clear manifestation of God's infinite wisdom in designing the world. Although Euler himself did not present his work explicitly in such metaphysical terms, he remained Maupertuis's most important and loyal supporter throughout the controversy.

Euler was a principal founder of complex analysis, and the field's fundamental relationship, eiθ = cosθ + isinθ, is known today as "Euler's formula." He contributed extensively to mathematical notation, introducing "f(x)" for a function, "e" for the base of natural logarithms, "i" for the square root of–1, and "Σ" for a sum. He worked extensively on number theory and many aspects of mathematical physics, including hydrodynamics and astronomy. Euler was a true polymath, and his deep mark on mathematics is evidenced today by the very numerous "Euler theorems" interspersed in a remarkably wide range of mathematical fields.

See also Alembert, Jean Le Rond d' ; Catherine II (Russia) ; Frederick II (Prussia) ; Lagrange, Joseph-Louis ; Mathematics ; Republic of Letters ; Voltaire .

BIBLIOGRAPHY

Primary Sources

Euler, Leonhard. Foundations of Differential Calculus. Translated by John D. Blanton. New York, 2000. Translation of first nine chapters of Institutiones Calculi Differentialis (1755).

——. Introduction to Analysis of the Infinite. 2 vols. Translated by John D. Blanton. New York, 1988, 1990. Translation of Introductio in Analysin Infinitorum (1748).