Leonhard Euler

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Euler, Leonhard

(lā`ônhärt oi`lər), 1707–83, Swiss mathematician. Born and educated at Basel, where he knew the Bernoullis, he went to St. Petersburg (1727) at the invitation of Catherine I, becoming professor of mathematics there on the departure of Daniel Bernoulli (1733). He was invited to Berlin (1741) by Frederick the Great and remained there until 1766, when he returned to St. Petersburg. Euler was the most prolific mathematician who ever lived; his collected works run to more than seventy volumes. He contributed to numerous areas of both pure and applied mathematics, including the calculus of variations, analysis, number theory, algebra, geometry, trigonometry, analytical mechanics, hydrodynamics, and the lunar theory (calculation of the motion of the moon). Euler was one of the first to develop the methods of the calculuscalculus,
branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.
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 on a wide scale. Though half-blind for much of his life and totally blind for the last seventeen years, he retained to the end a near-legendary skill at calculation. Among his results are the differential equation named for him, the formula relating the number of faces, edges, and vertices of a polyhedron (F + V = E + 2), and the famous equation eiπ + 1 = 0 connecting five fundamental numbers in mathematics.

Euler, Leonhard

 

Born Apr. 4 (15), 1707, in Basel, Switzerland; died Sept. 7 (18), 1783, in St. Petersburg. Mathematician, physicist, and mechanics specialist.

Euler received his initial education from his father, Paul Euler, a village pastor of modest means who as a young man had studied mathematics under Jakob Bernoulli. From 1720 to 1724, L. Euler studied at the University of Basel, where he attended the mathematics lectures of Johann Bernoulli.

In late 1726, Euler was offered a position at the St. Petersburg Academy of Sciences, and in May 1727 he arrived in St. Petersburg. The academy, which had just been organized, provided him with favorable conditions for scientific work, and he immediately undertook research in mathematics and mechanics. During the 14 years of the first St. Petersburg period of his life, he prepared approximately 80 works for publication and published more than 50. He learned Russian in this period.

Euler took part in various activities of the academy. He gave courses at the Academy University, was a member of several commissions of technical experts, and worked on the compilation of maps of Russia. In addition to writing the popular Einleitungin die Arithmetik (Introduction to Arithmetic; 1738–40; Russian translation, parts 1–2, 1740), he prepared, at the academy’s request, Scientia navalis (Naval Science; parts 1–2, 1749), a basic work on the theory of shipbuilding and navigation.

In 1741, Euler accepted an invitation from Frederick II, the king of Prussia, to join the Berlin Academy of Sciences, which Frederick was planning to form from the academy founded by G. W. von Leibniz. In Berlin, Euler was appointed director of the academy’s mathematics class and a member of the board. He ran the academy for several years after the death of its first president, P. L. de Maupertuis. In his 25 years in Berlin, Euler produced approximately 300 works, including several large monographs.

While living in Berlin, Euler continued to work intensively for the St. Petersburg Academy of Sciences, of which he remained an honorary member. He carried on an extensive correspondence dealing with scientific and organizational matters; in particular, he corresponded with M. V. Lomonosov, whom he highly esteemed. Euler edited the mathematics section of the Russian academy’s journal, in which he published during this period nearly as many papers as in the Mémoires of the Berlin Academy of Sciences. He helped train Russian mathematicians; the future academicians S. K. Kotel’nikov, S. Ia. Rumovskii, and M. Sofronov were sent to Berlin to study under his direction. Euler greatly assisted the St. Petersburg Academy of Sciences by, for example, acquiring scientific literature and equipment and negotiating with candidates for positions at the academy.

On July 17 (28), 1766, Euler and his family returned to St. Petersburg. Despite his age and the nearly total blindness that befell him, he worked productively to the end of his life. In the 17 years of his second stay in St. Petersburg he wrote approximately 400 works, including a number of large books. Euler continued to be involved in organizational work at the academy. In 1776 he was a member of the commission of experts appointed to study I. P. Kulibin’s design for a single-arch bridge across the Neva River; Euler was the only one on the commission to offer enthusiastic support for the project.

Euler’s accomplishments as a scientist and organizer of scientific research won wide recognition during his lifetime. In addition to the St. Petersburg and Berlin academies, he was a member of such prominent scientific institutions as the Académie des Sciences in Paris and the Royal Society of London.

Euler was extraordinarily prolific. Approximately 550 of his books and papers were published during his lifetime. He produced a total of approximately 850 works. In 1909 the Swiss Society of Natural Sciences undertook the publication of his complete collected works. The edition, which was completed in 1975, consists of 72 volumes. Euler’s immense scientific correspondence, which comprises approximately 3,000 letters, is also of considerable value; it has been published only in part.

The range of Euler’s interests was unusually broad. In addition to all branches of the mathematics of his time, it included mechanics, elasticity theory, mathematical physics, optics, music theory, the theory of mechanisms, ballistics, naval science, and insurance. Approximately three-fifths of Euler’s works pertain to mathematics, and the remaining two-fifths pertain chiefly to applications of mathematics.

Euler systematized his own findings and those of others in several classic monographs, which were written with remarkable clarity and provided with valuable examples. These monographs include Mechanica, sive motus scientia analytice exposita (Mechanics, or the Science of Motion Expounded Analytically; vols. 1–2, 1736); Introductio in analysin infinitorum (Introduction to the Analysis of Infinitesimals; vols. 1–2, 1748); Institutiones calculi differentialis (Principles of Differential Calculus; 1755): Theoria motus corporum solidorum seu rigidorum (Theory of the Motion of Solid or Rigid Bodies; 1765); Vollständige Anleitung zur Algebra (Complete Introduction to Algebra; first published in Russian translation, vols. 1–2, 1768–69), which was published in about 30 editions in six languages; and Institutiones calculi integralis (Principles of Integral Calculus; vols. 1–3,1768–70; vol. 4,1794). Euler’s Lettres à une princesse d’Allemagne sur quelques sujets de physique et de philosophie (Letters to a German Princess on Various Matters of Physics and Philosophy; parts 1–3, 1768–74) was published in more than 40 editions in ten languages. Lucidly written, it enjoyed great popularity in the 18th century and was still read in the 19th century. Much of the material in Euler’s monographs was subsequently incorporated into textbooks and study guides for higher educational institutions and, to some extent, for secondary schools.

It would be impossible to list all of Euler’s theorems, methods and formulas, only a few of which are known in the literature by his name. Among the various mathematical entities or expressions named for him are Euler’s constant, the Euler phi-function, Euler numbers, the Euler characteristic, Euler’s integrals, and Euler angles.

In Mechanica mathematical analysis is applied for the first time to the dynamics of a point. The first volume of the work deals with the free motion of a point under the action of various forces, both in a vacuum and in a medium that offers resistance. The second volume treats of the motion of a point along a given curve or surface. The chapter on the motion of a point under the action of a central force was of great importance in the development of celestial mechanics. In 1744, Euler gave the first correct formulation of the principle of least action and provided the first examples of the principle’s application. In Theoria motus corporum solidorum seu rigidorum he discussed the kinematics and dynamics of rigid bodies and gave the equations of their rotation about a fixed point, thereby laying the foundations of the theory of gyroscopes. Euler’s theory of ships made a valuable contribution to the theory of stability. Among his important discoveries in celestial mechanics are his contributions to the theory of lunar motion. In continuum mechanics he derived the fundamental equations of the motion of an ideal fluid in Eulerian and Lagrangian variables and studied the oscillations of a gas in tubes.

In optics Euler gave in 1747 an equation for a biconvex lens and proposed a method of calculating the refractive index of a medium. An adherent of the wave theory of light, he believed that to different colors there correspond different wavelengths of light. He suggested techniques of eliminating chromatic aberration in lenses and, in the third part of Dioptrica (Dioptrics), gave methods of designing the optical assemblies of microscopes.

In a large series of works begun in 1748, Euler studied such problems as the vibration of strings, plates, and membranes. In general his works dealing with physics and mechanics stimulated the development of such fields as the theory of differential equations, approximate methods of analysis, special functions, and differential geometry. They contain many of his mathematical discoveries.

Euler’s most important achievements as a mathematician were in the field of mathematical analysis. He laid the foundations of several areas of study that existed only in incipient form or were altogether absent in the infinitesimal calculus of I. Newton, Leibniz, Jakob Bernoulli, and Johann Bernoulli. For example, Euler was the first to introduce functions of a complex variable (Introductio in analysin, vol. 1) and to investigate the main elementary functions—that is, exponential, logarithmic, and trigonometric functions—of a complex variable. Of particular value was his derivation of formulas expressing relations between trigonometric functions and exponential functions. Euler’s work in this area gave rise to the theory of functions of a complex variable.

Euler was the inventor of the calculus of variations, which he expounded in Methodus inveniendi lineas curvas, maximi mini-mive proprietate gaudentes (A Method of Finding Curves That Enjoy Some Maximum or Minimum Property; 1744). After the appearance of works on the subject by J. Lagrange, Euler further developed the calculus of variations in Institutiones calculi integralis and several papers. The method by which he derived in 1744 the necessary condition for the extremum of a functional (Euler’s equation) was a prototype of the direct methods developed for the calculus of variations in the 20th century.

Euler created the theory of ordinary differential equations as an independent discipline and laid the foundations of the theory of partial differential equations. Here he made a tremendous number of discoveries. In addition to providing the classical method of solving linear equations with constant coefficients and the method of variation of arbitrary constants, he elucidated the main properties of the Riccati equation, developed the use of infinite series in integrating linear equations with variable coefficients, and established criteria for singular solutions. He also introduced the concept of an integrating factor, various approximate methods, and a number of procedures for solving partial differential equations. Many of his results are contained in Institutiones calculi integralis.

The differential and integral calculus in the narrow sense of the word were also enriched by Euler. In addition to the use of change of variables, his contributions include the proof of an important theorem on homogeneous functions, the introduction of the concept of the double integral, and the calculation of many special integrals. In Institutiones calculi differentialis he argued for the use of divergent infinite series and supplied his discussion with examples; he proposed generalized summation methods for series, thereby anticipating the ideas of the modern rigorous theory of divergent series developed at the turn of the 20th century. Moreover, Euler obtained a host of specific results in the theory of series. He discovered the Euler-Maclaurin summation formula, set forth the transformation of series that bears his name, determined the sums of an enormous number of series, and introduced into mathematics important new types of series, such as trigonometric series. His studies in the theory of continued fractions and other infinite processes may also be mentioned in this connection.

Euler was the founder of the theory of special functions. He was the first to treat the sine and cosine as functions rather than as line segments in a circle. He derived nearly all the classical expansions of elementary functions in infinite series and products. He originated the theory of the gamma function and investigated the properties of elliptic integrals, hyperbolic and cylindrical functions, the zeta function, some theta functions, the logarithmic integral, and important classes of special polynomials.

As P. L. Chebyshev notes, Euler was a pioneer in all the principal areas of the theory of numbers, to which more than 100 of his papers pertain. For example, he proved several assertions made by P. de Fermat (for example, Fermat’s lesser theorem), developed the foundations of the theory of power residues and the theory of quadratic forms, discovered (but did not prove) the law of quadratic reciprocity (seeQUADRATIC RESIDUE), and investigated a number of problems of Diophantine analysis. In his works on the expression of numbers as sums of other numbers and on the theory of primes, Euler was the first to use methods of analysis, thus becoming the founder of the analytic theory of numbers. In particular, he introduced the zeta function and proved a formula giving a relation between prime numbers and natural numbers [seeEULER FORMULA: (3)].

Euler’s contributions also were great in other areas of mathematics. In algebra he wrote papers on the solution by radicals of higher-degree equations, on equations in two variables, and on the sum of four squares [seeEULER FORMULA: (4)]. Euler substantially advanced analytic geometry, especiallly with respect to quadric surfaces. In differential geometry he investigated in detail the properties of geodesies. He was the first to use natural equations of curves. His most important contribution in this area was that he laid the foundations of the theory of surfaces. He introduced the concept of principal directions at a point on a surface, proved the orthogonality of the principal directions, derived a formula for the curvature of any normal section, and initiated the study of developable surfaces. In a work published posthumously in 1862, he partially anticipated the studies of K. F. Gauss on the intrinsic geometry of surfaces. Euler also worked on some problems of topology and proved, for example, an important theorem on convex polyhedra.

As a mathematician Euler is often described as a brilliant calculator. He was in fact an unsurpassed master of formal computations and transformations. In his works many mathematical formulas and symbols took on their modern form; for example, he introduced the symbols e and π. Euler, however, was not just a calculator of exceptional power. He set forth a number of profound ideas that since his day have been given rigorous substantiation and that serve as models of deep insight into the objects of scientific inquiry.

P. S. de Laplace called Euler the teacher of the mathematicians of the second half of the 18th century. Euler’s works provided starting points for investigations in various areas of mathematics by Laplace, Lagrange, G. Monge, A. M. Legendre, and Gauss and later by such mathematicians as A. Cauchy, M. V. Ostrogradskii, and P. L. Chebyshev. Russian mathematicians held his work in high regard. The members of the Chebyshev school viewed Euler as their intellectual predecessor because of his constant sense of the concrete and specific, his interest in concrete difficult problems requiring the development of new methods, and his desire to obtain the solutions of problems in the form of complete algorithms that would make it possible to find the answer to any required degree of accuracy.

WORKS

Opera omnia, series 1: Opera mathematica, vols. 1–29. Lausanne, 1911–56. Series 2: Opera mechanica et astronomica, vols. 1–30. Berlin-Leipzig, 1912–74. Series 3: Opera physica, miscellanae epistolae, vols. 1–12. Lausanne, 1911–73. Series 4: Commercium epistolicum, vol. 1,1975.
In Russian translation:
Universal’naia arifmetika, vols. 1–2. St. Petersburg, 1768–69.
Polnoe umozrenie stroeniia i vozhdeniia korablei, sochinennoe v pol’zu uchashchikhsia navigatsii. St. Petersburg, 1778.
Vvedenie v analiz beskonechnykh, vols. 1–2. Moscow, 1961.
Metod nakhozhdeniia krivykh linii, obladaiushchikh svoistvami mak-simuma, libo minimuma, Hi reshenie izoperimetricheskoi zadachi, vziatoi vsamom shirokom smysle. Moscow-Leningrad, 1934.
Osnovy dinamiki tochki. Moscow-Leningrad, 1938.
Novaia teoriia dvizheniia Luny. Leningrad, 1934.
Differentsial’noe ischislenie. Moscow-Leningrad, 1949.
Integral’noe ischislenie, vols. 1–3. Moscow, 1956–58.
Izbrannye kartograficheskie stat’i. Moscow, 1959.

REFERENCES

Erneström, G. Verzeichnis der Schriften Leonard Eulers, fascs. 1–2. (Jahresbericht der Deutschen Mathematiker-Vereinigung, supplementary vol. 4, fascs. 1–2.) Leipzig, 1910–13. (Contains bibliography.)
Fuss, N. Eloge de monsieur Léonard Euler. St. Petersburg, 1783. (Contains bibliography.) In Russian translation: ‘Pokhval’naia rech’ pokoinomu Leonardu Eileru.” In Akademicheskie sochineniia, vybrannye iz pervogo toma Deianii Akademii nauk, pod zaglaviem “Nova acta Academiae scientiarum imperialis Petropolitanae,” part 1. St. Petersburg, 1801.
Simonov, N. I. Prikladnye metody analiza u Eilera. Moscow, 1957.
Leonard Eiler: Sb. statei. Moscow, 1958.
Rukopisnye materialy L. Eilera v Arkhive Akademii nauk SSSR, vol. 1. Moscow-Leningrad, 1962.
Iushkevich, A. P. Istoriia matematiki v Rossii do 1917 goda. Moscow, 1968.

Adapted from EULER, LEONHARD in the 2nd edition of Bol’shaia Sovetskaia Entsiklopediia