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Euler is a scientist. Short biography of Leonhard Euler

Leonhard Euler, the son of a pastor, was born and took his first steps in the Swiss city of Basel in 1707 on April 15th.

The boy received his primary education at home. His father, pastor Pavel, prepared his son for the spiritual career from an early age.

The father put all kinds of knowledge into the boy, hoping for a comprehensive upbringing of his son. Abilities for the exact sciences manifested themselves in the child from the first steps of their study. Pavel, who was interested in mathematics, tried to pass on his knowledge to his young son.

The beginning of a genius career

The foundation of knowledge received by Leonard from his father turned out to be very voluminous and strong. Further studies at the Basel Gymnasium and admission to the University of Liberal Arts, at the age of 13, the result of home preparation.

All subjects were easy for Euler. At the lectures of Johann Bernoulli, Euler immediately attracted the attention of the teacher with his abilities. For a talented student, the Swiss mathematician of world renown, the most famous representative of the Bernoulli family, sets an individual course of study.

Bernoulli introduces Euler to the works of mathematical geniuses, teaches understanding and analysis of mathematical calculations. Thanks to Johann Bernoulli's teaching methodology, Leonhard Euler received his first Master of Arts degree at the age of sixteen. He presented the work of analytical comparison of the works of Descartes and Newton in Latin.

Euler's further scientific research is connected with the Bernoulli brothers. Their departure to the St. Petersburg Academy of Sciences helped Euler to take new steps. Bernoulli informed Leonard of the possibility of obtaining a position as a physiologist in the academy at the medical department. Euler enters the Faculty of Medicine at the University of Basel, while he does not leave mathematics.

Scientific activity in St. Petersburg and Berlin

The extraordinary breadth of interests and creative productivity served in St. Petersburg as the basis for the rise of the genius of Leonhard Euler. Living conditions allowed Euler to devote all his time to his favorite works in the field of mathematics and physics. During this period, the St. Petersburg Academy of Sciences received the status of the main center of mathematics of world importance.

The position of Leonhard Euler in the Academy of Sciences is improving: from 1727 to 1740, Euler, who took the post of head of the department of mathematics, publishes his works on geometry, analytical mechanics, and arithmetic. For the publication of a work on the tides of the sea, the scientist receives an award from the French Academy of Sciences.

The beginning of the revival of the Berlin Society of Sciences, whose progenitor was Leibniz, the German philosopher, mathematician, lawyer, diplomat, the Prussian King Frederick II began by inviting talented scientists. Euler was one of the first scientists to receive an invitation to the post of dean of the Department of Mathematics.

Leonhard Euler publishes several works on mathematics. The scientist devoted almost all his mathematical works to mathematical analysis. These treatises were formulated in such a simple and accessible way that mathematicians of the present day use them.

Return to Petersburg

Working in Berlin, Euler does not lose touch with Russia. He corresponded with Lomonosov, his friend Goldbach, an academician of the St. Petersburg Academy of Sciences. The scientist was not left thinking about Russia. In 1766, he accepted the invitation of the Empress and returned to St. Petersburg to the Academy of Sciences.

Euler's sons

senior Johann Albrecht as an academician in the field of physics,
Carl accepted a leading position in one of the bodies of medical management,
the youngest son Christopher came to his father from Berlin after the intervention of the empress. The Sestroretsk arms factory accepted a new director in the person of the youngest son of the great scientist.

The Last Days of a Genius

Continuous work, teaching students, writing papers affected the previously injured eye. The scientist began to lose his sight. However, the abilities of a genius, his unique memory helped him in his work. He dictated his articles and considerations on geometry and mathematics. Their number reached 380 from 1769 to 1793.

From the moment of becoming a scientist until his last days, he published over 900 scientific papers. Each of them consists of brilliant ideas and conclusions that are applied by modern users in their original writing. Works of recent years:

"On Orthogonal Trajectories", the most important in the mathematical field (1769);

the work "On bodies, the surface of which can be turned into a plane", (1771);

unique works on map projections, in which Euler was the first to scientifically substantiate the choice of section parallels in conic projections.

Euler's works concerned various fields of science. Only this genius without much difficulty managed to create a single system of such mathematical disciplines as algebra, trigonometry, geometry, number theory. Many scientific discoveries were added by Euler to this system. He created new mathematical disciplines, which to this day are taught to students unchanged.

His scientific research was extensive not only in mathematics. Astronomy, cartography, engineering also received many discoveries and developments thanks to Euler's research. Scientific research Leonhard Euler continued until the last days, being completely blind. Death occurred on September 18 (29), 1783 as a result of a stroke, surrounded by assistants close to him, professors Leksel and Kraft.

EILER, LEONARD(Euler, Leonhard) (1707–1783) is one of the top five greatest mathematicians of all time. Born in Basel (Switzerland) April 15, 1707 in the family of a pastor and spent his childhood in a nearby village, where his father received a parish. Here, in the bosom of rural nature, in the pious atmosphere of a modest pastor's house, Leonard received an initial upbringing that left a deep imprint on his entire subsequent life and worldview. Education in the gymnasium in those days was short. In the autumn of 1720, thirteen-year-old Euler entered the University of Basel, three years later he graduated from the lower - philosophical faculty and enrolled, at the request of his father, at the theological faculty. In the summer of 1724, at the annual university act, he read in Latin a speech on the comparison of Cartesian and Newtonian philosophy. Showing an interest in mathematics, he attracted the attention of Johann Bernoulli. The professor began to personally supervise the young man's independent studies and soon publicly admitted that he expected the greatest success from the insight and sharpness of the young Euler's mind.

Back in 1725, Leonhard Euler expressed a desire to accompany the sons of his teacher to Russia, where they were invited to the St. Petersburg Academy of Sciences, which was then opening at the behest of Peter the Great. The next year he received an invitation himself. He left Basel in the spring of 1727 and arrived in St. Petersburg after a seven-week journey. Here he was first enrolled as an adjunct in the department of higher mathematics, in 1731 he became an academician (professor), receiving the department of theoretical and experimental physics, and then (1733) the department of higher mathematics.

Immediately upon his arrival in St. Petersburg, he completely immersed himself in scientific work and at the same time impressed everyone with the fruitfulness of his work. Numerous of his articles in academic yearbooks, initially devoted mainly to the problems of mechanics, soon brought him worldwide fame, and later contributed to the fame of St. Petersburg academic publications in Western Europe. A continuous stream of Euler's writings has since been published in the Proceedings of the Academy for a whole century.

Along with theoretical research, Euler devoted a lot of time to practical work, fulfilling numerous assignments from the Academy of Sciences. So, he examined various devices and mechanisms, participated in the discussion of methods for raising a large bell in the Moscow Kremlin, etc. At the same time, he lectured at the academic gymnasium, worked at the astronomical observatory, collaborated in the publication of St. Vedomosti, did a lot of editorial work in academic publications, etc. In 1735, Euler took part in the work of the Geographical Department of the Academy, making a great contribution to the development of cartography in Russia. Euler's tireless work was not interrupted even by the complete loss of his right eye, which befell him as a result of an illness in 1738.

In the autumn of 1740, the internal situation in Russia became more complicated. This prompted Euler to accept the invitation of the Prussian king, and in the summer of 1741 he moved to Berlin, where he soon headed the mathematical class at the reorganized Berlin Academy of Sciences and Literature. The years Euler spent in Berlin were the most fruitful in his scientific work. During this period, his participation in a number of sharp philosophical and scientific discussions, including the principle of least action, also falls. The move to Berlin, however, did not interrupt Euler's close ties with the St. Petersburg Academy of Sciences. As before, he regularly sent his essays to Russia, participated in all kinds of examinations, taught students sent to him from Russia, selected scientists to fill vacant positions at the Academy, and carried out many other assignments.

Euler's religiosity and character did not correspond to the environment of the "free-thinking" Frederick the Great. This led to a gradual complication of relations between Euler and the king, who at the same time perfectly understood that Euler was the pride of the Royal Academy. In the last years of his life in Berlin, Euler actually performed the duties of president of the Academy, but he never received this position. As a result, in the summer of 1766, despite the resistance of the king, Euler accepted the invitation of Catherine the Great and returned to St. Petersburg, where he remained until the end of his life.

In the same year, 1766, Euler almost completely lost his sight in his left eye. However, this did not prevent the continuation of his activities. With the help of several students who wrote under his dictation and designed his works, the half-blind Euler prepared several hundred more scientific papers in the last years of his life.

In early September 1783, Euler felt a slight malaise. On September 18, he was still engaged in mathematical research, but suddenly lost consciousness and, in the apt expression of the panegyrist, "stopped calculating and living."

He was buried at the Smolensk Lutheran cemetery in St. Petersburg, from where his ashes were transferred in the fall of 1956 to the necropolis of the Alexander Nevsky Lavra.

The scientific legacy of Leonhard Euler is colossal. He owns the classical results in mathematical analysis. He advanced its substantiation, significantly developed integral calculus, methods of integrating ordinary differential equations and equations in partial derivatives. Euler owns the famous six-volume course of mathematical analysis, including Introduction to infinitesimal analysis, Differential calculus And Integral calculus(1748–1770). Many generations of mathematicians all over the world studied at this "analytical trilogy".

Euler received the basic equations of the calculus of variations and determined the ways of its further development, summing up the main results of his research in this area in the monograph Method for finding curved lines with maximum or minimum properties(1744). Euler's contributions to the development of function theory, differential geometry, computational mathematics, and number theory are significant. Euler's two-volume course The Complete Guide to Algebra(1770) went through about 30 editions in six European languages.

Fundamental results are due to Leonhard Euler in rational mechanics. He was the first to give a consistently analytical presentation of the mechanics of a material point, considering in his two-volume Mechanics(1736) the motion of a free and non-free point in a void and in a resisting medium. Euler later laid the foundations for kinematics and rigid body dynamics, deriving the corresponding general equations. The results of these studies of Euler are collected in his Theories of motion of rigid bodies(1765). The set of equations of dynamics representing the laws of momentum and angular momentum, the largest historian of mechanics Clifford Truesdell proposed to call the "Eulerian laws of mechanics".

Euler's article was published in 1752. Discovery of a new principle of mechanics, in which he formulated in a general form the Newtonian equations of motion in a fixed coordinate system, opening the way for the study of continuum mechanics. On this basis, he gave a derivation of the classical equations of hydrodynamics of an ideal fluid, finding a number of their first integrals. His works on acoustics are also significant. At the same time, he belongs to the introduction of both "Eulerian" (associated with the observer's frame of reference) and "Lagrangian" (in the frame of reference accompanying the moving object) coordinates.

Euler's numerous works on celestial mechanics are remarkable, among which his most famous New theory of the motion of the moon(1772), which significantly advanced the most important section of celestial mechanics for navigation of that time.

Along with general theoretical research, Euler is responsible for a number of important works in applied sciences. Among them, the first place is occupied by the theory of the ship. Questions of buoyancy, stability of the ship and its other seaworthiness were developed by Euler in his two-volume ship science(1749), and some questions of the structural mechanics of the ship - in subsequent works. He gave a more accessible presentation of the theory of the ship in A complete theory of the construction and driving of ships(1773), which was used as a practical guide not only in Russia.

Euler's comments on New beginnings of artillery B. Robins (1745), containing, along with his other works, important elements of external ballistics, as well as an explanation of the hydrodynamic "D'Alembert's paradox". Euler laid the foundation for the theory of hydraulic turbines, the impetus for the development of which was the invention of the jet "Segner wheel". He also created the theory of the stability of rods under longitudinal loading, which acquired particular importance a century later.

Many of Euler's works are devoted to various problems of physics, mainly geometric optics. Euler's three volumes deserve special mention. Letters to a German Princess on Various Subjects of Physics and Philosophy(1768-1772), which later went through about 40 editions in nine European languages. These "Letters" were a kind of textbook on the basics of science of that time, although their philosophical side did not correspond to the spirit of the Enlightenment.

Modern five-volume Mathematical Encyclopedia indicates twenty mathematical objects (equations, formulas, methods) that now bear the name of Euler. A number of fundamental equations of hydrodynamics and mechanics of a solid body also bear his name.

Along with numerous actual scientific results, Euler has the historical merit of creating a modern scientific language. He is the only author of the middle of the 18th century whose works are read even today without any difficulty.

The St. Petersburg Archive of the Russian Academy of Sciences also keeps thousands of pages of Euler's unpublished research, mainly in the field of mechanics, a large number of his technical expertise, mathematical "notebooks" and colossal scientific correspondence.

His scientific authority during his lifetime was unlimited. He was an honorary member of all the major academies and learned societies of the world. The influence of his works was very significant in the 19th century. In 1849, Karl Gauss wrote that "the study of all the works of Euler will forever remain the best, irreplaceable school in various branches of mathematics."

The total volume of Euler's writings is enormous. Over 800 of his published scientific papers amount to about 30,000 printed pages and consist mainly of the following: 600 articles in publications of the St. Petersburg Academy of Sciences, 130 articles published in Berlin, 30 articles in various European journals, 15 memoirs awarded prizes and encouragement from the Paris Academy Sciences, and 40 books of individual works. All this will amount to 72 volumes close to completion. Complete collection of works (Opera omnia) Euler, published in Switzerland since 1911. All works are printed here in the language in which they were originally published (i.e., in Latin and French, which were the main working languages in the middle of the 18th century, respectively, of the Petersburg and Berlin academies). To this will be added another 10 volumes of his scientific correspondence, which began publication in 1975.

Euler's special significance for the St. Petersburg Academy of Sciences, with which he was closely associated for over half a century, should be noted. “Together with Peter I and Lomonosov,” wrote academician S.I. Vavilov, “Euler became the good genius of our Academy, who determined its fame, its strength, its productivity.” It can be added that the affairs of the St. Petersburg Academy were conducted for almost a whole century under the guidance of Euler's descendants and students: from 1769 to 1855, his son, son-in-law and great-grandson were indispensable secretaries of the Academy from 1769 to 1855.

He raised three sons. The eldest of them was a St. Petersburg academician in the department of physics, the second was a court physician, and the youngest, an artilleryman, rose to the rank of lieutenant general. Almost all of Euler's descendants accepted in the 19th century. Russian citizenship. Among them were senior officers of the Russian army and navy, as well as statesmen and scientists. Only in the troubled times of the beginning of the 20th century. many of them were forced to emigrate. Today, Euler's direct descendants bearing his surname still live in Russia and Switzerland.

(It should be noted that the real pronunciation of Euler's name is "Oiler".)

Editions: Collection of articles and materials. M. - L.: Publishing House of the Academy of Sciences of the USSR, 1935; Digest of articles. M.: Publishing House of the Academy of Sciences of the USSR, 1958.

Gleb Mikhailov

1.Leonard Euler

2.Euler's works

2.1Euler-Maclaurin series

2.2The problem of string vibrations. wave equation

4Euler formula

1.Leonard Euler

This great scientist was undoubtedly the central figure in the science of the 18th century, and we will first of all get acquainted with his life and work.

Euler's scientific activity continued without interruption for almost sixty years. From 1726 to 1783 he conducted research in all areas of mathematics and mechanics of the 18th century, and in addition, in many departments of astronomy, physics and technology. He wrote about 850 scientific papers, including about two dozen voluminous monographs in one, two and three volumes. The publication of the complete collection of his works in three series and more than seventy volumes, begun in 1911, is not yet completely finished; it does not include the many hundreds of Euler's surviving scientific letters, often small papers, which are supposed to be published in the form of a fourth series. Euler was not only the greatest mathematician of his time, which in all fairness could be called the “Euler Age” in the history of physical and mathematical sciences, but also a major organizer of the work of two large academies: St. Petersburg and Berlin.

Leonhard Euler (1707-1783) was born in Basel and received his first lessons in mathematics from his father, pastor Paul Euler (1670-1745), who studied this subject with I. Bernoulli and in 1688 defended his dissertation on the theory of ratio and proportions. The father ordained his son also to the pastor, but the propensity for mathematics prevailed. During the years of study at the University of Basel (1720-1724), Leonhard Euler additionally studied mathematics and mechanics under the guidance of Johann Bernoulli. In 1725-1726. the young dealer came up with the first independent papers on isochronous curves in a resisting medium, on one special type of trajectory, on the best arrangement of masts on a ship (this work submitted to the competition of the Paris Academy was accepted for publication, although it did not receive a prize), on sound. The dissertation on sound was written in connection with Euler's intention to participate in the competition for the position of professor of physics at the University of Basel. Positions here were filled then by lot among the selected candidates. Euler was not admitted to the draw, probably due to his youth. As his Swiss biographer O. Spies writes, this was happiness for Euler: at that time, a broader perspective of activity opened up before him.

Indeed, when attempting to settle down in his homeland, Euler already had an invitation to the St. Petersburg Academy of Sciences, which was obtained for him by the sons of his mentor Daniil and Nicholas II Bernoulli, who had worked there since 1725. Euler followed this invitation and in the spring of 1727 arrived in the Russian capital. At first it was assumed that he would take the vacant post of adjunct, ie. junior academician, in physiology in order to apply mathematical methods to this science. Before the trip, Euler studied anatomy and medicine for several months, for which, however, he had no vocation. But in St. Petersburg everything worked out in the best possible way: he was given the opportunity to work in the field of mathematical sciences. Somewhat later it was formalized. In January 1731, Euler received a professorship, that is, an academician in physics, and in the summer of 1733 he replaced D. Bernoulli, who had left, at the department of mathematics.

In the favorable conditions of a large academy, in regular contact with other scientists - mathematicians, mechanics, astronomers, physicists - Euler's genius quickly manifested itself in its entirety. A man of exceptional energy, he took an active part in various academic activities that required the use of mathematics: compiling geographical maps, various technical examinations, solving numerous problems of shipbuilding and navigation, compiling textbooks and reviews of incoming essays, etc.

In the problems of practice, stimuli were also born for many of Euler's theoretical studies, which were the main subject of his tireless reflections.

Partly still in Basel, but mainly in the first years of his life in St. Petersburg, Euler outlined an extensive program of research in mathematics and mechanics, which he successfully carried out, constantly supplementing it, until the very last days. His discoveries, published in the academic "Notes" from their second volume for 1727 (1729) and often gaining fame even before publication thanks to his scientific correspondence, soon attracted the attention of the scientific world of Europe. His fame grew from year to year. This was expressed in a peculiar way in his letters to Euler by his former mentor Johann Bernoulli, calling him in 1728 "the most learned and gifted young man", in 1731 "the most glorious and most learned Mr. Professor, dearest friend" and, finally, in 1710 "head of mathematicians" (Mathematicorum princeps). At this time, Euler was a member of two academies - St. Petersburg and Berlin. A few years later he was elected as a foreign member by the Royal Society of London (1749) and the Paris Academy of Sciences (1755).

Euler lived in St. Petersburg for 14 years, marked by fundamental research in the theory of series, the theory of differential equations, the calculus of variations, the theory of numbers, the dynamics of a point, the theory of music, and ship science. Only a part of the manuscripts prepared by him at that time was then published; over the years, about 55 of them were published, including the two-volume Mechanics (1736). In the summer of 1741, Euler moved to Berlin, where you invite him! the Prussian King Frederick II, who wanted to raise the activities of the Berlin Academy of Sciences, which eked out the most miserable existence under his predecessor, to a high level. Euler accepted the invitation, since in the regency of Anna Leopoldovna, who ruled from November 1740 to December 1741, a very unstable and restless political situation developed in St. Petersburg, which was also reflected in the state of affairs in the Academy of Sciences.

Heading the Mathematical Class as its director, and in the absence of President Maupertuis and a number of years after his death, and the entire work of the Berlin Academy, Euler at the same time retained the title of honorary member of the St. Petersburg Academy (with a permanent pension), but in fact remained its full member from other cities. His strength was enough for a completely full-fledged "part-time job" in the two academies, he published his works almost equally in the editions of both, and even both together they could not cope with the timely publication of the inexhaustible stream of his works. In addition to the fact that he carried out the instructions of the Prussian government on hydraulic engineering, ballistics, organization of lotteries, etc., he edited the mathematical departments of the Berlin and St. Kotelnikova, S.Ya. Rumovsky, M. Sofronov (1729-1760), participated in the organization of scientific competitions of both academies, conducted live correspondence with German university professors and St. Petersburg academicians, including M.V. Lomonosov, looking for employees for our academy, purchasing tools and books for it. Euler's powers in his mature years seem inexhaustible. Continuing to implement the plans outlined in St. Petersburg, preparing or completing fundamental treatises in all departments of analysis, he includes new questions in algebra and number theory, elliptic integrals, equations of mathematical physics, trigonometric series, differential geometry of surfaces, problems of topology, and solid body mechanics. , hydrodynamics, the theory of the motion of the moon and planets, optics, magnetism, and in each of these areas receives significant and often paramount results.

At the same time, Euler had to participate in several important discussions, of which we will name at least three:

) the famous dispute over the nature of the functions included in the solution of the differential equation of an oscillating string, in which, besides him, first d'Alembert and D. Bernoulli participated, and then other prominent mathematicians got involved;

) a dispute with d'Alembert about the logarithms of negative numbers and, finally,

The years of Berlin life accounted for the publication of such large monographs by Euler as "A method for finding curved lines with the properties of a maximum or minimum" (Lausanne-Geneva, 1744), "New principles of artillery" (Berlin, 1745) two-volume "Introduction to the analysis of infinite" (Lausanne, 1748), the two-volume "Marine Science" (Petersburg, 1740), published in Berlin at the expense of the St. Greifswald, 1705) - a total of about 260 works.

The Petersburg Academy more than once raised before Euler the question of his return. In the 1960s, relations between Euler and Frederick II, who previously did not have mutual sympathy, deteriorated sharply. Euler, a Swiss burgher brought up in the Protestant tradition, and Frederick II, the Prussian absolute monarch, an admirer of Voltairian free-thinking, differed in many ways, including in relation to mathematics, which was for Euler the work of his whole life and in which the king, almost who did not know it at all, appreciated only immediate and immediate practical applications.

After the death of Maupertuis in 1759, the king offered d'Alembert the presidency, and when he refused, he instructed Euler to manage the academy without a presidential title and under his own personal leadership. Differences in some financial and administrative matters led to a rupture between the scientist and the king. Using his Swiss citizenship and the support of the Russian government, Euler forced his resignation and in the summer of 1760 returned to St. Petersburg forever.

Euler's ideological impulse in his young and mature years continued to give excellent results in his old age. We add that about 300 articles and fragments saw the light after his death.

With all the diversity of Euler's interests, the central place in them belongs to analysis. Of the 30 volumes of the mathematical series of his collected works, 19 are devoted to analysis, followed by number theory, geometry, algebra and combinatorics with probability theory. In addition, most of Euler's geometric works are devoted to the study of curves and surfaces using algebra and infinitesimal calculus, and many of his works on mechanics (there are also 30 volumes) contain new mathematical techniques for solving differential equations, integrating functions, etc. In our analysis courses a large number of formulas and methods still bear the name of Euler, and it is perhaps more common than other names. But, in addition to individual techniques and formulas, we owe to Euler the foundation of several large disciplines that existed only in embryonic form before: the theory of differential equations - ordinary and with partial derivatives, the calculus of variations, the elementary theory of functions of a complex variable. And he also laid the foundation for the theory of summation of series, expansions of functions into trigonometric series, the theory of special functions and definite integrals, differential geometry of surfaces, and, finally, number theory as a special science.

In a speech in memory of Euler given at the Paris Academy of Sciences, Condorcet, describing the last hours of Euler's life, said that he had finished "calculating and living." Euler, in fact, was a tireless "calculator" in both the narrow and broad sense of the word, and, perhaps, like no one else, he mastered the technique of calculations. This feature of his genius met the needs of the science of that time, which especially needed the rapid development of a formal analytical apparatus. But Euler was also a thinker who made a huge contribution to the development of the fundamental ideas of mathematics, without which all development was also impossible, such as the concepts of number, function, functional, sum of a series, integral, solution of a differential equation, etc.

At the same time, he created a new algebraic-arithmetic architecture of analysis. True, Euler was inferior in the construction of generalizing concepts to the younger Lagrange, who more clearly reflected in his theory of analytic functions and analytical mechanics the spiritual aspirations of the Enlightenment, which in other areas of thinking led to the creation of new large philosophical, historical, socio-political systems. However, it should not be forgotten that Lagrange followed Euler in many respects, deepening and improving his methods and concepts.

Euler's influence was exceptionally great. Laplace repeated to young mathematicians: read Euler, he is our common teacher. Euler had few direct students, but his works were desktop in the 18th century. and far beyond its borders for all creative mathematicians, and he directly directed the work of many through correspondence. Euler willingly and generously shared his thoughts, and Fontenelle's words about Leibniz apply to him: "he liked to watch the plants bloom in someone else's garden, the seeds of which he himself delivered."

It can be concluded that Euler's influence was very great.

euler mathematics physics astronomy

2.Euler's works

2.1Euler-Maclaurin series

Euler and independently, Maclaurin discovered a general summation technique, examples of which are the results of Newton and Stirling, and which expresses the partial sum of an infinite series s n = ∑ u (k) through another series whose terms contain the general term u (n), its integral and derivatives. For the first time, Euler gave a summation formula without proof and examples of use in the work of 1732. “The general method of summation of series” (Methodus generalis summandi progressiones. Commentarii, (1732 -1733) 1738), its derivation is given in the article “Finding the sum of a series by a given common term ”, presented by the St. Petersburg Academy in 1735 (Inventiosummae enjusque seriei ex dato termino generali. Commentarii, (1736-1741).

We mentioned this article in connection with the fact that in it the Taylor series was written in differential notation. Denoting the common term of the series X and the sum of its x terms S, Euler expanded S (x-1) into a Taylor series, and X into a series, from which he then obtained the expression S through X and its derivatives. To do this, he introduced dS/dx next to the indeterminate coefficients of the form, so that

(the constant of integration satisfies the condition that for x = 0 also X = 0 and S = 0). Further, by differentiation, he found an expression for d 2S/dx 2, d 3S/dx 3etc. and substituted them, together with the expression for dS / dx, into the expansion of the function X, after which, using the method of indefinite coefficients, he obtained equations that determine each of the numbers α, β, γ, δ, ε .... through all previous ones (counting after the first α ); this allows one to sequentially calculate

α = 1, β = 1/2, γ = 1/l2 ,δ = 0, e = - 1/720, etc.

Even earlier, Euler discovered that the ratio of two consecutive Bernoulli numbers B 2n+2 :B 2n as the index grows, it increases without limit in absolute value (Commentarii, (1739-1750). Therefore, the infinite Euler-Maclaurin series, generally speaking, diverges. However, the summation formula can provide excellent approximations if limited to partial sums of a series with an appropriate number of terms. In the paper just mentioned, Euler gave a new way of calculating π , based on the equality arctg, an approximate replacement of the integral by the sum and an estimate of the difference arctg t - S according to the summation formula.

Assuming t = 1, Euler obtained and at n = 5 calculated 12 correct decimal places. At the same time, he characterized the features of the behavior of the series in an exhaustive way and pointed out that for an approximate calculation, one should take the sum of those first terms of the series that decrease to the smallest inclusive. He even made an attempt to estimate the degree of approximation in this case by the number of terms used and the first discarded term, but he did not substantiate his estimate.

Asymptotic series also received important applications from Lagrange, Laplace, Legendre, who called these series semi-convergent (series demi-convergentes), and other scientists. Subsequently, they were studied by Cauchy, Poisson, who gave the first expressions for the remainder term, Jacobi, Lobachevsky, Ostrogradsky, etc. In a broad sense, L. Poincaré (1886) began to construct the theory of asymptotic expansions. The Euler-Maclaurin summation formula itself is now one of the main in the theory of finite differences and its applications.

2The problem of string vibrations. Wave equation (Euler solution).

In the paper just named, Euler first derives equation (1) for the vibration of a string. Then he formulates the requirement to find a general solution to this equation for an arbitrarily given string shape. The initial speed of the string is not directly mentioned, but from further calculations it follows that it is considered equal to the bullet. Under these conditions, Euler found a solution which, by his own admission, does not differ significantly in form from d'Alembert's solution. Euler solved equation (1) for any constant a, and therefore his solution has the form

y= φ (x + at) + ψ (x - at),(2)

Where φ And ψ - functions determined from the boundary and initial conditions of the problem in the same way as it was done by d'Alembert.

In 1766, Euler proposed a new method for solving the string vibration equation, which was then included in the third volume of his "Integral Calculus" (1770), and later in all textbooks on differential equations. Entering new coordinates:

u \u003d x + at, v \u003d x - at,

he transformed the equation (1) of string vibrations into an easily integrable form

According to modern terminology, Euler's coordinates u and v are called characteristic. In these coordinates, only the mixed derivative remains of the second derivatives of the function.

Euler was the first to understand that the equation for the vibration of a string reflects the process of wave propagation. In this case, a wave is called the process of movement of the deviation of a point of the string along the string.

3Euler's generalization of Fermat's theorem

In the last paper, Euler generalized Fermat's Theorem by stating (in notation derived from Gauss) that

a φ (m) ≡ 1 (mod m),

Where φ (m) is the number of numbers relatively prime to m and less than m. The number encountered here φ (m), which, according to Gauss' proposal, is now called the "Euler function", the latter presented in the same work in the form

φ (m)=m(1-1/p) (1-1/p ,)…,

where p, p ,,… are prime divisors of m. If m itself is a prime number, then the numbers 1, 2, 3, ..., (p - 1) will be coprime with it, and we get an important theorem stated by J. Wilson and published in 1770 by Waring in his "Algebraic Reflections" This theorem says that the quantity 1 * 2 * 3 ... (p-1) + 1 is divisible without remainder by p, where p, as everywhere here, is a prime number. This theorem, like Fermat's theorem, consists in the general congruence established by Lagrange

x p-1 - 1 ≡ (x + l) (x + 2)...(x + p - 1) (mod p)

for x = 0. It was also proved by Euler (Analytical Works, I, 1783) and Gauss (Arithmetical Investigations, 1801). A simplified proof of Fermat's theorem was given by I.G. Lambert, who was also willingly engaged in number theory (Nov. Acta Enid., 1769).

The most important achievements in the study of Euler's integers were led by efforts to prove another, already mentioned, Fermat's theorem that every prime number of the form 4m + 1 can be divided into the sum of two squares. Euler approached this theorem many times and from various angles and in doing so found a number of interesting propositions. Euler finally succeeded in proving it only in 1749, using the train of thought that he followed in the first proof of the comparison theorem a m ≡ 1 (mod p). This led him to consider the remainders of the division of the squares 1 2, 22, Z 2,..., (р-1) 2for a prime number p. Euler immediately saw that "many remarkable properties are obtained, the study of which sheds much light on the nature of numbers." Thus, he first raised the question of quadratic residues and understood their significance. The terms are already encountered here: residues (residua) and non-residues (non residua).

4Euler formula

Euler's formula is named after Leonhard Euler, who introduced it, and relates the complex exponential to trigonometric functions. Euler's formula states that for any real number x, the following equality holds:

e ix = cosx + isinx

Euler's formula was first given in the book "Harmony of Measures" by the English mathematician, Newton's assistant, Roger Cotes (1722, published posthumously). Cotes discovered the formula around 1714 and expressed it in logarithmic form:

ln(cosx + isinx) = ix

Euler published the formula in its usual form in a 1740 paper and in Introduction to the Analysis of Infinitesimals (1748), building the proof on the equality of the infinite power series expansions of the right and left sides. Neither Euler nor Kots imagined a geometric interpretation of the formula: the concept of complex numbers as points on the complex plane appeared about 50 years later.

The exponential and trigonometric forms of complex numbers are linked by Euler's formula.

Let the complex number z in trigonometric form have the form

z = r( cosφ + isinφ )

Based on the Euler formula, the expression in brackets can be replaced by an exponential expression. As a result, we get:

z=re iφ

This notation is called the exponential form of the complex number. Just like in trigonometric form, here

r = |z|, φ = argz

List of used sources

1.Yushkevich A.P., History of mathematics from ancient times to the beginning of the 19th century / A.P. Yushkevich. - M .: Nauka, 1972. - 496

2.Yushkevich A.P., History of mathematics from Descartes to the middle of the 19th century / A.P. Yushkevich. - M .: State publishing house of physical and mathematical literature, 1960. - 467

.http://en.wikipedia.org/wiki/Euler_Formula

A report about Leonhard Euler will tell you everything about the life of the great mathematician, physicist, mechanic and astronomer.

The life and work of Leonhard Euler briefly

The future scientist (the years of the life of Leonhard Euler 1707-1783) was born in Basel in Switzerland on April 15, 1707. After graduating from a local school, he attended Bernoulli's lectures at the University of Basel. He received his master's degree in 1723 and 3 years later he received an invitation from the St. Petersburg Academy of Sciences to the post of adjunct in mathematics.

In 1730 he took the chair of physics. In 1733, Euler received the title of academician. Euler spent 15 years in Russia and here he wrote the world's first textbook on theoretical mechanics and a course on mathematical navigation.

In 1741 Euler was invited by the Prussian King Frederick II to move to Berlin. Having accepted this offer, he changes his place of residence and issues 3 volumes of articles on the subject of ballistics. In 1747, a mathematician invented a complex lens.

In 1749, Euler published a two-volume work, in which he was the first to present the problems of navigation in mathematical form. He made many discoveries in the field of mathematical analysis, describing them in a book called "Introduction to the analysis of infinitesimal quantities." The great mathematician Leonhard Euler never ceases to explore differential, variational and integral calculus. He took up the issue of the passage of light through different media and how the effect of chromatism is connected with this.

He returned to Russia in 1766 and published his work "Elements of Algebra". By the way, he did not write it with his own hand, but dictated it, since by 1768 the mathematician was completely blind. But this illness did not prevent him from issuing several more publications and books, memoirs and volumes of integral calculus.

The Paris Academy of Sciences in 1775 accepted Euler as its 9th member of the society, while circumventing the laws of the academy and its statute, according to which only 8 people could be accepted into the society.

In general, mathematician Euler conducted more than 865 studies throughout his life, having a huge impact on the development of mathematics in Russia. He died in Petersburg on September 18, 1783.

Leonhard Euler interesting facts

In 1733, the scientist marries Katharina, the daughter of the artist Georg Gzel. During the 40 years of marriage, the wife gave Leonard 13 children. But only 5 of them survived - 2 daughters and 3 sons. In 1773, his beloved wife died and after 3 years Euler married a second time. On Katarina Salome, half-sister of the deceased wife.
In Russia, the scientist was called Leonty.
Euler was the first to systematically expound calculus. The mathematician is the founder of the scientific mathematical Russian school. He wrote many books on the theory of the motion of the planets and the Moon, on mechanics, geography, the theory of shipbuilding and the theory of music.
He didn't like theaters, and when his wife still managed to introduce him to the beautiful, Leonard mentally calculated complex mathematical schemes until the end of the performance, so as not to die of boredom.
He was a very capable person. Total at the age of 13 he became a student, and at 17 received a master's degree and received an invitation to head the Department of Physics at the Russian Academy of Sciences.
Despite his Swiss birth, Euler spent most of his adult life in St. Petersburg, Russia and in Berlin, Prussia.
Euler is remembered as the most important mathematician of the 18th century. He is remembered for his contributions to mechanics, fluid dynamics, optics, astronomy and music.
Leonhard Euler remained a faithful Calvinist all his life.
He lost sight in his right eye quite early, probably due to overwork.
He worked for 25 years at the Berlin Academy and then returned to Petersburg at the age of 59, during which time he lost sight in his other eye. Blindness did not stop him. In fact, he blindly completed a comprehensive analysis of the theory of the motion of the moon. All complex analysis was done entirely in his head.
In 1771 his house burned down. In 1776 his wife died. He died in 1783 at the age of 76.
It is known that he published more than 500 books and articles throughout his life, and another 400 were published posthumously. It has been estimated that he averaged about 800 pages per year.

Switzerland (1707-1727)

Basel University in the 17th-18th centuries

In the next two years, the young Euler wrote several scientific papers. One of them, "Thesis in Physics on Sound", which received a favorable review, was submitted to the competition to fill the unexpectedly vacant position of professor of physics at the University of Basel (). But, despite the positive feedback, the 19-year-old Euler was considered too young to be included in the number of candidates for a professorship. It should be noted that the number of scientific vacancies in Switzerland was quite small. Therefore, the brothers Daniel and Nikolai Bernoulli left for Russia, where the organization of the Academy of Sciences was in progress; they promised to work there for a position for Euler as well.

Euler was a phenomenal worker. According to contemporaries, for him to live meant doing mathematics. And the young professor had a lot of work: cartography, all kinds of expertise, consultations for shipbuilders and gunners, drafting training manuals, designing fire pumps, etc. They even require him to draw up horoscopes, which order Euler with all possible tact forwarded to the staff astronomer. But all this does not prevent him from actively conducting his own research.

During the first period of his stay in Russia, he wrote more than 90 major scientific papers. A significant part of the academic "Notes" is filled with the works of Euler. He made presentations at scientific seminars, gave public lectures, participated in the implementation of various technical orders of government departments.

All these dissertations are not only good, but also very excellent, for he [Lomonosov] writes about very necessary physical and chemical matters, which even the most witty people did not know and could not interpret today, what he did with such success that I am quite sure the validity of his explanations. In this case, Mr. Lomonosov must do justice, that he has an excellent talent for explaining physical and chemical phenomena. One should wish that other Academies would be able to produce such revelations, as Mr. Lomonosov showed.

Euler, in reply to His Excellency Mr. President, 1747

This high appraisal was not hindered even by the fact that Lomonosov did not write mathematical works and did not master higher mathematics.

Portrait from 1756 by Emanuel Handmann (Kunstmuseum, Basel)

According to contemporaries, Euler remained a modest, cheerful, extremely sympathetic person all his life, always ready to help another. However, relations with the king do not add up: Friedrich finds the new mathematician unbearably boring, completely unsecular, and treats him dismissively. Maupertuis, president of the Berlin Academy of Sciences, died in 1759. King Frederick II offered the post of president of the Academy to d'Alembert, but he refused. Friedrich, who did not like Euler, nevertheless entrusted him with the leadership of the Academy, but without the title of president.

Euler returns to Russia, now forever.

Again Russia (1766-1783)

Euler worked actively until his last days. In September 1783, the 76-year-old scientist began to feel headaches and weakness. On September 7 () after a dinner spent with his family, talking with Academician A.I. Leksel about the recently discovered planet Uranus and its orbit, he suddenly felt ill. Euler managed to say: "I'm dying," and lost consciousness. A few hours later, without regaining consciousness, he died of a brain hemorrhage.

“He stopped calculating and living,” Condorcet said at the mourning meeting of the Paris Academy of Sciences (fr. Il cessa de calculer et de vivre ).

Euler was a caring family man, willingly helping his colleagues and young people, generously sharing his ideas with them. There is a known case when Euler delayed his publications on the calculus of variations so that the young and then unknown Lagrange, who independently came to the same discoveries, could publish them first. Lagrange always admired Euler both as a mathematician and as a person; he said, "If you really love math, read Euler."

Contribution to science

Euler left important works on the most diverse branches of mathematics, mechanics, physics, astronomy, and a number of applied sciences. Mathematically, the 18th century is the age of Euler. If before him achievements in the field of mathematics were scattered and not always consistent, Euler was the first to link analysis, algebra, trigonometry, number theory, and other disciplines into a single system, and added many of his own discoveries. A significant part of mathematics has been taught since then "according to Euler".

Thanks to Euler, mathematics included the general theory of series, the amazingly beautiful “Euler formula”, the operation of comparison over an integer modulus, the complete theory of continued fractions, the analytical foundation of mechanics, numerous methods of integrating and solving differential equations, the number e, notation i for the imaginary unit , the gamma function with its environment, and much more.

In essence, it was he who created several new mathematical disciplines - number theory, calculus of variations, the theory of complex functions, differential geometry of surfaces, special functions. Other areas of his work: Diophantine analysis, astronomy, optics, acoustics, statistics, etc. Euler's knowledge was encyclopedic; in addition to mathematics, he deeply studied botany, medicine, chemistry, music theory, a variety of European and ancient languages.

Dispute with D "Alembert about the properties of the complex logarithm.
Dispute with English optician John Dollond about whether it is possible to create an achromatic lens.

In all the cases mentioned, Euler defended the correct position.

number theory

He refuted Fermat's conjecture that all numbers of the form are prime; turned out to be divisible by 641.

where is real. Euler deduced an expansion for it:

where the product is taken over all prime numbers. Thanks to this, he proved that the sum of a series of inverse primes diverges.

The first book on the calculus of variations

Geometry

In elementary geometry, Euler discovered several facts that were overlooked by Euclid:

The three altitudes of a triangle intersect at one point (orthocenter).
In a triangle, the orthocenter, the center of the circumscribed circle and the center of gravity lie on the same straight line - " Euler's line".
The bases of the three heights of an arbitrary triangle, the midpoints of its three sides, and the midpoints of the three segments connecting its vertices to the orthocenter all lie on the same circle (the Euler circle).
The number of vertices (B), faces (D) and edges (P) of any convex polyhedron are related by a simple formula: B + G = P + 2.

The second volume of "Introduction to Infinitely Small Analysis" () is the world's first textbook on analytic geometry and foundations of differential geometry. The term affine transformations is first introduced in this book along with the theory of such transformations.

When solving combinatorial problems, he deeply studied the properties of combinations and permutations, introduced the Euler numbers into consideration.

Other areas of mathematics

Graph theory began with Euler's solution of the seven-bridge problem of Königsberg.
Polyline method Euler.

Mechanics and mathematical physics

Many of Euler's works are devoted to mathematical physics: mechanics, hydrodynamics, acoustics, etc. In 1736, the treatise "Mechanics, or the science of motion, in an analytical presentation" was published, marking a new stage in the development of this ancient science. The 29-year-old Euler abandoned the traditional geometric approach to mechanics and laid a rigorous analytical foundation under it. Essentially, from that moment on, mechanics becomes an applied mathematical discipline.

Engineering

29 volumes in mathematics;
31 volumes on mechanics and astronomy;
13 - in physics.

Eight additional volumes will be devoted to Euler's scientific correspondence (over 3,000 letters).

Stamps, coins, banknotes

Bibliography

A new theory of the motion of the moon. - L .: Ed. Academy of Sciences of the USSR, 1934.
A method for finding curved lines that have the properties of either a maximum or a minimum. - M.-L.: GTTI, 1934.
Fundamentals of point dynamics. - M.-L.: ONTI, 1938.
Differential calculus. - M.-L., 1949.
Integral calculus. In 3 volumes. - M .: Gostekhizdat, 1956-58.
Selected cartographic articles. - M.-L.: Geodesizdat, 1959.
Introduction to the analysis of infinite. In 2 volumes. - M .: Fizmatgiz, 1961.
Ballistics research. - M .: Fizmatgiz, 1961.
Letters to a German princess about various physical and philosophical matters. - St. Petersburg. : Nauka, 2002. - 720 p. - ISBN 5-02-027900-5, 5-02-028521-8
Experience of a new theory of music, clearly stated in accordance with the immutable principles of harmony / transl. from lat. N. A. Almazova. - St. Petersburg: Ros. acad. Sciences, St. Petersburg. scientific center, publishing house Nestor-History, 2007. - ISBN 978-598187-202-0(Translation Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae (Tractatus de musica) . - Petropol.: Typ. Acad. Sc., 1739.)

Notes

References

Mathematics of the 18th century. Decree. op. - S. 32.
Glazer G.I. History of mathematics in the school. - M .: Education, 1964. - S. 232.
, With. 220.
Yakovlev A. Ya. Leonard Euler. - M .: Enlightenment, 1983.
, With. 218.
, With. 225.
, With. 264.
, With. 230.
, With. 231.
On the 150th anniversary of Euler's death: a collection. - Publishing House of the Academy of Sciences of the USSR, 1933.
A. S. Pushkin. Anecdotes, XI // Collected Works. - T. 6.
Marquis de Condorcet. Eulogy of Euler. History of the Royal Academy of Sciences (1783). - Paris, 1786. - P. 37-68.; see original text: fr. Madame, repondit-il, parce que je viens d'un pays où, quand on parle, on est pendu
Bell E.T. Decree. op. - S. 123.
Stroyk D. Ya. chapter VII // Brief essay on the history of mathematics / Translated by I. B. Pogrebyssky. - 3rd ed. - M., 1984.
Litvinova E.F. Euler // Copernicus, Galileo, Kepler, Laplace and Euler, Quetelet. Biographical stories. - Chelyabinsk, Ural, 1997. - T. 21. - S. 315. - (F. Pavlenkov's Library). - ISBN 5-88294-071-0