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Portrait by J. E. Handmann (1756) Leonhard Euler was born in 1707 into the family of Basel pastor Paul Euler, a friend of the Bernoulli family, and Marguerite Euler, née Brucker. Soon after Leonard's birth, the family moved to the village of Riechen (an hour's walk from Basel), where Paul Euler was appointed pastor; There the first years of the boy’s childhood passed. Leonard received his initial education at home under the guidance of his father (who at one time studied mathematics with Jacob Bernoulli). The pastor prepared his eldest son for a spiritual career, but he also studied mathematics with him, both as entertainment and for the development of logical thinking, and Leonard showed mathematical abilities early. When Leonard grew up, he was moved to his grandmother in Basel, where he studied at the gymnasium (while continuing to enthusiastically study mathematics). In 1720, a capable high school student was allowed to attend public lectures at the University of Basel; there he attracted the attention of Professor Johann Bernoulli (younger brother of Jacob Bernoulli). The famous scientist gave the gifted teenager mathematical articles to study, while allowing him to come to his home on Saturday afternoons to clarify difficult points. On October 20, 1720, 13-year-old Leonhard Euler became a student at the Faculty of Arts at the University of Basel. But Leonard's love for mathematics led him down a different path. While visiting his teacher's house, Euler met and became friends with his sons, Daniil and Nikolai, who, according to family tradition, also studied mathematics in depth. In 1723, Euler received (according to the custom at the University of Basel) the first award (primam lauream). On July 8, 1724, 17-year-old Leonhard Euler gave a speech in Latin about comparing the philosophical views of Descartes and Newton and was awarded the degree of Master of Arts. Over the next two years, young Euler wrote several scientific papers. One of them, “The dissertation in physics on sound,” was submitted to the competition for replacement unexpectedly in Switzerland 

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vacated the position of professor of physics at the University of Basel (1725). But, despite the positive review, 19-year-old Euler was considered too young to be included in the list of candidates for the professorship. It should be noted that the number of scientific vacancies in Switzerland was very small. Therefore, the brothers Daniel and Nikolai Bernoulli left for Russia, where the organization of the Academy of Sciences was just underway; they promised to work there for a position for Euler. University of Basel in the 17th-18th centuries At the beginning of the winter of 1726-1727. Euler received news from St. Petersburg: on the recommendation of the Bernoulli brothers, he was invited to the position of adjunct (assistant professor) in the department of physiology (this department was occupied by D. Bernoulli) with an annual salary of 200 rubles (a letter from Euler to the President of the Academy L. L. Blumentrost dated 9 has been preserved November 1726 with gratitude for admission to the Academy). Since Johann Bernoulli was a famous doctor, it was believed in Russia that Leonhard Euler, as his best student, was also a doctor. Euler postponed his departure from Basel, however, until spring, devoting the remaining months to the serious study of medical sciences, the deep knowledge of which he subsequently amazed his contemporaries. Finally, on April 5, 1727, Euler left Switzerland forever, although he retained Swiss (Basel) citizenship until the end of his life. Switzerland (continued) 

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On January 22 (February 2), 1724, Peter I approved the project for the organization of the St. Petersburg Academy. On January 28 (February 8), 1724, the Senate issued a decree on the creation of the Academy. Of the 22 professors and adjuncts invited in the first years, there were 8 mathematicians who also worked in mechanics, physics, astronomy, cartography, the theory of shipbuilding, and the service of weights and measures. Euler (whose route from Basel lay through Lubeck, Revel and Kronstadt) arrived in St. Petersburg on May 24, 1727; a few days earlier, Empress Catherine I, the patroness of the Academy, had died, and the scientists were despondent and confused. Euler, however, was helped to settle into his new place by fellow Basel residents: academicians Daniel Bernoulli and Jacob Hermann; the latter, who was a professor in the department of higher mathematics, was a distant relative of the young scientist and provided him with all kinds of patronage. Euler was made an adjunct of higher mathematics (and not physiology, as originally planned), although he was conducting research in the field of hydrodynamics of biological fluids in St. Petersburg, they gave him a salary of 300 rubles a year and provided him with a government apartment. To everyone’s surprise, the very next year after his arrival he began to speak Russian fluently. In 1728, the publication of the first Russian scientific journal “Comments of the St. Petersburg Academy of Sciences” (in Latin) began. Already the second volume contained three articles by Euler, and in subsequent years almost every issue of the academic yearbook included several of his new works. In total, more than 400 articles by Euler were published in this publication. In September 1730, the contracts concluded with academicians J. Herman and G. B. Bilfinger (the latter was a professor in the department of experimental and theoretical physics) expired. Their departments were headed by Daniel Bernoulli and Leonhard Euler, respectively; the latter received an increase in salary to 400 rubles, and on January 22, 1731, the official position of professor. Two years later (1733), Daniil Bernoulli returned to Switzerland, and Euler, leaving the department of physics, took over his department, becoming an academician and professor of higher mathematics with a salary of 600 rubles (however, Daniil Bernoulli received twice as much). On December 27, 1733, 26-year-old Leonhard Euler married his peer Katharina (German: Katharina Gsell), daughter of the academic painter Georg Gsell (a St. Petersburg Swiss). The newlyweds purchased a house on the Neva embankment, where they settled. 13 children were born into the Euler family, but 3 sons and 2 daughters survived. Russia 

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The young professor had a lot of work: cartography, all kinds of examinations, consultations for shipbuilders and artillerymen, drawing up training manuals, designing fire pumps, etc. He was even required to compile horoscopes, which Euler forwarded with all possible tact to the staff astronomer. A. S. Pushkin gives a romantic story: supposedly Euler compiled a horoscope for the newborn Ivan Antonovich (1740), but the result frightened him so much that he did not show it to anyone and only after the death of the unfortunate prince told Count K. G. Razumovsky about it. The reliability of this historical anecdote is extremely doubtful. During the first period of his stay in Russia, he wrote more than 90 major scientific works. A significant part of the academic “Notes” is filled with the works of Euler. He made reports at scientific seminars, gave public lectures, and participated in the implementation of various technical orders from government departments. During the 1730s, Euler led the work on mapping the Russian Empire, which (after Euler's departure, in 1745) culminated in the publication of an atlas of the country's territory. As N.I. Fuss said, in 1735 the Academy received the task of performing an urgent and very cumbersome mathematical calculation, and a group of academicians asked for three months to do this, and Euler undertook to complete the work in 3 days - and did it on his own; however, the overexertion did not pass without a trace: he fell ill and lost sight in his right eye. However, Euler himself, in one of his letters, attributed the loss of his eye to his work drawing up maps in the geographical department at the Academy. The two-volume work “Mechanics, or the science of motion, expounded analytically,” published in 1736, brought Euler fame throughout Europe. In this monograph, Euler successfully applied the methods of mathematical analysis to the general solution of problems of motion in vacuum and in a resistive medium. One of the most important tasks of the Academy was the training of domestic personnel, for which a university and a gymnasium were created at the Academy. Due to the acute shortage of textbooks in Russian, the Academy turned to its members with a request to compile such manuals. Euler compiled a very good “Manual to Arithmetic” in German, which was immediately translated into Russian and served for many years as an initial textbook. The translation of the first part was carried out in 1740 by the first Russian adjunct of the Academy, Euler's student Vasily Adodurov. Russia (continued) 

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Portrait of 1756 by Emanuel Handmann (Kunstmuseum, Basel) Euler submitted his resignation to the leadership of the St. Petersburg Academy: “For this reason I am forced, both for the sake of poor health and other circumstances, to seek a more pleasant climate and accept from His Royal Majesty the Prussian calling to me. For this reason, I ask the Imperial Academy of Sciences to most mercifully dismiss me and provide the necessary passport for travel for me and my family.” On May 29, 1741, permission from the Academy was received. Euler was “released” and confirmed as an honorary member of the Academy with a salary of 200 rubles. In June 1741, 34-year-old Leonhard Euler arrived in Berlin with his wife, two sons and four nephews. He spent 25 years there and published about 260 works. At first, Euler was received kindly in Berlin, even invited to court balls. The Marquis of Condorcet recalled that shortly after moving to Berlin, Euler was invited to a court ball. When asked by the Queen Mother why he was so taciturn, Euler replied: “I come from a country where whoever talks is hanged.” Euler had a lot of work to do. In addition to mathematical research, he directed the observatory and was involved in many practical matters, including the production of calendars (the main source of income for the Academy), the minting of Prussian coins, the laying of a new water supply system, the organization of pensions and lotteries. In 1742, a four-volume collected works of Johann Bernoulli was published. Sending him from Basel to Euler in Berlin, the old scientist wrote to his student: “I devoted myself to the childhood of higher mathematics. You, my friend, will continue her development into maturity.” During the Berlin period, Euler's works were published one after another: “Introduction to the Analysis of Infinitesimals” (1748), “Marine Science” (1749), “The Theory of the Motion of the Moon” (1753), “Instruction on Differential Prussia 

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calculus" (lat. Institutiones calculi differentialis, 1755). Numerous articles on specific issues are published in publications of the Berlin and St. Petersburg Academies. In 1744, Euler discovered the calculus of variations. His works use thoughtful terminology and mathematical symbolism, largely preserved to this day, and bring the presentation to the level of practical algorithms. Throughout his years in Germany, Euler maintained contact with Russia. Euler participated in the publications of the St. Petersburg Academy, purchased books and instruments for it, and edited the mathematical departments of Russian journals. For years, young Russian scientists sent on internships lived in his apartment, on full board. It is known that Euler had a lively correspondence with M.V. Lomonosov, in whose work he highly valued the “happy combination of theory and experiment.” In 1747, he gave a favorable review to the President of the Academy of Sciences, Count K. G. Razumovsky, about Lomonosov’s articles on physics and chemistry, stating: “All these dissertations are not only good, but also very excellent, for he [Lomonosov] writes about physical and chemical matters very necessary, which to this day did not know and the wittiest people could not interpret, which he did with such success that I am completely confident in the validity of his explanations. In this case, Mr. Lomonosov must be given justice that he has an excellent talent for explaining physical and chemical phenomena. We should wish that other Academies would be able to produce such revelations as Mr. Lomonosov showed.” This high assessment was not hindered even by the fact that Lomonosov did not write mathematical works and did not master higher mathematics. Euler's mother informed him of the death of his father in Switzerland (1745); she soon moved in with Euler (she died in 1761). In 1753, Euler bought an estate in Charlottenburg (a suburb of Berlin) with a garden and plot, where he settled his large family. According to contemporaries, Euler remained a modest, cheerful, extremely sympathetic person all his life, always ready to help others. However, relations with the king did not work out: Frederick found the new mathematician unbearably boring, completely unworldly and treated him dismissively. In 1759, Maupertuis, president of the Berlin Academy of Sciences and friend of Euler, died. 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. Prussia (continued) 

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Frederick II of Prussia During the Seven Years' War (1756-1763), Russian artillery destroyed Euler's house; Upon learning of this, Field Marshal Saltykov immediately compensated for the losses, and later Empress Elizabeth sent another 4,000 rubles from herself. In 1765, “The Theory of the Motion of Rigid Bodies” was published, and a year later, “Elements of the Calculus of Variations.” It was here that the name of the new branch of mathematics created by Euler and Lagrange first appeared. In 1762, Catherine II ascended the Russian throne and pursued a policy of enlightened absolutism. Well understanding the importance of science both for the progress of the state and for her own prestige, she carried out a number of important, favorable for science, transformations in the system of public education and culture. The Empress offered Euler management of a mathematical class, the title of conference secretary of the Academy and a salary of 1,800 rubles a year. “And if you don’t like it,” said the letter to her representative, “he would be pleased to communicate his conditions, so long as he doesn’t hesitate to come to St. Petersburg.” Euler responded with his conditions: a salary of 3,000 rubles a year and the post of vice-president of the Academy; annual pension of 1000 rubles to the wife after his death; paid positions for three of his sons, including the post of secretary of the Academy for the eldest. Prussia (continued) 

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The building of the St. Petersburg Academy of Sciences in the second half of the 18th century (Kunstkamera) On July 17 (28), 1766, 60-year-old Euler, his family and household (18 people in total) arrived in the Russian capital. Immediately upon arrival he was received by the empress. Catherine II greeted him as an august person and showered him with favors: she granted 8,000 rubles for the purchase of a house on Vasilievsky Island and for the purchase of furnishings, provided one of her cooks for the first time and instructed him to prepare ideas for the reorganization of the Academy. Unfortunately, after returning to St. Petersburg, Euler developed a cataract in his left eye - he stopped seeing. Probably for this reason, he never received the promised post of vice-president of the Academy (which did not prevent Euler and his descendants from participating in the management of the Academy for almost a hundred years). However, blindness did not affect the scientist’s performance; he only noticed that now he would be less distracted from doing mathematics. Before finding a secretary, Euler dictated his works to a tailor boy, who wrote everything down in German. The number of works he published even increased; During his second stay in Russia, Euler dictated more than 400 articles and 10 books, which constitutes more than half of his creative heritage. In 1768-1770, the two-volume classic monograph “Universal Arithmetic” was published (also published under the titles “Principles of Algebra” and “Complete Course of Algebra”). Initially, this work was published in Russian (1768-1769); the German edition was published two years later. The book was translated into many languages ​​and reprinted about 30 times (three times in Russian). All subsequent algebra textbooks were created under the strong influence of Euler's book. In the same years, the three-volume book “Dioptrica” (Latin: Dioptrica, 1769-1771) on lens systems and the fundamental “Integral Calculus” (Latin: Institutiones calculi integralis, 1768-1770), also in 3 volumes, were published. Euler's “Letters on Various Physical and Philosophical Matters, Written to a German Russia Again” gained enormous popularity in the 18th century, and partly in the 19th century.

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princess..." (1768), which went through over 40 editions in 10 languages ​​(including 4 editions in Russian). It was a wide-ranging popular science encyclopedia, written vividly and accessible to everyone. “Letters to a German Princess,” third edition (1780) In 1771, two serious events occurred in Euler’s life. In May, a large fire broke out in St. Petersburg, destroying hundreds of buildings, including Euler’s house and almost all of his property. The scientist himself was saved with difficulty. All manuscripts were saved from fire; Only part of the “New Theory of the Motion of the Moon” burned down, but it was quickly restored with the help of Euler himself, who retained a phenomenal memory into old age. Euler had to temporarily move to another house. Second event: in September of the same year, at the special invitation of the Empress, the famous German ophthalmologist Baron Wentzel arrived in St. Petersburg to treat Euler. After an examination, he agreed to perform surgery on Euler and removed a cataract from his left eye. Euler began to see again. The doctor ordered to protect the eye from bright light, not to write, not to read - just gradually get used to the new condition. However, just a few days after the operation, Euler removed the bandage and soon lost his sight again. This time it's final. 1772: "A New Theory of the Motion of the Moon." Euler finally completed his many years of work, having approximately solved the three-body problem. In 1773, on the recommendation of Daniel Bernoulli, Bernoulli's student, Nikolaus Fuss, came to St. Petersburg from Basel. This was a great success for Euler. Fuss, a gifted mathematician, immediately after his arrival took charge of Euler's mathematical works. Soon Fuss married Euler's granddaughter. In the next ten years - until his death - Euler mainly dictated his works to him, Russia Again (continued) 

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although sometimes he used the “eyes of his eldest son” and his other students. In the same 1773, Euler’s wife, with whom he lived for almost 40 years, died. The death of his wife was a painful blow for the scientist, who was sincerely attached to his family. Euler soon married Salome-Abigail, the half-sister of his late wife. In 1779, “General Spherical Trigonometry” was published, this is the first complete presentation of the entire system of spherical trigonometry. Euler worked actively until his last days. In September 1783, the 76-year-old scientist began to experience headaches and weakness. On September 7 (18), after lunch 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 cerebral hemorrhage. “He stopped calculating and living,” Condorcet said at the funeral meeting of the Paris Academy of Sciences (French: Il cessa de calculer et de vivre). He was buried at the Smolensk Lutheran cemetery in St. Petersburg. The inscription on the monument in German read: “Here lie the remains of the world-famous Leonhard Euler, a sage and a righteous man. Born in Basel on April 4, 1707, died on September 7, 1783." After Euler's death, his grave was lost and was found, in an abandoned state, only in 1830. In 1837, the Academy of Sciences replaced this tombstone with a new granite tombstone (still in existence today) with the Latin inscription “To Leonard Euler - St. Petersburg Academy” (Latin: Leonhardo Eulero - Academia Petropolitana). During the celebration of Euler’s 250th anniversary (1957), the ashes of the great mathematician were transferred to the “Necropolis of the 18th century” at the Lazarevsky cemetery of the Alexander Nevsky Lavra, where it is located near the grave of M.V. Lomonosov. Tombstone of L. Euler, granite sarcophagus Russia Again (continued) 

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Euler left important works in various branches of mathematics, mechanics, physics, astronomy and a number of applied sciences. Euler's knowledge was encyclopedic; in addition to mathematics, he deeply studied botany, medicine, chemistry, music theory, and many European and ancient languages. Euler willingly participated in scientific discussions, of which the most famous were: the dispute about the string; dispute with D'Alembert about the properties of the complex logarithm; debate with John Dollond about whether it was possible to create an achromatic lens. In all the cases mentioned, Euler's position is supported by modern science. Contribution to science 

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Euler's formula From the point of view of mathematics, the 18th century is the century of Euler. If before him achievements in the field of mathematics were scattered and not always coordinated, Euler was the first to link analysis, algebra, geometry, trigonometry, number theory and other disciplines into a single system, adding many of his own discoveries. Since then, a significant part of mathematics has been taught “according to Euler” almost unchanged. Thanks to Euler, mathematics included the general theory of series, the fundamental “Euler formula” in the theory of complex numbers, the operation of comparison modulo integer, the complete theory of continued fractions, the analytical foundation of mechanics, numerous techniques for integrating and solving differential equations, the number e, the designation i for the imaginary unit , a number of special functions and much more. Essentially, it was he who created several new mathematical disciplines - number theory, calculus of variations, theory of complex functions, differential geometry of surfaces; he laid the foundations of the theory of special functions. Other areas of his work: Diophantine analysis, mathematical physics, statistics, etc. Biographers note that Euler was a virtuoso algorithmist. He invariably tried to bring his discoveries to the level of specific computational methods and was himself an unsurpassed master of numerical calculations. J. Condorcet said that one day two students, independently performing complex astronomical calculations, received slightly different results in the 50th sign and turned to Euler for help. Euler did the same calculations in his head and indicated the correct result. Mathematics 

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One of Euler's main services to science is the monograph “Introduction to the Analysis of Infinitesimals” (1748). In 1755, the supplemented “Differential Calculus” was published, and in 1768-1770, three volumes of “Integral Calculus” were published. Taken together, this is a fundamental course, well illustrated with examples, with thoughtful terminology and symbolism. “We can say with confidence that a good half of what is now taught in courses of higher algebra and higher analysis is in the works of Euler” (N. N. Luzin). Euler was the first to give a systematic theory of integration and the techniques used in it. In particular, he is the author of the classical method of integrating rational functions by decomposing them into simple fractions and a method for solving differential equations of arbitrary order with constant coefficients. He introduced double integrals for the first time. Euler always paid special attention to methods for solving differential equations, both ordinary and partial differential ones, discovering and describing important classes of integrable differential equations. Explained Euler's “method of broken lines” (1768) - a numerical method for solving systems of ordinary differential equations. Simultaneously with A. K. Clairaut, Euler derived conditions for the integrability of linear differential forms in two or three variables (1739). Obtained serious results in the theory of elliptic functions, including the first theorems for the addition of elliptic integrals (1761). For the first time he studied the maxima and minima of functions of many variables. The first book on the calculus of variations The basis of natural logarithms has been known since the times of Napier and Jacob Bernoulli, but Euler carried out such a thorough study of this most important constant that since then it has been named after him. Another constant he studied: the Euler-Mascheroni constant. The base of natural logarithms has been known since Mathematical analysis 

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There is a surface that can be applied to a plane without folds or tears. Euler, however, gives here a completely general theory of the metric, on which the entire internal geometry of the surface depends. Later, the study of metrics became his main tool for the theory of surfaces. In connection with the problems of cartography, Euler deeply studied conformal mappings, using for the first time the means of complex analysis. Geometry (continued) 

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Euler's magic square Greco-Latin square of the fifth order Euler paid a lot of attention to the representation of natural numbers in the form of sums of a special form and formulated a number of theorems for calculating the number of partitions. When solving combinatorial problems, he deeply studied the properties of combinations and permutations and introduced Euler numbers into consideration. Euler studied algorithms for constructing magic squares using the chess knight traversal method. Two of his works (1776, 1779) laid the foundation for the general theory of Latin and Greco-Latin squares, the enormous practical value of which became clear after Ronald Fisher created methods for planning experiments, as well as in the theory of error-correcting codes. Combinatorics 

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The problem of circumventing the seven bridges of Königsberg Euler's 1736 paper “Solution of a problem related to the geometry of position” laid the foundation for graph theory as a mathematical discipline. The reason for the study was the problem of the seven bridges of Königsberg: is it possible to cross each bridge once and return to the starting place? Euler formalized it, reducing it to the problem of the existence in a graph (the vertices of which correspond to parts of the city separated by the channels of the Pregolya River, and the edges to bridges) of a cyclic route passing along each edge exactly once (in modern terminology - an Euler cycle). Solving the last problem, Euler showed: for there to be an Euler cycle in a graph, each vertex must have an even degree (the number of edges leaving the vertex) (but in the problem of the Königsberg bridges this is not the case: the degrees are 3, 3, 3 and 5 ). Euler made significant contributions to the theory and methods of approximate calculations. For the first time he applied analytical methods in cartography. He proposed a convenient method for graphically representing relations and operations on sets, called “Euler Circles” (or Euler-Venn). Other areas of mathematics 

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Many of Euler's works were devoted to various branches of mechanics and physics. Regarding Euler’s key role at the stage of formalizing mechanics into an exact science, K. Truesdell wrote: “Mechanics, as it is taught to engineers and mathematicians today, is to a large extent his creation.” Mechanics and physics 

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In 1736, Euler’s two-volume treatise “Mechanics, or the science of motion, in an analytical presentation” was published, which marked a new stage in the development of this ancient science and was dedicated to the dynamics of a material point. Unlike the founders of this branch of dynamics - Galileo and Newton, who used geometric methods, 29-year-old Euler proposed a regular and uniform analytical method for solving various problems of dynamics: drawing up differential equations of motion of a material object and their subsequent integration under given initial conditions. The first volume of the treatise examines the movement of a free material point, the second - a non-free one, and examines the movement both in emptiness and in a resisting medium. The problems of ballistics and the theory of the pendulum are considered separately. Here Euler first writes down the differential equation for the rectilinear motion of a point, and for the general case of its curvilinear motion he introduces natural equations of motion - equations in projections on the axis of the accompanying trihedron. In many specific problems he takes the integration of the equations of motion to completion; in cases where a point moves without resistance, he systematically uses the first integral of the equations of motion - the energy integral. In the second volume, in connection with the problem of the movement of a point on an arbitrarily curved surface, the differential geometry of surfaces created by Euler is presented. Euler returned to the dynamics of a material point later. In 1746, while studying the motion of a material point on a moving surface, he came (simultaneously with D. Bernoulli and P. Darcy) to a theorem on the change in angular momentum. In 1765, Euler, using the idea put forward in 1742 by C. Maclaurin about the expansion of velocities and forces along three fixed coordinate axes, wrote down for the first time the differential equations of motion of a material point in projections onto Cartesian fixed axes. The last result was published by Euler in his second fundamental treatise on analytical dynamics - the book “The Theory of the Motion of Rigid Bodies” (1765). Its main content is devoted, however, to another section of mechanics - the dynamics of a rigid body, the founder of which was Euler. The treatise, in particular, contains the derivation of a system of six differential equations of motion of a free rigid body. Of great importance for statics is the theorem stated in § 620 of the treatise on reducing a system of forces applied to a rigid body to two forces. Designing for Theoretical Mechanics 

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coordinate axes, the conditions for these forces to be equal to zero, Euler for the first time obtained the equilibrium equations of a rigid body under the action of an arbitrary spatial system of forces. Euler's angles The treatise of 1765 also sets out a number of fundamental results of Euler related to the kinematics of a rigid body (in the 18th century, kinematics had not yet been identified as a separate branch of mechanics). Among them, we highlight Euler’s formulas for the distribution of velocities of points of an absolutely rigid body (the vector equivalent of these formulas is Euler’s kinematic formula) and Euler’s kinematic equations, which give an expression for the derivatives of Euler’s angles (introduced by him in 1748; in mechanics they are used to specify the orientation of a rigid body) through projections of angular velocity on the coordinate axes. In addition to this treatise, two earlier works of Euler are important for rigid body dynamics: “Research on the mechanical knowledge of bodies” and “Rotational motion of solid bodies around a variable axis”, which were submitted to the Berlin Academy of Sciences in 1758, but published in her “Notes” later (in the same 1765 as the treatise). In them: the theory of moments of inertia was developed (in particular, the “Huygens-Steiner theorem” was proven for the first time); the existence of at least three axes of free rotation for any rigid body with a fixed point has been established; dynamic Euler equations were obtained that describe the dynamics of a rigid body with a fixed point; An analytical solution of these equations is given in the case when the main moment of external forces is equal to zero (Euler's case) - one of the three general cases of integrability in the problem of the dynamics of a heavy rigid body with a fixed point. In the article “General formulas for the arbitrary movement of a rigid body” (1775), Euler formulates and proves Euler’s fundamental rotation theorem, according to which the arbitrary movement of an absolutely rigid body with a fixed point is a rotation through a certain angle around one or another axis passing through the fixed point. Theoretical mechanics (continued) 

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A number of Euler's works are devoted to issues of machine mechanics. In his memoir “On the Most Advantageous Use of Simple and Complex Machines” (1747), Euler proposed studying machines not in a state of rest, but in a state of motion. Euler substantiated and developed this new, “dynamic” approach in his memoir “On Machines in General” (1753); in it, for the first time in the history of science, he pointed out the three components of a machine, which in the 19th century were defined as an engine, a transmission and a working element. In his memoir “Principles of the Theory of Machines” (1763), Euler showed that when calculating the dynamic characteristics of machines in the case of their accelerated motion, it is necessary to take into account not only the resistance forces and inertia of the payload, but also the inertia of all components of the machine, and gives (in relation to hydraulic engines ) an example of such a calculation. Euler also dealt with applied issues of the theory of mechanisms and machines: issues of the theory of hydraulic machines and windmills, the study of friction of machine parts, issues of profiling gears (here he substantiated and developed the analytical theory of involute gearing). In 1765, he laid the foundations for the theory of friction of flexible cables and obtained, in particular, Euler’s formula for determining cable tension, which is still used today in solving a number of practical problems (for example, when calculating mechanisms with flexible links). Mechanics of machines 

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The name of Euler is also associated with the consistent introduction into mechanics of the idea of ​​continuum, according to which a material body is represented, abstracting from its molecular or atomic structure, in the form of a continuous continuous medium. The continuum model was introduced by Euler in his memoir “The Discovery of a New Principle of Mechanics” (reported in 1750 to the Berlin Academy of Sciences and published in its “Memoirs” two years later). The author of the memoir based his consideration on Euler's principle of material particles - a position that is still given in many textbooks of mechanics and physics (often without mentioning Euler's name): a solid body can be modeled with any degree of accuracy by a system of material points, breaking it mentally into sufficiently small particles and treating each of them as a material point. Based on this principle, one can obtain certain dynamic relations for a solid body by writing their analogues for individual material particles (according to Euler, “taurus”) and summing them term by term (replacing summation over all points by integration over the volume of the region occupied by the body) . This approach allowed Euler to do without using such means of modern integral calculus (such as the Stieltjes integral), which were not yet known in the 18th century. Based on this principle, Euler obtained - by applying the theorem on the change in momentum to an elementary material volume - Euler's first law of motion (later Euler's second law of motion appeared - the result of applying the theorem on the change in angular momentum). Euler's laws of motion were in fact the basic laws of motion of continuum mechanics; To move to the currently used general equations of motion of such media, all that was needed was to express the surface forces through the stress tensor (this was done by O. Cauchy in the 1820s). Euler applied the results obtained in the study of specific models of solid bodies - both in the dynamics of a solid body (it was in the mentioned memoir that the equations of the dynamics of a body with a fixed point, related to arbitrary Cartesian axes, were first given), and in hydrodynamics, and in the theory of elasticity. Continuum mechanics 

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Euler is - along with D. Bernoulli and J. L. Lagrange - one of the founders of analytical hydrodynamics; here he is credited with creating the theory of motion of an ideal fluid (that is, a fluid that does not have viscosity) and solving a number of specific problems of fluid mechanics. In his work “Principles of the Motion of Fluids” (1752; published nine years later), he applied his equations of dynamics of an elementary material volume of a continuous medium to a model of an incompressible ideal fluid, and for the first time obtained the equations of motion for such a fluid, as well as the equation of continuity. While studying the irrotational motion of an incompressible fluid, Euler introduced the function S (later called the velocity potential by G. Helmholtz) and showed that it satisfies a partial differential equation - this is how the equation now known as Laplace's equation entered science. Euler significantly generalized the results of this work in his treatise “General Principles of the Motion of Fluids” (1755). Here he - already for the case of a compressible ideal fluid - presented (practically in modern notation) the continuity equation and the equations of motion (three scalar differential equations, which in vector notation correspond to the Euler equation - the basic equation of hydrodynamics of an ideal fluid). Euler noted that to close this system of four equations, a constitutive relation is needed to express the pressure p (which Euler called “elasticity”) as a function of density q and “another property r that affects elasticity” (in fact, he meant temperature). Discussing the possibility of the existence of non-potential motions of an incompressible fluid, Euler gave the first concrete example of its vortex flow, and for the potential motions of such a fluid he obtained the first integral - a special case of the now well-known Lagrange-Cauchy integral. Euler’s memoir “General Principles of the State of Equilibrium of Liquids” dates back to the same year, which contained a systematic presentation of the hydrostatics of an ideal liquid (including the derivation of the general equation of equilibrium of liquids and gases) and a barometric formula for an isothermal atmosphere was derived. Hydrodynamics 

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Euler collected the main achievements in this area in the three-volume Dioptrica (lat. Dioptrica, 1769-1771). Among the main results: rules for calculating the optimal characteristics of refractors, reflectors and microscopes, calculation of the highest image brightness, the largest field of view, the shortest instrument length, the highest magnification, and eyepiece characteristics. Newton argued that the creation of an achromatic lens was fundamentally impossible. Euler countered that a combination of materials with different optical characteristics could solve this problem. In 1758, Euler, after a long controversy, managed to convince the English optician John Dollond of this, who then made the first achromatic lens by connecting two lenses made of glasses of different compositions to each other, and in 1784, academician F. Apinus in St. Petersburg built the first in the world achromatic microscope. Optics 

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Euler worked a lot in the field of celestial mechanics. One of the pressing problems at that time was to determine the orbital parameters of a celestial body (for example, a comet) from a small number of observations. Euler significantly improved numerical methods for this purpose and practically applied them to the determination of the elliptical orbit of the comet of 1769; Gauss relied on these works and gave the final solution to the problem. Euler laid the foundations of perturbation theory, later completed by Laplace and Poincaré. He introduced the fundamental concept of osculating orbital elements and derived differential equations that determine their change over time. He built a theory of precession and nutation of the earth’s axis, predicted the “free movement of the poles” of the earth, discovered a hundred years later by Chandler. In 1748-1751, Euler published a complete theory of light aberration and parallax. In 1756, he published the differential equation of astronomical refraction and studied the dependence of refraction on pressure and air temperature at the observation site. These results had a huge impact on the development of astronomy in subsequent years. Euler outlined a very precise theory of the motion of the Moon, developing for this a special method of varying the orbital elements. Subsequently, in the 19th century, this method was expanded, applied to the model of the motion of large planets, and is used to this day. Mayer's tables, calculated on the basis of Euler's theory (1767), also turned out to be suitable for solving the pressing problem of determining longitude at sea, and the British Admiralty paid Mayer and Euler a special bonus for it. Euler's main works in this area: “The Theory of the Motion of the Moon,” 1753; “Theory of the motion of planets and comets”, 1774; “New Theory of the Motion of the Moon,” 1772. Euler studied the gravitational field of not only spherical but also ellipsoidal bodies, which represented a significant step forward. He was also the first in science to point out the secular shift in the inclination of the ecliptic plane (1756), and according to his proposal, the inclination was since adopted as a reference at the beginning of 1700. He developed the basic theory of the motion of the satellites of Jupiter and other highly compressed planets. In 1748, long before the work of P. N. Lebedev, Euler put forward the hypothesis that the tails of comets, auroras and zodiacal light have a common source of the influence of solar radiation on the atmosphere or matter of celestial bodies. Astronomy 

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All his life Euler was interested in musical harmony, striving to give it a clear mathematical basis. The purpose of his early work - “An Experience in a New Theory of Music” (Tentamen novae theoriae musicae, 1739) - was an attempt to mathematically describe how pleasant (euphonious) music differs from unpleasant (dissonant) music. At the end of Chapter VII of the “Essay,” Euler arranged the intervals according to “degrees of pleasantness” (gradus suavitatis), with the octave being ranked in class II (the most pleasant), and diaschism in the last, class XXVII (the most dissonant interval); some classes (including first, third, sixth) in Euler's pleasantness table were skipped. There was a running joke about this work that there was too much music for mathematicians and too much mathematics for musicians. In his declining years, in 1773, Euler read a report at the St. Petersburg Academy of Sciences, in which he finally formulated his lattice representation of the sound system; this representation was metaphorically designated by the author as the “mirror of music” (lat. speculum musicae). The following year, Euler's report was published as a short treatise, De harmoniae veris principiis per speculum musicum repraesentatis ("On the true foundations of harmony presented through speculum musicae"). Under the name “sound network” (German: Tonnetz), the Euler lattice was widely used in German music theory of the 19th century. Music theory 

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In 1749, Euler published a two-volume monograph, Naval Science, or a Treatise on Shipbuilding and Navigation, in which he applied analytical methods to practical problems of shipbuilding and navigation at sea, such as the shape of ships, issues of stability and balance, and methods for controlling the movement of a ship. The general theory of ship stability by A. N. Krylov is based on “Marine Science”. Euler's scientific interests also included physiology; in particular, he applied the methods of hydrodynamics to the study of the principles of blood movement in blood vessels. In 1742, he sent an article to the Dijon Academy on the flow of liquids in elastic tubes (considered as models of blood vessels), and in December 1775 he presented the memoir “Fundamentals of determining the movement of blood through arteries” to the St. Petersburg Academy of Sciences. This work analyzed the physical and physiological principles of blood movement caused by periodic contractions of the heart. Treating blood as an incompressible fluid, Euler found a solution to the equations of motion he composed for the case of rigid tubes, and in the case of elastic tubes he limited himself to only obtaining general equations of finite motion. Other areas of knowledge 

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Lunar Crater Euler Many concepts in mathematics and other sciences are named after Euler, see: list of objects named after Leonhard Euler; Euler Crater on the Moon; Asteroid 2002 Euler; International Mathematical Institute named after. Leonhard Euler Russian Academy of Sciences, founded in 1988 in St. Petersburg; Gold Medal named after Leonhard Euler of the USSR Academy of Sciences and the Russian Academy of Sciences; The Euler Medal, awarded annually since 1993 by the Canadian Institute of Combinatorics and Its Applications for achievements in this field of mathematics; International Charitable Foundation for the Support of Mathematics named after Leonhard Euler; Street in Almaty. The complete works of Euler, published since 1909 by the Swiss Society of Naturalists, are still not completed; it was planned to release 75 volumes, of which 73 were published: 29 volumes on mathematics; 31 volumes on mechanics and astronomy; 13 - in physics. Eight additional volumes will be devoted to Euler's scientific correspondence (over 3000 letters). Memory 

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According to contemporaries, Euler’s character was good-natured, gentle, and practically did not quarrel with anyone. Even Johann Bernoulli, whose difficult character was experienced by his brother Jacob and son Daniel, invariably treated him warmly. For a full life, Euler needed only one thing - the possibility of regular mathematical creativity. He could work intensively even “with a child on his lap and a cat on his back.” At the same time, Euler was cheerful, sociable, loved music and philosophical conversations. Academician P.P. Pekarsky, relying on the testimony of Euler’s contemporaries, recreated the image of the scientist in the following way: “Euler had the great art of not flaunting his learning, hiding his superiority and being on the level of everyone. Always an even disposition, gentle and natural gaiety, some mockery with an admixture of good nature, naive and playful conversation - all this made a conversation with him as pleasant as attractive.” As contemporaries note, Euler was very religious. According to Condorcet, every evening Euler gathered his children, servants and students who lived with him for prayer. He read them a chapter from the Bible and sometimes accompanied the reading with a sermon. In 1747, Euler published a treatise in defense of Christianity against atheism, “Defense of Divine Revelation against the Attacks of Freethinkers.” Euler's passion for theological reasoning became the reason for the negative attitude towards him (as a philosopher) of his famous contemporaries - D'Alembert and Lagrange. Frederick II, who considered himself a “freethinker” and corresponded with Voltaire, said that Euler “smells of a priest.” Euler was a caring family man, willingly helped colleagues and young people, and generously shared 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 mathematics, read Euler.” “Read, read Euler, he is our common teacher,” Laplace also liked to repeat (French Lisez Euler, lisez Euler, c "est notre maître à tous.). Euler’s works were also studied with great benefit by the “king of mathematicians” Carl Friedrich Gauss, and almost all famous scientists of the 18th-19th centuries.Personal qualities and assessments 

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Many facts in geometry, algebra and combinatorics, proven by Euler, are widely used in Olympiad mathematics. On April 15, 2007, an Internet Olympiad for schoolchildren in mathematics was held, dedicated to the 300th anniversary of the birth of Leonhard Euler, supported by a number of organizations. In December 2008 - March 2009, the Leonhard Euler Mathematical Olympiad was held for eighth-graders, designed to partially replace the loss of the regional and final stages of the All-Russian Mathematical Olympiad for 8th graders. Mathematical Olympiads 

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Historians have discovered more than a thousand direct descendants of Leonhard Euler. The eldest son Johann Albrecht became a prominent mathematician and physicist. The second son Karl was a famous doctor. The youngest son, Christopher, later became a lieutenant general in the Russian army and commander of the Sestroretsk arms factory. All of Euler's children accepted Russian citizenship (Euler himself remained a Swiss subject all his life). As of the late 1980s, historians counted about 400 living descendants, about half of them lived in the USSR. Some of Euler's famous descendants 

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New theory of the moon's motion. - L.: Ed. Academy of Sciences of the USSR, 1934. A method for finding curved lines that have the properties of a maximum or minimum or solving an isoperimetric problem taken in the broadest sense. - M.; L.: Gostekhizdat, 1934. - 600 p. Basics of point dynamics. - M.-L.: ONTI, 1938. Differential calculus. - M.-L.: Geodesizdat, 1949. Integral calculus. In 3 volumes. - M.: Gostekhizdat, 1956-1958. Variational principles of mechanics. Sat. articles: Fermat, Hamilton, Euler, Gauss, etc. / Polak L. (ed.). - M.: Fizmatlit, 1959. - 932 p. Selected cartographic articles. - M.-L.: Geodesizdat, 1959. Introduction to the analysis of infinite. In 2 volumes. - M.: Fizmatgiz, 1961. Research on ballistics. - M.: Fizmatgiz, 1961. Correspondence. Annotated index. - L.: Nauka, 1967. - 391 p. 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. An experience of a new theory of music, clearly presented in accordance with the immutable principles of harmony. - SPb.: Ross. acad. Sciences, St. Petersburg scientific center, publishing house Nestor-History, 2007. - ISBN 978-598187-202-0. Guide to arithmetic for use by the gymnasium of the Imperial Academy of Sciences. - M.: Onyx, 2012. - 313 p. - ISBN 978-5-458-27255-1, etc. Bibliography 

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Artemyeva T.V. Leonard Euler as a philosopher // Philosophy in the St. Petersburg Academy of Sciences of the 18th century. - St. Petersburg, 1999. - 182 p. Bashmakova I. G., Yushkevich A. P. Leonard Euler // Historical and mathematical studies. - M.: GITTL, 1954. - No. 7. - P. 453-512. Bell E.T. Creators of mathematics. - M.: Education, 1979. - 256 p. Bobylev D.K. Euler, Leonhard // Encyclopedic Dictionary of Brockhaus and Efron: in 86 volumes (82 volumes and 4 additional). - St. Petersburg, 1890-1907. Gindikin S.G. Stories about physicists and mathematicians. - 3rd ed., expanded. - M.: MTsNMO, 2001. - 465 p. - ISBN 5-900916-83-9. Delaunay B. N. Leonard Euler // Quantum. - 1974. - No. 5. History of mechanics in Russia / Rep. editors A. N. Bogolyubov, I. Z. Shtokalo. - Kyiv: Naukova Dumka, 1987. - 392 p. Kotek V.V. Leonard Euler. - M.: Uchpedgiz, 1961. - 106 p. Leonhard Euler 1707-1783. Collection of articles and materials for the 150th anniversary of his death. - Publishing House of the USSR Academy of Sciences, 1935. - 240 p. To the 250th anniversary of the birth of L. Euler. - Collection. - Publishing House of the USSR Academy of Sciences, 1958. Burya A. The Death of Leonhard Euler. - pp. 605-607. Chronicle of the Russian Academy of Sciences. - M.: Nauka, 2000. - T. 1: 1724-1802. - ISBN 5-02-024880-0. Mathematics of the 18th century // History of mathematics / Edited by A. P. Yushkevich, in three volumes. - M.: Nauka, 1972. - T. III. Moiseev N. D. Essays on the development of mechanics. - M.: Publishing house Mosk. University, 1961. - 478 p. Literature 

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The ideal mathematician of the 18th century is what Euler (1707-1789) is often called. He was born in small, quiet Switzerland. Around the same time, the Bernoulli family moved to Basel from Holland: a unique constellation of scientific talents led by the brothers Jacob and Johann. By chance, young Euler ended up in this company. But when the guys grew up, it turned out that there was not enough room for their minds in Switzerland. But in Russia the Academy of Sciences was established in 1725. There were not enough Russian scientists, and three friends went there. At first, Euler deciphered diplomatic dispatches, taught young sailors higher mathematics and astronomy, and compiled tables for artillery fire and tables for the movement of the Moon. At the age of 26, Euler was elected Russian academician, but after 8 years he moved from St. Petersburg to Berlin. The “king of mathematicians” worked there from 1741 to 1766; then he left Berlin and returned to Russia. Surprisingly, Euler’s fame did not fade even after the scientist was struck by blindness (shortly after moving to St. Petersburg). In the 1770s, the St. Petersburg mathematical school grew up around Euler, more than half consisting of Russian scientists. At the same time, the publication of his main book, “Fundamentals of Differential and Integral Calculus,” was completed. At the beginning of September 1783, Euler felt slightly unwell. On September 18, he was still engaged in mathematical research, but suddenly lost consciousness and “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.

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Leonhard Euler (1707-1783)

an outstanding mathematician who made a significant contribution to the development of mathematics, as well as mechanics, physics, astronomy and a number of applied sciences.

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Leonhard Euler was born in 1707 in Switzerland into the family of a Basel pastor. He discovered mathematical abilities early. The pastor prepared his eldest son for a spiritual career, but he also studied mathematics with him - both as entertainment and for the development of logical thinking. another way.
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Leonard Euler

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was founded in 1459
University of Basel
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Soon the capable boy attracted the attention of Professor Johann Bernoulli. He gave the gifted student mathematical articles to study, and on Saturdays he invited him to come to his home to jointly analyze the incomprehensible. On June 8, 1724, 17-year-old Leonhard Euler gave a speech in Latin about comparing the philosophical views of Descartes and Newton and was awarded a master's degree.
Johann Bernoulli
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Leonard Euler

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The number of scientific vacancies in Switzerland was very small. At the beginning of the winter of 1726, on the recommendation of the Bernoulli brothers, he was invited to the post of adjunct in physiology with a salary of 200 rubles. To everyone’s surprise, Euler began to speak Russian fluently the very next year after his arrival.
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Leonard Euler

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On January 22, 1724, Peter I approved the project for the organization of the St. Petersburg Academy. On January 28, the Senate issued a decree on the creation of the Academy.

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One of the most important tasks of the Academy was the training of domestic personnel. Euler compiled a very good “Manual to Arithmetic” in German, which was immediately translated into Russian and served for many years as an initial textbook. This was the first systematic presentation of arithmetic in Russian.
Leonard Euler

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In 1733, Euler became an academician and professor of pure mathematics with a salary of 600 rubles. On one of the last days of 1733, 26-year-old Leonard Euler married his peer, the daughter of a painter (a St. Petersburg Swiss) Katharina Gsell. The newlyweds purchased a house on the Neva embankment, where they settled. 13 children were born into the Euler family, but 3 sons and 2 daughters survived.
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Leonard Euler

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Euler was distinguished by his phenomenal efficiency. According to contemporaries, for him living meant doing mathematics. During the first period of his stay in Russia, he wrote more than 90 major scientific works.
In 1735, the Academy received the task of performing an urgent and very cumbersome astronomical calculation. A group of academicians asked for three months to complete this work, but Euler undertook to complete the work in 3 days - and did it on his own. However, the overexertion did not pass without a trace: he fell ill and lost sight in his right eye.
However, the scientist reacted to the misfortune with the greatest calm: “Now I will be less distracted from doing mathematics,” he noted philosophically

Leonard Euler

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After the death of Empress Anna in 1740, the Academy fell into disrepair. Euler is considering returning home. He accepts the offer of the Prussian King Frederick, who invited Euler to the Berlin Academy for the post of director of its Mathematics Department. The Russian Academy did not object. Euler was “released from the Academy” in 1741 and confirmed as an honorary academician with a salary of 200 rubles.
Leonard Euler

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While in Berlin, L. Euler never ceased to maintain contacts with the St. Petersburg Academy of Sciences. He purchased equipment and literature for the academy, edited the mathematical department, where he published as many articles as in the organ of the Berlin Academy of Sciences, and supervised the training of Russian mathematicians sent to Berlin.
It is said that when Frederick II asked Euler where he learned what he knew, the latter replied that he owed it all to his stay at the St. Petersburg Academy of Sciences. During the seven-year war with Prussia, when Russian troops occupied Berlin and Euler's house suffered, the Russian command apologized to him and compensated him for the loss, and Empress Elizabeth, in addition, sent him 4,000 rubles
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Leonard Euler

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In 1762, Catherine II ascended the Russian throne and pursued a policy of enlightened absolutism. The Empress offered Euler management of a mathematical class (department), the title of conference secretary of the Academy and a salary of 1800 rubles per year. “And if you don’t like it,” said the letter to her representative, “he would be pleased to communicate his conditions, so long as he doesn’t hesitate to come to St. Petersburg.”
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Leonard Euler

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Euler actually asked for more: a salary of 3,000 rubles a year and the post of vice-president of the Academy; annual pension of 1000 rubles to the wife after his death; paid positions for three of his sons, including the post of secretary of the Academy for the eldest. All these conditions were accepted.
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Leonard Euler

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. “In the current state of affairs, there is no money for a salary of 3,000 rubles, but for a person with such merits as Mr. Euler, I will add to the academic salary from state revenues, which together will amount to the required 3,000 rubles... I am sure that my Academy will be reborn from ashes from such an important acquisition, and I congratulate myself in advance for returning a great man to Russia.” (from Catherine’s letter to Chancellor Count Vorontsov)
Euler returns to Russia, now forever.
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Leonard Euler

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In July 1766, 60-year-old Euler, his family and household (18 people in total) arrived in the Russian capital. Immediately upon arrival he was received by the empress. Catherine greeted him as an august person and showered him with favors: she granted him 8,000 rubles to buy a house on Vasilievsky Island and to purchase furnishings, provided one of her cooks for the first time, and instructed him to prepare ideas for the reorganization of the Academy.
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Leonard Euler

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Unfortunately, after returning to St. Petersburg, Euler developed a cataract in his second, left eye - he stopped seeing. Probably for this reason, he never received the promised post of vice-president of the Academy. However, blindness did not affect his performance. Euler dictated his works to a tailor boy, who wrote everything down in German. The number of works he published even increased; during the decade and a half of his second stay in Russia, he dictated more than 400 articles and 10 books.
Surprisingly, the last years of his life turned out to be the most fruitful. A good half of what Euler accomplished occurred in the last decade of his life.
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Leonard Euler

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In May 1771, a large fire occurred in St. Petersburg, destroying hundreds of buildings, including Euler’s house and almost all of his property. The scientist himself was saved with difficulty. All manuscripts were saved from fire; Only part of the “New Theory of the Motion of the Moon” burned down, but it was quickly restored with the help of Euler himself, who retained a phenomenal memory into old age. Euler had to temporarily move to another house.
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Leonard Euler

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In September of the same year, at the special invitation of the Empress, the famous German ophthalmologist Baron Wentzel arrived in St. Petersburg to treat Euler. After an examination, he agreed to perform surgery on Euler and removed a cataract from his left eye. Euler began to see again. The doctor ordered to protect the eye from bright light, not to write, not to read - just gradually get used to the new condition. However, just a few days after the operation
Euler took off the bandage and soon lost his sight again. This time it's final.
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Leonard Euler

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Euler worked actively until his last days. In September 1783, the 76-year-old scientist began to experience headaches and weakness. On September 7 (18), after lunch spent with his family, talking with astronomer 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 cerebral hemorrhage.
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Leonard Euler

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Condorcet said at the funeral meeting of the Paris Academy of Sciences.
Euler himself joked at the end of his life that after his death the academy would publish his works for another 20 years. In fact, his archives were sorted by a whole generation of scientists, and his publications lasted for another 47 years.
During his lifetime he published 530 books and articles, and now more than 800 of them are known.
Statistical calculations show that Euler made on average one discovery per week. It is difficult to find a mathematical problem that was not addressed in the works of Euler. All mathematicians of subsequent generations studied with Euler in one way or another, and it was not without reason that the famous French scientist P.S. Laplace said: “Read Euler, he is the teacher of us all.”
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"Euler stopped living and calculating"

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He was buried at the Smolensk Lutheran cemetery in St. Petersburg. The inscription on the monument read: “Here lie the mortal remains of the wise, just, famous Leonhard Euler.” In 1955, the ashes of the great mathematician were transferred to the “Necropolis of the 18th century” at the Lazarevskoye cemetery of the Alexander Nevsky Lavra. The poorly preserved tombstone was replaced.
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Leonard Euler

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From the point of view of mathematics, the 18th century is the century of Euler.
“Read, read Euler, he is our common teacher” (Laplace)
«
“If you really love mathematics, read Euler.” (Lagrange)
“Together with Peter I and Lomonosov, Euler became the good genius of our Academy, who determined its glory, its strength, its productivity.” (S.I. Vavilov)
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Euler is one of the geniuses whose work has become the property of all mankind. Until now, schoolchildren in all countries study trigonometry and logarithms in the form that Euler gave them. Students study higher mathematics using manuals, the first examples of which were Euler's classical monographs. He was primarily a mathematician, but he knew that the soil on which mathematics flourishes is practical activity.
He left important works in various branches of mathematics, mechanics, physics, astronomy and a number of applied sciences. It is difficult to even list all the industries in which the great scientist worked.
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House of L. Euler (A. Gitshov) (Lieutenant Schmidt embankment, 15)
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The ideal mathematician of the 18th century is what Euler is often called. It was a short-lived Age of Enlightenment, wedged between eras of cruel intolerance. Just 6 years before Euler was born, the last witch was publicly burned in Berlin. And 6 years after Euler’s death - in 1789 - a revolution broke out in Paris. Euler was lucky: he was born in small, quiet Switzerland, where craftsmen and scientists came from all over Europe, who did not want to waste expensive working time on civil unrest or religious strife. This is how the Bernoulli family moved to Basel from Holland: a unique constellation of scientific talents led by the brothers Jacob and Johann. By chance, young Euler ended up in this company and soon became a worthy member of the “nursery of geniuses”
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Leonard Euler

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They became widely known thanks to the great mathematician Leonhard Euler, who, thanks to one riddle, created the theory of graphs. And the riddle was this: how to cross all seven bridges of Königsberg without passing over any of them twice. It turned out that in the case of the Königsberg bridges this is impossible. And Euler, in turn, was able to discover a rule, using which it was easy to determine whether such a problem had a solution or not.
seven bridges of Königsberg
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In a simplified diagram of parts of a city (graph), bridges correspond to lines (edges of the graph), and parts of the city correspond to points connecting lines (vertices of the graph). In the course of his reasoning, Euler came to the following conclusions: The number of odd vertices (vertices to which an odd number of edges lead) of the graph must be even. There cannot be a graph that has an odd number of odd vertices. If all the vertices of the graph are even, then you can draw a graph without lifting your pencil from the paper, and you can start from any vertex of the graph and end it at the same vertex. A graph with more than two odd vertices cannot be drawn with one stroke. The graph of Königsberg bridges had four odd vertices (i.e. all of them), therefore it is impossible to walk across all the bridges without passing over any of them twice.
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However, there were people who in their own way “solved” the insoluble problem. One of these people was Kaiser Wilhelm, who was famous for his straightforwardness, simplicity of thinking and soldierly narrow-mindedness.
One day, while at a social event, he almost became the victim of a joke that the learned minds present at the reception decided to play on him. They showed the Kaiser a map of Königsberg and asked him to try to solve this famous problem. To everyone's surprise, the Kaiser asked for a pen and a piece of paper, saying that he would solve the problem in a minute and a half. The stunned German establishment could not believe their ears, but paper and ink were quickly found. The Kaiser put the piece of paper on the table, took a pen, and wrote: “I order the construction of the eighth bridge on the island of Lomze.” So a new bridge appeared in Königsberg, which was called the Kaiser's Bridge. And now even a child could solve the problem with eight bridges.
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On the front side of the coin, in a circle framed by a beaded rim, there is a relief image of the emblem of the Bank of Russia - a double-headed eagle with lowered wings, under it there is an inscription in a semicircle “BANK OF RUSSIA”, and also along the circumference there are inscriptions separated by dots: indicating the denomination of the coin “TWO RUBLES” and the year of minting “2007”, between them are the designation of the metal according to the Periodic Table of Chemical Elements of D.I. Mendeleev, the alloy sample, the trademark of the Moscow Mint and the mass of the precious metal in purity. Reverse: On the reverse side of the coin there are relief images of the portrait of the mathematician L. Euler, on the right of the mathematical formula and below the celestial sphere, there are: at the top there is an inscription along the circle “LEONARD EULER” and to the left of the portrait in two lines the dates “1707” and “1783”.

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L. Euler Medal
The European Academy of Natural Sciences has developed and issued special awards, in particular, commemorative medals in honor of Nobel Prize laureates and outstanding European scientists. Today the Academy has more than 80 awards, which serve as moral and social support and encouragement for proactive and creative people.

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Swiss banknote with a portrait of the young Euler

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Postage Stamp. GDR 1983

The most important dates of life and activity April 4, 1707 - L. Euler was born in Basel (Switzerland) into a pastor's family 1720 - student of the junior faculty of philosophy at the University of Basel June 9, 1722 - received the degree "First Laurels" (bachelor) in philosophy 1723 - entered the theological faculty (at the insistence of his father) June 8, 1724 - received a Master of Arts degree (for a speech on a comparison of the philosophical views of Newton and Descartes) May 24, 1727 - adjunct of the St. Petersburg A.N. in mathematics 1731 - occupies the department of theoretical and experimental physics 1733 - academician of St. Petersburg A.N. in mathematics 1733 - marriage to the painter's daughter Ekaterina Gsell 1735 - work in the Geographical Department. – work in the Berlin A.N. – return to the St. Petersburg A.N. September 18, 1783 – death of L. Euler from cerebral hemorrhage

The main works of L. Euler 1. Introduction to arithmetic (German, two volumes, St. Petersburg). 2. Introduction to algebra (1770, German, St. Petersburg). 3. Introduction to the analysis of infinitesimals (1748, Latin, two volumes, Lausanne). 4. Differential calculus (1755, Latin, Berlin). 5. Integral calculus (Latin, three volumes, St. Petersburg). 6. Method for finding curved lines that have the properties of maximum or minimum (1744, Latin, Lausanne). 7. Mechanics in an analytical presentation (1736, Latin, two volumes, St. Petersburg). 8. Theory of motion of rigid bodies (1765, Latin, Rostock). 9. Mechanics of liquid bodies (the most important memoir dates back to 1769, Latin, St. Petersburg). 10. Resistance of columns (1757, French, Berlin).

11. Robins' New Principles of Artillery, translated from English and provided with the necessary explanations and many notes (1745, German, Berlin). 12. Theory of the motion of planets and comets (1744, Latin, Berlin). 13. The theory of the movement of the Moon (1753, Latin, Berlin) 14. The theory of the movement of the Moon, revised by a new method (1772, Latin, St. Petersburg) 15. The theory of ebbs and flows (1740, Latin, Paris). 16. Construction of lenses made of two glasses (achromatic, Latin, 1762, St. Petersburg). 17. Dioptrics (Latin, three volumes, St. Petersburg). 18. Theory of music (1739, Latin, St. Petersburg). 19. Dissertation on the magnet (Latin, Paris). 20. Marine science (1749, Latin, St. Petersburg) 21. Complete theory of the design and navigation of ships (1773, French, St. Petersburg). 22. Letters to a German princess about various subjects of physics and philosophy (French, three volumes, St. Petersburg).

Euler's main achievements Euler's importance for the development of mathematics, mechanics and many other sciences is very great; his works, paving new creative paths, are numerous. Currently, 865 of his works are known, of which 43 volumes are individual multi-page works. Contributed to such mathematical disciplines as calculus of variations, integration of ordinary differential equations, power series, special functions, differential geometry, number theory; He introduced double integrals, transformed trigonometry, giving it an almost modern form, and paid great attention to applied issues of mathematics;

He laid the foundations of mathematical physics, solid mechanics, hydrodynamics, hydraulics, and, in many respects, machine mechanics; He published a series of works on astronomy, systematically outlined the theory of elastic curves, obtained important results on the resistance of materials, and was actively involved in navigation, ballistics, and dioptrics; He created basic guides for universities in higher mathematics, wrote arithmetic and algebra textbooks for gymnasiums, expressed fundamental ideas for the development of school mathematics education...

Euler imparted a meaningful and methodological charge to mathematical education, which very quickly, by historical standards, brought domestic mathematical education closer to the European quality level. In Russia, he created and quickly put into action a mechanism for the patronage of mathematics as a science over mathematical education. This trend was embodied in a unique phenomenon in Russian history - the methodological school of L. Euler, which provided prompt access to the pedagogical and methodological ideas of Europe; enriched and rethought them; made it a priority to create original domestic mathematical literature rather than translated Western ones.

Euler's methodological ideas: the idea of ​​bringing together the content of mathematical education with modern mathematics; the idea of ​​isolating in school mathematical education the foundations of mathematical disciplines - arithmetic, geometry, trigonometry, and subsequently algebra; the idea of ​​constructing mathematical courses based on didactic principles such as systematic, scientific, accessible presentation of mathematical disciplines, taking into account the age characteristics of students.

Euler's theorem: The midpoints of the sides of a triangle, the bases of its altitudes and the midpoints of the segments of the altitudes of the triangle from the orthocenter to the vertex lie on the same circle; H – orthocenter of the triangle; K,Q,P – Euler points (the midpoints of the altitude segments of the triangle from the orthocenter to each of the vertices). This circle is called the nine-point circle or Euler circle. Its radius is equal to half the radius of the circle circumscribed about this triangle. The straight line connecting the orthocenter of the triangle with the center O of the circumscribed circle is called Euler's straight line.

Euler's theorem on polyhedra: For any simple polyhedron B - P + G = 2, where B is the number of vertices, P - the number of edges, G - the number of faces Euler's theorem on polyhedra: For any simple polyhedron B - P + G = 2, where B is the number of vertices, P is the number of edges, G is the number of faces. Using this theorem, it can be proven that there are no more than five types of regular polyhedra: tetrahedron, cube, octahedron, dodecahedron and icosahedron. Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Euler's function Continuing Fermat's work on number theory, Euler introduced the function φ(m), which is called the Euler function - the number of natural numbers less than a given m and relatively prime to it. Euler also generalized Fermat’s little theorem and proved that if a and m are coprime numbers, then a φ(m) – 1 is divisible by m. This proposition is called Euler's theorem (on comparisons).

Euler's integrals Trying to find a formula for the general expression of the sum of the hypergeometric series ... + 1 2 ...k + ... Euler came to the integrals, which were later called Euler integrals, and later - Euler's beta function and Euler's gamma function:

Euler's problem of seven bridges The problem solves the question: how can one walk across the seven Königsberg bridges over the Pregl River, passing over each bridge no more than once? On the “Order of the Seven Bridges” the dark places represent the river, and the white places represent the banks of the river and bridges. Euler proved that this was impossible and found general rules that govern problems of this type.

Euler's knight move problem The problem solves the question: How to place 64 numbers from 1 to 64 in 64 squares of a chessboard so that any two cells containing two consecutive numbers are connected by a knight move? Euler was the first to develop methods for solving this problem. Euler was buried in the St. Petersburg necropolis - Alexander Nevsky Lavra. The inscription on the monument read: “To Leonard Euler – St. Petersburg Academy.” Monument Without a doubt, the name of Leonard Euler is one of the most famous in the galaxy of outstanding mathematicians of all times, his works continue to have a decisive influence on the progress of all modern mathematics.
Literature Gnedenko B.V. Essays on the history of mathematics in Russia, Gostekhizdat, Kotek V.V. Leonard Euler. M.: Uchpedgiz, Polyakova T.S. History of domestic school mathematics education. Two centuries. Book 1: eighteenth century. Rostov n/d: publishing house Rost. ped. University, Prudnikov V.E. Russian mathematics teachers of the 18th-19th centuries. M.: Uchpedgiz, Stroik D. Ya. A brief outline of the history of mathematics. M.: Nauka, 1984. Yushkevich A.P. History of mathematics in Russia before 1917 M.: Nauka, 1968.