## Summary

Leonhard Euler (April 15, 1707, Basel, Switzerland – September 7 (18), 1783, St. Petersburg, Russian Empire) – Swiss, Prussian and Russian mathematician and mechanic, who made a fundamental contribution to the development of these sciences (as well as physics, astronomy and several applied sciences). Along with Lagrange – a major mathematician of the 18th century, he is considered one of the greatest mathematicians in history. Euler authored more than 850 works (including two dozen fundamental monographs) on mathematical analysis, differential geometry, number theory, approximate calculations, celestial mechanics, mathematical physics, optics, ballistics, shipbuilding, music theory, and other fields. He studied medicine, chemistry, botany, aeronautics, music theory, and many European and ancient languages. He was an academician of the St. Petersburg, Berlin, Turin, Lisbon, and Basel Academies of Sciences, and a foreign member of the Paris Academy of Sciences. First Russian member of the American Academy of Arts and Sciences.

Almost half of his life was spent in Russia, where he made a significant contribution to the formation of Russian science. In 1726 he was invited to work in St. Petersburg, where he moved a year later. From 1726 to 1741, and from 1766 he was an academician of the St. Petersburg Academy of Sciences (in 1741-1766 he worked in Berlin (at the same time remaining an honorary member of the St. Petersburg Academy). Already in a year of his stay in Russia he knew Russian well and some of his works (especially textbooks) were published in Russian. The first Russian academicians-mathematicians (S. K. Kotelnikov) and astronomers (S. Ya. Rumovsky) were pupils of Euler.

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## Switzerland (1707-1727)

Leonhard Euler was born in 1707 into the family of the Basel pastor Paul Euler, a friend of the Bernoulli family, and Marguerite Euler, née Brooker. Soon after Leonard was born, the family moved to Richeng, where the boy spent his early years. Leonard received his elementary education at home under the guidance of his father (the latter had once studied mathematics with Jakob Bernoulli). His pastor prepared his eldest son for a spiritual career, but he also did mathematics with him, both for fun and to develop logical thinking, and Leonard showed an early talent for mathematics.

When Leonard grew up, he was moved to his grandmother”s house in Basel, where he studied at the grammar school (while continuing to study mathematics with passion). In 1720, an able gymnasium student was allowed to attend public lectures at the University of Basel; there he caught the eye of Professor Johann Bernoulli (the younger brother of Jakob 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 areas.

On October 20, 1720, 13-year-old Leonhard Euler became a student in the Faculty of Arts at the University of Basel. But his love of mathematics led Leonard down a different path. Visiting the home of his teacher, Euler met and befriended his sons, Daniel and Nicholas, who also, in the family tradition, studied mathematics in depth. In 1723 Euler received (according to the custom at the University of Basel) his first award (primam lauream). On July 8, 1724, the 17-year-old Leonhard Euler gave a speech in Latin comparing the philosophical views of Descartes and Newton and was awarded the degree of Master of Arts.

In the next two years the young Euler wrote several scientific papers. One of them, “Dissertation on Physics of Sound,” was submitted to a competition to fill the unexpectedly vacant position of professor of physics at the University of Basel (1725). But despite the positive feedback, the 19-year-old Euler was considered too young to be included as a candidate for the professorship. At the time, the number of scientific vacancies in Switzerland was quite small. So the brothers Daniel and Nikolai Bernoulli left for Russia, where the Academy of Sciences was being organized; they promised to beg for a position for Euler there as well.

In the early winter of 1726-1727 Euler received news from Saint Petersburg: on the recommendation of the Bernoulli brothers he was invited to the post of Adjunct (assistant professor) in the department of physiology (this department was occupied by D. Bernoulli) with an annual salary of 200 rubles (we kept a letter of gratitude of November 9, 1726 to the president of the Academy L. Blumentrost for his acceptance at the Academy). Since Johann Bernoulli was a famous doctor, in Russia it was believed that Leonhard Euler, as his best pupil, was also a doctor. Euler, however, postponed his departure from Basel until spring, devoting the remaining months to a serious study of the medical sciences, whose profound knowledge of which he would later impress his contemporaries. Finally, on April 5, 1727, Euler left Switzerland forever, although he retained his Swiss (Basel) citizenship for the rest of his life.

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## Russia (1727-1741)

On January 22 (February 2), 1724 Peter I approved the project of the Petersburg Academy. On January 28 (February 8), 1724 issued a decree of the Senate on the establishment of the Academy. Out of 22 professors and associate professors invited in the first years there were 8 mathematicians who were also engaged in mechanics, physics, astronomy, cartography, theory of shipbuilding, service of measures and weights.

Euler (whose route from Basel lay through Lübeck, Revel and Kronstadt) arrived in St. Petersburg on May 24, 1727; a few days before, Empress Catherine I, the Academy”s patroness, had died, and the scientists were in despondency and confusion. Euler was helped to get accustomed to his new place by fellow countrymen: academicians Daniil Bernoulli and Jakob Hermann; the latter, being a professor in the chair of higher mathematics, was distant relative to the young scientist and offered him all kinds of patronage. Euler was made an associate professor of higher mathematics (not physiology, as originally planned), although he conducted research in St. Petersburg in the field of fluid dynamics. He received a salary of 300 rubles per year and was provided with an apartment.

Euler became fluent in Russian a few months after his arrival in St. Petersburg.

In 1728, the publication of the first Russian scientific journal, Commentaries 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 all, more than 400 articles by Euler were published in this edition.

In September 1730 the term of the contracts concluded with academicians J. Hermann (chair of mathematics) and G. B. Bilfinger (chair of experimental and theoretical physics). Daniil Bernoulli and Leonhard Ayler were approved for their vacancies, respectively; the latter received an increase in salary up to 400 rubles, and on January 22, 1731 – and the official post of a professor. Two years later (1733), Daniel Bernoulli returned to Switzerland, and Euler, leaving the chair of physics, took his place, becoming an academician and professor of higher mathematics with a salary of 600 rubles (however, Daniel Bernoulli received twice as much).

On December 27, 1733, 26-year-old Leonhard Euler married his age-mate Katharina (German: Katharina Gsell), daughter of the academic painter Georg Gsell (a Swiss from St. Petersburg). The newlyweds bought a house on the Neva embankment, where they settled. Thirteen children were born into the Euler family, but three sons and two daughters survived.

The young professor had a lot of work to do: cartography, all sorts of examinations, consultations for shipbuilders and artillerymen, drawing up training manuals, designing fire pumps, etc. He was even required to draw up horoscopes, which Euler with all possible tact referred to a staff astronomer. Alexander Pushkin cites a romantic story: supposedly Euler composed a horoscope for a newborn prince John Antonovich (1740), but the result so frightened him that he did not show it to anyone, and only after the death of the poor prince told about it to Count K. G. Razumovsky. The reliability of this historical anecdote is extremely doubtful.

During his first period in Russia, he wrote more than 90 major scientific papers. A large part of the academic “Notes” is filled with Euler”s writings. He gave papers at scientific seminars, gave public lectures, and participated in various technical orders of government departments. During the 1730s Euler led the work on mapping of the Russian Empire, which (after the departure of Euler, in 1745) was completed with the publication of the atlas of the country. As N. I. Fuss said, in 1735 the Academy was assigned to perform an urgent and very cumbersome mathematical calculation, and a group of academicians asked for three months, but Euler undertook the work in three days – and managed to do it himself; however the overstrain did not pass without leaving a trace: he fell ill and lost the sight in his right eye. However, Euler himself, in one of his letters, attributed the loss of his eye to his work in the geographical department at the Academy.

The two-volume work Mechanics, or the Science of Motion Presented Analytically, published in 1736, brought Euler Europe-wide fame. In this monograph Euler successfully applied methods of mathematical analysis to the general solution of problems of motion in a void and in a resisting medium.

One of the most important tasks of the Academy was the training of native personnel, for which a university and a gymnasium were established under the Academy. Because of an acute shortage of textbooks in Russian, the Academy asked its members to compile such manuals. Euler compiled in German a very good “Handbook of Arithmetic”, which was immediately translated into Russian and served for several years as a primary textbook. The translation of the first part was done in 1740 by Vasily Adodurov, the first Russian adjunct of the Academy and a student of Euler.

The situation worsened when Empress Anna Ioannovna died in 1740 and the minor John VI was declared emperor. “Something dangerous was about to happen,” Euler later wrote in his autobiography. – After the death of the venerable Empress Anna at the regency which then followed … the situation began to seem uncertain.” Indeed, during the regency of Anna Leopoldovna, the St. Petersburg Academy finally fell into disrepair. Euler began to consider the option of returning home or moving to another country. In the end he accepted the proposal of the Prussian king Friedrich, who invited him on very favorable terms to the Berlin Academy, to the post of director of its mathematical department. The Academy was created on the basis of the Prussian Royal Society, founded by Leibniz, but in those years it was in a dismal state.

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## Prussia (1741-1766)

Euler submitted his resignation to the leadership of the St. Petersburg Academy:

For this reason I am compelled, both for reasons of poor health and other circumstances, to seek a more pleasant climate and to accept his Royal Prussian Majesty”s call for me. For this reason I beg the Imperial Academy of Sciences to discharge me in all good grace and to provide me and my family with the necessary passport for my passage.

On May 29, 1741 permission of the Academy was received. Euler was “released” and approved as an honorary member of the Academy with a salary of 200 rubles. In June 1741, 34-year-old Leonhard Euler with his wife, two sons and four nephews arrived in Berlin. He spent 25 years there and published about 260 works.

At first Euler was well received in Berlin, even invited to court balls. The Marquis Condorcet recalled that soon after moving to Berlin Euler was invited to a court ball. Asked by the Queen Mother why he was so taciturn, Euler replied: “I have come from a country where whoever speaks is hanged.

Euler had a lot of work to do. In addition to mathematical research, he directed an observatory and was involved in many practical matters, including the production of calendars (the Academy”s main source of income), the minting of Prussian coins, the laying of a new water pipe, and the organization of pensions and lotteries.

In 1742, a four-volume collection of Johann Bernoulli”s works was published. Sending it from Basel to Euler in Berlin, the old scientist wrote to his pupil: “I devoted myself to the childhood of higher mathematics. You, my friend, will continue its formation in maturity.” During the Berlin period, one after another Euler”s works were published: “Introduction to the Analysis of Infinitesimals” (1748), “Science of the Sea” (1749), “Theory of Moon Motion” (1753), “Instruction on Differential Calculus” (Lat. Institutiones calculi differentialis, 1755). Numerous articles on selected issues were printed in the publications of the Berlin and St. Petersburg Academies. In 1744 Euler discovered calculus of variations. His works use elaborate terminology and mathematical symbolism, largely preserved to this day, and the narrative is brought to the level of practical algorithms.

All the years of his stay in Germany Euler kept in touch with Russia. Euler participated in the publications of the St. Petersburg Academy, purchased books and instruments for it, edited the mathematical sections of Russian journals. In his apartment, on a full board, for years lived young Russian scientists, sent for training. It is known of Euler”s lively correspondence with M. V. Lomonosov; in 1747 he gave a favorable review to the president of the Academy of Sciences, Count K. G. Razumovsky, of Lomonosov”s articles on physics and chemistry, stating

All these theses are not only good, but also very excellent, because he writes about the matter of physical and chemical very necessary, which hitherto was not known and could not be interpreted by the wittiest people, which he did with such success that I am quite sure of the justice of his explanations. In this case Mr. Lomonosov must be given credit for having an excellent gift for the explanation of the physical and chemical phenomena. I wish that the other Academies were able to produce such revelations, as Mr. Lomonosov has shown.

Even the fact that Lomonosov did not write mathematical works and had no knowledge of higher mathematics did not prevent this high appreciation. Nevertheless, in 1755, as a result of Lomonosov”s tactlessness, who published without Euler”s permission his private letter in his support, Euler terminated all relations with him. Relations were restored in 1761 because Lomonosov facilitated Euler”s return to Russia.

His mother notified Euler of his father”s death in Switzerland (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 a plot to house his large family.

According to contemporaries, Euler throughout his life remained a modest, cheerful, extremely responsive man, always ready to help another. However, the relationship with the king did not work out: Frederick found the new mathematician intolerably boring, completely unsocial and treated him contemptuously. In 1759 Mauperthuis, president of the Berlin Academy of Sciences and a friend of Euler, died. King Frederick II offered the post of president of the Academy to d”Alumbert, but he refused. Friedrich, who disliked Euler, nevertheless entrusted him with the leadership of the Academy, but without the title of president.

During the Seven Years” War (having learned about it, Field Marshal Saltykov immediately reimbursed the losses, and later the Empress Elisabeth sent from herself another 4,000 rubles.

In 1765, “Theory of Motion of Solids” was published, and a year later “Elements of Calculus of Variations” was published. It was here that the name of the new section 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 aware of the importance of science both for the progress of the state and for her own prestige, she carried out a number of important transformations in the system of public education and culture favorable to science. The empress offered Euler the 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 do not like it,” the letter to her representative said, “she will be pleased to inform you of her terms, if only you do not hesitate to come to St. Petersburg.

Euler communicated his terms in response:

All these conditions were accepted. On January 6, 1766 Catherine informed Count Vorontsov:

Mr. Euler”s letter to you has given me great pleasure, because I learn from it of his desire to rejoin my service. Of course, I find him perfectly worthy of the desirable title of Vice-President of the Academy of Sciences, but for this some measures must be taken before I establish this title – I say establish it, as hitherto it did not exist. In the present state of affairs there is no money for the salary of 3000 rubles, but for a man of such merit as Mr. Euler, I shall add to the academic salary from the state revenues, which together amount to the required 3000 rubles… I am sure that my Academy will rise from the ashes of such an important acquisition, and I congratulate myself in advance on having returned to Russia a great man.

Later Euler put forward a number of other conditions (an annual pension of 1,000 rubles for his wife after his death, compensation for travel expenses, a place for his medical son, and a rank for Euler himself). Catherine also satisfied these conditions of Euler, with the exception of the demand for rank, jokingly saying: “I would give him, when he wishes, the rank of… (in the draft of the letter in French the collegiate adviser is crossed out), if I did not fear that this rank would make him equal to the multitude of people who are not worthy of Mr. Euler. Truly, his fame is better than his rank for giving him due respect.”

Euler petitioned the king for dismissal from the service, but received no response. He submitted it again – but Frederick did not even want to discuss the question of his departure. Decisive support for Euler was provided by persistent petitions from the Russian representation on behalf of the empress. On May 2, 1766 Friedrich finally gave permission for the great scholar to leave Prussia, though he could not refrain from scathing jokes about Euler in his correspondence (for example, on July 25 he wrote to D”Alamber: “Mister Euler, who madly loves the Big Dipper and the Little Dipper, moved closer to the North so as to make their observation easier”). True, he served as lieutenant colonel of artillery (later, thanks to the intercession of Catherine II, he was still able to join his father and was promoted to lieutenant general in the Russian army. In the summer of 1766 Euler returned to Russia – now permanently.

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## Russia Again (1766-1783)

On July 17 (28), 1766, the 60-year-old Euler, his family and household (a total of 18 people) arrived in the Russian capital. Immediately upon his arrival he was received by the empress. Catherine II met him as an august person and showered him with favors: she granted 8000 rubles for the purchase of a house on Vasilievsky Island and for the purchase of furnishings, provided for the first time one of her cooks and instructed to prepare considerations for the reorganization of the Academy.

Unfortunately, after his return to St. Petersburg, Euler developed cataracts in his only remaining left eye and soon he permanently 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 ability to work; he only remarked that he would now be less distracted by mathematics. Before acquiring a secretary, Euler dictated his work to a portly boy, who wrote everything in German. The number of his published works even increased; during his second stay in Russia Euler dictated more than 400 articles and 10 books, more than half of his creative legacy.

In 1768-1770 he published a two-volume classic monograph, “Universal Arithmetic” (also published as “The Beginnings of Algebra” and “The Complete Course of Algebra”). This work was first published in Russian (1768-1769), and a German edition came out two years later. The book was translated into many languages and was reprinted about 30 times (three times in Russian). All subsequent algebra textbooks were created under the strongest influence of Euler”s book.

During the same years, three volumes of “Dioptrica” (lat. Dioptrica, 1769-1771) on lens systems and the fundamental “Integral Calculus” (lat. Institutiones calculi integralis, 1768-1770), also in three volumes, were published.

Euler”s “Letters on various physical and philosophical matters, written to a certain German princess…” (1768) became very popular in the 18th century, and partly in the 19th as well. (1768), which had more than 40 editions in 10 languages (including 4 editions in Russian). This was a popular science encyclopedia of wide scope, written vividly and generally accessible.

Two serious events occurred in Euler”s life in 1771. In May there was a great fire in St. Petersburg, which destroyed hundreds of buildings, including the house and almost all of Euler”s property. The scientist himself was barely saved. All the manuscripts were saved from the fire; only a part of his “New Theory of the Moon”s Motion” was burned, but it was quickly restored with the help of Euler, who had retained his phenomenal memory until his old age. Euler had to temporarily move to another house. The second event: in September of the same year, at the special invitation of the empress, the famous German oculist Baron Wentzel arrived in St. Petersburg to treat Euler. After an examination he agreed to perform an operation on Euler and removed a cataract from his left eye. Euler began to see again. The doctor prescribed to keep his eye from bright light, not to write, not to read – just gradually to get used to the new condition. But a few days after the operation Euler took the bandage off and soon lost his sight again. This time for good.

1772: “A New Theory of the Moon”s Motion. Euler finally completed his work of many years by solving the three-body problem approximately.

In 1773, on the recommendation of Daniel Bernoulli, a disciple of Bernoulli, Nikolaus Fuss, came to St. Petersburg from Basel. This was a great stroke of luck for Euler. Fuss, a gifted mathematician, immediately after his arrival took charge of Euler”s mathematical work. Fuss soon married Euler”s granddaughter. For the next ten years – until his death – Euler predominantly dictated his writings to him, although he sometimes used the “eyes of his eldest son” and his other students. In the same year, 1773, Euler”s wife, with whom he had lived for almost 40 years, died. The death of his wife was a painful blow to the scientist, who was sincerely attached to his family. Euler soon married Salome Abigail, half-sister of his late wife.

In 1779 the General Spherical Trigonometry was published, the first complete exposition of the entire system of spherical trigonometry.

Euler worked actively until his last days. In September 1783 the 76-year-old scientist began to feel headaches and weakness. On September 7 (18), after a dinner spent with his family, talking with Academician A. I. Lexel about the newly discovered planet Uranus and its orbit, he suddenly felt ill. Euler managed to utter: “I”m dying,” and lost consciousness. Several hours later, without regaining consciousness, he died of a cerebral hemorrhage.

“He ceased to calculate and to live,” said Condorcet at a mournful meeting of the Paris Academy of Sciences (fr. 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 reads: “Here rest the remains of the world-famous Leonhard Euler, sage and righteous man. Born April 4, 1707 in Basel, he died September 7, 1783. After Euler”s death, his grave was lost and was not found, in a derelict state, until 1830. In 1837 the Academy of Sciences replaced this tombstone with a new granite tombstone (still standing) with the inscription in Latin “Leonhard Euler – Academia Petropolitana” (lat. Leonhardo Eulero – Academia Petropolitana).

During the celebration of the 250th anniversary of Euler (1957) the ashes of the great mathematician were transferred to the “Necropolis of the 18th century” in the Lazarevsky cemetery of the Alexander Nevsky Lavra, where they are located close to the tomb of M. V. Lomonosov.

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, the theory of music, and many European and ancient languages.

Euler willingly participated in scientific discussions, of which the most famous were:

In all the cases mentioned, Euler”s position is supported by modern science.

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## Mathematics

In terms of mathematics, the 18th century is the age of Euler. Whereas before him advances in mathematics had been scattered and not always coordinated, Euler for the first time linked analysis, algebra, geometry, trigonometry, number theory, and other disciplines into a unified system, while adding many of his own discoveries. Much of mathematics has been taught since then “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 by integer modulo, the complete theory of continuous fractions, the analytic foundation of mechanics, numerous techniques of integration and solution of differential equations, the number e, the notation i for the imaginary unit, a number of special functions and much more.

In fact, it was Euler 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 include Diophantine analysis, mathematical physics, statistics, etc.

The historian of science Clifford Truesdell wrote: “Euler was the first scientist in Western civilization to write about mathematics in clear and easy-to-read language. Biographers note that Euler was a virtuoso algorithmicist. He always tried to bring his discoveries to the level of specific computational methods, and was a master of numerical calculations. J. Condorcet said that once two students, doing independently complex astronomical calculations, got slightly different results in the 50th sign and turned to Euler for help. Euler did the same calculations in his mind and stated the correct result.

П. L. Chebyshev wrote: “Euler was the beginning of all the investigations that make up the general theory of numbers. Most of the mathematicians of the 18th century were engaged in the development of analysis, but Euler carried the passion for ancient arithmetic through his entire life. Thanks to his writings, interest in number theory was revived by the end of the century.

Euler continued the research of Fermat, who had previously made (under the influence of Diophantus) a number of scattered hypotheses about natural numbers. Euler rigorously proved these hypotheses, generalized them considerably and combined them into a meaningful theory of numbers. He introduced the extremely important “Euler function” into mathematics and used it to formulate the “Euler theorem”. He disproved Fermat”s hypothesis that all numbers of the form Fn=22n+1{displaystyle F_{n}=2^{2^{n}}+1} are prime; it turned out that F5{displaystyle F_{5}} is divisible by 641. Proved Fermat”s statement about the representation of an odd prime number as a sum of two squares. Gave one of the solutions to the problem of four cubes. Proved that Mersenne”s number 231-1=2147483647{displaystyle 2^{31}-1=2147483647} is a prime number; it remained the largest known prime number for almost a hundred years (until 1867).

Euler created the basis for the theory of comparisons and quadratic deductions, specifying the solvability criterion for the latter. Euler introduced the notion of an original root and hypothesized that for any prime number p there is an original root modulo p; he could not prove this; later LeGendre and Gauss proved the theorem. Euler”s other hypothesis, the quadratic law of reciprocity, also proved by Gauss, was of great importance in the theory. Euler proved Fermat”s Grand Theorem for n=3{displaystyle n=3} and n=4{displaystyle n=4}, created a complete theory of continuous fractions, studied various classes of diophantine equations, and the theory of division of numbers into terms.

In the problem about the number of partitions of a natural number n{displaystyle n}, I got a formula that expresses the derivative function of the number of partitions p(n){displaystyle p(n)} through the infinite product

Euler defined the zeta function, a generalization of which was later named Riemann:

where s{displaystyle displaystyle s} is a real number (Riemann”s is complex). Euler derived a decomposition for it:

where the product is taken over all prime numbers p{displaystyle displaystyle p}. In this way he discovered that in number theory it is possible to apply methods of mathematical analysis, laying the foundation of analytic number theory, which is based on Euler”s identity and the general method of derivative functions.

One of Euler”s main services to science was his monograph “Introduction to the Analysis of Infinitesimals” (1748). In 1755 a supplemented Differential Calculus was published, and in 1768-1770 three volumes of the Integral Calculus were published. Taken together it is a fundamental, well-illustrated course, with elaborate terminology and symbolism. “It is safe to say that a good half of what is now taught in courses of higher algebra and higher analysis is in Euler”s writings” (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 integration of rational functions by decomposing them into simple fractions and the method of solving differential equations of arbitrary order with constant coefficients.

Euler always paid special attention to methods for solving differential equations, both ordinary and partial derivatives, having discovered and described important classes of integrable differential equations. He described Euler”s “method of broken lines” (1768), a numerical method for solving systems of ordinary differential equations. At the same time with A. C. Clero Euler derived conditions of integrability of linear differential forms of two or three variables (1739). He obtained serious results in the theory of elliptic functions, including the first theorems of addition of elliptic integrals (1761). First investigated maxima and minima of functions of many variables.

The basis of natural logarithms has been known since the days of Neper and Jacob Bernoulli, but Euler made such a profound study of this most important constant that it has been named after him ever since. Another constant he studied: the Euler-Mascheroni constant.

The modern definition of the exponential, logarithmic, and trigonometric functions is also his merit, as well as their symbolism and generalization to the complex case. The formulas often referred to in textbooks as “Cauchy-Riemann conditions” would more correctly be called “D”Alambert-Euler conditions”.

He shares with Lagrange the honor of discovering calculus of variations by writing out the Euler-Lagrange equations for the general variational problem. In 1744 Euler published his treatise “The Method of Finding Curves…” – the first work on calculus of variations (among other things, it contained the first systematic exposition of the theory of elastic curves and results on resistance of materials).

Euler greatly advanced the theory of series and extended it to the complex domain, giving the famous Euler formula that gives the trigonometric representation of the complex number. The mathematical world was greatly impressed by the series first summed up by Euler, including the inverse square series which no one had been able to do before him:

With the help of series Euler investigated transcendental functions, that is, those functions that are not expressed by an algebraic equation (for example, the integral logarithm). He discovered (1729-1730) the “Euler integrals” – special functions that entered science as the gamma and beta Euler functions – which now had a wide range of applications. In 1764, when solving the problem of the oscillations of an elastic membrane (which arose in connection with determining the pitch of kettledrums), Euler was the first to introduce Bessel functions for any natural index (a study by F. W. Bessel, whose name these functions bear today, refers to 1824).

From a later point of view, Euler”s actions with infinite series could not always be considered correct (the substantiation of the analysis was not carried out until half a century later), but his phenomenal mathematical intuition almost always suggested the correct result. At the same time, in many important respects his understanding was ahead of his time-for example, his proposed generalized understanding of the sum of divergent series and operations with them served as the basis for the modern theory of these series developed in the late 19th and early 20th centuries.

In elementary geometry Euler discovered several facts not noted by Euclid:

The second volume of Introduction to the Analysis of Infinitesimals (1748) was the world”s first textbook on analytic geometry and the foundations of differential geometry. Euler gave a classification of algebraic curves of the 3rd and 4th orders as well as surfaces of the second order. The term “affine transformations” was first introduced in this book along with the theory of such transformations. In 1732, Euler derived the general equation of geodesic lines on a surface.

In 1760 the fundamental “Investigations on the Curvature of Surfaces” was published. Euler discovered that at each point of a smooth surface there are two normal sections with minimum and maximum radii of curvature and that their planes are mutually perpendicular. He derived a formula for the relation between the curvature of the surface section and the principal curvatures.

In 1771, Euler published his work “On Bodies Whose Surface Can Be Unwrapped on a Plane”. This work introduces the notion of an unfoldable surface, that is, a surface that can be superimposed on a plane without folds or discontinuities. Euler, however, gives here a quite general theory of metrics, on which all the internal geometry of the surface depends. Later he makes the study of metrics the main tool of surface theory.

In connection with the tasks of cartography, Euler investigated conformal mappings in depth, applying for the first time the tools of complex analysis for this purpose.

Euler paid much attention to the representation of natural numbers as sums of a special kind 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 investigated algorithms for the construction of magic squares by chess-horse traversal. Two of his works (1776, 1779) laid the foundation for the general theory of Latin and Greek-Latin squares, whose great practical value became clear after Ronald Fisher created methods for planning experiments, as well as in the theory of error-correcting codes.

Euler”s 1736 article “Solutio problematis ad geometriam situs pertinentis” (Solutio problematis ad geometriam situs pertinentis) marked the beginning of graph theory as a mathematical discipline. The reason for the study was the problem of the Königsberg bridges: can one cross each bridge once and return to the original place? Euler formalized it by reducing it to the problem of the existence in a graph (whose vertices correspond to parts of the city, separated by branches of the Pregolya River, and the edges to bridges) of a cycle, or of a path, passing through each edge exactly once (in modern terminology, the Eulerian cycle and Eulerian path respectively). Solving the latter problem Euler showed that for an Eulerian cycle to exist in a graph its degree (the number of edges leaving the vertex) must be even for each vertex, and the Eulerian path must be even for all but two (in the problem about Koenigsberg bridges this is not so: degrees are 3, 3, 3 and 5).

Euler made a significant contribution to the theory and methods of approximate computation. He was the first to apply analytical methods to cartography. He proposed a convenient method of graphical representation of relations and operations on sets, called “Euler Circles” (or Euler-Vennes).

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## Mechanics and Physics

Many of Euler”s works are devoted to various branches of mechanics and physics. On the key role of Euler in shaping mechanics into an exact science, C. Truesdell wrote: “Mechanics, as it is taught to engineers and mathematicians today, is largely his creation.

In 1736 Euler”s two-volume treatise “Mechanics, or the Science of Motion, in an Analytical Statement” was published, which marked a new stage in the development of this ancient science and was devoted to the dynamics of the material point. In contrast to the founders of this section of dynamics – Galileo and Newton, who used geometric methods, the 29-year-old Euler proposed a regular and uniform analytical method for solving various problems of dynamics: the preparation of differential equations of motion of a material object and their subsequent integration under given initial conditions.

In the first volume of the treatise the motion of a free material point is considered, in the second – of a proprietary one, and the motion both in a void and in a resisting medium is investigated. The problems of ballistics and the theory of the pendulum are considered separately. Here Euler for the first time writes down the differential equation of rectilinear motion of a point, and for the general case of its curvilinear motion he introduces the natural equations of motion – equations in projections on the axes of the accompanying trihedron. In many specific problems he brings the integration of equations of motion to the end; in the cases of point motion without resistance he systematically uses the first integral of equations of motion – the integral of energy. In the second volume, in connection with the problem of the motion of a point on an arbitrarily curved surface, the differential geometry of surfaces created by Euler is presented.

Euler returned to the dynamics of the material point even later. In 1746, investigating the motion of a material point on a moving surface, he came (simultaneously with D. Bernoulli and P. Darcy) to the theorem on the change in angular momentum. In 1765 Euler, using the idea put forward in 1742 by C. McLaren of decomposition of velocities and forces along three fixed coordinate axes, for the first time wrote down the differential equations of motion of a material point in projections on the Cartesian fixed axes.

The last result was published by Euler in his second fundamental treatise on analytical dynamics – the book “Theory of motion of solids” (1765). Its main content, however, is devoted to another section of mechanics – the dynamics of solids, of which Euler was the founder. The treatise, in particular, contains the derivation of a system of six differential equations of motion of a free solid body. The theorem about the reduction of the system of forces applied to a solid body to two forces, stated in § 620 of the treatise, is of great importance for statics. By projecting the conditions of equality of these forces to zero onto the coordinate axes, Euler for the first time obtains the equations of equilibrium of a solid body under the action of an arbitrary spatial system of forces.

A number of Euler”s fundamental results related to the kinematics of solids (in the 18th century, kinematics was not yet distinguished as a separate section of mechanics) are also stated in the treatise of 1765. Among them, we can single out Euler”s formulas for the distribution of velocities of points of an absolutely solid body (the vector equivalent of these formulas is the Euler kinematic formula) and the kinematic Euler equations, which give the expression of derivatives of Euler angles (used in mechanics to specify the orientation of a solid body) through the projections of angular velocity on coordinate axes.

In addition to this treatise, two earlier works by Euler are important for the dynamics of solids: “Studies on the Mechanical Knowledge of Bodies” and “The Rotational Motion of Solids about a Variable Axis”, which were submitted to the Berlin Academy of Sciences in 1758, but were published in its “Notes” later (in the same 1765 as the treatise). In them: the theory of moments of inertia was developed (the existence of at least three axes of free rotation in any solid body with a fixed point was established; the Euler dynamic equations describing the dynamics of a solid body with a fixed point were obtained; the analytical solution of these equations was given in the case of equality to zero the main moment of external forces (Euler case) – one of three general cases of integrability in the problem of the dynamics of a heavy solid body with a fixed point.

In his article “General Formulas for the Arbitrary Movement of a Solid Body” (1775), Euler formulates and proves Euler”s fundamental rotation theorem, according to which the arbitrary movement of a perfectly solid body with a fixed point is a rotation by some angle around an axis passing through the fixed point.

Euler is credited with the analytical formulation of the principle of least action (proposed in 1744 – in a very fuzzy form – by P. L. Mauperthuis), the correct understanding of the conditions of applicability of the principle and its first proof (carried out in the same year of 1744 for the case of one material point moving under the action of a central force). The action here (it is the so-called shortened action, not the Hamiltonian action) with respect to a system of material points means the integral

where A{displaystyle A} and B{displaystyle B}} are two configurations of the system, mi,vi{displaystyle m_{i},;v_{i}} and dsi{displaystyle mathrm {d} s_{i}} – mass, algebraic velocity, and arc element of trajectory i{displaystyle i}th point, n{displaystyle n} is the number of points, respectively.

As a result, the Mauperthuis-Euler principle entered science, the first in the series of integral variational principles of mechanics; later this principle was generalized by J. L. Lagrange, and now it is usually treated as one of the forms (the Mauperthuis-Euler form, considered along with the Lagrange form and the Jacobi form) of the Mauperthuis-Lagrange principle. Despite his defining contribution, in the discussion that arose around the principle of least action, Euler strongly defended the priority of Mauperthuis and pointed out the fundamental importance of this principle in mechanics. This idea attracted the attention of physicists, who in the nineteenth and twentieth centuries figured out the fundamental role of variational principles in nature and applied the variational approach in many parts of their science.

A number of Euler”s works are devoted to mechanics of machines. In his memoir “On the Most Advantageous Application of Simple and Complex Machines” (1747), Euler proposed to study machines not at rest but in a state of motion. This new, “dynamic” approach Euler justified and developed in his memoir “On Machines in General” (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, transmission and working organ. In his memoir “Principles of the Theory of Machines” (1763), Euler showed that when calculating the dynamic characteristics of machines in 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 the components of the machine, and he gives (in relation to hydraulic engines) an example of such calculation.

Euler also dealt with applied questions of the theory of mechanisms and machines: questions of the theory of hydraulic machines and windmills, the study of friction of machine parts, questions of gear profiling (here he substantiated and developed the analytical theory of involute gearing). In 1765, he laid the foundations of the theory of friction of flexible cables and obtained, in particular, the Euler formula for determining the tension of the cable, which is still used in solving a number of practical problems (for example, in the calculation of mechanisms with flexible links).

Euler”s name is also associated with the consistent introduction into mechanics of the idea of the continuum, according to which a material body is represented, abstracting from its molecular or atomic structure, as a continuous continuous continuous medium. The continuum model was introduced by Euler in his memoir “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 statement that is still cited in many textbooks on mechanics and physics (often without mentioning Euler”s name): a continuous body can be modeled with any degree of accuracy by breaking it down mentally into small enough particles and treating each of them as a material point. Based on this principle, it is possible to obtain those or other dynamical relations for a solid body by writing down their analogues for individual material particles (in Euler”s terms, “corpuscles”) and summing them up by the summation over all points by integration over the volume of the area occupied by the body. This approach allowed Euler to avoid using such means of modern integral calculus (such as the Stiltjes integral), which were not yet known in the 18th century.

Based on this principle, Euler obtained – applying the theorem on the change of momentum to an elementary material volume – Euler”s first law of motion (later appeared the second Euler”s law of motion – the result of applying the theorem on the change of momentum). Euler”s laws of motion in fact constituted the basic laws of motion of continuum mechanics; for the transition to the currently used general equations of motion for such media, all that was missing was the expression of surface forces through the stress tensor (this was done by O. Cauchy in the 1820s). Euler applied the results obtained to the study of specific models of solid bodies – both in the dynamics of solid bodies (it was in the mentioned memoir that the equations of dynamics of a body with a fixed point, referred to arbitrary Cartesian axes, were first given), and in fluid dynamics, and in the theory of elasticity.

In the theory of elasticity, a number of Euler”s studies are devoted to the theory of bending of beams and rods; in his early works (1740s) he deals with the problem of longitudinal bending of an elastic rod, composing and solving the differential equation of the bent axis of the rod. In 1757, in his work “On the loading of columns”, Euler, for the first time in history, obtained the formula for determining the critical load in compression of an elastic rod, giving rise to the theory of stability of elastic systems. This formula found practical application much later, almost a hundred years later, when many countries (primarily, England) were engaged in the construction of railroads, which required calculations of the strength of railway bridges; it was at this time that engineers adopted – after some refinement – Euler”s model.

Euler is – along with D. Bernoulli and J. L. Lagrange – one of the founders of analytical fluid dynamics; here he is credited with creating the theory of motion of an ideal fluid (that is, a fluid with no viscosity) and solving a number of specific problems of fluid mechanics. In his work “Principles of motion of fluids” (published nine years later) he, applying his equations of dynamics of an elementary material volume of a continuous medium to the model of an incompressible perfect fluid, for the first time obtained for such a fluid the equations of motion, as well as (for the general three-dimensional case) the continuity equation. Studying the vortexless motion of an incompressible fluid, Euler introduced the function S{displaystyle S} (later called by H. Helmholtz the velocity potential) and showed that it satisfies the partial differential equation – thus science entered the equation that is now known as the Laplace equation.

Euler significantly generalized the results of this work in his treatise “General Principles of Fluid Motion” (1755). Here, for the case of a compressible ideal fluid, he presented (practically in modern notations) the continuity equation and equations of motion (three scalar differential equations, to which the Euler equation, the basic equation of hydrodynamics of an ideal fluid, corresponds in vector form). Euler noted that in order to close this system of four equations, a defining relation is needed to express the pressure p{displaystyle p} (which Euler called “elasticity”) as a function of density q{displaystyle q} and “another property r{displaystyle r}, which affects elasticity” (actually what was meant was temperature). Discussing the possibility of existence of non-potential motions of an incompressible fluid, Euler gave the first concrete example of its vortex flow, and for potential motions of such a fluid he obtained the first integral – a special case of the now well-known Lagrange-Cauchy integral.

The same year also includes Euler”s memoir “General Principles of Fluid Equilibrium State,” which contained a systematic presentation of the hydrostatics of an ideal fluid (including the derivation of the general equation of equilibrium of liquids and gases) and derived a barometric formula for an isothermal atmosphere.

In the above papers Euler, writing down equations of motion and equilibrium of a fluid, took as independent spatial variables the Cartesian coordinates of the current position of a material particle – Euler variables (D”Alambert was the first to use such variables in hydrodynamics). Later, in the work “On the Principles of Motion of Fluids. Section Two” (1770), Euler also introduced the second form of the equations of hydrodynamics, in which the Cartesian coordinates of the position of a material particle at the initial moment of time (now known as Lagrange variables) were taken as independent spatial variables.

The main achievements in this field Euler compiled in the three-volume Dioptrica (Latin: Dioptrica, 1769-1771). Among the main results: rules for calculating the optimal characteristics of refractors, reflectors, and microscopes, calculating the greatest image brightness, the greatest field of view, the shortest instrument length, the greatest magnification, and eyepiece characteristics.

Newton argued that creating an achromatic lens was fundamentally impossible. Euler argued that a combination of materials with different optical characteristics could solve the problem. In 1758 Euler, after a long polemic, managed to convince the English optician John Dollond, who then made the first achromatic lens by joining together two lenses made of glasses of different composition, and in 1784 Academician F. Epinus in St. Petersburg built the world”s first achromatic microscope.

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## Astronomy

Euler worked extensively in the field of celestial mechanics. One of the urgent tasks at that time was to determine the parameters of the orbit of a celestial body (such as 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; these works were relied on by Gauss, who gave the final solution of the problem.

Euler laid the foundations of perturbation theory, later completed by Laplace and Poincaré. He introduced the fundamental notion of orbital oscillating elements and derived differential equations determining their change with time. He constructed the theory of precession and nutation of the Earth”s axis, and predicted the “free movement of the poles” of the Earth, discovered a century 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 investigated the dependence of refraction on pressure and air temperature at the place of observation. These results had a great influence on the development of astronomy in the following years.

Euler set forth a very precise theory of the motion of the Moon, developing for this purpose a special method of variation of orbital elements. Subsequently, in the 19th century, this method was extended and applied to the model of the motion of the large planets and is still in use today. Mayer”s tables, calculated on the basis of Euler”s theory (1767), also proved suitable for solving the urgent problem of determining longitude at sea, and the British Admiralty paid a special prize to Mayer and Euler for it. Euler”s main works in this field:

Euler investigated 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 tilt of the ecliptic plane (1756), and at his suggestion the tilt at the beginning of 1700 has since been taken as a reference. He developed the basics of the theory of motion of the satellites of Jupiter and other strongly compressed planets.

In 1748, long before the work of P.N. Lebedev, Euler hypothesized that comet tails, auroras and zodiacal light have in common the influence of solar radiation on the atmosphere or substance of celestial bodies.

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## Music Theory

All his life Euler was interested in musical harmony, striving to give it a clear mathematical foundation. The aim of his early work, Tentamen novae theoriae musicae (Tentamen novae theoriae musicae, 1739), was to attempt to describe mathematically how pleasant (euphonious) music differs from unpleasant (unpleasant) music. At the end of chapter VII of the “Experiment”, Euler arranged the intervals according to “degrees of pleasantness” (gradus suavitatis), with the octave being assigned to II (some classes (including first, third and sixth) in Euler”s table of pleasantness were omitted. There was a joke about this work that it contained too much music for mathematicians and too much mathematics for musicians.

In his late years, in 1773, Euler delivered a paper at the St. Petersburg Academy of Sciences in which he formulated in final form 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 small treatise De harmoniae veris principiis per speculum musicum repraesentatis (“On the true foundations of harmony presented through speculum musicae”). Under the name of the “sound network” (German: Tonnetz), the Eulerian grid was widely used in 19th century German musical theory.

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## Other areas of expertise

In 1749 Euler published a two-volume monograph, “The Science of the Sea, or a Treatise on Shipbuilding and Ship Navigation,” in which he applied analytical methods to the practical problems of shipbuilding and navigation at sea, such as the shape of ships, questions of stability and equilibrium, and methods of controlling ship motion. Krylov”s general theory of ship stability 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 flow in vessels. In 1742 he sent to the Academy of Dijon an article on the flow of fluids in elastic tubes (regarded as models of vessels), and in December 1775 he presented to the Petersburg Academy of Sciences a memoir entitled Principia pro motu sanguines per arteria determinando (Principles of determining the movement of blood through arteries). 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 obtaining general equations of finite motion.

One of the main tasks assigned to Euler upon his arrival in Russia was the training of scientific personnel. Among Euler”s direct pupils:

One of Euler”s priorities was the creation of textbooks. He himself wrote “A Manual of Arithmetic for use in the gymnasium of the Imperial Academy of Sciences” (1738-1740), “Universal Arithmetic” (1768-1769). Euler, according to Fuss, resorted to an original technique – he dictated the textbook to a boy-servant, watching how he understood the text. As a result, the boy learned to solve problems and do calculations on his own

In honor of Euler are named:

The Complete Works of Euler, published since 1909 by the Swiss Society of Naturalists, is still incomplete; 75 volumes were planned, of which 73 were published:

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

In 1907 Russian and many other scientists celebrated the 200th anniversary of the great mathematician, and in 1957 the Soviet and Berlin Academies of Sciences dedicated solemn sessions to his 250th birthday. On the eve of Euler”s 300th birthday (2007) an international anniversary forum was held in St. Petersburg, and a film about Euler”s life was made. The same year a monument to Euler was unveiled in St Petersburg at the entrance to the International Euler Institute. The authorities of St. Petersburg, however, rejected all proposals to name a square or a street after the scientist; there is still not a single street in Russia named after Euler.

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## Personal qualities and grades

According to his contemporaries, Euler”s character was good-natured, not malicious, almost never quarrelled with anyone. Even Johann Bernoulli, whose hard character was experienced by his brother Jacob and his son Daniel, was unfailingly warm to him. Euler needed only one thing for the fullness of life – the possibility of regular mathematical creativity. He could work intensely even “with a child on his lap and a cat on his back.” At the same time Euler was cheerful, sociable, loved music, philosophical conversations.

Academician P. P. Pekarsky, based on the testimonies of Euler”s contemporaries, reconstructed the image of the scholar: “Euler had the great art of not flaunting his scholarship, hiding his superiority and being at everyone”s level. Always an even mood, mild and natural cheerfulness, some sneer with a touch of good-naturedness, naive and humorous conversation – all this made the conversation with him as pleasant as it was attractive.

As contemporaries note, Euler was very religious. According to Condorcet, every evening Euler would gather his children, servants, and students who lived with him to pray. 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 fascination with theological reasoning became the reason for the negative attitude to him (as a philosopher) of his famous contemporaries – D”Alamber and Lagrange. Frederick II, who considered himself a “freethinker” and corresponded with Voltaire, said that Euler “reeked of pope.

Euler was a caring family man, eager to help his colleagues and young people, and generously shared his ideas with them. It is well known that Euler delayed his publications on calculus of variations so that a young and then unknown Lagrange, who had independently arrived at the same discoveries, could publish them first. Lagrange always admired Euler both as a mathematician and as a man; he said, “If you really love mathematics, read Euler.

“Read, read Euler, he is our common teacher,” Laplace liked to repeat (Fr. Lisez Euler, lisez Euler, c”est notre maître à tous.). The works of Euler were studied with great benefit by the “king of mathematicians” Karl Friedrich Gauss, and almost all the famous scientists of the 18th and 19th centuries.

D”Alambert, in one of his letters to Lagrange, calls Euler “that devil” (frès se diable d”homme), as if wishing to indicate by this, according to commentators, that what Euler had done exceeded human power.

М. V. Ostrogradsky stated in a letter to N. N. Fuss: “Euler created modern analysis, enriched it alone more than all his followers put together, and made it the most powerful instrument of human reason. Academician S. I. Vavilov wrote: “Together with Peter I and Lomonosov, Euler became the good genius of our Academy, who determined its glory, its fortress, its productivity.

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## Addresses of residence

In 1743-1766, Euler lived in the house at 21 Berenstrasse

From 1766, Euler lived in an apartment building at 15 Nikolayevskaya Embankment (with an interruption caused by a major fire). In Soviet times, the street was renamed Lieutenant Schmidt Embankment. There is a memorial plaque on the house, and it now houses a high school.

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## Stamps, coins, banknotes

In 2007, the Russian Central Bank issued a commemorative coin to mark the 300th anniversary of L. Euler”s birth. Euler”s portrait was also placed on the Swiss 10-franc banknote (6th series) and on postage stamps of Switzerland, Russia and Germany.

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## Math Olympiads

Very many facts in geometry, algebra, and combinatorics proved by Euler are used everywhere in Olympiad mathematics.

On April 15, 2007 an Internet Olympiad for schoolchildren in mathematics was held in honor of the 300th anniversary of the birth of Leonhard Euler, which was supported by a number of organizations. Since 2008 the Leonhard Euler Mathematical Olympiad for eighth-graders has been held to partially replace the loss of the regional and final stages of the All-Russian Mathematical Olympiad for eighth-graders.

Historians have discovered only more than a thousand direct descendants of Leonhard Euler. The eldest son Johann Albrecht became a major mathematician and physicist. The second son Karl was a famous physician. Younger son Christopher was later 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 whom lived in the USSR.

Here is a brief genealogical tree of some of Euler”s known descendants (the last name is given if it is not “Euler”).

Other descendants of Euler include N. I. Gekker, V. F. Gekker and I. R. Gekker, V. E. Scalon, and E. N. Behrendts. Among the descendants are many scientists, geologists, engineers, diplomats, doctors, there are also nine generals and one admiral. A descendant of Euler is D. A. Shestakov, president of the St. Petersburg International Criminology Club.

**Sources**