Euler scientific works. Leonhard Euler interesting facts

During the existence of the Academy of Sciences in Russia, apparently one of its most famous members was the mathematician Leonhard Euler (1707-1783).

He became the first who began to build a consistent edifice of infinitesimal analysis in his works. Only after his research, set out in the grandiose volumes of his trilogy “Introduction to Analysis”, “Differential Calculus” and “Integral Calculus”, did analysis become a fully formed science - one of the most profound scientific achievements of mankind.

Leonhard Euler was born in the Swiss city of Basel on April 15, 1707. His father, Pavel Euler, was a pastor in Riechen (near Basel) and had some knowledge of mathematics. The father intended his son for a spiritual career, but he himself, being interested in mathematics, taught it to his son, hoping that it would later be useful to him as an interesting and useful activity. After finishing his home schooling, thirteen-year-old Leonard was sent by his father to Basel to listen to philosophy.

Among other subjects, elementary mathematics and astronomy were studied at this faculty, taught by Johann Bernoulli. Soon Bernoulli noticed the talent of the young listener and began to study with him separately.

Having received his master's degree in 1723, after delivering a speech in Latin on the philosophy of Descartes and Newton, Leonard, at the request of his father, began to study oriental languages ​​and theology. But he was increasingly attracted to mathematics. Euler began to visit his teacher’s house, and between him and the sons of Johann Bernoulli - Nikolai
Daniil - a friendship arose that played a very important role in Euler’s life.

In 1725, the Bernoulli brothers were invited to become members of the St. Petersburg Academy of Sciences, recently founded by Empress Catherine I. When leaving, Bernoulli promised Leonard to notify him if there was a suitable occupation for him in Russia. The following year they reported that there was a place for Euler, but, however, as a physiologist in the medical department of the academy. Upon learning of this, Leonard immediately enrolled as a medical student at the University of Basel. Studying diligently and successfully
Science Faculty of Medicine, Euler also finds time for mathematical studies. During this time, he wrote a dissertation on the propagation of sound and a study on the placement of masts on a ship, which was later published in 1727 in Basel.

In St. Petersburg there were the most favorable conditions for the flowering of Euler's genius: material security, the opportunity to do what he loved, the presence of an annual magazine for publishing works. The largest group of specialists in the field of mathematical sciences in the world at that time worked here, which included Daniel Bernoulli (his brother Nicholas died in 1726), the versatile H. Goldbach, with whom Euler shared common interests in number theory and other issues, the author of works in trigonometry F.Kh. Mayer, astronomer and geographer J.N. Delisle, mathematician and physicist G.V. Kraft and others. Since that time, the St. Petersburg Academy has become one of the main centers of mathematics in the world.

Euler's discoveries, which thanks to his lively correspondence often became known long before publication, make his name increasingly widely known. His position in the Academy of Sciences improved: in 1727 he began work with the rank of adjunct, that is, a junior academician, and in 1731 he became a professor of physics, that is, a full member of the Academy. In 1733 he received the chair of higher mathematics, which was previously occupied by D. Bernoulli, who returned the same year to Basel. The growth of Euler's authority was uniquely reflected in the letters to him from his teacher Johann Bernoulli. In 1728, Bernoulli addressed “the most learned and gifted young man, Leonhard Euler,” in 1737, “the most famous and witty mathematician,” and in 1745, “the incomparable Leonhard Euler, the leader of mathematicians.”

In 1735, the Academy needed to carry out a very difficult job of calculating the trajectory of a comet. According to academics, this required several months of labor. Euler undertook to do this in three days and completed the work, but as a result he fell ill with nervous fever with inflammation of his right eye, which he lost. Soon after this, in 1736, two volumes of his analytical mechanics appeared. The need for this book was great; Many articles were written on various issues of mechanics, but there was no good treatise on mechanics.

In 1738, two parts of an introduction to arithmetic appeared in German, and in 1739, a new theory of music. Then in 1840 Euler wrote an essay on the ebb and flow of the seas, which was awarded one-third of the prize of the French Academy; the other two thirds were awarded to Daniel Bernoulli and Maclaurin for essays on the same topic.

At the end of 1740, power in Russia fell into the hands of regent Anna Leopoldovna and her entourage. An alarming situation has developed in the capital. At this time, the Prussian king Frederick II decided to revive the Society of Sciences in Berlin, founded by Leibniz, which had been almost inactive for many years. Through his ambassador in St. Petersburg, the king invited Euler to Berlin. Euler, believing that “the situation began to seem quite
unsure,” accepted the invitation.

In Berlin, Euler first gathered a small scientific society around him, and then was invited to join the newly restored Royal Academy of Sciences and was appointed dean of the mathematical department. In 1743, he published five of his memoirs, four of them on mathematics. One of these works is remarkable in two respects. It indicates a way to integrate rational fractions by decomposing them into
partial fractions and, in addition, the now usual method of integrating linear ordinary equations of higher order with constant coefficients is presented.

In general, most of Euler's works are devoted to analysis. Euler so simplified and supplemented entire large sections of the analysis of infinitesimals, integration of functions, the theory of series, differential equations, begun before him, that they acquired approximately the form that they occupied to a large extent remains to this day. Euler, in addition, began a whole new chapter of analysis - the calculus of variations. This initiative of his was soon picked up by Lagrange and thus a new science was formed.

In 1744, Euler published three works in Berlin on the movement of luminaries: the first is the theory of the movement of planets and comets, which contains a statement of the method for determining orbits from several observations; the second and third are about the movement of comets.

Euler devoted seventy-five works to geometry. Some of them, although interesting, are not very important. Some simply made up an era. Firstly, Euler should be considered one of the founders of research on geometry in space in general. He was the first to give a coherent presentation of analytical geometry in space (in “Introduction to Analysis”) and, in particular, introduced the so-called Euler angles, which make it possible to study rotations
bodies around a point.

In his 1752 work, “Proof of certain remarkable properties to which bodies bounded by plane faces are subject,” Euler found a relationship between the number of vertices, edges, and faces of a polyhedron: the sum of the number of vertices and faces is equal to the number of edges plus two. This relationship was suggested by Descartes, but Euler proved it in his memoirs. This is, in a sense, the first major theorem in the history of mathematics of topology - the deepest part of geometry.

While studying questions about the refraction of light rays and having written many memoirs on this subject, Euler published an essay in 1762 in which he proposed the design of complex lenses to reduce chromatic aberration. The English artist Doldond, who discovered two types of glass with different refrangibility, following Euler's instructions, built the first achromatic lenses.

In 1765, Euler wrote an essay in which he solves differential equations for the rotation of a rigid body, which are called the Euler equations for the rotation of a rigid body.

The scientist wrote many essays on the bending and vibration of elastic rods. These questions are interesting not only mathematically, but also practically.

Frederick the Great gave the scientist instructions of a purely engineering nature. So, in 1749, he instructed him to inspect the Funo Canal between Havel and Oder and make recommendations for correcting the shortcomings of this waterway. Next he was tasked with fixing the water supply in Sans Souci.

This resulted in more than twenty memoirs on hydraulics, written by Euler at different times. First-order hydrodynamic equations with partial derivatives from the projections of velocity, density and pressure are called Euler hydrodynamic equations.

After leaving St. Petersburg, Euler retained the closest connection with the Russian Academy of Sciences, including the official one: he was appointed an honorary member, and he was given a large annual pension, and he, for his part, assumed obligations regarding further cooperation. He purchased books, physical and astronomical instruments for our Academy, selected employees in other countries, reporting detailed characteristics of possible candidates, edited the mathematical department of academic notes, acted as an arbiter in scientific
disputes between St. Petersburg scientists, sent topics for scientific competitions, as well as information about new scientific discoveries, etc. Students from Russia lived in Euler’s house in Berlin: M. Sofronov, S. Kotelnikov, S. Rumovsky, the latter later became academicians.

From Berlin, Euler, in particular, corresponded with Lomonosov, in whose work he highly valued the happy combination of theory and experiment. In 1747, he gave a brilliant review of Lomonosov’s articles on physics and chemistry sent to him for conclusion, which greatly disappointed the influential academic official Schumacher, who was extremely hostile to Lomonosov.

In Euler's correspondence with his friend Goldbach, an academician of the St. Petersburg Academy of Sciences, we find two famous “Goldbach problems”: to prove that every odd natural number is the sum of three prime numbers, and every even number is the sum of two. The first of these statements was proven using a very remarkable method already in our time (1937) by Academician I.M. Vinogradov, but the second has not been proven to this day.

Euler was drawn back to Russia. In 1766, through the ambassador in Berlin, Prince Dolgorukov, he received an invitation from Empress Catherine II to return to the Academy of Sciences on any terms. Despite persuasion to stay, he accepted the invitation and arrived in St. Petersburg in June.

The Empress provided Euler with funds to buy the house. The eldest of his sons, Johann Albrecht, became an academician in the field of physics, Karl took a high position in the medical department, Christopher, born in Berlin, was not released from military service by Frederick II for a long time, and it took the intervention of Catherine II so that he could come to his father. Christopher was appointed director of the Sestroretsk Armory
plant

Back in 1738, Euler went blind in one eye, and in 1771, after an operation, he almost completely lost his sight and could only write with chalk on a black board, but thanks to his students and assistants. I.A Euler, A I. Loksel, V.L. Kraft, S.K. Kotelnikov, M.E. Golovin, and most importantly N.I. Fuss, who arrived from Basel, continued to work no less intensively than before.

Euler, with his brilliant abilities and remarkable memory, continued to work and dictate his new memoirs. From 1769 to 1783 alone, Euler dictated about 380 articles and essays, and during his life he wrote about 900 scientific papers.

Euler's 1769 paper "On Orthogonal Trajectories" contains brilliant ideas about obtaining, using a function of a complex variable, from the equations of two mutually orthogonal families of curves on a surface (that is, lines such as meridians and parallels on a sphere) an infinite number of other mutually orthogonal families. This work turned out to be very important in the history of mathematics.

In his next work of 1771, “On bodies whose surface can be turned into a plane,” Euler proves the famous theorem that any surface that can be obtained only by bending a plane, but without stretching or compressing it, if it is not conical or cylindrical , is a set of tangents to some spatial curve.

Euler's work on map projections is equally remarkable.

One can imagine what a revelation Euler’s work on the curvature of surfaces and developable surfaces was for mathematicians of that era. The works in which Euler studies surface mappings that preserve similarity in the small (conformal mappings), based on the theory of functions of a complex variable,
should have seemed downright transcendental. And the work on polyhedra began a completely new part of geometry and, in its principles and depth, stood alongside the discoveries of Euclid.

Euler's tirelessness and perseverance in scientific research were such that in 1773, when his house burned down and almost all of his family's property was destroyed, even after this misfortune he continued to dictate his research. Soon after the fire, a skilled ophthalmologist, Baron Wentzel, performed cataract surgery, but Euler could not stand the appropriate time without reading and became completely blind.

Also in 1773, Euler's wife, with whom he lived for forty years, died. Three years later, he married her sister, Salome Gsell. Enviable health and a happy character helped Euler “withstand the blows of fate that befell him. Always an even mood, soft and natural cheerfulness, some kind of good-natured mockery, the ability to tell naively and funny stories made conversation with him so
as pleasant as it was desirable...” He could sometimes flare up, but “he was not
capable of harboring anger against someone for a long time...” recalled N I Fuss.

Euler was constantly surrounded by numerous grandchildren, often with a child sitting in his arms and a cat lying on his neck. He himself taught mathematics to the children. And all this did not stop him from working.

On September 18, 1783, Euler died of apoplexy in the presence of his assistants, professors Kraft and Leksel. He was buried at the Smolensk Lutheran Cemetery. The Academy commissioned the famous sculptor Zh.D. Rachette, who knew Euler well, received a marble bust of the deceased, and Princess Dashkova presented a marble pedestal.

Until the end of the 18th century, I.A. remained the conference secretary of the Academy. Euler, who was replaced by N.I. Fuss, who married the daughter of the latter, and in 1826 - Fuss's son Pavel Nikolaevich, so that the organizational side of the life of the Academy was in charge of the descendants of Leonhard Euler for about a hundred years. Euler's traditions had a strong influence on students
Chebysheva: A.M. Lyapunova, A.N. Korkina, E.I. Zolotareva, A.A. Markov and others, defining the main features of the St. Petersburg mathematical school.

There is no scientist whose name is mentioned in educational mathematical literature as often as the name of Euler. Even in high school, logarithms and trigonometry are still taught largely “according to Euler.”

Euler found proofs of all Fermat’s theorems, showed the falsity of one of them, and proved Fermat’s famous Last Theorem for “three” and “four”. He also proved that every prime number of the form 4n+1 always decomposes into the sum of the squares of the other two numbers.

Euler began to consistently build an elementary theory of numbers. Starting with the theory of power residues, he then took up quadratic residues. This is the so-called quadratic reciprocity law. Euler also spent many years solving indefinite equations of the second degree in two unknowns.

In all three of these fundamental questions, which for more than two centuries after Euler constituted the bulk of elementary number theory, the scientist went very far, but in all three he failed. The complete proof was obtained by Gauss and Lagrange.

Euler took the initiative to create the second part of the theory of numbers - the analytic theory of numbers, in which the deepest secrets of integers, for example, the distribution of prime numbers in the series of all natural numbers, are obtained from considering the properties of certain analytic functions.

The analytical theory of numbers created by Euler continues to develop today.

In 1707 in the Swiss city of Basel in the family of a priest Paul Euler A boy named Leonardo was born, who was destined to become one of the outstanding mathematicians of the time. Leonardo Euler showed a phenomenal memory, high ability to work, and a desire for new knowledge at an early age. At the age of 13, Leonardo Euler was enrolled in the art department of the University of Basel. The father dreamed of a career as a priest for his beloved son Leonardo. However, the extraordinary mathematical abilities that the boy possessed could not be buried in the ground. Leonardo would soon become a student of the famous Swiss mathematician Johann Bernoulli.

After some time, the sons of Johann Bernoulli were invited to St. Petersburg, and with them Leonardo Euler. An extremely talented young scientist is quickly becoming widely known. He is invited to the Academy of Sciences. In 1727, Leonardo Euler entered the Academy of Sciences with the rank of associate professor in physiology. In 1731 he received the title of professor of physics and became a full member of the Academy of Sciences. And two years later he already heads the department of higher mathematics.

Euler's erudition amazed his contemporaries. He was one of the most educated scientists: he knew Greek, Latin, and was fluent in German, French, Russian and other languages. In addition to mathematics, physics and astronomy, he had deep knowledge of geography, chemistry, botany, anatomy, medicine and other branches of science and technology. He was fond of music and literature, knew Virgil's Aeneid by heart.

A brilliant mathematician, an outstanding physicist, an engineer and astronomer, a geographer and a virtuoso calculator, Leonardo Euler made an invaluable contribution to the development of Russian scientific personnel. The “Manual to Arithmetic” compiled by him, translated into Russian, had a significant influence on world and Russian educational literature.

Euler's scientific work was admired by the depth of thought, diversity of interests, ideas and incredible productivity. Euler was simultaneously a member of many European academies and scientific schools. Euler's enormous and intense work had a negative impact on his health. In 1735, Leonardo Euler lost his right eye, and in 1766 he lost his second eye. Having lost his sight completely, Euler did not stop his scientific activities; his ability to work could only be envied. The blind scientist dictated part of his works to a scribe. Euler's scribe was a tailor boy whom the scientist sheltered and taught to read and write.

In the field of mathematical analysis, Euler wrote many works and made a huge number of discoveries. Euler's influence on the development of higher mathematics was significant. It was Leonardo Euler who brought trigonometry to the form we know and was one of the first to formulate the concept of function. Many mathematical concepts bear his name, including: Euler diagrams, Euler integrals, the broken line method, Euler circle, Euler substitutions, Euler's theorem and many others.

Euler's scientific heritage is extremely great. He managed to achieve brilliant results in mathematical analysis, geometry, number theory, calculus of variations, mechanics and other applications of mathematics. The complete collection of scientific works of Leonardo Euler consists of 72 volumes.

Euler is an example of a scientific genius whose work has become the property of all mankind. Schoolchildren all over the world study trigonometry and logarithms as Euler gave them. In higher education, students are taught higher mathematics according to the classical monographs of Euler. A brilliant mathematician, Euler knew that the fertile soil of science is, first of all, practical activity.

Perhaps there is not a single significant area of ​​mathematics in which one of the best mathematicians of the 18th century would not leave a mark. Leonard Euler.

It was Euler who directly proved the expansion of the exponential and arctangent series (an indirect proof through inverse power series was given by Newton and Leibniz between 1670 and 1680). His use of power series allowed him to solve the famous Basel problem in 1735 (he made a more rigorous proof in 1741):

The geometric meaning of Euler's formula Euler began to use exponentials and logarithms in analytical proofs. He managed to expand the logarithmic function into a power series and, using this schedule, determine logarithms for negative and complex numbers. He also extended the definition of the exponential function to complex numbers, and discovered the connection of the exponential function with trigonometric functions. Euler's formula states that for any real number x equality holds:

A special case of Euler's formula for x= ? is Euler's identity, which relates five fundamental mathematical constants:

e i ? + 1 = 0,

Called "the most wonderful mathematical formula" by Richard Feynman... In 1988, magazine readers Mathematical Intelligencer in the vote they called it “the most beautiful mathematical formula of all time.”
A corollary of Euler's Formula is Moivre's Formula.
In addition, Euler developed the theory of special transcendental functions by introducing the gamma function and introduced new methods for solving fourth-degree equations. He also found a way to evaluate integrals with complex limits, ahead of the development of modern complex analysis, and began the calculus of variations, including his famous result, the Euler-Lagrange equations.
Euler also pioneered the use of analytical methods to solve problems in number theory. In this way, he united two disparate areas of mathematics and introduced a new field of study, analytic number theory. The beginning was Euler's creation of the theory of hypergeometric series, Q-Series, hyperbolic trigonometric functions and the analytic theory of generalized fractions. For example, he proved the infinity of prime numbers using the harmonic series disagreement, and used methods of analysis to learn about the distribution of prime numbers. Euler's work in this area led to the appearance of the theorem on the distribution of prime numbers.
Number theory
Euler's interest in number theory can be explained by the influence of Christian Goldbach, second from the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre Fermat. Euler developed some of Fermat's ideas, and refuted some of his assumptions.
Euler connected the nature of the distribution of prime numbers with ideas on analysis. He proved that the sum of the inverses of prime numbers is divergent. In this way he discovered the connection between the Riemann zeta function and prime numbers, a result known as "Euler's identity in number theory".
Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made significant contributions to Lagrange's theorem on four squares. He also invented the Euler function? (N), equal to the number of positive numbers not exceeding the natural number N and which are relatively prime with N. Using the properties of this function, he generalized Fermat's little theorem to what is now called Euler's theorem. He made significant contributions to the theory of perfect numbers, which has fascinated mathematicians since the time of Euclid. Euler also made progress towards the prime number distribution theorem and proposed the quadratic reciprocity hypothesis. These two concepts are considered the fundamental theorems of number theory, and his ideas paved the way for the work of Gauss.
Before 1772, Euler proved that 2 31 – 1 = 2147483647 is a Mersenne number. It is likely that this number was the largest known prime number before 1867.
Graph theory
In 1736, Euler solved the problem known as the Seven Bridges of Königsberg. The city of Königsberg (today Kaliningrad) in Prussia is located on the Pregolya River and includes two large islands that were connected to each other and to the mainland by seven bridges. The problem is that you can find a path that passes each bridge exactly once and returns to the starting point. The answer is no: there is no Euler cycle. This statement is considered the first theorem of graph theory, in particular in the theory of planar graphs.
Euler also proved the formula V – E + F= 2, which connects the number of vertices, edges and faces of a convex polyhedron, and therefore planar graphs (for planar graphs V – E + F= 1). The left side of the formula, now known as the Euler characteristic of a graph (or other mathematical object), is associated with the concept of the genus of a surface.
The study and generalization of this formula, in particular by Cauchy and L'Huillier, were the beginnings of topology.
Applied Mathematics
Among Euler's greatest successes were the analytical solutions of practical problems, the description of numerous applications of Bernoulli numbers, Fourier series, Venn diagrams (also known as Euler circles), Euler numbers, constants e and?, continued fractions and integrals.
He combined Leibniz's differential calculus with Newton's method of fluxions, and created tools that made the application of analysis to physical problems easier. He made great strides in improving the numerical approximation of integrals, inventing what is now known as Euler's method and the Euler-Maclaurin formula. He also promoted the use of differential equations, in particular by introducing the Euler-Mascheroni constant:

One of Euler's most unusual interests was the application of mathematical ideas to music. In 1739 he wrote Tentamen novae theoriae musicae, hoping to finally incorporate music theory into mathematics. This part of his work, however, did not receive widespread attention and was once called "too mathematical for musicians and very musical for mathematicians."
Physics
Leonhard Euler made a significant contribution to the development of mechanics, in particular to solving the problem of the rotation of a rigid body. Euler's approach is associated with the concepts of Euler angles and Euler's kinematic equations. In 1757, Euler published a memoir “Principes generaux du mouvement des fluides” (General principles of fluid motion), in which he wrote down the equations of motion of an incompressible ideal fluid, called Euler’s equations. The result of work on the problem of beam deformation during loading was the Euler-Bernoulli equations, which subsequently found application in engineering science, in particular in the design of bridges.
Euler worked on general problems of mechanics, developing the Maupertuis principle. The equations of Lagrangian mechanics are often called the Euler-Lagrange equations.
Euler applied developed mathematical methods to solve problems of celestial mechanics. His work in this area received several awards from the Paris Academy of Sciences. Among his achievements are determining with great accuracy the orbits of comets and other celestial bodies, explaining the nature of comets, and calculating the parallax of the Sun. Euler's calculations made a significant contribution to the development of accurate latitude tables.
Euler's contribution to optics was important for his time. He denied Newton's then dominant corpuscular theory of light. Euler's work throughout the 1740s helped establish Christian Huygens' wave theory of light.
Astronomy
Most of Euler's astronomical works are devoted to issues of celestial mechanics that were relevant at that time, as well as spherical, practical and nautical astronomy, the theory of tides, the theory of astronomical climate, the refraction of light in the earth's atmosphere, parallax and aberration, and the rotation of the Earth. In the field of celestial mechanics, Euler made significant contributions to the theory of perturbed motion. Back in 1746, he calculated the excitations of the Moon and published lunar tables. Simultaneously with A.K. Clairaut and J.L.D "Alembert and independently of them, Euler developed general theories of the motion of the Moon, in which he was studied with very high accuracy. The first theory in which the method of expanding the desired coordinates into series in powers of small parameters and gave a partial development of an analytical method for varying the orbital elements, was published in 1753. This theory was used by T. I. Mayer in compiling high-precision tables of the motion of the Moon. A perfect analytical theory, in which the numerical development of the method is given and the tables are calculated, is presented in the work. published in St. Petersburg in 1772 in Latin. Its abbreviated translation into Russian entitled “New Theory of the Motion of the Moon” was carried out by A. N. Krylov and published in 1934. Computational methods proposed by Euler for obtaining accurate ephemerides of the Moon and planets, in particular. The rectangular coordinate axes he introduced were widely used by J.V. Gill. According to M.F. Subbotin, they became one of the most important sources of further progress in all celestial mechanics. Wide possibilities for the use of these methods arose with the advent of computers. The modern accurate and complete theory of the motion of the Moon was created in 1895-1908 by E. V. Brown. The work of Euler and Gill gave rise to the general theory of nonlinear oscillations, which plays an important role in modern science and technology.
Euler's work “On the Improvement of the Objective Glass of Telescopes” (1747) was important for astronomy, in which he showed that by combining two lenses of glass with different refractive powers, an achromatic lens can be created. Influenced by Euler's work, the first lens of this kind was made by the English optician J. Dollond in 1758.

On April 15, 1707, a son was born into the family of the Basel pastor Paul Euler, named Leonard. From early childhood, his father prepared him for a spiritual career. According to Paul, a good priest had to have clearly developed logic, so he attached great importance to mathematics. Not only did the pastor himself love this exact science, but he was also friends with the famous mathematician Jacob Bernoulli. When Leonard was barely 13 years old, Jacob's younger brother, university professor Johann Bernoulli, noticed extraordinary mathematical abilities in the boy and invited him to come to his house on Saturdays, where they, together with Johann's sons, Daniel and Nikolai, solved complex mathematical problems in an easy and relaxed atmosphere.

At the age of 17, Leonard received his master's degree. Soon his first serious scientific work, “Dissertation in Physics on Sound,” was published, which received very flattering reviews from serious scientists. In 1725, the young master tried to get a vacant position as a professor of physics at the University of Basel, but even despite Bernoulli’s patronage, the applicant was told that he was too young for such an honorable position. In general, scientific vacancies were so tight in Switzerland at that time that even the professor’s children could not find a worthy occupation. But scientific personnel were needed in neighboring Russia, where in 1724 Peter I established the country's first Academy. Daniil and Nikolai were the first to move to St. Petersburg, and already at the beginning of 1726, Leonard received a dispatch saying that, on the recommendation of the Bernoulli Herrs, he was invited to the position of adjunct in physiology with a salary of 200 rubles per year. Although this amount was not particularly large, it was significantly more than what the young mathematician could count on in his homeland. Therefore, already in April 1726, immediately upon receiving the advance, Euler left his native Switzerland. Then he still thought that it would be for a while.

In the capital of the Russian Empire, a young specialist who had learned to speak Russian quite fluently in less than a year was immediately loaded with work, not always related to mathematics. The shortage of specialists led to the fact that the scientist was either charged with tasks on cartography, or required written consultations for shipbuilders and artillerymen, or was entrusted with the design of fire pumps, or was even charged with drawing up court horoscopes. Euler carefully carried out all these tasks, and only requests regarding astrology were categorically forwarded to the court astronomers. Predictions in Russia have always been a matter of increased danger and require special caution.

In 1731, Leonard became an academician and received a position as a professor of physics with a salary double the previous one. And two years later he took the position of professor of pure mathematics. Now he was owed 600 rubles a year. With such an income, one could already think about a family. At the end of 1733, the 26-year-old scientist married his peer and compatriot Katharina, daughter of the artist Georg Gsell, and found a small house on the Neva embankment. During their marriage, the wife gave birth to 13 children to Leonard, but only five of them survived, two daughters and three sons.

In 1735, Euler independently, without any outside help, completed an urgent government cartographic (according to other sources - astronomical) task in three days, for which other academicians had been asking for several months. However, such intensity of work could not but affect the scientist’s health: due to extreme overexertion, Leonhard Euler became blind in his right eye.

By that time, his name was already widely known in Russia. And the treatise “Mechanics, or the science of motion, in an analytical presentation” written in 1736 brought the scientist truly worldwide fame. It was from him that theoretical mechanics became the applied part of mathematics.

Over the decade and a half he spent in Russia, Euler wrote and published more than 90 major scientific works. He was also the main author of the academic “Notes” - the central Russian scientific bulletin of that time. The mathematician spoke at scientific seminars, gave public lectures, and performed a wide variety of tasks. A former teacher, Johann Bernoulli, wrote to him: “I devoted myself to the childhood of higher mathematics. You, my friend, will continue her development into maturity.” The fame of Euler as an excellent mathematician grew to such an extent that when in 1740 the position of director of its mathematical department was vacant at the Berlin Academy, the Prussian King Frederick himself invited the scientist to take this position.

By that time, a time of stagnation had begun in the St. Petersburg Academy of Sciences. After the death of Empress Anna Ioannovna, the young John IV became king. The regent Joanna Anna Leopoldovna, who ruled the empire at that time, did not pay any attention to the sciences, and the Academy gradually fell into disrepair. “Something dangerous was foreseen,” Euler later wrote in his autobiography. - After the death of the illustrious Empress Anna during the regency that followed...

the situation began to seem uncertain.” Therefore, the scientist took Frederick’s invitation as a gift of fate and immediately submitted a petition in which he wrote: “For this reason, I am forced, both for the sake of poor health and other circumstances, to seek a pleasant climate and accept the call made to me from His Royal Majesty of Prussia. 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.” But, despite the general cool attitude towards science, the state administration was not at all eager to let go of an already recognized world luminary so easily. On the other hand, it was impossible not to let go. Therefore, as a result of short negotiations, we managed to obtain a promise from the mathematician, even while living in Berlin, to help Russia in every possible way. In return, he was awarded the title of honorary member of the Academy with a salary of 200 rubles. Finally, on May 29, 1741, all the documents were corrected, and already in June Euler, along with his entire family, his wife, children and four nephews, arrived in Berlin.

Here, as once in Russia, they also began to actively involve him in a variety of non-core work and projects. He was involved in organizing state lotteries, supervised the work of the mint, supervised the laying of a new water supply system and the organization of pensions. But Leonard’s relationship with King Frederick himself did not work out. The monarch did not like the mathematician, who was kind and smart, but not at all sociable. Indeed, Euler hated social receptions, balls and other entertainment events that interfered with scientific reasoning. When his wife managed to drag him into the theater, the mathematician would invent some complex example for himself, which he solved in his mind throughout the performance.

The scientist kept his word strictly, given before leaving Russia. He continued to publish his articles in Russian magazines, edited the works of Russian scientists, and purchased instruments and books for the St. Petersburg Academy. Young Russian scientists sent for internships lived in his house on full board. It was here that he met and became friends with a promising student of the Moscow “Spassky Schools” Mikhaila Lomonosov, in whom he most noted the “happy combination of theory and experiment.” When in 1747 the President of the Academy of Sciences, Count Razumovsky, asked him to give feedback on the articles of the young scientist, Euler rated them very highly. “All these dissertations,” he wrote in his report, “are not only good, but also very excellent, for he (Lomonosov) writes about very necessary physical and chemical matters, which to this day the wittiest people did not know and could not interpret, that he He did it 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.” It must be said that Mikhail Vasilyevich, very arrogant, proud and difficult to communicate with, also loved his Berlin teacher until the end of his days, wrote him friendly letters and considered him one of the greatest scientists in the world.

Most of the terms, concepts and techniques introduced by Euler almost three centuries ago are still used by mathematicians today. But all this did not in any way affect the cold attitude of the ruling royalty of Prussia towards him. When the president of the Berlin Academy of Sciences, Maupertuis, died in 1759, Frederick II could not find a replacement for him for a long time. The French encyclopedist and simply very clever Jean D'Alembert, to whom the king turned first, refused the tempting offer, believing that there was a more worthy candidate for this post in Berlin. Finally, Friedrich reconciled himself and gave Euler the leadership of the Academy. But he categorically refused to give him the title of president.

Meanwhile, in Russia, Euler's authority, on the contrary, is becoming increasingly stronger. During the Seven Years' War, Russian artillery accidentally destroyed the scientist's house in Charlottenburg (a suburb of Berlin). Field Marshal Saltykov, who learned about this, immediately compensated the scientist for all the losses caused. And when the news of the unsuccessful shelling reached Empress Elizabeth, she personally ordered another 4,000 rubles to be sent to her Berlin friend, which was a huge amount.

In 1762, Catherine II ascended the Russian throne, dreaming of establishing an “enlightened monarchy” in the country. She saw the return of a prominent mathematician to the country as one of her most important tasks. Therefore, Euler soon received a very interesting offer from her: to head the mathematics class, receiving the title of conference secretary of the Academy and a salary of 1800 rubles per year. “And if you don’t like it,” her instructions to the diplomatic representatives said, “she would be pleased to communicate her conditions, so long as she doesn’t hesitate to come to St. Petersburg.”

Euler, indeed, was pleased to put forward counter conditions:

The post of vice-president of the Academy with a salary of 3,000 rubles;

- an annual pension of 1000 rubles to the wife in the event of his death;

- paid positions for his three sons, including the post of secretary of the Academy for the eldest.

Such insolence on the part of some mathematician outraged the representative of the imperial administration, the prominent Russian diplomat Count Vorontsov. However, the empress herself thought differently. “Mr. Euler’s letter to you,” she wrote to the count, “gave me great pleasure, because I learn from it about his desire to re-enter my service. Of course, I find him completely worthy of the desired title of Vice-President of the Academy of Sciences, but for this, some measures must be taken before I establish this title - I say I will, since until now it has not existed. 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 the ashes from such an important acquisition, and I congratulate myself in advance on having returned a great man to Russia.”

Having received assurances that all his conditions were accepted at the highest level, Euler immediately wrote to Friedrich asking for his resignation. Perhaps because of the reluctance to let go of the prominent scientist, perhaps because of a negative attitude towards him, and most likely because of all this together, the king not only refused, but simply ignored Euler’s appeal without giving any answer to it. Euler wrote another petition. With the same result. Then the mathematician simply demonstratively stopped working at the Academy. Finally, Catherine herself turned to the King of Prussia with a request to release the scientist. Only after such high intervention did Frederick allow the mathematician to leave Prussia.

In July 1766, the scientist, along with 17 members of his household, arrived in St. Petersburg. Here he was immediately received by the Empress herself. And she not only accepted, but granted 8,000 rubles for the purchase of a house and furnishings, and even placed one of her best cooks at his complete disposal.

Already in Russia, Euler began work on one of his main works - “Universal Arithmetic”, also published under the titles “Principles of Algebra” and “Complete Course of Algebra”. Moreover, this book was initially published in Russian, and only two years later - in official scientific German. We can fully claim that all subsequent world algebra textbooks were based on this work. Immediately after him, Euler published two more large-scale monographs - “Optics” and “Integral Calculus”. When he was working hard on his new great work, “The New Theory of the Motion of the Moon,” tragedy happened. A large fire swept through St. Petersburg, destroying more than a hundred houses. Euler's house on Vasilyevsky Island also fell into this list. Fortunately, the scientist managed to save most of his manuscripts. What he could not save, he restored in a short time, dictating the texts from memory.

Precisely by dictating. For the vision of the scientist, who spent the day and night doing calculations and calculations, was in the most critical condition. Ophthalmologists long ago diagnosed Euler with rapidly progressing cataracts in his only working left eye. Therefore, he had long been “writing” most of his works with the hands of a nimble boy tailor. Empress Catherine, who knew about this, specially ordered the scientist from Berlin in 1771 to correct the vision of the scientist, the best specialist in this field - the personal ophthalmologist of the Austrian Emperor and the English King, Baron Wenzel. The operation was successful: Wenzel removed the cataract and warned the scientist that for the first few months he should stay away from bright light and stop reading so that the eye would get used to the new condition. But such torture was absolutely unbearable for the scientist. Within a few days, he, secretly from his family, took off the bandage and greedily attacked the latest scientific journals. The result was immediate: the scientist soon lost his sight again, this time completely. At the same time, his labor productivity not only did not decrease, but even increased. An incorrigible optimist, he sometimes said with a bit of humor that the loss of vision benefited him: he stopped being distracted by external beauties not related to mathematics.

Soon fate dealt him another serious blow. In 1773, his beloved wife Katharina, with whom he lived in a happy marriage for 40 years, died. But this loss did not knock him out of the saddle. Three years later he married a second time. On Katharina's half-sister Salome. She reminded Leonard of his late wife in everything and until the end of the scientist’s life she was his faithful assistant.

In the early 1780s, Euler increasingly began to complain of headaches and general weakness. On September 7, 1883, he had an afternoon conversation with academician Andrei Leksel. Both mathematicians and astronomers, they discussed the recently discovered planet Uranus and its orbit. Suddenly Euler felt ill. He only managed to say: “I’m dying,” after which he immediately lost consciousness. A few hours later he was gone. Doctors determined that death occurred from a cerebral hemorrhage.

The scientist was buried in St. Petersburg, at the Lutheran Smolensk cemetery. The words were carved on the tombstone: “Here lie the mortal remains of the wise, just, famous Leonhard Euler.”

The mathematician's children remained in Russia. The eldest son, also a talented mathematician and mechanic Johann Euler (1734-1800), as Empress Catherine promised, was secretary of the Imperial Academy of Sciences. The younger, Christopher (1743-1808), rose to the rank of lieutenant general and commanded the Sestroretsk arms factory. Grandson, Alexander Khristoforovich (1773-1849) became an artillery general, a hero of the Patriotic War of 1812. Another descendant, although he returned to the homeland of his ancestors, Sweden, Hans Karl August Simon von Euler-Helpin (1873-1964) became a famous biochemist, a foreign member of the USSR Academy of Sciences, and a Nobel Prize laureate in chemistry for 1929. Another Nobel Prize, only in 1970, was received by his son, Swedish biologist Ulf von Euler (1905-1983).

There are many monuments erected to Leonhard Euler. Institutes, streets, and scientific awards bear his name. Stamps and coins have been printed in his honor, and an asteroid and a crater on the Moon have been named. But perhaps the most original monument to the scientist can be found in children's notebooks. After all, schoolchildren often try to solve well-known problems: how to move a chess knight through all the cells of a drawn square without passing through the same cell twice, or how to cross several rivers using several bridges in the same way. At the same time, they often don’t even realize that it was the great Russian mathematician Leonhard Euler who came up with these problems, and not only thought of them, but also found an exhaustive algorithm for solving them almost three centuries ago. Whose name in Russia was Leonty.

Euler, the greatest mathematician of the 18th century, was born in Switzerland.
In 1727, at the invitation of the St. Petersburg Academy of Sciences, he came to Russia.
In St. Petersburg, Euler found himself in a circle of outstanding scientists: mathematicians, physicists, astronomers, and received great opportunities to create and publish his works.
He worked with passion and soon became, according to the unanimous recognition of his contemporaries, the first mathematician in the world.

Euler's scientific legacy is striking in its volume and versatility.
The list of his works includes more than 800 titles. The complete collected works of the scientist occupy 72 volumes.
Among his works are the first textbooks on differential and integral calculus.

In number theory, Euler continued the work of the French mathematician P. Fermat and proved a number of statements: Fermat's little theorem, Fermat's great theorem for exponents 3 and 4. He formulated problems that determined the horizons of number theory for decades.

Euler proposed using the means of mathematical analysis in number theory and took the first steps along this path. He realized that, moving further, it was possible to estimate the number of prime numbers not exceeding n, and he outlined a statement that would then be proven in the 19th century. mathematicians P. L. Chebyshev and J. Hadamard.

Euler also works a lot in the field of mathematical analysis.
The scientist was the first to develop a general doctrine of the logarithmic function, according to which all complex numbers, except zero, have logarithms, and each number corresponds to an infinite number of logarithmic values. In geometry, Euler laid the foundation for a completely new field of research, which later grew into an independent science - topology.

The formula is named after Euler,
connecting the number of vertices (B), edges (P) and faces (G) of a convex polyhedron:
B - P + G = 2.
Even the main results of Euler's scientific activities are difficult to list.
Here is the geometry of curves and surfaces, and the first presentation of the calculus of variations with numerous new concrete results.
He wrote works on hydraulics, shipbuilding, artillery, geometric optics, and even music theory.
For the first time, he gives an analytical presentation of mechanics instead of Newton's geometric presentation, and builds the mechanics of a solid matter, and not just a material point or a solid plate.

One of Euler's most remarkable achievements is related to astronomy and celestial mechanics.
He constructed an accurate theory of the movement of the Moon, taking into account the attraction of not only the Earth, but also the Sun.
This is an example of solving a very difficult problem.

The last 17 years of Euler's life were marred by almost complete loss of vision.
But he continued to create as intensely as in his youth.
Only now he no longer wrote himself, but dictated to his students, who carried out the most cumbersome calculations for him.
For many generations of mathematicians, Euler was a teacher.
Several generations studied from his mathematical manuals, books on mechanics and physics.
The main content of these books is included in modern textbooks.