Introduction

Euler is one of the geniuses whose work has become the property of all mankind. Until now, schoolchildren in all countries study trigonometry and logarithms in the form that Euler gave them. Students study higher mathematics using manuals, the first examples of which were Euler's classical monographs.<Рисунок 1 >.
Our lesson today is dedicated to this great man. First, I want to give the floor to N.P. Dolbilin, associate professor of physical and mathematical sciences, leading researcher at the Mathematical Institute of the Russian Academy of Sciences (a fragment of N.P. Dolbilin’s speech at the VI Moscow Pedagogical Marathon of academic subjects is shown, time 1.15 - 2.40).

We remember Euler's name when studying logarithms in the first year. It was in honor of the great Leonhard Euler that the number was named after the first letter of his last name. e. It was he who introduced the notation e for the base of natural logarithms.<Рисунок 2 >. Leonhard Euler introduced a lot of new things into the branches of mathematics studying trigonometry, logarithms, polyhedra, complex numbers, and graphs. He introduced many notations that we use today: 1734 - function notation f(x), 1736 – designation of the base of the natural logarithm e and the ratio of the circumference to the diameter of the circle, 1748 – designation of trigonometric functions sin x and cos x, 1753 – designation of the trigonometric function tg x, 1755 – sign of the sum, 1777 – designation of the imaginary unit i.<Рисунок 3 >.

Euler's formula

Euler’s name is given to the formula connecting the number of vertices (B), edges (P) and faces (G) of a convex polyhedron: B – P + G = ?.<Рисунок 4 >.

Task 1

Now you will see images of polyhedra: a triangular prism, a parallelepiped, a triangular pyramid, a truncated pentagonal pyramid, a regular octahedron, a regular dodecahedron. Your task is to count the number of vertices, edges and faces of these polyhedra and calculate for each of them В – Р + Г = ?. For each correct answer, the team receives 1 point. This task will take 10 minutes.
Images of polyhedra appear on the screen, and then after the teams submit their solutions to the jury, the answers:<Рисунок 5 >, <Рисунок 6 >, <Рисунок 7 >.
Leonhard Euler discovered this pattern in 1752 and later proved it.

Euler's childhood. Basel period of his life.

Leonhard Euler was born on April 4, 1707 in the family of a poor Protestant priest, Paul Euler, and Margarita Brucker in the Swiss city of Basel on the picturesque banks of the Rhine. At that time, Basel was a center of education and culture on a European scale.<Рисунок 8 >.
Leonardo was about a year old when the family moved to the town of Riechen, near Basel, where Leonardo's father was transferred as a pastor.
Leonard received his initial education from his father. The pastor prepared his son for a spiritual career, but also taught him mathematics, as entertainment and development of logical thinking. After home schooling, Leonard was sent to the Basel Latin Gymnasium.
In 1720, 13-year-old Leonhard Euler became an art student at the University of Basel. Having become a student, he easily mastered academic subjects, giving preference to mathematics. During these years he became friends with the Bernoulli family. Professor I. Bernoulli noticed talent in the young man and began to study individually with Leonard.
In 1724, 17-year-old Leonhard Euler gave a magnificent speech in Latin on the comparison of the philosophical views of Descartes and Newton and was awarded a master's degree (which now corresponds to the degree of Doctor of Philosophy). Over the next two years, young Euler wrote several scientific papers that received positive reviews. In 1725, he won a competition at the Paris Academy of Sciences for solving the problem of choosing the best place on a ship to install a mast; it is interesting that by this time he had never seen either the sea or seagoing ships.

Euler polynomial

Euler's polynomial is a polynomial X 2 – X+ 41. Leonhard Euler calculated its value for x from 1 to 40 and noticed a pattern.

Task 2

You need to calculate the value of this polynomial for x from 1 to 20. For each correct answer, the team receives 1 point. If you can guess the pattern, you will receive another 10 points.<Рисунок 9 >. This task will take 10 minutes.

Mathematicians have always been interested in prime numbers. Euclid also argued that there are infinitely many prime numbers in the natural series. In 1750, Leonhard Euler found the prime number 2 31 – 1. As a result of calculating the values ​​of this polynomial for x from 1 to 40, only prime numbers are obtained.<Рисунок 10 >

The first Petersburg period of life

In 1726, Empress Catherine I, on the recommendation of the Bernoulli brothers, invited the young Leonhard Euler to the Russian Academy of Sciences. Upon arrival in the Russian capital, Euler joined a group of mathematicians and physicists dealing with issues of applied mathematics. Scientists were also tasked with creating guidelines for the initial teaching of science.

On one of the last days of 1733, 26-year-old Leonard Euler married Ekaterina Gzel. Wedding, New Year - two holidays at once! The entire academy warmly congratulated the newlyweds. It turns out that a great mathematician can not only calculate and analyze, he is not alien to worldly life. They had 13 children, but only five survived childhood.

Euler was distinguished by his phenomenal efficiency. He simply could not help but study mathematics or its applications. In 1735, the Academy received the task of performing an urgent and very cumbersome astronomical calculation to calculate the trajectory of a comet. A group of academics asked for three months to do this work, but Euler undertook to complete the work in three days - and did it on his own. However, the overexertion did not pass without a trace: he fell ill and lost sight in his right eye. The scientist dealt with the misfortune with the greatest calm: “Now I will be less distracted from doing mathematics,” he noted philosophically.<Рисунок 11 >.

In 1736, Euler introduced the well-known notation . He calculated to 153 decimal places. This designation was first used by the English mathematician Johnson in 1706.

They say that Leonard Euler once, during insomnia, calculated the sixth power of the first 100 numbers, and repeated the results many days later. Another time, Euler, testing the series he had obtained, calculated the first 20 digits of the number within an hour.

Euler circles

One of Euler's works talks about circles, which are "very suitable for facilitating our thinking." These circles are usually called "Eulerian circles". Let's solve the next problem together.

Task: There are 40 people in the class. Of these, 19 people have “C” grades in Russian, 17 people in mathematics, and 22 people in physics. Only one subject has “C” grades: in Russian - 4 people, in mathematics - 4 people and in physics - 11 people. Seven people have C's in both mathematics and physics, five of them have C's in Russian. How many people study without grades? How many people have C grades in two out of three subjects? Let's look at the solution using the next slide<Рисунок 12 >.

Task 3

Count the mathematicians. There are 35 students in the class. Of these, 20 are involved in a math club, 11 in a biology club, 10 children do not attend these clubs. How many biologists are interested in mathematics? This task will take 5 minutes. Maximum score – 5 points.

The condition of the problem appears on the screen, and then its solution is considered<Рисунок 13 >.

Bridges in Konigsberg

Here is a translation of the Latin text, which is taken from Euler’s letter to the Italian mathematician and engineer Marinoni, sent from St. Petersburg on March 13, 1736 : “I was once asked a problem about an island located in the city of Königsberg and surrounded by a river across which seven bridges are thrown. The question is whether anyone can go around them continuously, passing only once through each bridge. And then I was informed that no one has yet been able to do this, but no one has proven that it is impossible. This question, although trivial, seemed to me, however, worthy of attention because neither geometry, nor algebra, nor combinatorial art are sufficient to solve it. ... After much thought, I found an easy rule, based on a completely convincing proof, with the help of which it is possible in all problems of this kind to immediately determine whether such a detour can be made through any number and any number of bridges located in any way or not.”

If the number of islands connected by bridges is more than two, then to solve the problem it is necessary to count how many bridges lead to each island. If there are an even number of bridges leading to each island, then a bypass is possible and you can start from any island. If there are an odd number of bridges leading to two islands, then a bypass is possible and should be started from any island that has an odd number of bridges leading to it. If there are more than two areas to which an odd number of bridges lead, then the specified transition is not possible.
In our problem there are 4 islands in total: A, B, C, D. The number of bridges leading to these sections is respectively: 5, 3, 3, 3, which means bypassing is impossible.<Рисунок 14 >.

Task 4

Find out if it is possible to walk around all the bridges by visiting each of them only once in the following cases.<Рисунок 15 >, <Рисунок 16 >. Each task takes 1 minute to complete. For each task - 2 points.

Graph theory

Graph theory is a relatively young science. The first work on graph theory belongs to Leonhard Euler. It appeared in 1736 in publications of the St. Petersburg Academy of Sciences and began with a consideration of the problem of the Koenigsberg bridges. The graphs made the conditions clearer, simplified the solution, and revealed the similarity of the problems. Nowadays, in almost every branch of science and technology you come across graphs: in electrical engineering when constructing electrical circuits, in chemistry when studying molecules and their chains, in economics when solving problems of choosing the optimal path for freight transport flows. A graph is a figure made up of points and lines.

Let's solve the following problem:

The school drama club decided to stage Gogol's The Inspector General. And then a heated argument broke out. It all started with Lyapkin-Tyapkin.

- I will be Lyapkin-Tyapkin! Dima said decisively. From early childhood I dreamed of bringing this image to life on stage.
“Well, okay, I agree to give up this role if they let me play Khlestakov,” Gena showed generosity.
“... And for me - Osipa,” Dima did not yield to him in generosity.
“I want to be Strawberry or Mayor,” said Vova.
“No, I will be the Governor,” Alik and Borya shouted in unison. - or Khlestakov, they added at the same time.

Will it be possible to distribute the roles like this? To keep the performers happy?<Рисунок 17 >.

Let's depict each participant in the drama circle as a dot, and all their wishes will be represented as lines. It can be seen that Osip will be played by Dima, Vova - Strawberry, Gen - Lyapkin - Tyapkin, Alik and Borya - Khlestakov and Gorodnichy.

Task 5

Use graphs to solve the following problem: There are 6 participants in the table tennis class championship: Andrey, Boris, Victor, Galina, Dmitry and Elena. The championship is held in a round robin system - each participant plays each of the others once. This task will take 5 minutes. Maximum score – 5 points.

The solution to the problem is displayed on the screen<Рисунок 18 >.

In 1736, Euler published two volumes of analytical mechanics. In this work, he applied the methods of mathematical analysis to the solution of problems of motion in vacuum and resistive media. This work was the first where differential and integral calculus were used to describe physical phenomena.<Рисунок 19 >.

In 1738, two volumes of the “Manual to Arithmetic” appeared in German, which was translated into Russian and published in 1740 as a textbook for high school students.

In 1739, Euler published a book on the theory of music, in which he considered music as a part of mathematics.

In 1740, Euler published a book on the ebb and flow of the seas, for which he received a prize from the Paris Academy of Sciences.

In just 14 years of the first period of his life in St. Petersburg, Euler prepared about 80 works for publication and published over 50. Euler participated in many areas of activity of the St. Petersburg Academy of Sciences. He lectured to students, participated in various technical examinations, and worked on compiling maps of Russia.

In 1741, Euler accepted the offer of the Prussian king Frederick II to move to Berlin.

Berlin period

While living in Berlin, Euler did not stop working intensively for the St. Petersburg Academy of Sciences, maintaining the title of its honorary member. He conducted extensive scientific correspondence, in particular he corresponded with Yas Lomonosov, whom he highly valued. With the money he received from Russia, Euler purchased books and instruments for the Academy, selected candidates for academic positions, and wrote reviews of scientific works.

Euler introduced symbolism close to what we are used to and completely clarified the question of the signs of trigonometric functions of any argument. Euler's predecessors understood trigonometric functions as images of lines in a circle of a certain radius, calling it the “full sine”. Now trigonometric functions have simply constituted a certain class of analytic functions, both real and complex arguments. In 1748, thanks to Euler, the familiar notation for sine and cosine came into use, and in 1753 for cotangent.

Task 6

Construct graphs of these functions in one coordinate system<Рисунок 20 >. This task will take 10 minutes. The maximum score is 10 points.

The figure shows that for values ​​of x close to unity, the graphs of these functions almost coincide<Рисунок 21 >. Euler received a representation of the trigonometric functions sine and cosine as a sum of functions, in the form of a polynomial.<Рисунок 22 >, <Рисунок 23 >.

At the Berlin Academy of Sciences, Leonhard Euler headed the observatory and botanical garden, and was involved in the publication of various geographical and calendars. During this period, Euler published 380 scientific papers, wrote books on mathematical analysis, shipbuilding and navigation, and on the movement of the Moon.<Рисунок 24 >.

The results obtained by Euler are used in space research. In particular, to control aircraft it is necessary to find the best (optimal) control. L. Euler developed it in 1726–1744. general method for solving extremal problems.

For example, moving along a cycloid, under the influence of gravity the body will descend from one point to another in the shortest possible time.

Euler discovered a formula that can be used to calculate the force, called critical, under the influence of which the column begins to bend and its axis takes the shape of a sinusoid.
The growth of Euler's authority was uniquely reflected in the letters to him from his teacher I. 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 head of mathematicians.”

Task 7

Find out by completing the necessary constructions on which line in an arbitrary triangle the following three points lie: the point of intersection of the heights, the point of intersection of the medians, the center of the circumscribed circle. This task will take 5 minutes. Maximum score – 5 points.
In an arbitrary triangle, the point of intersection of altitudes, the point of intersection of medians and the center of the circumscribed circle lie on the same straight line. This straight line is called Euler's straight line.<Рисунок 25 >.

Second Petersburg period of life

Euler returned to Russia in 1766. He brought many manuscripts to St. Petersburg that he did not manage to publish in Berlin. Despite his advanced age and the almost complete blindness that befell him, he worked productively until the end of his life.

In 1767, Euler wrote an algebra textbook, “Universal Arithmetic.” This book by Euler was published in Russian in 1768, in German in 1770. Translated into French, English, Spanish. Reprinted 30 times in 6 European languages.<Рисунок 26 >.

In 1776, Leonhard Euler was one of the experts on the project of a single-arch bridge across the Neva, proposed by I. Kulibin, and of the entire commission, he was the only one who broadly supported the project.

In 1777 Euler introduced the notation for the imaginary unit i and wrote down his famous formula, which Lagrange called one of the most beautiful inventions of the 18th century. Academician Krylov believes that this amazing formula combines arithmetic (–1), geometry (P), algebra (the square root of minus one is equal to imaginary one), analysis (e).<Рисунок 27 >.

Euler's range of activities, covering all departments of contemporary mathematics and mechanics,
theory of elasticity, mathematical physics, optics, music theory, machine theory, ballistics, marine science, insurance, etc.

Task 8

It is required to select 5 weights so that they can be used to weigh any load up to 30 kg, provided that the weights are placed on only one pan of the scale. Euler suggested taking the following weights: 1 kg, 2 kg, 4 kg, 8 kg, 16 kg. Try to “weigh” loads from 1 to 30 kilograms with these weights. For each correct answer 1 point. This task will take 5 minutes.

During 1777, Euler, being blind, prepared about 100 articles, i.e. almost 2 articles a week! During the 17 years of his second stay in St. Petersburg, Leonhard Euler prepared about 400 works.<Рисунок 28 >.

Euler's merits as a major scientist and organizer of scientific research were highly appreciated during his lifetime. In addition to the St. Petersburg and Berlin academies, he was a member of the largest scientific institutions: the Paris Academy of Sciences, the Royal Society of London and others.<Рисунок 29 >. 3/5 of Euler's works relate to mathematics, the remaining 2/5 to its applications.

Dominic Arago said: “Euler calculated, without any visible effort, how a person breathes or how an eagle soars above the earth.”

Task 9

Find out on which line in an arbitrary triangle lie: the bases of the heights, the bases of the medians, the midpoints of the segments connecting the point of intersection of the heights of the triangle with its vertices. This task will take 10 minutes. The maximum score is 10 points.

In an arbitrary triangle, the bases of the medians, the bases of the altitudes, as well as the midpoints of the segments connecting the point of intersection of the altitudes of the triangle with its vertices lie on the same circle. It is called the Euler circle.<Рисунок 30 >.

Leonhard Euler died on September 18, 1783. The French mathematician Condorcet said: “Euler stopped calculating and living.” He was buried at the Smolensk cemetery in St. Petersburg. The inscription on the monument read: “To Leonard Euler – St. Petersburg Academy.” Academician Vavilov would say later: “Together with Peter I and Lomonosov, Euler became the good genius of our academy, who determined its glory, its strength, its productivity.”<Рисунок 31 >. After 50 years, it was discovered that the grave was lost, and it was only found by chance. Later, Euler's remains were transferred to the necropolis of the Alexander Nevsky Lavra, where today you can see his grave.

The 18th century can rightfully be called the century of Euler. He had a great and fruitful influence on the development of mathematical education in Russia. A crater on the far side of the Moon is named after Euler. M. V. Ostrogradsky wrote that “Euler created modern analysis and made of it the most powerful apparatus of the human mind. He alone embraced analysis in its entirety and pointed out its many and varied applications.”

In 1909, the Swiss Natural Science Society began publishing Euler's complete works, which was completed in 1975. It consists of 72 volumes. The famous French scientist P. Laplace said: “Read, read Euler, he is our common teacher.” Several generations studied from Euler's books, and the main content of these books was included in modern textbooks.

In September 1983, the world celebrated the 200th anniversary of the death of the great St. Petersburg mathematician Leonardo Euler. The specially created Euler Committee at the Academy of Sciences of the GDR held a scientific conference with the participation of foreign mathematicians. A commemorative medal made of Meissen porcelain was issued for the opening of the conference.<Рисунок 32 >. A stamp was published with a portrait of Euler and one of his most famous formulas, as well as envelopes with a facsimile of his signature and an embossed portrait.<Рисунок 33 >.

In 2007, the 300th anniversary of the great mathematician Leonhard Euler was widely celebrated.

Summing up the game

The jury counts points and sums up the results

Literature:

"Mathematics". Educational and methodological newspaper. Special issue for the 300th anniversary of Leonhard Euler. No. 6, 2007.
Alkhova Z.N., Makeeva A.V. Extracurricular work in mathematics. – Saratov, JSC Lyceum, 2002.
Bavrin I.I., Fribus E.A. Antique math problems. – M.: Education, 1994.
Bavrin I.I., Fribus E.A. Fun math problems. – M.: Vlados, 2003.
Nikiforovsky V A. In the world of equations. – M.: Nauka, 1987.
Smyshlyaev V.K. About mathematics and mathematicians. – Yoshkar-Ola, Mari book publishing house, 1977.

Education at the gymnasium in those days was short. In the fall of 1720, thirteen-year-old Euler entered the University of Basel, three years later he graduated from the lower faculty of philosophy and, at the request of his father, enrolled in the theological faculty. In the summer of 1724, at a one-year university act, he read a speech in Latin on a comparison of Cartesian and Newtonian philosophy. Showing an interest in mathematics, he attracted the attention of Johann Bernoulli. The professor began to personally supervise the young man’s independent studies and soon publicly admitted that he expected the greatest success from the insight and sharpness of mind of young Euler.

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

Immediately upon his arrival in St. Petersburg, he completely immersed himself in scientific work and then amazed everyone with the fruitfulness of his work. His numerous articles in academic yearbooks, initially devoted primarily to problems in mechanics, soon brought him worldwide fame, and later contributed to the fame of St. Petersburg academic publications in Western Europe. A continuous stream of Euler's writings was published from then on in the proceedings of the Academy for a whole century.

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

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

Euler's religiosity and character did not correspond to the environment of the “freethinking” Frederick the Great. This led to a gradual deterioration in the relationship between Euler and the king, who was well aware that Euler was the pride of the Royal Academy. In the last years of his Berlin life, Euler actually acted as president of the Academy, but never received this position. As a result, in the summer of 1766, despite the king’s resistance, Euler accepted the invitation of Catherine the Great and returned to St. Petersburg, where he then remained until the end of his life.

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

At the beginning of September 1783, Euler felt slightly unwell. On September 18, he was still engaged in mathematical research, but suddenly lost consciousness and, in the apt expression of the panegyrist, “stopped calculating and living.”

Best of the day

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

The scientific legacy of Leonhard Euler is colossal. He is responsible for classic results in mathematical analysis. He advanced its rationale, significantly developed integral calculus, methods for integrating ordinary differential equations and partial differential equations. Euler authored the famous six-volume course on mathematical analysis, including Introduction to Infinitesimal Analysis, Differential Calculus, and Integral Calculus (1748–1770). Many generations of mathematicians around the world studied from this “analytic trilogy.”

Euler obtained the basic equations of the calculus of variations and determined the ways of its further development, summing up the main results of his research in this area in the monograph Method for Finding Curved Lines Having the Properties of Maximum or Minimum (1744). Euler's significant contributions were to the development of function theory, differential geometry, computational mathematics, and number theory. Euler's two-volume course Complete Guide to Algebra (1770) went through about 30 editions in six European languages.

Fundamental results belong to Leonhard Euler in rational mechanics. He was the first to give a consistent analytical presentation of the mechanics of a material point, having examined in his two-volume Mechanics (1736) the motion of a free and non-free point in emptiness and in a resisting medium. Later, Euler laid the foundations of the kinematics and dynamics of a rigid body, obtaining the corresponding general equations. The results of these studies by Euler are collected in his Theory of the Motion of Rigid Bodies (1765). The set of dynamic equations representing the laws of momentum and angular momentum was proposed by the greatest historian of mechanics, Clifford Truesdell, to be called “Eulerian laws of mechanics.”

In 1752, Euler’s article “Discovery of a new principle of mechanics” was published, in which he formulated in general form Newton’s equations of motion in a fixed coordinate system, opening the way for the study of continuum mechanics. On this basis, he derived the classical equations of hydrodynamics for an ideal fluid, finding a number of their first integrals. His work on acoustics is also significant. At the same time, he was responsible for the introduction of both “Eulerian” (associated with the observer’s reference system) and “Lagrangian” (in the reference system accompanying the moving object) coordinates.

Euler's numerous works on celestial mechanics are remarkable, among which the most famous is his New Theory of the Motion of the Moon (1772), which significantly advanced the most important branch of celestial mechanics for navigation of that time.

Along with general theoretical research, Euler contributed to a number of important works in applied sciences. Among them, the first place is occupied by the theory of the ship. Issues of buoyancy, stability of a ship and its other seaworthiness were developed by Euler in his two-volume Ship Science (1749), and some issues of the structural mechanics of a ship were developed in subsequent works. He gave a more accessible presentation of the theory of the ship in the Complete Theory of the Structure and Driving of Ships (1773), which was used as a practical guide not only in Russia.

Euler's comments to B. Robins's New Principles of Artillery (1745) were a significant success, containing, along with his other works, important elements of external ballistics, as well as an explanation of the hydrodynamic “D'Alembert's paradox”. Euler laid down the theory of hydraulic turbines, the impetus for the development of which was the invention of the reactive “Segner wheel”. He also created the theory of stability of rods under longitudinal loading, which acquired particular importance a century later.

Euler's many works were devoted to various issues of physics, mainly geometric optics. Of particular note are the three volumes of Letters to a German Princess on various subjects of physics and philosophy published by Euler (1768–1772), which subsequently went through about 40 editions in nine European languages. These “Letters” were a kind of educational manual on the basics of science of that time, although their philosophical side did not correspond to the spirit of the Enlightenment.

The modern five-volume Mathematical Encyclopedia lists twenty mathematical objects (equations, formulas, methods) that now bear Euler's name. A number of fundamental equations of hydrodynamics and solid mechanics also bear his name.

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

The St. Petersburg archive of the Russian Academy of Sciences also stores thousands of pages of Euler’s unpublished research, mainly in the field of mechanics, a large number of his technical examinations, mathematical “notebooks” and colossal scientific correspondence.

His scientific authority during his lifetime was limitless. He was an honorary member of all the largest academies and scientific societies in the world. The influence of his works was very significant in the 19th century. In 1849, Carl Gauss wrote that “the study of all of Euler’s works will forever remain the best, irreplaceable, school in various fields of mathematics.”

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

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

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

(It should be noted that Euler’s last name in its true pronunciation sounds like “Oyler.”)

Publications: Collection of articles and materials. M. – L.: Publishing House of the USSR Academy of Sciences, 1935; Collection of articles. M.: Publishing House of the USSR Academy of Sciences, 1958.

Great mathematician
jonny_doll 28.09.2010 10:52:50

I was “lucky” once in my life to meet the descendants of this truly great mathematician. They live in Moscow and still bear this surname. To my great regret, they turned out to be simply thieves.

The centuries-old world history of the classical exact natural sciences - mathematics, astronomy, physics, as well as the mountain ranges of the Earth, has its greatest peaks. In a short period of time compared to universal human history - just a couple of thousand years, such peaks in Europe were Archimedes, Hipparchus, Ptolemy, Copernicus, Kepler, Galileo, Newton... With Newton, a branching began: the appearance of not individual peaks, but entire mountain ranges. chains, in the form of scientific schools in mathematics and in mechanics, terrestrial and celestial, which combined the then physics and astronomy. The density of new peaks in these mountain ranges was astonishing, indicating a massive assault on the problems posed by Newton had begun. This was facilitated by traditional scientific competitions with considerable prize funds, announced by European academies.

The very first high-mountain peaks among Newton's heirs were Leonard Euler, Alexis Claude Clairaut, Jean le Rond D'Alembert. In the middle of the century, a new peak rose in this dense massif - the young J.L. Lagrange. The interaction of these brilliant minds, not inferior to each other, was reflected in their correspondence, through which there was an exchange of ideas and results. And yet, the most impressive peak, which not only amazed with its height, the abundance of spurs, but also for all that with its accessibility to climbing it (for understanding), was undoubtedly Euler (. 1).

This is, perhaps, the peak most visited by historical climbers. In 1957 our country, led by the Academy of Sciences, widely celebrated the 250th anniversary of his birth. (The memorial academic medal that I have retained since then has now become an exhibit in the Museum of the History of Astronomy at the old Krasnopresnenskaya Observatory of the SAI). In 1983 Two close memorable dates were no less widely celebrated: 275 years since the birth and 200 years since the death of Euler (the result was a voluminous collection of materials from the Moscow and Leningrad conferences held by the Academy of Sciences together with the Institute of the History of Natural Science and Technology (IIEiT) of the USSR Academy of Sciences, published in 1988).

This year 2007 - a special anniversary - April 15 (New Style) will mark exactly 300 years since the birth of Leonhard Euler. Celebrations are planned in St. Petersburg. At Moscow State University, almost all natural sciences departments have turned their traditional “Lomonosov Readings” into “Eulerians”. At the SAI, the anniversary meeting of the City-wide Seminar on the History of Astronomy, held on April 3 of this year, was dedicated to this event. Department of the History of Physical and Mathematical Sciences of the IIE&T RAS, Sector of the History of the Astronomical Observatory and SAI and Sector of the History of Astronomy of the “Astronomical Society” (International Public Organization - Astro). This electronic publication is an expanded text of the report made by Ph.D. physics and mathematics A.I. Eremeeva (Senior Researcher of the said Sector of the SAI, Chairman of the Sector of the History of Astronomy of Astro).

Due to the inability for one speaker to cover both the scientific merits of the hero of the day and the versatility of his interests, the author limited his message to a brief reminder of the main directions and the most impressive results of the activities of this unique genius. The main attention was focused on a lesser-known aspect of his scientific biography - the origins and conditions of the formation of Leonhard Euler as the first and greatest heir and successor to Newton's work in creating a new natural science, namely, new mathematics, mechanics and theoretical astronomy. Particularly noted was his lesser-known contribution to observational astronomy and to the science that was emerging already in the 18th century. astrophysics.

Leonhard Euler's anniversaries have been and are now celebrated all over the world. He, undoubtedly, is the pride and heritage of all Mankind. But it was in Russia that Euler received his “initial speed”, went through a scientific school, and then throughout his life, had nutritious soil for his work in it - even finding himself a quarter of a century outside its borders (from 1741 to 1766 he lived and worked in Berlin, heading the mathematical department of the Academy of Sciences, and for several years practically the academy). The St. Petersburg Academy of Sciences became such fertile ground for Euler, with which he never broke ties, remaining its honorary foreign member abroad, and then again becoming its full member. In 1766 he returned to St. Petersburg and remained here until the end. Having lost in 1738 vision in the right eye, and in 1766, having gone blind in both, Euler did not lose his unique ability to work. Possessing an equally unique memory, he could carry out the most complex calculations in his head and over the last decade of his life he published the largest (compared to previous such periods) number of works (34!), dictating them to his students and assistants, the main of whom were A.I. Leksel, N.I. Fus and M.E. Golovin (nephew of M.V. Lomonosov).

It can be said that the first two great peaks in the picture and in the history of the activities of our Academy - Euler and Lomonosov became the clearest expression of the fulfillment of the plans and testament of the transformer of Russia - Peter the Great.

The beginning of the biography.

Leonhard Euler was born on April 4/15, 1707. in the small village of Rigen (or Rien) 5 km. from the city of Basel (in the north of Switzerland, where it meets France and Germany) (Fig. 2), in the family of a poor Protestant pastor (there were four children in the family). L. Euler's ancestors - several generations (from the 13th century) of artisans, moved from Germany (Lindau) to Switzerland in the 16th century. His father was the first to change his profession, graduating in 1700. Basel University, where he attended lectures on mathematics by the famous Jacob Bernoulli, and became a pastor, receiving a small parish in Riegen. Hoping to guide his son along the same spiritual path, he, nevertheless, who was not averse to an interest in mathematics, taught it to little Leonard, convinced that this science puts the mind in order.

Friendship with the Bernoulli family lasted throughout L. Euler’s life. His surprisingly early abilities in mathematics led him at the age of 13 and a half to the University of Basel (Fig. 3) at the Faculty of “Liberal Arts” (where he enrolled, since the other three faculties in this ancient university of the 15th century were traditionally legal, theological and medical).[According to (Yushkevich, 1988), before that, after his father, he was taught mathematics by a home teacher-theologian. According to (Rybakov, 1957), Euler studied at the seminary and attended the university “in his spare time”] Lectures by another Bernoulli professor, Johann (Jacob’s brother), private conversations with him and self-education under his leadership quickly developed Euler’s innate mathematical talent. In 1723 he completed the course, receiving the title of Bachelor of Philosophy. A year later he became a “Master of Arts” (for a comparative study of the natural philosophy of Descartes and Newton). And although, following the wishes of his father, L. Euler continued his education at the Faculty of Theology, he soon left it and completely immersed himself in mathematics. However, getting a place at the small University of Basel in the only department of physics close to him turned out to be unrealistic. Even the sons of I. Bernoulli himself - like his father, outstanding mathematicians and mechanics - were forced to focus on acquiring additional, more “practical” specialties. As Euler himself later wrote, if he had remained in his homeland, then, even after waiting for the physics department to be vacated, he would have been simply a “digger” (university professor) there...

Peter the Great and the St. Petersburg Academy of Sciences.

Building of the St. Petersburg Academy of Sciences

And at the same time, in distant Russia, the rapid activity of the tsar-transformer, Peter the Great, was unfolding, “with an iron hand” he raised his huge power “on its hind legs” - virgin soil that had not been plowed for great deeds. The pinnacle of this transformative activity of Peter was his main plan - to make Russia a new scientific and industrial European center, to educate its scientists, and for this to create an Academy and to attract the most famous learned men of Europe to it, charging them with the responsibility of teaching Russian youths.

There was talk of creating an academy with its own university and gymnasium. Peter invited the famous French astronomer, geodesist and cartographer, an employee of the Paris Observatory, Joseph Nicolas Delisle (1688 - 1768), whom he met in Paris in 1717, to be its first professor. The Tsar's decree on the establishment of the Academy was signed on January 28 (02/08), 1724.

Peter died exactly a year later (02/8!), literally on the eve of the implementation of his grandiose plan. But his closest heirs, even despite their distance from science, feeling the reflections of his glory, had to zealously fulfill his behests. The Academy was opened in August 1725 by Catherine I, showing special attention to it and giving it complete freedom (.4). And although during the sad era for science of the reign (from 1730) of Anna Ioannovna and the omnipotence of her favorite Biron, the new academy found itself in decline (this partly forced Euler to leave for Berlin), but it was revived again (from 1742) under Peter’s daughter Elizabeth and reached its, perhaps, brightest flowering under the first educated empress of Russia, Catherine II the Great. The Academy became a fertile soil on which many domestic and, at first, Western European talents flourished in all fields of science - natural and humanities. Young people from small Western countries (and they were all geographically incomparable to the size of Russia) literally poured into this vast virgin land (although it also took courage to decide to go to a distant, little-known northern country...). But the benefit and conditions were decent: the state took upon itself not only to ensure scientific work, but also provided publication and everyday life (and this includes a house, firewood, and candles...), so that scientists would not be distracted from science and, as Peter himself bequeathed , "don't waste time idle."

One of Peter's first tasks was to ensure the growth of the sciences necessary to create a fleet and study the vast expanses of the empire, that is, astronomy, geodesy, and cartography. According to Peter's Academic Charter, the title of professor of astronomy was assigned to the highest, first class. The support for these sciences was mathematics and mechanics (otherwise physics). Therefore, out of 17 invited professors (as members of the Academy were then called), in its first composition there were, in addition to Delisle, seven mathematicians and physicists.

Zh.N. Delisle in Russia and the creation of his scientific school.

Delisle enthusiastically accepted the invitation of the Russian autocrat. One of the first Newtonians on the continent, he suffered a lot from the dominance in Paris of stubborn adherents of the already outdated Cartesianism, who did not recognize Newton’s new discoveries, led by the new director of the Paris Observatory, the son of J. Cassini. Already at the beginning of 1726. Delisle arrived in St. Petersburg with his detailed plan for the construction and equipment of the first state observatory in Russia, drawn up for the Tsar, which soon became widely known, causing admiration in Europe for both its thoughtful architecture and its rich equipment (Fig. 5, 6).

In addition to two large wall quadrants and sextants, she had several refracting telescopes. Of particular value was its unique exhibit - Halley's 5-foot sextant (with which he worked on St. Helena Island in 1676), purchased at one time by Y.V. Bruce for Peter and transferred to the observatory in 1735. according to the will of Ya.V. Bruce by his nephew and sole heir of A.R. Bruce.

Delisle's plans to create an astronomical, geodetic and physical scientific school in Russia were grandiose, and the program for training new personnel was carefully thought out (Nevskaya, 1984). The list of literature he recommended to his students alone included 500 titles of essays. Before being allowed to work at the observatory, the newcomer had to master his science according to Delisle’s program, “to help himself,” as he said, “hatch from the egg.” It was necessary not only to master the literature, but also to actively apply the acquired knowledge - solving problems, mastering observational techniques. The purpose of all the work was, first of all, to serve the needs of the state: the creation of an accurate Time Service, which was soon carried out by Delisle; conducting geodetic surveys and mapping the country. The latter led to the creation, on the initiative of Delisle, of the Geographical Department of the Academy, on the model of which the Bureau of Longitudes in Paris was subsequently created, etc. In the field of pure science, Delisle focused on solving scientific problems bequeathed by Newton.

Before arriving in St. Petersburg, Delisle visited the great scientist and received his famous “questions” to solve. They concerned both astronomy - the development of the theory of the movement of celestial bodies, and physics - the problem of chromatism of lenses, the problem of light diffraction.

Among Delisle's first students and collaborators was 26-year-old Daniel Bernoulli, who received a position as professor of physiology (i.e. medicine), but soon switched to mathematics and mechanics. - The St. Petersburg Academy invited foreigners to fill vacant positions as new members. But in the future it was free to choose the real field of activity. - Soon his “scientific squad” of brilliant young minds formed around Delisle. The average age of his students was 31 years old, Delisle himself was 38, the youngest, 20 years old, was Leonhard Euler. He was invited, on the recommendation of D. Bernoulli, as his assistant and at the end of 1726. He was also appointed in absentia as an adjunct in the class of physiology, in connection with which he began to study it in his homeland for the planned work on the problem of blood circulation.

Euler in Russia. First period.

Euler arrived in St. Petersburg in the spring of 1727. during the days of mourning for the just deceased Catherine I and during some instability of the court. But this no longer affected the work of the observatory and the Delisle school, which had settled into its rhythm. The observatory was still being completed, but astronomical and meteorological observations were already being carried out in it (in another “chamber”). Delisle was in dire need of mathematicians and computer scientists. And D. Bernoulli’s proposal for his young mathematician friend came at a very opportune time. By a lucky coincidence, when Euler arrived at the Academy, the position of an adjunct mathematician turned out to be vacant, which he immediately took (with a salary of 300 rubles per year. - Rybakov, 1957). Euler quickly got involved in the work (Fig. 7,8), made several reports at each meeting of the Academy, and soon his scientific articles began to stream into the academic “Comments” (Notes) (Fig. 9). But his studies in physiology also came in handy for Euler - he studied the structure of the eye as a multilayer lens and later used his knowledge to solve the problem of ridding refractor lenses of chromatic aberration. Based on his theory (1747), John Dollond by 1758 built the first high-quality achromat refractor. Euler's fundamental general work "Dioptrics" on the theory of achromatism of telescopes and microscopes was published in St. Petersburg in 1769. (Fig. 10). But in general, Euler also quickly switched to mathematics and mechanics. From January 1731 he was already a professor of physics, and since June 1733. and forever - higher mathematics.

At the same time, Euler took part from the very beginning, from 1733. almost daily, and in observations at the observatory. Thus, observations of the Sun included the exact determination of the moment of noon, which since then began to be marked, at the suggestion of Delisle, by the firing of a fortress cannon; the heights of the luminaries were measured (to determine the latitude of the observatory), the coverage of stars and planets by the Moon. Comets were observed appearing.

Theory and practice in Euler's works

Leonhard Euler entered the history of science, first of all, as one of the greatest mathematicians. At the same time, the peculiarity of his mathematical genius also manifested itself early. Back at home, he successfully and enthusiastically solved applied mathematical problems: for example, how best to equip a ship with masts. This first work of his, submitted to the competition of the Paris Academy in 1726 - 1727, although it did not receive a prize, was approved in 1728. published. Subsequently, he enthusiastically solved similar engineering problems in Russia, incl. as an expert: in the 1770s. he boldly supported (the only one from the academic commission) the project of the brilliant Russian self-taught mechanic I.P. Kulibin single-arch bridge across the Neva with an unprecedentedly large span of 298m. (applied no more than 60m); participated in calculating the amount of materials for the monument to Peter - the figure of the Bronze Horseman. And each time he combined the solution of a specific problem with the development of the most theoretical, primarily mathematical, apparatus. Among his mathematical works of the first St. Petersburg period, one was devoted to the theory of music (1739)

Euler in Berlin.

In Berlin, Euler focused primarily on the development of a new theory of infinitesimal calculus - the great invention of Newton and Leibniz - differential and integral calculus, which became the main and effective method of analytical - using differential and integral equations - description of natural processes (12).

Euler was one of the first who began to translate the mathematical description of processes into the analytical language of differential equations (instead of the cumbersome and labor-intensive ancient Greek geometric and graphical methods used by Newton and Halley). In astronomy, these new methods for the first time made it possible to take on the solution of a grandiose task - the study and creation of a theory of the perturbed motion of celestial bodies - the Moon, planets and comets. The incredible complexity of the picture that was revealed also gave rise to ingenious approximate (numerical, semi-empirical) mathematical methods for deciphering and describing the true celestial movements in the real Solar system, far from the ideal Keplerian-Newtonian model for a two-body system. In the general field of mutual gravitations of many bodies, elliptical orbits not only “came to life” and “breathed,” changing their Keplerian elements over time - eccentricities, inclinations, turning the apsidal axes, but also turned out to be open curves altogether!

On the way to solving these problems, Euler became the founder of entire new directions and sciences, both in the field of higher mathematics and in theoretical mechanics. Countless mathematical images and ingenious methods for solving problems contain the name of Euler: “Eulerian numbers”, "Eulerian equations", "Eulerian substitution". The most elegant ones were even reflected on commemorative stamps. For example, this is amazing "Eulerian characteristic" convex polyhedra: a o -a 1 +a 2 =2(the number of vertices minus the number of edges plus the number of faces in any such polyhedron is two)

True, as they say, this was already known to Descartes, but, apparently, it was forgotten and rediscovered by Euler. Or - a beautiful formula for the connection between exponential and trigonometric functions: e iφ =cosφ+isinφ.

By the way, let us recall that a whole series of mathematical symbols were proposed by Euler: i - for the imaginary unit; e is the base of natural logarithms; Σ - sum; Δ is a finite difference and even, it seems, the most famous symbol is π.

Euler was the first to apply higher mathematics in cartography, in the theory of map projections, using the functions of a complex variable for the first time. The founder of the theory of complex variables itself is also L. Euler. And his fundamental work on applied mechanics, written at the request of the Academy, “Marine Science, or a Treatise on Shipbuilding and Navigation” (started in 1740, published in 1749 in St. Petersburg) became a significant contribution to the development of general hydromechanics, as well as kinematics and dynamics of solids. bodies. But he wrote a school (for an academic gymnasium) textbook on arithmetic (1738), and a more accessible course for sailors on the construction and driving of ships (1773), translated into several languages ​​(including into Russian by Lomonosov’s nephew M.E. Golovin ).

Euler as the founder of analytical methods and theories in celestial mechanics.

Of the nearly 850 works of L. Euler (including 20 large monographs), more than 100 relate to astronomy. (Out of 72 volumes of the complete collection of his works - its publication by the Swiss Society of Natural Scientists, begun in 1907 by international subscription, took several decades - 10 volumes are devoted to astronomy. Only Laplace “surpassed” him by one volume, but his total collection consisted of only 14 volumes .). Euler's scientific notebooks (which he continuously kept from 1725 to 1783) amounted to 12 notebooks (about 4 thousand pages). Even his huge correspondence (about 3 thousand letters), which he carefully preserved, in his own words, mostly contained scientific reflections, ideas, results - i.e. also represented a special form of his scientific creativity. In the absence of scientific periodicals in those days (which could not be replaced by the voluminous collections of Commentaries published by the St. Petersburg Academy), private correspondence was the main way of quickly exchanging information between scientists. (By the way, considerable postage costs for it were also provided in Russia by the Academy of Sciences.)

In astronomy, Euler occupied the first place with celestial mechanics, which he himself proposed to call “astronomical mechanics” (this is embodied, one might say, in the modern term “astrodynamics” - a section that studies the movement of close satellites, for example satellites, in a complex gravitational field far from spherical shape of the real Earth).

The impetus for such research was also primarily practical problems: the urgent need to clarify methods for determining longitude at sea, to accurately count time, and to study the phenomenon of ebb and flow. All this required, first of all, the development of the theory of the motion of the Moon. To solve the first problem, we recall that kings and governments announced competitions for large prizes. They were announced: in 1603 - Henry IV; in 1604 - Spanish king; in 1714 - the English Parliament, at the instigation of Newton, appointed a prize for the method of determining longitude with an accuracy of half a degree at 20 thousand pounds sterling (then = 200 thousand rubles in gold); in France appointed in 1716 on behalf of the king the prize was 100 thousand livres.

Newton also drew attention to the inevitability of deviations in the movements of celestial bodies from Keplerian ones. The reason was the mutual influence of the bodies of the Solar System, which became more and more noticeable with increasing accuracy of observations. In this regard, Newton already faced an alarming question about the stability of our planetary system, since the most noticeable such deviations were of a “secular” nature, directed in one direction - acceleration or deceleration of the movement of a planet or satellite (they were discovered first of all by Saturn and Jupiter, and also on the Moon in the first half of the 17th century by Newton’s compatriot J. Horrocks). By the way, the clear division of disturbances into secular and periodic is also the merit of Euler. The problem of perturbed motion became the main one for celestial mechanics of the 18th century.

Euler was one of the first after Newton, simultaneously with the French celestial mechanics, to begin solving it and began to create an analytical theory of the motions of celestial bodies.

In 1740, he created the first theory of tides after Newton, receiving a prize for it in a competition from the Paris Academy of Sciences. (Literally on his heels was D'Alembert, who also discovered tides in the atmosphere.)

By the middle of the 18th century. Interest in comets especially increased in connection with the approach of the first return of a periodic comet predicted and pre-calculated by Halley (in 1758) (1682, the future “Halley’s Comet”). Delisle (1742) also set the task of clarifying its orbit, placing great hopes on Euler, with whom he was in intensive correspondence during his Berlin period. In cometary astronomy, Euler was responsible for the discovery of an equation that allows one to determine the basic parameters of the parabolic orbit of a comet. He also came up with a way to determine, based on four or five observations, what type of conical section the comet’s orbit has. In 1744 Euler constructed the first theory of the motion of planets and comets based on Newtonian gravity.

The complexity of the picture of perturbed motions made it practically impossible to obtain solutions to problems of celestial mechanics in a general analytical form, as an exact solution of differential and integral equations. Approximate methods have become a new invention of the human mind. Euler was among the first here, having invented in 1768. one of the simplest methods for such an approximate, numerical solution of differential equations (“Euler’s method of broken lines”).

But the main invention of the mathematical genius of Euler in celestial mechanics was a new method of describing the perturbed motion of celestial bodies using differential equations - the method of varying arbitrary constants, which were considered to be the previously considered constant values ​​of the Keplerian elements that determine the shape and size of the orbit of a celestial body. Celestial mechanics included new images - osculating (envelopes), intermediate orbits, osculating elements. Euler successfully applied his new “analytical theory of perturbed motion in osculating elements” to the study of the orbits of Jupiter, Saturn, Earth, Venus and other celestial bodies. The concept of "osculating elements" has become central to modern celestial mechanics. And the differential equation derived by Euler to determine their change over time was included in it as the “Euler equation”.

An effective new mathematical apparatus in celestial mechanics was the theory of expansion of various studied functions into series - sequences, where with an increase in the number of terms of the series (the so-called convergent ones), the result came closer and closer to displaying the true motion or true orbit of the body. Euler was the first (1777) to derive formulas for calculating the coefficients of expansion of a function into a trigonometric series, anticipating by decades the appearance of trigonometric Fourier series (1811) (Nowadays they are known as the “Euler-Fourier formulas.” The latter introduced them as a method for studying thermal conductivity. But How surprised he might have been to learn that in this powerful method a new, analytical expression was found and... the system of the ancient Ptolemaic epicycles and deferents was set by the great Greek one and a half thousand years earlier, using the mathematical invention of Apollonius of Perga, who lived another fifty years! years earlier, he “saved phenomena” - this is precisely the task that ancient Greek astronomers set themselves - for the first time managing to reflect in their system of the world the unevenness of the visible movement of the Sun, Moon and planets.)

At the same time, Euler’s scientific credo was the conviction that not even the most ideal mathematical theory can work for a long enough time without taking into account an increasing amount of observational data that makes it possible to control the theory and bring it closer to the real state of affairs. In this he was closer to reality than the idealist-determinists (the latter included Laplace). It was precisely this “semi-empirical” approach to solving problems that allowed Euler to create the two best (most effectively applicable in practice) theories of the motion of the Moon out of 20 proposed by his contemporaries.

Euler's ingenious discovery was that, using series expansion, he took into account the largest disturbances as a first approximation, and then proceeded to take into account smaller ones, which ensured better convergence of the series, etc. solution to the problem. His first analytical theory of the motion of the Moon (1753), in which he continued and significantly improved the similar theory of Clairaut (1752), became the basis for very accurate lunar tables compiled in 1755. T. Mayer (for these works the long-announced prize of the English Parliament was paid in 1765, divided between L. Euler, the widow of T. Mayer and the inventor of the chronometer J. Harrison, who received the main amount - the advent of the century of technical progress affected). By the way, both this work and most of the works of Euler, who lived in Berlin but remained an honorary foreign member of the St. Petersburg Academy, were printed at its expense. After returning to Russia, in the works of 1770 and 1772. Euler completed the development of his theory of the perturbed motion of the Moon. As was understood much later, Euler’s lunar theory of 1772 its accuracy was a hundred years ahead of its era.

He received a special, double prize from the Paris Academy of Sciences (in total, Euler was awarded 12 competitive prizes) for the theory of the perturbed motion of the Earth (1756). The extreme importance of this work was that the Earth - its annual movement and daily rotation - until very recently remained the only standard for measuring time on all time scales - from a year to seconds! Somewhat earlier, Euler, simultaneously with D'Alembert, built the first complete dynamic theory of precession and nutation of the Earth's axis (1749). In addition, Euler predicted a small additional, "free" (not associated with the Moon) oscillation of the Earth's axis (with a period of 305 days - "Euler period"), which should have caused a change in the position of the pole and, consequently, fluctuations in geographical latitudes (observationally discovered and studied for the first time in 1881-1891 by S.K. Chandler, USA, who also clarified the period: 428 days - "Chandler period" ).

The interest in the history of astronomy brought up at the Delisle school (together with others, L. Euler studied the works of Ulugbek and other Eastern scientists) led Euler (as a result of comparisons of star catalogs from different eras) to the conclusion that the position of the ecliptic plane itself had changed. In this regard, he pointed out the need to refer in catalogs to the era of their compilation (for example, to the era of the ecliptic at the beginning of 1700 - And, perhaps, for good reason: from “1 January 1700” the new accounting of time introduced by Peter I began in Russia, the new chronology is not from the “creation of the world”, but from the Nativity of Christ, “from R.H.”).

Studying the perturbed motion of the Earth allowed Euler to obtain for the first time a convincing estimate of the mass of the comet. Buffon also admitted (based on the appearance of the heads of comets) that their masses are comparable to the solar mass! After the passage of Comet Halley near the Earth in April - May 1759. Euler calculated that if its mass were equal, the terrestrial year on Earth would have to increase (due to the disturbance of the orbit from the comet) by 27 minutes, and with a mass 100 times greater than the earth’s, the increase in the year would be 45 hours! And since not the slightest disturbance from Halley’s comet was observed, its mass, according to Euler’s estimate, turned out to be many orders of magnitude less than the Earth’s!

While studying the perturbed motion of the Galilean satellites of Jupiter, it was Io (much closer to its planet than the Moon is to the Earth) that Euler discovered its secular motion of the line of apses and orbital nodes. This was essentially the first attempt to create a theory of the motion of a close satellite around a highly compressed planet and anticipated the work that appeared after the launch of the first satellite, and many modern theories turned out to be less accurate than Euler’s.

That. It should be emphasized that Leonhard Euler is not only a Man of the whole World, but also a Man of all times: mathematicians and mechanics of our time continue to struggle with the problems he set.

One cannot help but recall another important problem, to the solution and formulation of which Euler made a great contribution. Among the most difficult celestial-mechanical problems posed and partially solved by Euler himself is the famous problem of the motion of three bodies in a mutually common gravitational field. (Newton had already shown that due to the peculiarities of the structure of the Solar system, when considering the gravitational interaction of the Sun and the planet, the role of the remaining bodies can be replaced by their total gravity, as if by the action of an effective “third” body.) Euler was the first to show the unsolvability in general form and the “problem of three bodies", which was followed by the brilliant French mathematician and celestial mechanic J.L. Lagrange. But both left their names in her private decisions. Euler was the first to find a special case of a solution to the problem. (Although this appeared in printed form only in his writings in 1862, but, as already mentioned, scientific information was then disseminated through correspondence.) He showed that in the solar system for every two bodies revolving around a common center of mass in one plane and neglecting the mass of the “third” body (the Sun - a planet; a planet and its satellite), on a straight line passing through these bodies, there are three points (determined by the ratio of the masses of the main bodies) at which the bodies placed in them will stably maintain their position. They can only fluctuate a little, i.e. experience libration around these positions. These are the so-called collinear Euler libration points - L1, L2, L3. Two of them are located on one side of the central body - in the vicinity of the second, closer and behind it (L1 and L2), and the third - on the other side of the central body, near the orbit of the second on its inner side (L3) (See Kulikovsky, 2002 , p.75 and 268). Two more libration points were discovered later by Lagrange (1772): these are the most widely known “Lagrange triangular libration points” - the vertices of equilateral triangles, the common base of which is the straight line: the planet - the Sun. At such points, for example, in the orbit of Jupiter, well-known groups of asteroids were indeed discovered and are stable: the “Greeks” in front of the planet (near L4) and the “Trojans” behind it (near L5). Similar (but only dust) clusters were discovered in 1961. and in the Earth-Moon system. In turn, Euler noted that his libration points delimit the regions of planetary and satellite motions. Subsequently, these conclusions of his developed into such images as the “Hill sphere” - a region of instability, strong disturbances in the movement of bodies in the vicinity of a given center of gravity.

Another problem that was not solved in Euler’s time was posed by him as the problem of motion in the gravitational field of two fixed centers. When trying to apply it to a planetary system, Euler became convinced that such a field would create a cucumber-shaped body rotating around a major axis, which does not occur in reality, and therefore abandoned its study. And only in our time the task was posed again for a real - pole-compressed and non-spherical planet (Earth), again with an important applied goal - the creation of an accurate theory of the motion of satellites. Having generalized the problem to complex values ​​of the satellite motion parameters, Moscow celestial mechanics E.P. Aksenov, E.A. Grebenikov and V.G. Demin received her general decision (which was awarded the State Prize in 1971). The motion of a body relative to two fixed centers is now called “Eulerian motion.”

Euler as a representative of the early St. Petersburg astrophysical school.

The problem of longitude was solved by the method of lunar distances (by comparing the moments of a particular distance of the Moon from a bright star - tabulated for a certain longitude (where it was indicated, for example, every 3 hours) and observed on the spot) or by a similar comparison of the moments of the Moon covering a star or planet. This gave rise to a new task, relevant in the 18th century. with a general “obsession” with the idea of ​​a plurality of inhabited worlds. - Is there an atmosphere on other planets, on the Moon? The manifestation of the latter was suspected in the pattern of the light rim of the eclipsed Sun or even in the width of the light ring during an annular eclipse. In the end, Euler came to the conclusion that if the Moon has an atmosphere, it is much (according to his estimate, 200 times) more rarefied than that of the Earth (the next estimate after him by F.V. Bessel in 1834 was - in 2000 once!). On the other hand, the appearance of coloration (in additional colors) of the edges of Venus when it was covered by the Moon, which Delisle noticed back in Paris, aroused in him another suspicion - that diffraction of light was observed here. The study of diffraction became one of the topics of physical research at the observatory in St. Petersburg. The latter was important for resolving the dispute about the very nature of light - corpuscular, according to Newton, or wave, according to Huygens, whose supporters were Delisle and Euler (just like him, erroneously identifying light and sound as longitudinal vibrations of the world ether).

The first essentially astrophysical research at the St. Petersburg Observatory was observations (in a camera obscura on the top floor of the observatory) and the study of sunspots. In the 30s All of Delisle’s employees took part in this, incl. Euler. He developed methods for accurately determining the position and movement of sunspots, which made it possible to clarify the period of rotation of the Sun. But most importantly, perhaps for the first time, they revealed the connection between the abundance of sunspots and auroras and even weather changes.

In the northern capital of Russia, auroras attracted special attention from members of the Delisle astrophysical school. In 1748 Euler published a clearly astrophysical work, “Physical Investigation into the Cause of the Tail of Comets, Aurorae, and Zodiacal Light.” It was directed against the ideas of Zh.Zh. Dortu de Meran, the author of a work on the same topic, who considered all these phenomena to be effects in the atmosphere of the Sun. Considering the nature of these phenomena to be the same, Euler believed that their common cause was the “repulsive” effect of solar rays on light particles, respectively, the atmosphere of a comet, the Earth or the Sun itself (Nevskaya, 1969). This explanation of comet tails was given by Newton, which was natural for a supporter of the corpuscular theory of light. The same explanation is even more surprising for Euler, a supporter of the wave theory of light. He associated the shape of comet tails with the speed of particles leaving the comet's head, and the length and brightness with the distance of the comet from the Sun and the size of the atmosphere around the comet's solid body. Euler drew up a program for studying the movements of particles from the comet's nucleus and for the first time explained the phenomenon that was later called "synchronism" - the throwing of new portions of matter into the comet's tail in several stages, when the previous parts of the tail were still preserved. He relied on Euler's research in 1835. Bessel. The founder of the new mechanical theory of comet tails, F.A., can also be called his heir. Bredikhina.

Euler equated the phenomenon of zodiacal light with the phenomenon of the rings of Saturn. (However, here he was only at the level of advanced ideas, since a similar explanation for this phenomenon - as a collection of small particles-satellites - was given by Gian Cassini, who was one of the first to discover the phenomenon of zodiacal light in 1683.) In the polar lights, Euler also saw the manifestation of a certain “dust ring" around the Earth, which is affected by radiation from the Sun.

The search for the atmospheres of other planets and the Moon made it necessary to study the Earth's atmosphere. To this end, Delisle and Euler back in the 30s. conducted experimental firing from a vertically mounted cannon to determine the elasticity of the atmosphere by the speed of propagation of light and sound from the shot.

The focus of the Delisle school on searching for atmospheres around the Moon and planets later determined the specific task of Lomonosov (who also belonged to the Delisle school) in his famous observations of Venus in 1761. with physical intentions - to discover its atmosphere, the power of which Delisle had already spoken about (this was indicated by the absence of any details on the disk of Venus, while it was from them, considered to be surface details, that the rotation periods of other planets were determined: Mars, Jupiter, Saturn).

It can be said that the origins of astrophotometry also go back to the first astrophysical works of Euler. In 1752 He wrote the essay "Discourse on the various degrees of light of the Sun and other celestial bodies."

Finally, Euler devoted a lot of attention and effort to cartographic work in St. Petersburg as an assistant to Delisle, the first director of the Geographical Department (after returning to St. Petersburg in 1766, he himself became its director, replacing the deceased M.V. Lomonosov). Euler, together with Delisle, was directly involved in the labor-intensive work of compiling and drawing large geographical maps of Russia and was one of the co-authors of the large Russian Geographical Atlas (1745). Euler’s mathematical talent also manifested itself here - in the critical analysis and development of the theory of various cartographic projections (one of which he himself suggested).

Euler's unique ability to work was manifested in an extremely wide range of his activities. It included lecturing to academic students, technical examinations, and training of future academicians. Thus, in Berlin, future outstanding academicians, astronomers and mathematicians S.Ya. lived and studied with Euler. Rumovsky, S.K. Kotelnikov and others. With his advice and recommendations, Euler took a direct part in the activities of the St. Petersburg Academy. It was on his recommendation that he was invited to the St. Petersburg Academy in 1757. (in place of the tragically deceased G. Richman) young Berlin physics professor F.U.T. Aepinus, who clearly showed himself in Russia both in physics and astronomy (the idea of ​​the icy body of comets, the problem of comet danger, the first theory of lunar volcanism). Euler's activity in this regard did not decrease after returning to Russia. At the beginning of this article, Euler’s technical expertise on Kulibin’s project in the 1770s was already mentioned. etc.

Euler and Lomonosov.

Above, both of these geniuses were named the main peaks during the formation of Russian science and the St. Petersburg Academy of Sciences itself. It was they who determined the scientific face of the academy. They were almost the same age. Euler highly valued Lomonosov's talent, knowledge, and activities. And the “evil genius” of the St. Petersburg Academy (in reality, a clever official who seized power) I.D., who tried to cause a clash between them on scientific grounds. Schumacher suffered a complete fiasco in this: the work of Lomonosov, which he deliberately sent to Euler in Berlin, which contained certain ideas that did not coincide with Euler’s ideas, met here, on the contrary, with complete goodwill and received a very high assessment from Euler.

But, as far as we know, both scientists never met in real life. When the young master Euler began his career as an adjunct at the academy in St. Petersburg, Lomonosov (only four years younger than him) at the age of 19 made his way to the academic bench of his “academy” - the Slavic-Greek-Latin “secondary” school in Moscow, rapidly making up for lost time in distant Kholmogory years (when he nevertheless mastered a lot of self-taught knowledge of Smotritsky’s “Grammar” and Magnitsky’s “Arithmetic”). In 1736 sent in a group of the best graduates to St. Petersburg, in the fall he was sent abroad for several years to study metallurgy and physics. His return in 1741 coincided with Euler's departure to Berlin. And Euler, who returned to Russia, no longer found the first Russian scientist, Academician M.V. Lomonosov alive.

But fate brought both great names together once again, this time on the path to the development of education in Russia. The main work of Lomonosov's life here - the creation of Moscow University, in the first difficult period of its existence, especially after the early death of its founder, found unexpected support from the seemingly far from this Euler. In 1774 L. Euler together with S.Ya. Rumovsky, the new director of the academic observatory, supported the idea of ​​​​creating the first astronomical observatory at Moscow University and signed a decision to transfer to him a large number of astronomical instruments and instruments from the Academy.

Personality, family and descendants of L. Euler.

Leonhard Euler as a person embodied an extremely ordered, integral, perfect personality. Unlike most of his foreign colleagues, he deeply immersed himself in Russian culture, mastered the Russian language, in which he even wrote letters in his own clear handwriting. He was very kind and prudently thrifty, the support and guardian of the patriarchal structure of his large family. As in many generations of his ancestors, the family had many children. But the medicine of his era was powerless even for the royal family of Peter...

Of Euler's 13 children, only five survived infancy. Of his three sons, the eldest Johann-Albrecht also became a full member of the Academy, for many years he was its permanent secretary, and in the last years of his father’s life he acted as his co-author in some of his works. The middle one became a doctor, the youngest became a military man. Although two daughters left descendants, they did not survive their father, like his wife, who was the same age, with whom he lived since 1734. almost 40 years. It was precisely to preserve the family structure and comfort, the maintenance of which he could not imagine without a mistress, that Euler, already very old, married for the second time the half-sister of his deceased wife. A large family (16 people upon returning to Russia) together with his other relatives lived in a house specially built for Euler. Like all old cities, St. Petersburg often burned. In 1771 The fire practically destroyed Euler's house, which was rebuilt. But nothing could change the once and for all established rhythm of life and, most importantly, the work of the great mathematician.

The calmness and optimism of a thinker and worker who has not lost his creative energy emanates from his portraits in old age (Fig. 19 - 22). But the most amazing discovery was made in the Tretyakov Gallery: the portrait of the “unknown old man” that was there turned out to be the last lifetime portrait of Leonhard Euler, for which he posed for the German artist Darbes in 1778.

Euler had 45 grandchildren; by the end of his life, 26 were still alive. Tens and even hundreds of Euler’s descendants, including direct ones, with the same family name, live in Russia and other countries. (The results of the enormous work on compiling this family tree (traced to the 13th century), carried out by two of his distant descendants in the mid-20th century, were published in 1988 in a commemorative collection for the 275th anniversary of L. Euler. This publication itself became a kind of tribute to the memory of this family and its great representative, recognition of the enormous contribution of its branches to various areas of Russian life. This also erased a shameful stain from our state, where in previous years, especially during the Second World War, the descendants of the great Russian scientist - the pride of Russia, Leonhard Euler. ...for their German roots by stupid, overly zealous politicized official bodies...)

This extraordinary life of an extraordinary man, who harmoniously combined the greatest genius and an amazingly simple hard worker, able to concentrate in any situation, is clearly characterized from different sides by three catchphrases about Euler: About his life: “They said that he could work with a cat on back and surrounded by his grandchildren."

A widely known response to the sudden (from a stroke) death of Euler on September 7/18, 1783. became the words that could be his most expressive epitaph: “He stopped calculating and living.”

In contrast to this, Laplace’s visionary statement sounds, in which the future immortality of the genius was embodied: “Read, read Euler: we are all his students.”

A brilliant mathematician of Swiss origin, founder of the Russian mathematical school. The scientific legacy of Leonhard Euler is colossal. He is responsible for classic results in mathematical analysis. He advanced its rationale, significantly developed integral calculus, methods for integrating ordinary differential equations and partial differential equations. Euler authored the famous six-volume course on mathematical analysis, including Introduction to Infinitesimal Analysis, Differential Calculus, and Integral Calculus (1748–1770). Many generations of mathematicians around the world studied from this “analytic trilogy.”

Leonhard Euler (1707–1783) was a brilliant mathematician of Swiss origin, founder of the Russian mathematical school. Born in Basel (Switzerland) on April 15, 1707 in the family of a pastor, he spent his childhood in a nearby village where his father received a parish. Here, in the lap of rural nature, in the pious atmosphere of a modest parsonage, Leonard received his initial education, which left a deep imprint on his entire subsequent life and worldview. Education at the gymnasium in those days was short. In the fall of 1720, thirteen-year-old Euler entered the University of Basel, three years later he graduated from the lower faculty of philosophy and, at the request of his father, enrolled in the theological faculty. In the summer of 1724, at a one-year university act, he read a speech in Latin on a comparison of Cartesian and Newtonian philosophy. Showing an interest in mathematics, he attracted the attention of Johann Bernoulli. The professor began to personally supervise the young man’s independent studies and soon publicly admitted that he expected the greatest success from the insight and sharpness of mind of young Euler.

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

Immediately upon his arrival in St. Petersburg, he completely immersed himself in scientific work and then amazed everyone with the fruitfulness of his work. His numerous articles in academic yearbooks, initially devoted primarily to problems in mechanics, soon brought him worldwide fame, and later contributed to the fame of St. Petersburg academic publications in Western Europe. A continuous stream of Euler's writings was published from then on in the proceedings of the Academy for a whole century.

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

In the fall of 1740, the internal situation in Russia became more complicated. This prompted Euler to accept the invitation of the Prussian king, and in the summer of 1741 he moved to Berlin, where he soon headed a mathematical class at the reorganized Berlin Academy of Sciences and Letters. The years Euler spent in Berlin were the most fruitful in his scientific work. This period also marks his participation in a number of heated philosophical and scientific discussions, including the principle of least action. The move to Berlin did not, however, interrupt Euler’s close ties with the St. Petersburg Academy of Sciences. He continued to regularly send his works to Russia, taught students sent to him from Russia, selected scientists to fill vacant positions at the Academy, and carried out many other assignments.

Euler's religiosity and character did not correspond to the environment of the “freethinking” Frederick the Great. This led to a gradual deterioration in the relationship between Euler and the king, who was well aware that Euler was the pride of the Royal Academy. In the last years of his Berlin life, Euler actually acted as president of the Academy, but never received this position. As a result, in the summer of 1766, despite the king’s resistance, Euler accepted the invitation of Catherine the Great and returned to St. Petersburg, where he then remained until the end of his life.

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

At the beginning of September 1783, Euler felt slightly unwell. On September 18, he was still engaged in mathematical research, but suddenly lost consciousness and, in the apt expression of the panegyrist, “stopped calculating and living.”

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

The scientific legacy of Leonhard Euler is colossal. He is responsible for classic results in mathematical analysis. He advanced its rationale, significantly developed integral calculus, methods for integrating ordinary differential equations and partial differential equations. Euler authored the famous six-volume course on mathematical analysis, including Introduction to Infinitesimal Analysis, Differential Calculus, and Integral Calculus (1748–1770). Many generations of mathematicians around the world studied from this “analytic trilogy.”

Euler obtained the basic equations of the calculus of variations and determined the ways of its further development, summing up the main results of his research in this area in the monograph Method for Finding Curved Lines Having the Properties of Maximum or Minimum (1744). Euler's significant contributions were to the development of function theory, differential geometry, computational mathematics, and number theory. Euler's two-volume course Complete Guide to Algebra (1770) went through about 30 editions in six European languages.

Fundamental results belong to Leonhard Euler in rational mechanics. He was the first to give a consistent analytical presentation of the mechanics of a material point, having examined in his two-volume Mechanics (1736) the motion of a free and non-free point in emptiness and in a resisting medium. Later, Euler laid the foundations of the kinematics and dynamics of a rigid body, obtaining the corresponding general equations. The results of these studies by Euler are collected in his Theory of the Motion of Rigid Bodies (1765). The set of dynamic equations representing the laws of momentum and angular momentum was proposed by the greatest historian of mechanics, Clifford Truesdell, to be called “Eulerian laws of mechanics.”

In 1752, Euler’s article “Discovery of a new principle of mechanics” was published, in which he formulated in general form Newton’s equations of motion in a fixed coordinate system, opening the way for the study of continuum mechanics. On this basis, he derived the classical equations of hydrodynamics for an ideal fluid, finding a number of their first integrals. His work on acoustics is also significant. At the same time, he was responsible for the introduction of both “Eulerian” (associated with the observer’s reference system) and “Lagrangian” (in the reference system accompanying the moving object) coordinates.

Euler's numerous works on celestial mechanics are remarkable, among which the most famous is his New Theory of the Motion of the Moon (1772), which significantly advanced the most important branch of celestial mechanics for navigation of that time.

Along with general theoretical research, Euler contributed to a number of important works in applied sciences. Among them, the first place is occupied by the theory of the ship. Issues of buoyancy, stability of a ship and its other seaworthiness were developed by Euler in his two-volume Ship Science (1749), and some issues of the structural mechanics of a ship were developed in subsequent works. He gave a more accessible presentation of the theory of the ship in the Complete Theory of the Structure and Driving of Ships (1773), which was used as a practical guide not only in Russia.

Euler's comments to B. Robins's New Principles of Artillery (1745) were a significant success, containing, along with his other works, important elements of external ballistics, as well as an explanation of the hydrodynamic “D'Alembert's paradox”. Euler laid down the theory of hydraulic turbines, the impetus for the development of which was the invention of the reactive “Segner wheel”. He also created the theory of stability of rods under longitudinal loading, which acquired particular importance a century later.

Euler's many works were devoted to various issues of physics, mainly geometric optics. Of particular note are the three volumes of Letters to a German Princess on various subjects of physics and philosophy published by Euler (1768–1772), which subsequently went through about 40 editions in nine European languages. These “Letters” were a kind of educational manual on the basics of science of that time, although their philosophical side did not correspond to the spirit of the Enlightenment.

The modern five-volume Mathematical Encyclopedia lists twenty mathematical objects (equations, formulas, methods) that now bear Euler's name. A number of fundamental equations of hydrodynamics and solid mechanics also bear his name.

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

The St. Petersburg archive of the Russian Academy of Sciences also stores thousands of pages of Euler’s unpublished research, mainly in the field of mechanics, a large number of his technical examinations, mathematical “notebooks” and colossal scientific correspondence.

His scientific authority during his lifetime was limitless. He was an honorary member of all the largest academies and scientific societies in the world. The influence of his works was very significant in the 19th century. In 1849, Carl Gauss wrote that “the study of all of Euler’s works will forever remain the best, irreplaceable, school in various fields of mathematics.”

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

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

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

The world's greatest mathematician: Leonhard Euler

Abstract on the course “Mathematics”

Completed by student gr. 2g21 12/22/12

Checked

Tomsk – 2012

Introduction

Leonard Euler() - mathematician, mechanic, physicist and astronomer. Swiss by origin.

In 1726, Leonhard Euler was invited to the St. Petersburg Academy of Sciences and moved to Russia in 1727. He was an adjunct (1726), and in 1731-41 and from 1766 an academician of the St. Petersburg Academy of Sciences (in 1742-66 a foreign honorary member). In 1741-66 he worked in Berlin, a member of the Berlin Academy of Sciences.

L. Euler is a scientist of extraordinary breadth of interests and creative productivity. Author of over 800 works on mathematical analysis, differential geometry, number theory, approximate calculations, celestial mechanics, mathematical physics, optics, ballistics, shipbuilding, music theory and others, which had a significant impact on the development of science. During the existence of the Academy of Sciences in Russia, he is considered one of its most famous members.

Leonhard Euler became the first who in his works began to erect a consistent edifice of the analysis of infinitesimals. Only after his research, set out in the grandiose volumes of his trilogy “Introduction to Analysis”, “Differential Calculus” and “Integral Calculus”, analysis became a fully formed science - one one of humanity's most profound scientific achievements.

Biography

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 the next work of 1771, “On bodies whose surface can be turned into a plane,” Leonhard Euler proves the famous theorem that any surface that can be obtained only by bending the plane, but without stretching or compressing it, if it is not conical and not 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.

Leonhard 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 Leonhard 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 stories naively and amusingly made a conversation with him as pleasant as it was desirable...” He could sometimes flare up, but “was not able to harbor feelings against anyone for a long time.” or anger...” - recalled.

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.

Leonhard Euler died on September 18, 1783 from apoplexy in the presence of his assistants, professors Kraft and Leksel. He was buried at the Smolensk Lutheran cemetery. (Lutheranism is the largest branch of Protestantism. Founded by Martin Luther in the 16th century). The Academy commissioned a famous sculptor who knew Euler well to create a marble bust of the deceased, and Princess Dashkova donated a marble pedestal.

Until the end of the 18th century, he remained the conference secretary of the Academy, who was replaced by his son, who married the daughter of the latter, and in 1826, his son, 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 Chebyshev's students: A. M. Lyapunov and others, defining the main features of the St. Petersburg mathematical school.

Conclusion

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.”

Leonard 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.

L. 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 these three fundamental questions, which 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 Leonhard Euler continues to develop today.