Leonhard Euler, one of the greatest mathematicians of all time, was distinguished by an uncontrollable thirst for knowledge and irrepressible energy. Many classical theorems in all areas of mathematics are named after him.
Leonhard Euler was born in the Swiss city of Basel on April 15, 1707. Paul Euler, the boy's father, was a pastor and dreamed that his son would follow in his footsteps. From the first years of his life, he teaches Leonard all kinds of sciences, wanting to instill in him a thirst for new knowledge. Euler showed a special talent for precision objects, and his father immediately began to develop his abilities. Paul himself devoted almost all his free time to mathematics, and in his youth he even attended the lessons of the famous Jacob Bernoulli.
Home schooling became a solid foundation for the boy’s further education. When he entered the Basel gymnasium, all subjects were given to him with extraordinary ease. However, the level of teaching in high school left much to be desired, and Euler began to look for new opportunities to gain knowledge. At the age of 13, Leonard entered the University of Basel at the Faculty of Liberal Arts. This is how he ends up attending lectures on mathematics by Jacob Bernoulli’s younger brother, Johann.
The professor notices a capable student and assigns Euler individual lessons. Under the sensitive guidance of Bernoulli, the boy gets acquainted with the most complex works of great mathematicians, learns to understand and analyze them. This approach to learning allowed Leonard to receive his first academic degree at the age of 16, when he was able to conduct a comparative analysis of the works of Descartes and Newton in Latin. So Euler becomes a Master of Arts.
After graduating from university, Paul again intervened in his son’s education. Convinced that Leonard will become a priest, his father forces him to learn languages: Hebrew and Greek. Euler did not achieve much success, so his father had to come to terms with his passion for mathematics. However, the 17-year-old boy cannot find a job in his specialty - all places at the university are filled. He continues to visit Professor Bernoulli's house and develops close friendships with his sons: Daniel and Nikolai.
In 1727, following the Bernoulli brothers, the scientist left for St. Petersburg. Here Euler becomes an adjunct of higher mathematics. In 1730, Leonhard Euler was offered to head the department of physics, and in January 1731 he became a professor. Since 1733, under his leadership there was already a department of higher mathematics. During the 14 years he spent in St. Petersburg, he published works on hydraulics, navigation, mechanics, cartography and, of course, mathematics. In total, he has more than 70 scientific papers. In the West, Euler is recognized precisely as a Russian scientist. Leonard's Swiss roots remind of themselves only in his personal life - he marries a Swiss woman, Katerina Gsell.
The St. Petersburg Academy of Sciences at that time could boast of a unique teaching staff. Such famous scientists as J. Herman, D. Bernoulli, H. Goldbach and many others teach and conduct scientific activities here. Such a company allows Euler to go as deep as possible into his research, and the scientist publishes more and more new works in the Academy’s publications. The most significant of them is the two-volume “Mechanics”.
Frederick II, being the King of Prussia, decides to open the Berlin Academy on the basis of the Society of Sciences. He invites Euler to work in Berlin on very favorable terms. In 1841, the scientist decided to move, nevertheless, he maintained active correspondence with Russian scientists, in particular with Lomonosov. In Berlin, Leonard Euler meets the President of the Academy of Sciences, Moreau de Maupertuis, and actually becomes his deputy - Moreau is often ill, and Euler carries out his duties.
In Germany, the scientist continues to work in the field of number theory, mathematical analysis and calculus of variations, and applies a new approach to the study of geometry. The result of Euler's research is a new science - topology. At the same time, shipbuilding and celestial mechanics fell into Leonard’s field of interests. In the latter, he achieves unprecedented success - he creates a theory of the movement of the Moon, taking into account the gravity of the Sun.
Euler never received the long-awaited post of President of the Academy, which became one of the main reasons for his return to St. Petersburg. Here he is warmly received by the patroness of science herself, Catherine II. The scientist enthusiastically begins to work for the benefit of Russia.
Age takes its toll, and at the age of 60, Euler almost completely loses his sight, however, he does not stop his scientific activity. After returning, he manages to publish 200 essays in various fields of science.
Leonard's first wife dies soon after the move and, a couple of years later, the scientist marries her sister Salome-Abigail Gsell. His children accept Russian citizenship.
The government highly values the scientist’s achievements and his contribution to the development of science. Even after ceasing his scientific activities, Euler and his family were fully provided with everything they needed at the expense of the state. Leonhard Euler dies in 1783 in St. Petersburg at the age of 75. By this time he had 5 children and 26 grandchildren. He left behind 800 scientific articles and 72 volumes devoted to various fields of science.
During his scientific career, Leonhard Euler founded the theory of functions with complex variables, ordinary differential equations, and partial differential equations. He became a pioneer in the calculus of variations and topology, and applied new methods of integration. Many theorems of algebra and number theory, which later became classical, are named after him.
Using the results of Stirling and Newton, Euler in 1732 (at the same time as McLaren) discovered the general law of summation. In other words, he expressed the partial sum, integral and derivative of an infinite series sn= ∑ u (k) through a series with common terms u (n). By examining the data obtained, as well as the ratio of Bernoulli numbers B2n+2:B2n, Euler determined that this series was divergent, however, he was able to calculate its approximate value. To do this, the scientist used the sum of all terms of the series that decrease. This discovery led to the concept of an asymptotic series, to which many famous mathematicians subsequently devoted their works. Among them are Laplace, Legendre, Lagrange, Poisson and Cauchy. The Euler-McLaren formula became the basis of finite difference theory.
Fascinated by d'Alembert's work, Euler began to study string theory. In his article “On the Vibration of a String,” the scientist finds a general solution to the vibration equation, taking the initial velocity to be zero. It had the form y = φ (x + at) + ψ (x - at), where a is a constant, and differed little from d'Alembert's solution. However, in 1766 Euler also found his own method, which would later be included in his “Integral Calculus” (1770). To do this, he introduced new coordinates, which brought the equation to a simpler form for integration: u = x + at, v = x - at. In modern textbooks on differential equations, such coordinates are called characteristic and are widely used for various types of calculations.
One of Euler's main discoveries was the formula named after him. It says that for any real x the equality eix = cosx + isinx is true (i is the imaginary unit, e is the base of the natural logarithm). Thus, the scientist connected the trigonometric function and the complex exponential. The formula was published in the book "Introduction to the Analysis of Infinitesimals" (1748). Continuing his research in this area, Euler obtained an exponential form of a complex number of the form z = reiφ.
In addition, he significantly simplified and shortened mathematical notations - he introduced notations for trigonometric functions: tg x, ctg x, sec x, cosec x and was the first to consider them as functions of a numerical argument, which became the basis of modern trigonometry.
As Laplace later claimed, all mathematicians of the 18th century studied with Euler. However, even several centuries later, his mathematical methods are used in maritime affairs, ballistics, optics, music theory and insurance.
During the existence of the Academy of Sciences in Russia, apparently one of its most famous members was the mathematician Leonhard Euler (1707-1783).
He became the first who began to build a consistent edifice of infinitesimal analysis in his works. Only after his research, set out in the grandiose volumes of his trilogy “Introduction to Analysis”, “Differential Calculus” and “Integral Calculus”, did analysis become a fully formed science - one of the most profound scientific achievements of mankind.
Leonhard Euler was born in the Swiss city of Basel on April 15, 1707. His father, Pavel Euler, was a pastor in Riechen (near Basel) and had some knowledge of mathematics. The father intended his son for a spiritual career, but he himself, being interested in mathematics, taught it to his son, hoping that it would later be useful to him as an interesting and useful activity. After finishing his home schooling, thirteen-year-old Leonard was sent by his father to Basel to listen to philosophy.
Among other subjects, elementary mathematics and astronomy were studied at this faculty, taught by Johann Bernoulli. Soon Bernoulli noticed the talent of the young listener and began to study with him separately.
Having received his master's degree in 1723, after delivering a speech in Latin on the philosophy of Descartes and Newton, Leonard, at the request of his father, began to study oriental languages and theology. But he was increasingly attracted to mathematics. Euler began to visit his teacher’s house, and between him and the sons of Johann Bernoulli - Nikolai
Daniil - a friendship arose that played a very important role in Euler’s life.
In 1725, the Bernoulli brothers were invited to become members of the St. Petersburg Academy of Sciences, recently founded by Empress Catherine I. When leaving, Bernoulli promised Leonard to notify him if there was a suitable occupation for him in Russia. The following year they reported that there was a place for Euler, but, however, as a physiologist in the medical department of the academy. Upon learning of this, Leonard immediately enrolled as a medical student at the University of Basel. Studying diligently and successfully
Science Faculty of Medicine, Euler also finds time for mathematical studies. During this time, he wrote a dissertation on the propagation of sound and a study on the placement of masts on a ship, which was later published in 1727 in Basel.
In St. Petersburg there were the most favorable conditions for the flowering of Euler's genius: material security, the opportunity to do what he loved, the presence of an annual magazine for publishing works. The largest group of specialists in the field of mathematical sciences in the world at that time worked here, which included Daniel Bernoulli (his brother Nicholas died in 1726), the versatile H. Goldbach, with whom Euler shared common interests in number theory and other issues, the author of works in trigonometry F.Kh. Mayer, astronomer and geographer J.N. Delisle, mathematician and physicist G.V. Kraft and others. Since that time, the St. Petersburg Academy has become one of the main centers of mathematics in the world.
Euler's discoveries, which thanks to his lively correspondence often became known long before publication, make his name increasingly widely known. His position in the Academy of Sciences improved: in 1727 he began work with the rank of adjunct, that is, a junior academician, and in 1731 he became a professor of physics, that is, a full member of the Academy. In 1733 he received the chair of higher mathematics, which was previously occupied by D. Bernoulli, who returned the same year to Basel. The growth of Euler's authority was uniquely reflected in the letters to him from his teacher Johann Bernoulli. In 1728, Bernoulli addressed “the most learned and gifted young man, Leonhard Euler,” in 1737, “the most famous and witty mathematician,” and in 1745, “the incomparable Leonhard Euler, the leader of mathematicians.”
In 1735, the Academy needed to perform a very difficult job of calculating the trajectory of a comet. According to academics, this required several months of labor. Euler undertook to complete this in three days and completed the work, but as a result he fell ill with nervous fever with inflammation of his right eye, which he lost. Soon after this, in 1736, two volumes of his analytical mechanics appeared. The need for this book was great; Many articles were written on various issues of mechanics, but there was no good treatise on mechanics.
In 1738, two parts of an introduction to arithmetic appeared in German, and in 1739, a new theory of music. Then in 1840 Euler wrote an essay on the ebb and flow of the seas, which was awarded one-third of the prize of the French Academy; the other two thirds were awarded to Daniel Bernoulli and Maclaurin for essays on the same topic.
At the end of 1740, power in Russia fell into the hands of regent Anna Leopoldovna and her entourage. An alarming situation has developed in the capital. At this time, the Prussian king Frederick II decided to revive the Society of Sciences in Berlin, founded by Leibniz, which had been almost inactive for many years. Through his ambassador in St. Petersburg, the king invited Euler to Berlin. Euler, believing that “the situation began to seem quite
unsure,” accepted the invitation.
In Berlin, Euler first gathered a small scientific society around him, and then was invited to join the newly restored Royal Academy of Sciences and was appointed dean of the mathematical department. In 1743, he published five of his memoirs, four of them on mathematics. One of these works is remarkable in two respects. It indicates a way to integrate rational fractions by decomposing them into
partial fractions and, in addition, the now common method of integrating linear ordinary equations of higher order with constant coefficients is presented.
In general, most of Euler's works are devoted to analysis. Euler so simplified and supplemented entire large sections of the analysis of infinitesimals, integration of functions, the theory of series, differential equations, which had already begun before him, that they acquired approximately the form that they occupied to a large extent remains to this day. Euler, in addition, began a whole new chapter of analysis - the calculus of variations. This initiative of his was soon picked up by Lagrange and thus a new science was formed.
In 1744, Euler published three works in Berlin on the movement of luminaries: the first is the theory of the movement of planets and comets, which contains a statement of the method for determining orbits from several observations; the second and third are about the movement of comets.
Euler devoted seventy-five works to geometry. Some of them, although interesting, are not very important. Some simply made up an era. Firstly, Euler should be considered one of the founders of research on geometry in space in general. He was the first to give a coherent presentation of analytical geometry in space (in “Introduction to Analysis”) and, in particular, introduced the so-called Euler angles, which make it possible to study rotations
bodies around a point.
In his 1752 work, “Proof of certain remarkable properties to which bodies bounded by plane faces are subject,” Euler found a relationship between the number of vertices, edges, and faces of a polyhedron: the sum of the number of vertices and faces is equal to the number of edges plus two. This relationship was suggested by Descartes, but Euler proved it in his memoirs. This is, in a sense, the first major theorem in the history of mathematics of topology - the deepest part of geometry.
While studying questions about the refraction of light rays and having written many memoirs on this subject, Euler published an essay in 1762 in which he proposed the design of complex lenses to reduce chromatic aberration. The English artist Doldond, who discovered two types of glass with different refrangibility, following Euler's instructions, built the first achromatic lenses.
In 1765, Euler wrote an essay in which he solves differential equations for the rotation of a rigid body, which are called the Euler equations for the rotation of a rigid body.
The scientist wrote many essays on the bending and vibration of elastic rods. These questions are interesting not only mathematically, but also practically.
Frederick the Great gave the scientist instructions of a purely engineering nature. So, in 1749, he instructed him to inspect the Funo Canal between Havel and Oder and make recommendations for correcting the shortcomings of this waterway. Next he was tasked with fixing the water supply in Sans Souci.
This resulted in more than twenty memoirs on hydraulics, written by Euler at different times. First-order hydrodynamic equations with partial derivatives of the projections of velocity, density and pressure are called Euler hydrodynamic equations.
After leaving St. Petersburg, Euler maintained the closest connection with the Russian Academy of Sciences, including the official one: he was appointed an honorary member, and he was given a large annual pension, and he, for his part, assumed obligations regarding further cooperation. He purchased books, physical and astronomical instruments for our Academy, selected employees in other countries, reporting detailed characteristics of possible candidates, edited the mathematical department of academic notes, acted as an arbiter in scientific
disputes between St. Petersburg scientists, sent topics for scientific competitions, as well as information about new scientific discoveries, etc. Students from Russia lived in Euler’s house in Berlin: M. Sofronov, S. Kotelnikov, S. Rumovsky, the latter later became academicians.
From Berlin, Euler, in particular, corresponded with Lomonosov, in whose work he highly valued the happy combination of theory and experiment. In 1747, he gave a brilliant review of Lomonosov’s articles on physics and chemistry sent to him for conclusion, which greatly disappointed the influential academic official Schumacher, who was extremely hostile to Lomonosov.
In Euler's correspondence with his friend Goldbach, an academician of the St. Petersburg Academy of Sciences, we find two famous “Goldbach problems”: to prove that every odd natural number is the sum of three prime numbers, and every even number is the sum of two. The first of these statements was proven using a very remarkable method already in our time (1937) by Academician I.M. Vinogradov, but the second has not been proven to this day.
Euler was drawn back to Russia. In 1766, through the ambassador in Berlin, Prince Dolgorukov, he received an invitation from Empress Catherine II to return to the Academy of Sciences on any terms. Despite persuasion to stay, he accepted the invitation and arrived in St. Petersburg in June.
The Empress provided Euler with funds to buy the house. The eldest of his sons, Johann Albrecht, became an academician in the field of physics, Karl took a high position in the medical department, Christopher, born in Berlin, was not released from military service by Frederick II for a long time, and it took the intervention of Catherine II so that he could come to his father. Christopher was appointed director of the Sestroretsk Armory
plant
Back in 1738, Euler went blind in one eye, and in 1771, after an operation, he almost completely lost his sight and could only write with chalk on a black board, but thanks to his students and assistants. I.A Euler, A I. Loksel, V.L. Kraft, S.K. Kotelnikov, M.E. Golovin, and most importantly N.I. Fuss, who arrived from Basel, continued to work no less intensively than before.
Euler, with his brilliant abilities and remarkable memory, continued to work and dictate his new memoirs. From 1769 to 1783 alone, Euler dictated about 380 articles and essays, and during his life he wrote about 900 scientific papers.
Euler's 1769 paper "On Orthogonal Trajectories" contains brilliant ideas about obtaining, using a function of a complex variable from the equations of two mutually orthogonal families of curves on a surface (i.e., lines such as meridians and parallels on a sphere), an infinite number of other mutually orthogonal families. This work turned out to be very important in the history of mathematics.
In his next work of 1771, “On bodies whose surface can be turned into a plane,” Euler proves the famous theorem that any surface that can be obtained only by bending a plane, but without stretching or compressing it, if it is not conical or cylindrical , is a set of tangents to some spatial curve.
Euler's work on map projections is equally remarkable.
One can imagine what a revelation Euler’s work on the curvature of surfaces and developable surfaces was for mathematicians of that era. The works in which Euler studies surface mappings that preserve similarity in the small (conformal mappings), based on the theory of functions of a complex variable,
should have seemed downright transcendental. And the work on polyhedra began a completely new part of geometry and, in its principles and depth, stood alongside the discoveries of Euclid.
Euler's tirelessness and perseverance in scientific research were such that in 1773, when his house burned down and almost all of his family's property was destroyed, even after this misfortune he continued to dictate his research. Soon after the fire, a skilled ophthalmologist, Baron Wentzel, performed cataract surgery, but Euler could not stand the appropriate time without reading and became completely blind.
Also in 1773, Euler's wife, with whom he lived for forty years, died. Three years later, he married her sister, Salome Gsell. His enviable health and happy character helped Euler “withstand the blows of fate that befell him. Always an even mood, soft and natural cheerfulness, some kind of good-natured mockery, the ability to tell naively and funny stories made conversation with him so
as pleasant as it was desirable...” He could sometimes flare up, but “he was not
capable of harboring anger against someone for a long time...” recalled N I Fuss.
Euler was constantly surrounded by numerous grandchildren, often with a child sitting in his arms and a cat lying on his neck. He himself taught mathematics to the children. And all this did not stop him from working.
On September 18, 1783, Euler died of apoplexy in the presence of his assistants, professors Kraft and Leksel. He was buried at the Smolensk Lutheran Cemetery. The Academy commissioned the famous sculptor Zh.D. Rachette, who knew Euler well, received a marble bust of the deceased, and Princess Dashkova presented a marble pedestal.
Until the end of the 18th century, I.A. remained the conference secretary of the Academy. Euler, who was replaced by N.I. Fuss, who married the daughter of the latter, and in 1826 - Fuss's son Pavel Nikolaevich, so that the organizational side of the life of the Academy was in charge of the descendants of Leonhard Euler for about a hundred years. Euler's traditions had a strong influence on students
Chebysheva: A.M. Lyapunova, A.N. Korkina, E.I. Zolotareva, A.A. Markov and others, defining the main features of the St. Petersburg mathematical school.
There is no scientist whose name is mentioned in educational mathematical literature as often as the name of Euler. Even in high school, logarithms and trigonometry are still taught largely “according to Euler.”
Euler found proofs of all Fermat’s theorems, showed the falsity of one of them, and proved Fermat’s famous Last Theorem for “three” and “four”. He also proved that every prime number of the form 4n+1 always decomposes into the sum of the squares of the other two numbers.
Euler began to consistently build an elementary theory of numbers. Starting with the theory of power residues, he then took up quadratic residues. This is the so-called quadratic reciprocity law. Euler also spent many years solving indefinite equations of the second degree in two unknowns.
In all three of these fundamental questions, which for more than two centuries after Euler constituted the bulk of elementary number theory, the scientist went very far, but in all three he failed. The complete proof was obtained by Gauss and Lagrange.
Euler took the initiative to create the second part of the theory of numbers - the analytic theory of numbers, in which the deepest secrets of integers, for example, the distribution of prime numbers in the series of all natural numbers, are obtained from considering the properties of certain analytic functions.
The analytical theory of numbers created by Euler continues to develop today.
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Switzerland (1707-1727)
University of Basel in the 17th-18th centuries
Over the next two years, young Euler wrote several scientific papers. One of them, “Thesis in Physics on Sound,” which received a favorable review, was submitted to the competition to fill the unexpectedly vacant position of professor of physics at the University of Basel (). But, despite the positive review, 19-year-old Euler was considered too young to be included in the list of candidates for the professorship. It should be noted that the number of scientific vacancies in Switzerland was very small. Therefore, the brothers Daniel and Nikolai Bernoulli left for Russia, where the organization of the Academy of Sciences was just underway; they promised to work there for a position for Euler.
Euler was distinguished by his phenomenal efficiency. According to contemporaries, for him living meant doing mathematics. And the young professor had a lot of work: cartography, all kinds of examinations, consultations for shipbuilders and artillerymen, drawing up training manuals, designing fire pumps, etc. He was even required to compile horoscopes, which Euler forwarded with all possible tact to the staff astronomer. But all this does not prevent him from actively conducting his own research.
During the first period of his stay in Russia, he wrote more than 90 major scientific works. A significant part of the academic “Notes” is filled with the works of Euler. He made reports at scientific seminars, gave public lectures, and participated in the implementation of various technical orders from government departments.
| All these dissertations are not only good, but also very excellent, for he [Lomonosov] writes about very necessary physical and chemical matters, which the wittiest people still did not know and could not interpret, which he did with such success that I am absolutely sure the truth of his explanations. In this case, Mr. Lomonosov must be given justice that he has an excellent talent for explaining physical and chemical phenomena. One should wish that other Academies would be able to produce such revelations, as Mr. Lomonosov showed.
Euler, in response to His Excellency the President of 1747 |
This high assessment was not hindered even by the fact that Lomonosov did not write mathematical works and did not master higher mathematics.
Portrait of 1756 by Emanuel Handmann (Kunstmuseum, Basel)
According to contemporaries, Euler remained a modest, cheerful, extremely sympathetic person all his life, always ready to help others. However, relations with the king do not work out: Frederick finds the new mathematician unbearably boring, not at all secular, and treats him dismissively. In 1759, Maupertuis, president of the Berlin Academy of Sciences, died. King Frederick II offered the post of president of the Academy to D'Alembert, but he refused. Friedrich, who did not like Euler, nevertheless entrusted him with the leadership of the Academy, but without the title of president.
Euler returns to Russia, now forever.
Russia again (1766-1783)
Euler worked actively until his last days. In September 1783, the 76-year-old scientist began to experience headaches and weakness. On September 7 () after lunch spent with his family, talking with Academician A. I. Leksel about the recently discovered planet Uranus and its orbit, he suddenly felt unwell. Euler managed to say: “I’m dying,” and lost consciousness. A few hours later, without regaining consciousness, he died of a cerebral hemorrhage.
“He stopped calculating and living,” Condorcet said at the funeral meeting of the Paris Academy of Sciences (fr. Il cessa de calculer et de vivre ).
Euler was a caring family man, willingly helped colleagues and young people, and generously shared his ideas with them. There is a known case when Euler delayed his publications on the calculus of variations so that the young and then unknown Lagrange, who independently came to the same discoveries, could publish them first. Lagrange always admired Euler both as a mathematician and as a person; he said: “If you really love mathematics, read Euler.”
Contribution to science
Euler left important works in various branches of mathematics, mechanics, physics, astronomy and a number of applied sciences. From the point of view of mathematics, the 18th century is the century of Euler. If before him achievements in the field of mathematics were scattered and not always coordinated, Euler was the first to link analysis, algebra, trigonometry, number theory and other disciplines into a single system, and added many of his own discoveries. A significant part of mathematics has since been taught “according to Euler.”
Thanks to Euler, mathematics included the general theory of series, the amazingly beautiful “Euler formula”, the operation of comparison over an integer modulo, the complete theory of continued fractions, the analytical foundation of mechanics, numerous methods of integration and solving differential equations, number e, designation i for the imaginary unit, the gamma function with its environment and much more.
Essentially, it was he who created several new mathematical disciplines - number theory, calculus of variations, theory of complex functions, differential geometry of surfaces, special functions. Other areas of his work: Diophantine analysis, astronomy, optics, acoustics, statistics, etc. Euler's knowledge was encyclopedic; in addition to mathematics, he deeply studied botany, medicine, chemistry, music theory, and many European and ancient languages.
- Dispute with D'Alembert about the properties of the complex logarithm.
- Dispute with English optician John Dollond about whether it was possible to create an achromatic lens.
In all the cases mentioned, Euler defended the correct position.
Number theory
He refuted Fermat's hypothesis that all numbers of the form are prime; It turned out that it is divisible by 641.
where is real. Euler derived an expansion for it:
,where the product is taken over all prime numbers. Thanks to this, he proved that the sum of a series of inverse primes diverges.
The first book on the calculus of variations
Geometry
In elementary geometry, Euler discovered several facts not noticed by Euclid:
- The three altitudes of a triangle intersect at one point (orthocenter).
- In a triangle, the orthocenter, the center of the circumscribed circle and the center of gravity lie on one straight line - the “Euler straight line”.
- The bases of the three altitudes of an arbitrary triangle, the midpoints of its three sides and the midpoints of the three segments connecting its vertices with the orthocenter all lie on the same circle (Eulerian circle).
- The number of vertices (B), faces (G) and edges (P) of any convex polyhedron are related by the simple formula: B + G = P + 2.
The second volume of Introduction to Infinitesimal Analysis () is the world's first textbook on analytical geometry and the foundations of differential geometry. The term affine transformations was first introduced in this book along with the theory of such transformations.
When solving combinatorial problems, he deeply studied the properties of combinations and permutations and introduced Euler numbers into consideration.
Other areas of mathematics
- Graph theory began with Euler's solution to the problem of the seven bridges of Königsberg.
- Polyline method Euler.
Mechanics and mathematical physics
Many of Euler’s works are devoted to mathematical physics: mechanics, hydrodynamics, acoustics, etc. In 1736, the treatise “Mechanics, or the science of motion, in an analytical presentation” was published, marking a new stage in the development of this ancient science. 29-year-old Euler abandoned the traditional geometric approach to mechanics and laid a strict analytical foundation for it. Essentially, from this moment mechanics becomes an applied mathematical discipline.
Engineering
- 29 volumes on mathematics;
- 31 volumes on mechanics and astronomy;
- 13 - in physics.
Eight additional volumes will be devoted to Euler's scientific correspondence (over 3,000 letters).
Stamps, coins, banknotes
Bibliography
- New theory of the moon's motion. - L.: Publishing house. USSR Academy of Sciences, 1934.
- A method for finding curved lines that have the properties of either a maximum or a minimum. - M.-L.: GTTI, 1934.
- Basics of point dynamics. - M.-L.: ONTI, 1938.
- Differential calculus. - M.-L., 1949.
- Integral calculus. In 3 volumes. - M.: Gostekhizdat, 1956-58.
- Selected cartographic articles. - M.-L.: Geodesizdat, 1959.
- Introduction to the analysis of infinites. In 2 volumes. - M.: Fizmatgiz, 1961.
- Ballistics research. - M.: Fizmatgiz, 1961.
- Letters to a German princess about various physical and philosophical matters. - St. Petersburg. : Nauka, 2002. - 720 p. - ISBN 5-02-027900-5, 5-02-028521-8
- Experience of a new theory of music, clearly presented in accordance with the immutable principles of harmony / trans. from lat. N. A. Almazova. - St. Petersburg: Ros. acad. Sciences, St. Petersburg scientific center, publishing house Nestor-History, 2007. - ISBN 978-598187-202-0(Translation Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae (Tractatus de musica). - Petropol.: Typ. Acad. Sci., 1739.)
See also
- Astronomical Observatory of the St. Petersburg Academy of Sciences
Notes
Literature used
- Mathematics of the 18th century. Decree. op. - P. 32.
- Glazer G.I. History of mathematics in school. - M.: Education, 1964. - P. 232.
- , With. 220.
- Yakovlev A. Ya. Leonard Euler. - M.: Education, 1983.
- , With. 218.
- , With. 225.
- , With. 264.
- , With. 230.
- , With. 231.
- To the 150th anniversary of Euler's death: collection. - Publishing House of the USSR Academy of Sciences, 1933.
- A. S. Pushkin. Anecdotes, XI // Collected Works. - T. 6.
- Marquis de Condorcet. Eulogy of Euler. History of the Royal Academy of Sciences (1783). - Paris, 1786. - P. 37-68.; cm.
The achievements of Leonhard Euler, the great Swiss mathematician and physicist, are outlined in this article.
Leonhard Euler's contribution to science in brief
Achievements in mathematics were recognized during the mathematician’s lifetime. In addition to the fact that he headed the departments of the Berlin and St. Petersburg academies, Euler was a member of the Royal Society of London and the Paris Academy of Sciences. A distinctive feature of the scientist was his productivity. During his lifetime, more than 550 of his articles and books were published.
Leonard had a fairly wide range of activities - he studied modern mathematics and mechanics, mathematical physics, elasticity theory, optics, machine theory, music theory, ballistics, insurance and marine science. Euler first formulated the mechanical principle of small action and put it into practice. He is responsible for the development of rigid body dynamics and kinematics.
What did Leonard Euler discover?
The scientist made many discoveries in various fields of science. While exploring celestial mechanics, he put forward a theory of the movement of the Moon; in the field of optics, Leonard formulated the formula for a biconvex lens. He also proposed a calculation method for calculating the refractive indices of a medium. Calculated the optical components for the microscope.
He devoted a lot of time to researching the vibrations of strings, membranes and plates. But Leonardo Euler's main achievement was in the field of mathematics. He developed mathematical analysis and laid the foundation for the development of mathematical disciplines. The mathematician was the first to introduce the complex argument function and laid the foundation for the complex variable function.
He is also the creator of the calculus of variations and derived the extremum of the functional. He also owns the following achievements - the discovery of the classical method for solving linear equations with constant coefficients, the method of varying arbitrary ones, he identified the main properties of the Riccati equation, he integrated linear equations and created methods for solving them, and created the Euler-Maclaurin summation formula.
Euler is the founder of the theory of special functions. He was the first to consider cosine and sine as functions and began to study the properties of cylindrical, hyperbolic functions and elliptic integrals. He was the first to apply natural equations of curves and laid the foundation for the theory of surfaces.
Leonhard Euler's contribution to mathematics is reflected in his main works: “Mechanics, or the Science of Motion, Presented Analytically”, “Theory of Rigid Body Motion”, “Differential Calculus”, “Introduction to Analysis”, “Integral Calculus”, “Universal Arithmetic” , “Letters on various physical and philosophical matters, written to a certain German princess...”, “Mechanics”.
We hope that from this article you learned what the achievements of the Swiss mathematician Leonhard Euler are.
Leonhard Euler is an outstanding mathematician and physicist. The most accurate definition that can be used to characterize the works created by Euler is brilliant materials that have become the property of all mankind.
It is by his methods that students of many generations are taught in schools and higher educational institutions. Leonard made an enormous contribution to the development of mathematical and physical sciences and became the founder of a major series of scientific discoveries. Thanks to his achievements, Euler was an honorary academician in many countries around the world.
Euler's main focus was mathematics, but he worked in many fields of science, which allowed him to leave a huge amount of important work in astronomy, physics, mechanics and several types of applied sciences. Euler became not only the most important representative of history in the creation of educational literature for school and university students, but also was a teacher for many outstanding mathematicians of several generations who became followers of Euler's teachings. Many famous mathematicians, both past and present, based their studies of mathematical sciences largely on the work of Leonard. Among them are such “kings” of mathematics as Laplace and Carl Friedrich Gauss. Until now, after many years since Euler's death, he is an inspiration for many scientists from all over the world in achieving new heights in the field of mathematics and its branches.
Even in the modern world, in the age of high technology, Leonhard Euler's educational materials remain extremely in demand. In the branches of mathematics, Euler's concepts such as:
- straight;
- straight line in a circle;
- point;
- theorem for polyhedra;
- polyline method (method for solving differential equations);
- integral of beta function and gamma function;
- angle (in mechanics - to determine the movement of bodies);
- number (for working in hydrodynamics).
It is probably impossible to find at least one area in mathematical science that is not based on the teachings of such a brilliant scientist as Euler. He left a truly significant mark on science.
But it is not only the contributions of Leonhard Euler in various scientific fields that are interesting and significant. His life was no less interesting. Leonard was born on April 15, 1707 in Basel. He was raised by his father, a theologian by training and a clergyman by occupation. The boy received his initial education at home. His father Paul once studied mathematics with Jacob Bernoulli. And now he shared his knowledge with his son. While developing logical thinking in his child, Paul still hoped that Leonard would continue his spiritual career in the future. But the little genius was so passionate about exact science that he did not spend a day without learning more and more about this interesting science from his father.
However, when the time came to begin serious study and obtain a specialty, Leonard’s father sent him to the University of Basel, where the young man became a student at the Faculty of Arts. There they were supposed to make him a spiritual man and guide him along the path of his father, the pastor. But his love for mathematics since childhood changed all of Paul’s plans and sent the guy along a different path - the path of precise calculations, formulas and numbers. Leonard became the best student in his class, thanks to his impeccable memory and high abilities. And Bernoulli himself noticed the mathematical successes of the young genius. He invited Euler to study at his home, and these studies became weekly.
At the age of 17, Leonard was awarded a master's degree for excellently reading a lecture in Latin on the philosophy of views of Newton and Deckard. Euler was noted for several more outstanding works, one of which (in physics) won a competition at the University of Basel for the position of professor. His work caused a storm of admiration and a flurry of positive reviews. But despite the high recognition of the talent of the young talent, he was considered too young to take the responsible position of a university professor.
Soon, thanks to the recommendations of Bernoulli's sons, with whom Leonard developed warm and friendly relations, Euler got his chance to improve his skills. He was invited to St. Petersburg to head the department of physiology. Realizing that he will not reach significant heights in his hometown, Leonard accepts the invitation, leaves Switzerland and goes to St. Petersburg.
Meanwhile, science was actively developing in Europe. The brilliant Leibniz presented to the world a project designed to create scientific academies. Having learned about the development of this project, Peter I approved the plan for creating a St. Petersburg academy. Outstanding professors were invited to it. To promote science education and the development of Russian scientists, a university and gymnasium were built at the academy. The members of the academy were faced with the task of compiling methodological manuals for the initial study of mathematics, mechanics, physics and other specialties. Euler wrote a manual for studying arithmetic, which was soon translated into Russian. This recommendation became the first in Russian education, according to which they began to teach schoolchildren,
and she forever marked Euler in history as a person who made a colossal contribution to the development of society.
Soon the power changed, instead of Peter I the throne was taken by Anna Ioannovna. Politics have changed, views on the state have changed, including in terms of education. The training academy began to be seen as an institution that brought great losses and did not bring much benefit to the government. Rumors began to circulate about its closure.
But despite all the difficulties, the academy survived and continued its activities. Some professors left, afraid of the new government. Thanks to this, Leonard took the vacant position of professor of physics, which also allowed him to receive a fairly large salary. A couple of years later, Leonhard Euler became an academician at the mathematics department.
In addition to his brilliant career, Leonard also had a happy life. At the age of 26, he married the beautiful and sophisticated Ekaterina Gzel, the daughter of a famous painter. The wedding day was set for the New Year, and all the academy employees were invited guests. The two families of the great Euler gathered to celebrate two holidays. A family of relatives and a family from the Academy of Sciences. After all, for him, work became a second home, and his colleagues became close friends.
Euler's performance was amazing. He could not live without his scientific career. One day he took on a development assignment received by the academy. The peculiarity was that the task was incredibly large in scope. Three months were allocated for its implementation. However, Euler wanted to stand out, show his outstanding abilities, and completed this task in three days. This caused a storm of positive discussions and admiration for the professor’s talent. But severe overexertion had a negative impact on the scientist’s body - unable to withstand the powerful load, Leonard became blind in one eye. But Euler showed resilience and philosophical wisdom, declaring that he would now be able to devote more time to his family and personal life, since from now on he would be less distracted by mathematics.
After this, Euler became even more famous among the luminaries of science, and his grandiose work, which deprived him of half of his sight, brought him truly worldwide fame. His brilliant analytical presentation of mechanics as a method of movement was the discovery of a new milestone in the world of science.
As the world improved, so did science. Euler began studying the description of physical phenomena using integrals. The difficulty was that Leonard lived in St. Petersburg, where the scientific academy was not considered outstanding and did not have due respect. The development of science was further deteriorated by the fact that a new ruler was announced in Russia - the young John. According to Euler, the situation in the development of scientific research became unstable and did not have a developed bright future. Therefore, Euler gladly accepted the invitation to work for the Berlin Academy. But at the same time, the mathematician promised not to forget the St. Petersburg Academy, to which he devoted many years of his life, and to help as much as possible. In 25 years he will return to Russian soil. But for now he and his family, wife and children, are moving to Berlin. However, the entire time Euler stayed in Berlin, he continued to write works for the Russian Academy, edit new methods of Russian scientists, acquire Russian scientific books, and also receive in his home students from Russia sent for an internship with the great scientist. And most importantly, he remains an honorary member of the St. Petersburg Academy.
Soon the collected works of Bernoulli are published, which the old professor sends to his student in Berlin with a request to continue his works. And Euler did not let his teacher down. Despite his health problems, he began to actively produce works, which subsequently gained enormous success and recognition. These works were:
- “Introduction to the analysis of infinite”;
- “Manuals on differential calculus”;
- “Theory of the movement of the moon”;
- “Marine Science”;
- “Letters on various physical and philosophical matters.”
The last of these works was Euler's next great breakthrough, which was translated into dozens of languages and published in many publications around the world. In addition, Euler wrote many scientific articles that were very successful.
Despite his academic education, the professor did not strive to write abstruse articles. He always wrote in a language understandable to people of any level of knowledge. He described his works as if he were studying the topic at the same time as the reader, starting with the discovery of the topic, awareness of the purpose of the work, and reasoning leading to a logical conclusion. Having independently gone through the path of learning, going through all its difficult stages, Euler knew what people feel when they begin to delve into the complex structure of science. Therefore, he tried to make his works interesting and understandable.
A great achievement was the discovery of formulas that determine the critical load during compression of a rod. In those years, this work did not create a need for its use, but almost a century later, it became necessary in the construction of railway bridges in England.
Leonard carried out a huge amount of work based on his discoveries and calculations. About 1000 pages of his works were published per year. This is a serious scale even for literary works. But the fact that these pages contained numbers and formulas in such a volume... The genius of the professor is admirable!
The new Empress Catherine II allocated impressive sums for the development of science, and, drawing attention to the talented professor, invited him to return to St. Petersburg and head the management of the mathematics department at the academy. In her proposal, she indicated a fairly substantial salary, noting that if this amount turns out to be insufficient for the professor, she is ready to accept his conditions, if only he agrees to come to St. Petersburg. Euler agrees to this lucrative offer, but they do not want to let him go from service in Berlin. After several of his requests were rejected, Euler resorted to a trick and simply stopped publishing scientific works. This yielded results, and he was finally allowed to leave for Russia. Upon arrival in St. Petersburg, the Empress presented the professor with all sorts of benefits, including allocating funds for the purchase of a personal home and its comfortable furnishings. Catherine the Great's first request was a draft of ideas to modernize the academy.
Active work and intense stress finally deprived Leonhard Euler of his precious vision. But even this did not stop the scientific genius from improving the scientific world. He dictates all his thoughts, discoveries, and scientific works to a young boy, who carefully writes everything down in German.
Soon a terrible unforeseen situation occurred - a huge fire broke out in St. Petersburg, killing many buildings. Including the professor's house. It was with difficulty that he was saved. Fortunately, his scientific work was practically undamaged. Only one work burned down - “A New Theory of the Motion of the Moon.” But thanks to Leonard’s impeccable, phenomenal memory, which remained with him even in old age, the destroyed work was restored.
Euler was forced to move his family to a new house. This caused the professor, who had lost his sight, a lot of inconvenience, since everything in this house was unfamiliar to him, and it was difficult for him to navigate by touch. Soon an outstanding German ophthalmologist, Wenzel, arrived in St. Petersburg. He intended to restore the great professor's sight. The operation, which lasted only a few minutes, restored Euler's vision in his left eye. The doctor strongly recommended that Leonard take care of his eyes, avoid prolonged strain, and not write or read. But the professor’s obsessive love for science did not allow him to adhere to the recommendations of the ophthalmologist. He began to work actively again, which led to dire consequences - he finally lost his sight. To the surprise of those around him, the genius treats everything that happened with incredible calm. His scientific activity even increased - a clear stream of thoughts allowed him to comprehend a number of scientific achievements that appeared on paper thanks to his students who wrote from dictation.
Leonard's wife soon died, and this became a serious shock for him, a man insanely attached to his family. Having lived with his beloved wife for 40 years, Euler could no longer imagine life without her. Science helped him take his mind off his grief. Until the last days of his life, Euler continued to work actively and productively. His eldest son became his main assistant in writing, as well as several faithful students. All of them were the eyes of a professor, allowing him to present the latest thoughts of a genius to the scientific world.
In 1793, Leonard felt a sharp deterioration in his health; severe and regular headaches caused him serious anxiety and no longer allowed him to work productively. At one of the important meetings with Lexel, discussing the discovery of the new planet Uranus, Euler felt very dizzy. Having managed to utter the words “I’m dying,” the brilliant professor lost consciousness. A later medical examination revealed that he died of a cerebral hemorrhage.
The great mathematician Leonhard Euler was buried in the St. Petersburg Smolensk cemetery. The world has lost a talented, excellent scientist, professor and incredible human being. But he left behind an enormous amount of information necessary for humanity.
