Biography of Chebyshev Pafnuty Lvovich. Mathematician Chebyshev: biography, achievements, personal contribution to science

Pafnuty Lvovich Chebyshev (1821-1894)

Pafnuty Lvovich Chebyshev left an indelible mark on the history of world science and the development of Russian culture.

Numerous scientific works in almost all areas of mathematics and applied mechanics, works, deep in content and bright in the originality of research methods, made P. L. Chebyshev famous as one of the greatest representatives of mathematical thought. An enormous wealth of ideas is scattered in these works, and despite the fact that fifty years have passed since the death of their creator, they have not lost either their freshness or relevance, and their further development continues at the present time in all countries of the globe, where only the pulse of creative mathematical thought beats.

P. L. Chebyshev was available to everyone who wanted to work scientifically and had the data for this; he generously shared his ideas. Thanks to this, he left behind a large number of students who later became first-class scientists; among them are A. M. Lyapunov and A. A. Markov, essays on which are placed in this book. From him come the origins of many Russian mathematical schools in probability theory, number theory, the theory of approximation of functions, the theory of mechanisms, which successfully continue to work today.

The life of Pafnuty Lvovich Chebyshev is not rich in external events. He was born on May 26, 1821 in the village of Okatovo, Borovsky district, Kaluga province. He received his initial education and upbringing at home; he was taught literacy by his mother Agrafena Ivanovna, and arithmetic and French by his cousin Sukharev, a highly educated girl who, apparently, played a significant role in the education of the future mathematician. In 1832, the Chebyshev family moved to Moscow to prepare Pafnuty Lvovich and his older brother to enter the university. At the age of sixteen, he became a student at Moscow University and a year later he was awarded a silver medal for a mathematical essay on a topic proposed by the faculty. Since 1840, the financial situation of the Chebyshev family was shaken, and Pafnuty Lvovich was forced to live on his own earnings. This circumstance left an imprint on his character, making him prudent and thrifty; later, when he no longer experienced a lack of funds, he did not respect the economy in spending them only in the manufacture of models of various instruments and mechanisms, the ideas of which were often born in his head. At the age of twenty, P. L. Chebyshev graduated from the university, and two years later he published his first scientific work, which was soon followed by a number of others, more and more significant and quickly attracted the attention of the scientific world. At the age of twenty-five, P. L. Chebyshev defended his thesis for a master's degree at Moscow University on the theory of probability, and a year later he was invited to the department of St. Petersburg University and moved to St. Petersburg. Here began his professorial activity, to which P. L. Chebyshev devoted a lot of effort and which continued until he reached an advanced age, when he left lectures and devoted himself entirely to scientific work, which lasted literally until the last moment of his life. At the age of twenty-eight, he received a doctorate degree from St. The Academy of Sciences elected the thirty-two-year-old P. L. Chebyshev as an adjunct in the department of applied mathematics; six years later he had already become an ordinary academician. A year later he was elected a corresponding member of the Paris Academy of Sciences, and in 1874 the same academy elected him as its foreign member.

On December 8, 1894, Pafnuty Lvovich Chebyshev died in the morning, sitting at his desk. The day before was his reception day and he informed the students of the plans for his work and led them to think about topics for independent creativity.

To this external outline of P. L. Chebyshev’s life, we must add the characterization of him as a teacher and scientific educator, left by his contemporaries and students. The weight that the scientific school he founded has acquired in the history of mathematics already shows with maximum objectivity, regardless of personal opinions, that P. L. Chebyshev was able to kindle the scientific enthusiasm of his students. The main feature of this school, which is usually called the St. Petersburg school of mathematics, was the desire to closely connect the problems of mathematics with the fundamental questions of natural science and technology. Once a week, P. L. Chebyshev had a reception day, when the doors of his apartment were open to anyone who wanted to get some advice about their research. Few people left without enriching themselves with new thoughts and new plans. Contemporaries and, in particular, students of P. L. Chebyshev say that he willingly revealed the richness of his ideological world not only in conversations with the elite, but also in his lectures for a wide audience. To this end, he sometimes interrupted the course of the exposition in order to illuminate to his listeners the history and methodological significance of this or that fact or scientific position. He attached great importance to these retreats. They were quite long. Starting such a conversation, P. L. Chebyshev left the chalk and blackboard and sat down in a special chair that stood in front of the first row of listeners. Otherwise, the students characterize him as a pedantically accurate and accurate lecturer, who never missed, was never late, and never delayed the audience one minute longer than the allotted time. It is also interesting to note salient feature his lectures: he preceded any complex calculation with an explanation of its purpose and course in the most general terms, and then conducted it silently, very quickly, but in such detail that it was easy to follow him.

Against the background of this measured, prosperous life, not marked by any external shocks, in the quiet of the scientist’s calm office, great scientific discoveries which were destined not only to change and rebuild the face of Russian mathematics, but also to have a huge, invariably felt influence over the course of a number of generations on the scientific work of many outstanding scientists and scientific schools abroad. P. L. Chebyshev was not one of those scientists who, having chosen any one more or less narrow branch of their science, give it their whole life, first creating its foundations, and then carefully refining and improving its details. He belonged to those "wandering" mathematicians whom science knows among its greatest creators and who see their vocation in moving from one scientific field to another, in each of them leaving a number of brilliant basic ideas or methods, developing consequences or details of which they willingly provide their contemporaries and future generations. This does not mean, of course, that such a scientist annually changes the field of his scientific interests and, having published one or two articles in his chosen field, leaves it forever. No, we know that P. L. Chebyshev was engaged, for example, all his life developing more and more new problems of his famous theory of approximation of functions, that he addressed the main problems of probability theory three times - at the beginning, in the middle and at the very end of his creative way. But it is characteristic that he had many such chosen areas (the theory of integration, the approximation of functions by polynomials, the theory of numbers, the theory of probability, the theory of mechanisms, and a number of others) and that in each of them he was mainly attracted by the creation of basic, general methods, the expansion of the circle ideas, rather than a logical conclusion by carefully finishing all the details. And it is almost impossible to indicate a region where the seeds thrown by him would not give abundant and powerful shoots. His ideas were picked up and developed by a brilliant galaxy of students, and then became the property of wider scientific circles, including foreign ones, and everywhere they successfully recruited followers and successors. Among these ideas, there were those whose entire methodological significance could not be sufficiently realized by contemporaries and was revealed in its entirety only in the studies of subsequent generations of scientists.

as another the most important feature Scientific creativity of P. L. Chebyshev should be noted for his constant interest in practical issues. This interest was so great that, perhaps, it largely determines the originality of P. L. Chebyshev as a scientist. It can be said without exaggeration that most of his best mathematical discoveries were inspired by applied work, in particular his research on the theory of mechanisms. The presence of this influence was often emphasized by Chebyshev himself both in mathematical and applied works, but the idea of the fruitfulness of the connection between theory and practice was most fully expressed by him in the article "Drawing Geographical Maps". We will not retell the thoughts of the great scientist, but will quote his true words:

"The convergence of theory with practice gives the most beneficial results, and not only practice benefits from this; the sciences themselves develop under its influence, it opens up new subjects for research, or new aspects in subjects long known. Despite that high degree of development, up to which the mathematical sciences have been completed by the works of the great geometers of the last three centuries, practice clearly reveals their incompleteness in many respects, it proposes questions essentially new to science, and thus calls for the discovery of completely new methods. new developments of it, then it acquires even more by the discovery of new methods, and in this case science finds its true guide in practice. Among the huge number of tasks posed to a person by his practical activity, according to P. L. Chebyshev, one is of particular importance: "how to dispose of one's funds in order to achieve the greatest possible benefit?" That is why "most of the questions of practice are reduced to problems of the largest and smallest values, completely new to science, and only by solving these problems can we satisfy the requirements of practice," which everywhere seeks the best, the most profitable.

The above quotation for P. L. Chebyshev was the program of his entire scientific activity, was the guiding principle of his work.

Numerous applied works of P. L. Chebyshev, bearing far from mathematical names - "On a Mechanism", "On Gears", "On a Centrifugal Equalizer", "On the Construction of Geographic Maps", "On Dress Cutting" and many others, were combined one basic idea - how to dispose of cash to achieve the greatest benefit? So, in the work "On the construction of geographical maps" he sets himself the goal of determining such a projection of a map of a given country for which the scale distortion would be minimal. In his hands this task received exhaustive solution. For European Russia, he brought this solution to numerical calculations and found out that the most advantageous projection would give a scale distortion of no more than 2%, while the projections adopted at that time gave a distortion of at least 4-5% ( Part of the essay concerning the works of P. L. Chebyshev on the theory of mechanisms and marked at the beginning and end with asterisks, belongs to Acad. I. I. Artobolevsky)).

He spent a significant part of his efforts on the design (synthesis) of hinged mechanisms and on the creation of their theory. He paid special attention to the improvement of Watt's parallelogram - a mechanism that serves to turn a circular motion into a rectilinear one. The point was that this main mechanism for steam engines and other machines was very imperfect and gave curvilinear instead of rectilinear motion. Such a substitution of one movement for another caused harmful resistances that spoiled and wore out the machine. Seventy-five years have passed since Watt's discovery; Watt himself, his contemporaries and subsequent generations of engineers tried to fight this defect, but, groping, by trial, they could not achieve significant results. P. L. Chebyshev looked at the matter from a new point of view and posed the question as follows: to create mechanisms in which the curvilinear motion would deviate as little as possible from the rectilinear one, and at the same time determine the most advantageous dimensions of the machine parts. With the help of a specially developed apparatus of the theory of functions that deviate least from zero, he showed the possibility of solving the problem of approximately rectilinear motion with any degree of approximation to this motion.

On the basis of the method developed by him, he gave a number of new designs of approximate guiding mechanisms. Some of them still find practical application in modern devices.

But the interests of P. L. Chebyshev were not limited to consideration of only the theory of approximate-guiding mechanisms. He was engaged in other tasks that are also relevant for modern engineering.

Studying the trajectories described by individual points of links of hinged-lever mechanisms, P. L. Chebyshev stops at trajectories, the shape of which is symmetrical. By studying the properties of these symmetrical trajectories (crank curves), he shows that these trajectories can be used to reproduce many forms of movement that are important for technology. In particular, he shows that it is possible to reproduce rotational motion with different directions of rotation about two axes by hinged mechanisms, and these mechanisms will be neither parallelograms nor antiparallelograms, which have some remarkable properties. One of these mechanisms, later called paradoxical, is still the subject of surprise for all technicians and specialists. The gear ratio between the drive and driven shafts in this mechanism can vary depending on the direction of rotation of the drive shaft.

P. L. Chebyshev created a number of so-called mechanisms with stops. In these mechanisms, which are widely used in modern automation, the driven link performs intermittent movement, and the ratio of the idle time of the driven link to the time of its movement should change depending on the technological tasks assigned to the mechanism. P. L. Chebyshev for the first time gives a solution to the problem of designing such mechanisms. He has priority in the issue of creating mechanisms for "motion rectifiers", which, at the very recent times have been used in a number of designs of modern instruments, and such transmissions as progressive transmissions such as Vasant, Constantinescu and others.

Using his own mechanisms, P. L. Chebyshev built the famous stepping machine (step-walking machine), imitating the movement of an animal with its movement; he built the so-called rowing mechanism, which imitates the movement of the oars of a boat, a scooter chair, gave an original model of a sorting machine and other mechanisms. Until now, we have been observing the movement of these mechanisms with amazement and are amazed at the rich technical intuition of P. L. Chebyshev.

P. L. Chebyshev created over 40 different mechanisms and about 80 of their modifications. In the history of the development of the science of machines, it is impossible to point to a single scientist whose work would have produced such a significant number of original mechanisms.

But P. L. Chebyshev solved not only the problems of synthesis of mechanisms.

He, many years earlier than other scientists, derives the famous structural formula of flat mechanisms, which only due to a misunderstanding is called the Grübler formula - a German scientist who discovered it 14 years later than Chebyshev.

P. L. Chebyshev, independently of Roberts, proves the famous theorem on the existence of three-hinged four-link links describing the same connecting rod curve, and widely uses this theorem for a number of practical problems.

The scientific heritage of P. L. Chebyshev in the field of the theory of mechanisms contains such a wealth of ideas that paints the image of the great mathematician as a true innovator of technology.

For the history of mathematics, it is especially important that the design of mechanisms and the development of their theory served as the starting point for P. L. Chebyshev to create a new branch of mathematics - the theory of the best approximation of functions by polynomials. Here P. L. Chebyshev was a pioneer in the full sense of the word, having absolutely no predecessors. This is the area where he worked more than in any other, finding and solving more and more new problems and creating a new extensive branch of mathematical analysis by the totality of his research, which continues to develop successfully even after his death. The original and simplest formulation of the problem began with the study of Watt's parallelogram and consisted in finding a polynomial of a given degree, which would deviate less than all other polynomials of the same degree from zero in some given interval of change of the argument. Such polynomials were found by P. L. Chebyshev and were called "Chebyshev polynomials". They have many remarkable properties and are currently one of the most widely used research tools in many questions of mathematics, physics and technology.

The general formulation of P. L. Chebyshev’s problem is related to the main problems of the application mathematical methods to natural science and technology. It is known that the concept of functional dependence between variables is fundamental not only in mathematics, but also in all natural and technical sciences. The question of calculating the values of a function for each given value of the argument arises before anyone who studies the relationship between various quantities that characterize a particular process, a particular phenomenon. However, direct calculation of the values of functions can be performed only for a very narrow class of functions of polynomials and a quotient of two polynomials. Therefore, the problem of replacing a computed function close to it by a suitable polynomial arose long ago. Of particular interest has always been the problem of interpolation, i.e., finding polynomial n-th degree that takes exactly the same values as the given function for n + 1 given values of the argument. The formulas proposed by the famous mathematicians Newton, Lagrange, Gauss, Bessel and others solve this problem, but have a number of drawbacks. In particular, it turns out that the addition of one or more new values of the function requires redoing all calculations anew, and more importantly, an increase in the number n, i.e., the number of coinciding values of the function and the polynomial, does not guarantee unlimited convergence of their values for all values of the argument. Moreover, it turns out that there are such functions for which, in case of an unsuccessful choice of the values of the argument, for which the values of the function and the polynomial coincide, the removal of the polynomial from the approximated function can even be obtained.

P. L. Chebyshev could not reconcile himself to such a serious shortcoming in a question that plays an outstanding role both in theory and in practice, and approached it from his own point of view. In his statement, the interpolation problem was transformed as follows: among all polynomials of a given degree, find the one that gives the smallest absolute values of the differences between the values of the function and the polynomial for all values of the argument in a given interval of its change. This setting was extremely fruitful and had an exceptional influence on the work of subsequent mathematicians. At present, there is a huge literature devoted to the development of the ideas of P. L. Chebyshev, at the same time, the range of problems in which the methods developed by P. L. Chebyshev are of invaluable benefit is expanding.

We will stop at brief description achievements of P. L. Chebyshev are still only in two areas - number theory and probability theory.

It is difficult to point to another concept that is as closely connected with the emergence and development of human culture as the concept of number. Take away this concept from mankind and see how much our spiritual life and practical activity will become impoverished: we will lose the opportunity to make calculations, measure time, compare distances, and sum up the results of labor. No wonder the ancient Greeks attributed to the legendary Prometheus, among his other immortal deeds, the invention of the number. The importance of the concept of number prompted the most prominent mathematicians and philosophers of all times and peoples to try to penetrate the secrets of the arrangement of prime numbers. Of particular importance in ancient greece received a study of prime numbers, i.e., numbers divisible without remainder only by itself and by one. All other numbers are, therefore, products of prime numbers, and hence prime numbers are the elements from which every whole number is formed. However, results in this area were obtained with the greatest difficulty. Ancient Greek mathematics, perhaps, knew only one general result about prime numbers, now known as Euclid's theorems. According to this theorem, there are an infinite number of primes in the series of integers. On the same questions about how these numbers are located, how correctly and how often, Greek science did not have an answer. About two thousand years that have passed since the time of Euclid have not brought any changes in these problems, although many mathematicians have dealt with them, among them such luminaries of mathematical thought as Euler and Gauss. Empirical calculations made by Legendre and Gauss led them to the conclusion that within the tables of primes known to them, the number of primes among all the first n numbers is approximately In n times less than the number l. This statement remained a purely empirical fact, established only for numbers within a million. There was no reason to carry it over to large values of n, and there were no ways for a rigorous proof. In the 40s of the last century, the French mathematician Bertrand made another hypothesis about the nature of the arrangement of prime numbers: between n and 2n, where n is any integer greater than one, there must be at least one prime number. For a long time this hypothesis remained only an empirical fact, for the proof of which there was absolutely no way.

The analysis of Euler's scientific heritage awakened Chebyshev's interests in number theory and made it possible to manifest here the strength of his mathematical talent. Having taken up the theory of numbers, P. L. Chebyshev, using absolutely elementary methods, established an error in the Legendre-Gauss hypothesis and corrected it.

Soon, P. L. Chebyshev proved a proposition from which Bertrand's postulate followed immediately, as a simple consequence, using a completely elementary and exceptionally witty trick. It was the greatest triumph of mathematical thought. The greatest mathematicians of the time said that in order to obtain further advances in the distribution of prime numbers, an intelligence as much superior to Chebyshev's was required as Chebyshev's was superior to the mind of an ordinary person. We will not dwell on other results of P. L. Chebyshev in number theory; what has already been said sufficiently shows how powerful his genius was.

We now turn to that section of mathematical science in which the ideas and achievements of P. L. Chebyshev were of decisive importance for its entire further development and determined for many decades, up to the present day, the direction of the most relevant research in it. This branch of mathematics is called probability theory. Threads literally stretch from all areas of knowledge to the theory of probability. This science deals with the study of random phenomena, the course of which cannot be predicted in advance and the implementation of which, under exactly the same conditions, can proceed in completely different ways, depending on the case. The two basic laws of this science are the law big numbers and the central limit theorem - those two laws around which, until very recently, almost all research has been grouped and which continue to be the subject of the efforts of a large number of specialists today. Both of these laws in their modern interpretation originate from P. L. Chebyshev.

We will not dwell on the substantive content of these laws. Created by P. L. Chebyshev, the famous elementary method allowed him to prove with amazing ease the law of large numbers in such broad assumptions that even the incomparably more complex analytical methods of his predecessors could not master. To prove the central limit theorem, P. L. Chebyshev created his own method of moments, which continues to play a significant role in modern mathematical analysis, but he did not have time to complete the proof; it was later completed by a student of P. L. Chebyshev, Academician A. A. Markov. Perhaps even more importance than the actual results of Chebyshev, for the theory of probability is the fact that he aroused the interest of his students in it and created the school of his followers, as well as the fact that it was he who first gave it the face of a real mathematical science. The fact is that in the era when P. L. Chebyshev began his work, the theory of probability as a mathematical discipline was in its infancy, without having its own fairly general problems and research methods. It was P. L. Chebyshev who first created the missing ideological and methodological core for her and taught his contemporaries and followers to treat her with the same severe exactingness (in particular, with regard to the logical rigor of her conclusions) and the same careful and serious attentiveness and care, like in any other mathematical discipline. Such an attitude, now shared by the entire scientific world and even the only one conceivable, was new and unusual for the last century, and the foreign world learned it from the Russian scientific school, in which it has become an unshakable tradition since the time of Chebyshev.

World science knows few names of scientists whose creations in various branches of their science would have had such a significant impact on the course of its development, as was the case with the discoveries of P. L. Chebyshev. In particular, the vast majority of Soviet mathematicians still feel the beneficial influence of P. L. Chebyshev, which reaches them through the scientific traditions he created. All of them with deep respect and warm gratitude honor the blessed memory of their great compatriot.

The main works of P. L. Chebyshev: Experience in elementary analysis of probability theory. An essay written for a master's degree, M., 1845; Theory of Comparisons (Doctoral dissertation), St. Petersburg, 1849 (3rd ed., 1901); Works, St. Petersburg, 1899 (vol. I), 1907 (vol. II), a biographical sketch written by K. A. Posse is attached. Complete works, vol. 1 - Theory of numbers, M. - L., 1944; Selected mathematical works (On determining the number of prime numbers not exceeding a given value; On prime numbers; On the integration of irrational differentials; Drawing geographical maps; Questions about the smallest values associated with an approximate representation of functions; On quadratures; On the limiting values of integrals; On approximate expressions the square root of the variable through simple fractions; On two theorems regarding probabilities), M. - L., 1946.

About P. L. Chebyshev:Lyapunov A. M., Pafnutii L'vovich Chebyshev, "Communications of the Kharkov Mathematical Society", series II, 1895, vol. IV, nos. 5-6: Steklov V. A., Theory and practice in Chebyshev's research. Speech delivered at the solemn celebration of the centenary of the birth of Chebyshev by the Russian Academy of Sciences. Petrograd, 1921; Bernstein S. N., 0 mathematical works P. L. Chebyshev, "Nature", L., 1935, No. 2; Krylov A, N., Pafnuty Lvovich Chebyshev, Biographical sketch, M. - L., 1944.

Pafnuty Lvovich Chebyshev

Mathematician, mechanic.

He received his primary education in the family.

Chebyshev was taught literacy by his mother, and French and arithmetic by his cousin, an educated woman who played a big role in the scientist's life. Her portrait hung in Chebyshev's house until the scientist's death.

In 1832 the Chebyshev family moved to Moscow.

Since childhood, Chebyshev limped, often used a cane. This handicap prevented him from becoming an officer, which he longed for some time. Perhaps, thanks to Chebyshev's lameness, world science received an outstanding mathematician.

In 1837 Chebyshev entered Moscow University.

Only the uniform that students were required to wear, and the strict inspector PS Nakhimov, brother of the famous admiral, reminded of military schools at the university. Meeting a student in a uniform unbuttoned out of shape, the inspector shouted: “Student, button up!” And he said one thing to all excuses: “Did you think? Nothing to think! What a habit you have to think! I have been serving for forty years and never thought about anything, that I would be ordered, and that's what I did. Only geese think, and Indian roosters. It is said - do it!

Chebyshev lived in the house of his parents on full support. This gave him the opportunity to fully devote himself to mathematics. Already in the second year of study, he received a silver medal for the essay "Calculation of the roots of an equation."

In 1841, famine struck Russia.

The financial situation of the Chebyshevs deteriorated sharply.

Chebyshev's parents were forced to move to live in the countryside and could no longer financially provide for their son. However, Chebyshev did not drop out of school. He simply became prudent and economical, which remained in him for the rest of his life, sometimes quite surprising those around him. It is known that in later years, already having a considerable income from the position of academician and professor, as well as from the publication of his works, Chebyshev used most of the money he earned to buy land. These operations were handled by its manager, who then profitably resold the purchased lands. Apparently, it was not in vain that Chebyshev argued that, perhaps, the main question that a person should pose to science should be this: “How to dispose of one’s funds in order to achieve the greatest possible benefit?”

In 1841 Chebyshev graduated from the university.

He began his scientific activity (together with V. Ya. Bunyakovsky) with the preparation for publication of the works of the Russian academician Leonhard Euler, devoted to number theory. Since that time, his own works devoted to various problems of mathematics began to appear.

In 1846, Chebyshev defended his master's thesis "An attempt at elementary analysis of probability theory." The purpose of the dissertation, as he himself wrote, was "... to show, without the mediation of transcendental analysis, the basic theorems of the calculus of probabilities and their main applications, which serve as the basis for all knowledge based on observations and evidence."

In 1847, Chebyshev was invited to St. Petersburg University as an adjunct. There he defended his doctoral thesis "Theory of Comparisons". Published as a separate book, this work by Chebyshev was awarded the Demidov Prize. The Theory of Comparisons has been used by students as a valuable tool for almost fifty years.

The well-known work of Chebyshev "Theory of Numbers" (1849) and the no less famous article "On Prime Numbers" (1852) were devoted to the question of the distribution of prime numbers in the natural series.

“It is difficult to point out another concept that is as closely connected with the emergence and development of human culture as the concept of number,” wrote one of Chebyshev's biographers. “Take away this concept from humanity and see how much poorer our spiritual life and practical activity are because of this: we will lose the opportunity to make calculations, measure time, compare distances, and sum up the results of labor. No wonder the ancient Greeks attributed to the legendary Prometheus, among his other immortal deeds, the invention of the number. The importance of the concept of number prompted the most prominent mathematicians and philosophers of all times and peoples to try to penetrate the mysteries of the arrangement of prime numbers. Of particular importance already in ancient Greece was the study of prime numbers, that is, numbers that are divisible without a remainder only by themselves and by one. All other numbers are the elements from which each integer is formed. However, results in this area were obtained with the greatest difficulty. Ancient Greek mathematics, perhaps, knew only one general result about prime numbers, now known as Euclid's theorems. According to this theorem, there are an infinite number of primes in a series of numbers. On the same questions about how these numbers are located, how correctly and how often, Greek science did not have an answer. About two thousand years that have passed since the time of Euclid did not bring any changes in these problems, although many mathematicians dealt with them, among them such luminaries of mathematical thought as Euler and Gauss ... In the forties of the XIX century, the French mathematician Bertrand spoke about the nature of the arrangement of prime numbers even one hypothesis: n and 2 n, where n– any integer greater than one, at least one prime number must be found. For a long time this hypothesis remained only an empirical fact, for the proof of which the ways were not felt at all ... "

Turning to number theory, Chebyshev quickly established an error in the well-known Legendre-Gauss conjecture, and, using a witty trick, proved his own proposition, from which Bertrand's postulate followed immediately, as a simple consequence.

This work of Chebyshev made an extraordinary impression on mathematicians. One of them quite seriously argued that in order to obtain new results in the distribution of prime numbers, it would be necessary to have an intelligence that was probably as superior to Chebyshev's as Chebyshev's was to the average person.

Number theory became one of the important areas of the famous mathematical school founded by Chebyshev. A significant contribution to it was made by students and followers of Chebyshev - famous mathematicians E. I. Zolotorev, A. N. Korkin, A. M. Lyapunov, G. F. Voronoi, D. A. Grave, K. A. Posse, A. A. Markov and others.

Chebyshev's works on the analysis of number theory, probability theory, the theory of approximation of functions by polynomials, integral calculus, the theory of synthesis of mechanisms, analytic geometry and other areas of mathematics received worldwide recognition.

In each of these areas, Chebyshev was able to create a number of basic, general methods and put forward deep ideas.

“In the mid-1950s,” recalled Professor K. A. Posse, “Chebyshev moved to live in the Academy of Sciences, first to a house overlooking the 7th line of Vasilyevsky Island, then to another house of the Academy, opposite the university, and finally again in a house on the 7th line, in a large apartment. Neither the change in the situation nor the increase in material resources affected Chebyshev's way of life. At home, he did not collect guests; his visitors were people who came to him to talk about questions of a scientific nature or on the affairs of the Academy and the University. Chebyshev constantly sat at home and studied mathematics ... "

Long before the physicists of the 20th century, who made such seminars the main field for developing new ideas, Chebyshev began to study with students in an informal setting. At the same time, Chebyshev never limited himself to narrow topics. Putting aside the chalk, he stepped away from the blackboard, sat down in a special chair intended only for him, and with pleasure plunged into the discussion of any distraction that was interesting to him and his opponents. In all other respects, he remained a rather dry, even pedantic person. By the way, he strongly disapproved of reading the current mathematical literature. He believed, perhaps not without reason, that such reading was unfavorable for the originality of his own work.

In 1859, Chebyshev was elected an ordinary academician.

While doing a great deal of work at the Academy, Chebyshev taught analytic geometry, number theory, and higher algebra at the university. From 1856 to 1872, in parallel with his main studies, he also worked in the Academic Committee of the Ministry of Public Education.

Chebyshev achieved a lot in the field of probability theory.

Probability theory is connected with all areas of human knowledge.

This science deals with the study of random phenomena, the course of which cannot be predicted in advance and the implementation of which, under completely identical conditions, can proceed in completely different ways, really, depending on the case. Studying the application of the law of large numbers, Chebyshev introduced the concept of "expectation" into science. It was Chebyshev who first proved the law of large numbers for sequences and gave the so-called central limit theorem of probability theory. These studies are still not only the most important components of the theory of probability, but also the fundamental basis of all its applications in the natural, economic and technical disciplines. Chebyshev, on the other hand, is credited with the systematic introduction to the consideration of random variables and the creation of a new technique for proving the limit theorems of probability theory - the so-called method of moments.

Dealing with complex problems of mathematics, Chebyshev always had an interest in solving practical problems.

“The convergence of theory with practice,” he wrote in the article “On the Construction of Geographic Maps,” “gives the most beneficial results, and not only practice benefits from this; the sciences themselves develop under its influence. It opens up new subjects for them to explore, or new aspects of things that have been known for a long time. Despite the high degree of development to which the mathematical sciences have been brought by the works of the great geometers of the last three centuries, practice clearly reveals their incompleteness in many respects; it proposes questions which are essentially new to science, and thus calls into question entirely new methods. If theory gains a lot from new applications of the old method or from its new development, then it gains even more by the discovery of new methods, and in this case science finds its true guide in practice ... "

Purely practical include such works by Chebyshev as - "On a Mechanism", "On Gears", "On a Centrifugal Equalizer", "On the Construction of Geographic Maps", and even such a completely unexpected one, read by him on August 28, 1878 at meeting of the French Association for the Development of Science, - "On the cutting of dresses."

In the “Reports” of the Association, the following was said about this report by Chebyshev:

“... Pointing out that the idea of this report arose from him after a report on the geometry of the weaving of matter, which Mr. Lucas made two years ago in Clermont-Ferrand, Mr. Chebyshev establishes general principles to determine the curves following which various pieces of matter must be cut in order to make them into a tight-fitting sheath, the purpose of which is to cover an object of any shape. Taking as a starting point the principle of observation that the change in the fabric must first be noticed as a first approximation, as a change in the angles of inclination of the warp and weft threads, while the length of the threads remains the same, he gives formulas that allow you to determine the contours of two, three or four pieces of matter assigned to cover the surface of the sphere with the most desirable approximation. G. Chebyshev presented to the section a rubber ball covered with cloth, two pieces of which were cut according to his instructions; he noticed that the problem would change significantly if skin were taken instead of matter. The formulas proposed by Mr. Chebyshev also give a method for tight fitting of parts when sewing. The rubber ball, covered with cloth, walked over the hands of those present, who examined and examined it with great interest and animation. This is a well-made ball, well-cut, and members of the section even tested it in a game of rounders in the lyceum yard.

Chebyshev devoted a lot of time to the theory of various mechanisms and machines.

He made suggestions to improve the steam engine of J. Watt, which prompted him to create new theory highs and lows. In 1852, having visited Lille, Chebyshev examined the famous windmills of this city and calculated the most advantageous form of mill wings. He built a model of the famous plant-walking machine imitating the gait of animals, built a special rowing mechanism and a scooter chair, and finally, he created an adding machine - the first continuous calculating machine.

Unfortunately, most of these instruments and mechanisms remained unclaimed, and Chebyshev presented his adding machine to the Paris Museum of Arts and Crafts.

In 1893, the World Illustration newspaper wrote:

“For many years in a row, in the public, not initiated into all the mysteries of mechanics and mathematics, there were vague rumors that our venerable mathematician, academician P. L. Chebyshev, invented the perpetuum mobile, that is, realized the cherished dream with which they rush dreamers for almost a thousand years, just as the alchemists once rushed with their philosopher's stone and elixir eternal life, and mathematicians - with the squaring of the circle, dividing the angle into three parts, etc. Others argued that Mr. Chebyshev built some kind of wooden "man", who supposedly walks by himself. The basis of all these tales was the not at all fantastic works of the venerable scientist on the development of possible simplified engines from cranked levers, which engines were built by him in a timely manner and are applicable to various projectiles: a scooter chair, sorting for grain, to a small boat. All these inventions of Mr. Chebyshev are currently being reviewed by visitors at the world exhibition in Chicago ... "

Engaged in the development of the most advantageous form of oblong projectiles for smooth-bore guns, Chebyshev very soon came to the conclusion that it was necessary to switch artillery to rifled barrels, which significantly increased the accuracy of fire, its range and efficiency.

Contemporaries called Chebyshev a "wandering mathematician."

It meant that he was one of those scientists who see their vocation, first of all, in moving from one field of science to another, in each leaving a number of brilliant ideas or methods that affect the imagination of researchers for a long time. Chebyshev's original ideas were instantly picked up by his numerous students, becoming the property of the entire scientific world.

In June 1872, twenty-five years of Chebyshev's professorship were celebrated at St. Petersburg University.

According to the rules in force at that time, a professor who had served for twenty-five years was dismissed from his post. But this time the University Council filed before the Ministry public education petition, so that the term of Chebyshev's professorship was extended by five years.

“The big name of the scientist that I have to talk about,” wrote in memo Professor A. N. Korkin, - forces me to be very brief in the present case. The general fame that Pafnuty Lvovich acquired for himself makes listing and analyzing his numerous works superfluous; they don't need criticism; suffice it to say that, being considered classical, they became an indispensable subject for every mathematician and that his discoveries in science entered the courses along with the studies of other famous geometers.

The general respect enjoyed by the works of Pafnuty Lvovich was expressed by his election to the membership of many academies and learned societies. It is known that he is a full member of the local academy, a corresponding member of the Paris and Berlin Academies, the Paris Philomatic Society, the London Mathematical Society, the Moscow Mathematical and Technical Society, etc.

To give an idea of the high opinion that Chebyshev has in the scientific world, I will point to a report on the recent progress in mathematics in France, presented by Acad. Bertrand to the Minister of Public Education on the occasion of the Paris World Exhibition in 1867. Here, evaluating the work of French mathematicians, Bertrand considered it necessary to mention those foreign geometers whose research had a particularly important influence on the course of science and was in close connection with the works he analyzed. Of the foreigners, only three were mentioned. The name of Chebyshev is placed along with the name of the brilliant Gauss.

By his peculiar choice of questions and the originality of the methods of solving them, Chebyshev sharply separates himself from other geometers. Some of his studies deal with the solution of certain questions, the difficulty of which stopped the most famous European scientists; with others it opened the way to vast new areas of analysis, hitherto untouched, the further development of which belongs to the future. In these studies of Chebyshev, Russian science acquires its own special, original character; to follow in the direction he created is the task of Russian mathematicians, and in particular of his many students, whom he educated during his 25 years of professorship. Many of them hold chairs at various universities in various departments of the exact sciences. In one of our universities, six students of Chebyshev teach: three mathematicians and three physicists.

Petersburg University, despite its relatively short existence, considers the most famous scientists among its leaders; in Chebyshev he has a first-class geometer, whose name will forever be associated with his fame.

As a result of these troubles, Chebyshev finally retired only in 1882.

In 1890, the President of France presented Chebyshev with the Order of the Legion of Honor.

On this occasion, the mathematician S. Hermit wrote to Chebyshev:

“My dear brother and friend!

I took great liberty in regard to you, taking the liberty, as President of the Academy of Sciences, to apply to the Minister of Foreign Affairs with a request to apply for awarding you with an order: the Commander's Cross of the Legion of Honor, which was granted to you by the President of the Republic. This difference is only a small reward for the great and wonderful discoveries with which your name is forever associated and which have long ago put you in the forefront of the mathematical science of our era ...

All the members of the Academy, to whom the petition I initiated was presented, supported it with their signatures and took the opportunity to testify to the warm sympathy that you inspire in them. They all joined me, assuring me that you are the pride of science in Russia, one of the first geometers in Europe, one of the greatest geometers of all time...

Can I hope, my dear brother and friend, that this token of respect coming to you from France will give you some pleasure?

At the very least, I ask you not to doubt my fidelity to the memories of our scientific closeness and that I have not forgotten and will never forget our conversations during your stay in Paris, when we talked about so many subjects that are far from Euclid ... "

With some traits of his character, Chebyshev often amazed those around him.

“... I will tell you about one observation made by my brother,” O. E. Ozarovskaya recalled. – He spent the summer in 1893 in Revel. The window of his room overlooked the flat roof of the neighboring house, which served as a kind of veranda for one attic. In it, the inhabitant of the attic, a bald and bearded old man, spent whole days in fine weather, writing sheets of paper.

With the kind of curiosity young man, abandoned by chance in a strange city, with a portion of leisure and boredom that prepared this curiosity, my brother took a closer look at the old man's writings and guessed the continuous outlines of integrals from the movements of the pen. The mathematician wrote all day long. My brother got used to him and during the day he asked himself questions and solved them: the mathematician, it is true, sleeps after dinner, the mathematician walks, how many sheets he wrote down today, etc.

But then the sun began to warm the venerable bald head too much, and the old man, instead of writing, one day took up sewing six sheets. After dinner, my brother went into a brush shop and ran into an old man who was buying six fine floor brushes. My brother was highly interested: why did a mathematician need such a large number of brushes?

The next morning, when my brother woke up, he saw an old man working in the shade under a white awning. The awning was fixed on six yellow sticks, and the brushes themselves lay right there under the bench.

This old man turned out to be none other than the great mathematician Pafnuty Lvovich Chebyshev.

He sketched out a plan of work with students who visited his house every week.

Votyagova Svetlana

Pafnuty Lvovich Chebyshev left an indelible mark on the history of world science and the development of Russian culture.

P.L. Chebyshev the glory of one of the greatest representatives of mathematical thought. A huge wealth of ideas is scattered in these works, they still have not lost either their freshness or relevance, and their further development continues at the present time in all countries of the globe, where the pulse of creative mathematical thought is only beating.

The aim of my research was to recover life path P.L. Chebyshev and consider his contribution to the development of mathematical science.

To do this, it is necessary to solve the following tasks:

Study bibliographic information about P.L. Chebyshev
Focus on the unique aspects of his life story
Determine the significance of the scientific activity of P.L. Chebyshev for mathematical science

The main stage of my work was the study of selected literature. After that, in my work, I tried to highlight the issues of the life and scientific work of P.L. Chebyshev, to show its significance in the development of “national Russian mathematical science”. I was very interested in the plots described in the biography books of the great mathematician. Much has been written about Chebyshev's life, but I have chosen only the most important and interesting information.

Chebyshev's scientific activity deserves attention because it is the basis, the beginning of the rapid development of mathematics in the second half of the 19th century in St. Petersburg. Chebyshev and his students formed the core of a scientific team of mathematicians, which was given the name of the St. Petersburg School of Mathematics.

P.L. Chebyshev was available to everyone who wanted to work scientifically and had the data for this; he generously shared his ideas. Thanks to this, he left behind a large number of students who later became first-class scientists; among them: A.M. Lyapunov, A.A. Markov. From him come the origins of many Russian mathematical schools - the theory of numbers, the theory of approximation of functions, the theory of mechanisms, which successfully continue to work today.

Interesting, in my opinion, are his works on applied mechanics. His invariable interest in questions of practice was so great that, perhaps, he largely determined the originality of P. L. Chebyshev as a scientist. It can be said without exaggeration that most of his best mathematical discoveries were inspired by applied work, in particular his research on the theory of mechanisms. The presence of this influence was often emphasized by Chebyshev himself, both in mathematical and applied works.

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MOU Sergeihinskaya secondary school

Kameshkovsky district

Vladimir region

Life and scientific achievements

P.L. Chebyshev

Research work

Made by a 8th grade student

Votyagova Svetlana Igorevna

Scientific adviser -

mathematic teacher

Toropova Galina Vasilievna

village Lubentsy, 2011

1. Introduction

2. Main part. Life and scientific achievements of P.L. Chebyshev

2.1. Childhood of a scientist.

2.2. Youth.

2.3. Work at St. Petersburg University.

2.4.Chebyshev-teacher.

3.Conclusion

4. Bibliographic list.

5.App.

1. Introduction

Pafnuty Lvovich Chebyshev left an indelible mark on the history of world science and the development of Russian culture.

The purpose of my research was to restore the life path of P.L. Chebyshev and consider his contribution to the development of mathematical science.

To do this, it is necessary to solve the following tasks:

Study bibliographic information about P.L. Chebyshev
Focus on the unique aspects of his life story
Determine the significance of the scientific activity of P.L. Chebyshev for mathematical science

P.L. Chebyshev was available to everyone who wanted to work scientifically and had the data for this; he generously shared his ideas. Thanks to this, he left behind a large number of students who later became first-class scientists; among them: A.M. Lyapunov, A.A. Markov. From him come the origins of many Russian mathematical schools - the theory of numbers, the theory of approximation of functions, the theory of mechanisms, which successfully continue to work today.

2. The main part. Life and scientific achievements of P.L. Chebyshev

2.1. Childhood of a scientist.

Pafnuty Lvovich was born on May 4 (16), 1821 in the village of Okatovo of the Kaluga viceroy, on the very border of the Moscow and Kaluga provinces. It was one of the usual middle class estates. An old windmill on a hill and a beautiful manor's pond, in the waters of which white swans still swim, are picturesquely blended into the Okatovsky landscape.

The father of the future mathematician Lev Pavlovich (Appendix 2), at the age of twenty he was a dashing cavalry cornet, participated in battles against the French. Then he retired, settled in his estate and took up farming. People around him considered him a good person. But Agrafena Ivanovna, the mother of Pafnuty, was not loved for her cruelty and arrogance, and even close relatives, especially those who were poorer, never counted on her favor.

The boy was born on May 16, 1821 and was the first of the sons of the Okatov master. At baptism he was given the name of St. Paphnutius, the great Russian miracle worker, soothsayer and healer, famous for his virtues, the main of which are generosity, mercy and humility.

It is very likely that the baby baptized in the family church of the Transfiguration of the Savior received such a rare name because 20 km from Okatovo is the famous Borovsky St. Pafnutiev Monastery, one of the most famous in Russia (Appendix 3). The Chebyshevs considered it almost their own home, making generous contributions and setting an example in this for the entire district nobility.

Pafnuty Lvovich's childhood passed in an old huge house (Appendix 3). There seemed to be an innumerable number of rooms in it, and the long, semi-dark corridors in the evening inspired awe in the boys, which in the morning seemed ridiculous and absurd to them. This house grew decrepit from year to year, then it was dismantled and a new one was built. And in the place where he stood for almost a century and a half, Pafnuty Lvovich and his younger brothers will later install a huge granite block, on which the words will be carved: "Here, Lev Pavlovich and Agrafena Ivanovna Chebyshev had five sons and four daughters." The stone is still there.

Pafnuty's parents would like to see their eldest son as a cavalry officer, if it were not for his physical handicap - a slight lameness, due to which the boy from the very early childhood I was forced to stay at home more, sometimes avoiding outdoor games with peers. However, the impressionable and diligent boy did not sit idle and was known in the family as a “big dreamer”, creating various mechanical devices with great love.

He molested his father a lot, having planned to conduct water into the master's bath with the help of an Archimedean screw, tortured adults with endless questions on practical mechanics, and tormented home teachers with this topic. He borrowed books from the local priest, rummaged for hours in his father's library, ordered all kinds of collections for relatives. Read about technical innovations, firmly and permanently stuck in his young head.

Very little is known about specific facts about the childhood of Pafnuty Lvovich. The scientist himself, unfortunately, did not leave behind any memories, much less autobiographical notes. It is only known that his mother taught him to read and write, and his cousin taught him French and arithmetic. Pafnuty also studied music, though unsuccessfully, but not without a trace: these studies, as he later believed, taught him "to precision and analysis." Young Pafnutiy spent especially much time reading books. Chebyshev retained this love for a solitary life, for intense mental work until his death.

2.2. Youth

In order to prepare him and his brother Pavel for entering the university, the Chebyshevs moved to the capital in 1832. The best teachers were invited to work with children.

For example, the teacher of mathematics was Platon Nikolaevich Pogorelsky, the famous director of the 3rd Moscow Real Gymnasium. He presented his material in an extremely clear and accessible form, he considered the ability to explain the subject an art. Until his last days, Chebyshev will remember him right words: "Go down lower, speak easier if you want to be understood." There is no doubt that the first seeds of love for mathematics, for a concise, clear and accessible presentation of its foundations, rigor and high demands on knowledge - all this was sown in Chebyshev's mind during Pogorelsky's lessons. The most difficult tasks that usually baffle many strong students are Pafnutiy solved easily and freely, and sat on more difficult ones for several days, finding particular pleasure in solving such problems.

Latin was taught to the Chebyshev brothers by a talented student of the medical faculty of Moscow University, Alexei Terentyevich Tarasenkov, an excellent connoisseur of the ancient language.

In 1837, the 16-year-old Pafnuty, after successfully passing the exams, became a student of the Physics and Mathematics Department of the Faculty of Philosophy of Moscow University, he was an excellent student. What kind of student was he? There are no special details about this. It seems that at the university he did not stand out among his comrades: he wore a strict uniform, fastened to the very chin with all the shining buttons, and the invariable student cocked hat with a cockade. He had the best behavior and never received any comments, he was always ready for classes, in all subjects he only managed to "excellently".

In 1838, participating in a student competition, he received a silver medal for his work on finding the roots of an equation of the nth degree. The original work was completed as early as 1838 and based on Newton's algorithm. For his work, Chebyshev was noted as the most promising student.

One of the teachers who influenced him the most in the future was Nikolai Brachman, who introduced him to the work of the French engineer Jean-Victor Poncelet.

Since 1840, the financial situation of the Chebyshev family was shaken, and Pafnuty Lvovich was forced to live on his own earnings. This circumstance left an imprint on his character, making him prudent and thrifty; later, when he no longer experienced a lack of funds, he did not respect the economy in spending them only in the manufacture of models of various instruments and mechanisms, the ideas of which were often born in his head.

In 1841, Pafnuty graduated with honors from the university, in 1846, being left at the university, he defended his dissertation for a master's degree on the topic "On the Application of Methods of Mathematical Analysis in Probability Theory."

2.3 Work at St. Petersburg University.

In 1847 he moved to St. Petersburg, where he successfully defended his dissertation at the university and began lecturing on algebra and number theory. In 1849, he defended his doctoral dissertation, which was awarded the Demidov Prize by the St. Petersburg Academy of Sciences in the same year; in 1850 he becomes a professor (Appendix 4).

It was here that he began his professorship, which

P.L. Chebyshev gave a lot of energy, and which continued until he reached old age, when he left lectures and devoted himself entirely to scientific work, which continued literally until the last moment of his life. At the age of twenty-eight, he received a doctorate degree from St. For thirty-two years, the Academy of Sciences elected P.L. Chebyshev as an adjunct in the Department of Applied Mathematics, and thirty-eight as an ordinary academician.

The growth of Chebyshev's scientific authority was later marked by his election to the number of academicians (1856). In 1871, Chebyshev was elected a foreign member of the Berlin Academy of Sciences, in 1873 the Bologna Academy of Sciences, in 1874 the Paris Academy of Sciences, in 1893 the Royal Swedish Academy of Sciences, and in 1877 the Royal Society of London.

Chebyshev's scientific heritage includes more than 80 works. It had a huge impact on the development of mathematics, especially on the formation of the St. Petersburg School of Mathematics. Chebyshev's works are characterized by a close connection with practice, a wide scope of scientific problems, rigor of presentation, and the economy of mathematical means in achieving major results. Chebyshev's mathematical achievements were mainly obtained in the following areas: number theory, probability theory, the problem of the best approximation of functions and the general theory of polynomials, the theory of integration of functions.

In 1863, a special "Chebyshev Commission" took an active part from the Council of St. Petersburg University in the development of the University Charter. The university charter, signed by Alexander II on June 18, 1863, granted autonomy to the university as a corporation of professors. This charter lasted until the era of counter-reforms of the government of Alexander III and was considered by historians as the most liberal and successful university regulations in Russia in the 19th and early 20th centuries.

For a long time P.L. Chebyshev takes an active part in the work of the artillery department of the Military Scientific Committee of the Military Department and the Scientific Committee of the Ministry of Public Education of Russia.

At the end of November 1894, P. L. Chebyshev suffered the flu on his feet - he was not used to going to bed, he had not complained to doctors before - and suddenly fell ill. The day before, he was still accepting students. And the next day he got up and dressed. He made tea himself, poured a glass. There was no one in the dining room. A few minutes later the servant, who entered the room, found him sitting at the table, but already dead. And the glass was hot, and a whitish steam rose from it ... A hundred kilometers from Moscow and five from the Balobanovo Kievskaya station railway, in a picturesque area near the Istya River, there is a small village of Spas on Prognanyi. It has a church built by Chebyshev's ancestors. Chebyshev's father and mother are buried on the north side of the churchyard. P. L. Chebyshev and his two brothers were buried under the bell tower in a tightly walled crypt.

2.4. Chebyshev is a teacher.

The merit of Chebyshev as a teacher is great. The weight that the scientific school he created in the history of mathematics has acquired shows that P.L. Chebyshev knew how to kindle the scientific enthusiasm of his students. The main feature of the St. Petersburg School of Mathematics was the desire to closely connect the problems of mathematics with the fundamental issues of natural science and technology.

Once a week at P.L. Chebyshev had a reception day when the doors of his apartment were open to anyone who wanted to get advice about their research. Few people left without enriching themselves with new thoughts and new plans. During such receptions, the scientist, in a calm and relaxed atmosphere of home comfort, led frank and lengthy conversations about classical music, opera, fashion artists, writers of the historical genre, about theology and European politics, diluting this mosaic with original findings in the field of mathematics and mechanics.

To this end, he sometimes interrupted the course of the exposition in order to illuminate to his listeners the history and methodological significance of this or that fact or scientific position. He attached significant importance to this retreat. They were quite long. Starting such a conversation, Chebyshev left the chalk and blackboard and sat down in a special chair that stood in front of the first row of listeners. Contemporaries and, in particular, students of P.L. Chebyshev, they say that he willingly revealed the richness of his ideological world not only in conversations with the elite, but also in his lectures for a wide audience.

So long before the mathematicians of the twentieth century, a wonderful Russian teacher began to study with students in an informal setting.

Otherwise, the students characterize him as a pedantically accurate and accurate lecturer, who never missed, was never late, and never delayed the audience one minute longer than the due time. It is interesting to note another characteristic feature of his lectures: he preceded any complex calculation with an explanation of its purpose and course in the most general terms, and then conducted it silently, very quickly, but in such detail that it was easy to follow him.

Lectures by P.L. Chebyshev were so fascinating that many came to listen to them twice. Cases are known when free places there were not enough classrooms for everyone, so they were occupied in advance, sometimes even an hour before the start of the lecture.

Dozens of students from the Faculty of Law sought to attend Chebyshev's lectures: they were eager to take a course in his Theory of Probability. Lawyers came here to learn, in their words, “from Professor Chebyshev the logic of drawing conclusions and the amazing evidence of speech”, i.e. logic and rhetoric.

The merit of P.L. Chebyshev in his many years of work on the methodological improvement of the teaching of mathematics at universities, secondary and primary schools.

Participating in the affairs of the Scientific Committee of the Department of Public Education, he actively reviewed textbooks in mathematics, protecting schools from the penetration of deliberately bad, or, as he liked to say, "limited" textbooks. Compiling a catalog of textbooks on arithmetic for elementary and secondary schools, he especially highly valued and considered useful the following: Busse's "Guide to Arithmetic", Loeve's "Arithmetic", Mikhailov's "Arithmetic" and Zolotov's "Arithmetic", and preferred Russian originals. textbooks.

2.5. Scientific achievements in mathematics.

The largest number of Chebyshev's works is devoted to mathematical analysis. In his 1847 dissertation for the right to lecture, Chebyshev investigates the integrability of certain irrational expressions in algebraic functions and logarithms. In the work of 1853. "On the integration of differential binomials" Chebyshev, in particular, proves his famous theorem on conditions for the integrability of a differential binomial in elementary functions. Several papers by Chebyshev are devoted to the integration of algebraic functions.

Chebyshev began working in number theory in the 1940s. It began with the fact that Academician Bunyakovsky involved him in commenting and publishing Euler's works on number theory. At the same time, Chebyshev was preparing a monograph on the theory of comparisons and its applications in order to present it as a doctoral dissertation. By 1849, both of these tasks were completed and the corresponding papers were published.

In number theory, Chebyshev became the founder of the Russian school, the glory of which was the work of his students G.F. Voronoi, E.I. Zolotarev, A.N. Korkin, A.A. Markov. Chebyshev managed to obtain important results in solving the problem of the distribution of prime numbers - to clarify the number of prime numbers that do not exceed a given number x [“On determining the number of prime numbers that do not exceed a given value” (1849); "On Prime Numbers" (1852)]. In the work "On an Arithmetic Question" (1866), Chebyshev considered the problem of approximating numbers by rational numbers, which played an important role in the development of the theory of Diophantine approximations.

Chebyshev's works on the theory of probability ["Experience of elementary analysis of the theory of probability" (1845); "An elementary proof of one general position theory of probability” (1846); "On Averages" (1867); "On two theorems concerning probabilities" (1887)] marked an important stage in the development of the theory of probability. P.L. Chebysheev began to systematically use random variables. He proved the inequality that now bears the name of Chebyshev, and - in a very general form- the law of large numbers.

2.6. Applied works of P.L. Chebyshev.

The most important feature of P. L. Chebyshev's scientific work is his constant interest in practical issues, most of his best mathematical discoveries were inspired by applied work.

Numerous applied works of P. L. Chebyshev, bearing far from mathematical names - “On a Mechanism”, “On Gears”, “On a Centrifugal Equalizer”, “On the Construction of Geographic Maps”, “On Cutting Dresses” and many others, were combined one basic idea - how to dispose of cash to achieve the greatest benefit? So, in the work “On the construction of geographical maps”, he sets himself the goal of determining such a projection of a map of a given country for which the scale distortion would be minimal. In his hands, this task has received an exhaustive solution. For European Russia, he brought this solution to numerical calculations and found out that the most advantageous projection would give a scale distortion of no more than 2%, while the projections adopted at that time gave a distortion of at least 4-5%.

The works of the scientist in mechanics make up about a quarter of his scientific research.

The great theoretician, who glorified himself with brilliant discoveries in mathematics, enthusiastically solved the urgent problems of industrial practice. Chebyshev visited plants and factories, he listened with interest to the opinions of engineers on technical questions that could not be resolved, and as a mathematician he often suggested a brilliant way out of a difficulty.

Here is one example. Mechanical engineers were unhappy with Watt's straightening mechanism, the so-called Watt's parallelogram. This mechanism, designed to transform a circular motion into a rectilinear one, performed its task unsatisfactorily. The motion could only be considered rectilinear in a rough approximation. And because of such an imperfection of Watt's parallelogram, harmful resistances arose in the machines.

Chebyshev came to the aid of the engineers. A method has appeared for the theoretical calculation of straightening mechanisms, that is, mechanisms capable of "straightening" the rotational movement, turning it into a rectilinear one. Today, such mechanisms have become the basis of many advanced designs.

Work on the straightening mechanism was for Chebyshev the starting point in his work on the creation of the theory of mechanisms and machines.

In an effort to more fully demonstrate the power of mechanics, Chebyshev himself became an engineer. He creates a variety of mechanisms that can accurately reproduce complex movements, work with stops, turn continuous movement into intermittent movement. Over forty mechanisms and eighty of their modifications were designed by the scientist.

Pafnuty Lvovich made many devices and mechanisms from wood with his own hands. Most of these models have survived to this day.

With his own hands, he built 40 working models of articulated mechanisms, including models: a single-cylinder steam engine, a centrifugal regulator, a scooter chair (Appendix 5), a rowing machine that repeats the movements of oars in a boat (Appendix 7), an automatic adding machine (Appendix 8). He builds his famous stepping machine, which accurately reproduces the movements of a walking animal.

He knew how and loved to work with his hands: having quickly mastered carpentry and turning, he could make home furniture (the chair he made - the chair has survived to this day) (Appendix 7), finally, like a real tailor, furrier or shoemaker, by several standards, sew for clothes, hat or shoes.

One of the scientist's memoirs, published in 1878 in Paris and called by him completely unscientifically "On cutting clothes." In this basic geometric work of Chebyshev, which he himself did not take very seriously, a sketch of an original solution of interesting problems in surface theory is given. Helping aeronautics enthusiasts a lot (designer Mozhaisky A.M. and others), Chebyshev asked himself the question: what curves should be used to cut out parts of thin matter in order to sew a case from them that fits snugly to a body of some shape, for example, to a ball (speech could go about the balloon). Here Chebyshev applied his theory of functions that deviate least from zero. In dealing with such questions, the scientist advanced in a completely unknown area. On this path he had no predecessors. It is interesting that modern textbooks for technical colleges such as “Fundamentals of Clothing Design” contain dozens of pages devoted to the presentation of methods for designing clothing patterns in “Chebyshev networks”, and the great fashion designers of our time Vyacheslav Zaitsev, Yves Saint Laurent or Pierre Cardin hardly guess which of the brilliant scientists they owe part of their success.

Few people know that he excelled in another technical area. The pinnacle of all his ideas as a watchmaker is a striking clock (Appendix 9). The design clearly reflected the ideas of the scientist on the synthesis of mechanisms.

A cupid with a bow and a cup sits at ease on a large black ball. Every hour, the Chebyshev clock beat off a strictly defined number of strokes, played the melody of the anthem, and the little Cupid, using a hinged-lever mechanism, threw up his hand with a healthy bowl. Time did not spare the dial, but the mechanism of the miracle watch remained intact and today delights specialists.

And finally, it should be mentioned in conclusion that Chebyshev's discoveries in the field of probability theory and interpolation greatly contributed to the development of our theory of shooting and sighting, they almost immediately entered the textbooks of artillery and ballistics (the formula for the range of a projectile in the air). For forty years, Chebyshev took an active part in the work of the military artillery department and worked to improve the range and accuracy of artillery fire. In ballistics courses, the Chebyshev formula for calculating the range of a projectile has been preserved to this day.

Through his work, Chebyshev had a great influence on the development of Russian artillery science. Engaged in the development of the most advantageous form of oblong projectiles for smooth-bore guns, Chebyshev very soon came to the conclusion that it was necessary to switch artillery to rifled barrels, which significantly increased the accuracy of fire, its range and efficiency.

3. Conclusion.

In the process of research, I came to the conclusion: only major historical figures and their path in science set cultural patterns of professionalism and scientific service.

World science knows few names of scientists whose creations in various branches of their science would have such a significant impact on the course of its development, as was the case with the discoveries of P. L. Chebyshev.

The range of his scientific research is wide, but in each of them he left an indelible mark: this is the theory of probability, the theory of interpolation, the theory of functions, integral calculus, the theory of mechanisms and others. Chebyshev's laws, Chebyshev's polynomials, Chebyshev's formulas, Chebyshev's functions, Chebyshev's inequalities have forever remained in mathematics. For forty-two years Chebyshev worked at the Academy of Sciences, increasing its fame and pride. For thirty-five years he headed the mathematical sciences at St. Petersburg University, created one of the most significant Russian mathematical schools. His brilliant ideas, results and methods, his books lived, are alive and will live in the works of numerous successors of his scientific and pedagogical work.

Numerous students of Chebyshev spread the ideas of their teacher throughout Russia and far beyond its borders.

The services of P.A. Chebyshev to the Fatherland were highly appreciated.

Named after P. L. Chebyshev:

a crater on the moon;

asteroid 2010 Chebyshev;

mathematical journal "Chebyshevsky Collection";

supercomputer SKIF MGU "CHEBYSHEV";

many objects in modern mathematics.

In 1944, the USSR Academy of Sciences established the P. L. Chebyshev Prize "for the best research in the field of mathematics and the theory of mechanisms."

4. Bibliographic list.

1. The Great Soviet Encyclopedia. Ed. 2nd. M.; Ch. scientific Publishing House "Great Soviet Encyclopedia", 1954. T. 47.

2. Glazer G.I. History of mathematics at school: IV - VI class. A guide for the teacher. M.; Enlightenment, 1981. - 239p., ill.

3. Kolesnikov N.N. Pafnuty Lvovich Chebyshev. Magazine "Quantum", 1971, No. 5

4.Lebedev S. "Chebyshev's adding machine". Newspaper "Mathematics", 2001,

№ 19,

5. Lebedev S. "Chebyshev Pearls". Newspaper "Mathematics", 2001,

№ 19.

6.Lebedev S. "Chebyshev's aphorisms". Newspaper "Mathematics", 2001,

№ 20.

7. Encyclopedic dictionary of a young mathematician. /Comp. A.P. Savin. M.; Pedagogy, 1985. - 352p.

Remarkable mathematicians were put forward by Russian science in the middle of the 19th century.

Pafnuty Lvovich Chebyshev (1821 - 1894) was the first in this glorious cohort both in time of activity and in importance.

Pafnuty Lvovich Chebyshev.

Chebyshev's life was calm, measured, outwardly monotonous. But how stormy and intense was the work of this great rebel and innovator of science! Chebyshev's ideas are still helping science move forward.

Like Euler and Ostrogradsky, Chebyshev did not shy away from practice. “The convergence of theory with practice,” said the scientist, “gives the most beneficial results, and not only practice benefits from this; the sciences themselves develop under its influence, it opens up new subjects for research or new aspects in subjects that have long been known.

These ideas were the motto of all Chebyshev's activities. Many of his works even have names that are not at all mathematical: “On the construction of geographical maps”, “On the cutting of clothes”, “On gear wheels”. In these works, Chebyshev, by means of mathematics, finds a solution to questions of the best, most economical and rational use of cash, which are extremely important for practice. Chebyshev writes: Most of questions of practice is reduced to problems of the largest and smallest magnitudes, completely new to science, and only by solving these problems can we satisfy the requirements of practice, which everywhere seeks the best, the most profitable.

In the work “On the construction of geographical maps”, the scientist gives an exhaustive answer to the question of how to determine such a projection in which the scale distortion will be the least. For European Russia, Chebyshev even brings the solution to a numerical calculation and shows that with drawing methods that correspond to the result he found, the distortion will be halved.

His interest in practice is so great that he even sets out to the Parisian tailors the results of the research he carried out in his work “On the Cutting of Clothes”, teaches them the most reasonable and economical way to line the fabric for cutting.

The methods discovered by Chebyshev are now used in the cutting of parachutes and in the construction of various apparatuses.

Having developed a special geometric network, P. L. Chebyshev used it to project the surface of complex bodies onto a plane. Above - the “Chebyshev network”.
The following shows how this network wraps around a complex geometric body- pseudosphere.

Chebyshev takes the requests of practice for himself as a creative order. He comes to the aid of engineers who have been trying for a long time to improve the "Watt parallelogram" - a mechanism for converting translational motion into rotational motion, and gives them a method for calculating this mechanism. Starting with Watt's parallelogram, Chebyshev created his remarkable theory of mechanisms, equipping technicians with the ability to calculate and design the most ingenious joints of levers, connecting rods, gears, and wheels. (We will talk about these works of Chebyshev in the chapter "Mechanics and Builders".)

The Watt parallelogram problem required the researcher to develop completely new mathematical methods, and he creates a mathematical theory of the best approximation of functions.

A function of mathematics is a variable that changes depending on changes in another variable - the argument. Functional dependence is constantly encountered in nature, science and technology. The circumference of a circle is a function of the radius; the path taken by a moving body depends on time; the speed of gas molecules is determined by the temperature; sine is a function of angle, etc.

The study of functions, functional dependence is the basis of the foundations of higher mathematics.

Often, when studying the problems of natural science and technology, researchers have to deal with very complex functional dependencies.

Chebyshev succeeded in simplifying the study of such functions. He found a way to express complex functions arbitrarily exactly using the sum of simple algebraic expressions. Algebraic series - Chebyshev polynomials - is a tool for solving a wide variety of problems.

Of exceptional importance are Chebyshev's works on probability theory, a branch of mathematics that studies the laws that govern random phenomena.

Many scientists then looked at this theory, the beginnings of which were laid by Pascal, Fermat, J. Bernoulli, Moivre, Laplace, Gauss and Poisson, as a semi-science, a kind of mathematical entertainment. This theory cannot be given such rigor, they argued, that it can be used as a method of knowledge and research.

The Russian mathematician refuted the statements of these scientists by his activities. Chebyshev rigorously proved the "law of large numbers", which states that the arithmetic mean of a large number of random variables that vary independently of each other is equal to a constant value. This basic law governing random phenomena makes it possible to calculate the total effect of a large number of random variables. The law of large numbers is of exceptional importance for natural science, technology, and statistics. Using it, one can see the patterns of this movement in apparent chaos, such as, for example, the movement of gas molecules, and display them in strict mathematical formulas. Chebyshev's law also serves as a basis in such a purely practical matter as the assessment of product quality. In elevators, the quality of a huge pile of grain is judged by examining the grain scooped up by a relatively small measure. The quality of cotton is judged by small bundles plucked at random from a huge bale. Selective control methods are based on the conclusions from this law.

With his law, Chebyshev laid a solid foundation for the theory of probability, gave it the right to be called a science no less rigorous than all other mathematical disciplines.

Chebyshev also worked fruitfully in such an important area of mathematics as number theory.

Chebyshev's method, ingenious in simplicity and wit, proved Bertrand's postulate on the distribution of prime numbers (that is, divisible only by itself and by one) among other numbers.

The postulate, empirically established by the French mathematician Bertrand, stated that between any number and a number twice its size, there must be at least one prime number.

Chebyshev's work was the greatest victory of mathematical thought. Ways to prove Bertrand's postulate were not even outlined; mathematicians around the world despaired of being able to substantiate this postulate. Having become acquainted with the work of Chebyshev, an English mathematician said that in order to move further in the study of the distribution of prime numbers, one needs an intelligence that is as much superior to Chebyshev's mind as Chebyshev's mind is superior to an ordinary mind.

Chebyshev Pafnuty Lvovich Chebyshev Pafnuty Lvovich

(pronounced Chebyshev) (1821-1894), mathematician, founder of the St. Petersburg scientific school, academician of the St. Petersburg Academy of Sciences (1856). Chebyshev's work is characterized by a variety of areas of research, the ability to find fundamental results by elementary means, and the desire to connect the problems of mathematics with the fundamental questions of natural science and technology. Many of Chebyshev's discoveries were due to applied research, mainly in the theory of mechanisms. He created the theory of the best approximation of functions with the help of polynomials, in probability theory he proved, in a very general form, the law of large numbers, in number theory - the asymptotic law of distribution of prime numbers, etc. Chebyshev's works laid the foundation for the development of many new branches of mathematics.

CHEBYSHEV Pafnuty Lvovich

CHEBYSHEV Pafnuty Lvovich (1821-94), Russian mathematician, founder of the St. Petersburg scientific school, academician of the St. Petersburg Academy of Sciences (1856). Chebyshev's work is characterized by a variety of areas of research, the ability to achieve fundamental results by elementary means, and the desire to connect the problems of mathematics with the fundamental questions of natural science and technology. Many of Chebyshev's discoveries were due to applied research, mainly in the theory of mechanisms. He created the theory of the best approximation of functions with the help of polynomials, in probability theory he proved, in a very general form, the law of large numbers, in number theory - the asymptotic law of distribution of prime numbers, etc. Chebyshev's works laid the foundation for the development of many new branches of mathematics.
* * *
CHEBYSHEV Pafnuty Lvovich, Russian mathematician and mechanic, member of the St. Petersburg Academy of Sciences (since 1856), founder of the St. Petersburg School of Mathematics. Member of the Berlin Academy of Sciences (1871), Bologna Academy of Sciences (1873), Parisian Academy of Sciences (1874; corresponding member from 1860), Royal Society of London (1877), Swedish Academy of Sciences (1893) and honorary member of many Russian and foreign learned societies, academies, universities.
Chebyshev on problems of mathematics
AT scientific creativity P. L. Chebysheva practical work were inextricably linked with high science and stemmed from a philosophical attitude, which he formulated with the greatest completeness in the report “Drawing Geographical Maps” at a solemn act on February 8, 1856 at St. Petersburg University: “Mathematical sciences from the most ancient times attracted special attention; now they have received even more interest in their influence on the arts and industry. The convergence of theory with practice yields the most favorable results, and practice alone does not benefit from this; the sciences themselves develop under its influence: it opens up new subjects for research or new aspects in subjects that have long been known. Despite the high degree of development to which the mathematical sciences have been brought by the works of the great geometers of the last three centuries, practice clearly reveals their incompleteness in many respects; it proposes questions that are essentially new to science and thus calls for the discovery of entirely new methods. If theory gains much from new applications of the old method or from new developments of it, then it gains still more by the discovery of new methods, and in this case the sciences find their true guide in practice.
The practical activity of man is extremely diverse, and to satisfy all its requirements, of course, science lacks many and various methods. But of these, of particular importance are those that are necessary for solving various modifications of one and the same problem common to the whole practical life of a person: how to dispose of one’s means in order to achieve the greatest possible benefit.?
Childhood, education
As was customary in the noble families of that time, P. L. Chebyshev received his initial education at home. At the age of sixteen he entered Moscow University. His work "Calculation of the roots of equations", presented on the topic announced by the faculty, is awarded a silver medal. In the same 1841, Chebyshev graduated from Moscow University, where in 1846 he defended his master's thesis "An Experience in Elementary Analysis of Probability Theory".
Moving to Petersburg
In 1847, after moving to St. Petersburg, he defended his dissertation "On Integration by Logarithms" at St. Petersburg University for the right to lecture, and after being approved as an assistant professor, he began lecturing on algebra and number theory. In 1849 he defended his doctoral thesis "Theory of Comparisons" at St. Petersburg University, which was awarded the Demidov Prize in the same year. From 1850 to 1882 - professor at St. Petersburg University. After his retirement, Chebyshev was engaged in scientific work until the end of his life.
Mathematical analysis
The largest number of Chebyshev's works is devoted to mathematical analysis. In his 1847 dissertation for the right to lecture, Chebyshev investigates the integrability of certain irrational expressions in algebraic functions and logarithms. In his 1853 work "On the Integration of Differential Binomials" Chebyshev, in particular, proves his famous theorem on conditions for the integrability of a differential binomial in elementary functions. Several papers by Chebyshev are devoted to the integration of algebraic functions.
Theory of mechanisms
During a business trip abroad in May-October 1852 (to France, England and Germany), Chebyshev got acquainted with the steam engine regulator - a parallelogram of James Watt (cm. WATT James). The “Report of the extraordinary professor of St. Petersburg University Chebyshev on traveling abroad” says the following: “Of the many subjects of research that presented themselves to me when considering and comparing various mechanisms for transmitting motion, especially in a steam engine, fuel, and the strength of the machine depends a lot on the methods of transferring the work of steam, I especially occupied myself with the theory of mechanisms known as parallelograms. Looking for various means to extract the most work from the steam in the case when it is necessary to have a rotational movement, as is most often the case, Watt invented a special mechanism for converting the rectilinear movement of the piston into the rotational (movement) of the rocker - a mechanism known as a parallelogram. From the history of practical mechanics, it is only known that the idea of the possibility of such a mechanism, the great converter of steam engines, was induced by examining a special projectile, where, through the copulation of various rotational movements various curved lines, some close to a straight line, were obtained. But we do not know in what way he reached the most advantageous form of his mechanism and the size of its elements. The rules that Watt followed in the construction of parallelograms could serve as a guide for practice only as long as there was no need to change its shape; with the change in the form of this mechanism, new rules were required. These rules and practices, and modern theory extracted from the beginning, which, apparently, was followed by Watt when constructing his parallelograms. The judgments which are advanced in proof of this principle cannot, obviously, withstand any scrutiny; even in practice, it often turns out to be inconvenient to use the elements of parallelograms, which are necessary according to this beginning, so that special tables were needed to correct them. From what I have said, it is clear to what extent it was necessary to subject Watt's parallelogram and its modifications to rigorous analysis, replacing the above-mentioned principle essential properties this mechanism and the conditions encountered in practice. To this end, I paid special attention to the circumstances that determine some of its elements both in factory machines and on steamships, and on the other hand, to the harmful effects of irregularities in its course, of which traces can be seen on machines that have been in use for a long time. .
Assuming to derive the rules for the construction of parallelograms directly from the properties of this mechanism, I encountered questions of analysis about which I knew very little until now. All that has been done in this respect belongs to Mr. Poncelet, a member of the Paris Academy. (cm. Poncelet Jean Victor), a well-known scientist in practical mechanics; the formulas he found are used very much in calculating the harmful resistances of machines. For the theory of Watt's parallelogram, more general formulas are needed, and their application is not limited to the study of these mechanisms.
In practical mechanics and other applied sciences, there are a number of issues for the solution of which they are necessary.
For Chebyshev, who thought in depth on the problems of the mathematical theory of parallelograms, machines made under the direct supervision of James Watt were of particular interest. The lucky opportunity that Chebyshev had been looking for came soon after his arrival in England. The Report describes this as follows: “On arrival in London, I turned to two famous English geometers, Sylvester and Cayley. I am indebted to the disposition of these scientists, on the one hand, for interesting conversations on various branches of mathematics, for which I used evenings and Sundays, during which all factories are closed, and on the other hand, the opportunity to get acquainted with the famous English mechanical engineer Gregory. Having learned about the purpose of my journey, and in particular about those questions of practical mechanics, the solution of which was the subject of my studies, he volunteered to assist me in finding in London factories the items that were most necessary for me. To this end, he traveled with me to various factories, where he believed to find various machines arranged by Watt himself. These machines were of particular interest to me as data on the rules that Watt followed in constructing his parallelograms, rules against which I was to compare the results of my investigations mentioned above. Unfortunately, it turned out that one of Watt's oldest machines, which had been preserved for a long time, was sold for scrap; but Mr. Gregory managed to find two machines, which, as can be seen from the patents, were very recently altered by Watt and are now preserved as a memorial.
P.L. Chebyshev outlined the results of his research in an extensive memoir “The Theory of Mechanisms Known as Parallelograms” (1854), laying the foundations for one of the most important sections of the constructive theory of functions - the theory of the best approximation of functions. It was in this work that P.L. Chebyshev introduced orthogonal polynomials, which now bear his name. In addition to approximation by algebraic polynomials, P.L. Chebyshev considered approximation by trigonometric polynomials and rational functions.
Least square method
From the problem of constructing polynomials that deviate least from zero, Chebyshev moved on to constructing a general theory of orthogonal polynomials, starting from the problem of integration using parabolas using the least squares method.
Work in the artillery department of the military scientific committee, a member of which long time was Chebyshev, led to the need to solve some problems related to quadrature formulas [the work "On quadratures" (1873) is devoted to them] and the theory of interpolation.
Mechanism design
In addition to Watt's parallelogram, Chebyshev was also interested in other hinged mechanisms, as evidenced, for example, by such of his works as "On some modification of Watt's cranked parallelogram" (1861), "On parallelograms" (1869), "On parallelograms consisting of three -or elements” (1879), etc. He himself was engaged in the design of mechanisms, built the famous “foot-walking machine”, reproducing the movement of an animal when walking, an automatic adding machine, mechanisms with stops and many other mechanisms.
In his work “On the Construction of Geographical Maps” (1856), Chebyshev set the task of finding such a cartographic projection of the country that would preserve similarity in small parts so that the greatest difference in scales in the vicinity of various points would be minimal.
Number theory works
In number theory, Chebyshev became the founder of the Russian school, the glory of which was the work of his students G. F. Voronoi (cm. VORONOY Georgy Feodosevich), E. I. Zolotareva, A. N. Korkina, (cm. Korkin Alexander Nikolaevich) A. A. Markova (cm. MARKOV Andrei Andreevich (1856-1922)). Chebyshev managed to obtain important results in solving the problem of the distribution of prime numbers - to clarify the number of prime numbers that do not exceed a given number x [“On determining the number of prime numbers that do not exceed a given value” (1849); "On Prime Numbers" (1852)]. In the work "On an Arithmetic Question" (1866), Chebyshev considered the problem of approximating numbers by rational numbers, which played an important role in the development of the theory of Diophantine approximations.
Works on the theory of probability
Chebyshev's works on the theory of probability ["Experience in elementary analysis of the theory of probability" (1845); "An elementary proof of one general proposition of the theory of probability" (1846); "On Averages" (1867); "On two theorems concerning probabilities" (1887)] marked an important stage in the development of the theory of probability. PL Chebyshev began to systematically use random variables. He proved the inequality that now bears the name of Chebyshev, and - in a very general form - the law of large numbers.
In 1944, the Academy of Sciences established the P.L. Chebyshev Prize.

encyclopedic Dictionary. 2009 .