Great Soviet Encyclopedia: Chebyshev (pronounced Chebyshev) Pafnuty Lvovich, Russian mathematician and mechanic; adjunct (1853), from 1856 extraordinary, from 1859 - ordinary academician of the St. Petersburg Academy of Sciences. He received his primary education at home; At the age of 16 he entered Moscow University and graduated in 1841. In 1846 he defended his master's thesis at Moscow University. In 1847 he moved to St. Petersburg, where in the same year he 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 became a professor at St. Petersburg University. long time took part in the work of the artillery department of the military-scientific committee and the scientific committee of the Ministry of Public Education. In 1882, he stopped lecturing at St. Petersburg University and, after retiring, completely engaged in scientific work. Ch. - the founder of the St. Petersburg mathematical school, the most prominent representatives of which were A.N. Korkin, E.I. Zolotarev, A.A. Markov, G.F. Voronoi, A.M. Lyapunov, V.A. Steklov, D.A. Grave.
The characteristic features of C.'s work are a variety of areas of research, the ability to obtain great scientific results through elementary means, and a constant interest in practical issues. Research Ch. related to the theory of approximation of functions by polynomials, integral calculus, number theory, probability theory, theory of mechanisms and many other branches of mathematics and related fields of knowledge. In each of the above sections, Ch. managed to create a number of basic, general methods and put forward ideas that outlined the leading directions in their further development. The desire to link the problems of mathematics with the fundamental issues of natural science and technology largely determines his originality as a scientist. Many of Ch.'s discoveries are inspired by applied interests. This was repeatedly emphasized by Ch. himself, saying that in the creation of new research methods “... the sciences find their true guide in practice” and that “... the sciences themselves develop under its influence: it opens up new subjects for them to study .. .” (Poln. sobr. soch., vol. 5, 1951, p. 150).
In the theory of probability, Ch. belongs to the merit of a 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 (1845, 1846, 1867, 1887). They have proven big numbers law in a very general form; At the same time, his proof is striking in its simplicity and elementarity. Ch. did not complete his study of the conditions for the convergence of distribution functions of sums of independent random variables to the normal law. However, through some additions to Ch.'s methods, A.A. managed to do this. Markov. Without rigorous conclusions, Ch. also outlined the possibility of refinements of this limit theorem in the form of asymptotic expansions of the distribution function of the sum of independent terms in powers of n?1/2, where n is the number of terms. Work Ch. on the theory of probability constitute an important stage in its development; in addition, they were the basis on which the Russian school of probability theory grew up, which at first consisted of direct students of Ch.
In the theory of numbers, Ch., for the first time after Euclid, significantly advanced (1849, 1852) the study of the question of the distribution of prime numbers ... The study of the arrangement of prime numbers in the series of all integers led Ch. also to the study of quadratic forms with positive determinants. Ch.'s work on the approximation of numbers by rational numbers (1866) played an important role in the development of the theory of Diophantine approximations. He was the creator of new areas of research in number theory and new research methods.
The most numerous works of Ch. in the field mathematical analysis. He was, in particular, devoted to the thesis for the right to lecture, in which Ch. investigated the integrability of certain irrational expressions in algebraic functions and logarithms. Ch. also devoted a number of other works to the integration of algebraic functions. In one of them (1853), a well-known theorem on integrability conditions in elementary functions of a differential binomial was obtained. An important area of ​​research in mathematical analysis is his work on the construction of a general theory of orthogonal polynomials. The reason for its creation was parabolic interpolation by the least squares method. Ch.'s research on the problem of moments and on quadrature formulas adjoins this circle of ideas. With the reduction in calculations in mind, Ch. proposed (1873) to consider quadrature formulas with equal coefficients (see Approximate integration). Studies on quadrature formulas and on the theory of interpolation were closely connected with the tasks that were set for Ch. in the artillery department of the military scientific committee.
Ch. - the founder of the so-called. constructive theory of functions, the main constituent element of which is the theory of the best approximation of functions (see Approximation and interpolation of functions, Chebyshev polynomials) ...
The theory of machines and mechanisms was one of those disciplines that Ch. systematically interested in all his life. Especially numerous are his works devoted to the synthesis of hinged mechanisms, in particular the Watt parallelogram (1861, 1869, 1871, 1879, etc.). He paid much attention to the design and manufacture of concrete mechanisms. Interesting, in particular, are his plantigrade machine, which imitates the movement of an animal when walking, as well as an automatic adding machine. The study of Watt's parallelogram and the desire to improve it prompted Ch. to formulate the problem of the best approximation of functions (see above). The applied works of Ch. also include an original study (1856), where he set the task of finding such a cartographic projection of a given country that preserves similarity in small parts so that the greatest difference in scale at different points on the map is the smallest. Ch. expressed the opinion, without proof, that for this the mapping must preserve the constancy of scale on the boundary, which was later proved by D.A. Grave.
Ch. left a bright mark on the development of mathematics and their own research, and the formulation of relevant questions to young scientists. So, on his advice, A.M. Lyapunov began a cycle of research on the theory of equilibrium figures of a rotating fluid, the particles of which are attracted according to the law of universal gravitation.
The works of Ch. during his lifetime found wide recognition not only in Russia, but also abroad; he was elected a member of the Berlin Academy of Sciences (1871), the Bologna Academy of Sciences (1873), the Parisian Academy of Sciences (1874; corresponding member 1860), the Royal Society of London (1877), the Swedish Academy of Sciences (1893) and an honorary member of many other Russian and foreign scientific societies, academies and universities.
In honor of the Ch. Academy of Sciences, the USSR established in 1944 a prize for the best research in mathematics.

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. As a result, he left behind big number 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 energy and which continued until he reached an advanced 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. 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 interesting to note another characteristic feature of his lectures: he prefaced 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 backdrop of this measured, prosperous life, not marked by any external shocks, in the quiet of the scientist’s calm study, great scientific discoveries were made, which were destined not only to change and rebuild the face of Russian mathematics, but also to have a huge, invariably felt influence on scientific research for a number of generations. the 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 important feature of P. L. Chebyshev's scientific work, one should note his invariable interest in questions of practice. 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 he most fully expressed the idea of ​​the fruitfulness of the connection between theory and practice in the article "Drawing geographical maps". We will not retell the thoughts of the great scientist, but will give 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. old methods or from new developments of it, then it acquires even more by the discovery of new methods, and in this case science finds a true leader in practice. "Among the huge number of tasks that his practical activity poses to a person, according to P. L. Chebysheva, one: “how to dispose of one’s funds 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 seeks everywhere the best, the most advantageous.

For P. L. Chebyshev, the above citation was the program of all his scientific activities, 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 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% ( 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 have recently been used in a number of designs of modern devices, 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 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 the problem of P. L. Chebyshev is connected with the main problems of the application of 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 the polynomial nth degree, which takes exactly the same values ​​as the given function when n + 1 argument values ​​are given. 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 - the law of large numbers and the central limit theorem - are the two laws around which almost all research has been grouped until very recently and which continue to be the subject of 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. The famous elementary method created by P. L. Chebyshev 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 important 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, as in any other mathematical discipline. This attitude, now shared by all scientific world and even the only thing conceivable was new and extraordinary 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 square root of a variable in terms of simple fractions; On two theorems on 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 of P. L. Chebyshev, "Nature", L., 1935, No. 2; Krylov A, N., Pafnuty Lvovich Chebyshev, Biographical sketch, M. - L., 1944.

Mathematician, mechanic.

Born on May 16, 1821 in the small village of Okatovo, Borovsky district, Kaluga province.

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.

Pursuing difficult problems 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 a new theory of maximums and minimums. 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 once the alchemists rushed about with their philosopher's stone and the elixir of eternal life, and mathematicians - with the squaring of the circle, dividing the angle into three parts, etc. Others asserted that Mr. Chebyshev built some kind of a wooden "man" who seems to walk 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. original ideas Chebyshev was 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 a petition with the Ministry of Public Education, so that the term of Chebyshev's professorship was extended by five years.

“The big name of the scientist about whom I have to speak,” Professor A. N. Korkin wrote in a memo, “forces me to be very brief in this 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 flat roof neighboring house, which served as a 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.

G. Prashkevich

Ministry of Education Russian Federation

Secondary school №6

Essay

on the topic of:

P.L. Chebyshev -

father of the Petersburg Mathematical School.

Made by 8th grade student

Maltsev M. M.

Checked by math teacher

Malova T.A.

Work plan

Introduction

1. Main body

1.1. Number theory.

1.2. Distribution of prime numbers.

1.3. Bertrand's postulate.

1.4. Probability theory

1.5. The theory of approximation of functions.

1.6. Chebyshev's scientific activity

1.7. The contribution of the St. Petersburg School of Mathematics to the development of the country

2. Conclusion

3. List of used literature

Introduction

This year marks 190 years since the birth of the great mathematician and mechanic Pafnuty Lvovich Chebyshev, a remarkable scientist and teacher who brought the domestic mathematical science to the world level. 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.

I decided to choose this topic because I like mathematics and I respect the scientists who developed it, so my essay is on this topic.

Russian science in the middle of the 19th century brought forward a whole galaxy of remarkable mathematicians. And the world-famous Pafnuty Lvovich Chebyshev was the first among them both in time of activity and in scientific significance in this glorious cohort. P.L. Chebyshev was born on May 16, 1821 in the village of Okatovo, Borovsky district, Kaluga province, on the noble estate of his father, Lev Pavlovich Chebyshev. Having entered the mathematical department of Moscow University, Chebyshev immediately attracted the attention of the famous mathematician Professor Brashman. The latter was one of the few professors at Moscow University who sought to use science to develop the economy. Brashman had a significant influence on the formation of the scientific views of P.L. Chebyshev. Noticing in Chebyshev a serious attitude to studies, love and ability for science, he began to diligently supervise his studies and persuade him to devote himself exclusively to mathematics. Although the financial situation of a promising young man, due to the frustrated affairs of his father, became extremely poor, Chebyshev nevertheless followed the advice of his teacher, and, having graduated from the university course in 1841 with honors, he devoted himself entirely to scientific work. In 1845, Chebyshev submitted to Moscow University as a master's thesis the essay "An Experience in Elementary Analysis of Probability Theory" and the mathematical department of the university recognized him as worthy of a master's degree. In 1849, Chebyshev, after successfully defending his dissertation on the topic "Theory of Comparisons", received a doctorate in mathematics and astronomy. In 1856 he was elected an extraordinary academician, and in 1859 Chebyshev was elected an ordinary academician in the department of applied mathematics. In 1872, Pafnuty Lvovich was awarded the title of Honored Professor of St. Petersburg University. In 1882, Chebyshev left teaching at St. Petersburg University and switched completely to scientific work at the Academy of Sciences. Chebyshev's mathematical research relates to integral calculus, number theory, probability theory, mechanism theory, and many other branches of mathematics. P.L. Chebyshev, with his multifaceted and fruitful activity, determined the paths and directions for the development of mathematics in Russia for many years to come and had a huge impact on the world of mathematical science. The works of Pafnuty Lvovich found wide recognition during his lifetime, both in Russia and abroad. He was elected a member of the Berlin, Bologna, Paris and Swedish Academy of Sciences, a corresponding member of the Royal Society of London and an honorary member of many other Russian and foreign scientific societies, academies and universities. Chebyshev is the founder of the Petersburg School of Mathematics. Died P.L. Chebyshev in his St. Petersburg apartment, at the age of 74 from heart failure in 1894. Obituaries were placed in most Russian newspapers, which emphasized “Russian science has suffered a heavy loss in the person of the deceased ordinary academician P.L. Chebyshev, who has long gained fame as an outstanding mathematician and fame as one of the first geometers in Europe by scientific merit. Chebyshev was born in the Kaluga province, studied in Moscow, lived, worked and died in St. Petersburg, and yet we Izmalkovites have the right to consider him to some extent our countryman. Since Pafnuty Lvovich for many years came to summer time to the estate of his younger brother, General and Honored Professor of the Artillery Academy Vladimir Lvovich Chebyshev, which was located within the boundaries of the current village of Znamenka of the Ponomarevsky Village Council. Pafnuty Lvovich lived there from 2 to 6 months in each of his visits to the village of Chebyshev, and in total he spent more than 5 years in the village of Chebyshev. Pafnuty Lvovich willingly communicated with the peasants of the village of Chebyshev, his circle of acquaintance with them was quite wide and he always treated all the inhabitants of the village very kindly. During the stay of Pafnuty Lvovich in the village of Chebyshev, more than one brilliant scientific work. In the village of Chebyshev, there are still people who personally knew P.L. Chebyshev, who speak very warmly of the scientist and respectfully call him none other than our Pafnuty Lvovich.

After Euler's death in 1783, the level of mathematical research in

Petersburg has declined significantly. A new rise emerged only in the 20s of the XIX century. It was determined by the scientific and organizational activities of M. V. Ostrogradsky (1801-1861) and V. Ya. Bunyakovsky (1804-1889), and later P. L. Chebyshev (1821-1894). By the middle of the 19th century, the activities of Ostrogradsky and Bunyakovsky, their students, many of whom became prominent specialists in various fields of mathematics and technology, determined a new upsurge in mathematics in Russia, especially in St. Petersburg. A team of creatively working mathematicians began to take shape, in which P. L. Chebyshev took the leading place by the end of Ostrogradskii's life. 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, behind which

the name of the Petersburg Mathematical School was fixed.

Pafnuty Lvovich Chebyshev graduated from Moscow University in 1841. At the competition of student works for an essay on the topic "Calculation of the roots of the equation" he was awarded a silver medal. Being left at the university, in 1846 he defended his master's thesis "An attempt at an elementary analysis of probability theory." The following year, Chebyshev moved to St. Petersburg and began working at the university. Here in 1849 he defended his doctoral thesis: "Theory of Comparisons" and worked as a professor for many years, until 1882. At the St. Petersburg Academy of Sciences, Chebyshev's activity began in 1853, when he was elected an adjunct.

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 coverage of scientific problems, rigor of presentation, and economical use of mathematical means to achieve 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.

Chebyshev's research relates to the theory of approximation of functions by polynomials, integral calculus, theory of numbers, probability theory, theory of mechanisms, and many other branches of mathematics and related fields of knowledge. Chebyshev created a number of basic, general methods and put forward ideas that outlined the leading directions in these areas of science and their further development. He sought to link the problems of mathematics with the fundamental issues of the development of natural science and technology, leaving numerous works in the field of mathematical analysis, the theory of machines and mechanisms, etc. For a long time, Chebyshev participated in the work of the artillery department of the military scientific committee, solving problems with which his research was closely related. on quadrature formulas and on the theory of interpolation, which was important for the development of artillery sciences. Chebyshev's works have found wide recognition all over the world. He was elected a member of many Academies of Sciences: Berlin (1871), Bologna (1873), Paris (1874), Swedish (1893), Royal Society of London (1877) and an honorary member of other Russian and foreign scientific societies, academies and universities. In honor of Chebyshev, the Academy of Sciences of the USSR established a prize in 1941.

number theory .

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. As an appendix to his Theory of Comparisons, Chebyshev published his memoirs On Determining the Number of Primes Not Exceeding a Given Value.

Distribution of prime numbers.

The problem of the distribution of prime numbers in a series of natural numbers is one of the oldest in number theory. It has been known since ancient Greek mathematics. Euclid took the first step towards its solution by proving the theorem that there are infinitely many prime numbers in the natural series. As long as Euler did not involve the means of mathematical analysis, its solution practically did not advance. The new proof, in essence, did not give a new result, but included new methods. The idea of ​​Euler's proof is as follows: the convergence of the harmonic series follows from the finiteness of the set of primes, since it is then represented as a product of a finite number of geometric progressions. It was only in 1837 that Dirichlet generalized Euclid's theorem, proving that any arithmetic progression (a + nb), where a and b are coprime, contains infinitely many primes. In the period 1798-1808, Legendre, having studied tables of prime numbers up to a million, empirically deduced that the number of prime numbers in the segment p(x) is expressed by the formula x/p(x)=ln x - 1.08366.

Chebyshev proved that the Legendre formula is inaccurate by examining the properties of the function p(x) and showed that the true order of growth of this function is the same as that of the function x/ln x. Moreover, he found clarifications: the relation

concluded between 0.92129 and 1.10555.

Chebyshev's discovery made a very big impression. Many mathematicians worked to improve his results. Sylvester, in his papers of 1881 and 1892, narrowed the gap to . Schur (1929) and Breish (1932) achieved further narrowing.

Chebyshev also found integral estimates for the values ​​of p(x). He managed to prove that as x increases, the value of p(x) fluctuates around. It was not until 1896 that Hadamard and de la Vallée-Poussin proved the following limit theorem. Already in the time close to us (1949), Selberg found another proof of this asymptotic regularity. In 1955, A. G. Postnikov and N. P. Romanov simplified Selberg’s cumbersome reasoning.

Bertrand's postulate.

The French mathematician Bertrand in his works (1845) relied on the following statement: for any natural number n>1, there is a prime number between n and 2n. Bertrand used it without proof. The statement was proved by Chebyshev (1850), so it is sometimes called Chebyshev's theorem. The main idea of ​​​​the proof is the estimation of the powers of prime numbers into which the binomial coefficient is divided by writing in it in the p-ary number system (there is a beautiful analogy with the sign of divisibility by 9 in the decimal system - however, it is quite possible to do without such a notation). In fact, the estimate can be strengthened: for n>5, there are two integer primes between n and 2n. Stronger inequalities can also be obtained.

Studies on the arrangement of prime numbers in the natural series also led to the appearance of Chebyshev's works on the theory of quadratic forms. In 1866, his article "On an Arithmetic Question" was published, devoted to Diophantine approximations, i.e. integer solutions of Diophantine equations using the apparatus of continued fractions.

Probability theory

Chebyshev turned to the theory of probability in his youth, devoting his master's thesis to it. In those days, a kind of crisis took place in the theory of probability. The fact is that the basic laws of this science were basically found back in the 18th century. This refers to the law of large numbers; limit theorem of Moivre-Laplace - the limit law of probabilities of deviation of the number x of occurrences of a random event from the mathematical expectation, a of this number in n experiments with probability p; introduction of the concept of dispersion. Awareness of the wide applicability of these regularities led to attempts to apply them even to the social practice of people, i.e. outside the reasonable area of ​​valid applications. This led to a large number of confused, unfounded and erroneous conclusions, which affected the scientific reputation of the theory of probability. Without a solid substantiation of the concepts and results, the further development of this science became impossible.

Chebyshev wrote only 4 works on the theory of probability (1845, 1846, 1867, 1887), but, by all accounts, it was these works that brought the theory of probability back to the rank of mathematical sciences, served as the basis for the creation of a new mathematical school. Chebyshev's initial positions appeared already in his master's thesis. He set himself the goal of giving such a construction of the theory of probability, which would least involve the apparatus of mathematical analysis. He achieved this by refusing passages to the limit and replacing them with systems of inequalities in which all relations are contained. Numerical estimates of deviations and errors remained characteristic features and subsequent works of Chebyshev on the theory of probability.

However, Chebyshev managed to find a sufficiently general and rigorous proof of the central limit theorem only by 1887. To prove it, Chebyshev had to find a method known in modern literature as the method of moments. Chebyshev's proof had a logical gap, which was eliminated by Chebyshev's student A.A. N. Kolmogorov, now their works are everywhere perceived as the starting point for all further development of the theory of probability, not excluding the modern one. In their works, the method of moments (Markov) and the method of characteristic functions (Lyapunov) were developed. Particularly worth noting is the theory of Markov chains.

The theory of approximation of functions.

A significant place in the works of Chebyshev is occupied by the theory of approximation of functions. This group of works is notable for its great theoretical consequence, which led to the emergence of the modern constructive theory of functions. The latter studies, as is known, the dependences between the properties of various classes of functions and the nature of their approximation by other, simpler functions in a finite or unbounded domain.

During a scientific trip abroad in 1852, Chebyshev became interested in various types of hinged mechanisms, with the help of which the rectilinear translational motion of a steam engine piston is converted into a circular motion of a flywheel (or vice versa). One of the varieties of such mechanisms is the well-known Watt's parallelogram.

Chebyshev built many mechanisms during his life and studied their kinematics. The extremal problems that arise in this case (such as calculating a mechanism with a minimum deviation of some part of it from the vertical) lead to mathematical problems in the theory of approximation of functions. The most convenient function for operating in mathematics is a polynomial. From this follow the problems of determining polynomials that deviate from zero, as well as approximating functions by polynomials (1854, "The theory of mechanisms known as parallelograms").

Consider, for example, the following problem: among all polynomials of a fixed degree with the highest coefficient equal to 1, find a polynomial with a minimum of the maximum modulus on the interval [-1,1].

Solution: this is the Chebyshev polynomial Pn = cos(n arccos x)/(2n-1). The fact that its leading coefficient is equal to 1 (and, in general, that it is a polynomial) follows from the recurrent formula Pn+1(x)= x Pn(x)-1/4 Pn-1(x), and that it has a minimum maximum modulus, - estimating the number of sign changes - and, consequently, the roots - of the polynomial Pn(x)-Q(x), where Q(x) is the polynomial with the maximum value of the modulus l/2n-1, l<1.

Chebyshev found a class of special polynomials that bear his name to this day. The Chebyshev, Chebyshev - Laguerre, Chebyshev - Hermite polynomials and their varieties play an important role in mathematics and in various applications. Chebyshev's theory of the best approximation of functions by polynomials is applied to geodesic and cartographic problems (1856, "On the construction of geographical maps"), approximate quadratures, interpolations, the solution of algebraic equations, not to mention the kinematics of mechanisms, which served as its starting point. The Chebyshev theory under consideration contains ideas of the general theory of orthogonal polynomials, the theory of moments, and quadrature methods. Chebyshev connected orthogonal polynomials with the method of least squares.

Chebyshev's scientific activity

Chebyshev left a deep and bright mark on the development of mathematics, gave impetus to the creation and development of many of its sections, both by his own research and by posing relevant questions to young scientists. So, on his advice, A. M. Lyapunov began a series of studies on the theory of equilibrium figures of a rotating fluid, the particles of which are attracted according to the law of universal gravitation. Of course, the scientific interests of the St. Petersburg mathematicians, and Chebyshev himself, were much broader. Of the areas of mathematics not mentioned in the abstract, the most intensive work was carried out on problems in the theory of differential equations (Lyapunov, Imshenetsky, Sonin, and others) and the theory of functions of a complex variable (especially Sokhotsky).

Petersburg mathematics by the beginning of our century was a wide association of many scientific directions. They have had and are having a significant impact on the development of mathematics in our country and abroad. Ties with other scientific associations, especially in recent times, have become so entrenched, and scientific interests are so intertwined that the term "Petersburg Mathematical School" has lost its isolating meaning.

In 1867, another very remarkable memoir by Chebyshev, On Mean Values, appeared in the second volume of the Moscow Mathematical Collection, in which a theorem is given that underlies various problems in probability theory and includes the famous theorem of Jacob Bernoulli as a special case.

These two works would be enough to perpetuate the name of Chebyshev. In integral calculus, the memoir of 1860 is especially remarkable, in which, for a given polynomial x4 + αx3 + βx2 + γx + δ with rational coefficients, an algorithm is given for determining such a number A that the expression is integrated in logarithms, and calculating the corresponding integral.

The most original, both in terms of the essence of the issue and the method of solution, are the works of Chebyshev "On functions that deviate least from zero." The most important of these memoirs is an 1857 memoir entitled "Sur les questions de minima qui se rattachent à la représentation approximative des fonctions" (On the question of the minimum standards that apply to an approximate idea of ​​a function).

(in "Mem. Acad. Sciences"). Professor Klein, in his lectures at the University of Göttingen in 1901, called this memoir "wonderful" (wunderbar). Its content was included in the classic work of I. Bertrand Traité du Calcul diff. et integral. In connection with the same questions is the work of Chebyshev "On the drawing of geographical maps." This series of works is considered the foundation of the theory of approximations. In connection with the questions "on functions that deviate least from zero", there are also Chebyshev's works on practical mechanics, which he studied a lot and with great love.

Also remarkable are Chebyshev's works on interpolation, in which he gives new formulas that are important both in theoretical and practical respects.

One of Chebyshev's favorite tricks, which he used especially often, was the application of the properties of algebraic continued fractions to various problems of analysis.

The works of the last period of Chebyshev's activity include the research "On the limiting values ​​of integrals" ("Sur les valeurs limites des intégrales", 1873). The completely new questions posed here by Chebyshev were then worked out by his students. Chebyshev's last memoir of 1895 belongs to the same field.

Chebyshev's social activities were not limited to his professorship and participation in the affairs of the Academy of Sciences. As a member of the Academic Committee of the Ministry of Education, he reviewed textbooks, drafted programs and instructions for primary and secondary schools. He was one of the organizers of the Moscow Mathematical Society and the first mathematical journal in Russia - "Mathematical Collection".

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.

Based on the traditions of the St. Petersburg mathematical school, Leningrad scientists worked fruitfully in many areas of mathematics and mechanics. The theory of functions of a complex variable and the theory of differential equations were developed in the works of V. I. Smirnov. The five-volume "Course of Higher Mathematics" created by V.I. Smirnov became a reference book for students of natural sciences and technical universities. A significant contribution to number theory was made by a student of Ya. V. Uspensky, I. M. Vinogradov. The works of A. D. Aleksandrov were devoted to problems of geometry and topology, N. M. Gunther and S. L. Sobolev - to problems of mathematical physics. The largest achievements in the prewar period were obtained in various fields of physics. The efforts of many physicists have been concentrated on the problem of the physics of the atomic nucleus. In 1932, D. D. Ivanenko developed a proton neutron model of the nucleus. GN Flerov and Yu. B. Khariton carried out in 1939 the classic work on the chain reaction of uranium fission. At the Physicotechnical Institute, work on nuclear physics was led by I. V. Kurchatov. On the eve of the war, I. V. Kurchatov and A. I. Alikhanov worked on the creation of a 100-ton cyclotron, the launch of which was scheduled for 1942 (the first cyclotron in Europe began to work at the Radium Institute in Leningrad). In 1940, the Academic Commission on the uranium problem was organized in Leningrad. The development of nuclear physics at the Physico-Technical Institute did not proceed smoothly: A. F. Ioffe and his institute were severely criticized for their enthusiasm for fundamental research and their detachment from production. Nuclear physics was one of the areas under attack.

The contribution of the St. Petersburg mathematical school to the development of the country.

Based on the traditions of the St. Petersburg mathematical school, Leningrad scientists worked fruitfully in many areas of mathematics and mechanics. The theory of functions of a complex variable and the theory of differential equations were developed in the works of V. I. Smirnov. Based on the traditions of the St. Petersburg mathematical school, Leningrad scientists worked fruitfully in many areas of mathematics and mechanics. The theory of functions of a complex variable and the theory of differential equations were developed in the works of V. I. Smirnov. The five-volume "Course of Higher Mathematics" created by V.I. Smirnov became a reference book for students of natural sciences and technical universities. A significant contribution to number theory was made by a student of Ya. V. Uspensky, I. M. Vinogradov. The works of A. D. Aleksandrov were devoted to problems of geometry and topology, N. M. Gunther and S. L. Sobolev - to problems of mathematical physics. The largest achievements in the prewar period were obtained in various fields of physics. The efforts of many physicists have been concentrated on the problem of the physics of the atomic nucleus. In 1932, D. D. Ivanenko developed a proton neutron model of the nucleus. GN Flerov and Yu. B. Khariton carried out in 1939 the classic work on the chain reaction of uranium fission. At the Physicotechnical Institute, work on nuclear physics was led by I. V. Kurchatov. On the eve of the war, I. V. Kurchatov and A. I. Alikhanov worked on the creation of a 100-ton cyclotron, the launch of which was scheduled for 1942 (the first cyclotron in Europe began to work at the Radium Institute in Leningrad). In 1940, the Academic Commission on the uranium problem was organized in Leningrad. The development of nuclear physics at the Physico-Technical Institute did not proceed smoothly: A. F. Ioffe and his institute were severely criticized for their enthusiasm for fundamental research and their detachment from production. Nuclear physics was one of the areas under attack.

Conclusion

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 merits of Chebyshev were appreciated by the scientific world in a worthy way. He was elected a member of the St. Petersburg (1853), Berlin and Bologna Academies, the Paris Academy of Sciences in 1860 (Chebyshev shared this honor with only one more Russian scientist, the famous Baer, ​​who was elected in 1876 and died the same year), corresponding member of the London Royal society, the Swedish Academy of Sciences, etc., in total 25 different Academies and scientific societies. Chebyshev was also an honorary member of all Russian universities.

The characteristics of his scientific merits are very well expressed in the note of Academicians A. A. Markov and I. Ya. Sonin, read at the first meeting of the Academy after Chebyshev's death. This note, among other things, says:

Chebyshev's works bear the imprint of genius. He invented new methods for solving many difficult questions that had been posed for a long time and remained unresolved. At the same time, he raised a number of new questions, on the development of which he worked until the end of his days.

The famous mathematician Charles Hermite stated that Chebyshev "is the pride of Russian science and one of the greatest mathematicians of Europe," and Professor Mittag-Leffler of Stockholm University claimed that Chebyshev is a brilliant mathematician and one of the greatest analysts of all time.

Named after P. L. Chebyshev:

* crater on the moon;
* asteroid 2010 Chebyshev;
* mathematical journal "Chebyshevsky Collection"
* many objects in modern mathematics.

Bibliography

|Golovinsky IA On the justification of the method of least squares in PL Chebyshev. // Historical and mathematical research. Kolmogorov A. N., Yushkevich A. P. (ed.) Mathematics of the 19th century. M.: Science.

Volume 1 Mathematical logic. Algebra. Number theory. Probability Theory. 1978.

Chebyshev (pronounced Chebyshev) Pafnuty Lvovich (1821-1894), Russian mathematician and mechanic.

Born on May 26, 1821 in the village of Okatov, Kaluga province, in a noble family. In 1837 he entered Moscow University.

In 1846 he defended his master's thesis on the topic "An attempt at an elementary analysis of the theory of probability." In 1847 he was invited to the Department of Mathematics at St. Petersburg University, where he lectured on algebra and number theory. In 1849, Chebyshev's "Theory of Comparisons" was published, according to which the author defended his doctoral dissertation in the same year at St. Petersburg University.

In 1850 he became a university professor. In 1882 he retired to devote himself to scientific work. Chebyshev managed to create new directions in various scientific fields: probability theory, the theory of approximation of functions by polynomials, integral calculus, number theory, etc.

In the theory of probability, the scientist introduced the method of moments; proved the law of large numbers by applying the inequality (the Bieneme-Chebyshev inequality).

In number theory, Chebyshev is responsible for a number of papers on the distribution of prime numbers. The works of the scientist in the field of mathematical analysis are known, in particular the study "On the limiting values ​​of integrals" (1873).

Chebysheva “on functions that deviate least from zero” are original both in terms of the essence of the issue and the method of solution. In 1878, he invented a calculating machine (kept in the Museum of Arts and Crafts in Paris). Chebyshev's works made his name famous not only in Russia, but also abroad.

The scientist was a member of the St. Petersburg, Berlin and Paris Academies of Sciences and the Bologna Academy, a corresponding member of the Royal Society of London and the Royal Swedish Academy of Sciences.

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Thank you!!! good for a report