Life and work of Pl Chebyshev. Life and scientific achievements of P.L. Chebyshev

one of the founders of modern approximation theory

Pafnuty Lvovich Chebyshev(very widespread mispronunciation Surnames with stress on the first syllable "Chebyshev") (4 (May 16), Okatovo, Kaluga province - November 26 (December 8), St. Petersburg) - Russian mathematician and mechanic. Honorary Member of the Academic Council of IMTU.

Biography

Chebyshev was born in the village of Okatovo, Borovsky district, Kaluga province, in the family of a wealthy landowner Lev Pavlovich. He received his initial upbringing and education at home, his mother Agrafena Ivanovna taught him to read and write, arithmetic and French - cousin Avdotya Kvintilanovna Sukhareva. In addition, since childhood Pafnuty Lvovich studied music.

Scientific activity

Chebyshev's scientific activity, which began in 1843 with the appearance of a small note "Note sur une classe d'intégrales dé finies multiples" ("Journ. de Liouville", vol. VIII), did not stop until the end of his life. His last memoir, "On Sums Depending on positive values any function", was published after his death (, "Mem. de l'Ac. des sc. de St.-Peters.").

Of the numerous discoveries of Chebyshev, first of all, works on number theory should be mentioned. Their beginning was laid in the additions to Chebyshev's doctoral dissertation: "Theory of Comparisons", published in the city. The famous "Mémoire sur les nombres premiers" appeared in the city, where two limits are given, which contain the prime numbers lying between two given numbers.

These two works would be enough to perpetuate the name of Chebyshev. In integral calculus, the memoir of 1860, Sur l'intégration de la différentielle, is especially remarkable, in which a way is given to find out by means of a finite number of operations, in the case rational coefficients radical polynomial, is it possible to determine the number A so that this expression is integrated in logarithms and, if possible, find the 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 Mr.'s memoir entitled "Sur les questions de minima qui se rattachent à la représentation approximative des fonctions" (in Mem. Acad. Sciences). This work is especially appreciated by scientists in Germany and France; for example, Professor Klein, in his lectures at the University of Göttingen in 1901, called this memoir "amazing" (wunderbar). Its content was included in the classic work of I. Bertrand, “Traité du Calcul diff. et integral". In connection with the same questions, there is Chebyshev's work “On drawing geographical maps". This series of works is considered the foundation of approximation theory.

Further, Chebyshev's works on interpolation are remarkable, 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 issues 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", 3873). The completely new questions posed here by Chebyshev were then developed by his students. The last memoir of Chebyshev in 1895 belongs to the same field. In connection with the questions "about functions that deviate least from zero", there are also Chebyshev's works on practical mechanics, which he studied a lot and with great love.

Chebyshev continued to teach his students even after they completed their university course, guiding their first steps in the scientific field, through conversations and precious indications of fruitful questions. Chebyshev created a school of Russian mathematicians, many of whom are known today.

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, it has survived to this day Chebyshev formula to calculate the range of the projectile. Through his work, Chebyshev provided big influence on the development of Russian artillery science.

Chebyshev's students

For Chebyshev, the task of creating and developing the Russian mathematical school has always been no less important than specific scientific results.

Among the direct students of Chebyshev are such well-known mathematicians as:

• Sokhotsky, Yulian Vasilievich

Publications

• Chebyshev P. L. On sums composed of the values ​​of the simplest monomials multiplied by a function that remains positive. - St. Petersburg, 1891. - 67s. - Zap. Imp. Acad. Nauk, T. 64, No. 7.
• Chebyshev P. L. On functions that slightly deviate from zero for some values ​​of the variable. - St. Petersburg, 1881. - 29 p. - Zap. Imp. Acad. Nauk, T. 40. No. 3.
• Chebyshev P. L. On the ratio of two integrals extended to the same values ​​of a variable. - St. Petersburg, 1883. - 33 p. - Zap. Imp. Acad. Nauk, T. 44. No. 2.
• Chebyshev P.L. On approximate expressions for the square root of a variable in terms of simple fractions. - St. Petersburg, 1889. - 22 p. - Zap. Imp. Acad. Nauk, T. 61, No. 1.

Grades and memory

The merits of Chebyshev were appreciated by the scientific world in a worthy way. He was elected a member of the St. Petersburg (), Berlin and Bologna Academies, the Paris Academy of Sciences (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 Royal Society of London, Swedish Academy of Sciences, etc., a total of 25 different Academies and learned 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.

see also

• Chebyshev set
• Chebyshev system of functions

Notes

Literature

• Prudnikov V. E. Pafnuty Lvovich Chebyshev, 1821-1894. L.: Nauka, 1976.
• Golovinsky I. A. To the justification of the method least squares P.L. Chebyshev. // Historical and mathematical research, M.: Nauka, vol. XXX, 1986, pp. 224-247.

Links

• Glazer G.I. History of mathematics in the school. - M.: Enlightenment, 1964. - 376 p.
• Kolmogorov A. N., Yushkevich A. P. (ed.) Mathematics of the 19th century. M.: Science.

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

• K. Posse. Chebyshev Pafnuty Lvovich // Critical and Biographical Dictionary of S. A. Vengerov.
• Pafnuty Lvovich Chebyshev - short biography and main works

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Pafnuty Lvovich Chebyshev Date of birth: 4 (May 16) 1821 Place of birth: Okatovo, Kaluga province ... Wikipedia

Chebyshev, Pafnuty Lvovich- (1821 1894) mathematician and mechanic, founder of the St. Petersburg scientific school. From 1847 he taught at St. Petersburg University (in 1850 he became 82 professors). long time took part in the work of the artillery department of the military scientific committee. ... ... Pedagogical terminological dictionary

OUTSTANDING COUNTRYMANS

GREAT MATHEMATIC P. L. CHEBYSHEV

In terms of contribution to world mathematics, the works of our fellow countryman Pafnutiy Lvovich Chebyshev are only comparable with those of Lobachevsky. He can rightfully be called a genius of mathematics. Peru owns outstanding works on analytic geometry, number theory, higher algebra, etc. Pafnuty Lvovich wrote about 100 scientific papers on number theory, probability theory, integral calculus, and the theory of mechanisms. He was the first in the world to prove "Bertrand's postulate", the theory of the distribution of prime numbers in the natural series. Chebyshev is the founder of a new branch of mathematics - the constructive theory of functions.

Pafnuty Lvovich Chebyshev was born in 1921 in the village Akatov(Okatovo) of the Borovsky district of the Kaluga province in the family of the Borovsky landowner, marshal of the nobility Lev Pavlovich Chebyshev. Elementary education young Pafnuty received houses from his mother Agrafena Ivanovna, nee. Pozdnyakova; At the age of 16 he entered Moscow University. The young man immediately discovered a huge talent in mathematics. While still a student, he received a silver medal for the essay "Calculation of the roots of an equation", and in 1846 he defended his master's thesis "An attempt at an elementary analysis of probability theory." In 1847, the young scientist was invited to work at St. Petersburg University, where he worked for 35 years. Here in 1849 he defended his doctoral thesis "Theory of Comparisons", awarded the Demidov Prize by the St. Petersburg Academy of Sciences. In 1850 Chebyshev was elected professor. He was entrusted with lecturing on analytic geometry, number theory, higher algebra, etc. Soon Chebyshev became an adjunct at St. Petersburg University. Simultaneously engaged scientific work in Russian Academy Sciences. Since 1856 Pafnuty Lvovich - extraordinary, since 1859 - ordinary academician of the St. Petersburg Academy of Sciences. Oleg MOSIN,

He was one of the first to connect the problems of mathematics with the fundamental questions of natural science and technology. He created more than 40 new and improved more than 80 machine mechanisms. Many of them were shown at exhibitions in Paris (1878) and Chicago (1893), having won the interest of world scientific thought.

For a long time, Pafnuty Lvovich took part in the work of the artillery department of the military scientific committee and the scientific committee of the Ministry of Public Education. And this is no coincidence. His younger brother, Vladimir Lvovich, is an artillery general, a professor at the artillery academy, and is engaged in mathematical calculations of shooting. Subsequently, these calculations will make him the founder of the arms business in Russia. He designed barrel mortars made at the Tula plant. Of all the brothers, it was he who was especially close to P. L. Chebyshev, with whose financial support the first two-volume collected works were published in 1900.

Chebyshev can rightly be called the second Lobachevsky; he is the founder of the St. Petersburg scientific school of mathematicians and mechanics, the most major representatives which were prominent scientists A. N. Korkin, E. I. Zolotarev, A. A. Markov, G. F. Voronoi, A. M. Lyapunov, V. A. Steklov, D. A. Grave. Characteristic features of Chebyshev's work are the variety of areas of research and constant interest in practical issues. Pafnutiy Lvovich's researches relate to number theory, algebra, integral calculus, probability theory, theory of mechanisms and many other sections of mathematics and related fields of knowledge.

The desire to connect the problems of mathematics with the fundamental issues of natural science and technology largely determines his originality as a scientist. Many of Chebyshev's discoveries are inspired by applied interests. This was repeatedly emphasized by Pafnuty Lvovich himself, saying that even 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 ... ". In the theory of probability, Chebyshev is credited with the systematic 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. They've proven the law big numbers in a very general form; at the same time, his proof is striking in its simplicity and elementary nature even to a person who is little versed in science.

The work of Pafnuty Lvovich on the theory of probability constitutes an important stage in its development; in addition, they were the basis on which the Russian school of probability theory grew up, which consisted of direct students of the scientist. In number theory, Chebyshev, for the first time after Euclid, significantly advanced the study of the question of the distribution of prime numbers. He was the first in the world to prove "Bertrand's postulate", the theory of the distribution of prime numbers in the natural series. These brilliant works of the scientist played an important role in the development of the theory of approximations, putting him on the same level with Euclid and Lobachevsky.

Chebyshev's most numerous works are in the field of mathematical analysis. He was also the subject of a dissertation in which he investigated the integrability of irrational expressions in algebraic functions and logarithms. Chebyshev also devoted a number of other works to this interesting problem. In one of them, a well-known theorem on integrability conditions in elementary functions of a differential binomial was obtained. An important line of research on mathematical analysis are his works on the construction of the theory of orthogonal polynomials. All these studies were closely related to the tasks that were set for Chebyshev in the artillery department of the military scientific committee.

Pafnuty Lvovich - the founder of the so-called constructive theory of functions, the creator of new areas of research in number theory and new research methods. The theory of machines and mechanisms was one of those disciplines in which Chebyshev was systematically interested throughout his life. Especially numerous are his works devoted to hinged mechanisms, in particular, Watt's parallelogram and others. He paid much attention to the design and manufacture of mechanisms. He designed and improved more than 100 new machines and mechanisms, which won first place at exhibitions in Paris (1878) and Chicago (1893). Very interesting and original are the plantigrade machine he created, which imitates the movement of a person when walking, as well as an automatic adding machine. The study of Watt's parallelogram and the desire to improve it prompted Chebyshev to solve the problem of the best approximation of functions. The applied work of the scientist also includes an original study, 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 scales in different points cards was the smallest. Chebyshev suggested that in order to do this, the mapping must maintain a constant scale on the boundary, which was later proved by the mathematician D. A. Grave.

The scientist left a bright mark in the development of mathematics both by his own research and by setting priority questions for young scientists. So, on his advice, A. M. Lyapunov began to work on the theory of equilibrium of a rotating fluid, the particles of which are attracted according to the law gravity thus creating a new science.

Chebyshev's works, even during his lifetime, found wide recognition not only in Russia, but also abroad; he was elected a member of the Berlin (1871), Bologna (1873), Paris (1874), Swedish (1893) academies of sciences, the Royal Society of London and many other foreign societies, academies and universities. Awarded with the Order of the Blessed Prince. Alexander Nevsky, the French Order of the Legion of Honor. In honor of Chebyshev, the USSR Academy of Sciences established in 1944 a prize for the best research in mathematics.

Pafnuty Lvovich died in 1894. He was buried in the village of Spas-Prognan, Borovsky district, Kaluga province, in a family crypt under the church. In the village of Akatovo, a monument was erected on the site of the house where the scientist grew up.

Svetlana MOSINA

Literature: Scientific legacy of P. L. Chebyshev. M.-L., 1945. Prudnikov V. E. P. L. Chebyshev. L., 1976; Chebyshev P.L. Complete Works. M. - L., 19441951; Chebyshev P. L. Selected Works. M., 1955; Khromienkov N. A., Chebysheva K. V. P. L. Chebyshev. L., 1976; Scientific legacy of P. L. Chebyshev. Issue. 1. - M. - L., 1945; P. L. Chebyshev: (Obituary) // KGV. 1894. No. 129; Chebysheva K. V. P. L. Chebyshev. - M., 1979; Prudnikov V. E. Pafnuty Lvovich Chebyshev. 1821-1894. - L., 1976; Zelenov V. S. Tourist trails Kaluga region. Tula, 1990.

Sym-metric-noy from-but-si-tel-but direct-my, passing through the fixed red ball-nir. You can say that in such a case, tra-ek-to-riya si-ne-go shar-ni-ra will be the same sim-met-rich-on from-but -si-tel-but some-swarm straight-my, passing through a motionless ball-nir. Russian ma-te-ma-tik Pa-f-nu-tiy Lvo-vich Che-by-shev is-sled-to-val-question, how can this tra -ek-to-riya.

An important particular case of a gray tra-ek-to-rii is a circle. In practice, he is re-a-li-zu-et-sya to-add-le-ni-em one-and-no-moving-no-go (red-no-go) ball-no- ra and the leading link for a certain length.

For the blue-it, the tra-ek-the-rii is two-important cases-cha-I-mi is-la-is-there is a similarity of its tin with the direct cut , whether with a circle or its arc. Che-by-shev p-shet: “Here we’ll take a look at the cases, the most simple and most pre- becoming-la-yu-shchih-sya on prak-ti-ke, but name-but when-has-to mean-to-be-chit the movement along the curve, someone - a swarm of some kind of paradise part, more or less significant, a little different from the arc of the circle or from the straight line.

Namely, to you-yav-le-niyu of the best-pair-ra-meters of this me-ha-niz-ma, re-sha-yu-sche-go-re-number-len -nye for-yes-chi, Pa-f-nu-tiy Lvo-vich for the first time himself applies the theory of approximation of functions, times-ra-bo-tan they didn’t have long before this while studying para-ral-le-lo-gram-ma Wat-ta.

Under-bi-paradise distance between-for-fortified-len-us-mi shar-ni-ra-mi, the length of the leading link, as well as the angle between the links, Pa-f-nu-tiy Lvo-vich in-lu-cha-et for-mknu-tuyu tra-ek-to-ryu, ma-lo bias-nya-yu-shchu -yu-Xia from straight-mo-li-her-but-go from-cut. Bias-non-blue-tra-ek-to-rii from direct-mo-li-her-noy can be reduced, from me-not-nyaya pa-ra-met-ry me-ha- low-ma. However, at the same time, it will decrease and the length of the ho-yes si-not-go ball-ni-ra. But this is about-is-ho-dit honey-len-nee than a decrease from-clo-non-niya from my direct one, therefore, for practical tasks, we can -but in-to-take pleasurable-your-ri-tel-nye para-meters. This is one of the options for near-wife-no-go straight-mi-la, pre-lo-female-no-go Che-by-she-vym.

Pe-rey-dem to the case of similarity of the blue curve with the circle.

Ras-smat-ri-vaya case, when the links make up a straight line, we come to the me-ha-bottom-mu, in the same way on Greek letter-wu "lamb-da". With some-ry-mi pa-ra-met-ra-mi Che-by-shev used-pol-zo-shaft him to build the first in the world "one hundred-po- ho-dya-schey ma-shi-ny ". At the same time, the blue crooked would look like a hat of a white mushroom. Pod-bi-rai pa-ra-met-ry lamb-da-me-ha-niz-ma in a different way, you can-but-be-cheat tra-ek-to-ryu, in a way -ryod-but ka-sa-yu-shu-yu-sya of two end-cen-three-che-circle-stay and remain-yu-shu-yu-sya all the time between them . From-me-pa-ra-meter-ry me-ha-niz-ma, you can reduce the distance between the end-cen-three-che-ski-mi around -stya-mi, inside-ri-ryh races-on-lo-same-on the blue tra-ek-to-rya.

Do-stro-im lamb-da-me-ha-nism, do-ba-viv motionless ball-nir and two links, the sum of the lengths of some-ry equals-on ra-di- y-su of a larger circle, and the difference is ra-di-u-su of lesser necks.

Better-chiv-she-e-sya device has bi-fur-ka-tion points or, as they say, syn-gu-lar-nye or special points ki. Being in such a point, with the same movement of the lamb-da-me-ha-niz-ma along the cha-so-howling arrow to-add-len -nye links can start to rotate either according to the clockwise arrow, or against. There are six such checks of bi-fur-ka-tions in our me-ha-niz-me - when the added links are on-ho-dyat-sya on one straight.

There is a pain and an important on-right-le-tion in ma-te-ma-ti-ke - the theory of especially-ben-no-stay - research-sle-to-va -nie pre-me-ta through the study of its special to-checks. A very simple special case is the study of the function through the study of the check of its mac- si-mu-ma and mi-ni-mu-ma.

In order for our mechanism to go through all six special to-checks in one-on-a-forward, you-branded-on-right-le-ni, a little link connection-zy-va-yut with ma-ho-vi-com, someone-swarm, bu-duchi ras-ru-chen-nym in some kind of hundred-ro-well, you-in-dit me -ha-nism from a special point, rotating in the same hundred-ro-well.

If, from the point of bi-fur-ka-tion, spread the ma-ho-vik as well as the leading link, according to the hour of the arrow, then in one the turn of the ve-du-shche-th link-on ma-ho-vik will make two turn-of-ta.

If, from a special point, give the ma-ho-vi-ku the movement against the hour of the arrow, then in one turn we-du-sche- the first link according to the cha-so-howling arrow-ke ma-ho-vik will make a whole four-you-re ob-ro-ta!

This is the key-cha-et-pa-ra-doc-sal-ness of this me-ha-niz-ma, with-du-man-no-go and done-lan-no-go Pa -f-well-ti-em Lvo-vi-than Che-by-she-vym. Ka-for-moose would be, a flat ball-nir-ny mechanism-ha-nism should work one-but-meaning-but, one-on-one, as you can see, this is not all -when so. And at the same time, there are special points.

Chebyshev Pafnuty Lvovich (1821-1894) Russian mathematician and mechanic, member of the St. Petersburg Academy of Sciences (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 since 1860), Royal Society of London (1877), Swedish Academy of Sciences (1893) and honorary member of many Russian and foreign scientific societies, academies, universities .

He was born on May 4, 1821 in the village of Okatovo, Kaluga province, in the family of a landowner. In the summer of 1837, Pafnuty Lvovich began studying mathematics at Moscow University in the second philosophical department. Among his teachers who influenced him the most in the future: Nikolai Brachman, who introduced him to the work of the French engineer Jean-Victor Poncelet. 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. In 1841, there was a famine in Russia, and the Chebyshev family could no longer support him. However, Pafnuty Lvovich was determined to continue his studies. He successfully graduates from the university and defends his dissertation. In 1847, Chebyshev was approved as an associate professor and began to lecture on algebra and number theory at St. Petersburg University. At the age of twenty-eight, he received a doctorate degree from St.

The scientific interests of P. L. Chebyshev are distinguished by great diversity and breadth. He left behind brilliant research in the field of mathematical analysis, especially in the theory of approximation of functions by polynomials, in integral calculus, number theory, probability theory, geometry, ballistics, theory of mechanisms and other fields of knowledge.

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.

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. 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 is 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.

P. L. Chebyshev's research in number theory was of outstanding importance for science. For the first time after Euclid, he received the most important results in the problem of the distribution of prime numbers in the works "On the determination of the number of prime numbers not exceeding a given value" and "On prime numbers". Chebyshev's works on the theory of probability ["Experience in 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. 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.

One of the sciences that Pafnuty Lvovich was interested in all his life was the theory of mechanisms and machines, and Chebyshev was engaged not only in theoretical research in this area, but also paid great attention to the direct design of specific mechanisms. 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 with hinged mechanisms rotary motion with different directions of rotation about two axes, 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 owns 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 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.

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 and machines.

The main works of P L. Chebyshev: Experience of 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), appended biographical sketch, written by K. A. Posse. Complete works, vol. I - 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 variable through simple fractions; On two theorems regarding probabilities), M. - L., 1946.

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.

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 the questions of the best, most economical, and rational use cash. 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 corresponding 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.

Chebyshev takes the requests of practice for himself as a creative order. He comes to the aid of the engineers, for a long time trying to improve "Watt's parallelogram" - a mechanism for turning forward movement into rotational, 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- 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. With the help of it, in apparent chaos, as, for example, the movement of gas molecules seems to be, to see the patterns of this movement 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 greatest victory 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.