Pafnuty Lvovich Chebyshev was a Russian mathematician. He is known for his work in the fields of probability, statistics, mechanics, and number theory.
Background
Chebyshev was born on May 16, 1821, in the village of Okatovo in the district of Borovsk. One of nine children, he was born into a family which traced its roots back to a 17th-century Tatar military leader named Khan Chabysh. His father, Lev Pavlovich Chebyshev, was a retired army officer who had participated in the war against Napoleon, and his mother was Agrafena Ivanovna Pozniakova Chebysheva.
Trendelenburg's gait affected Chebyshev's adolescence and development. From childhood, he limped and walked with a stick and so his parents abandoned the idea of his becoming an officer in the family tradition. His disability prevented his playing many children's games and he devoted himself instead to mathematics.
Education
In 1832 the Chebyshevs moved to Moscow, where Pafnuty completed his secondary education at home. He was taught mathematics by P. N. Pogorelski, one of the best tutors in Moscow and author of popular textbooks in elementary mathematics.
In 1837 Chebyshev enrolled in the department of physics and mathematics (then the second section of the department of philosophy) of Moscow University. Mathematical disciplines were then taught brilliantly by N. D. Brashman and N. E. Zernov. Brashman, who always directed his pupils toward the most essential problems of science and technology (such as the theory of integration of algebraic functions or the calculus of probability, as well as recent inventions in mechanical engineering and hydraulics), was especially important to Chebyshev’s scientific development.
As a student Chebyshev wrote a paper, “Vychislenie komey uravneny” (“Calculation of the Roots of Equations”). This paper, written by Chebyshev for a competition on the subject announced by the department of physics and mathematics for the year 1840-1841, was awarded a silver medal, although it undoubtedly deserved a gold one.
In the spring of 1841 Chebyshev graduated from Moscow University with a candidate (bachelor) of mathematics degree.
Career
Chebyshev published an article on the theory of multiple integrals in Liouville’s Journal des mathématiques pures et appliquées and in 1844 an article on the convergence of Taylor series in Crelle’s Journal fur die reine und angewandte Mathematik. Shortly afterward he submitted as his master’s thesis, Opyt elementarnogo analiza teorii veroyatnostey (“An Essay on an Elementary Analysis of the Theory of Probability”). The thesis was defended in the summer of 1846 and was accompanied by “Démonstration élémentaire d’une proposition générale de la théorie des probabilités” (Journal für die reine und angewandte Mathematik, 1846), which was devoted to Poisson’s law of large numbers. These works aimed at a strict but elementary deduction of the principal propositions of the theory of probability.
It was almost impossible to find an appropriate teaching job in Moscow, so Chebyshev willingly accepted the offer of an assistant professorship at Petersburg University. As a thesis he submitted “Ob integrirovanii pomoshchyu logarifmov” (“On Integration by Means of Logarithms”), written, at least in the first draft, as early as the end of 1843. The thesis, defended in the spring of 1847, incidentally solved a problem of integration of algebraic irrational functions in the final form that had been posed shortly before by Ostrogradski. The thesis was published posthumously, as late as 1930, but Chebyshev included its principal results in his first publication on the subject in 1853. In September 1847, at Petersburg University, Chebyshev began lecturing on higher algebra and the theory of numbers. Later he lectured on numerous other subjects, including integral calculus, elliptic functions, and calculus of finite differences; but he taught the theory of numbers as long as he was at the university (until 1882).
From 1860 he regularly lectured on the theory of probability, which had previously been taught for a long time by V. Y. Bunyakovski. A. M. Lyapunov, who attended Chebyshev’s lectures in the late 1870’s, thus characterized them. His courses were not voluminous, and he did not consider the quantity of knowledge delivered; rather, he aspired to elucidate some of the most important aspects of the problems he spoke on. These were lively, absorbing lectures; curious remarks on the significance and importance of certain problems and scientific methods were always abundant. Sometimes he made a remark in passing, in connection with some concrete case they had considered, but those who attended always kept it in mind. Consequently, his lectures were highly stimulating; students received something new and essential at each lecture; he taught broader views and unusual standpoints.
Soon after Chebyshev moved to St. Petersburg, he was hired by Bunyakovski to work on the new edition of Euler’s works on the theory of numbers that had been undertaken by the Academy of Sciences. This edition comprised not only all of Euler’s previously published papers on the subject but also numerous manuscripts from the Academy’s archives; in addition Bunyakovski and Chebyshev contributed a valuable systematic review of Euler’s arithmetical works. Probably this work partly inspired Chebyshev’s own studies on the theory of numbers; these studies and the investigations of Chebyshev’s disciples advanced the theory of numbers in Russia to a level as high as that reached a century before by Euler. Some problems of the theory of numbers had been challenged by Chebyshev earlier, however, in his thesis pro venia legendi. He devoted to the theory of numbers his monograph Teoria sravneny (“Theory of Congruences”), which he submitted for a doctorate in mathematics. He defended it at Petersburg University on 27 May 1849 and a few days later was awarded a prize for it by the Academy of Sciences. Chebyshev’s systematic analysis of the subject was quite independent and contained his own discoveries; it was long used as a textbook in Russian universities. It also contained the first of his two memoirs on the problem of distribution of prime numbers and other relevant problems; the second memoir, submitted to the Academy of Sciences in 1850, appeared in 1852.
Through these two works, classics in their field, Chebyshev’s name became widely known in the scientific world. Later he returned only seldom to the theory of numbers. In 1850 Chebyshev was elected extraordinary professor of mathematics at Petersburg University; in 1860 he became a full professor. This was a decade of very intensive work by Chebyshev in various fields. First of all, during this period he began his remarkable studies on the theory of mechanisms, which resulted in the theory of the best approximation of functions. From his early years, Chebyshev showed a bent for construction of mechanisms; and his studies at Moscow University stimulated his interest in technology, especially mechanical engineering. In 1849-1851 he undertook a course of lectures on practical (applied) mechanics in the department of practical knowledge of Petersburg University (this quasi-engineering department existed for only a few years); he gave a similar course in 1852-1856 at the Alexander Lyceum in Tsarskoe Selo (now Pushkin), near St. Petersburg.
Chebyshev’s mission abroad, from July to November 1852, was another stimulus to his technological and mathematical work. In the evenings he talked with the best mathematicians of Paris, London, and Berlin or proceeded with his scientific work; morning hours were devoted to the survey of factories, workshops, and museums of technology. He paid special attention to steam engines and hinge-lever driving gears. He began to elaborate a general theory of mechanisms and in doing so met, according to his own words, certain problems of analysis that were scarcely known before. These were problems of the theory of the best approximation of functions, which proved to be his outstanding contribution; in this theory, his technological and mathematical inclinations were synthesized. Back in St. Petersburg, Chebyshev soon submitted to the Academy of Sciences his first work on the problem of the best approximation of functions, prepared mainly during his journey and published in 1854. This was followed by another work on the subject, submitted in 1857 and published in 1859. These two papers marked the beginning of a great cycle of work in which Chebyshev was engaged for forty years.
While in Europe, Chebyshev continued his studies on the integration of algebraic functions. His first published work on the problem, far surpassing the results at which he had arrived in the thesis pro venia legendi, appeared in 1853. He published papers on this type of problem up to 1867, the object of them being to determine conditions for integration in the final form of different classes of irrational functions. Here, as in other cases, research was associated with university teaching; Chebyshev lectured on elliptic functions for ten years, until 1860. In 1853 Chebyshev was voted an adjunct (i.e., junior academician) of the Petersburg Academy of Sciences with the chair of applied mathematics. Speaking for his nomination, Bunyakovski, Jacobi, Struve, and the permanent secretary of the Academy, P. N. Fuss, emphasized that Chebyshev’s merits were not restricted to mathematics; he had also done notable work in practical mechanics. In 1856 he was elected an extraordinary academician and in 1859 ordinary academician (the highest academic rank), again with the chair of applied mathematics.
From 1856 Chebyshev was a member of the Artillery Committee, which was charged with the task of introducing artillery innovations into the Russian army. In close cooperation with the most eminent Russian specialists in ballistics, such as N. V. Maievski, Chebyshev elaborated mathematical devices for solving artillery problems. He suggested (1867) a formula for the range of spherical missiles with initial velocities within a certain limit; this formula was in close agreement with experiments. Some of his works on the theory of interpolation were the result of the calculation of a table of fire effect based on experimental data. Generally, he contributed significantly to ballistics. Simultaneously Chebyshev began his work with the Scientific Committee of the Ministry of Education. Like Lobachevski, Ostrogradski, and a number of other Russian scientists, Chebyshev was active in working for the improvement of the teaching of mathematics, physics, and astronomy in secondary schools. For seventeen years, up to 1873, he participated in the elaboration of syllabi for secondary schools. His concise but solid reviews were of great value to the authors of textbooks that the Scientific Committee was supposed, as one of its principal functions, to approve or reject. From the middle of the 1850’s, the theory of the best approximation of functions and the construction of mechanisms became dominant in Chebyshev’s work. Studies on the theory of functions embraced a very great diversity of relevant problems: the theory of orthogonal polynomials, the doctrine of limiting values of integrals, the theory of moments, interpolation, methods of approximating quadratures, etc. In these studies, the apparatus of continued fractions, brilliantly employed by Chebyshev in many studies, was further improved.
From 1861 to 1888 Chebyshev devoted over a dozen articles to his technological inventions, mostly in the field of hinge-lever gears. Examples of these devices are preserved in the Mathematical Institute of the Soviet Academy of Sciences in Moscow and in the Conservatoire des Arts et Métiers in Paris. In the 1860’s Chebyshev returned to the theory of probability. One of the reasons for this new interest was, perhaps, his course of lectures on the subject started in 1860. He devoted only two articles to the theory of probability, but they are of great value and designate the beginning of a new period in the development of this field. In the article of 1866 Chebyshev suggested a very wide generalization of the law of large numbers. In 1887 he published (without extensive demonstration) a corresponding generalization of the central limit theorem of Moivre and Laplace. Besides the above-mentioned mathematical and technological fields, which were of primary importance in Chebyshev’s life and work, he showed lively interest in other problems of pure and applied mathematics.
Chebyshev investigated a problem of binding a surface with cloth that is formed in the initial flat position with two systems of nonextensible rectilinear threads normal to one another. When the surface is bound with cloth the “Chebyshev net,” whose two systems of lines form curvilinear quadrangles with equal opposite sides, appears. Wrapping a surface in cloth is a more general geometrical transformation than is deformation of a surface, which preserves the lengths of all the curved lines; distances between the points of the wrapped cloth that are situated on different threads are, generally speaking, changed in wrapping. In recent decades Chebyshev’s theory of nets has become the object of numerous studies. Theoretical mechanics also drew Chebyshev’s attention. Thus, in 1884 he told Lyapunov of his studies on the problem of the ring-shaped form of equilibrium of a rotating liquid mass the particles of which are mutually attracted according to Newton’s law. It is hard to know how far Chebyshev advanced in this field, for he published nothing on the subject. Still, the very problem of the form of equilibrium of a rotating liquid mass, which he proposed to Lyapunov, was profoundly investigated by the latter, who, along with Markov, was Chebyshev’s most prominent disciple. Among his technological inventions was a calculating machine built in the late 1870’s.
Between 1868 and 1880 he read twelve reports at the congresses of Russian naturalists and physicians, and sixteen at the sessions of the Association Française pour l’Avancement des Sciences between 1873 and 1882; it was at these sessions that he read “Sur la coupe des vêtements” and reported on the calculating machine. He gave numerous demonstrations of his technological inventions both at home and abroad. Chebyshev was in contact with the Moscow and St. Petersburg mathematical societies and with the Moscow Technological College (now Bauman Higher Technological College). In the summer of 1882, after thirty-five years of teaching at Petersburg University, Chebyshev retired from his professorship, although he continued his work at the Academy of Sciences. Nonetheless, he was constantly in touch with his disciples and young scientists. When Chebyshev was over sixty, he could not work at his former pace; nevertheless, he published some fifteen scientific papers, including a fundamental article on the central limit theorem. He submitted his last work to the Academy of Sciences only a few months before his death at the age of seventy-three.
Personality
Chebyshev was highly endowed with the ability to attract beginners to creative work, setting them tasks demanding a profound theoretical investigation to solve and promising brilliant results.