# Pafnutii Lvovich Chebyshev

## Chebyshev, Pafnutii L’vovich

Born May 14 (26), 1821, in the village of Okatovo, Kaluga Province, now Kaluga Oblast; died Nov. 26 (Dec. 8), 1894, in St. Petersburg. Russian mathematician and specialist in mechanics. Academician of the St. Petersburg Academy of Sciences (1859; adjunct, 1853; corresponding member [academician extraordinary], 1856).

After receiving his initial education at home, Chebyshev entered Moscow University at the age of 16. He graduated from the university in 1841 and defended his master’s dissertation there in 1846. In 1847 he moved to St. Petersburg and, after defending a dissertation, began teaching algebra and number theory at the University of St. Petersburg. In 1849, Chebyshev defended his doctoral dissertation, which was awarded the Demidov Prize of the St. Petersburg Academy of Sciences in the same year. He became a professor at the University of St. Petersburg in 1850. Chebyshev was long associated with the artillery division of the military academic committee and the academic committee of the Ministry of Public Education. In 1882 he gave up his teaching position at the University of St. Petersburg to devote himself fully to scientific work.

Chebyshev was the founder of the St. Petersburg mathematical school, the most prominent members of which were A. N. Korkin, E. I. Zolotarev, A. A. Markov, G. F. Voronoi, A. M. Liapunov, V. A. Steklov, and D. A. Grave.

Chebyshev contributed to a variety of fields. He was able to obtain important scientific results through elementary means, and he exhibited an unflagging interest in practical problems. The many branches of mathematics and related areas of knowledge in which he worked included probability theory, number theory, integral calculus, the approximation of functions by polynomials, and the theory of mechanisms. In each field mentioned, Chebyshev created a number of fundamental, general methods and set forth ideas establishing the principal directions of its subsequent development. Chebyshev’s genius as a scientist consisted largely in his tendency to link problems of mathematics to fundamental problems of the natural sciences and engineering. Many of his discoveries were inspired by the needs of applied sciences. He himself noted this fact several times. He wrote that in the creation of new methods of investigation “the sciences find themselves a reliable guide in practice” and that “the sciences are advancing under the influence of practice as it discovers new objects of study for them” (*Poln. sobr. soch*., vol. 5,1951, p. 150).

In probability theory Chebyshev introduced random variables into study on a systematic basis and devised a new procedure for proving limit theorems—the method of moments (1845, 1846, 1867, 1887). He provided a proof of striking simplicity and elegance for the law of large numbers in an extremely general form. He investigated the conditions under which the distribution of sums of independent random variables converges to a normal distribution. By supplementing Chebyshev’s methods, A. A. Markov subsequently provided a satisfactory proof of such convergence. Chebyshev also suggested, without rigorous proof, the possibility of refinements in this limit theorem in the form of asymptotic expansions of the distribution function of a sum of independent terms in powers of *n*^{–½}, where *n* is the number of terms. Chebyshev’s works on probability theory constitute an important stage in the field’s development. They provided the basis for the Russian school of probability theory, which initially consisted of Chebyshev’s own students.

In number theory Chebyshev was responsible for the first substantial advances since Euclid in the study of the distribution of prime numbers (1849, 1852). He proved that the function π(*x*)—the number of primes that do not exceed *x*—satisfies the inequalities

where *a* < 1 and *b* > 1 are constants that he calculated (*a* = 0.921, *b* = 1.06). An investigation of the distribution of primes in the sequence of integers led him to the investigation of quadratic forms with positive determinants. A paper by him that was devoted to approximating numbers by rational numbers (1866) played an important part in the development of the theory of Diophantine approximations. Chebyshev was the creator of new fields of study in number theory and of new methods of investigation.

Chebyshev’s works in mathematical analysis are the most numerous. In particular, the dissertation he submitted for the right to teach at the University of St. Petersburg deals with this field: in it he investigated the integrability of some irrational expressions in terms of algebraic functions and logarithms. He also devoted a number of other papers to the integration of algebraic functions; in one of them (1853) he derived the well-known theorem on the conditions of integrability of a binomial differential in terms of elementary functions. Chebyshev’s works on the construction of a general theory of orthogonal polynomials played a seminal role in the development of this area of mathematical analysis. His starting point in the creation of the theory was parabolic interpolation by the least squares method. Chebyshev’s studies on the problem of moments and on quadrature formulas draw on the same range of ideas. Wishing to shorten calculations, he proposed in 1873 the examination of quadrature formulas with equal coefficients, or weights (*see*APPROXIMATE INTEGRATION). His research on quadrature formulas was closely connected with the problems he faced in the artillery division of the military academic committee.

Chebyshev was the founder of the constructive theory of functions, the principal component of which is the theory of best approximation of functions (*see*APPROXIMATION AND INTERPOLATION OF FUNCTIONS and CHEBYSHEV POLYNOMIALS). His simplest statement of the problem (1854) is as follows: given a continuous function *f*(*x*), find among all polynomials of degree *n* the polynomial *P*(*x*) such that in a given interval [*a, b*] the expression

is a minimum. In addition to the described best uniform approximation, Chebyshev studied approximation in the quadratic mean, and in addition to approximation by algebraic polynomials he investigated approximation by means of trigonometric polynomials and rational functions.

The theory of machines and mechanisms interested Chebyshev throughout his life. He devoted many papers to linkages, particularly Watt’s parallel motion (for example, 1861, 1869, 1871, 1879), and he designed and built several devices. Of particular interest are his plantigrade machine, which simulated the motion of a walking animal, and his automatic calculating machine. While studying Watt’s linkage and seeking to improve it, Chebyshev formulated the problem of the best approximation of functions.

Chebyshev’s works of an applied nature include an original study (1856) in which he posed the problem of finding for a given country the conformal cartographic projection under which the maximum difference in scale at different points of the map is a minimum. He voiced without proof the opinion that in such a projection the scale at the border must be constant. This conjecture was subsequently proved by Grave.

Chebyshev’s position in the development of mathematics rests not only on his research but on the assistance he provided to young mathematicians, to whom he suggested problems for study. For example, it was on Chebyshev’s advice that A. M. Liapunov began his series of studies on the figures of equilibrium of a rotating fluid whose particles are attracted according to the law of universal gravitation.

During his lifetime Chebyshev’s works won broad recognition both in Russia and abroad. He was elected a member of the Berlin Academy of Sciences in 1871, of the Bologna Academy of Sciences in 1873, of the Académie des Sciences in 1874 (correspondent, 1860), of the Royal Society of London in 1877, and of the Swedish Academy of Sciences in 1893. In addition, he was named ’. an honorary member of many Russian and foreign scientific societies, academies, and universities.

An award for outstanding research in mathemetics was established in honor of Chebyshev by the Academy of Sciences of the USSR in 1944.

### WORKS

*Sochineniia*, vols. 1–2. St. Petersburg, 1899–1907.

*Poln. sobr. soch*., vols. 1–5. Moscow-Leningrad, 1944–51. (Contains bibliography.)

*Izbr. trudy*. Moscow, 1955.

### REFERENCES

Liapunov, A. M. “Pafnutii L’vovich Chebyshev.” In the book P. L. Chebyshev,*Izbr. matematicheskie trudy*. Moscow-Leningrad, 1946.

Steklov, V. A.

*Teoriia i praktika v issledovaniiak Chebysheva: Rech’*. Paris, 1921.

Krylov, A. N.

*Pafnutii L’vovich Chebyshev: Biograficheskii ocherk*. Moscow-Leningrad, 1944.

*Nauchnoe nasledie P. L. Chebysheva*, fascs. 1–2. Moscow-Leningrad, 1945.

Delone, B. N.

*Peterburgskaia shkola teorii chisel*. Moscow-Leningrad, 1947. (Contains bibliography.)

B. V. GNEDENKO