 Pafnuty Chebyshev is widely known as a mathematician and an engineer, and his formulas are still used today, and using Lego every school kid can make the walking machine of Chebyshev.

Despite the difficulty in understanding of mathematical discoveries of Chebyshev by ordinary people, we will try (at least for people with higher education) to present them in simple terms, and then will move on to the mechanics, where the images of mechanisms will tell you about the achievements of Paphnutius Chebyshev.

# Chebyshev Mathematics

In each of the branches of mathematics, which Chebyshev worked in, he proposed general methods and put forward ideas that outlined the main directions of their future development.

## Theory of Numbers

In 1951Chebyshev published his paper on the definition of prime numbers that are part of an arbitrarily chosen number. In the framework of the Legendre hypothesis, he developed a better solution to find the number of prime numbers. That work immediately brought him worldwide fame.

## Probability Theory

Chebyshev published only four papers, but they all made a significant contribution to the theory of probability. For example, he formulated the “Chebyshev’s inequality”, which limits the probability of deviations. Chebyshev used inequality to substantiate the law of large numbers.

Chebyshev gave a definition of a random variable, which is still used today.

It is believed that it was Chebyshev who first developed the method which later became known as the method of determining the probability distribution by its moments.

## Functions Approximation Theory

It is believed that it was Chebyshev who first defined this area of mathematics.

Chebyshev came to the conclusion that Taylor’s formula was not effective enough for the approximation of analytic functions. To do this, he developed the mathematical apparatus of the polynomial, which describes such function very well. Now these polynomials are part of a standard approach in computer programming to describe the functions. The concepts that were named after Chebyshev: Chebyshev polynomials, Chebyshev norm of the functions differences (this approach is particularly useful in mapping and creating of accurate watch movements), point of Chebyshev alternance.

With regards to mapping Chebyshev formulated the theorem of the best display of the Earth’s surfaceon the map: a theorem of Chebyshev-Grave, Chebyshev projections.

## Mathematical Analysis and Geometry

Chebyshev formulated inequalities for monotone functions – Chebyshev inequalities.

In the theory of orthogonal polynomials, Chebyshev introduced two new systems of classical orthogonal polynomials (Chebyshev-Hermite polynomials and Chebyshev-Laguerre polynomials).

Chebyshev also introduced a new class of grids (Chebyshev net) in differential geometry, which found direct practical application in creating patterns for clothes (the paper was even called “On patterns in making clothes”). Creating patterns for clothes