# Pafnuty Chebyshev

The greater the dark, the easier to be a star.
Stanisław Jerzy Lec

Pafnuty Lvovich Chebyshev (Пафнутий Львович Чебышёв) (1821-1894) was a prominent Russian mathematician, who is considered to be a founding father of Russian mathematics. Besides mathematics, Chebyshev spent much of his time working on questions of mechanical engineering, and in the early 1870s, he designed and manufactured very interesting calculating machines.

In 1876 Chebyshev made a report to the French Association of Assistance to Prosperity of Sciences about his first calculating machine. The report was called “An Adding Machine of Continuous Motion”. In the same 1876, the device was probably exhibited at the scientific meeting in Clermont-Ferrand.

It was a ten decimal places adding machine with a continuous tens carry, the first known machine with this type of carry mechanism. In a regular calculation with discrete carry, the wheel of higher rank moves on by one point, while the lower rank wheel moves from 9 to 0. During the continuous tens carry, the wheel of higher rank moves from one point to the next gradually and continuously, while the lower rank wheel turns by one revolution. Hence, it is necessary to have a construction such that the wheel of the tens moves ten times slower than that of the units, the wheel of the hundreds moves ten times slower than that of the tens, etc. This can be achieved with gears in a 1 to 10 ratio between the units and the tens, between the tens and the hundreds, and so on. This solution then totally dispenses with the usually complex mechanisms required for an intermittent motion. It also makes it possible to do calculations much faster.

Chebyshev reached this effect of gradual movement by implementing a planetary transmission (the same principle will be used some ten years later on by Eduard Selling).

Two years later Chebyshev created a second improved model of his adding device, which was presented in 1878 to Conservatoire des Arts et Métiers in Paris. Later, he made a dividing-multiplying extension unit for the machine, which was also sent to the Paris museum in 1881. So the machine became a real arithmometre (it can be used for all arithmetical operations), which has two separate blocks—one for addition and subtraction, and one for multiplication and division.

In 1882, Chebyshev returned to France and had a second machine built in Paris by the firm Gautier. Another copy of the machine was built after Chebyshev’s death for exhibition in Moscow.

The main purpose of the machine was to demonstrate the new principle of continuous tens carry. The dividing-multiplying unit also had some innovations, e.g. the automatic shifting of the carriage from decimal place to decimal place. The unit itself served as the carriage part, that is the moving part of the mechanism. It was mounted on the adding machine, thus imposing one single device. To perform multiplication, the operator only needed to turn the handle. The number of turns was equal to the sum of numbers of the multiplication factor, added to the number of its decimal places minus one. After multiplying by a number (digit) of one decimal place, the mechanism automatically stopped multiplication and shifted the carriage to the next decimal place. This was repeated with the next decimal place digit, etc. The number of handle turns was automatically controlled by means of a special counter, connected to the mechanism for setting the factor.

Since the donation of the machine to the museum was not followed by any publications in western media, this invention didn’t become famous (but the arithmometre was described in some Russian sources, for example, this one from 1894, and this article by von Bool.). As late as 1890 the French mathematician François Édouard Anatole Lucas (one of the creators of the famous Genaille-Lucas rulers) displayed a variety of Chebyshev’s mechanisms, including the arithmometre, on a special stand at the Paris Museum and gave several lectures about Chebyshev. Later on, the French historian Maurice d’Ocagne (1862-1938) contacted Chebyshev for a description of the machine and published an article.

In fact, both machines of Chebyshev were made only for demonstration purposes. He never seriously thought of creating a device for practical or commercial use. His personal innovations are continuous tens carry and automatic shifting of the carriage from decimal place to decimal place during multiplication (Chebyshev proved that these principles can help build extremely efficient machines). Both inventions became popular and were widely implemented in 1930s when electromotive drives came into use in the quickly growing generation of automatic and semi-automatic keyboard calculating machines.

The adding machine of Chebyshev is an unusual calculating machine. Using gradual motion, it dispenses with carry levers and such, and allows for faster calculations. It is also a modular machine, in which an independent addition component becomes a sliding carriage of a larger multiplication machine. Chebyshev also automated the multiplication by one digit and the shift of the carriage, so that the user merely has to rotate the crank and not worry about shifting when multiplying by numbers greater than 9.

#### Biography of Pafnuty Chebyshev

Pafnuty Lvovich Chebyshev was born in his father’s estate in the village of Okatovo, district of Borovsk, province of Kaluga, on 16 May 1821, into an upper-class family with an impressive history. He was the first son of Lev Pavlovich Chebyshev (1789-1861), a military officer and landowner, who in 1815 left the army as a cornet in the cavalry, and in 1817 married Agrafena Ivanovna Pozniakova (1790-1867) from Kaluga. The family had (from the beginning of the 18th century) a big estate in Spas-Prognanie village, Kaluga area. Lev and Agrafena had five sons and four daughters: Elisaveta (1819-1888), Pafnuty (1821-1894), Pavel (1822-1869), Piotr (1824-1891), Ekaterina (1827-1910), Nadezhda (1819-1910), Nikolai (1830-1875), Vladimir (1831-1905), and Olga (1836-1908).

Pafnuty Lvovich’s early education was at home where both his mother and his cousin Avdotia Kvintillianova Soukhareva were his teachers. From his mother, he learned the basic skills of reading and writing, while his cousin acted as a governess to the young boy and taught him French and arithmetic. Chebyshev mentioned that his music teacher also played an important role in his education, for she “raised his mind to exactness and analysis.” One of the children’s hobbies of Pafnuty was the study of the mechanisms of toys and machines, and he himself invented and made them. All was not easy for the young boy, however, for with one leg longer than the other he had a limp which prevented him from taking part in many of the normal childhood activities, so his parents abandoned the idea of his becoming an officer in the family tradition.

In 1832 the family moved to Moscow, mainly to attend to the education of their eldest sons (Pafnuty and Pavel, who would become lawyers). His parents engaged teachers of excellent reputation, including Platon Pogorelski, who was considered the best elementary mathematics tutor in Moscow, and Pafnuty studied mathematics, physics, and Latin. In 1837 he entered Moscow University, where he studied mathematics under professor Nikolaus Braschmann, and obtained a bachelor’s degree in 1841, and a master’s degree in 1846. In 1847 Pafnuty was appointed as a professor at St Petersburg University and kept this position until 1882.

Chebyshev was a remarkable inventor, who devised multiple machines, presented and prized at many expositions all over the world, e.g. London Exposition of 1876, Philadelphia Exposition of 1876, Paris Exposition of 1876, and Chicago Exposition of 1893. His inventions inlcude: a steam machine, bicycle, sorting machine, rowing machine, scooter, press, and many other devices.

As to Chebyshev’s personal life, he never married and lived alone in a large house with ten rooms. He was rich, spending little on everyday comforts but he had one great love, namely that of buying property and paying for models of his machines. It was on this that he spent most of his money but he did financially support a daughter whom he refused to officially acknowledge. He did spend time with this daughter, especially after she married a colonel.

Pafnuty Lvovich Chebyshev died in his home in St Petersburg on 26 November 1894, of complications after influenza, and was buried at his family estate in Spas-Prognanie village.