This world's first walking mechanism, invented by a Russian mathematician, received universal approval at the World Exhibition in Paris in 1878.

Pafnuty Lvovich Chebyshev is an outstanding Russian mathematician whose research covered a wide range of scientific problems.

In his works, he sought to combine mathematics with the fundamentals of natural science and technology. A number of Chebyshev's discoveries are associated with applied research, primarily related to the theory of mechanisms. In addition, Chebyshev is one of the founders of the theory of best approximation of functions using polynomials. He proved in general form the law of large numbers in probability theory, and in number theory the asymptotic law of distribution of prime numbers, etc. Chebyshev’s research was the basis for the development of new branches of mathematical science.

The future world-famous mathematician was born on May 26, 1821 in the village of Okatovo, Kaluga province. His father, Lev Pavlovich, was a wealthy landowner. The mother, Agrafena Ivanovna, was involved in the upbringing and education of the child. When Paphnutius turned 11 years old, the family moved to Moscow to continue their children’s education. Here Chebyshev met some of the best teachers - P. N. Pogorevsky, N. D. Brashman.

In 1837, Paphnutius entered Moscow University. In 1841, Chebyshev wrote the work “Calculating the Roots of Equations,” and it was awarded a silver medal. In the same year, Chebyshev graduated from the university.

In 1846, Pafnuty Lvovich defended his master's thesis, and a year later he moved to St. Petersburg. Here he began teaching at St. Petersburg University.

In 1849, Chebyshev defended his doctoral dissertation “The Theory of Comparisons” (it was awarded the Demidov Prize). From 1850 to 1882, Chebyshev was a professor at St. Petersburg University.

A significant number of Chebyshev's works are related to problems of mathematical analysis. Thus, the scientist’s dissertation for the right to give lectures is devoted to the integrability of some irrational expressions in algebraic functions and logarithms. The proof of the famous theorem on the conditions for the integrability of a differential binomial in elementary functions is presented in the 1853 work “On the integration of differential binomials.” Several more of Chebyshev's works are devoted to the integration of algebraic functions.

In 1852, during a trip to Europe, Chebyshev became acquainted with the device of the steam engine regulator - the J. Watt parallelogram. The Russian scientist set out to “derive the rules for the arrangement of parallelograms directly from the properties of this mechanism.” The results of research concerning this problem were presented in the work “The Theory of Mechanisms Known as Parallelograms” (1854). This work simultaneously laid the foundations for one of the branches of the constructive theory of functions - the theory of best approximation of functions.

In The Theory of Mechanisms, Chebyshev introduced orthogonal polynomials, which were later named after him. It should be noted that, in addition to approximation by algebraic polynomials, the scientist studied approximation by trigonometric polynomials and rational functions.

Subsequently, Chebyshev began developing a general theory of orthogonal polynomials based on integration using parabolas using the method of least squares - one of the methods of error theory used to estimate unknown quantities from measurement results that contain random errors. This method is used when processing observations.

As a member of the artillery department of the military scientific committee, Chebyshev solved a number of problems related to quadrature formulas - the results are presented in the work “On Quadratures” (1873) - and the theory of interpolation. Quadrature formulas are used to approximately calculate integrals over the values ​​of the integrand at a finite number of points.

Interpolation in mathematics and statistics is a method of finding intermediate values ​​of a quantity based on some of its known values.

Chebyshev's cooperation with the artillery department was aimed at improving the range and accuracy of artillery fire. Chebyshev's formula is known, designed to calculate the flight range of a projectile. Chebyshev's works had a significant influence on the development of Russian artillery science.

Chebyshev's research interest was attracted not only by Watt's parallelograms, but also by other hinged mechanisms. A number of the scientist’s works are devoted to their study: “On a certain modification of Watt’s cranked parallelogram” (1861), “On parallelograms” (1869), “On parallelograms consisting of any three elements” (1879), etc.

Chebyshev not only studied existing mechanisms, but also designed them himself; in particular, he created the so-called “plantigrade machine,” which reproduces the movements of an animal when walking, an automatic adding machine, mechanisms with stops, etc.

In 1868, Chebyshev proposed a special device - a flat four-bar hinge mechanism for reproducing the movement of a certain point of the link in a straight line without the use of guides. This device was named after the Russian mathematician Chebyshev's parallelogram.

The scientist was also interested in issues of cartography and the search for ways to obtain an optimal cartographic projection of the country, allowing the relationships of objects to be reproduced as accurately as possible. Chebyshev’s work “On the Construction of Geographical Maps” (1856) is devoted to this problem.

Chebyshev made significant progress in solving the problem of distribution of prime numbers. He presented the results of his research in the works: “On the determination of the number of prime numbers not exceeding a given value” (1849) and “On prime numbers” (1852).

Pafnutiy Lvovich Chebyshev was very interested in teaching. He organized a school of Russian mathematicians, the graduates of which became famous mathematicians - D. A. Zolotarev, A. N. Lyapunov, K. A. Sokhotsky and others.

Further, in his work “On an Arithmetic Question” (1866), the scientist analyzed the problem of approximating numbers by rational numbers, which played a significant role in the development of the theory of Diophantine approximations. It should be noted that in number theory, Chebyshev was the founder of an entire school of Russian scientists.

Chebyshev's works in this direction marked an important stage in the development of probability theory. The Russian mathematician began to systematically use random variables, proved the inequality that was later named after him, developed a new technique for proving limit theorems in probability theory, the so-called method of moments, and also substantiated the law of large numbers in a general form.

Chebyshev owns a number of works on probability theory. Among them are “An Experience in Elementary Analysis of the Theory of Probability” (1845), “Elementary Proof of a General Statement of the Theory of Probability” (1846), “On Average Values” (1867), “On Two Theorems Regarding Probabilities” (1887). However, he failed to complete the study of the conditions for the convergence of distribution functions of sums of independent random variables to the normal law. This was done by A. A. Markov, one of the scientist’s students. Chebyshev's research in the field of probability theory was a significant stage in its development and became the basis for the formation of the Russian school of probability theory, which initially consisted of Chebyshev's students.

Chebyshev also worked on the theory of approximation. This is the name of the branch of mathematics that studies the possibilities of approximate representation of some mathematical objects by others, usually of a simpler nature, as well as the problem of estimating the error introduced by this.

Approximate formulas for calculating functions such as roots or constants were developed in ancient times.

However, the beginning of modern approximation theory is considered to be Chebyshev’s work “Sur les questions de minima qui se rattachent a la representation approximative des fonctions” (1857), which is devoted to polynomials that least deviate from zero, currently called “Chebyshev polynomials of the first kind.”

Approximation theory has found application in the construction of numerical algorithms, as well as in information compression. Currently, several scientific journals are published in English and devoted to the problems of approximation theory: Journal on Approximation Theory (USA), East Journal on Approximation (Russia and Bulgaria), Constructive Approximation (USA).

Chebyshev made a great contribution to the development of artillery. To this day, textbooks on ballistics contain the formula derived by Chebyshev to calculate the flight range of a projectile.

For his services, Chebyshev was elected a member of the St. Petersburg, Berlin and Bologna, Paris Academies of Sciences, a corresponding member of the Royal Society of London, the Swedish Academy of Sciences, etc. In addition, the outstanding mathematician was an honorary member of all universities in the country.

In the fall of 1894, Chebyshev fell ill with the flu and soon died. However, the name of the outstanding Russian mathematician has not yet been forgotten.

In 1944, the Academy of Sciences established the P. L. Chebyshev Prize.

THEORY OF MECHANISMS

During the period under review in Russia, the beginning of the theory of one of the most important departments of applied mechanics was laid - the theory of mechanisms. This was done in the middle of the 19th century. P.L. Chebyshev. In the field of mathematics, he owns fundamental results in number theory, probability theory, integration of irrational functions and the creation of a new theory of best approximation of functions. Chebyshev came to this theory starting from some practical problems in the theory of mechanisms. For a mechanic, the name of Chebyshev is associated primarily with his work in this direction and, to a lesser extent, with work on ballistics.

Pafnuty Lvovich Chebyshev (1821-1894) was born in the village. Okatov, Kaluga province, studied at home, and then entered Moscow University, where he listened to lectures by N.D. Brashman, who attracted a talented student to independent scientific work. In 1841, Chebyshev graduated from the university, two years later his first scientific work was published, and in 1845 he defended his master's thesis on probability theory. Since 1847, Chebyshev began giving lectures at St. Petersburg University. Here he became close to V.Ya. Bunyakovsky and his earlier acquaintance I.I. Somov. The mathematical sciences at St. Petersburg University owe their flourishing to the three of them (and most of all to Chebyshev). Chebyshev worked at the university for 35 years, until 1882, and here he educated a galaxy of remarkable students who formed the core of the famous St. Petersburg mathematical school.

Soon after his arrival in St. Petersburg, Chebyshev defended his doctoral dissertation - “The Theory of Comparison” (1849). After this, Chebyshev’s articles began to appear regularly in Notes of the Academy of Sciences and other journals, which quickly brought him wide fame. In 1853, he was elected a member of the St. Petersburg Academy of Sciences, then a foreign member of the Berlin and Paris Academies (the first Russian after Peter I), the Royal Society of London, etc.

Chebyshev did not limit himself to intensive activities at the Academy of Sciences and the university. He worked actively for many years in the Artillery Department of the Military Scientific Committee and in the Scientific Committee of the Ministry of Public Education. He did not stop his scientific work almost until his death.

Chebyshev's work is characterized by an organic combination of applied and strictly theoretical interests. As noted by V.A. Steklov, his great interest in practical issues sometimes surprised people who knew Chebyshev as a scientist working in the field of abstract knowledge: probability theory, integration of functions, number theory. But this circumstance receives a natural explanation if we delve into the foundations of those guiding ideas that served as the primary source of Chebyshev’s discoveries. Chebyshev himself wrote: “Bringing theory closer to practice produces the most beneficial results, and it is not only practice that benefits from this; the sciences themselves develop under its influence, it opens up for them new subjects for research or new aspects in subjects that have long been known.” (212)

In the 19th century In connection with the growth of industry in Western Europe and Russia, new problems arose in the field of design and improvement of machines. These problems were partially solved experimentally, through persistent repeated searches, and groping for the best technical solutions. However, the very breadth of the tasks posed in connection with the emergence of new areas of technology required theoretical generalizations. There is a need to develop general methods for designing individual mechanisms and assemblies that transform movement of one type into movement of another type, to improve known and create new hinge mechanisms, as well as methods for designing guide mechanisms of various types.

PAFNUTY LVOVICH CHEBYSHEV (1821-1894)

Russian mathematician and mechanic. He made classical discoveries in number theory, probability theory, and the theory of mechanisms. All his scientific activities are characterized by the desire to closely connect the solution of mathematical problems with fundamental issues of natural science and technology. P.L. Chebyshev is the founder of the St. Petersburg Mathematical School

The appearance in Russia in the second half of the 19th century was directly related to advances in technology. fundamental works on the theory of mechanisms, and above all the works of P.L. Chebysheva. Chebyshev took interest in this range of problems from Moscow University under the influence of Brashman and partly Ershov. Chebyshev tirelessly got acquainted with various industries, talked with the most prominent engineers and selected material for the course of practical mechanics, which he taught at the university, as well as at the Alexander Lyceum.

Chebyshev was an unsurpassed master of solving specific problems and carried them out with exceptional clarity and rigor. He sought - and found - not only a general solution to the problem, but also indicated effective practical methods for its implementation. He translated his results into numbers, carried out specific numerical calculations, and, if necessary, compiled tables.

Chebyshev understood that the introduction of machines into Russian technology, which at that time was significantly behind Western technology, was of great importance. That is why he studied steam engines, turbines, etc. with particular interest. From the program of his course in practical mechanics at St. Petersburg University it is clear that he was especially interested in the theory of gears, machine dynamics, impacts in parts of mechanisms, etc.

As an object of scientific research, Chebyshev chose one of the most difficult problems in the theory of mechanisms, the problem of synthesizing mechanisms, i.e., constructing mechanisms that perform a given movement, a problem whose solution cannot be considered complete at the present time. In this area, he took on the most complex and at that time almost unstudied problem of the synthesis of hinge mechanisms. P.L. Chebyshev created a new school of mechanism synthesis. His work in this area was far ahead of its time and remains important to this day. These works brilliantly demonstrated the peculiarity of Chebyshev's scientific genius, which consisted in the ability to combine the most abstract areas of mathematical analysis with the consideration of directly technical problems. This is exactly how the metric synthesis according to Chebyshev arose in the theory of mechanisms.

Of Chebyshev's fifteen studies on the theory of mechanisms, most are devoted to issues of synthesis of mechanisms. His general idea was this. If a certain mechanism satisfies the given conditions exactly only approximately, then its links should be selected so that the largest resulting error is the smallest of all that are possible for a mechanism of this type. Guided by this idea and starting from the properties of the so-called Watt parallelogram, used in steam engines to convert the rectilinear motion of a piston into the rotational motion of a shaft, Chebyshev created a new branch of mathematical analysis - the theory of best approximation of functions (or the theory of functions that least deviate from zero).

In the study “The Theory of Mechanisms Known as Parallelograms” (1853), Chebyshev gave rational grounds for determining the dimensions of rectilinear guide mechanisms, which for 75 years, starting with Watt, were selected empirically by engineers.

In addition to guiding mechanisms, Chebyshev synthesized and built a number of others. The most interesting of them are: a mechanism for converting the rotational movement of the crank into the oscillatory movement of the rocker arm with two swings per revolution of the crank; rocker mechanism of a steam engine; mechanism for measuring curvature; grain sorting machine mechanism; scooter and bicycle mechanism; the rowing mechanism of a boat, etc. A very ingenious mechanism is known as the “foot-walking machine,” which imitates the movement of a horse.

Among the mechanisms built by Chebyshev, the so-called paradoxical mechanism, consisting of six links connected by hinges, stands out. As Chebyshev showed, it is possible to select such link sizes that if the driving link is rotated clockwise, the driven link will make two revolutions, and if the drive link is rotated counterclockwise, the driven link will make four revolutions.

By studying those parts of the trajectories described by various points of the connecting rod that differ little from the circles, and by adding additional links, Chebyshev created mechanisms with stops, in which individual links stop for a while, although the leading link continues to rotate.

This is a short and far from complete list of Chebyshev’s works on the synthesis of mechanisms.

In 1870, in his work “On Parallelograms,” Chebyshev investigated the same problem and for the first time gave the so-called structural formula of mechanisms.

Let us add to this that Chebyshev built a new adding machine with continuous motion.

In the obituary dedicated to P.L. Chebyshev, A.M. Lyapunov wrote: “Brilliant ideas scattered in the works of P.L. Chebyshev, without a doubt, not only are not exhausted in all their conclusions, but can bear proper fruit only in the future, and only then will it be possible to obtain a correct idea of ​​the great significance of the scientist, which science has recently lost” (213).

Ideas by P.L. Chebyshev could indeed be assessed in the light of their further development. This development occurred in all scientific centers of the world, and especially in Russia. We will not dwell here on the history of the theory of mechanisms in Russia in the last quarter of the 19th and early 20th centuries, but will note only a few works.

An interesting series of studies in this direction was carried out at Novorossiysk (now Odessa) University, founded in 1865. A number of books and articles on the kinematics of systems with applications to technical problems were published by professor of mechanics V.N. Ligin (1846-1900). Ligin’s student, associate professor Kh. I. Gokhman, gave in “Kinematics of Machines” (Odessa, 1890) a classification of kinematic pairs according to degrees of freedom and a division of mechanisms into six categories depending on the number of possible movements. Gokhman’s somewhat earlier work “Theory of gears, generalized and developed through analysis” (Odessa, 1886) also retained interest. At Odessa University, a graduate of Moscow University N.B. defended his master’s thesis “Transmission of rotation and mechanical drawing of curves by hinged lever mechanisms” (1894). Delaunay (1856-1931), from 1906 he occupied the department of mechanics at the Kiev Polytechnic Institute. To more widely popularize Chebyshev’s work on hinge mechanisms abroad, Delaunay in 1900 published the book “Chebyshev’s Work on the Theory of Hinge Mechanisms” in Leipzig in German.

Special merits in the theory of mechanisms belong to Ivan Alekseevich Vyshnegradsky (1831-1895), a student of Ostrogradsky at the Main Pedagogical Institute in St. Petersburg, the physics and mathematics department of which he graduated in 1851. After defending his master's thesis “On the motion of a system of material points determined by complete differential equations "(1854) Vyshnegradsky taught mathematics and applied mechanics at the Artillery Academy, and then began working at the St. Petersburg Institute of Technology. In addition to the above courses, he taught others, the theory of elasticity, thermodynamics, various parts of mechanical engineering, etc. In 1862 he was approved as a professor of mechanics, and in 1888 he was elected an honorary member of the Academy of Sciences.

Vyshnegradsky was an outstanding design engineer and theorist. His main contribution to science was the creation of the theory of automatic control, the foundations of which he outlined in two essays - “On Direct-Acting Regulators” (1877) and “On Indirect-Acting Regulators” (1878). Vyshnegradsky then published his discoveries in French and German journals.

The concepts and methods introduced by Vyshnegradsky have been widely used in modern regulation theory, which is becoming increasingly important in a wide variety of areas of production. For example, the stability criterion of a regulatory system bears Vyshnegradsky’s name.

In 1909, a study by N.E. was published. Zhukovsky “Reduction of dynamic problems about a kinematic chain to problems about a lever.” It contains a theorem of deep fundamental significance. The essence of this theorem is that the question of the equilibrium of a mechanism, i.e., a system of bodies, is reduced to the simpler problem of the equilibrium of one rigid body rotating around a given center. Zhukovsky's method made it possible to solve the general problem of the dynamics of mechanisms (for mechanisms with one degree of freedom), consisting in determining the movement of mechanisms under the action of given forces, i.e., it made it possible to carry out kinetostatic calculations of the mechanism taking into account inertial forces.

In 1914-1917 works by professor of the St. Petersburg Polytechnic Institute L.V. Assur (1878-1920), who gave a new general system of classification of planar kinetic chains, on which the methodology for studying planar mechanisms is based, and each class has its own method of analysis. Assur's classification and a number of concepts introduced by him (“Assur's points”, etc.) play an important role in the modern theory of mechanisms and machines.

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The future great mathematician was born in 1821 to his father, a veteran of the Patriotic War, and his mother, a strict and domineering landowner typical of that time. Wanting to make their children educated people, the Chebyshev family moves from near Kaluga to Moscow, closer to the university. Today, perhaps, you will not find such stern teachers as Chebyshev had in childhood. Very little Paphnutius was taught to read and write by his iron mother, and French and arithmetic by his cousin, who was also probably not a muslin young lady. Having matured a little, the capable boy fell into the hands of a man-machine, known for his manic pedantry and harshness towards his students. The outstanding mathematician and supporter of stick discipline Platon Nikolaevich Pogorelsky firmly implanted his science in the minds of teenagers, and soon young Chebyshev began to solve complex problems faster than a squirrel nuts. By the way, the formidable Platon Nikolaevich taught mathematics to the future writer Turgenev.

A boat driven by a Chebyshev rowing mechanism. In total, at least three such waterfowl were made.

A graduate of Moscow University, he conducted his scientific activities at the University of St. Petersburg. Here he became a professor at only 29 years old, and here he created the later famous St. Petersburg School of Mathematics. While teaching mathematics, Professor Chebyshev was famous for his punctuality - he was never late for lectures, started them at a strictly appointed time and finished them exactly on the clock, even if he had to stop his story mid-sentence - there was definitely something of a robot in him.
Several of Chebyshev's students subsequently became equally famous mathematicians themselves. According to the online database “Mathematical Genealogy,” which calculates the academic pedigree of famous mathematicians, by the fall of 2013, Chebyshev, who died in 1894, had 9,609 “descendants” all over the world—people whose PhD thesis supervisors were students of his students’ students. The calculation is based on six students of Chebyshev, who defended their dissertation with him back in the 19th century. To remain in the history of mathematics as a world-famous figure, Pafnutiy Chebyshev would only need two works published by him. The first, published in 1850 in French “Memoriesurlesnombrespremiers,” took the theory of prime numbers (those that are divisible by themselves and one without a remainder) to a new level. In his 1867 work “On Average Values,” he presented calculations known today as Chebyshev’s theorem. It became one of the foundations of probability theory - the main tool of modern statistics. However, prime numbers and probability theory were drops in the ocean of Pafnutiy Lvovich’s mathematical and near-mathematical interests. Being not just a genius, but a generalist, he explored a variety of dissimilar areas of mathematics, much like Pushkin wrote frivolous poetry, poems, and historical novels with equal success.

In 1881, Chebyshev designed the world's first automatic machine for calculations, which was far ahead of all calculating machines that existed at that time. This machine, by coincidence, did not become widespread, but gave impetus to the improvement of “machine mathematics”, and then to the emergence of cybernetics.

In addition to mathematicians, mechanics and roboticists, geographers, artillerymen and... feminists consider Chebyshev to be “their people”. The first two categories pay tribute to the memory of Pafnutiy Lvovich for his contributions to improving cartography techniques and his active work on improving the range and accuracy of artillery fire. Fighters for the rights of the weaker sex remember that it was he who proposed to the physics and mathematics department of the St. Petersburg Academy to elect a woman mathematician Sofya Vasilievna Kovalevskaya as a corresponding member of the academy.

With your left foot - march in step! How the footwalker moves, see the website www.tcheb.ru

How are the mathematical works of the St. Petersburg professor and his plantigrade machine connected? Pafnuty Lvovich believed that any mathematical calculations can and should be tested in practice. So the machine designed by Chebyshev turned out to be the embodiment of two theories he developed - approximation of functions and synthesis of mechanisms. Practical mechanics was for him a continuation of his mathematical research, when numbers and symbols turn into tangible hinges and links. Chebyshev's plantigrade machine does not stand still like an idol, but walks thanks to the so-called lambda mechanisms. One of the hinges of the mechanism rotates around the axis in a circle, pushing the driven hinge, which, in turn, moves the leg with the “foot”.
One axis drives two mechanisms, that is, two legs. Accordingly, two axes - four legs. The first plantigrade machine, created by Chebyshev himself, can be seen today in the Polytechnic Museum in Moscow. A real professor can always surprise and confuse others. Chebyshev had one mechanism for this, which moved in a very mysterious way even for modern researchers. It’s called a paradoxical mechanism. Chebyshev was a true innovator, much earlier than others, he deduced the structural formula of flat mechanisms and proved the famous theorem about the existence of three-jointed four-bar mechanisms. He built a rowing mechanism that imitated the movement of boat oars, a scooter chair, and an original model of a sorting machine. In total, he created about 40 mechanisms and about 80 of their modifications, on the construction of which he spent most of his professorial salary. Without knowing it, we can still see many of the mechanisms invented by Chebyshev in modern devices today.
In addition to living heirs, Professor Chebyshev has one worthy iron descendant - the supercomputer “SKIF MSU Chebyshev” built in 2008. Today Chebyshev is one of the most powerful computing complexes in Eastern Europe. The peak performance of the supercomputer, built on 1250 quad-core processors, is 60 teraflops.

There are two objects in space named after the Russian mathematician - the Chebyshev crater on the Moon and the asteroid 2010-Chebyshev.

Since the invention of the steam engine by James Watt, the task has been to build a hinged mechanism that converts circular motion into linear motion.

The great Russian mathematician Pafnutiy Lvovich Chebyshev was unable to accurately solve the original problem, however, while studying it, he developed the theory of approximation of functions and the theory of synthesis of mechanisms. Using the latter, he selected the dimensions of the lambda mechanism so that... But more on that below.

Two fixed red hinges, three links have the same length. Because of its appearance, similar to the Greek letter lambda, this mechanism got its name. The loose gray hinge of the small driving link rotates in a circle, while the driven blue hinge describes a trajectory similar to the profile of a porcini mushroom cap.

Let us place marks at equal intervals on the circle along which the driving joint rotates uniformly and the corresponding marks on the trajectory of the free joint.

The lower edge of the “cap” corresponds to exactly half the time the driving link moves around the circle. In this case, the lower part of the blue trajectory differs very little from movement strictly in a straight line (the deviation from the straight line in this section is a fraction of a percent of the length of the short driving link).

What else, besides a mushroom cap, does the blue trajectory look like? Pafnuty Lvovich saw the similarity with the trajectory of a horse’s hoof!

Let's attach a “leg” with a foot to the lambda mechanism. Let's attach another one to the same fixed axes in the opposite phase. For stability, we will add a mirror copy of the already constructed bipedal part of the mechanism. Additional links coordinate their rotation phases, and the axes of the mechanism are connected by a common platform. We have received, as they say in mechanics, the kinematic diagram of the world's first walking mechanism.

Pafnutiy Lvovich Chebyshev, being a professor at St. Petersburg University, spent most of his salary on the manufacture of invented mechanisms. He embodied the described mechanism “in wood and iron” and called it the “Poligrade Machine.” This world's first walking mechanism, invented by a Russian mathematician, received universal approval at the World Exhibition in Paris in 1878.

Thanks to the Polytechnic Museum of Moscow, which preserved Chebyshev’s original and provided the opportunity for “Mathematical Etudes” to measure it, we have the opportunity to see in motion an accurate 3D model of Pafnuty Lvovich Chebyshev’s plantigrade machine.

Original articles by P. L. Chebyshev:

• On the transformation of rotational motion into motion along certain lines using articulated systems / According to the book: Complete works of P. L. Chebyshev. Volume IV. Theory of mechanisms. - M.-L.: Publishing House of the USSR Academy of Sciences. 1948. pp. 161–166.

Museums and archives:

• The mechanism is kept in the Polytechnic Museum (Moscow); Automation Department; PM No. 19472.
• Two wooden draft models of a plantigrade machine with notes by P. L. Chebyshev are kept at the Department of Theoretical and Applied Mechanics of St. Petersburg State University.

Research:

• I. I. Artobolevsky, N. I. Levitsky. Mechanisms of P. L. Chebyshev / In the book: Scientific heritage of P. L. Chebyshev. Vol. II. Theory of mechanisms. - M.-L.: Publishing House of the USSR Academy of Sciences. 1945. pp. 52–54.
• I. I. Artobolevsky, N. I. Levitsky. Models of mechanisms by P. L. Chebyshev / In the book: Complete works of P. L. Chebyshev. Volume IV. Theory of mechanisms. - M.-L.: Publishing House of the USSR Academy of Sciences. 1948. pp. 227–228.

Chebyshev mechanism- a mechanism that converts rotational motion into motion that is close to linear.

Description

The Chebyshev mechanism was invented in the 19th century by mathematician Pafnuty Chebyshev, who conducted research on theoretical problems of kinematic mechanisms. One of these problems was the problem of converting rotational motion into something approximating linear motion.

Rectilinear movement is determined by the movement of point P - the midpoint of the link L 3, located in the middle between the two extreme coupling points of this four-bar mechanism. ( L 1 , L 2 , L 3, and L 4 are shown in the illustration). When moving along the area shown in the illustration, point P deviates from ideal linear movement. The relationships between the lengths of the links are as follows:

$L\_1:\; L\_2:\; L\_3\; =\; 2:\; 2.5:\; 1\; =\; 4:\; 5:\; 2.$

Point P lies in the middle of the link L 3. The given relations show that the link L 3 is positioned vertically when it is in the extreme positions of its movement.

The lengths are related mathematically as follows:

$L\_4=L\_3+\backslash sqrt(L\_2^2\; -\; L\_1^2).$

Based on the described mechanism, Chebyshev produced the world's first walking mechanism, which enjoyed great success at the World Exhibition in Paris in 1878.

see also

Other ways to convert rotational motion into approximately linear motion are the following:

• Heuken mechanism is a type of Chebyshev mechanism;

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Rotational Straight-line ...approximately Progressive Complex movement

An excerpt characterizing the Chebyshev Mechanism

-About...the wolf!...hunters! - And as if not deigning to deign the embarrassed, frightened count with further conversation, he, with all the anger he had prepared for the count, hit the sunken wet sides of the brown gelding and rushed after the hounds. The Count, as if punished, stood looking around and trying with a smile to make Semyon regret his situation. But Semyon was no longer there: he, taking a detour through the bushes, jumped the wolf from the abatis. Greyhounds also jumped over the beast from both sides. But the wolf walked through the bushes and not a single hunter intercepted him.

Nikolai Rostov, meanwhile, stood in his place, waiting for the beast. By the approach and distance of the rut, by the sounds of the voices of dogs known to him, by the approach, distance and elevation of the voices of those arriving, he felt what was happening on the island. He knew that there were arrived (young) and seasoned (old) wolves on the island; he knew that the hounds had split into two packs, that they were poisoning somewhere, and that something untoward had happened. Every second he waited for the beast to come to his side. He made thousands of different assumptions about how and from which side the animal would run and how it would poison it. Hope gave way to despair. Several times he turned to God with a prayer that the wolf would come out to him; he prayed with that passionate and conscientious feeling with which people pray in moments of great excitement, depending on an insignificant reason. “Well, what does it cost you,” he said to God, “to do this for me! I know that You are great, and that it is a sin to ask You for this; but for the sake of God, make sure that the seasoned one comes out on me, and that Karai, in front of the “uncle” who is watching from there, slams into his throat with a death grip.” A thousand times during these half-hours, with a persistent, tense and restless gaze, Rostov looked around the edge of the forest with two sparse oak trees over an aspen underhang, and the ravine with a worn edge, and the uncle’s hat, barely visible from behind a bush to the right.
“No, this happiness will not happen,” thought Rostov, but what would it cost? Will not be! I always have misfortune, both in cards and in war, in everything.” Austerlitz and Dolokhov flashed brightly, but quickly changing, in his imagination. “Only once in my life would I hunt down a seasoned wolf, I don’t want to do it again!” he thought, straining his hearing and vision, looking to the left and again to the right and listening to the slightest shades of the sounds of the rut. He looked again to the right and saw something running towards him across the deserted field. “No, this can’t be!” thought Rostov, sighing heavily, like a man sighs when he accomplishes something that has been long awaited by him. The greatest happiness happened - and so simply, without noise, without glitter, without commemoration. Rostov could not believe his eyes and this doubt lasted more than a second. The wolf ran forward and jumped heavily over the pothole that was on his road. It was an old beast, with a gray back and a full, reddish belly. He ran slowly, apparently convinced that no one could see him. Without breathing, Rostov looked back at the dogs. They lay and stood, not seeing the wolf and not understanding anything. Old Karai, turning his head and baring his yellow teeth, angrily looking for a flea, clicked them on his hind thighs.