Plantigrade machine

Muse­ums and archives

Mech­a­nism is stored in the Poly­tech­nic Museum (Moscow, Rus­sia); fund repos­i­tory, PM № 19472.

Two rough wooden mod­els of the machine (with Tcheby­shev’s marks) are stored at the The­o­ret­i­cal and Applied Mechan­ics Depart­ment of the Saint Peters­burg State Uni­ver­sity (Rus­sia).

Orig­i­nal papers by Tcheby­shev

On trans­for­ma­tion of rotary move­ment into move­ment along some lines using joined sys­tems / After: The com­plete works of P. L. Tcheby­shev. Vol. IV. The­ory of mech­a­nisms. — Moscow-Leningrad: AS USSR. 1948. P. 161–166. (Russ­ian)


I. I. Arto­bolevsky, N. I. Lev­it­sky. Tcheby­shev's Mech­a­nisms / In: P. L. Tcheby­shev’s Sci­en­tific Her­itage. Iss. 2. The­ory of mech­a­nisms. — Moscow-Leningrad: AS USSR. 1945. P. 52–54. (Russ­ian)

I. I. Arto­bolevsky, N. I. Lev­it­sky. Mod­els of mech­a­nisms of P. L. Tcheby­shev / In: The com­plete works of P. L. Tcheby­shev. Vol. IV. The­ory of mech­a­nisms. — Moscow-Leningrad: AS USSR. 1948. P. 227–228. (Russ­ian)

Other sources

Planti­grade Machine (In metal)

Tcheby­shev’s Planti­grade machine. Vsemir­naya Illus­trat­sia (World Illus­tra­tion). 1886. № 889. P. 100. (Russ­ian)

A let­ter of the 9th Feb­ru­ary 1889 from the direc­tor of the National Con­ser­va­tory of Arts and Crafts (Con­ser­va­toire National des Arts et Métiers) to P. L. Tcheby­shev (about the dona­tion of the engrav­ing rep­re­sent­ing Planti­grade machine; In french; Archive of the RAS, fonds 505, opis 1, № 107).

The engrav­ing of a planti­grade machine (stored in the Musée des arts et métiers du Con­ser­va­toire national des arts et métiers; Paris, France); CNAM № 10475-0000.


Since James Watt invented the steam engine there has been a prob­lem to build a hinge mech­a­nism that trans­form cir­cu­lar motion to lin­ear.

A great Russ­ian math­e­mati­cian Pafnu­tiy Lvovich Tcheby­shev couldn't solve the ini­tial prob­lem, but inves­ti­gat­ing it he devel­oped a the­ory of approx­i­ma­tion and of mech­a­nism syn­the­sis. Apply­ing the lat­ter he could choose the para­me­ters of a lambda-mech­a­nism so that… Well, we'll talk about it below.

Two fixed red hinges, three edges of the same length. As it looks like the greek let­ter lambda, this mech­a­nism was named after it. The loose gray hinge of the lit­tle dri­ving edge turns around form­ing a cir­cle while the slave blue hinge has a tra­jec­tory that looks like a mush­room's cap.

Put on the cir­cle of the dri­ving hinge marks at reg­u­lar inter­vals and the cor­re­spond­ing marks on the tra­jec­tory of the slave one.

The lower part of the «cap» cor­re­sponds exactly to a half of the period of the dri­ving hinge's motion. At the same time, the lower part of the blue curve doesn't dif­fer much from a straight lin­ear motion (the dif­fer­ence is less that a per­cent of the short dri­ving edge length).

What else does this blue tra­jec­tory look like? Pafnu­tiy Lvovich could see sim­i­lar­ity with the horse hoof's motion tra­jec­tory!

Let's attach a mir­ror copy of the two-leg part we've already made. Addi­tional links coor­di­nate the phases of rota­tion and a com­mon plat­form con­nects the axes. As they say in mechan­ics, we've got a kine­mat­i­cal scheme of the first walk­ing machine in the world.

Being a pro­fes­sor in Saint-Peters­burg uni­ver­sity, Pafnu­tiy Lvovich spent most part of his salary on con­struc­tion of the mech­a­nisms he invented. He built the described one «in wood and steel» and called it a «Planti­grade machine». The first walk­ing machine in the world invented by a Russ­ian math­e­mati­cian was greatly approved dur­ing the Wold exhi­bi­tion in Paris, 1878.

Thanks to the Poly­tech­nic Museum of Moscow that pre­served the Tcheby­shev's orig­i­nal and let «Math­e­mat­i­cal etudes» mea­sure it, we have an oppor­tu­nity to see in action an exact 3D-model of a planti­grade machine of Pafnu­tiy Lvovich Tcheby­shev.


Geo­met­ric 3D model by Math­e­mat­i­cal Etudes.

All Mechanisms

Original model dimensions
Model by Tchebyshev (Polytechnic Museum)
Draft model by Tchebyshev
Draft model by Tchebyshev
Kinematic scheme
Kinematic scheme
Kinematic scheme
Kinematic scheme
Kinematic scheme
Kinematic scheme
Kinematic scheme