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 Poly­tech­nical 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.