Leonardo's gears

The Madrid Codices I and II are two collections of Leonardo da Vinci's manuscripts found in a collection in the National Library of Madrid at the end of the 1960s. In particular, the Madrid Codex I consists of 382 pages of notes accompanied by something like 1600 between sketches and drawings and addresses a problem for which Leonardo is in some way particularly known as an engineer and designer: gears.
Leonardo's starting point is the study of friction. This is a force that opposes motion, but thanks to its opposition it is possible for us to walk without slipping or losing balance. As we all know today, however, the intensity of the frictional force depends on the surfaces that are in contact with each other, on how smooth or rough they are, which is independent of the area in contact and which can be reduced by using, for example, a lubricant or cylinders. All this, however, was already known to Leonardo, as it is possible to observe from the reading of the Madrid Code I. Furthermore, it is always Leonardo who introduced the concept of friction coefficient, defining it as the ratio between the force required to slide two surfaces horizontally on top of each other and the pressure between the two surfaces. Leonardo also estimated the value of this friction coefficient in 1/4, consistent with the materials best known to the florentine and with which he could carry out experiments (wood on wood, bronze on steel, etc.)⁽¹⁾.
At this point Leonardo is ready to develop a series of gears capable of carrying mechanical energy and producing motion, minimizing friction with the use of spheres and cylinders, as can be seen from his numerous drawings. In particular, however, it is Leonardo's mechanical use of two particular geometric shapes that is striking, because it anticipates their actual adoption by centuries: the epicloidal teeth and the globoidal gear.

Teeth and epicycles

The epicycloid is a particular plane curve generated by rolling one curve over another. In particular, the epicycloid is defined by the positions assumed by a point placed on a circumference that is rolled outside another.
The first to deal with this kind of curves was the greek mathematician Hipparchus in his model for the motion of the Moon. Subsequently it was the greek astronomer Ptolemy who used the epicyclics to correct the mathematical errors of the geocentric model with respect to the observed motions of celestial bodies. The first to describe how to build an epicycloid was, however, the German artist Albrecht Durer in 1525 in the first of the four volumes of his geometric work Underweysung der Messung mit dem Zirckel und Richtscheyt or Instructions for Measuring with Compass and Ruler, while in 1640⁽²⁾ the french engineer Girard Desargues used the epicycloid within the parisian water system.
The application of the epicyclioid to gears is credited to the danish astronomer Olaus Roemer, although perhaps only in theoretical terms⁽²⁾, while the first practical application seems to date back to 1694 due to Philippe de La Hire⁽¹⁾. The comparison between Leonardo's drawings (on the left in the image below) and those of de La Hire (on the right) shows Leonardo's ability to be able to find optimal solutions to mechanical problems (such as finding the most suitable form to minimize the effects of friction) in a fairly intuitive way⁽¹⁾.

Leo Gearloose

Globoidal curves are the lines external to the surface generated by the rotation of a circumference around a straight line placed on a plane different from that of the circumference itself. Very similar to the epicycloids, the globoids, so called by the engineer Franz Reuleaux in 1876, were used around 1740 by the clockmaker Henry Hindley to build the globoidal gears, obviously without knowing that something like this had already been designed by Leonardo da Vinci all about 250 years earlier⁽¹⁾!

Obviously, all the other projects by Leonardo contained in the Madrid Codex I are equally astonishing, even if perhaps the most amazing design of all is that of the toilet with flush toilet, at least according to the interpretation provided by Augusto Macaroni, who observed how this project was in the section dedicated to hydraulic devices⁽³⁾.

Finally, I remember that the Madrid Code also contains the project for a perpetual motion machine (later also built), which Leonardo designed to show how perpetual motion was impossible.

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1. Reti, L. (1971). Leonardo on bearings and gears. Scientific American, 224(2), 100-111. doi:10.1038/scientificamerican0271-100 (jstor) ↩︎ ↩︎ ↩︎ ↩︎

2. The epicycloid by Dennis & Emily Astley ↩︎ ↩︎

3. Gardner, M. (1975). Mathematical Games: The curious magic of anamorphic art. Scientific American, 232(4), 126-133. doi:10.1038/scientificamerican0475-126 (jstor) ↩︎