### Perpetual motion

Popular Science's cover by Norman Rockwell, October 1920 - via commons
Like the research on the philosopher's stone, the mysterious alchemical material which should allow the transmutation of the elements, particularly of base metals into precious gold, there is the search for a tool that can generate perpetual motion, or a gear capable to move indefinitely without any need of power supply from the outside.
As we will see this research has well over a thousand years and continues today among people who genuinely (and a little naively!) looking to get what would be a considerable technological leap and scammers themselves. The best way to deal with all of these is to remember what Richard Feynman said some students who invited him to a demonstration for an engine running unless perpetual but rather long:
You have to ask yourself, 'Where is the power supply?'(1)
The magic wheel

The first tool would have to create the perpetual motion was the so called magic wheel, a wheel that turns on its axis the movement of which would have to be powered by a lot of magnets. This instrument made its first appearance in the eighth century in Bavaria: designed to rotate in perpetuity was defeated in the long run, by friction, so that the magic wheel was overcome by the inevitable thermodynamic end. Although the times don't match, someone say around that this magic wheel from Bavaria is based on an earlier project proposed by the Indian mathematician and astronomer Bhaskara II, wholived in 12th century.
His most important work is the Siddhanta-Shiromani, the Crown of treatises, a poem where, among others results, he comes to approximate the derivative for the sine function: $\frac{\text{d}}{\text{d} y} \sin y = \cos y$ He also made a demonstration of the Pythagorean theorem, and his path is crossed, as it can only in the tortuous paths of mathematics, with Pierre de Fermat, the amateur mathematician known to throw challenges to more titles colleagues, as in the case the best known Fermat's last theorem or for the following Diophantine equation: $61 x^2 + 1 = y^2$ The latter, proposed in 1657, was resolved in 18th century by Euler, unless we consider the solution discovery by Bhaskara II already 6 centuries before.
As astronomer most of his contributions are contained in the aforementioned Siddhanta-Shiromani, where, as we have seen, he has developed some concepts about trigonometry, a branch of mathematics important, if not necessary to make observations as accurate as possible.
Bhaskara II, astronomically speaking, was heir of Aryabhata (fourth century) and Brahmagupta (seventh century) who they developed, about a thousand years in advance on European astronomers, a heliocentric model. Drawing on these theoretical and observational basis, Bhaskara II made a series of observations on celestial bodies, first of all on moon and sun.
As an engineer, however, it is best known for Bhaskara's wheel, a wheel whose spokes were partially filled with mercury. According Bhaskara it would be just that mercury to ensure the perpetual motion of the wheel(2).
The speculators of perpetual motion

Taccola's wheel - via commons
In Europe, two of the most famous perpetual motion machine was designed by Villard de Honnecourt and Mariano di Iacopo, called Taccola (jackdaw).
Both scholars, as well as many of their contemporaries and successors, focused on the wheel. In particular, the French proposed a wheel that is constantly biased by a series of equidistant hammers places around the circumference. In de Honnecourt's notes there is also a reference to mercury as a substitute for hammers, proving that he knew the project of Bhaskara(2).
The project by Taccola, however, is more a variation on hammers' wheel de Honnecourt that oin the Bhaskara's wheel: the Italian, in fact, proposes to use radius with a joint in the middle. The idea is to induce and support the movement by using the gravitation on the not fixed edge of the radius. About the Taccola's draft, I would observe that it is one of the most beautiful and precise of its age.
In the diatribe was inserted also Leonardo da Vinci. Multifaceted character, is one of the Alhazen's heirs, traces of which we can be found throughout his works, also the artistic one. And the perpetual motion machine, designed and built to demonstrate the impossibility, it is the most striking example. The design of the project, in the Codex Madrid, was reconstructed and preserved in the Museum of Leonardo da Vinci in Florence, and it is well described in the notes of Leonardo:
(...) any weight will be applied to the wheel, which weight is due to the motion of the wheel, without a doubt the center of this weight will stop in the center of his pole; and no instrument that human ingenuity can build that with its pole faces is, to that effect will repair.
Leonardo, then, doesn't spare a joke against seekers of perpetual motion, which basically compares to the alchemists in constant search of the philosopher's stone:
Dear speculators of perpetual motion, how many vain chimeras have you pursued? Go away with them gold seekers.
Pots and mills
The rejection of Leonardo (and some decades after the ones by Planck), was never considered final and thus attempts to create perpetual motion continued through the centuries to the present day. The parade would be quite long and boring (I linked you the wiki page that serves as the main inspiration for the post), from which, however, I like to share with you a couple of tools like the Robert Boyle's flask, who perhaps should be built as a bottle Klein in order to work!

Another machine, more complex than the first but in some sense also based on the vessels (albeit rather large!) was proposeb by Vittorio Zonca(2).
Zonca had imagined a mill in perpetual motion with gigantic vessel partially filled with water, as explained in the treaty Novo teatro di machine et edificci per varie et sicure operationi (Theater of machines):

From the modern age, starting from the 19th century, I would cite Nikola Tesla who claimed to have discovered a principle on which it would build a perpetual motion machine of the second kind (read the classification in the last section):
A departure from known methods - possibly of a "self-acting" engine or machine, inanimate, yet capable, like a living being, of deriving energy from the medium - the ideal way of obtaining motive power.
This is the era of the patents: the tools, although they don't perpetual motion machines, was mechanisms that lose energy very slowly. Obviously these same devices are revealed for most fake (a separate discussion to decide between the fraud and the inability of the designer). An example could be the proposals by Steorn Ltd .: its demonstration on the 4th July, 2007 turned out ruinous, canceled due to technical problems, while the jury of scientists selected in 2006, in June 2009 stated that the technology was not working.
On the feasibility of perpetual motion
Now come back in time: at the beginning of the 19th century George Biddell Airy observed that the impossibility of perpetual motion depends on the integration of the following differential equation $x \text{d}x + y \text{d}y + z \text{d}z$
(...) as in all the forces of which we have an accurate knowledge this expression is a complete differential, it follows that perpetual motion is incompatible with those forces.(3)
It is clear what is the Airy's idea: he would understand the mathematical limits of validity about the perpetual motion.
The conditions for this are well defined (my explanation is an adjustment of Airy's one(3)): take a pendulum; the speed of the weight at the end of the pendulum string will be maximum at the lowest point, but at each new oscillation this speed is reduced due to the friction, making inevitable motion to stop. Pendulum clocks operate, in principle, precisely thanks to the correction that the gears are of the loss due to the friction, although this does not prevent them from having to be periodically reloaded (in some way this correction implies an expenditure of energy that must be restored). To have a perpetual motion, there will be a force that must act just like the gears of the pendulum clock: exactly counterbalance the friction in such a way that the speed at the lowest point of oscillation is constant in time.
A system of this kind is described by the mathematician through a differential equation, whose analytical resolution is not at all simple. So to get a result, Airy is forced to neglect one of the two terms. Without discussing its solution (on the other hand even Airy is interested in it, but about the possibility that, conceptually, there might be a solution to the mathematical problem), from the point of view of physics, to find an approximate solution coincides with to find the actual conditions to make a motion, albeit for a limited time, compatible with a perpetual motion. Or also to understand, for example, for which angles the motion of a pendulum could be described with the approximate equations discovered by Galileo Galilei.
In general, we can consider the Airy's paper as one of the first, if not the first attempt to deal in a scientific manner the problem of perpetual motion.
The thermodynamics of perpetual motion
Max Planck wrote in the Treatise on Thermodynamics that:
It is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion, i.e. it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.
Starting from the thermodynamics point of view, we can define three types of perpetual motion.
From a termodynamical point of view we can define two different types of perpetual motion. The first type, implies the violation of the first law of thermodynamics, or conservation of energy: it would be carried out using a machine capable of producing an amount of energy greater than that consumed, so it would be able to feed itself. Machines that would make this kind of perpetual motion use magnets as energy source, but although they can move for a long time, they are still destined to stop because they cannot in any way extract free energy.
On the other hand, the second type of perpetual motion implies the violation of the second law of thermodynamics, since systems of this kind would imply the transfer of energy from a cold to a warm body without spending work. Said in simpler terms, such machines would have a 100% efficiency, since they would have to convert the thermal energy extracted from a given source completely into mechanical energy, for example. In this case the first principle would not be violated.
In this way, the research of a perpetual motion became the research of a violation of one of the first two law of thermodynamics and therefore a much more fundamental research than that which perpetual motion machine manufacturers have so far led, beyond their motivations.
Based on History of perpetual motion machines.
(1) Richard Feynman, Mr. Papf's Perpetual Motion
(2) Donald E. Simanek, A perpetual futility
(3) Airy, G. B. (1830). On certain conditions under which a perpetual motion is possible. Cambridge Philosophical Transactions.