In 1635, the first patent on a perpetual motion was granted in England, and to this day, people are churning them out. The laws of thermodynamics mean that a closed system can only lose energy, not produce it, and the various schemes that have been proposed all overlook certain basic facts of physics that prevent them churning out free energy forever. Tidal turbines and Stirling engines are not enough; these tireless inventors want to start a revolution. And can you really blame them? Given the panoply of human desires, it’s hard to take umbrage at people who want to get rich and famous off a revolutionary discovery that will give everyone clean, cheap power. And what could be more romantic than a grand windmill-till? Consider the torrents of ink wasted throughout the ages in failed attempts to square the circle. The problem, as stated given by Greek mathematicians, is to create a square with the same area as that of a given circle, using only a compass and an unmarked straightedge. Squaring the circle was, along with doubling the cube and trisecting the angle, one of the three construction problems of classical Greek mathematics that haunted mathematicians (both professional and amateur) down through the ages.
The proof that one cannot trisect the angle or double the cube using just a compass and straight-edge is fairly simple. It relies on an analagous statement of the problem and a few results from Galois theory. The proof of pi’s irrationality (that is, the proof that no whole-number ratio is equal to pi) isn’t terribly difficult. But the proof of pi’s transendence (that is, the proof that there is no polynomial equation using integer coefficients for which pi is a solution, which settles the matter once and for all) is rather complex; in my current state of mathematical rustiness, I’m not sure that I could follow it. The proof was not discovered until 1882, by which time hundreds of mathematicians both talented and less so had taken a crack at it.
The successful circle-squarer would earn no financial rewards. A large cash prize led John Harrison to invent the chronometer, but the chronometer was an immensely useful device that made longitudinal readings practical. To square the circle would bring only glory. But the problem consumed small intellects as well as large, and people submitted proofs until the experts were sick of reading them:
[I]n 1775 the Paris Academy passed a resolution which meant that no further attempted solutions submitted to them would be examined. A few years later the Royal Society in London also banned consideration of any further ‘proofs’ of squaring the circle as large numbers of amateur mathematicians tried to achieve fame by presenting the Society with a solution. This decision of the Royal Society was described by [British mathematician Augustus] De Morgan about 100 years later as the official blow to circle-squarers… De Morgan suggests that St Vitus be made the patron saint of circle-squarers. This is a reference to St Vitus’ dance, a wild leaping dance in which people screamed and shouted and which led to a kind of mass hysteria. De Morgan also suggested the term ‘morbus cyclometricus’ as being the ‘circle squaring disease’.
For a long time, the question was unsettled, but as with so many things, the amount of writing produced was usually directly proportional to the author’s status as a crank. (In Fads & Fallacies in the Name of Science, Martin Gardner notes that de Morgan’s Budget of Paradoxes singled out James Smith, "the the Liverpool merchant who wrote book after book to prive that pi was exactly 3 1/8." The idea that pi could be declared a rational number by legislative fiat is similar).
The fashion for kook mathematics has largely died out (although not completely, as a stroll through sci.math archives would reveal). Physics has replaced mathematics as a crank science repository. But I feel a certain admiration for James Smith and his false-proof-writing spiritual kin. Maybe someday someone will bolt together a machine (in the basement, perhaps, or the garage) that puts out more juice than it takes in. And that’ll show all of us laughing boys what’s what. I don’t think it will ever happen, but I don’t think it needs to. I can’t think of anything that demonstrates more dedication to knowledge (even if it’s of a slightly cracked sort) than a relentless attempt to prove to the world what you know to be true, to invent something impossible, to discover something that just isn’t there.