A controversy over gears

In the introduction to his Traité des épicycloïdes et de leur usage dans les méchaniques (Treatise on Epicycloids and Their Use in Mechanics), dated 1694, Philippe de La Hire neatly sums up the state of gearing at the end of the 17^th century: "Since no one gives any thought to the shape of gear teeth, leaving it entirely to the craftsman who makes them, most mechanisms simply do not work!" In his treatise, he proposed a solution: tooth profiles shaped like cycloids. Was he the first to think of it? In the same introduction, he explained that he had once rebuilt the wheel of a pump at the Château de Beaulieu in Essonne; the pump used this cycloidal profile, and the wheel had been designed by the "excellent geometer" Girard Desargues. Since Desargues left no written account of the subject, he had probably arrived at the profile by trial and error…

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Investigating a water pump ---------------------------