On the Teeth of Wheels
Converting minutes into days is a problem that gears can solve exactly, but what if the ratio of two speeds is π? Here the first law of gearwork fails because it runs up against the zeroth law—that the number of teeth on a gear must be an integer. No ratio of integers can be equal to π. The best one can hope for is a good rational approximation. This is where Merritt's "Brocot table" enters the story, and it put me back on the trail of Brocot's original paper.
A visit to the New York Public Library proved tantalizing; I found several volumes of the Revue Chronométrique, but not the volume I needed. On the other hand, the library was able to supply the enigmatic Camus on the Teeth of Wheels. Camus turned out to be Charles-Étienne-Louis Camus, 1699–1768, author of a Cours de Mathématique published in 1749. The section of this textbook dealing with gearwork was extracted and translated into English by John Isaac Hawkins, a civil engineer, and published under the title A Treatise on the Teeth of Wheels, Demonstrating the Best Forms Which Can Be Given to Them for the Purposes of Machinery; Such as Mill-work and Clock-work, and the Art of Finding Their Numbers.
The first part of Camus's treatise deals with a geometrical rather than a number-theoretical question: What is the ideal shape for a gear tooth? This issue engaged the talents of mathematicians and other savants for generations. Robert Hooke, Thomas Young, Leonhard Euler and Josiah Willard Gibbs all debated the merits of epicycloids and involutes. It's a fascinating problem, but I turned to Part II of the treatise, where Camus takes up the numerical aspects of gear design.
For cases where an exact solution is possible, Camus explains the method of reducing a number to its prime factors and then partitioning the factors into as many groups as there are pairs of gears. He then turns to the task of approximating a ratio when the numbers have no convenient factorization. As an example he offers this problem: "To find the number of the teeth...of the wheels and pinions of a machine, which being moved by a pinion, placed on the hour wheel, shall cause a wheel to make a revolution in a mean year, supposed to consist of 365 days, 5 hours, 49 minutes." Multiplying out the days and hours yields a target ratio of 720/525949. The numerator of this fraction factors conveniently, but the denominator is a prime. Thus the aim is to find another fraction, as close as possible in value to 720/525949, but with both a numerator and a denominator that have small factors. Camus remarks: "In general this is done by repeated trials; but as this method is defective, we shall here propose another, by which the problem may be solved with more certainty."
But the next 20 pages, which set forth the method through worked examples, leave the impression that it's hardly much better than trial and error. Camus's procedure for finding ratios close to the target is a fairly arduous algebraic process, made worse by awkward and verbose notation. Furthermore, trial-and-error is still required, because there is no guarantee that a ratio generated by the method will be factorable. Camus reports seven failures before he hits on the ratio 196/143175, which can be factored as 4/25 x 7/69 x 7/83. It was Brocot, a century later, who found a better way.
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