On the Teeth of Wheels
Just as Stern mentions no practical applications of his number-theoretical function, Brocot gives little attention to the mathematical foundations of his gear-train algorithm. (In a longer essay, which I have still not been able to lay hands on, Brocot claims to provide more theoretical background.) There is no sign that either man knew of the other's work. After the fact, however, connections between them are easy to see; they are doing the same thing but describing it differently.
There is even a connection between Brocot's algorithm and the Fibonacci series, where this whole quest began. To see the relation, just try using Brocot's method to find ratios approximating the constant known as phi, or the golden ratio, an irrational number with a value of approximately 1.618. The series of approximants begins 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,.... Hidden within these ratios is the complete sequence of Fibonacci numbers.
Working through examples of Brocot's process by hand, and leafing through the pages of the printed Brocot table, leaves me feeling wistful and uneasy. The ingenuity and diligence on exhibit here are certainly admirable, and yet from a modern point of view they are also tinged with a horrifying futility. I am reminded of those prodigies who spent years of their lives calculating digits of the decimal expansion of π—a task that is now a mere warmup exercise for computer software. I cannot help wondering which of my own labors will appear equally quaint and pathetic to some future reader who ransacks libraries for old volumes of American Scientist.
The fact is, the design of simple gear trains is no longer a computationally interesting problem, because computation itself has overwhelmed it. With so much calculating power at your fingertips, it's hardly worth the bother of being clever. You can solve gearing problems by brute-force, using methods that would have been unthinkable for Camus or Brocot, or even for Merritt, who was writing hardly more than 50 years ago. If you need to approximate some ratio, just have the computer try all pairs of gears with no more than 100 teeth. There are only 10,000 combinations; you can churn them out in an instant. For a two-stage compound train, running through the 100 million possibilities is a labor of minutes.
The whirling gears of progress have put the gearmakers out of work.
© Brian Hayes