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On the Teeth of Wheels

Brian Hayes

The Professor and the Watchmaker

Thus the situation might have remained but for the prompting of a friend and longtime reader of this department, Horacio A. Caruso of La Plata, Argentina. Caruso and his colleague Sebastián M. Marotta were sufficiently interested in the Vibonacci phenomenon to undertake investigations of their own. For example, they applied the Vibonacci algorithm to complex numbers, creating an intriguing series of fractal images. When Caruso inquired about the works of Stern and Brocot, I was belatedly inspired to go look them up.

Stern's paper was not hard to find. Moritz Abraham Stern (my earlier spelling Moriz was erroneous) was a prominent figure in the mathematical world of his day, a colleague of Carl Friedrich Gauss who succeeded Gauss as Ordinary Professor of Mathematics at the University of Göttingen. According to Peter Pulzer, Stern was "the first Jew to be appointed to a full professorship at a German university without first converting to Christianity."

Stern's paper appeared in the Journal für die Reine und angewandte Mathematik, also known as Crelle's Journal, after its first editor, August Leopold Crelle. When the journal was founded in 1826, its title reflected the growing division in mathematics between Reine and angewandte—"pure" and "applied." Stern's paper, "On a number-theoretical function," is of the pure persuasion. He discusses several versions of the procedure for forming mediants and relates the sequence of mediants to other ways of constructing the set of rational numbers, such as continued fractions. Nowhere does he hint that his number-theoretical function might be of use to the makers of gears.

Finding Brocot's contribution was more challenging. His article was published in the Revue Chronométrique, a French journal that commenced publication in 1855 and ceased in 1914. Only when I began searching for the Revue did it occur to me to wonder why a work on number theory was appearing in a clockmakers' journal.

None of the libraries within easy reach had the Revue Chronométrique, but a catalogue search at Duke University did return one hit for the term "Brocot." I was referred to a 1947 work by Henry Edward Merritt, titled Gear Trains: Including a Brocot Table of Decimal Equivalents and a Table of Factors of All Useful Numbers up to 200,000. A Brocot table? Useful numbers? What was this all about? I walked from the mathematics library to the engineering library next door and soon had a worn blue volume in my hands. When I opened to the preface, I knew I would have to read the rest of the book. Merritt begins:

Prefaces are not what they were. Who could resist the opening phrases of the editor's preface to the second edition of Camus on the Teeth of Wheels, published in 1836—
"Always feeling annoyed at meeting with a long preface to a book, labouring as it were to beget a prepossession in favour of the author, and standing between the reader and the subject, like an impertinent porter, who detains a visitor at the gate, instead of giving him admission to the presence of the master, the editor will confine himself to five pages of preliminary remarks...."

Indeed, who could resist? And, furthermore, what is this mysterious Camus on the Teeth of Wheels? The title would have been an apt one for the tormented existentialist Albert Camus, but it comes from the wrong century.

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