Given a desired ratio r not expressible as a proper fraction, limits r1 and r2 are set within which ratios afforded by feasible tooth-number combinations must lie. From the properties of conjugate fractions developed by Brocot, Rasche and the author, a pair of conjugate fractions approximating to r1 and r2 are first derived. Fractions intermediate in value are then expressed as series, the numerators and denominators of which are successively registered on a desk calculating machine. These are tested for factors, and potential solutions containing suitable factors are recorded. Best solutions are usually accurate to about 1 in 106.
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