The object sought to be gained by this tabulation may be best seen by considering a specific case. When we wish to produce, by help of toothed wheels, a certain ratio between the rotations of two axes, we must have that ratio represented by integer numbers, and these numbers must contain no prime factors exceeding the limit of the number of teeth which can conveniently be cut in any wheel. If, for example, we wish to represent the mean motions of the planets, we have to express the ratios of their periodic times by decomposable numbers; and as this, in general, can only be done approximately, we have to prescribe some limit of error on either side, and have to seek for all fractions between those limits, and then among these to search for those that are suitable.

In my treatise on wheel-teeth a solution of this problem is given, complete so far as the discovery of the fractions is concerned, but imperfect in this, that it does not place those fractions in the order of their magnitudes. I propose now to supply this deficiency by explaining an exceedingly simple process, which enables us to make a complete list, arranged in the order of their values, of all irreducible fractions whose denominators shall not exceed some specified number; and also to take up and interpolate any portion of the list. This process is founded on a general theorem flowing directly from the doctrine of continued fractions, but of which the following simple demonstration may be given.