III.—On the Extension of Brouncker's Method to the Comparison of Several Magnitudes.

Extract

The discovery of those numbers which shall, either truly or approximately, represent the ratio of two magnitudes, necessarily attracted the attention of the earliest cultivators of exact science. The definition of the equality of ratios given in Euclid's compilation clearly exposes the nature of the process used in his time. This process consisted in repeating each of the two magnitudes until some multiple of the one agreed perfectly or nearly with a multiple of the other; the numbers of the repetitions, taken in inverse order, represented the ratio. Thus, if the proposed magnitudes were two straight lines, Euclid would have opened two pairs of compasses, one to each distance, and, beginning at some point in an indefinite straight line, he would step the two distances along, bringing up that which lagged behind, until he obtained an exact or a close coincidence.