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



Magnitudes. By Edward Sang, Esq. 



(Read 7th February 1870.) 



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. 



He seems to have assumed that, in the case of incommensurable magnitudes, 

 the further continuation of the process must give still closer approximations ; 

 but we do not find any indication of a knowledge of the fact that, in the course 

 of that continuation, we shall certainly come upon coincidences still more close 

 than any which we have already obtained. 



This process for finding the numerical expression for a ratio is inconvenient 

 from its bulkiness ; it is also unnatural, for the mind, in comparing two unequal 

 magnitudes, is rather inclined to regard them as made up each of so many 

 measures, than to consider how many times the one must be augmented in order 

 that the result may be a multiple of the other ; it prefers the direct to the 

 inverse comparison. 



Lord Brouncker's method of continued fractions enables us with great 

 rapidity and within the compass of the magnitudes themselves, to determine 

 directly their ratio. It is one of the great landmarks in the progress of the 

 science of numbers. 



By one or two slight improvements in the mode of calculation, the chain or 

 continued fraction became a ready tool in the hands of arithmeticians. It placed 

 in a clear light the whole doctrine of indeterminate equations of the first degree, 

 leaving scarcely anything further to be desired in this branch of the Diophantine 

 analysis. 



VOL. XXVI. PART I. Q 



