62 MR EDWARD SANG ON THE EXTENSION OF BROUNCKER'S METHOD 



of the one fraction becomes the denominator of the other, the approximation is 

 toward that root of the equation which is farthest from zero ; and that if the 

 progression be carried backwards, the approximation is then toward the root 

 nearest to zero. 



These remarks may suffice to show that this branch of the theory of numbers 

 promises to yield important results. Now, the whole doctrine of quadratic 

 recurrence sprung from the comparison of two magnitudes ; and so the com- 

 parison of three magnitudes must be the true foundation on which to build 

 the doctrine of cubic recurrence. I propose, therefore, in the present paper, to 

 discuss the elementary operation by which the ratios of three incommensurable 

 magnitudes may be approximately ascertained. 



Let there be three homogeneous quantities, A, B, C, arranged in the order 

 of their magnitudes, and let it be proposed, if possible, to find their common 

 measure. 



By repeatedly subtracting the second B from the greatest A, we obtain a 

 remainder less than B ; this remainder may or may not be greater than C ; if it 

 be greater, we may take C from it until we obtain a remainder D less than the 

 least of the three proposed quantities. In this way we have an equation of 

 the form 



A=p 1 B + q 1 C +D, 



in which p cannot be zero, while q may. 



Treating now the three quantities B, C, D, in the same way we have a new 

 equation 



B=p 2 C+q 2 B + E } 



and we may proceed in this way until there be no remainder, or until the 

 remainder be so small as to pass the limits of exactitude demanded by the 

 nature of the case in hand. 



We are now able, by means of successive substitutions, to obtain values of 

 A, B, C, in terms of the ultimate remainders ; and our first business is to devise 

 some convenient arrangement for this purpose. 



For the sake of giving greater generality to our investigations, let us put 

 the successive equations in the form 



A = p 1 B + q x C + r x D , 



B = p 2 C + q 2 D ■ + r 2 E , 

 C = P s D + q a E + r s F , 



Q = p n E + q n S + r a T , 



in which r lt r s , . . . . have been written for the unit of the usual process. 



