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



On applying Brouncker's method to two incommensurable quantities of the 

 second degree, it was found that the denominators eventually came to be 

 repeated or circulated indefinitely; and Lagrange showed that while every cir- 

 culating chain-fraction was known to represent the root of a quadratic equation, 

 the roots of all such equations were developable in such a fraction. Hence the 

 conclusion was drawn that the root of no equation of a higher order can possibly 

 be represented by a circulating chain-fraction. 



Although, to the mind of Brouncker, the continued fraction presented the 

 readiest way of expounding his idea, it is not essential thereto ; a much clearer 

 view of the true nature of the process may be obtained without it. The opera- 

 tion consists, essentially, in deducting, as often as possible, the less from the 

 greater ; the remainder again from the preceding subtrahend, and so on ; in 

 keeping note of the numbers of the subtractions ; and in computing from these 

 numbers the value of the magnitudes in terms of the ultimate subtrahend. The 

 chain-fraction is merely one way of representing the final computation. By 

 stopping at the first denominator, then at the second, afterwards at the third, 

 and so on, we obtain a series of fractions alternately too great and too small, 

 but approaching rapidly to each other and to the true expression for the ratio. 

 Now this series may be deduced directly from the equations representing the 

 various subtractions ; wherefore, in our subsequent investigations, we may put 

 the idea of the chain-fraction entirely aside, without thereby changing the in- 

 trinsic character of the Brounckerian process. 



On examining the two series converging to the two roots of a quadratic 

 equation, I observed that the circulating quotients are the same for both, but that 

 their order is inverted. This observation led me to a singular law, which some 

 years ago I submitted to the Society. It is this, that if we continue the forma- 

 tion of the series for one root beyond the non-circulating quotients, obliterate 

 these and the fractions adjoining them, and then, using only the circulators, 

 compute the series backwards, we shall obtain the other root of the equation; 

 so that both roots are given by a, so to speak, two-headed progression. 



The periodical recurrence of the quotients enables us to approximate as 

 closely as may be desired to the roots of equations of the second order, with 

 very little labour ; and a kind of regret accompanied the conviction that the 

 same facilities cannot be obtained for equations of higher degrees. On con- 

 sidering the arguments on which this conviction rested, it appeared to me that 

 the whole circumstances of the case had not been taken into account ; one, and 

 a most influential one, had been concealed under the notation employed, that is, 

 under the scheme of continued fractions. If we assume any two fractions to 

 take the place of two contiguous terms in a Brounckerian progression, and 

 operate upon these in the usual way, that is, by adding to a multiple of each 

 member of the second fraction the corresponding member of the preceding ; and 



