ECONOMY OF LABOUR IN MATHEMATICS 213 



write — " I suspect all progressing series except when I know 

 the producing function, and I trust all alternating series even 

 when I do not know the producing function," and he speaks 

 of divergent series as " wicked infinities in a finite disguise, 

 pretending to be finites in an infinite disguise." 



Having considered the separation of the roots into real 

 and imaginary ; positive and negative ; and the determination 

 of the limits between which the real roots lie ; we have next 

 to consider the numerical approximation to their values. 



Many methods have been discovered, which vary greatly 

 both in their theoretical perfection and practical application. 

 Dary in his first method uses tables or " canons " he has 

 constructed of " Potestates " arising from a binomial, but 

 after his discovery of iteration he uses logarithms for the 

 evaluation of the successive roots. But the method of arrang- 

 ing the work for the continuous reduction of an equation 

 discovered by Ruffini in 1804 and rediscovered by Horner, 1819, 

 is so expeditious that it seems as though further simplification is 

 not to be expected. 



It is true that Budan in 1807 also obtained a method of 

 transformation of which the numerical work is very simple, 

 but the number of steps required takes away from the value. 

 Thus by the Ruffini-Horner method, the equation would be 

 reduced to another having its roots less than the roots of 

 the original equation by 9, in one step, while it would require 

 nine steps by the Budan method. 



One disadvantage of the Ruffini-Horner method is that in 

 equations of advanced degrees in which several terms are 

 absent, the method of reduction introduces significant numbers 

 in place of the zero coefficients. A method of avoiding this 

 has been introduced by Mr. Weddle of Newcastle-on-Tyne, 

 the peculiarity of whose method is, that it conducts its steps 

 by aid of transformations, through all of which the zero co- 

 efficients are transmitted. 



If we ask why it is that Dary and his work have passed 

 into oblivion, a little consideration will perhaps show us that 

 the only person likely to be remembered by posterity is the 

 man who is popularly supposed to have completed something. 



Cardan's name lives in connection with the cubic, and 

 Ferrari with the biquadratic, because these men completed 

 a step in the history of mathematics. 



