ECONOMY OF LABOUR IN MATHEMATICS 209 



consists of those parcels that have the secondary root in 

 them." 



In other words, the method used by him at the date 1664 

 consists of the usual one of finding an equation whose roots 

 are less than those of the proposed equation by a given amount /. 



On the morning of August 15, 1674, Dary however has a 

 new idea, that of iteration ; he is so pleased with it, that 

 although he had sent Newton three papers the day before, 

 he says : " I cannot refrain from sending you this," and adds, 

 " Truly it pleaseth me well, but yet I do hereby submit it to 

 your censure." 



The example he gives in his letter is : 



+ z p = az q ± n 



and the rule is this — " first guess at the root as nearly as you 

 can, the nearer the better (not for necessity but for accommo- 

 dation), and suppose that guess to be z," etc. 

 He then gets — 



V+ az q + n = b y+ ab q -f n = c 



V+ ac q + n = d V+ ad q + n = e 



V+ ae q + n = / V '+ af q + n = g, 



etc. etc. 



The examples of the quintic which he solves are given in 

 the Bring-Jerrard form, and the evaluation of the fifth root is 

 performed by the aid of logarithms. 



Let us consider for a moment in what way the iterative 

 methods of modern writers differ from that of Dary. The 

 most important difference is this : Dary does not show he was 

 aware that his method was not universally applicable ; that 

 he might for instance have such an equation, and make such 

 an initial guess, that instead of approaching the root, he might 

 oscillate about it. He does indeed experience some difficulty, 

 and says of one equation, " But this soure crabb I cannot deale 

 with by no method." 



The modern mathematician, with his greater familiarity 

 with curves and their singular points, and his wider idea of 

 functions, looks at the problem in a much more general way. 

 For instance : given any algebraic equation, he knows that 



