4 o 4 SCIENCE PROGRESS 



nounced by him in a letter to Newton in 1674 ; has been subse- 

 quently rediscovered and somewhat elaborated by Legendre, 

 Vogel, Lemeray, W. Heymann, C. Isenkrahe (1897), myself 

 (1908, 1909), and others ; but is not mentioned in the text- 

 books which I have searched for it. Newton's method, which I 

 think may have been derived from Dary's, is well known, but 

 still very inadequately dealt with in such books. 



I will use Newton's equation to illustrate both methods. 

 For Dary's method write it in the form 



x = >/$ + 2X. 



Take any number x , substitute it for x in the right side of 

 this equation, and let the result be x x — that is, let 



Xi = -\/f( -f- 2x0. 



Again substitute x x for x in the right side of this equation and 

 let the result be x 2 : and so on, ad infinitum. Then, in this 

 case, whatever real number x may be, and whatever slips we 

 may make in our numerical work, the iterants x , x if x 2 . . . x m 

 will infallibly approach the real root of the equation, namely 

 2*09455. . . . For examples let x be 100, o, or — 100 ; then we 

 obtain the following : 



100, 5*896, 2*560, 2*163, 2*0959, 2*0946, 2*0945, • • • ; 



o, 1*710, 2*034, 2*085, 2*093, 2*0943, 2*0945, • • • ; 



— 100, - 5798, - 1*515, i*253j i*94i, 2*071, 2*091, 2*094, • • • 



The figures are roughly calculated by the aid of Barlow's 

 Tables, and purposely contain some errors. It is curious that 

 this surprising and beautiful result should not be deemed 

 worthy even of mention in many books on the Theory of 

 Equations. 



For Newton's method (which was however stated by him 

 in another way), write the given equation in the form 



X\ = x — 4rr~ : which is in this case 



5 + 2X — x s 



Xi — #0 ; • 



2 — 3X 2 



Substitute for x any number greater than </f and proceed as 

 in Dary's method. If x = 1, or = 100, we have 



*> 7, 5, 2*5, 2*163, 2*097, 2*0946, . . . ; 



100, 77, 51, 34, 23, 15, 10, 7, 5, . . . 2*0946, . . . 



