208 SCIENCE PROGRESS 



Fourier, in his Theory of Heat (English translation, p. 272), 

 uses the method for the solution of — 



e = arctan e/ 



and gives graphical illustrations. 



Vogel, Isenkrahe, and Lemeray have given numerous 

 papers on the subject, and the present Astronomer Royal for 

 Ireland sent me a paper in which he had used it, and in a 

 letter said that although he thought the method must have 

 been previously used, he could not remember having seen it 

 anywhere. In a paper now appearing in the Proceedings of 

 the London Mathematical Society by Mr. E. H. Neville, we 

 find the same method is used. 



Perhaps a few details of the process may help to a clearer 

 understanding of the subject, and enable us to comprehend 

 in what way Dary's method differs from the methods of those 

 who have rediscovered it. 



The method of Dary to which the name of iteration has 

 been given consists in the transformation of an equation to 

 such a form that the continual repetition of some functional 

 operation, such as the finding of a square, cube, or higher root, 

 may lead us by successive steps to the value sought. 



In The General Doctrine of Equations (printed in 1664), we 

 find that the method pursued by Dary is the well-known 

 continuous depression of an equation by substituting for the 

 unknown a binomial / •+■ n. 



Dary gives a summary of the known properties of equa- 

 tions, and in simultaneous equations eliminates, or, as he calls 

 it, purges the equations of all unknowns except the one desired 

 to be known. 



He then deals with the " clearing and trimming of an 

 equation whereby to know how many and what kind of roots 

 it hath." He further shows that the coefficients are symmetric 

 functions of the roots. 



For the solution he substitutes in place of the unknown a 

 binomial, and says : " Put / -f- n for unknown, which Potestates 

 of / -f- n being orderly placed do make up a canon for the equation 

 proposed. Which canon may be considered to consist of two 

 parts, the head and the tail. The head consists of those 

 parcels that have not the secondary root n in them, the tail 



