OPERATIVE DIVISION 219 



of Newton, in or before 1674, and has since then been redis- 

 covered by Legendre, W. Heymann, and others (who did not, 

 however, add much to the original conception) ; while, in 

 my opinion, Newton, to whom Dary communicated his inven- 

 tion, may even have used it in devising his own famous method 

 of approximation, which is, indeed, nothing but the quickest 

 form of " directed iteration." However this may be, it now 

 became obvious to me that my original method of solution 

 by operative division gives the general algebraic expression 

 for Dary's arithmetical solution by iteration ; and I mentioned 

 this in a third and brief paper (Nature, February 4, 1909). 

 Still more recently I have found that it gives the general 

 algebraic expression for Newton's method also — thus com- 

 pleting the story. 



My paper on Verb Functions has, I believe, received no 

 attention beyond a short approval in Nature, which, however, 

 did not refer to operative division ; and, so far as I can ascer- 

 tain, not even Dary's, Legendre's, and Heymann's ideas 

 receive so much as the honour of mention even in the most 

 comprehensive and recent textbooks on the theory of equations 

 — doubtless because the mathematicians who write these books 

 are fully occupied by graver matters. But the methods are 

 so pretty that they will be of interest to modern " philomaths " 

 — not to mention those who, like myself, scarcely dare aspire 

 even to this grade of mathematical knowledge and are content to 

 call ourselves amateurs. Moreover, workers in many branches 

 of science often require to solve equations ; and a restatement 

 and completion of the whole subject may, therefore, be not out 

 of place in the pages of Science Progress. 



II. I will begin by describing the notation suggested by 

 me for " operative algebra." 



Let be an operator such that 0" denotes the operation 

 of raising a number or other subject to the nth algebraic 

 power ; and let square brackets be used to denote operation 

 upon as distinct from multiplication into.* Thus for example : 



[o n ] x = x n 



[ao° + bo 1 -1- co*] x = a + bx + cx 1 



1 I used /3 for o in my first paper ("Verb Functions," 1905), but the latter is 

 more convenient. The square brackets [] may also be replaced by a double dot 

 for short, as in «° : x — e*. A single dot now expresses multiplication, as in 



€° . X = X(°. 



