SECT. I.] MODE OF APPROXIMATION. 273 



successively closer limits between the which magnitude sought is 

 always included. Either approximation is represented by the 

 formula 



= . . . arc tan 



- arc tan j - arc tan f- arc tan - ) I \. 



When several of the operations indicated have been effected, 

 the successive results differ less and less, and we have arrived at 

 an approximate value of e. 



288. We might attempt to apply the two equations 

 e = arc tan u and u = - 



A. 



in a different order, giving them the form u = tan e and e = \n. 

 We should then take an arbitrary value of e, and, substituting it 

 in the first equation, we should find a value of u, which being 

 substituted in the second equation would give a second value of 

 e; this new value of e could then be employed in the same 

 manner as the first. But it is easy to see, by the constructions 

 of the figures, that in following this course of operations we 

 depart more and more from the point of intersection instead of 

 approaching it, as in the former case. The successive values of e 

 which we should obtain would diminish continually to zero, or 

 would increase without limit. We should pass successively from 

 e&quot; to u&quot;, from u&quot; to e , from e to u , from u to e, and so on to 

 infinity. 



The rule which we have just explained being applicable to the 

 calculation of each of the roots of the equation 



tan e 



which moreover have given limits, we must regard all these roots 

 as known numbers. Otherwise, it was only necessary to be as 

 sured that the equation has an infinite number of real roots. 

 We have explained this process of approximation because it is 

 founded on a reinarkable construction, which may be usefully 

 employed in several cases, and which exhibits immediately the 

 nature and limits of the roots ; but the actual application of the 

 process to the equation in question would be tedious ; it would be 

 easy to resort in practice to some other mode of approximation. 



F. H. 18 



