358 THREE GEOMETRICAL PROBLEMS [CH. XVI 



done in England, found the value of it to 333 places* of 

 decimals (330 correct); in 1854 he carried the approximation 

 to 400 places - !* > anc - i n 1855 carried it to 500 placesj. 



Of the series and formulae by which these approximations 

 have been calculated, those used by Machin and Dase are perhaps 

 the easiest to employ. Other series which converge rapidly 

 are the following: 



7T = 1 1 _1_ 1^3 J_ 



6 2 + 2"3.2 3 + 2.4 , 5.2 5 + "'" 

 and 



T = 22 tan" 1 ^ + 2 tan" 1 ^ - 5 tan" 1 ^JU - 10 tan" 1 - j 



4 28 ^ 443 1393 11018 ' 



the latter of these is due to Mr Escott§. 



As to those writers who believe that they have squared the 

 circle their number is legion and, in most cases, their ignorance 

 profound, but their attempts are not worth discussing here. 

 " Only prove to me that it is impossible," said one of them, 

 "and I will set about it immediately"; and doubtless the 

 statement that the problem is insoluble has attracted much 

 attention to it. 



Among the geometrical ways of approximating to the truth 

 the following is one of the simplest. Inscribe in the given 

 circle a square, and to three times the diameter of the circle 

 add a fifth of a side of the square, the result will differ from 

 the circumference of the circle by less than one-seventeen- 

 thousandth part of it. 



An approximate value of it has been obtained experimentally 

 by the theory of probability. On a plane a number of equi- 

 distant parallel straight lines, distance apart a, are ruled ; and 

 a stick of length I, which is less than a, is dropped on the 

 plane. The probability that it will fall so as to lie across one of 

 the lines is 2l/ira. If the experiment is repeated many hundreds 



* Oriinert't Archiv, vol. xxi, p. 119. 



t Ibid., vol. xxin, p. 476: the approximation given in vol. xxn, p. 473, is 

 eonect only to 330 places. 



% Ibid., vol. zzv, p. 472; and Elbinger Anzeigen, No. 85. 



§ L' Intermediate des MathSmaticiem, Paris, Dec. 1896, vol. in, p. 276. 



