CH. XVI] THREE GEOMETRICAL PROBLEMS 357 



the formula £tt = tan" 1 £ + tan~ l £ or £tt = 5 tan" 1 \ + 2 tan" 1 7 3 ,, 

 but neither carried the approximation as far as had been done 

 previously. 



Vega, in 1789* gave the value of ir to 143 places of 

 decimals (126 correct); and, in 1794 f, to 140 places (136 

 correct). 



Towards the end of the eighteenth century F. X. von Zach 

 saw in the Radcliffe Library, Oxford, a manuscript by an 

 unknown author which gives the value of ir to 154 places of 

 decimals (152 correct). 



In 1837, the result of a calculation of it to 154 places of 

 decimals (152 correct) was published^. 



In 1841 Rutherford § calculated it to 208 places of decimals 

 (152 correct), using the formula ^7r=4 tan -1 £— tan -1 ,^+tan -1 fa. 



In 1844 Dase || calculated it to 205 places of decimals (200 

 correct), using the formula \ir = tan -1 \ + tan -1 £ + tan -1 £. 



In 1847 Clausen IT carried the approximation to 250 places 

 of decimals (248 correct), calculating it independently by the 

 formulae \tt = 2 tan -1 £ + tan -1 \ and \ir = 4 tan -1 \ — tan -1 ^$. 



In 1853 Rutherford** carried his former approximation to 

 440 places of decimals (all correct), and William Shanks pro- 

 longed the approximation to 530 places. In the same year 

 Shanks published an approximation to 607 places ft : and in 

 1873 he carried the approximation to 707 places of decimalsJJ. 

 These were calculated from Machin's formula. 



In 1853 Richter, presumably in ignorance of what had been 



* Nova Acta Academiae Scientiarum Petropolitanae for 1790, Petrograd, 

 1795, vol. ix, p. 41. 



+ Thesaurus Logarithmorum (logarithmisch-trigonometrischer Tafeln), Leipzig, 

 1794, p. 633. 



X J. !?• Callet's Tables, etc., Prtcis 'faUmentaire, Paris, tirage, 1837. Tirage, 

 1894, p. 96. 



§ Philosophical Transactions, 1841, p. 283. 



|| Crelle's Journal, 1844, vol. xxvn, p. 198. 



IT Schumaoher, Astronomische Nachrichten, vol. xxv, ool. 207. 



*• Proceedings of the Royal Society, Jan. 20, 1853, vol. vi, pp. 273 — 275. 



j-f Contributions to Mathematics, W. Shanks, London, 1853, pp. 86 — 87. 



%X Proceedings of the Royal Society, 1872-3, vol. xxi, p. 318; 1873-4, vol. 

 xxii, p. 45. 



