356 THREE GEOMETRICAL PROBLEMS [CH. XVI 



but neither of these theorems was used to any large extent for 

 calculation. 



Subsequent calculators have relied on converging infinite 

 series, a method that was hardly practicable prior to the in- 

 vention of the calculus, though Descartes* had indicated a 

 geometrical process which was equivalent to the use of such 

 a series. The employment of infinite series was proposed by 

 James Gregory f, who established the theorem that 



= tan0-£tan»0 + -£tan l! 0-..., 

 the result being true only if 8 lies between — \ir and \ir. 



The first mathematician to make use of Gregory's series 

 for obtaining an approximation to the value of ir was Abraham 

 Sharp]:, who, in 1699, on the suggestion of Halley, determined 

 it to 72 places of decimals (71 correct). He obtained this 

 value by putting 6 = \ir in Gregory's series. 



Machin§, earlier than 1706, gave the result to 100 places 

 (all correct). He calculated it by the formula 

 \ -rr = 4 tan" 1 i - tan- 1 ^. 



De Lagny||, in 1719, gave the result to 127 places of 

 decimals (112 correct), calculating it by putting 6 = ^nr in 

 Gregory's series. 



Huttonlf, in 1776, and Euler**, in 1779, suggested the use of 



* See Euler'B paper in the Novi Commentarii Academiae Scientiarum, 

 Petrograd, 1763, vol. vin, pp. 157—168. 



+ See the letter to Collins, dated Feb. 16, 1671, printed in the Commereium 

 Epistolicum, London, 1712, p. 25, and in the Macclesfield Collection, Carre- 

 tpondence of Scientific Men of the Seventeenth Century, Oxford, 1841, vol. n, 

 p. 216. 



X See Life of A. Sharp by W. Cudworth, London, 1889, p. 170. Sharp's 

 work is given in one of the preliminary discourses (p. 53 et eeq.) prefixed to 

 H. Sherwin's Mathematical Tables. The tables were issued at London in 1705 : 

 probably the discourses were issued at the same time, though the earliest copies 

 I have seen were printed in 1717. 



§ W. Jones's Synopsis Palmariorum, London, 1706, p. 243; and Maseres, 

 Scriptores Logarithmici, London, 1796, vol. m, pp. vii — ix, 155 — 164. 



|| Histoire de V Academic for 1719, Paris, 1721, p. 144. 



1 Philosophical Transactions, 1776, vol. lxvi, pp. 476 — 492. 



** Nova Acta Academiae Scientiarum Petropolitanae for 1793, Petrograd, 

 1798, vol. xi, pp. 133—149 : the memoir was read in 1779. 



