CH. XVI] THEEE GEOMETRICAL PROBLEMS 



349 



The use of a single symbol to denote this number 3-14159... 

 seems to have been introduced about the beginning of the 

 eighteenth century. William Jones* in 1706 represented it by tt; 

 a few years later f John Bernoulli denoted it by c; Euler in 

 1734 used p, and in 1736 used c; Christian Goldback in 1742 

 used tr; and after the publication of Euler's Analysis the 

 symbol it was generally employed. 



The numerical value of tt can be determined by either of 

 two methods with as close an approximation to the truth as 

 is desired. 



The first of these methods is geometrical. It consists in 

 calculating the perimeters of polygons inscribed in and circum- 

 scribed about a circle, and assuming that the circumference of 

 the circle is intermediate between these perimeters^. The ap- 

 proximation would be closer if the areas and not the perimeters 

 were employed. The second and modern method rests on the 

 determination of converging infinite series for it. 



We may say that the ^-calculators who used the first 

 method regarded it as equivalent to a geometrical ratio, but 

 those who adopted the modern method treated it as the symbol 

 for a certain number which enters into numerous branches of 

 mathematical analysis. 



It may be interesting if I add here a list of some of the 

 approximations to the value of it given by various writers§. 

 This will indicate incidentally those who have studied the sub- 

 ject to the best advantage. 



* Synopsis Palmariorum Matheseos, London, 1706, pp. 243, 263 et seq. 



t See notes by G. Enestrom in the Bibliotheca Mathematica, Stockholm, 

 1839, vol. m, p. 28; Ibid., 1890, vol. iv, p. 22. 



J The history of this method has been written by K. E. I. Selander, Historik 

 gjver Ludolphska Talet, Upsala, 1868. 



g For the methods used in classical times and the results obtained, see the 

 notices of their authors in M. Cantor's Geschiehte der Mathematik, Leipzig, 

 vol. I, 1880. For medieval and modern approximations, see the article by 

 A. De Morgan on the Quadrature of the Cirole in vol. xrx of the Penny 

 Cyclopaedia, London, 1841 ; with the additions given by B. de Haan in the 

 Verhandelingen of Amsterdam, 1858, vol. iv, p. 22 : the conclusions were 

 tabulated, corrected, and extended by Dr J. W. L. Glaisher in the Messenger of 

 Mathematics, Cambridge, 1873, vol. n, pp. 119—128 ; and Ibid., 1874, vol. w, 

 pp. 27—46. 



