CH. XVI] THREE GEOMETRICAL PROBLEMS 359 



of times, the ratio of the number of favourable cases to the 

 whole number of experiments will be very nearly equal to 

 this fraction: hence the value of it can be found. In 1855 

 Mr A. Smith* of Aberdeen made 3204 trials, and deduced 

 7r = 31553. A pupil of Prof. De Morgan*, from 600 trials, 

 deduced 7T = 3-137. In 1864 Captain Foxf made 1120 trials 

 with some additional precautions, and obtained as the mean 

 value it = 3-1419. 



Other similar methods of approximating to the value of ir 

 have been indicated. For instance, it is known that if two 

 numbers are written down at random, the probability that 

 they will be prime to each other is 6/Tr 1 . Thus, in one case J 

 where each of 50 students wrote down 5 pairs of numbers at 

 random, 154 of the pairs were found to consist of numbers prime 

 to each other. This gives Q/ti 3 = 154/250, from which we get 



7T = 312. 



* A. De Morgan, Budget of Paradoxes, London, 1872, pp. 171, 172 [quoted 

 from an article by De Morgan published in 1861]. 



+ Messenger of Mathematics, Cambridge, 1873, vol. n, pp. 113, 114. 



X Note on t by R. Chartres, Philosophical Magazine, London, series 6, 

 vol. xixix, March, 1901, p. 315. 



