THE SQUARING OF THE CIRCLE. 



139 



where a is the length of the arc belonging to any central angle in a 

 circle of radius i, and a the tangent to this angle. From this we 

 derive the following : 



where a, b, c . . . are the tangents of angles whose sum is 45*. De- 

 termining, therefore, the values of a, b, c . . . , which are equal to 

 small and convenient fractions and fulfil the conditions just men- 

 tioned, we obtain series of powers which are adapted to the com- 

 putation of n. 



The first to add by the aid of series of this description addi- 

 tional decimal places to the old 35 in the number n was the Eng- 

 lish arithmetician Abraham Sharp, who, following Halley's instruc- 

 tions, in 1700 worked out n to 72 decimal places. A little later 

 Machin, professor of astronomy in London, computed n to 100 

 decimal places, by putting, in the series given above, a = b = c = d 

 = % and e= ^J-g-; that is, by employing the following series: 



= A ___ 



4 "15 3-5 



5-5 



7-5 



, 



239 3-239 5-239 



In the year 1819, Lagny of Paris outdid the computation of 

 Machin, determining in two different ways the first 127 decimal 

 places of n. Vega then obtained as many as 140 places, and the 

 Hamburg arithmetician Zacharias Dase went as far as 200 places. 

 The latter did not use Machin's series in his calculation, but the 

 series produced by putting in the general series above given = J, 

 &=%, t = $. Finally, at a recent date, n has been computed to 

 500 places.* 



The computation to so many decimal places may serve as an 

 illustration of the excellence of the modern methods as contrasted 

 with those anciently employed, but it has otherwise neither a theo- 



*In 1873 the approximation was carried by Shanks to 707 places of decimals. 

 Trans. 



