138 THE SQUARING OF THE CIRCLE. 



and Leibnitz, and known as the differential and the integral cal- 

 culus. 



About the middle of the seventeenth century, before Newton 

 and Leibnitz represented rr by series of powers, the English mathe- 

 maticians Wallis and Lord Brouncker, Newton's predecessors in 

 certain lines, succeeded in representing ?r by an infinite series of 

 figures combined according to the first four rules of arithmetic. A 

 new method of computation was thus opened. Wallis found that 

 the fourth part of TT is represented by the regularly formed product 



I XfXfXf XfXfXfX etc. 



more and more exactly the farther the multiplication is continued, 

 and that the result always comes out too small if we stop at a proper 

 fraction but too large if we stop at an improper fraction. Lord 

 Brouncker, on the other hand, represents the value in question by 

 a continued fraction in which the denominators are all 2 and the 

 numerators are the squares of the odd numbers. Wallis, to whom 

 Brouncker had communicated his elegant result without proof, de- 

 monstrated the same in his Arithmetic of Infinites. 



The computation of n could scarcely have been pushed to a 

 greater degree of exactness by these results than that to which Lu- 

 dolf and others had carried it by the older and more laborious 

 methods. But the series of powers derived from the differential 

 calculus of Newton and Leibnitz furnished a means of computing 

 n to hundreds of decimal places. 



Gregory, Newton, and Leibnitz found that the fourth part of 

 n was equal exactly to 



if we conceive this series, which is called the Leibnitz series, con- 

 tinued indefinitely. This series is wonderfully simple but is not 

 adapted to the computation of ?r, for the reason that entirely too 

 many members have to be taken into account to obtain n accurately 

 to a few decimal places only. The original formula, however, from 

 which this series is derived, gives other formulae which are excel- 

 lently adapted to the actual computation. The original formula is 

 the general series : 



