THE SQUARING OF THE CIRCLE. 117 



l — l-Li — i-Ll ^1_4-JL_ .... 



if we conceive tbis series, which is called the Leibuitzian, indefioitely 

 continued. This series is indeed wonderfully simple, but is not adapted 

 to the computation of -, for the reason that entirely too many members 

 have to be taken into account to obtain 7: accurately to a few decimal 

 places only. The original formula, however, from which this series is 

 derived, gives other formulas which are excellently adapted to the ac- 

 tual computation. This formula is the general series : 



a=a—^a-'-\-l(v'—\a'+ . . ., 



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

 circle of radius 1, and where a is the tangent to this augle. From this 

 we derive the following : 



'L={a + h+c+ . . . )-i^{a^+h'+c'+ • • • ) + J {a'j^h'+c'+ ...)-..., 



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

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

 and easy fractions and fulfill the condition just mentioned, we obtain 

 series of powers which are adapted to the computation of r. The first 

 to add by the aid of series of this description additional decimal places 

 to the old 35 in the number - was the English arithmetician Abraham 

 Sharp, who, following Halley's instructions, in 1700, worked out r to 

 72 decimal places. A little later Machin, professor of astronomy in 

 London, computed n to 100 decimal places; putting, in the series given 

 above, a=b=c=d=l and e——^^,^, that is employing the following- 

 series : 



1=" 



1 1 11 



5~3.5^ +5.5= ~7.5' 



+ ir^-i +i .1 



J L^"^^ 3.239^^5.239^ J 



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 arithme- 

 tician, 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 a=^, b=}, 0=1. Finally, at a recent 

 date, TT has been computed to 500 places. 



The computation to so many decimal places may serve as an illustra- 

 tion of the excellence of the modern method as contrasted with those 

 anciently employed, but otherwise it has neither a theoretical nor a 

 practical value. That the computation of r to say 15 decimal places 

 more than sufficiently satisfies the subtlest requirements of practice may 

 be gathered from a concrete example of the degree of exactness thus 

 obtainable. 



