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XX. Computation of the Ratio of the Diameter of a Circle to its circumference to 

 208 places of figures. By William Rutherford, Esq., of the Royal Military 

 Academy. Communicated by S. Hunter Christie, Esq., M.A., Sec. R.S. S^c. 8gc. 



Received April 16, — Read May 6, 1841. 



Before the time of Machin, the approximation to the ratio of the circumference 

 of a circle to its diameter had been carried as far as seventy-two places of decimals 

 by Abraham Sharp, by means of the series 



T = T^3 (^ ""¥* F+T'^-yis'^^-J- 

 By employing the series arising from the formula, 



T _i 1 X —1 1 



-J = 4 tan -5- -^ tan 239' 



Machin extended the approximation to 100 places. By the same means M. De Lagny 

 carried this approximation to 127 places ; and in an Oxford manuscript it is extended 

 to 152 places, which, as far as I am aware, is the greatest extent to which the ap- 

 proximation has ever been pushed. 



The processes employed in these approximations may be greatly simplified by 



replacing tan~^ g^ ^Y *^^"^ 70 "" ^^^"^99' i^^smuch as the calculation of the terms 



of the series involving inverse powers of 70 and 99 may be effected by arithmetical 

 processes of very great facility. By employing the synthetic process of division, the 

 division by 99 (100 — 1) becomes even more simple than that by 9 or 11, since it is 

 effected by adding together two numbers each less than 10. 

 By means of the formula 



J = 4 tan"^-^- — tan"^ yQ + tan"^ gg*, 



I have computed the value of -^, and thence that of rr to 208 places of decimals. 

 Previously to entering upon the calculations I considered whether it would be simpler 



* When the calculations for determining the value of * were presented to the Royal Society, it was presumed 



that the formula — = 4 tan~"^ -r- — tan~^ — + tan"*-— had not before been investigated. I have since 

 4 D /U yy 



found that Euler, in an article entitled " De progressionibus arcuum circularium quorum tangentes secundum 



certam legem procedunt," obtained the very same formula. — Novi Commentarii Petropol., torn. ix. 1764. 



