274 



ference of a circle to its Diameter." By William Rutherford, Esq., 

 F.R.A.S. Communicated by S. Hunter Christie, Esq., Sec. R.S. &.c. 

 Received November 17, 1852. 



The author, referring to a former communication on this subject, 

 published in the Phil. Trans. 1841, states that, in the value of tt 

 there given to 208 places of decimals, there exists, in the latter 

 part of one of the terms of the series for determining the value 



of tan~^i-, a transposition of the figures of a recurring decimal, 



which vitiates a considerable number of the figures in the latter part 

 of the value. This error had been detected in consequence of Pro- 

 fessor Schumacher having observed that in the value of tt which had 

 been given him by M. Dase, who had calculated it to 200 places, 



from the formula -7= tan ~^^+ tan~'^ — +tan~^^, the fio:ures from 



4 2 o S 



the 153rd to the 200th differed entirely from those given by the 

 author. The accuracy of M. Dase's result was confirmed by a 

 double computation of Dr. Clausen of Dorpat, who deduced the 

 value of - to 250 places of decimals, both by Ixlachin's formula 



_i 1 _i 1 



^=4 tan ^-tan 239. 



and by the formula 



TT ,1 ,1 



— = 2tan-'Y+tan 'y; 



and the author's result was shown to differ from the coiTect value 



by the periodic decimal '36. 



Having been informed by W. Shanks of Houghton-le- Spring, 

 that he had pushed his computation of the value of - to the extent 

 of 318 decimals, the author resolved to extend his operations to 

 upwards of 400 decimals. As ]\Ir. Shanks had employed Machin's 

 formula, the author resolved to make use of the same. At his request 

 Mr. Shanks resumed his calculation?, and has not odIy verified the 

 author's value of 77 to 440 places of decimals, but has carried his 

 own to the extent of 530 places. The author states that the values of 



tan^'^-^and tan~^ ^^ipr, as well as the value of tt, which are here 

 o 2.jy 



subjoined, have been obtained by the independent computations of 

 Mr. Shanks and himself, and that they both feel confident that 

 these values are correct in every figure as far as 440 decimals. 



tau -19739 5559S 75S37 CC49- 65194 -9C29 34475 85103 787S5 



5 21015 17688 94C24 1C339 699-S 245-S 57326 97828 03728 80441 

 12628 11807 36913 6cic4 45647 oS£67 94239 35574 75^54 95216 

 30327 00522 107^.- czi^o --5C15 56CC6 12861 85526 63325 73186 

 92806 64389 68c6i S9528 4C5S2 59311 24251 61329 73139 93397 

 11323 35378 21796 08417 66483 10525 47303 96657 25650 48887 

 81553 C9384 29057 93116 95934 19285 18063 64919 69751 94017 

 08560 94952 73686 73738 50840 08123 67856 15800 93298 22514 

 02324 66755 49211 02670 45743 78815 47483 9^799 7 



