Royal Society. 213 



Jan. 20. — The following papers were read : — 



1. "On the Extension of the value of the ratio of the Circum- 

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

 F.R.A.S. 



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



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



here 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" ^ — , 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 j = tan ~ ^ q + tan~ ^ t 4- tan~ ^ ^ , the figures from 



the 153rd to the 200th diff^ered 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 TT to 250 places of decimals, both by Machin's formula 



TT ,1 _i 1 



-^=4 tan -5— tan 239. 



and by the formula 



~=2tan 'Y+tan 'y; 



and the author's result was shown to diflfer from the correct value 



by the periodic decimal '36. 



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

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

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

 upwards of 400 decimals. As Mr. Shanks had employed Machin's 

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

 Mr. Shanks resumed his calculations, and has not only verified the 

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

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



tan~^-^andtan~^^c^7r, as well as the value of tt, which are here 



O £iOP 



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. 



tan I .j^^^^ j.^^^8 ^^ggQ ^^g^^ 00^^^ 6^1^^ ^^029 34475 85103 78785 

 5 21015 17688 94024 10339 6997S 24378 57326 97828 03728 80441 

 12628 11807 36913 60104 45647 98867 94239 35574 75654 95216 

 30327 00522 10747 00156 45015 56006 12861 85526 63325 73186 

 92806 64389 68c6i 89528 40582 59311 24251 61329 73139 93397 

 11323 35378 21796 08417 66483 10525 47303 96657 2565048887 

 81553 09384 29057 93116 95934 19285 18063 64919 69751 94017 

 08560 94952 73686 73738 50840 08123 67856 15800 93298 22514 

 02324667554921102670457437881547483907997 



