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Mr. W. Shanks on the Calculation of 



[April 11, 



numerical value of Euler's constant, which is largely employed in " Infi- 

 nitesimal Calculus," to a greater extent than has hitherto been found, and 

 free from error. 



In Crelle's Journal for 1860, vol. Ix. p. 375, M. Oettinger has contri- 

 buted an article on Euler's constant, and especially on certain discre- 

 pancies" in the value given by former mathematical writers. Adopting 

 the formula there employed, as being well adapted for the purpose, the 

 writer of this paper has both corrected and extended what has been pre- 

 viously done ; and as very great care has been bestowed upon the calcu- 

 lations, so as to exclude error, he confidently believes that his results are, 

 as far as they go, absolutely correct. He may remark that, since the 

 separate values of n in the formula (which, see below) produce identical 

 results as far as they go, and the higher the value of n the more nearly 

 we can approximate to the value of the constant, we thus have sufficient 

 proof aiforded of the correctness of the value found when n is 10, 20, 50, 

 or 100. If the writer can command sufficient leisure, he may resume the 

 calculation by and by, and, making n 1000, he may thus verify, as well as 

 extend, the value of Euler's constant given in this paper. The numbers 

 10, 20, 50, 100, 200, and 1000, especially 10 and its integral powers, are 

 more easily handled than others, particularly in those terms of the formula 

 which contain Bernoulli's numbers. The harmonic progression is here 

 " summed" much further than was requisite for finding E to 50 or 55 de- 

 cimals ; but this was of some importance in ensuring correctness in the 

 decimal expression of each of the higher terms of S^^q and S^qq. It may 

 be observed that the numbers of decimal places in E, obtained from n being 

 10, 20, 50, 100, and 200, are nearly proportional to 10*, 20^ 50* 100*» 



and 200* — a rather curious coincidence. 



The formula for Euler's constant, employed by M. Oettinger, as above 

 stated, is — 



IB B B B 

 Constant = S?2— logsW-^-l-^^-^-f g^— ~-f . . . .&c., where 



Sn= l + i + J + ? + 4- + ..i, and B,, B„, B3, &c. are Bernoulli's numbers. 



n 



1+1 + ^-+... .-J^=2 -928968253 



l_l-i4.i_|....._V=3-59773 96571 43681 91148 37690 68908 38779 38367 

 10245 37897 60291 30827 89243 77995 58542 59259 

 83201 52499 71906 31865 55446 61736 61220 10084 

 85838 20720 04363 02603 48526 602. 



1-1-1+ 2V = 3-81595 81777 53506 86913 48136 76474 73449 06181 



89635 55401 83780 86220 32609 11027 77063 20242 

 32778 03544 32662 95332 52228 09675 62970 52433 

 81377 45056 57669 24455 54624 85197 30818 82894 

 77830 46585 03877 77968 87905 92007 71409 68243 

 4- (circulating period consists of 1584 decimal places). 



