432 



On the Numerical Value of Euler^s Constant. [xipril 11^, 



E = -57721 56649 01532 86060 65120 90082 40243 10421 

 59335 93995 35988 05771 53865 48677 (last term 



~28.500^V' 



When w = 1 000, we have 



+ .^j_^=7-48547 08605 50344 91265 65182 04333 90017 65216 



79169 70883 36657 73626 74995 76993 49165 20244 

 09599 34437 41184 50813+ 



E= -57721 56649 01532 86060 65120 90082 40243 10421 



59335 93995 35988 05772 02455 61942 00508 15825 



last term 



50.1000^ 



Hence we see that 54 decimal places are correct in the value of E (n being 

 200) last given in the paper dated March 2, 1867, — also that 59 decimals 

 are correct in the value of E when w=500. When n= 1000, probably 65 

 decimals in the value of E are correct. 



When n=\, we readily find E='57 ^last term 



57721 (last term — 



= 2, 



5, 



» E = 



•57721 



56649 015 l^last term + 



When 



n= 1, E consists of 2 decimals. 



i) 



= 10, 



j> 



21 decimals. 



>} 



= 100, 



JJ 



46 decimals. 



JJ 



= 1000, 



JJ 



65 decimals, probably. 



f» 



71= 2, 



>j 



5 decimals. 



i> 



20, 



»j 



28 decimals. 



)> 



200, 



JJ 



54 decimals. 



j> 



n= 5, 



JJ 



13 decimals. 



3» 



50, 



JJ 



39 decimals. 



JJ 



500, 



JJ 



59 decimals. 



From the above we may fairly infer that when n is increased in a geome- 

 trical ratio, the corresponding number of decimals obtained in the value 

 of E increases only in something like an arithmetical one, and that pro- 

 bably from 50,000 to 100,000 terms in the Harmonic Progression would 

 require to be summed in order to obtain 100 places of decimals in the value 

 of E, Euler's constant. 



