187L] 



Calculation of Euler's Constant. 



521 



Mr. Shanks computed log 2 and log 5 from formulae of the form 

 Log 2 = 2 (7 P+5 Q + 3 R), 

 Log5 = 2(l6P-f-12Q+7R), 



P, Q, and R being infinite series (Proc. Roy. Soc. vol. yi. p. 397; ' Rectifica- 

 tion of the Circle,' p. 88). 



If, therefore, any error was made in the calculation of P say, it would 

 produce errors in log 2 and log 5 proportional to 7 and 16. On trial it 

 was found that sixteen times (E) was equal to seven times (F), the difference 

 being only such as an error of a unit in the 100th decimal of (E) or (F) would 

 produce. This afforded a moral proof that Mr. Shanks had made an 

 error equal to one-fourteenth of (E) in the calculation of P, or 



_1 _L_ 1 



31 + 3.31 8 + 5.3'1 5 Jf * 



which has rendered his values of log 2 and log 5 incorrect, and that (with 

 the exception of the error previously noticed) the harmonic series was 

 summed correctly. 



Since log 3 was calculated from the formula 



Log 3 = 2 (11 P + 8 Q + 5 R), 



its value is also incorrect, as also is that of log 10 (log 2 + log 5) and the 

 modulus (the reciprocal of log 10). 



The values of all these quantities are given to 205 decimal places in 

 vol. vi. Proceedings of the Royal Society, p. 397; but all the figures after 

 the 59th decimal place are incorrect in each case. 



The correct values to 100 places are : — 



Log 2-. 



= •69314 



71805 



59945 



30941 



72321 





21458 



17656 



80755 



00134 



36025 





52541 



20680 



00949 



33936 



21969 





G9471 



56058 



63326 



99641 



86875 



Log 3 = 



; 1-09861 



22886 



68109 



69139 



52452 





36922 



52570 



46474 



90557 



82274 





94517 



34694 



33363 



74942 



93218 





60896 



68736 



15754 



81373 



20888 



Log 5 = 



160943 



79124 



341C0 



37460 



07593 





33226 



18763 





01354 



26851 





77219 



12647 



89147 



41789 



87707 





65776 



46301 



33878 



09317 



96108 



Log 10 = 



: 2-30258 



50929 



94045 



68401 



79914 





54684 



36420 



76011 



01488 



62877 





29760 



33327 



90096 



75726 



09677 





35248 



02359 



97205 



08959 



S2Q&2 



It is to be observed that the above value of log 3 is not quite as well 

 determined as the others ; the calculations in regard to Euler's constant 



