468 ON THE VALUES OF THE NAPIERIAN LOGARITHMS, ETC. [57 



Of course in working with such a large number of interminable decimals, 

 the necessary neglect of decimals of higher orders causes an uncertainty in 

 a few of the last decimal places, but when due care is taken, this uncer- 

 tainty ought not to affect more than two or three of the last figures. 



The Napierian logarithm of 10 is equal to 23a 6& + lOc, and the Modulus 

 of common logarithms is the reciprocal of this quantity. 



Since the value found for the logarithm of 10 cannot be depended 

 upon beyond 262 places of decimals, a corresponding uncertainty will affect 

 the value of the Modulus found from it. 



In the operation of dividing unity by the assumed value of log 10, 

 however, the quotient was carried to 282 places of decimals. 



This was done for the purpose of supplying the means of correcting 

 the value found for the Modulus, without the necessity of repeating the 

 division, when I should have succeeded in tracing the errors of calculation 

 alluded to above, and thus finding a value of log 1 which might be 

 depended upon to a larger number of decimal places. 



Through inadvertence, the values of the logarithms concerned, and the 

 resulting value of the Modulus, were printed in my paper in the Pro- 

 ceedings above referred to exactly as they resulted from the calculations, 

 without the suppression of the decimals of higher orders, which in the 

 case of the logarithms were uncertain, and in the case of the Modulus 

 were known to be incorrect. 



Although it was unlikely that this oversight would lead to any mis- 

 apprehension as to the degree of accuracy claimed for my results in the 

 mind of a reader of the paper itself, there might be a danger of such 

 misapprehension if my printed results were quoted in full unaccompanied 

 by the statement that the later decimal places were not to be depended on. 



My attention has been recalled to this subject by the circumstance 

 that in the excellent article on Logarithms which Mr Glaisher has con- 

 tributed to the new edition of the Encyclopedia Britannica, he has quoted 

 my value of the Modulus, and has given the whole of the 282 decimals 

 as printed in the Proceedings of the Royal Society, without expressly stating 

 that this value does not claim to be accurate beyond 262 or 263 places of 

 decimals. 



