1887.] Values of the Napierian Logarithms, Sfc. 23 



than was there claimed for them, jet I was not entirely satisfied with 

 the resnlt, since the calculation of the fundamental quantities had 

 been carried to 269 places of decimals, and therefore the above-cited 

 equation of condition showed that some errors, which I had not suc- 

 ceeded in tracing, had crept into the calculations so as to vitiate the 

 results beyond the 263rd place of decimals. 



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 uncertainty ought not to affect more than two or three of 

 the last figures. 



The Napierian logarithm of 10 is equal to 23a — 66 + 10c, 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 10 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 

 ' Proceedings ' above referred to exactly as they resulted from the cal- 

 culations, 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 un- 

 accompanied 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 

 contributed to the new edition of the ' Encyclopaedia 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. 



I have now succeeded in tracing and correcting the errors which 

 vitiated the later decimals in my former calculations, and have 

 extended the computations to a few more decimal places. The com- 



