398 



Mr. A. J. Ellis. On the Potential Radix [Feb. 3, 



it follows that the number of decimal places in the quotient k + l, will 

 be two less than in the given logarithm. Moreover, as the last decimal 

 place is always approximate, it follows that the number e cannot be 

 found with certainty to more than three decimal places less than the 

 number of decimal places in the given tab. logarithm. Hence, in the 

 present case, although tab. log e is known to eighteen places of deci- 

 mals, e is known with certainty only to fifteen places of decimals (and 

 sixteen digits). But the error in the next place (or digit) will not 

 probably exceed one unit. 



Having found e, we have to multiply it in succession by the num- 

 bers corresponding to the logarithms subtracted in the preparations 

 in this example, 1/014, 1*9, and 10 11 . This is most readily done in the 

 way sufficiently explained by the notes in the example. The resulting 

 number is accidentally correct to seventeen digits, but only sixteen 

 can be used with certainty. Hence, if we use this bimodular method 

 of finding logarithms and anti-logarithms, we should always find the 

 logarithms to two or three places of decimals more than we require 

 digits in the final number to be found. 



V. " On the Potential Radix as a Means of Calculating Loga- 

 rithms to any Required Number of Decimal Places, with a 

 Summary of all Preceding Methods Chronologically Ar- 

 ranged.*' By Alexander J. Ellis, B.A.. F.R.S., F.S.A. 

 Received January 17, 1881. 



In the tables attached to my paper " On an Improved Bimodular 

 Method of Computing Logarithms, &c." (" Proc. Roy. Soc," vol. 31, 

 p. 381), the logarithms used were all taken direct, or immediately 

 calculated, from the tables of Wolframm and Gray. But a complete 

 method of calculating logarithms should be independent of extraneous 

 aid and be applicable to the first construction of tables of logarithms. 

 I shall here show that my improved bimodular method is capable of 

 furnishing a practical means of calculating natural logarithms, and 

 hence logarithms to any base and to any number of places of 

 decimals. 



By the term positive numerical radix I shall understand a table of 

 the numbers r, l"r, 1«+ '0 m r, with their corresponding natural loga- 

 rithms, where r varies from 1 to 9, m means a series of m zeroes, and 

 m varies from 1 to any required number. The word Radix in this 

 sense is adopted from R. Flower, 1771, mentioned below. By the 

 term negative numerical radix I mean a similar table of 1 — 'Q m r, and 

 the negatives of their corresponding logarithms. When these radixes 

 (forming an English plural, as radices would be misleading) have been 



