1881.] Mr. A. J. Ellis. On the Potential Radix. 379 



the equivalent of Hoppe's Table II, and (with the substitution of natural 

 for tabular logarithms) of Thoman's Table I, col. last. Atwood's 

 first case is therefore precisely the same as Hoppe's method, having 

 identical tables but a different arrangement of computations and 

 notation. 



Taking the second case, where the reduced number is greater than 1, 

 Atwood multiplies it " down to 1." The process of finding the 

 factors is precisely that of Weddle, 1845; Hearn, 1847; Thoman, 

 1867 ; and Wace, 1873 ; exemplified in my former paper (vol. 31, 

 p. 400). The product of the factors (l-'04), (l--0 3 6), (l-'0 4 l), is 

 mm + 



written 1*04061, for which might be substituted 1 — '04061, where 

 — "04061 is the sum of the first terms of the natural logarithms of the 

 factors, which in this case are all negative. Hence on calculating a 

 table of — '0 n m — nat. log (l — '0 n m), which is easily done by the 

 first case, we find the corrections. These are given in Atwood's 

 Table II, which has not been given by any subsequent writer, because 

 Hearn and Thoman, using tabular logarithms, gave the complete 

 value of the tab. log of the factors, instead of corrections for the 

 first term. 



Hence it appears that At wood, 1786, rediscovered Flower's method, 

 1771, but transformed it in the manner carried out ninety years later 

 by Hoppe, 1876; and not only anticipated Weddle's method, 1845, 

 but showed the connexion of the two methods, as that of multiplying 

 the reduced number " up to 1 " in the first case, and " down to 1 " in 

 the second. To Atwood, then, must be attributed the real discovery 

 of these methods, their correlation, and their subordination to a 

 general principle of an " arithmetic of factors," and the merit of 

 having calculated his tables, as he says (p. 36), "without reference to 

 the hyperbola, binomial theorem, or any fluxional process, by multi- 

 plication only, the elementary properties of ratios and of powers and 

 their indices being granted." 



Eeeata. — The following slight errors escaped correction in my former papers : 

 vol. 31, p. 397, 1. l,for The preparation read In Ex. 2 the preparation ; same page, 

 1. 18, for Here read In Ex. 3 ; p. 398, 1. 15 from bottom, for Wolframm read 

 Wolfram ; p. 399, 1. 6, read paper, by the positive numerical radix, and dele T. 9 ; 

 p. 405, 1. 9 from bottom, read (supra, p. 382) ; p. 406, 1. 6, read *Gray ; ibid., 1. 19 

 from bottom, read fPineto; ibid., 1. 17 from bottom, read Petersburg; and 1. 9 

 from bottom, read Thesaurus, and p. 409, 1. last, read gives (on p. 412). 



VOL. XXXII. 



