378 



Mr. A. J. Ellis. On the Potential Radix. [June 16, 



paper, or, as Atwood writes 1 . [njk or 1 . \_n\h) then l=A~M.pqrst, 

 to any required number of decimal places. Atwood works to 13 

 places to secure 12, and applies the result to extraction of roots, &c, 

 as well as to the computation of logarithms and anti-logarithms, 

 which can alone be noticed here. It is evident that when these factors 

 are once found — log A=log M + log p + log q+ . . . to any base, and 

 that p, q, &c, and log p, log q, &c, will be numbers and logarithms 

 in a positive or negative numerical radix (vol. 31, p. 398) according 

 &s p, q, &c, are of the form 1 + •()»& or 1— •()„&. 



If B=the first or two first figures of 10'*-"- A, then we may take 

 M=B-f-10*, where n is the number of places in the integer of A, or 

 — n is the number of zeroes between the decimal point and the first 

 significant digit. This determination of M is an anticipation of 

 the "reciproques approches " of Thoman, 1867, and Atwood's Table 

 IV of the values of B is equivalent to the two first columns of 

 Thoman's Table I, and of Hoppe's Table IV. The number is thus 

 reduced to the form *9 . . . or l'O . . . 



Taking the first case, where the reduced number is less than 1, 

 Atwood multiplies it "up to 1," which is a revival of Flower's 

 reflected rule, with which Atwood was clearly not acquainted. But 

 instead of retaining the reduced number in the form - 9 . . ., and 

 finding the factors l'O n m as Flower did, by taking m = the excess of 

 9 over the corresponding digit in the number, Atwood transforms it 

 into the form 1— '0 . . ., by subtracting each digit, except the last, 

 from 9, and the last from 10. This is an anticipation of Hoppe's 

 artifice, 1876, In this case the values of m in V0 n m are the succes- 

 sive digits of 1 — *0 . . . themselves. 



The product of the resulting factors, say, l - 04, 1'0 3 6, l'Oil (or in 

 Atwood's notation 1 . [1]4, 1 . [3] 6, 1 . [4]1), were not written as a pro- 

 duct but as 1*04061, the line underneath indicating that it is not a 

 decimal fraction, and the position of each digit indicating the number 

 of preceding zeroes in the corresponding factor. The figures under- 

 lined correspond precisely to Flower's "line of ratios." To find the 

 nat. log of this product F04061, Atwood observes that it is very 

 nearly '04061, which, however, is too large, and requires correction, 

 being in fact the sum of the first terms of the nat. logs of the factors. 

 This correction he finds by assuming that log l'Ogl^'Ogl to 13 

 places, and similarly for all smaller numbers, which determines the 

 kind of logarithm, and then he finds the logarithms of larger num- 

 bers in his radix by resolution into factors. Finally he tabulates only 

 the differences or '0 n m — nat. log (l + '0 M m) in his Table I, which is 

 thus identical with Hoppe's Table III to the first 13 places. By this 

 table and resolution into factors, Atwood calculates log B (where B 

 consists of two digits, and is between 1 and 10), and his Table III, 



