1881.] as a Means of Calculating Logarithms, fyc. 409 



The first pair would give a potential radix determining logarithms 

 to at least twelve decimal places without correction. The second 

 pair would give one determining logarithm with at least forty-two 

 places correct, and generally many more. 



Having found nat. log (1 + '0,»1) for the extreme value of m, proceed 

 thus: — Form (l + (h,„l) r up to r=10, and the corresponding loga- 

 rithms. Both operations are performed by simple addition or sub- 

 traction. Then find nat. log (l + "0 m _ 1 l) either direct from the 

 series or by the improved bimodular method from the next least 

 (l + - r/i l) 9 and next greater (l + 'O^l) 10 , of which the latter will give 

 more places, or by both methods, to check the work. Then find the 

 numerical radix for the stage 1 + '0 m r from the potential radix to this 

 extent. Next proceed with the potential radix for the stage 

 (l + '0 m _il) r whence derive the numerical radix for this stage, and so 

 on till we obtain (1*1 ) 8 , which is just a little larger than 2, and from 

 which nat. log of 2 and its powers may be found, which will include 

 any number, however great. 



In order to make this clear, I give a short positive and negative 

 jDotential radix to twenty-one places of decimals up to 1"0 2 1, which 

 with corrections (obtained from a table of cubes, like Barlow's, of 

 numbers of four digits, from the formula 12c=x$ where x is the quotient 

 to four significant places, and c is calculated also to four significant 

 places) will give fourteen decimal places at least, and sometimes 

 more. I then show the mode of calculating the numerical radix 

 from it. It must be remembered that the nat. log. l'OOl is approxi- 

 mate, the digits after the eighteenth are 16681 in place of 167. 

 Hence the tenth multiple will not have after the eighteenth place 

 670, but 668, and similarly in other cases. The three last places are 

 given to avoid such errors and make eighteen places perfectly correct. 



Here we may suppose that nat. logs of 1'0 2 1, l'Ol, 1*1, and 1 — -0. 3 1, 

 1 — "01, have been calculated directly from the formula. Then the 

 powers of these numbers, and the multiples of their logarithms are 

 obtained by simple addition and subtraction, and the potential radix 

 is constituted except as regards 2 and 10. The calculation of nat. 

 logs of 2, 3, 5, 7, 10 and 1 -4- nat. log 10 to upwards of 260 decimal 

 places, by independent methods, by Professor J. C. Adams (" Pro- 

 ceedings," vol. 27, p. 92, for 7th February, 1878) obviates any necessity 

 for the separate calculation of them by the present or any other 

 method, but they could be calculated by this method, if it were 

 necessary. 



The next step is to calculate the numerical radix, or nat. log 

 (l + '0 ?n ?*). Take for example 1*004, which is very slightly less than 

 (l"001) 4 of which the nat. log is known. Then my improved bimo- 

 dular method, suppressing the details of the division, gives — 



2 g 2 



