412 



Mr. A. J. Ellis. On the Potential Radix, Sfc. [Feb. 8, 



1-004 



1 -004 006 004 001 



a 



b = (l '001) 4 , next greater 



b + a, divisor 



2(6 — <x), dividend 



2(b — a) -7- (& + «), quotient 



correction= T V X C0 5 59 80) 3 



log b — log a 



log b 



log a— log b — (log b — log a) 



2 -008 006 004 001 

 12 008 002 



•000 005 980 062 796 661 85 

 •0 1fi 17 82 



•000 005 980 062 796 679 67 

 •003 998 001 332 334 132 67 



•003 992 021 269 537 453 00 



The result is correct to the last or twentieth place. If we had 

 formed log a from the next less or (l'OOl) 3 , the difference between 

 the numbers would have been so large that the result would have been 

 correct to thirteen places only, and we should have required higher 

 stages in the radix to obtain twenty places. Hence the nearest 

 number should always be selected. 



To find— nat. log (1 — "004) = — nat. log *996, we should deduce it 

 from — nat. log. (1— '001) 4 , and as the difference in this case, which 

 is always the approximate quotient, and hence logarithm, is less than 

 •0 5 6, and T x ¥ x (•0 5 6) 3 ='0 16 18, we should obtain sixteen places without 

 correction, and four with correction, or twenty places in all. 



We thus proceed to form the whole of this stage of the numerical 

 radix, but we cannot obtain twenty places in all cases. Thus for 

 1-009, the difference from (1-001) 9 is -0 4 36084, the quotient is 

 •0 4 3508, and the correction = T V x (-0 4 3508) 3 =-0 14 3594, so that we 

 should obtain only eighteen places, that is to say, although the 

 potential radix is calculated to twenty-one places, it will not furnish 

 a numerical radix of more than eighteen places when we begin with 

 the stage 1'0 2 1, and hence will not give logarithms of general numbers 

 to more than sixteen places certain. 



In the stage l'Ol the radix of that stage will not furnish so many 

 places, and we have to reduce to the preceding stage, which is now 

 supposed to be fully calculated for both the positive and negative 

 numerical radixes. Thus for nat. log 1*04 as derived from 4 nat. log 

 1*01, the difference is ^36040, giving correction *0 10 1836, and hence 

 fourteen places. But on dividing by 1'04 we obtain 1"0 3 58 . . . and 

 the difference from 1'0 3 6, of which the natural logarithm in a pre- 

 ceding stage is known, is "0 4 2 . . ., the logarithm of which can be 

 found to eighteen places. If more still were required we should divide 

 by 1'0 3 5, obtaining 1"0 4 8 ... of which we can find the logarithm 

 through that of 1*0 4 8, to twenty places at least. Hence if the potential 

 radix has been commenced at a sufficiently high stage and to a sufficient 

 number of decimal places, a numerical radix for natural logarithms can 

 be calculated to any number of places, and from it the natural logarithm 



