1881.] as a Means of Calculating Logarithms, Sfc. 407 



exceptions, which were taken from Callet. The process is essentially 

 the same as Weddle's and Hearn's, but was discovered independently. 



1876. * Gray j Peter. " Tables, &c, to Twenty-four Places, with ex- 

 planatory introduction and historical preface." See above, 1865. The 

 tables are an extended positive numerical radix, containing Vr, I'O^r 

 from r=001 to ?-=999, and m=l to ra=4, and by inference to m—7, 

 and twenty-four places of decimals. The tables were calculated to 

 twenty-seven places, and verified to twenty-four by laborious processes 

 fully described, bat as far as possible, Abraham Sharpe's and 

 Wolframm's tables were employed. The process is that of 1848, 

 adapted to periods of three digits. Mr. Thomas Warner, who assisted 

 Mr. Gray to publish these tables, showed how they might be applied 

 to Flower's rule in periods of three figures by means of *Crelle's 

 Rechentafelu for the multiplication by three digits. This is the latest 

 simplification of Flower's rule. I have been much indebted to 

 Mr. Gray's historical preface, and to the loans of papers and books 

 from him in the compilation of this list, but I have personally 

 examined every process I have described. 



1876. fHoppe, Professor Dr. Reinhold. " Tafeln zur dreissigstellig- 

 en logarithmischen Rechnung," Leipzig. For natural logarithms and 

 anti-logarithms to thirty places of decimals, the tables giving thirty- 

 three places, independently calculated and verified. This is a most 

 ingenious transformation of the positive numerical radix effected by 

 subtracting tbe logarithmic series from its first term, so that instead 

 of placing nat. log l'0 m r against the number l'0 m r in the radix, 

 Professor Hoppe places '0 m r— nat. log V0 m r against it. This trans- 

 formed radix is calculated from r=l to r=9, and m=l to m=\h to 

 thirty-three places. The calculation is consequently an alteration of 

 Flower's reflected rule, adapted to natural logarithms, by which many 

 figures are saved. It is probably, therefore, the shortest rule yet dis- 

 covered. A reversed process gives the number, Table IY gives a 

 multiplier of at most two digits, or a divisor of at most one digit, by 

 which any number can be reduced to the form - 9 . . ., which over- 

 comes the principal difficulty in the use of Flower's reflected rule. 



1877. Namur, A. " Tables de Logarithmes a 12 Decimales jusqu'a 

 434 milliards, avec preuves," Brussels. This is for tabular logarithms 

 only, and depends upon the properties of logarithms nearly equal to 

 the modulus, to which all others are reduced by appropriate factors. 

 After this reduction the work is simple, no division being required, 

 but I find the tables complicated, and very likely to produce error in 

 consultation. The process is adapted only to tabular logarithms. 



From this list it will be clear that the improved bimodular method 

 of my former paper, and the potential radix which follows, have not 

 been previously proposed. 



VOL. XXXI. 



