1881.] as a Means of Calculating Logarithms, Src. 401 



sum (in this case 2) would amount to multiplying the significant 

 figures by the modulus only, which is Thoman's rule. In that case 

 there would be no correction. The completion consisting of adding 

 the logarithms of the divisors and subtracting those of the multipliers, 

 is the usual one, but as the negative radix gives —tab. log (1 — "0 m r) 

 direct, there is no occasion for using arithmetical complements. We 

 might, also, have continued the process till all the decimal places 

 were zero, and then have made the whole work one of completion. 

 The last half of the process need not be gone through, as the mul- 

 tipliers in the negative radix can be taken from it at sight. This is 

 the process of Mr. Weddle, the inventor of the negative radix. 



The number may be recovered from the logarithms in various ways 

 from the positive radix, and, among others, by my improved bi- 

 modular method, or in Mr. Weddle's method, from the negative radix. 

 Hence the problem is reduced to finding a simple way of calculating 

 the positive and negative radixes. Before explaining the method 

 proposed in this paper, however, it will be best to prefix a chrono- 

 logical summary of the methods actually proposed for calculating 

 logarithms with or without a radix, and with or without an annexed 

 indication of the means employed for calculating the radix.* 



Chronological Summary of Methods. 



1624. *Briggs, H. "Arithmetica Logarithmica," p. 32, contains 

 the first positive numerical radix, under the name of § " Tabella 

 inventioni Logarithmorum inserviens," giving r, l"r, l'0 m r from r=l 

 to r=9, and m=l to m=8 with their tabular logarithms to fifteen 

 places of decimals. The fifteenth place is often more than one unit 

 wrong, and two other errors occur, namely, tab. log 4= '60205 9991 

 .... for -60205 9990 . . ., corrected in the chiliads, and tab. log 

 r0 3 5 = -0 3 21 700 .. . for -0 3 21 709 . . ., which last error is re- 

 produced in the chiliads. Briggs does not explain how he calculated 

 this table. He uses it to interpolate in his chiliads, and finds the 

 logarithm by means of a series of divisions with a continually 

 augmented divisor, which is in fact the product of the successive 

 factors into which the number is gradually resolved, but he does not 

 explain this contrivance. He finds the number corresponding to the 

 logarithm by subtracting the next less logarithms successively and 

 multiplying by the corresponding numbers, a method generally 

 adopted. 



* The works marked * are in the library of the Royal Society, the originals of 

 those marked f> an< A transcripts of the whole of the necessary portions of those 

 marked §, were given to the Royal Society by the author of this paper when it was 

 read, so as to put a tolerably complete collection of all the papers bearing upon the 

 subject in the possession of the Royal Society. 



