1881.] 



as a Means of Calculating Logarithms, fyc. 



405 



1848. §Gray, Peter. "A Table for the Easy Formation of Anti- 

 Logarithms, with its Application to the Converse Problem of the 

 Formation of Logarithms," in the " Mechanics' Magazine " for the 12th 

 and 26th February, 1848. This was founded on Hearn's paper, whence 

 Mr. Gray obtained his first knowledge of a positive numerical radix, 

 never having seen Briggs's or Flower's, and it contained the first of his 

 enlarged positive numerical radixes for twelve decimal places, containing 

 log r from r=l to r=9, log Vr, and logl , 2 m?", from m=l to m=5, 

 and r=01 to r=99. This he used only as an anti-logarithm process, 

 proposing, for the discovery of logarithms, a continually augmenting 

 divisor, which was, like Leonelli's, an independent discovery of 

 Briggs's method, proceeding, however, by periods of two places instead 

 of one. 



1848. ^Orchard, W., in the "Mechanics' Magazine" for 26th 

 February, 1848, referring to Hearn's positive arithmetical radix, 

 showed how it might be applied to finding the logarithm by a process 

 amounting in fact, to an independent re-discovery of Flower's reflected 

 rule, using, however, Mr. Gray's tables of the 12th February, 1848, 

 just mentioned. He also suggested another method derived from 

 Manning's, by using factors of the form 1 + "0 OT 1, which would amount 

 to an anticipation of my potential positive radix described below, but 

 it was differently conceived, and was worked out by the binomial 

 theorem. 



1849. f Byrne, Oliver. " Practical, Short, and Direct Method of 

 Calculating the Logarithm of any Given Number, and the Number 

 corresponding to any Given Logarithm," London (Appleton), 1849. 

 This is an independent method. Mr. Byrne finds ten numbers between 

 1 and 10 10 , the tab. logarithms of which, including the index, contain 

 the same digits as the numbers themselves, to sixteen digits (except 

 one which holds only for fourteen digits). Then, taking these as 

 constants, he multiplies any number up to one of these numbers, by 

 successive powers of 1*0»1, using binomial coefficients, and subtracts 

 the tab. logs of these powers from the tab. log of the constant. He 

 finds the number from the logarithm by a similar process. 



1851. fKoralek, Philippe. "Methode Nouvelle pour calculer les 

 Logarithmes des Nombres," Paris. This is a bimodular method, de- 

 pending upon series (4) in my former paper (supra, p. 392). By a 

 series of multipliers, he reduces all numbers to others lying between 

 800 and 1000, for which the first term of the series gives him seven 

 places accurately, without corrections. He then calculates the suc- 

 ceeding terms of the series by a somewhat laborious process, and finds 

 logarithms to twenty-seven places. His process differs entirely from 

 mine, except in being originally bimodular. 



1865. f Steinhduser, A. " Kurze Hilfstafel zur bequemen Rechnung 

 fiinfzehnstelliger Logarithmen zu gegebenen Zahlen, und umgekehrt," 



