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LXIII. On Fractions for the Value o/M. 

 By R. Templeton, Esq* 



HAVING had occasion to make use of the modulus of com- 

 mon logarithms carried to a considerable number of figures, 

 I attempted the verification of the figures given in Hutton's 

 Tables (Introduction, Hyperbolic Logarithms) by the methods to 

 be met with in ordinary works, but soon gave it up, embarrassed 

 by the endless rows of figures which the modes of procedure 

 required. After a little consideration I was led to have recourse 

 to an expedient which had already been resorted tof for avoid- 

 ing this irksome calculation — that of deducing the logarithm of 

 10 from the logarithms of a moderate number of small fractions, 

 the denominators of which differ by unity from the numerators, 

 and are composed of a sufficient number of figures to ensure rapid 

 convergence, and, if practicable, of digits easily worked with. 

 The following consideration indicates the mode by which these 

 small fractions may be found. If by any process the denomi- 

 nator of a fraction can be doubled, the series into which that 

 fraction is introduced will converge with more than twice the 

 rapidity, since the higher powers will be proportionally of larger 



X-\- 1 



dimensions. Now the square of the fraction will be, when 



X 



x exceeds unity, very nearly equal to g — or . Let y be 



the small fraction required to restore equality, then 



whence 

 and 



_ x* + 2x 

 y ~x 2 + 2x + l' 



x + 2 /ar+.lY # a + 2a? 



\ x ) x 2 -i 



+ 2.27+1 



As an example, let x = 48; then 



48 + 2 _ 25 _ /49V 2400 

 48 ~24 \48/ 2401* 



We have here the denominator of the fraction || doubled, thus 

 making the convergence much more rapid at the expense of the 

 labour imposed by the introduction of another fraction, of, how- 



* Communicated by the Author. 



t Vide paper by the Rev. John Hellins in the Phil. Trans. 1/96, p. 135. 



