338 NEW SERIES for the 



nite product, we get from it 



log* =(*-!) _ 



2 



- + 1 — + I ~r+ I — + I 



A 4 ,T 8 ^ ,3 " 



<?*/ infinitum. 



and the produd of any finite number of factors of this expref- 

 fion will always be lefs than log.r. 



These formulae, which are analogous to that given by Eu- 

 ler for an arch of a circle, (fee Art. $$.), are not inelegant, 

 confidered as analytical transformations. It does not feem, 

 however, that without fome analytical artifice, they can be ap- 

 plied with advantage to the actual calculation of logarithms, 

 by reafon of the great labour which would be necefTary to ob- 

 tain a refult tolerably accurate. 



66. I shall now conclude this Paper, with fome examples 

 of the application of the formulae to the computation of the 

 length of one-fourth the circumference of a circle whofe radius 

 is unity, (which is the extreme and the moll unfavourable 

 cafe), and to the computation of a logarithm ', as alfo of the 

 modulus of the common fyftem of logarithms, which is the re- 

 ciprocal of Napier's logarithm of io. 



Example 



