■Jt 



"K 



THE COMPUTATION OF LOGARITHMS, &c. 223 



.„ , -1 24478 , „^ ^ _i 685601 

 = ''^" 873121 + ^'^" 69049993 ^^S) 



-1 3 „„ _i 685601 .„.. 



: 88 tan-- -20 tan ^—^-^3 (24) 



These all converge rapidly, and more especially (20), (21), (23), and (24). 

 Such, indeed, is the convergence in the last mentioned formula, that, in one of 

 the series arising from it, each term is less than a seven-hundredth part, and in 

 the other less than a ten-thousandth part, of the term preceding it ! The value 



of k, however, for the tangent g^Q^gggg , is inconvenient in practice, being the un- 

 manageable decimal 0.0000985763636735639552 : and though this inconvenience 

 might be obviated in a considerable degree by tabulating the multiples of k by 2, 

 3, &c., up to 9, and thus facilitating its multiplication by the preceding terms of 

 the series in employing formula (11), there is little doubt but the computation 

 would be found to be easier by means of some of the other formulae already 

 given. 



