THE COMPUTATION OF LOGARITHMS, &c. 221 



Yet, even in the very unfavourable case in which a; = 1, and the arc = 45", we 

 should have, by series (9), 



4^ = 2 + 3(2) +3:5(2) +^^-' 



less than twenty terms of which would give the circumference true for six places 

 of decimals ; while many thousand terms of the series, 



derived from (12), would be required to effect the same object. 



In the actual computation, however, of the circumference to a great degree 

 of accuracy, the series found above is applied with most advantage in connexion 

 with the curious and elegant principle first employed by Machin, and afterwards 

 extended by Euler, — that of finding arcs whose tangents are rational, and are 

 smaU known fractions, and the sum or difference of which arcs, or of their mul- 

 tiples, is a known part of the circumference. Such arcs are innumerable ; and, 

 by taking them sufficiently small, any degree of convergence whatever may be ob- 

 tained. Rapidity of convergence, however, is far from being the sole important 

 consideration. The convergence may be very great, and yet the fraction k may be 

 of such a form as to render the computation laborious and difficult. No arc, in- 



10™ 

 deed, answers well, unless p^ + q^ be of the form — ^, m and n being whole positive 



numbers ; and even of arcs having this property, many are, in other respects, in- 

 convenient. Of a great number of tangents which I have tried, those which seem 



121 3 



to answer best are 3, yy, y, and ^ ; which give respectively for the values of ;?:, 0.1, 



1 2 



0.032, 0.02, and 0.00144: and, since it is easy to shew that[3 tan~^ 3 — tan"' yy = 1 % 



we get, by quadrupling, 



«• = 12tan-ii — 4 tan-i A (13) 



3 II 



In a similar manner, it would appear that 



5r= 8tan-ii +4tan-ii (14» 



3^7 



ir= 10 tan-'i _ 2tan-iA (15) 



, 3 79 



5r = 8 tan-1 A-|-I2tan-il (16) 



11 7 



T = 20 tan-i-?-— I2tan-iA (17) 



II 79 ^ 



!r = 20tan-ii + 8tan-iA (18) 



