I 



THE COMPUTATION OF LOGARITHMS, &c. 221 



Yet, even in the very unfavourable case in which x = 1, and the arc = 45°, we 

 should have, by series (9), 



1 I , 2/l\2 . 2.4 /1\» , . 



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



less than twenty tei-ms of which would give the circumference true for six places 

 of decimals ; whUe 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 appMed with most advantage in connexion 

 with the cm-ious and elegant principle first employed by Machin, and afterwards 

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

 small 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 ^^ -I- ^ be of the fonn -^, m and n being whole positive 



numbex-s ; 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, 7, and =5 ; which give respectively for the values of A, 0.1, 



auu, «mce Kx« easy .o«new wax., .ar-^ - "' ' 

 we get, by quadrupling, 



0.032, 0.02, and 0.00144: and, since it is easy to shew that|3 tan ' 3 — tan"' yy = ^ t. 



^ = 12tan-il — 4 tan-iA (13) 



3 11 



In a similar manner, it would appear that 



3-= 8tan-'i +4tan-»i (14) 



3 ^ 7 



^= lOtan-'i — 2tan-'i- (15) 



3 79 



«•= 8tan-ii--t-I2tan-'i (16) 



11 7 



T=20tan-il— 12tan-iA (17) 



11 79 



ir = 20tan-'l -I- 8tan-> — (18) 



7 79 



