350 Prof. Young on the Summation 



we shall presently see, may be easily narrowed. By consi- 

 dering die series to terminate at the former of the above terms, 

 it will be the same as that investigated by the late Mr. Baron 

 Maseres, in his Scriptores Logaritkmici, vol. iii. p. 219, and 

 which the Baron considered to be different from that deduced 

 by Simpson, from a formula still more general, in his Mathe- 

 matical Dissertations, p. 62. The two forms are, however, 

 mutually interchangeable, regard being paid to the signs of 

 the differences. 



Suppose in the series S that a = ; then, dividing both 

 sides of (A.) by — .r, we shall have 



S = b — ex + dx^ — ex^ -{■ &c. = 

 A^ A^^^ A" x^ 



n+l ^,n + l 



-&c (B.) 



which when x ■= \ becomes 



S = Z> — c + (/— e+&c. = 



2 ^ 4 ^ 8 ^^^ + 2"+i ••• ^ ^ 



A" 

 This series, supposing it to terminate at — ri, furnishes the 



method proposed by Dr. Hutton (Mathemat. Tracts, vol. i. 

 p. 176). lii Dr.Hutton's process the value of S is approached 

 by determining in succession the values of 



h b ^ A ^ , A _^ A^ . 



which is done by finding first the successive sums of a few 

 of the leading terms in the proposed series, taking the arith- 

 metical means between each pair of consecutive results, then 

 the means between each pair of these means, and so on. But 

 simple as this operation undoubtedly is, yet, when the num- 

 bers consist of several places of figures, it cannot safely be 

 performed mentally; so that Dr. Button's numerical examples, 

 which exhibit only the results of these operations, do not fairly 

 present the whole amount of numerical labour. Still his me- 

 thod is, upon the whole, more easy and convenient than any 

 which 1 have elsewhere seen, although, like all the other me- 

 thods, it leaves us in doubt about the accuracy of the final 

 decimal or two in the results determined by it, as indeed, 

 every process of approximation must do which does not fur- 

 nish limits both above and below the value sought. 



