282 MR. RUTHERFORD ON THE COMPUTATION OF THE RATIO OF THE DIAMETER 



to combine the respective terms of the three series, or to combine only the terms of the 

 two series arising from tan""^ Sq and tan~^ g^, computing the value of 4 tan~^ y se- 

 parately. The reciprocals of the powers of 5 being terminate decimals, and the results 

 of the several terms in the series for tan~^ -r- circulating in small periods, induced 



me to compute its value apart from that of the other two. 



It is unnecessary to give a lengthened description of the mode of computation, and 

 I shall only briefly state the method of constructing the auxiliary tables which I 

 employed. 



The first of these Tables marked (A), comprises the reciprocals of the successive 

 odd powers of 5, and is formed by dividing continuously by 25, or multiplying suc- 

 cessively by '04. Tables (B) and (C) contain the reciprocals of the successive powers 

 of 70 tind 99 respectively, the former being obtained by continuous divisions by 70, 

 and the latter by dividing continuously by 99, by the very simple process already 

 adverted to. Tables (D) and (E) contain the values of the several terms of the first 

 series, viz. 



5 3 ' 5^ + 5 * 5^ 7*57+9*59 11 ' 511 "r 



the former table comprising the values of the positive terms, and the latter those of 

 the negative terms, both being derived from the subsidiary table (A). Table (F) is 

 formed in like manner from the subsidiary tables (B) and (C), Part I. comprising the 

 negative terms, and Part II. the positive terms of the series 



\7o "" 99/ T \W " 99"^ + T \W "~ 99^/ ~" T \W ~ 997/ + 



And for the readier verification of the summation of the several columns in Tables 

 (D), (E), (F), the sum of each column is written in full in a diagonal position, pre- 

 serving the local values of the several figures in each sum, from which by a second 

 summation the total sum is finally obtained. The excess of the sum of the positive 

 terms in Table (D) above that of the negative terms in Table (E) is then multiplied 



by 4 to obtain the value of 4 tan~^ -^, and the excess of the sum of the positive terms 

 in Table (F) above that of the negative terms gives the value of tan~^ ^^ — tan~^ ^. 

 This value is then transferred to Table (D), and subtracted from that of 4 tan"^ -r-? 



which gives the value of ^, and thence, finally, by multiplying by 4, the value of t, 



or the ratio of the diameter of a circle to the circumference to 208 places of decimals. 



In conclusion I have only to remark, that the computations have been very carefully 



conducted, and that almost every part of the work has been verified by myself or the 



