83 REPORT — 1873. 



A very good collection is given by Matnaed (described below), and 

 also by Byrne, 1849. This portion of the present Eeport is very far from 

 complete, as the values of mathematical constants have, as a rule, appeared in 

 periodical publications, while those only that are most used by the general 

 computer are to be found in collections of mathematical tables. We refrain, 

 therefore, from giving references to several periodicals we have met with 

 containing constants, as they belong properly to a subsequent portion of the 

 Report ; and it is hoped that, after the completion of the examination of 

 the memoirs, a pretty complete list, either of the constants themselves, or at 

 all events of the places where they are to be found, will be given. 



We may, however, notice a paper of Paucker's in the first volume of 

 ' Grunert's Archiv der Mathematik und Physik,' in which a number of 

 constants involving tt are given to a great many places, and Gauss's 

 memoirs on the lemniscate-functions (' Werke,' t. iii. pp. 426 &c.), where 



e~^, e~^^, e~^'^, &c. are calculated to about fifty places. On Euler's con- 

 stant, see ' Proceedings of the Eoyal Society,' t. xv. p. 429 ; t. xvi. pp. 154, 

 299 ; t. xviii. p. 49 (Shanks) ; t. xix. p. 514 (Glaisher) ; t. xx. pp. 27, 29 

 (Shanks). On e, the base of the Napierian logarithms, log^2, log^S &c., see, 

 besides the places just referred to, ' Poy. Soc. Proc' t. ■vi. p. 397, and ' Brit. 

 Assoc. Report' (Sections) 1871, p. 16, and also Shanks 1853 (§ 4). Several 

 constants are to be found in the different works of Maseres. Mr. Maynard 

 and Mr. Merrifield have independently calculated log^M and log^??i (M and m 

 being the modulus and its reciprocal) to 30 places (' Assurance Magazine,' 

 t. vi. p. 298). 



The value of it has been calculated to 500 places of decimals by Shanks 

 and Richter independently, and to 707 places by the former alone : see 

 references, ' Messenger of Mathematics,' December 1872 and July 1873. Mr. 

 Shanks's latest value appears in the ' Roy. Soc. Proc' t. xxi. p. 319. 

 It is proper here to remark that Rutherford's 208-decimal value of tt, given 

 in the ' Phil. Trans.' 1841, p. 283, is erroneous after the 152nd place : this 

 value is reproduced in BniNE, 1849 (§ 4), and in Matnakd; so that it is 

 erroneous also in both of these works. 



[Maynard.] A good table of constants involving tt, such as tt V2, tt "^ 

 Vtt, &c., and some few involving e &c., to a great many (generally 30) 

 places. There are also other constants not included in the subjects of this 

 Report. 



The copy of these constants that we examined consisted of six leaves 

 without a cover, and which were evidently extracted from some work. Mr^ 

 C. W. Merrifield, E.R.S., subsequently called our attention to the particil- 

 larly good collection of constants in ' The Millwright and Engineers' Pocket 

 Companion ; . . . . By William Templeton .... Corrected by Samuel May- 

 nard. . . . Eifteenth edition, carefully revised,' London, 1871, 8vo, and lent 

 us a copy ; and on examination it appeared that it was to this work that 

 Maynard's collection belonged, where it occupies pp. 169-180. There are, 

 altogether, 58 constants involving tt, and their logarithms, given generally to 

 30 places, and 13 others that may also be properly styled mathematical. It 

 is mentioned that part of the table had previously appeared in Keith's 

 ' Measurer ' (twenty-fourth edition, 1846). Templeton's work contains several 

 other tables (areas of circles, &c.), and square roots which would have been 

 included in this Report had we seen the book earlier ; as it is they will be 

 noticed in the Supplement. On Rutherford's value of tt, quoted by May- 

 nard, see introductory remarks to this article. 



