12 REPORT 1873. 



formula, tt — IG tan ~i g-— 4 tan "i ^i^. The terms in the expansion both of 

 tan ~i i and tan ~i ^iy are given separately to 530 places. The former 



occupy 60 pp. and extend to ^ ^_r-^, ; and the latter cccujiy 24 pp. and cx- 



teiid to Qio.osn^ifi ' While the Tvork was passing through the press Mr, 



Shanks extended his value of tt to 607 decimals ; and to this number of 

 places it is given on pp. 86 and 87 of the book. 



[T. II.] (pp. 90-!)5) gives every twelfth power of 2 (viz. 2", 2=% &c.) as far 

 as 2^^' (which contains 212 figures). 



On p. 89 are given the values of <;, log, 2, log^3, log, 5, and log, 10, to 137 

 places, and the modulus to 130. Values of these quantities were given also 

 by Mr. Shanks to 205 places (I'roc. Roy. Soc. vol. vi. p. 397). The value of e 

 was verified by the reporter to 137 places by calculation from a continued 

 fraction (see Erit. Assoc, lleport, 1871, pp. 16-18, sectional proceedings). 

 The same writer also showed in vol. xix. p. 521 of the ' Proceedings of the 

 lloyal Society,' that Mr. Shanks's values of log 2, 3, 5, and 10 were inaccurate 

 after the 59th place (all owing to one error in a series on which they depended), 

 and deduced the correct values to 100 places. These results were verified by 

 Mr. Shanks, who has recalculated the values of these logarithms, as well as 

 that of the modulus, to 205 places : they are published in vol. xx. p. 27 of 

 the 'Proceedings of the Royal Society' (1871). 



Mr. Shanks's 607-place value is given in Knight's 'English Cyclojjsedia,' 

 (Art. "Quadrature of the Circle") copied from the work under notice ; and it 

 has been verified by a subsequent calculation of llicliter to 500 places. A 

 list of the calculators of tt, the number of places, &c. to which they have 

 extended their calculations, with references to the places where they aro 

 to be found, is given by Bierens de Haan on a page at the beginning of his 

 " Tables d'lntegrales Definies " in t. iv. of the Amsterdam Transactions. 

 This page, however, does not appear in the separate copies of the tables 

 (the ' Nouvelles Tables,' Leyden, 1867). For an extended and corrected copy 

 of this list, see ' Messenger of Mathematics,' December 1872, and some addi- 

 tional corrections in the same Journal for July 1873 (t. iii. pp. 45, 46). 



Some years ago Mr. Slianks calculated the reciprocal of the prime number 

 17389 so as to exhibit the complete circulating period, consisting of 17388 

 figures, and placed a copy of it in the Archives of the Royal Society. Quite 

 recently he has extended his calculation of w to 707 decimal places (Proc. 

 Roy. Soc. vol. xxi. p. 318). Mr. Shanks has sent us three corrections to this 

 paper : viz. the 459th, 460th, and 461st decimals in n should be 962 instead 

 of 834, and the 513th, 514th, and 515th decimals should be 065 instead of 

 193; also the 75th decimal of tan "'4- should be 8 instead of 7. The two 

 corrections in tt applj' also to the work under notice. 



Sharp, 1717. [T. I.] (p. 40). The first hundred multiples of |t, to 21 places. 



[T. II.] jb-eas of segments of circles. The area of the whole circle is 

 taken as unity ; and the argument is the vei-sed sine (or height of the 

 segment), the diameter being taken as unity. The table then gives areas to 

 17 places for arguments -0001 to -5000 at intervals of -0001, with difierences. 

 Thus, strictly, the argument is the ratio of the height of the segment to the 

 diameter, and the tabular result the ratio of the area of the segment to that 

 of the whole circle. The table occupies 50 pp., and is the largest of the kind 

 we have seen. 



[T. III.] Tahle for computing the solidifi/ of the xipright hiipcrhoVic section 

 of a cone, viz. for facilitating the calculation of the volumes of segments of 



