-------------------------------------+
|1842 | Rutherford | 208 | 152 | _Trans. Roy. Soc._ (London, 1841), p. 283. |
|1844 | Dase | 205 | 200 | _Crelle's Journ._. xxvii. 198. |
|1847 | Clausen | 250 | 248 | _Astron. Nachr._ xxv. col. 207. |
|1853 | Shanks | 318 | 318 | _Proc. Roy. Soc._ (London, 1853), 273. |
|1853 | Rutherford | 440 | 440 | Ibid. |
|1853 | Shanks | 530 | .. | Ibid. |
|1853 | Shanks | 607 | .. | W. Shanks, _Rectification of the Circle_ |
| | | | | (London, 1853). |
|1853 | Richter | 333 | 330 | _Grunert's Archiv_, xxi. 119. |
|1854 | Richter | 400 | 330 | Ibid. xxii. 473. |
|1854 | Richter | 400 | 400 | Ibid. xxiii. 476. |
|1854 | Richter | 500 | 500 | Ibid. xxv. 472. |
|1873 | Shanks | 707 | .. | _Proc. Roy. Soc._ (London), xxi. |
+-----+------------+--------+--------+--------------------------------------------+

By these computers Machin's identity, or identities analogous to it, e.g.

[pi]/4 = tan^{-1} (1/2) + tan^{-1} 1/5 + tan^{-1} 1/8 (Dase, 1844),
[pi]/4 = 4tan^{-1} (1/5) - tan^{-1} 1/70 + tan^{-1} 1/99 (Rutherford),

and Gregory's series were employed.[29]

A much less wise class than the [pi]-computers of modern times are the
pseudo-circle-squarers, or circle-squarers technically so called, that
is to say, persons who, having obtained by illegitimate means a
Euclidean construction for the quadrature or a finitely expressible
value for [pi], insist on using faulty reasoning and defective
mathematics to establish their assertions. Such persons have flourished
at all times in the history of mathematics; but the interest attaching
to them is more psychological than mathematical.[30]

It is of recent years that the most important advances in the theory of
circle-quadrature have been made. In 1873 Charles Hermite proved that
the base [eta] of the Napierian logarithms cannot be a root of a
rational algebraical equation of any degree.[31] To prove the same
proposition regarding [pi] is to prove that a Euclidean construction for
circle-quadrature is impossible. For in such a construction every po