π⁄4 = tan-1 1⁄7 + 2 tan-1 1⁄3 (Hutton, 1776),
neither of which was nearly so advantageous as several found by Charles Hutton, calculated π correct to 136 places.[28] This achievement was anticipated or outdone by an unknown calculator, whose manuscript was seen in the Radcliffe library, Oxford, by Baron von Zach towards the end of the century, and contained the ratio correct to 152 places. More astonishing still have been the deeds of the π-computers of the 19th century. A condensed record compiled by J.W.L. Glaisher (Messenger of Math. ii. 122) is as follows:—
| Date. | Computer. | No. of fr. digits calcd. | No. of fr. digits correct. | Place of Publication. |
| 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.
π⁄4 = tan-1 ½ + tan-1 1⁄5 + tan-1 1⁄8 (Dase, 1844),
π⁄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 π-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 π, 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 η of the Napierian logarithms cannot be a root of a rational algebraical equation of any degree.[31] To prove the same proposition regarding π is to prove that a Euclidean construction for circle-quadrature is impossible. For in such a construction every point of the figure is obtained by the intersection of two straight lines, a straight line and a circle, or two circles; and as this implies that, when a unit of length is introduced, numbers employed, and the problem transformed into one of algebraic geometry, the equations to be solved can only be of the first or second degree, it follows that the equation to which we must be finally led is a rational equation of even degree. Hermite[32] did not succeed in his attempt on π; but in 1882 F. Lindemann, following exactly in Hermite’s steps, accomplished the desired result.[33] (See also [Trigonometry].)
References.—Besides the various writings mentioned, see for the history of the subject F. Rudio, Geschichte des Problems von der Quadratur des Zirkels (1892); M. Cantor, Geschichte der Mathematik (1894-1901); Montucla, Hist. des. math. (6 vols., Paris, 1758, 2nd ed. 1799-1802); Murhard, Bibliotheca Mathematica, ii. 106-123 (Leipzig, 1798); Reuss, Repertorium Comment. vii. 42-44 (Göttingen, 1808). For a few approximate geometrical solutions, see Leybourn’s Math. Repository, vi. 151-154; Grunert’s Archiv, xii. 98, xlix. 3; Nieuw Archief v. Wisk. iv. 200-204. For experimental determinations of π, dependent on the theory of probability, see Mess. of Math. ii. 113, 119; Casopis pro pïstováni math. a fys. x. 272-275; Analyst, ix. 176.