int
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 [pi]; 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 (Goettingen,
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 [pi], dependent on the theory of probability, see _Mess. of Math._
ii. 113, 119; _Casopis pro pistovani math. a fys._ x. 272-275;
_Analyst_, ix. 176. (T. MU.)

FOOTNOTES:

[1] Eisenlohr, _Ein math. Handbuch d. alten Aegypter, uebers. u.
erklaert_ (Leipzig, 1877); Rodet, _Bull. de la Soc. Math. de France_,
vi. pp. 139-149.

[2] H. Hankel, _Zur Gesch. d. Math. im Alterthum_, &c., chap, v
(Leipzig, 1874); M. Cantor, _Vorlesungen ueber Gesch. d. Math._ i.
(Leipzig, 1880); Tannery, _Mem. de la Soc._, &c., _a Bordeaux_;
Allman, in _Hermathena_.

[3] Tannery. _Bull. des sc. math._ [2], x. pp. 213-226.

[4] In modern trigonometrical notation, 1 + sec [theta]:tan
[theta]::1:tan 1/2[theta].

[5] Tannery, "Sur la mesure du cercle d'Archimede," in _Mem....
Bordeaux_[2], iv. pp. 313-339; Menge, _Des Archimedes
Kreismessung_ (Coblenz, 1874).

[6] De Morgan, in _Penny Cyclop_, xix. p. 186.

[7] Kern, _Aryabhattiyam_ (Leiden, 1874), trans. by Rodet
(Paris,1879).

[8] De Morgan, art. "Quadrature of the Circle," in _English
Cyclop._; Glaisher, _Mess. of Math._ ii. pp. 119-128, iii. pp.
27-46; de Haan, _Nieuw Archief v. Wisk._ i. pp. 70