A condensed record compiled by J. W. L. Glaisher (Messenger of Math. ii. 122) is as follows:—
|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||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 1 + tan–1 1 + tan–1 1 (Dase, 1844),
π/4 = 4tan–1 1 − tan–1 1 + tan–1 1 (Rutherford),
and Gregory’s series were employed.
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.
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. 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 did not succeed in his attempt on π; but in 1882 F. Lindemann, following exactly in Hermite’s steps, accomplished the desired result. (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 pĭstování math. a fys. x. 272-275; Analyst, ix. 176. (T. Mu.)
CIRCLEVILLE, a city and the county-seat of Pickaway county, Ohio, U.S.A., about 26 m. S. by E. of Columbus, on the Scioto river and the Ohio Canal. Pop. (1890) 6556; (1900) 6991 (551 negroes); (1910) 6744. It is served by the Cincinnati & Muskingum Valley (Pennsylvania lines) and the Norfolk & Western railways, and by the Scioto Valley electric line. Circleville is situated in a farming region, and its leading industries are the manufacture of straw boards and agricultural implements, and the canning of sweet corn and other produce. The city occupies the site of prehistoric earth-works, from one of which, built in the form of a circle, it derived its name. Circleville, first settled about 1806, was chosen as the county-seat in 1810. The court-house was built in the form of an octagon at the centre of the circle, and circular streets were laid out around it; but this arrangement proved to be inconvenient, the court-house was destroyed by fire in 1841, and at present no trace of the ancient landmarks remains. Circleville was incorporated as a village in 1814, and was chartered as a city in 1853.
CIRCUIT (Lat. circuitus, from circum, round, and ire, to go), the act of moving round; so circumference, or anything encircling or encircled. The word is particularly known as a law term, signifying the periodical progress of a legal tribunal for the purpose of carrying out the administration of the law in the several provinces of a country. It has long been applied to the journey or progress which the judges have been in the habit of making through the several counties of England, to hold courts and administer justice, where recourse could not be had to the king’s court at Westminster (see Assize).
In England, by sec. 23 of the Judicature Act 1875, power was conferred on the crown, by order in council, to make regulations respecting circuits, including the discontinuance of any circuit, and the formation of any new circuit, and the appointment of the place at which assizes are to be held on any circuit. Under this power an order of council, dated the 5th of February 1876, was made, whereby the circuit system was remodelled. A new circuit, called the North-Eastern circuit, was created, consisting of Newcastle and Durham taken out of the old Northern circuit, and York and Leeds taken out of the Midland circuit. Oakham, Leicester and Northampton, which had belonged to the Norfolk circuit, were added to the Midland. The Norfolk circuit and the Home circuit were abolished and a new South-Eastern circuit was created, consisting of Huntingdon, Cambridge, Ipswich, Norwich, Chelmsford, Hertford and Lewes, taken partly out of the old Norfolk circuit and partly out of the Home circuit. The counties of Kent and Surrey were left out of the circuit system, the assizes for these counties being held by the judges remaining in London. Subsequently Maidstone and Guildford were united under the revived name of the Home circuit for the purpose of the summer and winter assizes, and the assizes in these towns were held by one of the judges of the Western circuit, who, after disposing of the business there, rejoined his colleague in Exeter. In 1899 this arrangement was abolished, and Maidstone and Guildford were added to the South-Eastern circuit. Other minor changes in the assize towns were made, which it is unnecessary to particularize. Birmingham first became a circuit town in the year 1884, and the work there became, by arrangement, the joint property of the Midland and Oxford circuits. There are alternative assize towns in the following counties, viz.:—On the Western circuit, Salisbury and Devizes for Wiltshire, and Wells and Taunton for Somerset; on the South-Eastern, Ipswich and Bury St Edmunds for Suffolk; on the North Wales circuit, Welshpool and Newtown for Montgomery; and on the South Wales circuit, Cardiff and Swansea for Glamorgan.
According to the arrangements in force in 1909 there are four assizes in each year. There are two principal assizes, viz. the winter assizes, beginning in January, and the summer assizes, beginning at the end of May. At these two assizes criminal and civil business is disposed of in all the circuits. There are two other assizes, viz. the autumn assizes and the Easter assizes. The autumn assizes are regulated by acts of 1876 and 1877 (Winter Assizes Acts 1876 and 1877), and orders of council made under the former act. They are held for the whole of England and Wales, but for the purpose of these assizes the work is to a large extent “grouped,” so that not every county has a separate assize. For example, on the South-Eastern circuit Huntingdon
- On the calculations made before Shanks, see Lehmann, “Beitrag zur Berechnung der Zahl π,” in Grunert’s Archiv, xxi. 121-174.
- See Montucla, Hist. des rech. sur la quad. du cercle (Paris, 1754, 2nd ed. 1831); de Morgan, Budget of Paradoxes (London, 1872).
- “Sur la fonction exponentielle,” Comples rendus (Paris), lxxvii. 18, 74, 226, 285.
- See Crelle’s Journal, lxxvi. 342.
- See “Über die Zahl π,” in Math. Ann. xx. 213.