Chapter III: Part 3

Up to this point the credit of most that had been done may be set down to Archimedes. A new departure, however, was made by Willebrord Snell of Leiden in his _Cyclometria_, published in 1621. His achievement was a closely approximate geometrical solution of the problem of rectification (see fig. 9): ACB being a semicircle whose centre is O, and AC the arc to be rectified, he produced AB to D, making BD equal to the radius, joined DC, and produced it to meet the tangent at A in E; and then his assertion (not established by him) was that AE was nearly equal to the arc AC, the error being in defect. For the purposes of the calculator a solution erring in excess was also required, and this Snell gave by slightly varying the former construction. Instead of producing AB (see fig. 10) so that BD was equal to r, he produced it only so far that, when the extremity D' was joined with C, the part D'F outside the circle was equal to r; in other words, by a non-Euclidean construction he trisected the angle AOC, for it is readily seen that, since FD' = FO = OC, the angle FOB = (1/3)AOC.[14] This couplet of constructions is as important from the calculator's point of view as it is interesting geometrically. To compare it on this score with the fundamental proposition of Archimedes, the latter must be put into a form similar to Snell's. AMC being an arc of a circle (see fig. 11) whose centre is O, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC < HK, but > AC. In modern trigonometrical notation the propositions to be compared stand as follows:--

2 tan 1/2[theta] > [theta] > 2sin 1/2 [theta] (Archimedes);

3 sin [theta]
tan (1/3)[theta] + 2sin (1/3)[theta] > [theta] > --------------- (Snell).
2 + cos [theta]

It is readily shown that the latter gives the best approximation to [theta]; but, while the former requires for its application a knowledge of the trigonometrical ratios of only one angle (in other words, the ratios of the sides of only one right-angled triangle), the latter requires the same for two angles, [theta] and (1/3)[theta]. Grienberger, using Snell's method, calculated the ratio correct to 39 fractional places.[15] C. Huygens, in his _De Circuli Magnitudine Inventa_, 1654, proved the propositions of Snell, giving at the same time a number of other interesting theorems, for example, two inequalities which may be written as follows[16]--

4chd [theta] + sin [theta] 1
chd [theta] + --------------------------- . --- (chd [theta] - sin [theta]) >
2chd [theta] + 3sin [theta] 3

1
[theta] > chd [theta] + --- (chd [theta] - sin [theta]).
3

As might be expected, a fresh view of the matter was taken by Rene Descartes. The problem he set himself was the exact converse of that of Archimedes. A given straight line being viewed as equal in length to the circumference of a circle, he sought to find the diameter of the circle. His construction is as follows (see fig. 12). Take AB equal to one-fourth of the given line; on AB describe a square ABCD; join AC; in AC produced find, by a known process, a point C1 such that, when C1B1 is drawn perpendicular to AB produced and C1D1 perpendicular to BC produced, the rectangle BC1 will be equal to 1/4ABCD; by the same process find a point C2 such that the rectangle B1C2 will be equal to 1/4BC1; and so on _ad infinitum_. The diameter sought is the straight line from A to the limiting position of the series of B's, say the straight line AB[oo]. As in the case of the process of Archimedes, we may direct our attention either to the infinite series of geometrical operations or to the corresponding infinite series of arithmetical operations. Denoting the number of units in AB by 1/4c, we can express BB1, B1B2, ... in terms of 1/4c, and the identity AB[oo] = AB + BB1 + B1B2 + ... gives us at once an expression for the diameter in terms of the circumference by means of an infinite series.[17] The proof of the correctness of the construction is seen to be involved in the following theorem, which serves likewise to throw new light on the subject:--AB being any straight line whatever, and the above construction being made, then AB is the diameter of the circle circumscribed by the square ABCD (self-evident), AB1 is the diameter of the circle circumscribed by the regular 8-gon having the same perimeter as the square, AB2 is the diameter of the circle circumscribed by the regular 16-gon having the same perimeter as the square, and so on. Essentially, therefore, Descartes's process is that known later as the process of _isoperimeters_, and often attributed wholly to Schwab.[18]

In 1655 appeared the _Arithmetica Infinitorum_ of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian co-ordinates, and algebra playing an important part. In a very curious manner, by viewing the circle y = (1 - x squared)^1/2 as a member of the series of curves y = (1 - x squared)1, y = (1 - x squared) squared, &c., he was led to the proposition that four times the reciprocal of the ratio of the circumference to the diameter, i.e. 4/[pi], is equal to the infinite product

3 . 3 . 5 . 5 . 7 . 7 . 9 ...
-----------------------------;
2 . 4 . 4 . 6 . 6 . 8 . 8 ...

and, the result having been communicated to Lord Brounker, the latter discovered the equally curious equivalent continued fraction

1 squared 3 squared 5 squared 7 squared
1 + --- --- --- --- ...
2 + 2 + 2 + 2

The work of Wallis had evidently an important influence on the next notable personality in the history of the subject, James Gregory, who lived during the period when the higher algebraic analysis was coming into power, and whose genius helped materially to develop it. He had, however, in a certain sense one eye fixed on the past and the other towards the future. His first contribution[19] was a variation of the method of Archimedes. The latter, as we know, calculated the perimeters of successive polygons, passing from one polygon to another of double the number of sides; in a similar manner Gregory calculated the areas. The general theorems which enabled him to do this, after a start had been made, are

_____
A2n = \/AnA'n (Snell's _Cyclom._),

2 An A'n 2 A'n A2n
A'2n = ---------- or ----------- (Gregory),
An + A'2n A'n + A2n

where An, A'n are the areas of the inscribed and the circumscribed regular n-gons respectively. He also gave approximate rectifications of circular arcs after the manner of Huygens; and, what is very notable, he made an ingenious and, according to J.E. Montucla, successful attempt to show that quadrature of the circle by a Euclidean construction was impossible.[20] Besides all this, however, and far beyond it in importance, was his use of infinite series. This merit he shares with his contemporaries N. Mercator, Sir I. Newton and G.W. Leibnitz, and the exact dates of discovery are a little uncertain. As far as the circle-squaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the secant in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries formed an epoch in the history of mathematics generally, and had, of course, a marked influence on after investigations regarding circle-quadrature. Even among the mere computers the series

[theta] = tan - (1/3) tan^3 [theta] + (1/5) tan^5 [theta] - ...,

specially known as Gregory's series, has ever since been a necessity of their calling.

The calculator's work having now become easier and more mechanical, calculation went on apace. In 1699 Abraham Sharp, on the suggestion of Edmund Halley, took Gregory's series, and, putting tan [theta] = (1/3) sqrt(3), found the ratio equal to

__ / 1 1 1 \
\/12 ( 1 - ----- + ------ - ------ + ... ),
\ 3 . 3 5 . 3 squared 7 . 3 cubed /

from which he calculated it correct to 71 fractional places.[21] About the same time John Machin calculated it correct to 100 places, and, what was of more importance, gave for the ratio the rapidly converging expression

16 / 1 1 1 \
-- ( ---- + ----- - ----- + ... ) -
5 \ 3.5 squared 5.5^4 7.5^6 /

4 / 1 1 \
--- ( 1 - ------ + ------- - ... ),
239 \ 3.239 squared 5.239^4 /

which long remained without explanation.[22] Fautet de Lagny, still using tan 30 deg., advanced to the 127th place.[23]

Leonhard Euler took up the subject several times during his life, effecting mainly improvements in the theory of the various series.[24] With him, apparently, began the usage of denoting by [pi] the ratio of the circumference to the diameter.[25]

The most important publication, however, on the subject in the 18th century was a paper by J.H. Lambert,[26] read before the Berlin Academy in 1761, in which he demonstrated the irrationality of [pi]. The general test of irrationality which he established is that, if

a1 a2 a2
-- -- -- ...
b1 +- b2 +- b3 +-

be an interminate continued fraction, a1, a2, ..., b1, b2 ... be integers, a1/b1, a2/b2, ... be proper fractions, and the value of every one of the interminate continued fractions

a1 a2
-- , -- , ... be < 1,
b1 +- ... b2 +- ...

then the given continued fraction represents an irrational quantity. If this be applied to the right-hand side of the identity

m m m squared m squared
tan --- = --- ---- ---- ...
n n - 3n - 5n

it follows that the tangent of every arc commensurable with the radius is irrational, so that, as a particular case, an arc of 45 deg., having its tangent rational, must be incommensurable with the radius; that is to say, [pi]/4 is an incommensurable number.[27]

This incontestable result had no effect, apparently, in repressing the [pi]-computers. G. von Vega in 1789, using series like Machin's, viz. Gregory's series and the identities

[pi]/4 = 5tan^{-1} (1/7) + 2tan^{-1} (3/79) (Euler, 1779),
[pi]/4 = tan^{-1} (1/7) + 2tan^{-1} ( 1/3) (Hutton, 1776),

neither of which was nearly so advantageous as several found by Charles Hutton, calculated [pi] 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 [pi]-computers of the 19th century. A condensed record compiled by J.W.L. Glaisher (_Messenger of Math._ ii. 122) is as follows:--

+-----+------------+-----------------+--------------------------------------------+
| | |No. of fr. digits| |
|Date.| Computer. +--------+--------+ Place of Publication. |
| | | calcd. |correct.| |
+-----+------------+--------+--------+--------------------------------------------+
|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 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 [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. 70-86, 206-211.

[9] Vieta, _Opera math._ (Leiden, 1646); Marie, _Hist. des sciences
math._ iii. 27 seq. (Paris, 1884).

[10] Kluegel, _Math. Woerterb._ ii. 606, 607.

[11] Kaestner, _Gesch. d. Math._ i. (Goettingen, 1796-1800).

[12] But see _Les Delices de Leide_ (Leiden, 1712); or de Haan,
_Mess. of Math._ iii. 24-26.

[13] For minute and lengthy details regarding the quadrature of the
circle in the Low Countries, see de Haan, "Bouwstoffen voor de
geschiedenis, &c.," in _Versl. en Mededeel. der K. Akad. van
Wetensch._ ix., x., xi., xii. (Amsterdam); also his "Notice sur
quelques quadrateurs, &c.," in _Bull. di bibliogr. e di storia delle
sci. mat. e fis._ vii. 99-144.

[14] It is thus manifest that by his first construction Snell gave
an approximate solution of two great problems of antiquity.

[15] _Elementa trigonometrica_ (Rome, 1630); Glaisher, _Messenger of
Math._ iii. 35 seq.

[16] See Kiessling's edition of the _De Circ. Magn. Inv._
(Flensburg, 1869); or Pirie's tract on _Geometrical Methods of
Approx. to the Value of [pi]_ (London, 1877).

[17] See Euler, "Annotationes in locum quendam Cartesii," in _Nov.
Comm. Acad. Petrop._ viii.

[18] Gergonne, _Annales de math._ vi.

[19] See _Vera Circuli et Hyperbolae Quadratura_ (Padua, 1667); and
the _Appendicula_ to the same in his _Exercitationes geometricae_
(London, 1668).

[20] _Penny Cyclop._ xix. 187.

[21] See Sherwin's _Math. Tables_ (London, 1705), p. 59.

[22] See W. Jones, _Synopsis Palmariorum Matheseos_ (London, 1706);
Maseres, _Scriptores Logarithmici_ (London, 1791-1796), iii. 159
seq.; Hutton, _Tracts_, i. 266.

[23] See _Hist. de l'Acad._ (Paris, 1719); 7 appears instead of 8 in
the 113th place.

[24] _Comment. Acad. Petrop._ ix., xi.; _Nov. Comm. Ac. Pet._ xvi.;
_Nova Acta Acad. Pet._ xi.

[25] _Introd. in Analysin Infin._ (Lausanne, 1748), chap. viii.

[26] _Mem. sur quelques proprietes remarquables des quantites
transcendantes, circulaires, et logarithmiques._

[27] See Legendre, _Elements de geometrie_ (Paris, 1794), note iv.;
Schloemilch, _Handbuch d. algeb. Analysis_ (Jena, 1851), chap. xiii.

[28] _Nova Acta Petrop._ ix. 41; _Thesaurus Logarithm. Completus_,
633.

[29] On the calculations made before Shanks, see Lehmann, "Beitrag
zur Berechnung der Zahl [pi]," in _Grunert's Archiv_, xxi. 121-174.

[30] See Montucla, _Hist. des rech. sur la quad. du cercle_ (Paris,
1754, 2nd ed. 1831); de Morgan, _Budget of Paradoxes_ (London,
1872).

[31] "Sur la fonction exponentielle," _Comples rendus_ (Paris),
lxxvii. 18, 74, 226, 285.

[32] See _Crelle's Journal_, lxxvi. 342.

[33] See "Ueber die Zahl [pi]," in _Math. Ann._ xx. 213.

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 is grouped with Cambridge; on the Midland, Rutland is grouped with Lincoln; on the Northern, Westmorland is grouped with Cumberland; and the North Wales and South Wales circuits are united, and no assizes are held at some of the smaller towns. At these assizes criminal business only is taken, except at Manchester, Liverpool, Swansea, Birmingham and Leeds. The Easter assizes are held in April and May on two circuits only, viz. at Manchester and Liverpool on the Northern and at Leeds on the North-Eastern. Both civil and criminal business is taken at Manchester and Liverpool, but criminal business only at Leeds.

Other changes were made, with a view to preventing the complete interruption of the London sittings in the common law division by the absence of the judges on circuit. The assizes were so arranged as to commence on different dates in the various circuits. For example, the summer assizes begin in the South-Eastern and Western circuits on the 29th of May; in the Northern circuit on the 28th of June; in the Midland and Oxford circuits on the 16th of June; in the North-Eastern circuit on the 6th of July; in the North Wales circuit on the 7th of July; and in the South Wales circuit on the 11th of July. Again, there has been a continuous development of what may be called the single-judge system. In the early days of the new order the members of the court of appeal and the judges of the chancery division shared the circuit work with the judges in the common law division. This did not prove to be a satisfactory arrangement. The assize work was not familiar and was uncongenial to the chancery judges, who had but little training or experience to fit them for it. Arrears increased in chancery, and the appeal court was shorn of much of its strength for a considerable part of the year. The practice was discontinued in or about the year 1884. The appeal and chancery judges were relieved of the duty of going on circuit, and an arrangement was made by the treasury for making an allowance for expenses of circuit to the common law judges, on whom the whole work of the assizes was thrown. In order to cope with the assize work, and at the same time keep the common law sittings going in London, an experiment, which had been previously tried by Lord Cairns and Lord Cross (then home secretary) and discontinued, was revived. Instead of two judges going together to each assize town, it was arranged that one judge should go by himself to certain selected places--practically, it may be said, to all except the more important provincial centres. The only places to which two judges now go are Exeter, Winchester, Bristol, Manchester, Liverpool, Nottingham, Stafford, Birmingham, Newcastle, Durham, York, Leeds, Chester, and Cardiff or Swansea.

It could scarcely be said that, even with the amendments introduced under orders in council, the circuit system was altogether satisfactory or that the last word had been pronounced on the subject. In the first report of the Judicature Commission, dated March 25th, 1869, p. 17 (_Parl. Papers_, 1868-1869), the majority report that "the necessity for holding assizes in every county without regard to the extent of the business to be transacted in such county leads, in our judgment, to a great waste of judicial strength and a great loss of time in going from one circuit town to another, and causes much unnecessary cost and inconvenience to those whose attendance is necessary or customary at the assizes." And in their second report, dated July 3rd, 1872 (_Parl. Papers_, 1872, vol. xx.), they dwell upon the advisability of grouping or a discontinuance of holding assizes "in several counties, for example, Rutland and Westmorland, where it is manifestly an idle waste of time and money to have assizes." It is thought that the grouping of counties which has been effected for the autumn assizes might be carried still further and applied to all the assizes; and that the system of holding the assizes alternately in one of two towns within a county might be extended to two towns in adjoining counties, for example, Gloucester and Worcester. The facility of railway communication renders this reform comparatively easy, and reforms in this direction have been approved by the judges, but ancient custom and local patriotism, interests, or susceptibility bar the way. The Assizes and Quarter Sessions Act 1908 contributed something to reform by dispensing with the obligation to hold assizes at a fixed date if there is no business to be transacted. Nor can it be said that the single-judge system has been altogether a success. When there is only one judge for both civil and criminal work, he properly takes the criminal business first. He can fix only approximately the time when he can hope to be free for the civil business. If the calendar is exceptionally heavy or one or more of the criminal cases prove to be unexpectedly long (as may easily happen), the civil business necessarily gets squeezed into the short residue of the allotted time. Suitors and their solicitors and witnesses are kept waiting for days, and after all perhaps it proves to be impossible for the judge to take the case, and a "remanet" is the result. It is the opinion of persons of experience that the result has undoubtedly been to drive to London much of the civil business which properly belongs to the provinces, and ought to be tried there, and thus at once to increase the burden on the judges and jurymen in London, and to increase the costs of the trial of the actions sent there. Some persons advocate the continuous sittings of the high court in certain centres, such as Manchester, Liverpool, Leeds, Newcastle, Birmingham and Bristol, or (in fact) a decentralization of the judicial system. There is already an excellent court for chancery cases for Lancashire in the county palatine court, presided over by the vice-chancellor, and with a local bar which has produced many men of great ability and even eminence. The Durham chancery court is also capable of development. Another suggestion has been made for continuous circuits throughout the legal year, so that a certain number of the judges, according to a rota, should be continuously in the provinces while the remaining judges did the London business. The value of this suggestion would depend on an estimate of the number of cases which might thus be tried in the country in relief of the London list. This estimate it would be difficult to make. The opinion has also been expressed that it is essential in any changes that may be made to retain the occasional administration by judges of the high court of criminal jurisdiction, both in populous centres and in remote places. It promotes a belief in the importance and dignity of justice and the care to be given to all matters affecting a citizen's life, liberty or character. It also does something, by the example set by judges in country districts, to check any tendency to undue severity of sentences in offences against property.

Counsel are not expected to practise on a circuit other than that to which they have attached themselves, unless they receive a special retainer. They are then said to "go special," and the fee in such a case is one hundred guineas for a king's counsel, and fifty guineas for a junior. It is customary to employ one member of the circuit on the side on which the counsel comes special. Certain rules have been drawn up by the Bar Committee for regulating the practice as to retainers on circuit. (1) A special retainer must be given for a particular assize (a circuit retainer will not, however, make it compulsory upon counsel retained to go the circuit, but will give the right to counsel's services should he attend the assize and the case be entered for trial); (2) if the venue is changed to another place on the same circuit, a fresh retainer is not required; (3) if the action is not tried at the assize for which the retainer is given, the retainer must be renewed for every subsequent assize until the action is disposed of, unless a brief has been delivered; (4) a retainer may be given for a future assize, without a retainer for an intervening assize, unless notice of trial is given for such intervening assize. There are also various regulations enforced by the discipline of the circuit bar mess.

In the United States the English circuit system still exists in some states, as in Massachusetts, where the judges sit in succession in the various counties of the state. The term _circuit courts_ applies distinctively in America to a certain class of inferior federal courts of the United States, exercising jurisdiction, concurrently with the state courts, in certain matters where the United States is a party to the litigation, or in cases of crime against the United States. The circuit courts act in nine judicial circuits, divided as follows: _1st circuit_, Maine, Massachusetts, New Hampshire, Rhode Island; _2nd circuit_, Connecticut, New York, Vermont; _3rd circuit_, Delaware, New Jersey, Pennsylvania; _4th circuit_, Maryland, North Carolina, South Carolina, Virginia, West Virginia; _5th circuit_, Alabama, Florida, Georgia, Louisiana, Mississippi, Texas; _6th circuit_, Kentucky, Michigan, Ohio, Tennessee; _7th circuit_, Illinois, Indiana, Wisconsin; _8th circuit_, Arkansas, Colorado, Oklahoma, Iowa, Kansas, Minnesota, Missouri, Nebraska, New Mexico, North Dakota, South Dakota, Utah, Wyoming; _9th circuit_, Alaska, Arizona, California, Idaho, Montana, Nevada, Oregon, Washington, and Hawaii. A circuit court of appeals is made up of three judges of the circuit court, the judges of the district courts of the circuit, and the judge of the Supreme Court allotted to the circuit.

In Scotland the judges of the supreme criminal court, or high court of justiciary, form also three separate circuit courts, consisting of two judges each; and the country, with the exception of the Lothians, is divided into corresponding districts, called the Northern, Western and Southern circuits. On the Northern circuit, courts are held at Inverness, Perth, Dundee and Aberdeen; on the Western, at Glasgow, Stirling and Inveraray; and on the Southern, at Dumfries, Jedburgh and Ayr.

Ireland is divided into the North-East and the North-West circuits, and those of Leinster, Connaught and Munster.

CIRCULAR NOTE, a documentary request by a bank to its foreign correspondents to pay a specified sum of money to a named person. The person in whose favour a circular note is issued is furnished with a letter (containing the signature of an official of the bank and the person named) called a letter of indication, which is usually referred to in the circular note, and must be produced on presentation of the note. Circular notes are generally issued against a payment of cash to the amount of the notes, but the notes need not necessarily be cashed, but may be returned to the banker in exchange for the amount for which they were originally issued. A forged signature on a circular note conveys no right, and as it is the duty of the payer to see that payment is made to the proper person, he cannot recover the amount of a forged note from the banker who issued the note. (See also LETTER OF CREDIT.)

CIRCULUS IN PROBANDO (Lat. for "circle in proving"), in logic, a phrase used to describe a form of argument in which the very fact which one seeks to demonstrate is used as a premise, i.e. as part of the evidence on which the conclusion is based. This argument is one form of the fallacy known as _petitio principii_, "begging the question." It is most common in lengthy arguments, the complicated character of which enables the speaker to make his hearers forget the data from which he began. (See FALLACY.)

CIRCUMCISION (Lat. _circum_, round, and _caedere_, to cut), the cutting off of the foreskin. This surgical operation, which is commonly prescribed for purely medical reasons, is also an initiation or religious ceremony among Jews and Mahommedans, and is a widespread institution in many Semitic races. It remains, with Jews, a necessary preliminary to the admission of proselytes, except in some Reformed communities. The origin of the rite among the Jews is in Genesis (xvii.) placed in the age of Abraham, and at all events it must have been very ancient, for flint stones were used in the operation (Exodus iv. 25; Joshua v. 2). The narrative in Joshua implies that the custom was introduced by him, not that it had merely been in abeyance in the Wilderness. At Gilgal he "rolled away the reproach of the Egyptians" by circumcising the people. This obviously means that whereas the Egyptians practised circumcision the Jews in the land of the Pharaohs did not, and hence were regarded with contempt. It was an old theory (Herodotus ii. 36) that circumcision originated in Egypt; at all events it was practised in that country in ancient times (Ebers, _Egypten und die Buecher Mosis_, i. 278-284), and the same is true at the present day. But it is not generally thought probable that the Hebrews derived the rite directly from the Egyptians. As Driver puts it (_Genesis_, p. 190): "It is possible that, as Dillmann and Nowack suppose, the peoples of N. Africa and Asia who practised the rite adopted it from the Egyptians, but it appears in so many parts of the world that it must at any rate in these cases have originated independently." In another biblical narrative (Exodus iv. 25) Moses is subject to the divine anger because he had not made himself "a bridegroom of blood," that is, had not been circumcised before his marriage.

The rite of circumcision was practised by all the inhabitants of Palestine with the exception of the Philistines. It was an ancient custom among the Arabs, being presupposed in the Koran. The only important Semitic peoples who most probably did not follow the rite were the Babylonians and Assyrians (Sayce, _Babyl. and Assyrians_, p. 47). Modern investigations have brought to light many instances of the prevalence of circumcision in various parts of the world. These facts are collected by Andree and Ploss, and go to prove that the rite is not only spread through the Mahommedan world (Turks, Persians, Arabs, &c.), but also is practised by the Christian Abyssinians and the Copts, as well as in central Australia and in America. In central Australia (Spencer and Gillen, pp. 212-386) circumcision with a stone knife must be undergone by every youth before he is reckoned a full member of the tribe or is permitted to enter on the married state. In other parts, too (e.g. Loango), no uncircumcised man may marry. Circumcision was known to the Aztecs (Bancroft, _Native Races_, vol. iii.), and is still practised by the Caribs of the Orinoco and the Tacunas of the Amazon. The method and period of the operation vary in important particulars. Among the Jews it is performed in infancy, when the male child is eight days old. The child is named at the same time, and the ceremony is elaborate. The child is carried in to the godfather (_sandek_, a hebraized form of the Gr. [Greek: sunteknos], "godfather," post-class.), who places the child on a cushion, which he holds on his knees throughout the ceremony. The operator (_mohel_) uses a steel knife, and pronounces various benedictions before and after the rite is performed (see S. Singer, _Authorized Daily Prayer Book_, pp. 304-307; an excellent account of the domestic festivities and spiritual joys associated with the ceremony among medieval and modern Jews may be read in S. Schechter's _Studies in Judaism_, first series, pp. 351 seq.). Some tribes in South America and elsewhere are said to perform the rite on the eighth day, like the Jews. The Mazequas do it between the first and second months. Among the Bedouins the rite is performed on children of three years, amid dances and the selection of brides (Doughty, _Arabia Deserta_, i. 340); among the Somalis the age is seven (Reinisch, _Somalisprache_, p. 110). But for the most part the tribes who perform the rite carry it out at the age of puberty. Many facts bearing on this point are given by B. Stade in _Zeitschrift fuer die alttest. Wissenschaft_, vi. (1886) pp. 132 seq.

The significance of the rite of circumcision has been much disputed. Some see in it a tribal badge. If this be the true origin of circumcision, it must go back to the time when men went about naked. Mutilations (tattooing, removal of teeth and so forth) were tribal marks, being partly sacrifices and partly means of recognition (see MUTILATION). Such initiatory rites were often frightful ordeals, in which the neophyte's courage was severely tested (Robertson Smith, _Religion of the Semites_, p. 310). Some regard circumcision as a substitute for far more serious rites, including even human sacrifice. Utilitarian explanations have also been suggested. Sir R. Burton (_Memoirs Anthrop. Soc._ i. 318) held that it was introduced to promote fertility, and the claims of cleanliness have been put forward (following Philo's example, see ed. Mangey, ii. 210). Most probably, however, circumcision (which in many tribes is performed on both sexes) was connected with marriage, and was a preparation for connubium. It was in Robertson Smith's words "originally a preliminary to marriage, and so a ceremony of introduction to the full prerogative of manhood," the transference to infancy among the Jews being a later change. On this view, the decisive Biblical reference would be the Exodus passage (iv. 25), in which Moses is represented as being in danger of his life because he had neglected the proper preliminary to marriage. In Genesis, on the other hand, circumcision is an external sign of God's covenant with Israel, and later Judaism now regards it in this symbolical sense. Barton (_Semitic Origins_, p. 100) declares that "the circumstances under which it is performed in Arabia point to the origin of circumcision as a sacrifice to the goddess of fertility, by which the child was placed under her protection and its reproductive powers consecrated to her service." But Barton admits that initiation to the connubium was the primitive origin of the rite.

As regards the non-ritual use of male circumcision, it may be added that in recent years the medical profession has been responsible for its considerable extension among other than Jewish children, the operation being recommended not merely in cases of malformation, but generally for reasons of health.

AUTHORITIES.--On the present diffusion of circumcision see H. Ploss,
_Das Kind im Brauch und Sitte der Voelker_, i. 342 seq., and his
researches in _Deutsches Archiv fuer Geschichte der Medizin_, viii.
312-344; Andree, "Die Beschneidung" in _Archiv fuer Anthropologie_,
xiii. 76; and Spencer and Gillen, _Tribes of Central Australia_. The
articles in the _Encyclopaedia Biblica_ and _Dictionary of the Bible_
contain useful bibliographies as well as historical accounts of the
rite and its ceremonies, especially as concerns the Jews. The _Jewish
Encyclopedia_ in particular gives an extensive list of books on the
Jewish customs connected with circumcision, and the various articles
in that work are full of valuable information (vol. iv. pp. 92-102).
On the rite among the Arabs, see Wellhausen, _Reste arabischen
Heidentums_, 154. (I. A.)

CIRCUMVALLATION, LINES OF (from Lat. _circum_, round, and _vallum_, a rampart), in fortification, a continuous circle of entrenchments surrounding a besieged place. "Lines of Contravallation" were similar works by which the besieger protected himself against the attack of a relieving army from any quarter. These continuous lines of circumvallation and contravallation were used only in the days of small armies and small fortresses, and both terms are now obsolete.

CIRCUS (Lat. _circus_, Gr. [Greek: kirkos] or [Greek: krikos], a ring or circle; probably "circus" and "ring" are of the same origin), a space, in the strict sense circular, but sometimes oval or even oblong, intended for the exhibition of races and athletic contests generally. The circus differs from the theatre inasmuch as the performance takes place in a central circular space, not on a stage at one end of the building.

1. _In Roman antiquities_ the circus was a building for the exhibition of horse and chariot races and other amusements. It consisted of tiers of seats running parallel with the sides of the course, and forming a crescent round one of the ends. The other end was straight and at right angles to the course, so that the plan of the whole had nearly the form of an ellipse cut in half at its vertical axis. Along the transverse axis ran a fence (_spina_) separating the return course from the starting one. The straight end had no seats, but was occupied by the stalls (_carceres_) where the chariots and horses were held in readiness. This end constituted also the front of the building with the main entrance. At each end of the course were three conical pillars (_metae_) to mark its limits.

The oldest building of this kind in Rome was the _Circus Maximus_, in the valley between the Palatine and Aventine hills, where, before the erection of any permanent structure, races appear to have been held beside the altar of the god Consus. The first building is assigned to Tarquin the younger, but for a long time little seems to have been done to complete its accommodation, since it is not till 329 B.C. that we hear of stalls being erected for the chariots and horses. It was not in fact till under the empire that the circus became a conspicuous public resort. Caesar enlarged it to some extent, and also made a canal 10 ft. broad between the lowest tier of seats (_podium_) and the course as a precaution for the spectators' safety when exhibitions of fighting with wild beasts, such as were afterwards confined to the amphitheatre, took place. When these exhibitions were removed, and the canal (_euripus_) was no longer necessary, Nero had it filled up. Augustus is said to have placed an obelisk on the _spina_ between the _metae_, and to have built a new _pulvinar_, or imperial box; but if this is taken in connexion with the fact that the circus had been partially destroyed by fire in 31 B.C., it may be supposed that besides this he had restored it altogether. Only the lower tiers of seats were of stone, the others being of wood, and this, from the liability to fire, may account for the frequent restorations to which the circus was subject; it would also explain the falling of the seats by which a crowd of people were killed in the time of Antoninus Pius. In the reign of Claudius, apparently after a fire, the _carceres_ of stone (tufa) were replaced by marble, and the _metae_ of wood by gilt bronze. Under Domitian, again, after a fire, the circus was rebuilt and the carceres increased to 12 instead of 8 as before. The work was finished by Trajan. See further for seating capacity, &c., ROME: _Archaeology_, Sec. "Places of Amusement."

The circus was the only public spectacle at which men and women were not separated. The lower seats were reserved for persons of rank; there were also various state boxes, e.g. for the giver of the games and his friends (called _cubicula_ or _suggestus_). The principal object of attraction apart from the racing must have been the _spina_ or low wall which ran down the middle of the course, with its obelisks, images and ornamental shrines. On it also were seven figures of dolphins and seven oval objects, one of which was taken down at every round made in a race, so that spectators might see readily how the contest proceeded. The chariot race consisted of seven rounds of the course. The chariots started abreast, but in an oblique line, so that the outer chariot might be compensated for the wider circle it had to make at the other end. Such a race was called a _missus_, and as many as 24 of these would take place in a day. The competitors wore different colours, originally white and red (_albata_ and _russata_), to which green (_prasina_) and blue (_veneta_) were added. Domitian introduced two more colours, gold and purple (_purpureus et auratus pannus_), which probably fell into disuse after his death. To provide the horses and large staff of attendants it was necessary to apply to rich capitalists and owners of studs, and from this there grew up in time four select companies (_factiones_) of circus purveyors, which were identified with the four colours, and with which those who organized the races had to contract for the proper supply of horses and men. The drivers (_aurigae, agitatores_), who were mostly slaves, were sometimes held in high repute for their skill, although their calling was regarded with contempt. The horses most valued were those of Sicily, Spain and Cappadocia, and great care was taken in training them. Chariots with two horses (_bigae_) or four (_quadrigae_) were most common, but sometimes also they had three (_trigae_), and exceptionally more than four horses. Occasionally there was combined with the chariots a race of riders (_desultores_), each rider having two horses and leaping from one to the other during the race. At certain of the races the proceedings were opened by a _pompa_ or procession in which images of the gods and of the imperial family deified were conveyed in cars drawn by horses, mules or elephants, attended by the colleges of priests, and led by the presiding magistrate (in some cases by the emperor himself) seated in a chariot in the dress and with the insignia of a triumphator. The procession passed from the capitol along the forum, and on to the circus, where it was received by the people standing and clapping their hands. The presiding magistrate gave the signal for the races by throwing a white flag (_mappa_) on to the course.

Next in importance to the Circus Maximus in Rome was the _Circus Flaminius_, erected 221 B.C., in the censorship of C. Flaminius, from whom it may have taken its name; or the name may have been derived from Prata Flaminia, where it was situated, and where also were held plebeian meetings. The only games that are positively known to have been celebrated in this circus were the _Ludi Taurii_ and _Plebeii_. There is no mention of it after the 1st century. Its ruins were identified in the 16th century at S. Catarina dei Funari and the Palazzo Mattei.

A third circus in Rome was erected by Caligula in the gardens of Agrippina, and was known as the _Circus Neronis_, from the notoriety which it obtained through the Circensian pleasures of Nero. A fourth was constructed by Maxentius outside the Porta Appia near the tomb of Caecilia Metella, where its ruins are still, and now afford the only instance from which an idea of the ancient circi in Rome can be obtained. It was traced to Caracalla, till the discovery of an inscription in 1825 showed it to be the work of Maxentius. Old topographers speak of six circi, but two of these appear to be imaginary, the Circus Florae and the Circus Sallustii.

Circus races were held in connexion with the following public festivals, and generally on the last day of the festival, if it extended over more than one day:--(1) The _Consualia_, August 21st, December 15th; (2) _Equirria_, February 27th, March 14th; (3) _Ludi Romani_, September 4th-19th; (4) _Ludi Plebeii_, November 4th-17th; (5) _Cerialia_, April 12th-19th; (6) _Ludi Apollinares_, July 6th-13th; (7) _Ludi Megalenses_, April 4th-10th; (8) _Floralia_, April 28th-May 3rd.

In addition to Smith's _Dictionary of Antiquities_ (3rd ed., 1890),
see articles in Daremberg and Saglio's _Dictionnaire des antiquites_,
Pauly-Wissowa's _Realencyclopaedie der classischen
Altertumswissenschaft_, iii. 2 (1899), and Marquardt, _Roemische
Staatsverwaltung_, iii. (2nd ed., 1885), p. 504. For existing remains
see works quoted under ROME: _Archaeology_.

2. _The Modern Circus._--The "circus" in modern times is a form of popular entertainment which has little in common with the institution of classical Rome. It is frequently nomadic in character, the place of the permanent building known to the ancients as the circus being taken by a tent, which is carried from place to place and set up temporarily on any site procurable at country fairs or in provincial towns, and in which spectacular performances are given by a troupe employed by the proprietor. The centre of the tent forms an arena arranged as a horse-ring, strewn with tan or other soft substance, where the performances take place, the seats of the spectators being arranged in ascending tiers around the central space as in the Roman circus. The traditional type of exhibition in the modern travelling circus consists of feats of horsemanship, such as leaping through hoops from the back of a galloping horse, standing with one foot on each of two horses galloping side by side, turning somersaults from a springboard over a number of horses standing close together, or accomplishing acrobatic tricks on horseback. These performances, by male and female riders, are varied by the introduction of horses trained to perform tricks, and by drolleries on the part of the clown, whose place in the circus is as firmly established by tradition as in the pantomime.