Chapter II: Part 2

CIPRIANI, GIOVANNI BATTISTA (1727-1785), Italian painter and engraver, Pistoiese by descent, was born in Florence in 1727. His first lessons were given him by an Englishman, Ignatius Heckford or Hugford, and under his second master, Antonio Domenico Gabbiani, he became a very clever draughtsman. He was in Rome from 1750 to 1753, where he became acquainted with Sir William Chambers, the architect, and Joseph Wilton, the sculptor, whom he accompanied to England in August 1755. He had already painted two pictures for the abbey of San Michele in Pelago, Pistoia, which had brought him reputation, and on his arrival in England he was patronized by Lord Tilney, the duke of Richmond and other noblemen. His acquaintance with Sir William Chambers no doubt helped him on, for when Chambers designed the Albany in London for Lord Holland, Cipriani painted a ceiling for him. He also painted part of a ceiling in Buckingham Palace, and a room with poetical subjects at Standlynch in Wiltshire. Some of his best and most permanent work was, however, done at Somerset House, built by his friend Chambers, upon which he lavished infinite pains. He not only prepared the decorations for the interior of the north block, but, says Joseph Baretti in his _Guide through the Royal Academy_ (1780), "the whole of the carvings in the various fronts of Somerset Place--excepting Bacon's bronze figures--were carved from finished drawings made by Cipriani." These designs include the five masks forming the keystones to the arches on the courtyard side of the vestibule, and the two above the doors leading into the wings of the north block, all of which are believed to have been carved by Nollekens. The grotesque groups flanking the main doorways on three sides of the quadrangle and the central doorway on the terrace appear also to have been designed by Cipriani. The apartments in Sir William Chambers's stately palace that were assigned to the Royal Academy, into which it moved in 1780, owed much to Cipriani's graceful, if mannered, pencil. The central panel of the library ceiling was painted by Sir Joshua Reynolds, but the four compartments in the coves, representing Allegory, Fable, Nature and History, were Cipriani's. These paintings still remain at Somerset House, together with the emblematic painted ceiling, also his work, of what was once the library of the Royal Society. It was natural that Cipriani should thus devote himself to adorning the apartments of the academy, since he was an original member (1768) of that body, for which he designed the diploma so well engraved by Bartolozzi. In recognition of his services in this respect the members presented him in 1769 with a silver cup with a commemorative inscription. He was much employed by the publishers, for whom he made drawings in pen and ink, sometimes coloured. His friend Bartolozzi engraved most of them. Drawings by him are in both the British Museum and Victoria and Albert Museum. His best autograph engravings are "The Death of Cleopatra," after Benvenuto Cellini; "The Descent of the Holy Ghost," after Gabbiani; and portraits for Hollis's memoirs, 1780. He painted allegorical designs for George III.'s state coach--which is still in use--in 1782, and repaired Verrio's paintings at Windsor and Rubens's ceiling in the Banqueting House at Whitehall. If his pictures were often weak, his decorative treatment of children was usually exceedingly happy. Some of his most pleasing work was that which, directly or indirectly, he executed for the decoration of furniture. He designed many groups of nymphs and _amorini_ and medallion subjects to form the centre of Pergolesi's bands of ornament, and they were continually reproduced upon the elegant satin-wood furniture which was growing popular in his later days and by the end of the 18th century became a rage. Sometimes these designs were inlaid in marqueterie, but most frequently they were painted upon the satin-wood by other hands with delightful effect, since in the whole range of English furniture there is nothing more enchanting than really good finished satin-wood pieces. There can be little doubt that some of the beautiful furniture designed by the Adams was actually painted by Cipriani himself. He also occasionally designed handles for drawers and doors. Cipriani died at Hammersmith in 1785 and was buried at Chelsea, where Bartolozzi erected a monument to his memory. He had married an English lady, by whom he had two sons.

CIRCAR, an Indian term applied to the component parts of a _subah_ or province, each of which is administered by a deputy-governor. In English it is principally employed in the name of the NORTHERN CIRCARS, used to designate a now obsolete division of the Madras presidency, which consisted of a narrow slip of territory lying along the western side of the Bay of Bengal from 15 deg. 40' to 20 deg. 17' N. lat. These Northern Circars were five in number, Chicacole, Rajahmundry, Ellore, Kondapalli and Guntur, and their total area was about 30,000 sq. m.

The district corresponds in the main to the modern districts of Kistna, Godavari, Vizagapatam, Ganjam and a part of Nellore. It was first invaded by the Mahommedans in 1471; in 1541 they conquered Kondapalli, and nine years later they extended their conquests over all Guntur and the districts of Masulipatam. But the invaders appear to have acquired only an imperfect possession of the country, as it was again wrested from the Hindu princes of Orissa about the year 1571, during the reign of Ibrahim, of the Kutb Shahi dynasty of Hyderabad or Golconda. In 1687 the Circars were added, along with the empire of Hyderabad, to the extensive empire of Aurangzeb. Salabat Jang, the son of the nizam ul mulk Asaf Jah, who was indebted for his elevation to the throne to the French East India Company, granted them in return for their services the district of Kondavid or Guntur, and soon afterwards the other Circars. In 1759, by the conquest of the fortress of Masulipatam, the dominion of the maritime provinces on both sides, from the river Gundlakamma to the Chilka lake, was necessarily transferred from the French to the British. But the latter left them under the administration of the nizam, with the exception of the town and fortress of Masulipatam, which were retained by the English East India Company. In 1765 Lord Clive obtained from the Mogul emperor Shah Alam a grant of the five Circars. Hereupon the fort of Kondapalli was seized by the British, and on the 12th of November 1766 a treaty of alliance was signed with Nizam Ali by which the Company, in return for the grant of the Circars, undertook to maintain troops for the nizam's assistance. By a second treaty, signed on the 1st of March 1768, the nizam acknowledged the validity of Shah Alam's grant and resigned the Circars to the Company, receiving as a mark of friendship an annuity of L50,000. Guntur, as the personal estate of the nizam's brother Basalat Jang, was excepted during his lifetime under both treaties. He died in 1782, but it was not till 1788 that Guntur came under British administration. Finally, in 1823, the claims of the nizam over the Northern Circars were bought outright by the Company, and they became a British possession.

CIRCASSIA, a name formerly given to the north-western portion of the Caucasus, including the district between the mountain range and the Black Sea, and extending to the north of the central range as far as the river Kuban. Its physical features are described in the article on the Russian province of KUBAN, with which it approximately coincides. The present article is confined to a consideration of the ethnographical relations and characteristics of the people, their history being treated under CAUCASIA.

The Cherkesses or Circassians, who gave their name to this region, of which they were until lately the sole inhabitants, are a peculiar race, differing from the other tribes of the Caucasus in origin and language. They designate themselves by the name of Adigheb, that of Cherkesses being a term of Russian origin. By their long-continued struggles with the power of Russia, during a period of nearly forty years, they attracted the attention of the other nations of Europe in a high degree, and were at the same time an object of interest to the student of the history of civilization, from the strange mixture which their customs exhibited of chivalrous sentiment with savage customs. For this reason it may be still worth while to give a brief summary of their national characteristics and manners, though these must now be regarded as in great measure things of the past.

In the patriarchal simplicity of their manners, the mental qualities with which they were endowed, the beauty of form and regularity of feature by which they were distinguished, they surpassed most of the other tribes of the Caucasus. At the same time they were remarkable for their warlike and intrepid character, their independence, their hospitality to strangers, and that love of country which they manifested in their determined resistance to an almost overwhelming power during the period of a long and desolating war. The government under which they lived was a peculiar form of the feudal system. The free Circassians were divided into three distinct ranks, the princes or _pshi_, the nobles or _uork_ (Tatar _usden_), and the peasants or _hokotl_. Like the inhabitants of the other regions of the Caucasus, they were also divided into numerous families, tribes or clans, some of which were very powerful, and carried on war against each other with great animosity. The slaves, of whom a large proportion were prisoners of war, were generally employed in the cultivation of the soil, or in the domestic service of some of the principal chiefs.

The will of the people was acknowledged as the supreme source of authority; and every free Circassian had a right to express his opinion in those assemblies of his tribe in which the questions of peace and war, almost the only subjects which engaged their attention, were brought under deliberation. The princes and nobles, the leaders of the people in war and their rulers in peace, were only the administrators of a power which was delegated to them. As they had no written laws, the administration of justice was regulated solely by custom and tradition, and in those tribes professing Mahommedanism by the precepts of the Koran. The most aged and respected inhabitants of the various _auls_ or villages frequently sat in judgment, and their decisions were received without a murmur by the contending parties. The Circassian princes and nobles were professedly Mahommedans; but in their religious services many of the ceremonies of their former heathen and Christian worship were still preserved. A great part of the people had remained faithful to the worship of their ancient gods--Shible, the god of thunder, of war and of justice; Tleps, the god of fire; and Seosseres, the god of water and of winds. Although the Circassians are said to have possessed minds capable of the highest cultivation, the arts and sciences, with the exception of poetry and music, were completely neglected. They possessed no written language. The wisdom of their sages, the knowledge they had acquired, and the memory of their warlike deeds were preserved in verses, which were repeated from mouth to mouth and descended from father to son.

The education of the young Circassian was confined to riding, fencing, shooting, hunting, and such exercises as were calculated to strengthen his frame and prepare him for a life of active warfare. The only intellectual duty of the _atalik_ or instructor, with whom the young men lived until they had completed their education, was that of teaching them to express their thoughts shortly, quickly and appropriately. One of their marriage ceremonies was very strange. The young man who had been approved by the parents, and had paid the stipulated price in money, horses, oxen, or sheep for his bride, was expected to come with his friends fully armed, and to carry her off by force from her father's house. Every free Circassian had unlimited right over the lives of his wife and children. Although polygamy was allowed by the laws of the Koran, the custom of the country forbade it, and the Circassians were generally faithful to the marriage bond. The respect for superior age was carried to such an extent that the young brother used to rise from his seat when the elder entered an apartment, and was silent when he spoke. Like all the other inhabitants of the Caucasus, the Circassians were distinguished for two very opposite qualities--the most generous hospitality and implacable vindictiveness. Hospitality to the stranger was considered one of the most sacred duties. Whatever were his rank in life, all the members of the family rose to receive him on his entrance, and conduct him to the principal seat in the apartment. The host was considered responsible with his own life for the security of his guest, upon whom, even although his deadliest enemy, he would inflict no injury while under the protection of his roof. The chief who had received a stranger was also bound to grant him an escort of horse to conduct him in safety on his journey, and confide him to the protection of those nobles with whom he might be on friendly terms. The law of vengeance was no less binding on the Circassian. The individual who had slain any member of a family was pursued with implacable vengeance by the relatives, until his crime was expiated by death. The murderer might, indeed, secure his safety by the payment of a certain sum of money, or by carrying off from the house of his enemy a newly-born child, bringing it up as his own, and restoring it when its education was finished. In either case, the family of the slain individual might discontinue the pursuit of vengeance without any stain upon its honour. The man closely followed by his enemy, who, on reaching the dwelling of a woman, had merely touched her hand, was safe from all other pursuit so long as he remained under the protection of her roof. The opinions of the Circassians regarding theft resembled those of the ancient Spartans. The commission of the crime was not considered so disgraceful as its discovery; and the punishment of being compelled publicly to restore the stolen property to its original possessor, amid the derision of his tribe, was much dreaded by the Circassian who would glory in a successful theft. The greatest stain upon the Circassian character was the custom of selling their children, the Circassian father being always willing to part with his daughters, many of whom were bought by Turkish merchants for the harems of Eastern monarchs. But no degradation was implied in this transaction, and the young women themselves were generally willing partners in it. Herds of cattle and sheep constituted the chief riches of the inhabitants. The princes and nobles, from whom the members of the various tribes held the land which they cultivated, were the proprietors of the soil. The Circassians carried on little or no commerce, and the state of perpetual warfare in which they lived prevented them from cultivating any of the arts of peace.

CIRCE (Gr. [Greek: Kirke]), in Greek legend, a famous sorceress, the daughter of Helios and the ocean nymph Perse. Having murdered her husband, the prince of Colchis, she was expelled by her subjects and placed by her father on the solitary island of Aeaea on the coast of Italy. She was able by means of drugs and incantations to change human beings into the forms of wolves or lions, and with these beings her palace was surrounded. Here she was found by Odysseus and his companions; the latter she changed into swine, but the hero, protected by the herb _moly_ (q.v.), which he had received from Hermes, not only forced her to restore them to their original shape, but also gained her love. For a year he relinquished himself to her endearments, and when he determined to leave, she instructed him how to sail to the land of shades which lay on the verge of the ocean stream, in order to learn his fate from the prophet Teiresias. Upon his return she also gave him directions for avoiding the dangers of the journey home (Homer, _Odyssey_, x.-xii.; Hyginus, _Fab._ 125). The Roman poets associated her with the most ancient traditions of Latium, and assigned her a home on the promontory of Circei (Virgil, _Aeneid_, vii. 10). The metamorphoses of Scylla and of Picus, king of the Ausonians, by Circe, are narrated in Ovid (_Metamorphoses_, xiv.).

_The Myth of Kirke_, by R. Brown (1883), in which Circe is explained
as a moon-goddess of Babylonian origin, contains an exhaustive summary
of facts, although many of the author's speculations may be proved
untenable (review by H. Bradley in _Academy_, January 19, 1884); see
also J.E. Harrison, _Myths of the Odyssey_ (1882); C. Seeliger in W.H.
Roscher's _Lexikon der Mythologie_.

CIRCEIUS MONS (mod. _Monte Circeo_), an isolated promontory on the S.W. coast of Italy, about 80 m. S.E. of Rome. It is a ridge of limestone about 31/2 m. long by 1 m. wide at the base, running from E. to W. and surrounded by the sea on all sides except the N. The land to the N. of it is 53 ft. above sea-level, while the summit of the promontory is 1775 ft. The origin of the name is uncertain: it has naturally been connected with the legend of Circe, and Victor Berard (in _Les Pheniciens et l'Odyssee_, ii. 261 seq.) maintains in support of the identification that [Greek: Ahiaie], the Greek name for the island of Circe, is a faithful transliteration of a Semitic name, meaning "island of the hawk," of which [Greek: nesos Kirkes] is the translation. The difficulty has been raised, especially by geologists, that the promontory ceased to be an island at a period considerably before the time of Homer; but Procopius very truly remarked that the promontory has all the appearance of an island until one is actually upon it. Upon the E. end of the ridge of the promontory are the remains of an enceinte, forming roughly a rectangle of about 200 by 100 yds. of very fine polygonal work, on the outside, the blocks being very carefully cut and jointed and right angles being intentionally avoided. The wall stands almost entirely free, as at Arpinum--polygonal walls in Italy are as a rule embanking walls--and increases considerably in thickness as it descends. The blocks of the inner face are much less carefully worked both here and at Arpinum. It seems to have been an acropolis, and contains no traces of buildings, except for a subterranean cistern, circular, with a beehive roof of converging blocks. The modern village of S. Felice Circeo seems to occupy the site of the ancient town, the citadel of which stood on the mountain top, for its medieval walls rest upon ancient walls of Cyclopean work of less careful construction than those of the citadel, and enclosing an area of 200 by 150 yds.

Circei was founded as a Roman colony at an early date--according to some authorities in the time of Tarquinius Superbus, but more probably about 390 B.C. The existence of a previous population, however, is very likely indicated by the revolt of Circei in the middle of the 4th century B.C., so that it is doubtful whether the walls described are to be attributed to the Romans or the earlier Volscian inhabitants. At the end of the republic, however, or at latest at the beginning of the imperial period, the city of Circei was no longer at the E. end of the promontory, but on the E. shores of the Lago di Paola (a lagoon--now a considerable fishery--separated from the sea by a line of sandhills and connected with it by a channel of Roman date: Strabo speaks of it as a small harbour) one mile N. of the W. end of the promontory. Here are the remains of a Roman town, belonging to the 1st and 2nd centuries, extending over an area of some 600 by 500 yards, and consisting of fine buildings along the lagoons, including a large open _piscina_ or basin, surrounded by a double portico, while farther inland are several very large and well-preserved water-reservoirs, supplied by an aqueduct of which traces may still be seen. An inscription speaks of an amphitheatre, of which no remains are visible. The transference of the city did not, however, mean the abandonment of the E. end of the promontory, on which stand the remains of several very large villas. An inscription, indeed, cut in the rock near S. Felice, speaks of this part of the _promunturium Veneris_ (the only case of the use of this name) as belonging to the city of Circei. On the S. and N. sides of the promontory there are comparatively few buildings, while, at the W. end there is a sheer precipice to the sea. The town only acquired municipal rights after the Social War, and was a place of little importance, except as a seaside resort. For its villas Cicero compares it with Antium, and probably both Tiberius and Domitian possessed residences there. The beetroot and oysters of Circei had a certain reputation. The view from the highest summit of the promontory (which is occupied by ruins of a platform attributed with great probability to a temple of Venus or Circe) is of remarkable beauty; the whole mountain is covered with fragrant shrubs. From any point in the Pomptine Marshes or on the coast-line of Latium the Circeian promontory dominates the landscape in the most remarkable way.

See T. Ashby, "Monte Circeo," in _Melanges de l'ecole francaise de
Rome_, XXV. (1905) 157 seq. (T. As.)

CIRCLE (from the Lat. _circulus_, the diminutive of _circus_, a ring; the cognate Gr. word is [Greek: kirkos], generally used in the form [Greek: krikos]), a plane curve definable as the locus of a point which moves so that its distance from a fixed point is constant.

The form of a circle is familiar to all; and we proceed to define certain lines, points, &c., which constantly occur in studying its geometry. The fixed point in the preceding definition is termed the "centre" (C in fig. 1); the constant distance, e.g. CG, the "radius." The curve itself is sometimes termed the "circumference." Any line through the centre and terminated at both extremities by the curve, e.g. AB, is a "diameter"; any other line similarly terminated, e.g. EF, a "chord." Any line drawn from an external point to cut the circle in two points, e.g. DEF, is termed a "secant"; if it touches the circle, e.g. DG, it is a "tangent." Any portion of the circumference terminated by two points, e.g. AD (fig. 2), is termed an "arc"; and the plane figure enclosed by a chord and arc, e.g. ABD, is termed a "segment"; if the chord be a diameter, the segment is termed a "semicircle." The figure included by two radii and an arc is a "sector," e.g. ECF (fig. 2). "Concentric circles" are, as the name obviously shows, circles having the same centre; the figure enclosed by the circumferences of two concentric circles is an "annulus" (fig. 3), and of two non-concentric circles a "lune," the shaded portions in fig. 4; the clear figure is sometimes termed a "lens."

The circle was undoubtedly known to the early civilizations, its simplicity specially recommending it as an object for study. Euclid defines it (Book I. def. 15) as a "plane figure enclosed by one line, all the straight lines drawn to which from one point within the figure are equal to one another." In the succeeding three definitions the centre, diameter and the semicircle are defined, while the third postulate of the same book demands the possibility of describing a circle for every "centre" and "distance." Having employed the circle for the construction and demonstration of several propositions in Books I. and II. Euclid devotes his third book entirely to theorems and problems relating to the circle, and certain lines and angles, which he defines in introducing the propositions. The fourth book deals with the circle in its relations to inscribed and circumscribed triangles, quadrilaterals and regular polygons. Reference should be made to the article GEOMETRY: _Euclidean_, for a detailed summary of the Euclidean treatment, and the elementary properties of the circle.

_Analytical Geometry of the Circle._

Cartesian co-ordinates.

In the article GEOMETRY: _Analytical_, it is shown that the general equation to a circle in rectangular Cartesian co-ordinates is x^2+y^2+2gx+2fy+c=0, i.e. in the general equation of the second degree the co-efficients of x^2 and y^2 are equal, and of xy zero. The co-ordinates of its centre are -g/c, -f/c; and its radius is (g^2+f^2-c)^1/2. The equations to the chord, tangent and normal are readily derived by the ordinary methods.

Consider the two circles:--

x^2+y^2+2gx+2fy+c=0, x^2+y^2+2g'x+2f'y+c'=0.

Obviously these equations show that the curves intersect in four
points, two of which lie on the intersection of the line, 2(g - g')x +
2(f - f')y + c - c' = 0, the radical axis, with the circles, and the
other two where the lines x squared + y squared = (x + iy) (x - iy) = 0 (where i =
sqrt -1) intersect the circles. The first pair of intersections may be
either real or imaginary; we proceed to discuss the second pair.

The equation x squared + y squared = 0 denotes a pair of perpendicular imaginary
lines; it follows, therefore, that circles always intersect in two
imaginary points at infinity along these lines, and since the terms
x squared + y squared occur in the equation of every circle, it is seen that all
circles pass through two fixed points at infinity. The introduction of
these lines and points constitutes a striking achievement in geometry,
and from their association with circles they have been named the
"circular lines" and "circular points." Other names for the circular
lines are "circulars" or "isotropic lines." Since the equation to a
circle of zero radius is x squared + y squared = 0, i.e. identical with the circular
lines, it follows that this circle consists of a real point and the
two imaginary lines; conversely, the circular lines are both a pair of
lines and a circle. A further deduction from the principle of
continuity follows by considering the intersections of concentric
circles. The equations to such circles may be expressed in the form
x squared + y squared = [alpha] squared, x squared + y squared = [beta] squared. These equations show that the
circles touch where they intersect the lines x squared + y squared = 0, i.e.
concentric circles have double contact at the circular points, the
chord of contact being the line at infinity.

In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry. In this article the equations to the more important circles--the circumscribed, inscribed, escribed, self-conjugate--will be given; reference should be made to the article TRIANGLE for the consideration of other circles (nine-point, Brocard, Lemoine, &c.); while in the article GEOMETRY: _Analytical_, the principles of the different systems are discussed.

Trilinear co-ordinates.

The equation to the circumcircle assumes the simple form
a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0, the centre
being cos A, cos B, cos C. The inscribed circle is cos 1/2A sqrt([alpha])
cos 1/2B sqrt([beta]) + cos 1/2C sqrt([gamma]) = 0, with centre [alpha] =
[beta] = [gamma]; while the escribed circle opposite the angle A is
cos 1/2A sqrt(-[alpha]) + sin 1/2B sqrt([beta]) + sin 1/2C sqrt([gamma]) =
0, with centre -[alpha] = [beta] = [gamma]. The self-conjugate circle
is [alpha] squared sin 2A + [beta] squared sin 2B + [gamma] squared sin 2C = 0, or the
equivalent form a cos A [alpha] squared + b cos B [beta] squared + c cos C [gamma] squared =
0, the centre being sec A, sec B, sec C.

The general equation to the circle in trilinear co-ordinates is
readily deduced from the fact that the circle is the only curve which
intersects the line infinity in the circular points. Consider the
equation

a[beta][gamma] + b[gamma][alpha] + C[alpha][beta] + (l[alpha] +
m[beta] + n[gamma]) (a[alpha] + b[beta] + c[gamma]) = 0 (1).

This obviously represents a conic intersecting the circle
a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0 in points on the
common chords l[alpha] + m[beta] + n[gamma] = 0, a[alpha] + b[beta] +
c[gamma] = 0. The line l[alpha] + m[beta] + n[gamma] is the radical
axis, and since a[alpha] + b[beta] + c[gamma] = 0 is the line
infinity, it is obvious that equation (1) represents a conic passing
through the circular points, i.e. a circle. If we compare (1) with the
general equation of the second degree u[alpha] squared + v[beta] squared + w[gamma] squared
+ 2u'[beta][gamma] + 2v'[gamma][alpha] + 2w'[alpha][beta] = 0, it is
readily seen that for this equation to represent a circle we must have

-kabc = vc squared + wb squared - 2u'bc = wa squared + uc squared - 2v'ca = ub squared + va squared - 2w'ab.

Areal co-ordinates.

The corresponding equations in areal co-ordinates are readily derived
by substituting x/a, y/b, z/c for [alpha], [beta], [gamma]
respectively in the trilinear equations. The circumcircle is thus seen
to be a squaredyz + b squaredzx + c squaredxy = 0, with centre sin 2A, sin 2B, sin 2C; the
inscribed circle is sqrt(x cot 1/2A) + sqrt(y cot 1/2B) + sqrt(z cot 1/2C) =
0, with centre sin A, sin B, sin C; the escribed circle opposite the
angle A is sqrt(-x cot 1/2A) + sqrt(y tan 1/2B) + sqrt(z tan 1/2C)=0, with
centre - sin A, sin B, sin C; and the self-conjugate circle is x squared cot
A + y squared cot B + z squared cot C = 0, with centre tan A, tan B, tan C. Since in
areal co-ordinates the line infinity is represented by the equation x
+ y + z = 0 it is seen that every circle is of the form a squaredyz + b squaredzx +
c squaredxy + (lx + my + nz)(x + y + z) = 0. Comparing this equation with ux squared
+ vy squared + wz squared + 2u'yz + 2v'zx + 2w'xy = 0, we obtain as the condition
for the general equation of the second degree to represent a
circle:--

(v + w - 2u')/a squared = (w + u - 2v')/b squared = (u + v - 2w')/c squared.

Tangential co-ordinates.

In tangential (p, q, r) co-ordinates the inscribed circle has for its
equation (s - a)qr + (s - b)rp + (s - c)pq = 0, s being equal to 1/2(a +
b + c); an alternative form is qr cot 1/2A + rp cot 1/2B + pq cot 1/2C = 0;
the centre is ap + bq + cr = 0, or p sin A + q sin B + r sin C = 0.
The escribed circle opposite the angle A is -sqr + (s - c)rp + (s -
b)pq = 0 or -qr cot 1/2A + rp tan 1/2B + pq tan 1/2C = 0, with centre -ap +
bq + cr = 0. The circumcircle is a sqrt(p) + b sqrt(q) + c sqrt(r) =
0, the centre being p sin 2A + q sin 2B + r sin 2C = 0. The general
equation to a circle in this system of co-ordinates is deduced as
follows: If [rho] be the radius and lp + mq + nr = 0 the centre, we
have [rho] = (lp1 + mq1 + nr1)/(l + m + n), in which p1, q1, r1 is a
line distant [rho] from the point lp + mq + nr = 0. Making this
equation homogeneous by the relation [Sigma]a squared(p - q) (p - r) =
4[Delta] squared (see GEOMETRY: _Analytical_), which is generally written
{ap, bq, cr} squared = 4[Delta] squared, we obtain {ap, bq, cr} squared[rho] squared =
4[Delta] squared{(lp + mq + nr)/(l + m + n)} squared, the accents being dropped, and
p, q, r regarded as current co-ordinates. This equation, which may be
more conveniently written {ap, bq, cr} squared = ([lambda]p + [mu]q +
[nu]r) squared, obviously represents a circle, the centre being [lambda]p +
[mu]q + [nu]r = 0, and radius 2[Delta]/([lambda] + [mu] + [nu]). If we
make [lambda] = [mu] = [nu] = 0, [rho] is infinite, and we obtain {ap,
bq, cr} squared = 0 as the equation to the circular points.

_Systems of Circles._

_Centres and Circle of Similitude._--The "centres of similitude" of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the "external centre," the transverse tangents to the "internal centre." It may be readily shown that the external and internal centres are the points where the line joining the centres of the two circles is divided externally and internally in the ratio of their radii.

The circle on the line joining the internal and external centres of similitude as diameter is named the "circle of similitude." It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the radii of the two circles.

With a system of three circles it is readily seen that there are six centres of similitude, viz. two for each pair of circles, and it may be shown that these lie three by three on four lines, named the "axes of similitude." The collinear centres are the three sets of one external and two internal centres, and the three external centres.

_Coaxal Circles._--A system of circles is coaxal when the locus of points from which tangents to the circles are equal is a straight line. Consider the case of two circles, and in the first place suppose them to intersect in two real points A and B. Then by Euclid iii. 36 it is seen that the line joining the points A and B is the locus of the intersection of equal tangents, for if P be any point on AB and PC and PD the tangents to the circles, then PA.PB = PC squared = PD squared, and therefore PC = PD. Furthermore it is seen that AB is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the radii. The line AB is termed the "radical axis." A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis--considered as a circle of infinite radius. In the case of two non-intersecting circles it may be shown that the radical axis has the same metrical relations to the line of centres.

There are several methods of constructing the radical axis in this
case. One of the simplest is: Let P and P' (fig. 5) be the points of
contact of a common tangent; drop perpendiculars PL, P'L', from P and
P' to OO', the line joining the centres, then the radical axis bisects
LL' (at X) and is perpendicular to OO'. To prove this let AB, AB1 be
the tangents from any point on the line AX. Then by Euc. i. 47, AB squared =
AO squared - OB squared = AX squared + OX squared + OP squared; and OX squared = OD squared - DX squared = OP squared + PD squared - DX squared.
Therefore AB squared = AX squared - DX squared + PD squared. Similarly AB' squared = AX squared - DX squared + DP' squared.
Since PD = PD', it follows that AB = AB'.

To construct circles coaxal with the two given circles, draw the
tangent, say XR, from X, the point where the radical axis intersects
the line of centres, to one of the given circles, and with centre X
and radius XR describe a circle. Then circles having the intersections
of tangents to this circle and the line of centres for centres, and
the lengths of the tangents as radii, are members of the coaxal
system.

In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I', where the orthogonal circle to the system intersects the line of centres; these points are named the "limiting points." In the case of a coaxal system having real points of intersection the limiting points are imaginary. Analytically, the Cartesian equation to a coaxal system can be written in the form x squared + y squared + 2ax +- k squared = 0, where a varies from member to member, while k is a constant. The radical axis is x = 0, and it may be shown that the length of the tangent from a point (0, h) is h squared +- k squared, i.e. it is independent of a, and therefore of any particular member of the system. The circles intersect in real or imaginary points according to the lower or upper sign of k squared, and the limiting points are real for the upper sign and imaginary for the lower sign. The fundamental properties of coaxal systems may be summarized:--

1. The centres of circles forming a coaxal system are collinear;

2. A coaxal system having real points of intersection has imaginary
limiting points;

3. A coaxal system having imaginary points of intersection has real
limiting points;

4. Every circle through the limiting points cuts all circles of the
system orthogonally;

5. The limiting points are inverse points for every circle of the
system.

The theory of centres of similitude and coaxal circles affords elegant demonstrations of the famous problem: To describe a circle to touch three given circles. This problem, also termed the "Apollonian problem," was demonstrated with the aid of conic sections by Apollonius in his book on _Contacts_ or _Tangencies_; geometrical solutions involving the conic sections were also given by Adrianus Romanus, Vieta, Newton and others. The earliest analytical solution appears to have been given by the princess Elizabeth, a pupil of Descartes and daughter of Frederick V. John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his _Sequel to Euclid_; an analytical solution by Gergonne is given in Salmon's _Conic Sections_. Here we may notice that there are eight circles which solve the problem.

_Mensuration of the Circle._

All exact relations pertaining to the mensuration of the circle involve the ratio of the circumference to the diameter. This ratio, invariably denoted by [pi], is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number. Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 2[pi]r and [pi]r squared. The exact geometrical evaluation of the second quantity, viz. [pi]r squared, which, in reality, is equivalent to determining a square equal in area to a circle, engaged the attention of mathematicians for many centuries. The history of these attempts, together with modern contributions to our knowledge of the value and nature of the number [pi], is given below (_Squaring of the Circle_).

The following table gives the values of this constant and several
expiessions involving it:--

+--------------+-----------+-----------+
| | Number. | Logarithm.|
+--------------+-----------+-----------+
| [pi] | 3.1415927 | 0.4971499 |
| 2 [pi] | 6.2831858 | 0.7981799 |
| 4 [pi] |12.5663706 | 1.0992099 |
| (1/2) [pi] | 1.5707963 | 0.1961199 |
| (1/3) [pi] | 1.0471976 | 0.0200286 |
| (1/4) [pi] | 0.7853982 | 1.8950899 |
| (1/6) [pi] | 0.5235988 | 1.7189986 |
| (1/8) [pi] | 0.3926991 | 1.5940599 |
| (1/12) [pi] | 0.2617994 | 1.4179686 |
| (4/3) [pi] | 4.1887902 | 0.6220886 |
| | | |
| [pi] | | |
| ------ | 0.0174533 | 2.2418774 |
| 180 | | |
| | | |
| 1 | | |
| ------ | 0.3183099 | 1.5028501 |
| [pi] | | |
| | | |
| 4 | | |
| ------ | 1.2732395 | 0.1049101 |
| [pi] | | |
| | | |
| 1 | | |
| ------ | 0.0795775 | 2.9097901 |
| 4 [pi] | | |
| | | |
| 180 | | |
| ------ |57.2957795 | 1.7581226 |
| [pi] | | |
| | | |
| [pi] squared | 9.8696044 | 0.9942997 |
| | | |
| 1 | | |
| -------- | 0.0168869 | 2.2275490 |
| 6 [pi] squared | | |
| | | |
| _____ | | |
| \/ [pi] | 1.7724539 | 0.2485750 |
| | | |
| _____ | | |
| \ cubed/ [pi] | 1.4645919 | 0.1657166 |
| | | |
| | | |
| 1 | | |
| -------- | | |
| _____ | 0.5641896 | 1.7514251 |
| \/ [pi] | | |
| | | |
| 2 | | |
| -------- | | |
| _____ | 1.1283792 | 0.0524551 |
| \/ [pi] | | |
| | | |
| 1 | | |
| ---------- | | |
| _____ | 0.2820948 | 1.4503951 |
| 2 \/ [pi] | | |
| | | |
| _____ | | |
| / 6 | | |
| \ cubed/ ---- | 1.2407010 | 0.0936671 |
| V [pi] | | |
| | | |
| ______ | | |
| / 3 | | |
| \ cubed/ ------- | 0.6203505 | 1.7926371 |
| V 4 [pi] | | |
| | | |
| log e [pi] | 1.1447299 | 0.0587030 |
+--------------+-----------+-----------+

Useful fractional approximations are 22/7 and 355/113.

A synopsis of the leading formula connected with the circle will now
be given.

1. _Circle._--Data: radius = a. Circumference = 2[pi]a. Area = [pi]a squared.

2. _Arc_ and _Sector_.--Data: radius = a; [theta] = circular measure
of angle subtended at centre by arc; c = chord of arc; c2 = chord of
semi-arc; c4 = chord of quarter-arc.

Exact formulae are:--Arc = a[theta], where [theta] may be given
directly, or indirectly by the relation c = 2a sin 1/2[theta]. Area of
sector = 1/2a squared[theta] = 1/2 radius x arc.

Approximate formulae are:--Arc = (1/3)(8c2 - c) (Huygen's formula);
arc = (1/45)(c - 40c2 + 256c4).

3. _Segment._--Data: a, [theta], c, c2, as in (2); h = height of
segment, i.e. distance of mid-point of arc from chord.

Exact formulae are:--Area = 1/2a squared([theta] - sin [theta]) = 1/2a squared[theta]
-1/4c squared cot 1/2[theta] = 1/2a squared - 1/2c sqrt(a squared - 1/4c squared). If h be given, we can use
c squared + 4h squared = 8ah, 2h = c tan 1/4[theta] to determine [theta].

Approximate formulae are:--Area = (1/15)(6c + 8c2)h; = (2/3) sqrt(c squared +
(8/5)h squared).h; = (1/15)(7c + 3[alpha])h, [alpha] being the true length of
the arc.

From these results the mensuration of any figure bounded by circular
arcs and straight lines can be determined, e.g. the area of a _lune_
or _meniscus_ is expressible as the difference or sum of two segments,
and the circumference as the sum of two arcs. (C. E.*)

_Squaring of the Circle._

The problem of finding a square equal in area to a given circle, like all problems, may be increased in difficulty by the imposition of restrictions; consequently under the designation there may be embraced quite a variety of geometrical problems. It has to be noted, however, that, when the "squaring" of the circle is especially spoken of, it is almost always tacitly assumed that the restrictions are those of the Euclidean geometry.

Since the area of a circle equals that of the rectilineal triangle whose base has the same length as the circumference and whose altitude equals the radius (Archimedes, [Greek: Kyklou metresis], prop. 1), it follows that, if a straight line could be drawn equal in length to the circumference, the required square could be found by an ordinary Euclidean construction; also, it is evident that, conversely, if a square equal in area to the circle could be obtained it would be possible to draw a straight line equal to the circumference. Rectification and quadrature of the circle have thus been, since the time of Archimedes at least, practically identical problems. Again, since the circumferences of circles are proportional to their diameters--a proposition assumed to be true from the dawn almost of practical geometry--the rectification of the circle is seen to be transformable into finding the ratio of the circumference to the diameter. This correlative numerical problem and the two purely geometrical problems are inseparably connected historically.

Probably the earliest value for the ratio was 3. It was so among the Jews (1 Kings vii. 23, 26), the Babylonians (Oppert, _Journ. asiatique_, August 1872, October 1874), the Chinese (Biot, _Journ. asiatique_, June 1841), and probably also the Greeks. Among the ancient Egyptians, as would appear from a calculation in the Rhind papyrus, the number (4/3)^4, i.e. 3.1605, was at one time in use.[1] The first attempts to solve the purely geometrical problem appear to have been made by the Greeks (Anaxagoras, &c.)[2], one of whom, Hippocrates, doubtless raised hopes of a solution by his quadrature of the so-called _meniscoi_ or _lune_.[3]

[The Greeks were in possession of several relations pertaining to the quadrature of the lune. The following are among the more interesting. In fig. 6, ABC is an isosceles triangle right angled at C, ADB is the semicircle described on AB as diameter, AEB the circular arc described with centre C and radius CA = CB. It is easily shown that the areas of the lune ADBEA and the triangle ABC are equal. In fig. 7, ABC is any triangle right angled at C, semicircles are described on the three sides, thus forming two lunes AFCDA and CGBEC. The sum of the areas of these lunes equals the area of the triangle ABC.]

As for Euclid, it is sufficient to recall the facts that the original author of prop. 8 of book iv. had strict proof of the ratio being <4, and the author of prop. 15 of the ratio being >3, and to direct attention to the importance of book x. on incommensurables and props. 2 and 16 of book xii., viz. that "circles are to one another as the squares on their diameters" and that "in the greater of two concentric circles a regular 2n-gon can be inscribed which shall not meet the circumference of the less," however nearly equal the circles may be.

With Archimedes (287-212 B.C.) a notable advance was made. Taking the circumference as intermediate between the perimeters of the inscribed and the circumscribed regular n-gons, he showed that, the radius of the circle being given and the perimeter of some particular circumscribed regular polygon obtainable, the perimeter of the circumscribed regular polygon of double the number of sides could be calculated; that the like was true of the inscribed polygons; and that consequently a means was thus afforded of approximating to the circumference of the circle. As a matter of fact, he started with a semi-side AB of a circumscribed regular hexagon meeting the circle in B (see fig. 8), joined A and B with O the centre, bisected the angle AOB by OD, so that BD became the semi-side of a circumscribed regular 12-gon; then as AB:BO:OA::1: sqrt(3):2 he sought an approximation to sqrt(3) and found that AB:BO > 153:265. Next he applied his theorem[4] BO + OA:AB::OB:BD to calculate BD; from this in turn he calculated the semi-sides of the circumscribed regular 24-gon, 48-gon and 96-gon, and so finally established for the circumscribed regular 96-gon that perimeter:diameter < (3-1/7):1. In a quite analogous manner he proved for the inscribed regular 96-gon that perimeter:diameter > 3-(10/71):1. The conclusion from these therefore was that the ratio of circumference to diameter is < 3-1/7 and > 3-(10/71). This is a most notable piece of work; the immature condition of arithmetic at the time was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever.[5]

No advance of any importance was made upon the achievement of Archimedes until after the revival of learning. His immediate successors may have used his method to attain a greater degree of accuracy, but there is very little evidence pointing in this direction. Ptolemy (fl. 127-151), in the _Great Syntaxis_, gives 3.141552 as the ratio[6]; and the Hindus (c. A.D. 500), who were very probably indebted to the Greeks, used 62832/20000, that is, the now familiar 3.1416.[7]

It was not until the 15th century that attention in Europe began to be once more directed to the subject, and after the resuscitation a considerable length of time elapsed before any progress was made. The first advance in accuracy was due to a certain Adrian, son of Anthony, a native of Metz (1527), and father of the better-known Adrian Metius of Alkmaar. In refutation of Duchesne(Van der Eycke), he showed that the ratio was < 3-(17/120) and > 3-(15/106), and thence made the exceedingly lucky step of taking a mean between the two by the quite unjustifiable process of halving the sum of the two numerators for a new numerator and halving the sum of the two denominators for a new denominator, thus arriving at the now well-known approximation 3-(16/113) or 355/113, which, being equal to 3.1415929..., is correct to the sixth fractional place.[8]

The next to advance the calculation was Francisco Vieta. By finding the perimeter of the inscribed and that of the circumscribed regular polygon of 393216 (i.e. 6 X 2^16) sides, he proved that the ratio was > 3.1415926535 and < 3.1415926537, so that its value became known (in 1579) correctly to 10 fractional places. The theorem for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form 1 - cos [theta] = 2 sin squared 1/2[theta]. With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the _Universales Inspectiones_, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying _Canon mathematicus_ a table of sines, tangents and secants.[9] Further, in comparing the labours of Archimedes and Vieta, the effect of increased power of symbolical expression is very noticeable. Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a "rule" in words; in the 16th century it could be condensed into a "formula." Accordingly, we find in Vieta a formula for the ratio of diameter to circumference, viz. the interminate product[10]--

___________________
__________ / ___________
___ / ___ / / ___
1/2 \/ 1/2 . \/ 1/2 + 1/2\/ 1/2 . \/ 1/2 + 1/2 \/ 1/2 + 1/2 \/ 1/2 ...

From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy; the problem being reduced to an arithmetical computation. Thus in connexion with the subject a genus of workers became possible who may be styled "[pi]-computers or circle-squarers"--a name which, if it connotes anything uncomplimentary, does so because of the almost entirely fruitless character of their labours. Passing over Adriaan van Roomen (Adrianus Romanus) of Louvain, who published the value of the ratio correct to 15 places in his _Idea mathematica_ (1593),[11] we come to the notable computer Ludolph van Ceulen (d. 1610), a native of Germany, long resident in Holland. His book, _Van den Circkel_ (Delft, 1596), gave the ratio correct to 20 places, but he continued his calculations as long as he lived, and his best result was published on his tombstone in St Peter's church, Leiden. The inscription, which is not known to be now in existence,[12] is in part as follows:--

... Qui in vita sua multo labore circumferentiae circuli proximam
rationem ad diametrum invenit sequentem--

quando diameter est 1
tum circuli circumferentia plus est

quam 314159265358979323846264338327950288
1OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

et minus
quam 314159265358979323846264338327950289
1OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ...

This gives the ratio correct to 35 places. Van Ceulen's process was essentially identical with that of Vieta. Its numerous root extractions amply justify a stronger expression than "multo labore," especially in an epitaph. In Germany the "Ludolphische Zahl" (Ludolph's number) is still a common name for the ratio.[13]