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circle

circle, closed plane curve consisting of all points at a given distance from some fixed point, called the center. A circle is a conic section cut by a plane perpendicular to the axis of the cone. The term circle is also used to refer to the region enclosed by the curve, more properly called a circular region. The radius of a circle is any line segment connecting the center and a point on the curve; the term is also used for the length r of this segment, i.e., the common distance of all points on the curve from the center. Similarly, the circumference of a circle is either the curve itself or its length of arc. A line segment whose two ends lie on the circumference is a chord; a chord through the center is the diameter. A secant is a line of indefinite length intersecting the circle at two points, the segment of it within the circle being a chord. A tangent to a circle is a straight line touching the circle at only one point, the point of contact, or tangency, and is always perpendicular to the radius drawn to this point. A circle is inscribed in a polygon if each side of the polygon is tangent to the circle; a circle is circumscribed about a polygon if all the vertices of the polygon lie on the circumference. The length of the circumference C of a circle is equal to π (see pi) times twice the radius distance r, or C=2πr. The area A bounded by a circle is given by Ar2. Greek geometry left many unsolved problems about circles, including the problem of squaring the circle, i.e., constructing a square with an area equal to that of a given circle, using only a straight edge and compass; it was finally proved impossible in the late 19th cent. (see geometric problems of antiquity). In modern mathematics the circle is the basis for such theories as inversive geometry and certain non-Euclidean geometries. The circle figures significantly in many cultures. In religion and art it frequently symbolizes heaven, eternity, or the universe.

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circle

cir·cle / ˈsərkəl/ (abbr.: cir. or circ.) • n. 1. a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center). ∎  the line enclosing such a figure. ∎  something in the shape of such a figure: the lamp spread a circle of light. ∎  a group of people or things arranged to form such a figure: they all sat around in a circle. ∎  a movement or series of movements that follows the approximate circumference of such a figure: the astrological houses rotate in a circle. ∎  a dark circular mark below each eye, typically caused by illness or tiredness. ∎  a curved upper tier of seats in a theater.See also dress circle. 2. a group of people with a shared profession, interests, or acquaintances: she did not normally move in such exalted circles. • v. [tr.] move all the way around (someone or something), esp. more than once: the two dogs circle each other with hackles raised | [intr.] we circled around the island. ∎  [tr.] (from the air) move in a ring-shaped path above (someone or something), esp. more than once: they were circling the airport. ∎  [intr.] (circle back) move in a wide loop back toward one's starting point. ∎  (often be circled) form a ring around. ∎  draw a line around: circle the correct answers. PHRASES: come (or turn) full circle return to a past position or situation, esp. in a way considered to be inevitable. go around (or around and around) in circles inf. do something for a long time without achieving anything but purposeless repetition. run around in circles inf. be fussily busy with little result.

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Circle

Circle

A circle can be defined as a closed curved line on which every point is equally distant from a fixed point within it. Following is some of the terminology used in referring to a circle:

1. The fixed point is called the center of the circle (C in Figure 1).

2. A line segment joining the center to any point on the circle is the radius of the circle (CA in Figure 1).

3. A line segment passing through the center of the circle and joining any two points on the circle is the diameter of the circle (DB in Figure 1). The diameter of a circle is twice its radius.

4. The distance around the circle is called the circumference of the circle.

5. Any portion of the curved line that makes up the circle is an arc of the circle (for example, AB or DA in Figure 1).

6. A straight line inside the circle joining the two end points of an arc is a chord of the circle (DE in Figure 1).

Mathematical relationships

One of the interesting facts about circles is that the ratio between their circumference and their diameter is always the same, no matter what size the circle is. That ratio is given the name pi (π ) and has the value of 3.141592+. Pi is an irrational number. That is, it cannot be expressed as the ratio of two whole numbers. The + added at the end of the value above means that the value of pi is indeterminate: you can continue to divide the circumference of any circle by its diameter forever and never get an answer without a remainder.

The area of any circle is equal to its radius squared multiplied by π, or: A = π r2. The circumference of a circle can be found by multiplying its diameter by π (C = π D) or twice its radius by π (C = 2πr).

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Circle

Circle

a set or series of parts connected to form a whole; a company assembled about a central point or topic of interest; a circular ring of persons or things. See also company, ring.

Examples: circle of action, 1752; of admirers, 1793; of acquaintances, 1752; of doctrine, 1531; of fallacy, 1646; of foliages, 1713; of glory, 1595; literary circle; circle of onlookers, 1875; of pleasures, 1759; of passion, 1768; of possibilities, 1644; of probability, 1851; of sciences, 1854; of stars, 1611.

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circle

circle circle the wagons unite in defence of a common interest (originally North American, with reference to the defensive position of a wagon train under attack).
the wheel has come full circle the situation has returned to what it was in the past, as if completing a cycle, with reference to Shakespeare's King Lear, by association with the wheel fabled to be turned by Fortune and representing mutability.

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circle

circle sb. XIV. ME. cercle — (O)F. :- L. circulus, dim. of circus ring (see CIRCUS); later respelt after L. OE. circul was an adoption directly from the Latin, but did not survive.
So circle vb. XIV. — L. circulāre, or from the sb.

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circle

circle Plane geometric figure that is the locus of points equidistant from a fixed point (the centre). This distance is the radius (r). The area of a circle is πr2 and its perimeter (circumference) is 2πr.

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circle

circlecackle, crackle, grackle, hackle, jackal, mackle, shackle, tackle •ankle, rankle •Gaskell, mascle, paschal •tabernacle • ramshackle •débâcle, diarchal, matriarchal, monarchal, patriarchal, sparkle •rascal •deckle, freckle, heckle, Jekyll, shekel, speckle •faecal (US fecal), treacle •chicle, fickle, mickle, nickel, pickle, prickle, sickle, strickle, tickle, trickle •besprinkle, crinkle, sprinkle, tinkle, twinkle, winkle, wrinkle •fiscal •laical, Pharisaical •vehicle • stoical • cubicle • radical •medical, paramedical •Druidical, juridical, veridical •syndical •methodical, periodical, rhapsodical, synodical •Talmudical • graphical • pontifical •magical, tragical •strategical •alogical, illogical, logical •dramaturgical, liturgical, metallurgical, surgical •anarchical, hierarchical, monarchical, oligarchical •psychical •angelical, evangelical, helical •umbilical • biblical • encyclical •diabolical, follicle, hyperbolical, symbolical •dynamical, hydrodynamical •academical, agrochemical, alchemical, biochemical, chemical, petrochemical, photochemical, polemical •inimical • rhythmical • seismical •agronomical, anatomical, astronomical, comical, economical, gastronomical, physiognomical •botanical, Brahmanical, mechanical, puritanical, sanicle, tyrannical •ecumenical •geotechnical, pyrotechnical, technical •clinical, cynical, dominical, finical, Jacobinical, pinnacle, rabbinical •canonical, chronicle, conical, ironical •tunicle • pumpernickel • vernicle •apical • epical •atypical, prototypical, stereotypical, typical •misanthropical, semi-tropical, subtropical, topical, tropical •theatrical •chimerical, clerical, hemispherical, hysterical, numerical, spherical •calendrical •asymmetrical, diametrical, geometrical, metrical, symmetrical, trimetrical •electrical • ventricle •empirical, lyrical, miracle, panegyrical, satirical •cylindrical •ahistorical, allegorical, categorical, historical, metaphorical, oratorical, phantasmagorical, rhetorical •auricle • rubrical • curricle •classical, fascicle, neoclassical •farcical • vesicle •indexical, lexical •commonsensical, nonsensical •bicycle, icicle, tricycle •paradoxical • Popsicle • versicle •anagrammatical, apostatical, emblematical, enigmatical, fanatical, grammatical, mathematical, piratical, prelatical, problematical, sabbatical •impractical, practical, syntactical, tactical •canticle •ecclesiastical, fantastical •article, particle •alphabetical, arithmetical, heretical, hypothetical, metathetical, metical, parenthetical, poetical, prophetical, reticle, synthetical, theoretical •dialectical •conventicle, identical •sceptical (US skeptical) • testicle •analytical, apolitical, critical, cryptanalytical, diacritical, eremitical, geopolitical, hypercritical, hypocritical, political, socio-political, subcritical •deistical, egoistical, logistical, mystical, papistical •optical, synoptical •aeronautical, nautical, vortical •cuticle, pharmaceutical, therapeutical •vertical • ethical • mythical • clavicle •periwinkle • lackadaisical •metaphysical, physical, quizzical •whimsical • musical •Carmichael, cervical, cycle, Michael •unicycle • monocycle • motorcycle •cockle, grockle •corncockle • snorkel •bifocal, focal, local, univocal, varifocal, vocal, yokel •archducal, coucal, ducal, pentateuchal •buckle, chuckle, knuckle, muckle, ruckle, suckle, truckle •peduncle, uncle •parbuckle • carbuncle • turnbuckle •pinochle • furuncle • honeysuckle •demoniacal, maniacal, megalomaniacal, paradisiacal, zodiacal •manacle • barnacle • cenacle •binnacle • monocle • epochal •reciprocal •coracle, oracle •spectacle •pentacle, tentacle •receptacle • obstacle • equivocal •circle, encircle •semicircle

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Circle

CIRCLE

CIRCLE . The circle is used as a polyvalent symbol (commonly representing the cosmos and cosmic movement) as well as a pattern of ritual action (in which macrocosmic realities are transformed into microcosmic space with various meanings). As a symbol and as a ritual pattern, the circle is a cross-cultural form occurring in the round shapes of houses, public buildings, tombs, cult objects (such as altars), and ritual spaces. The circle is used ritually by inscribing circles on the ground, on amulets, or other objects and as a pattern for processions around altars, temples, spaces, and towns for various reasons. Circular shapes are often understood as patterned after the solar and lunar disks, and circular movements are frequently thought to replicate the circular motion of heavenly bodies; both circular shapes and motions are frequently assigned a sacred or religious function. In the past, scholars frequently indulged in the vain pursuit of the origins of the ritual use of circular shapes and motions, falsely assuming that such traditions have a unified origin providing a key to understanding their meaning. However, the meaning of such ritual patterns is probably polyvalent and must be based on contextual analysis, combining the emic explanations found in ancient interpretations and the etic explanations arrived at through cross-cultural comparison by modern theorists.

Neolithic and Bronze Age

The megalithic passage tombs in eastern Ireland, at Newgrange, Knowth, and Dowth (c. 3200 bce), built by Neolithic farming communities, exhibit a ritualistic architecture. They are laid out in large circular shapes that have clear astronomical alignments, such as the winter solstice sunrise at Newgrange and the equinox sunrise at Loughcrew. According to ancient Roman sources, the Gauls associated the moon with death, and it may well be that the shape of the moon with this symbolic significance is replicated in circular megalithic tomb architecture. The same is true of the megalithic stone circle constructed somewhat later at Stonehenge in the vicinity of Salisbury, Wiltshire, England (constructed in three stages during the Late Neolithic and Early Bronze Age, c. 18001400 bce). Stonehenge IIIa (c. 1600 bce) consists of a circle of thirty upright monoliths capped by a continuous ring of carefully dressed stone lintels. The solstitial alignment of the various phases of the construction of Stonehenge suggest that it functioned as a place of worship involving the sun and moon, though little more is known. John North, in Stonehenge: Neolithic Man and the Cosmos, argues, "The aim [of the Neolithic builders] was not to discover the patterns of behaviour of the sun, Moon or stars but to embody those patterns, already known in broad outline, in a religious architecture" (North, 1996, p. xxxvi).

Unusual evidence for the cults of prehistoric Cyprus is in a clay sanctuary model of polished red earthenware found in a dromos tomb, part of an extensive necropolis dating to the Early Bronze, c. 2000 bce. Described in detail in "The Excavations at Vounous-Bellapais in Cyprus, 193132" (Dikaios, 1938), the model consists of an open-air temenos or sacred precinct enclosed by a circular wall with a large arched entrance. On the floor is a semicircular curb that separates three statues of divinities from the rest of the temenos. Numerous seated and standing figures suggest that a ceremony of some type is depicted, which somehow involves the symbolic significance of the bulls, heads, and snakes that decorate the wall opposite the entrance. The circular temenos wall contrasts with other Early and Middle Bronze domestic architecture in which the rectangle predominates. Since the fundamental architectural principle in the Neolithic and Chalcolithic periods in Cyprus is the circle, the round temenos of this clay model represents a survival from an earlier period (the circular form also characterizes some Early Bronze age tombs at Vounous).

Ancient Greece and Rome

The Greeks had several words for "circle," including gyros (a trench around trees, used for the circle of the heaven or earth in LXX Iob 22.14 and Is 40.22), kyklos (the circle of the sky in Herodotos 1.131), and trochos (wheel, circular race); the terms kirkos and krikos both mean circle in the sense of a "ring" or "hoop." The preposition peri is prefixed to a number of verbs with variations on the meaning "to encircle," "surround" (e.g., periechō, periistēmi, perikykloō, peritechizō, peritithēmi, peritrechō ). The primary Latin word for "circle" or "circular course" is circus and its diminutive form circulus, which describes "a circular figure or form." The preposition circum (around, about) is used as a prefix for a large number of verbs to describe various types of circular movement. According to the dominant pre-Hellenistic cosmology, the earth was shaped like a circular disk, encircled by Ocean (Herodotos 4.36), flowing in one direction (clockwise); the river Acheron, further out, flowed in the opposite direction, and Tartarus, the land of the dead, was located below the earth (Plato, Phaedo 112e). Concentric circles dominate this cosmology. According to Plato (Philebus 62a), Socrates speaks of the person who has knowledge of the divine circle and sphere (kyklou men kai sphairas )based on the Platonic doctrine of ideasbut is ignorant of the human sphere and circle, even when building a circular house. Here Plato's theory of ideas provides a basis for distinguishing between microcosmic imitations of the macrocosmic circle or sphere.

In the lengthy description (ekphrasis ) of the shield of Achilles in Iliad 18.483608, one design depicts a city at peace where a dispute has arisen between two men. Heralds keep back the crowd from the area where elders sit upon polished stones "in a sacred circle" (Iliad 18.504). The scholiast explains that "the law courts are sacred." The reason they are sacred, thus rendering the circle of stone seats as sacred, is the belief that Zeus presides over judicial proceedings (Iliad 9.9899). Eustathius, expanding on the scholiast, comments in Eustathii Commentarii ad Homeri Iliadem Pertinentes, "A sacred circle is the kind in the agora, where because of local law and custom, such a circle is understood to be sacred" (Eustathius, 19971987, p. 4.236). This reflects a link between the "sacred sircle" consisting of "smooth stones," sometimes with a sacred hearth or pit at the center (Odyssey 6.266), where the speaking and debating was done, and the "encircling agora" (Euripides, Orestes 919), where the assembly was gathered. Speakers within the "sacred circle" customarily held a scepter and enjoyed a limited immunity.

In Plutarch's narrative of the founding of Rome by Romulus (Romulus 9), he relates how a circular trench (bothros kykloteres ) was dug around what later became the Comitium (a place of public assembly for the Comitia Curiata, which by the third century ce became a circular amphitheatre), into which each participant placed fruits and some earth from his native land. This trench was called mundus, reflecting a conscious cosmic symbolism. With this center, the city was marked out in a circle indicating the pomerium, leaving unplowed the places for the gates. The wall was thought to be sacred, for the city was the dwelling place of gods and people. The mythical character of this story is underlined by the fact that, although the city was traditionally called Roma Quadrata, the plowed trench is described as circular.

Round Altars and Temples

Vitruvius has a brief discussion of circular temples (aedae rutundae ) in which he focuses exclusively on architectural matters (On Architecture 4.8). Servius claims that round temples were usually dedicated to Vesta, Diana, Hercules, and Mercury (Commentarii in Aeneidem 4.8.3). While there appears to be a close association between circular spatial and architectural forms and hero cults, there are no hard and fast rules or associations. In the Greek world of the classical period, round temples were commonly found in connection with the cult of Hestia, where the hearth of the polis was located. The hearth of individual homes as well as the hearths of cities clearly symbolized the sacred center of both.

The Greek term tholos generally referred to a round building with a conical roof in the archaic and classical periods, but in the Hellenistic period the same term is used for a variety of complex round architectural forms. In Athens, the term tholos was used of the rotunda or prytaneion, called the "Skias" in inscriptions, in which the magistrates dined (Plato Apology 32c; Andocides 1.45; Demosthenes 19.249; Aristotle Athenian Constitution 43.3; Pausanias 1.5.1). The prytaneion at Epidauros was also called the Thymela in inscriptions and tholos by Pausanias (2.27.25), constructed in the 380s bce. Important tholoi of the Hellenistic period include the Rotunda of Arsinoe in Samothrake in the sanctuary of the Great Gods, built in the 280s bce; the tholos near Kepoi on the Black Sea; and the round court with three annexed tholoi in Pella. The temple of Vesta (aedes Vestae ) in Rome was a rotunda where the city hearth was located. It contained no image of the goddess and was part of a complex of buildings called the Atrium Vestae. It was circular and thought to have originated as a structure of wattles with a thatched roof, for example, preserving the tradition of a primitive Italic round hut (Ovid Fasti 6.261266).

The most famous round temple in Rome is the Pantheon, actually the third in a succession of three buildings, the last built after 118 ce by Hadrian. Rather than a temple sensu stricto, the Pantheon in its three reincarnations was a dynastic monument (Hadrian reportedly held court there), a templum mundi (i.e., a "temple of the world") with Rome and its emperor at the center of the Roman world. The cosmic symbolism of the enormous dome as representing the celestial home of the gods struck Dio Cassius (59.27.24). The oculus ("eye") at the top of the dome provides all the illumination for the building, which would have spotlighted different parts of the floor and walls with the movement of the sun.

Two round temples of Hercules were erected in Republican Rome, a temple of Hercules Victor ad Forum Boarium (Livy 10.23.3) and a temple of Hercules Victor ad Portam Trigeminam (Macrobius Sat. 3.6.10). Though Augustus did not erect any significant cultic rotundas in Rome, the arch and exedra shapes were used extensively during his principate. In Augustan temples, apses framed cult statues delimiting a divine realm.

Round shapes were closely associated with the graves, shrines, and temples of heroes in Greco-Roman antiquity, though there is little evidence that either a consistent association with heroes or with cosmic symbolism determined the architectural use of circles, apses, and domes. The grave of Aeptytus was reportedly a mound surrounded by a circular base of stone (Iliad 2.592; Pausanias 8.16.3); the oracle of Trophonius was a circular construction of white marble (Pausanias 9.39.9), and Osiris reportedly had a circular tomb (Herodotos 2.170). The Maussolleion, the monumental tomb of Maussollos of Caria (d. 353 bce) and his wife Artemisia was constructed in the vicinity of Halicarnassus (Strabo 14.656; Diodorus 16.45; Pliny hist. nat. 36, 3031). The Mausoleum Augusti, the first Augustan building on the Campus Martius was begun in 28 bce but not completed until several years later. The circular marble base measured more than eighty-five meters in diameter and surrounded a mound about forty-five meters high (Suetonius Aug. 100.4; Strabo 5.3.8 [236]). A bronze statue of Augustus was located at the summit. Alexander the Great also had a circular tomb. The Mausoleum Hadriani was constructed with a square base eighty-seven meters on each side and ten meters high. Mounted on this base is a drum sixty-four meters in diameter and perhaps twenty-one meters high. The Mausoleum Hadriani had only the circular shape in common with the Mausoleum Augusti.

The ritual of marching around a sacred place, often carrying ritual objects, whether an altar or shrine, as a preliminary means of setting such a sacred place apart for cultic purposes was widespread throughout the ancient world. One aspect of the protocol of classical Hellenic sacrifice involved the ritual encirclement of the sacred space containing the altar, the worshipers, and the victim before the killing of the victim. Two ritual objects which are frequently mentioned as being carried around the altar are a basin containing lustral water and a basket containing barley corns, a fillet, and a knife (Aristophanes Peace 948962, 971, Birds 850, 958; Lysistrata 11291131). In a festival called Laphria in honor of Artemis, logs of green wood were arranged in a circle around the altar (Pausanias 7.18.11).

Ancient Mediterranean Magic

The ritual circle, when used by individuals for private and antisocial purposes, becomes a magic circle. The ouroboros the figure of a snake "biting" (bora ) its "tail" (oura ), thus forming a circleis a polyvalent ancient Egyptian symbol representing many things, including the sun, the moon, an earth-surrounding boundary, rejuvenation and rebirth, eternity, or a cartouche for the names of kings with claims to be world rulers. Two ouroboroi were incised on the walls of a shrine of Tutankhamen (13571349 bce), one encircling his feet and the other his head. The serpent about the head is named Menen the Enveloper. In a papyrus of the twenty-first dynasty, the deceased woman (named Her-Uben) adores the solar disk surrounded by an ouroboros representing eternity. The "Book of Overthrowing Apep," from the Ptolemaic period in Egypt, describes one use of the ouroboros figure, which is pierced with a knife and thrown to the ground to destroy the evil beings associated with Apep.

While the ouroboros is rarely mentioned in classical and Hellenistic Greek texts, Plato relates a cosmology in which he describes certain rivers as coiling around the earth one or more times in a circle like serpents (Phaedo 112e7), which seems to reflect the ouroboros mythology. The ouroboros is commonly found on magical amulets, typically functioning as a border providing sanctity to that which is depicted within it, sometimes functioning as a symbol for the universe, eternity, or the year. Such an amulet is described in The Greek Magical Papyri in Translation : "And engraved on the stone is: Helios as a lion-faced figure, holding in the left hand a celestial globe and a whip, and around him in a circle is a serpent biting its tail [ouroboros ]" (Betz, 1992, p. 7). Another magical text gives instructions for an amulet on a lamella or papyrus containing a sequence of magical words, magical characters, and an inscription: "Protect my body and the entire soul of me [insert name]," all written inside an ouroboros serpent (Betz, 1992, p. 134). This protective charm placed the bearer in the protective cosmic circle framed by the ouroboros, symbolizing protective encirclement. The ouroboros continued to be popular through the Middle Ages. A fourteenth-century ce Venetian alchemical manuscript is pictured in Gnosis: The Nature and History of Gnosticism, depicting the ouroboros encircling an inscription in Greek meaning "the All is One" (Rudolf, 1983, p. 70).

Acts of Thomas 32 (a third-century ce Christian document) refers to the ouroboros serpent, for the snake speaking to Thomas claims to be related to "the one who is outside the ocean, whose tail is set in his own mouth." A similar ouroboros conception is in the Coptic Gnostic Pistis Sophia (126): "The outer darkness is a great serpent whose tail is in its mouth, and it is outside the whole world, and it surrounds the whole world." In Christian Coptic magical texts, mention is made of the drawing of a magic circle around a person to prevent demons from entering. The comparative rarity of these references suggests that the ritual and magical use of the circle played only a minor role in early Christian ritual practices.

Rings were often used as magical objects because of the inherent power of the circular shape. Greek Magical Papyri in Translation (Betz, 1992) contains instructions for preparing a defixio in which the inside and outside of an iron ring (kirkos ) are traced on papyrus with formulas and symbols to be inscribed within the outline of the ring as well as inside and outside the outline. In another text in the Greek Magical Papyri (Betz, 1992, XII, pp. 270350), there are instructions for making a ring, also called a kyklos or "circle," on which an ouroboros serpent is engraved on heliotrope stone. This ring is said to be useful for opening doors, breaking chains, and performing exorcisms. Some magical procedures are written, like the script on magical bowls, in a tight spiral from inside to outside with a figure in the middle. Magical formulas can also be written in a circle on various materials, including an olivewood table (Betz, 1992, III, p. 292), the hide of an ass (Betz, 1992, IV, p. 2016), a papyrus sheet (Betz, 1992, IV, p. 2070), and a shell (Betz, 1992, VII, pp. 468470). The one performing a magical procedure can also stand in the center of a protective circle drawn on the ground with chalk (Betz, 1992, VII, p. 858).

Ancient Israel and Early Judaism

The verb chûgh belongs to the semantic field of "circles and circular motion" and is distinguished from other lexemes in this field by its geometrical meaning "to draw a circular line [with a compass]." The term is used in cosmological contexts for describing two concentrical circular boundaries, the earth disk and the heavenly mountain island. The circle of the horizon is described in Proverbs 8:27: "When he established the heavens, I was there, when he drew a circle on the face of the deep." Isaiah 40:22 refers to God as "he who sits above the circle of the earth," while Job 26:10 says that God "has described a circle upon the face of the waters." The heavens also are circular in Job 22:14, which describes God as "walking on the circle of heaven." These important texts attest to the Israelite perception of the circle as a cosmological shape, which can serve as a pattern for ritual imitation. There is meager evidence for Israelite ritual circumambulation of the altar prior to sacrifice in Psalms 26:6: "I wash my hands in innocence and go about thy altar, O Lord," where the purpose is to enclose a sacred area so that evil influences cannot penetrate. Joshua 6:34, a fragment of a liturgical or ceremonial text, contains divine instructions to march around Jericho once each day for six days, culminating in seven encirclements on the seventh day. The Septuagint Joshua 6:3 is much shorter than the Masoretic text: "And you arrange the fighting men in a circle [kyklōi ] around it [the city]." This circumambulation ritual, in which the number seven plays an important role, can be interpreted as a ritual means of laying claim to territory or as a ceremony of ritual cursing.

In Mishnah Taanith 3:8, the story is told of how Honi ha-Me'aggel ("the Circle-Drawer"), a first century bce holy man, prayed for rain. When his prayer was not answered, he drew a circle (Hebrew, me'aggel), swearing an oath by God's great name that he would not step outside the circle until God sends rain. When a few drops fell, he complained that this was not enough. When it rained torrentially, he complained that it was too much. It then began to rain moderately. This story is summarized in Josephus (Ant. 14:22), who calls him Onias, but with the magical features suppressed. The fact that he is addressed in rabbinic sources as "the circle-drawer" suggests that this epithet reflected a fixed feature of his prayer ritual.

Circular and spherical shapes were combined in the magical bowls made and used by Jews, Mandaeans, Christians, and Manichaeans from the fifth to the seventh centuries ce in Mesopotamia for apotropaic and exorcistic purposes. The ouroboros is occasionally found in the center of Aramaic incantation bowls, however, in these cases the circle does not circumscribe a place of protection but is rather a place for trapping a demon, specifically depicted within the ouroboros, at the bottom center of the bowl. The adjurations are written in a long, tight spiral beginning from the bottom of the bowl and ending up near the rim. These inscriptions are often framed by two circles, one at the bottom of the bowl (sometimes replaced with the ouroboros ) and near the rim of the bowl.

Native American Cultures

The Adena culture, an Early Woodland culture of eastern North America, consisting of numerous small communities of ancient North American Indians who occupied the middle Ohio River Valley (c. 800 bce to c. 200 ce), used circular architecture in the construction of houses of poles and bark and ceremonial circles constructed of earth. There are approximately five hundred extant Adena cites, three hundred in the central Ohio Valley and the rest scattered in Pennsylvania, West Virginia, Kentucky, and Indiana. Some of the larger Adena sites consist of large earthworks in the shapes of circles and other geometric figures. The Grave Creek Mound (Moundsville, West Virginia), the largest Adena burial mound, is 240 feet in diameter and 62 feet high, with an encircling ditch (4045 feet wide; 45 feet deep). The Dominion site (located in present-day Columbus, Ohio) is the oldest Adena circle (dated some time after 500 bce). Other Adena circles include the Mount Horeb site in Kentucky and the Anderson site in Indiana. These circular structures have no obvious or certain interpretation, though astronomical and cosmological symbolism is highly likely.

The term medicine wheel was first applied to the stone circle, cairn, and spoke configuration called the Big Horn Medicine Wheel ten thousand feet above sea level on Medicine Mountain in Wyoming. Medicine wheels were originally small decorative hoops3 inches to 4 inches in diametermade by the Cheyenne or Ojibwa with a web in the middle or two or more spokes bridging the circle and with several bird feathers attached to the lower perimeter. The term medicine indicates the ritual significance of both the miniature hoops and by extension the stone circle configurations. The Big Horn Medicine Wheel, which is about eighty-five feet in diameter with a central circular cairn thirteen feet in diameter, has twenty-seven stone spokes corresponding to the twenty-seven days of the lunar month. Following the identification of the Big Horn Medicine Wheel, about seventy-five similar artificially constructed stone surface forms were identified (in the northern Plains from Wyoming to South Dakota and north into Canada), characterized by a variety of stone circle, central cairn, and spoke configurations. These medicine wheels have several common features: they are made of unmodified chunks of stone, and they include a central cairn, two or more stone spokes, and one or more concentric stone rings arranged in a symmetrical form. The medicine wheels were frequently added to, making dating difficult. The earliest medicine wheels were constructed by the Oxbow complex in southeastern Saskatchewan (and extending somewhat into Alberta, Manitoba, Montana, and North Dakota), dating from 2750 to 1050 bce. These were later embellished by subsequent cultures. Some of the medicine wheels clearly had solar and calendrical functions, and all of them were the sites of special ceremonies.

The Lakota and several other Plains tribes depict the sacred cosmic order represented by Wakan-Tanka ("Great Spirit") with the circle, and they understand the circle as a key symbol representing the whole of the universe and their part in it. The centrally important significance of the circle is emphasized in the following statement by Black Elk, in Black Elk Speaks :

You have noticed that everything an Indian does is in a circle, and that is because the Power of the World always works in circles, and everything tried to be round. In the old days when we were a strong and happy people, all our power came to us from the sacred hoop of the nation, and so long as the hoop was unbroken, the people flourished. The flowering tree was the living center of the hoop, and the circles of the four quarters nourished it. (Neihardt, 1979, p. 194)

The largest representation of the circle among a number of Plains tribes (including the Arapaho, Kiowa, Cheyenne, and Sioux) were their encampments, called "the sacred hoop," just as they speak of the "hoop of the world." Everything within the camp circle was Lakota, while outside were enemies, evil spirits, and eventually the white people. Tribal divisions among the Cheyenne were located at the same place in the circular encampment, and the lodge of medicine arrows and the lodge of the buffalo cap were placed within the circle at predetermined locations. According to Black Elk, Buffalo Calf Woman gave the peace pipe (with its round bowl) and prescribed the camp circle, both of which share comparable degrees of sanctity. The pipe bowl was decorated with seven circles, representing the seven major rituals of the Lakota. The centers of the sweat lodges and of the bowls of the sacred pipes, where fires were made, represented the sun at the center of a circular cosmos. When the world is perceived as disordered, it can only be restored to its proper balance by sacred ritual, which the Lakota refer to as "making roundness."

The Teotihuacan civilization of Mesoamerica (c. 400800 ce) oriented major urban sites using astronomical observations. Evidence for this survives in the form of "pecked" crosses (so-called because cross petroglyphs are "pecked" using a percussive device producing cuplike depressions in stone or plaster floors of important buildings and in rock outcroppings outside of buildings), indicating astronomical orientations. The form of pecked crosses typically consists of a double circle (sometimes single or triple) centered on a pair of orthogonal axes (Aveni, 1980, p. 227). In Teotihuacan (which was the location for such important buildings as the Pyramid of the Sun and the Pyramid of the Moon), the axes of the crosses align with the grid of that ceremonial city (the peculiar clockwise deviation from true north in the axial plan of Teotihuacan is shared by sites all over Mesoamerica), some apparently functioning as architectural benchmarks. The combination of pecked crosses and circles (reminiscent of calendar wheels) seems to unite spatial and calendrical with religious functions, though no single hypothesis can account for their origin.

See Also

Circumambulation.

Bibliography

Allcroft, A. H. The Circle and the Cross. London, 1927. A rich and lavishly illustrated collection of circular architectural, somewhat marred by a fixation on the problem of origins.

Altmann, Walter. Die italischen Rundbauten. Berlin, 1906. A classical discussion of round architectural forms in ancient Italy, particularly Rome.

Aveni, Anthony F. Skywatchers of Ancient Mexico. Austin, Tex., and London, 1980.

Betz, Hans Dieter, ed. The Greek Magical Papyri in Translation. 2d ed. Chicago, 1992.

Bonner, Campbell. Studies in Magical Amulets, Chiefly Graeco-Egyptian. Ann Arbor, Mich., 1950. A large collection of magical amulets including pictures and descriptions of many contained the ouroboros ("snake biting its tail").

Brown, Joseph Epes. The Gift of the Sacred Pipe. Norman, Okla., 1982. A transcription of some of the revelations of Black Elk, the Lakota medicine man.

Brumley, John H. Medicine Wheels on the Northern Plains: A Summary and Appraisal. Manuscript Series no. 12. Edmonton, Canada, 1988. An authoritative anthropological analysis of the native American stone circles in the northern Plains states and Canada.

Burl, Aubrey. Great Stone Circles: Fables, Fictions, Facts. New Haven, Conn., and London, 1999. A comprehensive treatment of stone circles in Britain, Ireland, and Brittany.

Castleden, Rodney. The Making of Stonehenge. London and New York, 1993.

Dikaios, P. "The Excavations at Vounous-Bellapais in Cyprus, 193132." Archaeologia 88 (1938): 1174.

Eustathius. Eustathii Commentarii ad Homeri Iliadem Pertinentes. Edited by M. van der Valk. Leiden, 19711987.

Gaster, T. H. Myth, Legend, and Custom in the Old Testament. New York, 1969. An important work focusing on many of the magical traditions of the Hebrew Bible, including the use of circular shapes and circular processions.

Grinnell, George Bird. The Cheyenne Indians: Their History and Ways of Life. 2 vols. New Haven, Conn., 1924.

Montgomery, James A. Aramaic Incantation Texts from Nippur. Philadelphia, 1913. An early and comprehensive publication of the corpus of magical bowls.

Naveh, Joseph, and Shaul Shaked. Amulets and Magic Bowls. Jerusalem, 1985. The publication of some incantation bowls discovered after Montgomery's 1913 publication.

Neihardt, John Gnelsenau. Black Elk Speaks. Lincoln, Neb., and London, 1979.

North, John. Stonehenge: Neolithic Man and the Cosmos. London, 1996. Important work on the cosmological symbolism of Stonehenge.

Paper, Jordan. Offering Smoke: The Sacred Pipe and Native American Religion. Moscow, Idaho, 1988.

Rakob, Friedrich, and Wolf-Dieter Heilmeyer. Der Rundtempel am Tiber in Rom. Mainz am Rhein, Germany, 1973. Architectural and archaeological analysis of one of the ancient round temples in Rome.

Richardson, L. A New Topographical Dictionary of Ancient Rome. Princeton, N.J., 1992.

Robert, Fernand. Thymélè: Recherches sur la signification et la destination des Monuments circulaires dans l'Architecture religieuse de la Grèce. Paris, 1939. A comprehensive discussion of circular architecture and its symbolism in the world of ancient Greece.

Rudolf, Kurt. Gnosis: The Nature and History of Gnosticism. San Francisco, 1983.

Seidenberg, Abraham. "The Ritual Origin of the Circle and Square." Archive for History of Exact Sciences 25, no. 4 (1981): 270321. An important article on the ritual uses of the circle, flawed by an emphasis on unknowable origins.

David E. Aune (2005)

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Circle

Circle

In the language of Euclidean geometry, a circle is the locus of points in a plane that are all at an equal distance from a single point, called the center of the circle. The fixed distance is called the radius of the circle. A line segment with each of its endpoints on the circle, which passes through the center of the circle, is called a diameter of the circle. The length of a diameter is twice the radius.

The distance around a circle, called its circumference, is the length of the line segment that would result if the circle were broken at a point and straightened out. This length is given by 2πr, where r is the radius of the circle and π (the Greek letter pi, pronounced pie) is a mathematical constant equal to approximately 3.14159. The circles arc is any portion of the circle lying between two points on the circle.

The history of the circle began before recorded history. Some historians think that the circle was likely first considered after the wheel was first invented. One of the first descriptions of the circle goes back to Egyptian scribe A hmes (16801620 BC) when he proposed a rule for finding the area of a circle. Later, Greek philosopher Thales of Miletus (624547 BC) developed the first recorded theorems involving the properties of circles.

Points lying outside the circle are those points whose distance from the center is greater than the radius of the circle, and points lying in the circle are those points whose distance from the center is less than the radius of the circle. Any line that intersects the circumference at one point is tangent to the circle, while if the line intersects it at two points, then it is called a secant. The area covered by a circle, including all the points within it, is called the area of the circle. The area of a circle is also related to its radius by the formula A = πr2, where A is the area, r is the radius, and π is the same constant as that in the formula for the circumference. Concentric circles are all circles that possess the same center but different lengths for diameters.

In the language of algebra, a circle corresponds to the set of ordered pairs (x, y) such that (x a)2 + (y b)2 = r2, where the point corresponding to the ordered pair (a, b) is the center of the circle and the radius is equal to r. Based on this equation, circles are classified within the family of curves called conic sections. As such, circles are expressed as the intersection of a right circular cone with a plane that is perpendicular to the cones axis.

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Circle

Circle

In the language of geometry , a circle is the locus of points in a plane that are all at an equal distance from a single point , called the center of the circle. The fixed distance is called the radius of the circle. A line segment with each of its endpoints on the circle, that passes through the center of the circle, is called a diameter of the circle. The length of a diameter is twice the radius. The distance around a circle, called its circumference, is the length of the line segment that would result if the circle were broken at a point and straightened out. This length is given by 2πr, where r is the radius of the circle and π (the Greek letter π, pronounced "pie") is a constant equal to approximately 3.14159. Points lying outside the circle are those points whose distance from the center is greater than the radius of the circle, and points lying in the circle are those points whose distance from the center is less than the radius of the circle. The area covered by a circle, including all the points within it, is called the area of the circle. The area of a circle is also related to its radius by the formula A = πr 2, where A is the area, r is the radius, and π is the same constant as that in the formula for the circumference. In the language of algebra , a circle corresponds to the set of ordered pairs (x,y) such that (x - a)2 + (y - b)2 = r2, where the point corresponding to the ordered pair (a,b) is the center of the circle and the radius is equal to r.

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