vertical circle, in astronomy, the great circle on the celestial sphere that passes from the observer's zenith through a given celestial body. In the altazimuth coordinate system the altitude of a body is measured along its vertical circle.

squaring the circle: see geometric problems of antiquity.

meridian circle: see transit instrument.

hour circle, in astronomy, a secondary axis in the equatorial coordinate system. The hour circle of a celestial body is the great circle on the celestial sphere that passes through both the body and the north celestial pole. A star's hour circle is used in determining its right ascension and declination.

diurnal circle, apparent path followed by a star due to the earth's rotation on its axis. The stars appear to move on the celestial sphere in concentric circular paths centered at the celestial poles. Since the earth rotates west to east, the stars appear to move from east to west along their diurnal circles. Stars whose diurnal circles lie completely above the horizon are called circumpolar stars.

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 A=πr². 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.

Knights of the Golden Circle, secret order of Southern sympathizers in the North during the Civil War. Its members were known as Copperheads. Dr. George W. L. Bickley, a Virginian who had moved to Ohio, organized the first "castle," or local branch, in Cincinnati in 1854 and soon took the order to the South, where it was enthusiastically received. Its principal object was to provide a force to colonize the northern part of Mexico and thus extend proslavery interests, and the Knights became especially active in Texas. Secession and the outbreak of the Civil War prompted a shift in its aims from filibustering in Mexico to support of the new Southern government. Appealing to the South's friends in the North, particularly in areas that were suffering economic dislocation, the order soon spread to Kentucky, Indiana, Ohio, Illinois, and Missouri. Its membership in these states, where it became strongest, was largely composed of Peace Democrats, who felt that the Civil War was a mistake and that the increasing power of the federal government was leading toward tyranny. They did not, however, at this time engage in any treasonable activity. In late 1863 the Knights of the Golden Circle was reorganized as the Order of American Knights and again, early in 1864, as the Order of the Sons of Liberty, with Clement L. Vallandigham, most prominent of the Copperheads, as its supreme commander. Only a minority of its membership was radical enough—in some localities—to discourage enlistments, resist the draft, and shield deserters. Numerous peace meetings were held. A few extreme agitators, some of them encouraged by Southern money, talked of a revolt in the Old Northwest, which, if brought about, would end the war. Southern newspapers wishfully reported stories of widespread disaffection, and John Hunt Morgan's raid (1863) into Kentucky, Indiana, and Ohio was undertaken in the expectation that the disaffected element would rally to his standard. Gov. Oliver P. Morton of Indiana and Gen. Henry B. Carrington effectively curbed the Sons of Liberty in that state in the fall of 1864. With mounting Union victories late in 1864, the order's agitation for a negotiated peace lost appeal, and it soon dissolved.

Arctic Circle, imaginary circle on the surface of the earth at 661/2°N latitude, i.e., 231/2° south of the North Pole. It marks the northernmost point at which the sun can be seen at the winter solstice (about Dec. 22) and the southernmost point of the northern polar regions at which the midnight sun is visible.

Antarctic Circle, imaginary circle on the surface of the earth at 661/2°S lat., i.e., 231/2° north of the South Pole. It marks the southernmost point at which the sun can be seen at the winter solstice (about June 22) and the northernmost point of the southern polar regions at which the midnight sun is visible.

Parallel of latitude approximately 66°30' north of the Equator that circumscribes the northern frigid zone. It marks the southern limit of the area within which, for one day or more each year, the sun does not set or rise. The length of continuous day or night increases northward from the Arctic Circle, mounting to six months at the North Pole.

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Parallel of latitude approximately 66°30' south of the Equator that circumscribes the southern frigid zone. It marks the northern limit of the area within which, for one day or more each year, the sun does not set or rise. The length of continuous day or night increases southward from the Antarctic Circle, mounting to six months at the South Pole.

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