sphere of influence, term formerly applied to an area over which an outside power claims hegemony with the intention of subsequently gaining more definite control, as in colonization, or with the intention of securing an economic monopoly over the territory without assuming political control. A sphere of influence was usually claimed by an imperialistic nation over an underdeveloped or weak state that bordered an already existing colony. The expression came into common use with the colonial expansion of European powers in Africa during the late 19th cent. A sphere of influence was formalized by treaty, either between two colonizing nations who agreed not to interfere in one another's territory, or between the colonizing nation and a representative of the territory. Theoretically, the sovereignty of a nation was not impaired by the establishment of a sphere of influence within its borders; in actuality, the interested power was able to exercise great authority in the territory it dominated, and if disorders occurred it was in a position to seize control. Thus the creation of spheres of influence was frequently the prelude to colonization or to the establishment of a protectorate. The term in this sense is no longer recognized in international law, however. Currently, it is used by the more powerful nations of the world to denote the exclusive or predominant interest they may have in certain areas of the globe, especially for the purposes of national security.

sphere, in geometry, the three-dimensional analogue of a circle. The term is applied to the spherical surface, every point of which is the same distance (the radius) from a certain fixed point (the center), and also to the volume enclosed by such a surface. The curve formed by a plane cutting a sphere is a circle. If the plane goes through the center of the sphere, the circle is called a great circle of the sphere. It is the largest circle that can be drawn upon the sphere, and all great circles of the same or equal spheres are of equal size. The shortest distance between two points on a spherical surface, measured on the surface, is the distance along the great circle through those points. A plane cutting a sphere in a great circle divides the sphere into two equal segments called hemispheres. The diameter of a sphere is the diameter of one of its great circles. The formula for the area of the surface of a sphere is S=4πr², and for the volume it is V=4/3 πr³, where r is the radius of the sphere. Spherical geometry and spherical trigonometry are methods of determining magnitudes and figures on a spherical surface.

celestial sphere, imaginary sphere of infinite radius with the earth at its center. It is used for describing the positions and motions of stars and other objects. For these purposes, any astronomical object can be thought of as being located at the point where the line of sight from the earth through the object intersects the surface of the celestial sphere. In astronomical coordinate systems, the coordinate axes are great circles on the celestial sphere. In most systems of this type, the reference points are fixed on the sphere, so the two coordinates needed to locate a body are relatively constant.

In international politics, a state's claim to exclusive or predominant control over a foreign area or territory. Beginning in the late 1880s, European colonial powers undertook legal agreements consisting of promises not to interfere with each other's actions in mutually recognized spheres of influence in Africa and Asia. After colonial expansion ceased, geopolitical rather than legal claims to spheres of influence became common, examples being the U.S. claim to dominance in the Western Hemisphere under the much-earlier Monroe Doctrine and the Soviet Union's expansion of its sphere of influence to eastern Europe following World War II. Seealso Iron Curtain.

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Apparent surface of the heavens, on which the stars seem to be fixed. For the purpose of establishing celestial coordinate systems to mark the positions of heavenly bodies, it can be thought of as a real sphere at an infinite distance from Earth. Earth's rotational axis, extended to infinity, touches this sphere at the northern and southern celestial poles, around which the heavens seem to turn. The intersection of the plane of Earth's Equator with the sphere marks the celestial equator.

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