1911 Encyclopædia Britannica/Sphere

SPHERE (Gr. σφαῖρα, a ball or globe), in geometry, the solid or surface traced out by the revolution of a semicircle about its diameter; this is essentially Euclid's definition;^[1] in the modern geometry of surfaces it is defined as the quadric surface passing through the circle at infinity. Every point is equidistant from a fixed point within the surface; this point is the “centre,” the constant distance the “radius,” and any line through the centre and intersecting the sphere is a “diameter.” All sections of the sphere are necessarily circles; if the cutting plane contains the centre, the section is said to be “meridional,” the curve of intersection is a “great circle,” and the solid cut off a “hemisphere.” If the plane does not contain the centre, the curve of intersection is a “small circle,” and the solid cut off is a “segment.” “Great” circles may also be defined as circles on a sphere which pass through the extremities of a diameter; they are familiar as the meridians or lines of longitude of geographers; lines of latitude are “small circles.” The shortest distance between two points on a sphere is the arc of the great circle containing the points. This proposition is the basis of the “great circle sailing” of navigators, and the arc of the great circle is called the “rhumbline” or “loxodromic curve.” The determination of the shortest distance between two small circles on a sphere is given in the article Variations, Calculus of. The extremities of the diameter perpendicular to a small circle are called the “poles” of that circle, and the distance from the pole to the circle, measured by the arc of the great circle through the pole, is the “polar distance” of the small circle. The solid enclosed by a small circle and the radii vectores from the centre of the sphere is a “spherical sector”; and the solid contained between two spherical sectors standing on copolar small circles is a “spherical cone.” A “spherical sector” and “spherical cone” may be also regarded as the solids of revolution of a circular sector about one of its bounding radii, and about any other line through the vertex respectively. The solid intercepted between two parallel planes is a “zone.”

The geometry of the sphere was studied by the Greeks; Euclid, in book xii. of his Elements, discusses various properties of the sphere, and in book xiii. he shows how to inscribe the five regular polyhedra within it. But with the sole exception of proving that the volumes of spheres are in the triplicate ratio of their diameters, a theorem probably due to Eudoxus, no mention is made of its mensuration. This subject was investigated by Archimedes, who, by his “method of exhaustions,” derived the principal results. He showed that the surface of a segment is equal to the area of the circle whose radius equals the distance from the vertex to the base of the segment; that the surface of the entire sphere is equal to the curved surface of the circumscribing cylinder, and to four times the area of a great circle of the sphere; and that the volume is twothirds that of the circumscribing cylinder. To Zenodorus (c. 200–100 B.C.) is due the important problem in maxima and minima that for a given surface the sphere is the solid of maximum volume. Calling the radius r, and denoting by π the ratio of the circumference to the diameter of a circle, the volume is ⁴/₃πr³, and the surface 4πr².

Archimedes gave his results in the treatise Περὶ τῆς σφαίρας καὶ τοῦ κυλίνδρου: he left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio. A solution by means of the parabola and hyperbola was given by Dionysodorus of Amisus (c. 1st century B.C.), and a similar problem—to construct a segment equal in volume to a given segment, and in surface to another segment—was solved by the Arabian mathematician and astronomer, Al Kuhi.

In analytical geometry, the equation to the sphere takes the forms x²+y²+z² = a², and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere. If the centre be (α, β, γ), the Cartesian equation becomes (x − α)² + (y − β)² + (z − γ)² = a²; consequently the general equation is x²+y²+z² + 2Ax + 2By + 2Cz + D=0, and it is readily shown that the co-ordinates of the centre are (−A, −B, −C), and the radius A² + B² + C² − D. A sphere can therefore be described so as to satisfy four given conditions. Systems of spheres have characters analogous to those of systems of circles. If r, r₁ be the radii of two spheres, d the distance between the centres, and φ the angle at which they intersect, then d² = r² + r₁² + 2rr₁cos φ; hence 2rr₁cos φ = d² − r² − r₁². This function is named the “power” of the two spheres, and it is important in the investigation of systems of spheres. If the sphere r₁ degenerate to a point, the function 2rr₁cos φ has the limit d² − r²; this is the square of the tangent to the sphere from the point, and is named the “power of the sphere at the point,” or the “power of the point with respect to the sphere.” Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x² + y² + z² + 2Ax + D = O, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres. Corresponding to the radical centre of three circles, it may be shown that four spheres have a radical centre, i.e. that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts the four spheres at right angles; this “orthotomic” sphere corresponds to the orthogonal circle of a system of circles.

The investigation of triangles and other figures drawn upon the surface of a sphere is all-important in the sciences of astronomy, geodesy and geography. In astronomy, we are principally concerned with the orientation of points on a sphere—the so-called celestial sphere—with regard to certain planes and points within the sphere; this subject is treated in the article Astronomy (Spherical). In “geodesy," and the cognate subject “figure of the earth,” the matter of greatest moment with regard to the sphere is the determination of the area of triangles drawn on the surface of a sphere—the so-called “spherical triangles”; this is a branch of trigonometry, and is studied under the name of spherical trigonometry. In mathematical geography the problem of representing the surface of a sphere on a plane is of fundamental importance; this subject is treated in the article Map.

1. The surfaces formed by revolving a circle about any chord also received attention at the hands of the Greeks. According to Heron and Geminus they were discussed under the name spire by Perseus (c. 200-100 B.C.), their sections were termed spiral sections, and are probably the same as the hippopede of Eudoxus. The surface and solid traced by the revolution of the lesser segment of a circle is termed a “spindle.” An “anchor ring” or “tore” results when a circle revolves about an axis in its plane.