See also:geometry, the solid or
See also:surface traced out by, the revolution of a semicircle about its diameter; this is essentially Euclid's definition;' in the
See also:modern geometry of surfaces it is defined as the
See also: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
See also:constant distance the "
See also:radius," and any
See also:line through the centre and intersecting the sphere is a " diameter." All sections of the The surfaces formed by revolving a circle about any chord also received
See also:attention at the hands of the Greeks . According to
See also:Heron and Geminus they were discussed under the name
See also:spire by
See also:Perseus (c . 200—100 B.C.), their sections were termed
See also:spiral sections, and are probably the same as the hippopede of
See also:Eudoxus . The surface and solid traced by the revolution of the lesser segment of a circle is termed a " spindle." An " anchor
See also:ring " or " tore " results when a circle revolves about an
See also:axis in its
See also:plane.sphere are necessarily circles; if the cutting plane contains the centre, the section is said to be " meridional," the
See also:curve of inter-section is a "
See also: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
See also: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 rhumb-line " 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
See also: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
See also:standing on copolar small circles is a " spherical
See also: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." 4ar2 . Archimedes gave his results in the
See also:treatise IIepl rue sotiaipas Kett
See also:roD KvatvIpou: he
See also:left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio . A solution by means of the
See also:parabola and
See also:hyperbola was given by Dionysodorus of Amisus (c . 1st century s.c), and a similar problem—to construct a segment equal in
See also:volume to a given segment, and in surface to another segment—was solved by the Arabian mathematician and astronomer, Al Kuhi .
See also:analytical geometry, the equation to the sphere takes the forms x2+y2+z2=
See also: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 (a, p, y), the Cartesian equation becomes (x — a)2 + (y — R)2 + (z -7)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
See also:spheres have characters analogous to those of systems of circles . If r, ri be the radii of two spheres, d the distance between the centres, and ¢ the
See also:angle at which they inter-
See also:sect, then d2=
See also:r2+'rig + 2rri cos (g; hence 2rri cos p=d2— r2 rig . This
See also:function is named the " power " of the two spheres, and it is important in the investigation of systems of spheres . If the sphere ri degenerate to a point, the function 2rri cos 4, 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
See also:system of spheres which intersect in a
See also: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 ye being the plane of intersection, and the axis of x the line of centres . Corresponding to the
See also: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 The geometry of the sphere was studied by the Greeks; Euclid, in
See also:book xii. of his Elements, discusses various properties of the sphere, and in book xiii. he shows how to inscribe the five
See also:regular polyhedra within it . But with the
See also: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
See also:principal results . He showed that the surface of a segment is equal to the
See also:area of the circle whose radius equals the distance from the vertex to the
See also:base of the segment; that the surface of the entire sphere is equal tp 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 two-thirds that of the circumscribing cylinder . To Zenodorus (c .
200—100 B.c.) is due the important problem in
See also:maxima and minima that for a given surface the sphere is the solid of maximum volume . Calling the radius r, and denoting by it the ratio of the circumference to the diameter of a circle, the volume is 3irr', and the surface 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
See also: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
See also:orientation of points on a sphere—the so-called
See also:celestial sphere—with regard to certain planes and points within the sphere; this subject is treated in the article ASTRONOMY (Spheri- • cal) . In " geodesy," and the cognate subject " figure of the
See also:earth," the
See also: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 .
SPHENODON, or TUATARA
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