See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

SPHERE (Gr. vclsa?pa, a See also:

• BALL (in Mid. Eng. bal; the word is probably cognate with " bale," Teutonic in origin, cf. also Lat. falls, and Gr. 7raXXa)
• BALL, JOHN (1585-1640)
• BALL, JOHN (1818-1889)
• BALL, JOHN (d. 1381)
• BALL, SIR ALEXANDER JOHN, BART
• BALL, THOMAS (1819- )

ball or globe)  , in See also:

• GEOMETRY

geometry, the solid or See also:

• SURFACE

surface traced out by, the revolution of a semicircle about its See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter; this is essentially See also:

• EUCLID
• EUCLID [EucLEIDES]

Euclid's See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition;' in the See also:

• MODERN

modern geometry of surfaces it is defined as the See also:

• QUADRIC

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, BENJAMIN (1845-1902)

constant distance the " See also:

• RADIUS

radius," and any See also:

• LINE

line through the centre and intersecting the See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

sphere is a " diameter." All sections of the The surfaces formed by revolving a circle about any chord also received See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention at the hands of the Greeks . According to See also:

• HERON (Fr. heron; Ital. aghirone, airone; Lat. ardea; Gr. ipw&bs: A.-S. hragra; Icelandic, hegre; Swed. hager; Dan. heire; Ger. Heiger, Reiher, Heergans; Dutch, reiger)

Heron and Geminus they were discussed under the name See also:

• SPIRE

spire by See also:

• PERSEUS

Perseus (c . 200—100 B.C.), their sections were termed See also:

• SPIRAL

spiral sections, and are probably the same as the hippopede of See also:

• EUDOXUS

Eudoxus . The surface and solid traced by the revolution of the lesser segment of a circle is termed a " spindle." An " See also:

• ANCHOR (from the Greek ayrcvpa, which Vossius considers is from 6yi17 , a crook or hook)

anchor See also:

• RING (O.E. hring; a word common to Teutonic languages; and probably cognate with the Lat. circus, Gr. KtpKOS or KpLKOS, Skt'. chakra, wheel, circle, cf. also " harangue "); in art, a band of circular shape of varying sizes, made of any material and used f

ring " or " tore " results when a circle revolves about an See also:

• AXIS (Lat. for " axle ")

axis in its See also:

• PLANE

plane.sphere are necessarily circles; if the cutting plane contains the centre, the See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section is said to be " meridional," the See also:

• CURVE (Lat. curvus, bent)

curve of inter-section is a " See also:

• GREAT

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 (through the Fr. familier, from Lat. familiaris, of or belonging to the familia, family)

familiar as the . meridians or lines of See also:

• LONGITUDE (from Lat. longitudo, " length ")

longitude of geographers; lines of See also:

• LATITUDE (Lat. latitudo, latus, broad)

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 See also:

• ARTICLE (from Lat. articulus, a joint)

article See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

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 (FAMILY)
• POLE, REGINALD (1500-1558)
• POLE, RICHARD DE LA (d. 1525)
• POLE, WILLIAM (1814—1900)

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

standing on copolar small circles is a " spherical See also:

• CONE (Gr. Kwvos)

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 " See also:

• ZONE (Gr. 'WV77, a girdle, from 'wvvLvay to gird)

zone." 4ar2 . See also:

• ARCHIMEDES (c. 287–212 B.C.)
• ARCHIMEDES, SCREW OF

Archimedes gave his results in the See also:

• TREATISE

treatise IIepl See also:

• RUE (Fr. rue, Lat. ruta, from Gr. pur)

rue sotiaipas Kett See also:

• ROD (O.E. rodd, probably related to Norw. rudda, stick, rodda, stake)
• ROD, EDOUARD (1857-1910)

roD KvatvIpou: he See also:

• LEFT

left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio . A See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution by means of the See also:

• PARABOLA

parabola and See also:

• HYPERBOLA

hyperbola was given by Dionysodorus of Amisus (c . 1st See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century s.c), and a similar problem—to construct a segment equal in See also:

• VOLUME

volume to a given segment, and in surface to another segment—was solved by the Arabian mathematician and astronomer, Al Kuhi .

In See also:

• ANALYTICAL

analytical geometry, the See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation to the sphere takes the forms x2+y2+z2=See also:

• A2(z)

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 See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

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, MUSIC OF THE

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 (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle at which they inter-See also:

• SECT

sect, then d2=See also:

• R2(rt-s2)

r2+'rig + 2rri See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos (g; hence 2rri cos p=d2— r2 rig . This See also:

• FUNCTION

function is named the " See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

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

system of spheres which intersect in a See also:

• COMMON

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 (Lat. radix, a root)

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

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

regular polyhedra within it . But with the See also:

• SOLE (Solea)

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 See also:

• MENSURATION (Lat. mensura, a measure)

mensuration . This subject was investigated by Archimedes, who, by his " method of exhaustions," derived the See also:

• PRINCIPAL

principal results . He showed that the surface of a segment is equal to the See also:

• AREA

area of the circle whose radius equals the distance from the vertex to the See also:

• BASE

base of the segment; that the surface of the entire sphere is equal tp the curved surface of the circumscribing See also:

• CYLINDER (Gr. KvAwSpos, from KvXivaety, to roll)

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

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

drawn upon the surface of a sphere is all-important in the sciences of See also:

• ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

astronomy, See also:

• GEODESY (from the Gr. y, the earth, and SatEty, to divide)

geodesy and See also:

• GEOGRAPHY (Gr. yil, earth, and ypiickty, to write)

geography . In astronomy, we are principally concerned with the See also:

• ORIENTATION

orientation of points on a sphere—the so-called See also:

• CELESTIAL, OR STELLAR

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
• EARTH (a word common to Teutonic languages, cf. Ger. Erde, Dutch aarde, Swed. and Dan. jord; outside Teutonic it appears only in the Gr. 'pq'e, on the ground; it has been connected by some etymologists with the Aryan root ar-, to plough, which is seen in
• EARTH, FIGURE OF
• EARTH, FIGURE OF THE

earth," the See also:

• MATTER

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 See also:

• BRANCH (from the Fr. branche, late Lat. branca, an animal's paw)

branch of See also:

• TRIGONOMETRY (from Gr. rpiywvov, a triangle, /ATpov, measure)

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 See also:

• MAP
• MAP (or MAPES), WALTER (d. c. 1208/9)

MAP .