See also:

• CUBE (Gr. K46os, a cube)

CUBE (Gr. K46os, a cube)  , in See also:

• GEOMETRY

geometry, a solid bounded by six equal squares, so placed that 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 between any pair of adjacent faces is a right angle . This solid played an all-important See also:

• PART

part in the geometry and cosmology of the Greeks . See also:

• PLATO

Plato (See also:

• TIMAEUS (c. 345—c. 250 B.e.)

Timaeus) described the figure in the following terms:—" The isosceles triangle which has its See also:

• VERTICAL (from Lat. vertex, highest point)

vertical angle a right angle . . combined in sets of four, with the right angles See also:

• MEETING (from " to meet," to come together, assemble, 0. Eng. metals ; cf. Du. moeten, Swed. mota, Goth. gamotjan, &c., derivatives of the Teut. word for a meeting, seen in O. Eng. Wit, moot, an assembly of the people; cf. witanagemot)

meeting at the centre, See also:

• FORM (Lat. forma)

form a single square . Six of these squares joined together formed eight solid angles, each produced by three See also:

• PLANE

plane right angles: and the shape of the See also:

• BODY

body thus formed was cubical, having six square planes for its surfaces." In his cosmology Plato assigned this solid to " 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," for " ` earth ' is the least See also:

• MOBILE

mobile of the four (elements—' See also:

• FIRE (in O. Eng. f j'r; the word is common to West German languages, cf. Dutch vuur, Ger. Feuer; the pre-Teutonic form is seen in Sanskrit pu, pavaka, and Gr. irup; the ultimate origin is usually taken to be a root meaning to purify, cf. Lat. purus)

fire," See also:

• WATER

water," See also:

• AIR (from an Indo-European root meaning " breathe," " blow ")
• AIR, or ASBEN

air ' and ` earth ') and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most See also:

• STABLE

stable." The See also:

• MENSURATION (Lat. mensura, a measure)

mensuration of the See also:

• CUBE (Gr. K46os, a cube)

cube, and its relations to other geometrical solids are treated in the See also:

• ARTICLE (from Lat. articulus, a joint)

article See also:

• POLYHEDRON (Gr. rain, many, ESpa, a base)

POLYHEDRON; in the same article are treated the Archimedean solids, the truncated and snub-cube; reference should be made to the article See also:

• CRYSTALLOGRAPHY
• CRYSTALLOGRAPHY (from the Gr. Kpbo TaXXO , ice, and -ypachew, to write)

CRYSTALLOGRAPHY for its significance as a crystal form . A famous problem concerning the cube, namely. to construct a cube of twice the See also:

• VOLUME

volume of a given cube, was attacked with See also:

• GREAT

great vigour by the Pythagoreans, See also:

• SOPHISTS (from Gr. codtoris, literally, man of wisdom)

Sophists and Platonists . It became known as the " Delian problem " or the " problem of the duplication of the cube," and ranks in See also:

• HISTORICAL

historical importance with the problems of " trisecting an angle " and " squaring the circle." The origin of the problem is open to conjecture . The See also:

• PYTHAGOREAN

Pythagorean See also:

• DISCOVERY

discovery of " squaring a square," i.e. constructing a square of twice the See also:

• AREA

area of a given square (which follows as a corollary to the Pythagorean See also:

• PROPERTY

property of a right-angled triangle, viz. the square of the hypotenuse ,equals the sum of the squares on the sides), may have suggested the strictly analogous problem of doubling a cube . Eratosthenes (c . 200 B.c.), however, gives a picturesque origin to the problem . In a See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter to See also:

• PTOLEMY (CLAUDIUS PTOLEMAEUS)

Ptolemy Euergetes he narrates the See also:

• HISTORY

history of the problem . The Delians, suffering a dire pestilence, consulted their oracles, and were ordered to See also:

• DOUBLE
• DOUBLE (from the Mid. Eng. duble, the form which gives the present pronunciation, through the Old Er. duble, from Lat. duplus, twice as much)

double the volume of the See also:

• ALTAR (Lat. altare, from alias, high; some ancient etymological guesses are recorded by St Isidore of Seville in Etymologiae xv. 4)

altar to their tutelary See also:

• GOD

god, See also:

• APOLLO (Gr. 'A2r6XXwv,'Airkkhwv)

Apollo .

An altar was built having an edge double the length of the See also:

• ORIGINAL

original; but the See also:

• PLAGUE (in Gr. Xotµos; in Lat. pestis, pestilentia)

plague was unabated, the oracles not having been obeyed . The See also:

• ERROR (Lat. error, from errare, to wander, to err)

error was discovered, and the Delians applied to Plato for his See also:

• ADVICE (Fr. avis, from Lat. ad, to, and visum, viewed)

advice, and Plato referred them to See also:

• EUDOXUS

Eudoxus . This See also:

• STORY, JOHN (c. 1510-1571)
• STORY, JOSEPH (1779-1845)
• STORY, ROBERT HERBERT (1835-1907)
• STORY, WILLIAM WETMORE (1819—1895)

story is See also:

• MERE
• MERE, ADALBERT (1838-1909)

mere See also:

• FABLE (Fr. fable, Lat. fabula)

fable, for the problem is far older than Plato . See also:

• HIPPOCRATES

Hippocrates of See also:

• CHIOS

Chios (c . 430 B.C.), the discoverer of the square of a lune, showed that'the problem reduced to the determination of two mean proportionals between two given lines, one of them being twice the length of the other . Algebraically expressed, if x and y be the required mean proportionals and a, 2a, the lines, we have a: x:: x : y :: y : 2a, from which it follows that x3= 2a'', . Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the See also:

• LANGUAGE (adapted from the Fr. langage, from langue, tongue, Lat. lingua)

language of algebraic geometry, is to solve geometrically a cubic See also:

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

equation . According to See also:

• PROCLUS, or PROCULUS (A.D. 410-485)

Proclus, a See also:

• MAN
• MAN, ISLE OF (anc. Mona)

man named Hippias, probably Hippias of Ells (c . 46o B.c.), trisected an angle with a See also:

• MECHANICAL

mechanical See also:

• CURVE (Lat. curvus, bent)

curve, named the See also:

• QUADRATRIX (from Lat. quadrator, squarer)

quadratrix (q.v.) . See also:

• ARCHYTAS (c. 428—347 B.C.)

Archytas of See also:

• TARENTUM
• TARENTUM (Gr. rapas)

Tarentum (c . 430 B.c.) solved the problems by means of sections of a See also:

• HALF

half See also:

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

cylinder; according to Eutocius, Menaechmus solved them by means of the inter-sections of conic sections; and Eudoxus also gave a See also:

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

solution . All these solutions were condemned by Plato on the ground that they were mechanical and not geometrical, i.e. they were not effected by means of circles and lines .

However, no proper geometrical solution, in Plato's sense, was obtained; in fact it is now generally agreed that, with such a restriction, the problem is insoluble . The pursuit of mechanical methods furnished a stimulus to the study of mechanical loci, for example. the See also:

• LOCUS (Lat. for " place "; in Gr. rinros)

locus of a point carried on a See also:

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

rod which is caused to move according to a definite See also:

• RULE

rule . Thus Nicomedes invented the See also:

• CONCHOID (Gr. «oyXn, shell, and ethos, form)

conchoid (q.v.); Diodes the See also:

• CISSOID (from the Gr. rcunr6s, ivy, and ethos, form)

cissoid (q.v.); Dinostratus studied the quadratrix invented by Hippias; all these curves furnished solutions, as is also the See also:

• CASE
• CASE, JOHN (d. 1600)

case with the See also:

• TRISECTRIX

trisectrix, a See also:

• SPECIAL

special form of See also:

• PASCAL, BLAISE (1623-1662)
• PASCAL, JACQUELINE (1625-T661)

Pascal's limagon (q.v.) . These problems were also attacked by the Arabian mathematicians; See also:

• TOBIT, THE BOOK OF

Tobit See also:

• BEN (from Old Eng. bennan, within)

ben Korra (836–901) is credited with a solution, while Abul Gud solved it by means of a See also:

• PARABOLA

parabola and an equilateral See also:

• HYPERBOLA

hyperbola . In See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

algebra, the " cube " of a quantity is the quantity multiplied by itself twice, i.e. if a be the quantity a X a X a.(= a3) is its cube . Similarly the " cube See also:

• ROOT (late O.E. rot, adopted from Scand., cf. Norw. and Swed. rot, Dan. rod; the true O.E. word was wyrt, plant, represented in Ger. Wurz or Wurzel; the ultimate root is the same in both words, and is seen in Lat. radix)
• ROOT, ELIHU (1845– )

root " of a quantity is another quantity which when multiplied by itself twice gives the original quantity; thus ai is the cube root of a (see See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

ARITHMETIC and ALGEBRA) . A " cubic equation " is one in which the highest See also:

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

power of the unknown is the cube (see EQUATION); similarly, a " cubic curve " has an equation containing no See also:

• TERM

term of a power higher than the third, the See also:

• POWERS, HIRAM (1805-1873)

powers of a See also:

• COMPOUND
• COMPOUND (from Lat. componere, to combine or put together)

compound term being added together . In mensuration, " cubature " is sometimes used to denote the volume of a solid; the word is parallel with " See also:

• QUADRATURE (from Lat. quadratura, a making square)

quadrature, ", to de termine the area of a See also:

• SURFACE

surface (see MENSURATION; INFINITESIMAL CALCULUS) .