Now, the Greeks believed the world to be composed of four elements--earth, air, fire, water,--and to the Greek mind the conclusion was inevitable(2a) that the shapes of the particles of the elements were those of the regular solids. Earth-particles were cubical, the cube being the regular solid possessed of greatest stability; fire-particles were tetrahedral, the tetrahedron being the simplest and, hence, lightest solid. Water-particles were icosahedral for exactly the reverse reason, whilst air-particles, as intermediate between the two latter, were octahedral. The dodecahedron was, to these ancient mathematicians, the most mysterious of the solids: it was by far the most difficult to construct, the accurate drawing of the regular pentagon necessitating a rather elaborate application of PYTHAGORAS' great theorem.(1) Hence the conclusion, as PLATO put it, that "this (the regular dodecahedron) the Deity employed in tracing the plan of the Universe."(2b) Hence also the high esteem in which the pentagon was held by the Pythagoreans. By producing each side of this latter figure the five-pointed star (fig. 9), known as the pentagram, is obtained. This was adopted by the Pythagoreans as the badge of their Society, and for many ages was held as a symbol possessed of magic powers. The mediaeval magicians made use of it in their evocations, and as a talisman it was held in the highest esteem. (2a) _Cf_. PLATO: The Timaeus, SESE xxviii--xxx. (1) In reference to this matter FRANKLAND remarks: "In those early days the innermost secrets of nature lay in the lap of geometry, and the extraordinary inference follows that Euclid's _Elements_, which are devoted to the investigation of the regular solids, are therefore in reality and at bottom an attempt to 'solve the universe.' Euclid, in fact, made this goal of the Pythagoreans the aim of his _Elements_."--_Op. cit_., p. 35. (2b) _Op. cit_., SE xxix. Music played an important part in the curriculum of the Pythagorean Brotherhood, and the important discovery that the relations between the notes of musical scales can be expressed by means of numbers is a Pythagorean one. It must have seemed to its discoverer--as, in a sense, it indeed is--a striking confirmation of the numerical theory of the Cosmos. The Pythagoreans held that the positions of the heavenly bodies were governed by similar numerical relations, and that in consequence their motion was productive of celestial music. This concept of "the harmony of the spheres" is among the most celebrated of the Pythagorean doctrines, and has found ready acceptance in many mystically-speculative minds. "Look how the floor of heaven," says Lorenzo in SHAKESPEARE'S _The Merchant of Venice_--