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