PYTHAGORAS’ foremost achievement in mathematics I have already mentioned.  Another notable piece of work in the same department was the discovery of a method of constructing a parallelogram having a side equal to a given line, an angle equal to a given angle, and its area equal to that of a given triangle.  PYTHAGORAS is said to have celebrated this discovery by the sacrifice of a whole ox.  The problem appears in the first book of EUCLID’S Elements of Geometry as proposition 44.  In fact, many of the propositions of EUCLID’S first, second, fourth, and sixth books were worked out by PYTHAGORAS and the Pythagoreans; but, curiously enough, they seem greatly to have neglected the geometry of the circle.

The symmetrical solids were regarded by PYTHAGORAS, and by the Greek thinkers after him, as of the greatest importance.  To be perfectly symmetrical or regular, a solid must have an equal number of faces meeting at each of its angles, and these faces must be equal regular polygons, i.e. figures whose sides and angles are all equal.  PYTHAGORAS, perhaps, may be credited with the great discovery that there are only five such solids.  These are as follows:—­

The Tetrahedron, having four equilateral triangles as faces.

The Cube, having six squares as faces.

The Octahedron, having eight equilateral triangles as faces.

The Dodecahedron, having twelve regular pentagons
(or five-sided figures) as faces.

The Icosahedron, having twenty equilateral triangles as faces.[1]

[1] If the reader will copy figs. 4 to 8 on cardboard or stiff paper, bend each along the dotted lines so as to form a solid, fastening together the free edges with gummed paper, he will be in possession of models of the five solids in question.

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.