me 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) He