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.