born in these latter centuries, would surely have rejoiced
in this, enlargement, whereby the continuous as well as
the discontinuous is brought, if not under the rule of number,
under the rule of mathematics indeed.

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.