orded by Sir WILLIAM HAMILTON'S discovery, or, rather, invention of
Quarternions, but no better example of the utilitarian triumph of the
theoretical over the so-called practical mind can be adduced than that
afforded by PYTHAGORAS. Given this rule for constructing a right angle,
about whose reason the Egyptian who used it never bothered himself, and
the mind of PYTHAGORAS, searching for its full significance, made that
gigantic geometrical discovery which is to this day known as the Theorem
of PYTHAGORAS--the law that in every right-angled triangle the square
on the side opposite the right angle is equal in area to the sum of the
squares on the other two sides.(1) The importance of this discovery
can hardly be overestimated. It is of fundamental importance in most
branches of geometry, and the basis of the whole of trigonometry--the
special branch of geometry that deals with the practical mensuration of
triangles. EUCLID devoted the whole of the first book of his _Elements
of Geometry_ to establishing the truth of this theorem; how PYTHAGORAS
demonstrated it we unfortunately do not know.

(1) Fig. 3 affords an interesting practical demonstration of the truth
of this theorem. If the reader will copy this figure, cut out the
squares on the two shorter sides of the triangle and divide them along
the lines AD, BE, EF, he will find that the five pieces so obtained can
be made exactly to fit the square on the longest side as shown by the
dotted lines. The size and shape of the triangle ABC, so long as it
has a right angle at C, is immaterial. The lines AD, BE are obtained
by continuing the sides of the square on the side AB, _i.e_. the side
opposite the right angle, and EF is drawn at right angles to BE.

After absorbing what knowledge was to be gained in Egypt, PYTHAGORAS
journeyed to Babylon, where he probably came into contact with even
greater traditions and more potent influences and sources of knowledge
than in Egypt, for there is reason for believing that the ancient
Chaldeans were the builders of the Pyramids and in many ways the
intellectual superiors of the Egyptians.

At last, after having travelled still further East, probably as far as
India, PYTHAGORAS returned to his birthplace to teach the men of his
native land the knowledge he had gained. But CROESUS was tyrant over
Samos, and so oppressive was his rule that none had leisure in which to
learn. Not a student came to PYTHAGORAS, until, in despair, so the s