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