but based his calculations upon measurements of the actual side
of that figure. Nevertheless, he had learned to square the circle
with a close approximation to the truth, and, in general, his
measurement sufficed for all his practical needs. Just how much
of the geometrical knowledge which added to the fame of Thales
was borrowed directly from the Egyptians, and how much he
actually created we cannot be sure. Nor is the question raised in
disparagement of his genius. Receptivity is the first
prerequisite to progressive thinking, and that Thales reached out
after and imbibed portions of Oriental wisdom argues in itself
for the creative character of his genius. Whether borrower of
originator, however, Thales is credited with the expression of
the following geometrical truths:

1. That the circle is bisected by its diameter.

2. That the angles at the base of an isosceles triangle are
equal.

3. That when two straight lines cut each other the vertical
opposite angles are equal.

4. That the angle in a semicircle is a right angle.

5. That one side and one acute angle of a right-angle triangle
determine the other sides of the triangle.

It was by the application of the last of these principles that
Thales is said to have performed the really notable feat of
measuring the distance of a ship from the shore, his method being