Anaxagoras, born at Clazomenae which is near Smyrna, was a credible
mathematician with stronger interests in philosophy. He was one of the first
philosophers to settle in Athens, but he is of interest to us primarily because of
his reported interest in squaring the circle. Whether this is true or not, it
sets the earliest recorded time that the squaring the circle problem was
recognized as a purely mathematical problem.
This is also the time of the formation of the three great problems of antiquity
--- squaring the circle, trisecting an angle, and doubling the cube.
A brief biography:
Anaxagoras was imprisoned for asserting that the sun was not a diety but a huge red-hot stone as large as all of Peloponnesus and the moon borrowed its light from the sun.
At the ripe age of seventy-two he was condemned to death for advocating the Persian cause.
Anaxagoras represents bold, rational inquiry. He represented the Greek trademark: ``the desire to know". His principle interest was in philosophy, where his main belief was that ``reason rules the world."
He wrote the book On Nature, the first widely circulated book on scientific subjects. Cost: 1 drachma.
Anaxagoras was the teacher and friend of Pericles.
Anaxagoras was mostly a natural philosopher rather than a mathematician.
But while in prison attempted to square the circle using only straight edge and compass.
Note that it was probably known that regular polygons, of an arbitrarily large number of sides, inscribed in a circle could be squared using only the straight edge and compass. From this preliminary work, most likely, the general ``squaring the circle" conjecture was made.
This attempt conveys a remarkable amount of information.
It clearly shows that that Greeks were more than just casually interested in
non-practical problems and that they had a very clear distinction between the exact and the approximate.
He lived in the age where the great problems of antiquity were formed:
Doubling the cube. (Delian problem)
Trisecting an angle.
Squaring the circle.
These are geometric problems to be solved using standard constructions, i.e. only with a compass and straight
edge. As we will see, mathematicians of the day found a variety of ingenious ways to solve these
problems using non-standard constructions.
The Heroic Age
This has been called the heroic age of Greek mathematics
for the reason that the Greeks attempted to solve these very difficult problems. That they are difficult is evidenced from that fact that it would be two millennia before their resolution was complete. They would tempt, perplex, and ultimately resist the efforts of the very best mathematicians of every age until the century. Attempts to solve these problems would drive the development of mathematics up to modern times.