Antiphon’s Contribution in Mathematics


Antiphon, the first Sophist (one who exercises wisdom or learning) of Athens was born in 480 BC. He was an orator, writer of defence speeches, and philosopher. Some historians doubted, that if there was one Antiphon who was a sophist philosopher or there were two separate individuals. Some experts even claimed three distinct Antiphons. Unfortunately, nothing is known about his life. Didymus Chalcenterus, a grammarian, mentioned two Antiphons in the 5th century BC. One Antiphon was the orator, and author of the Tetralogies, which contains several speeches written by Antiphon. Three of these speeches were written by Antiphon as the prosecutor in murder trials and were real. Twelve speeches were specimen speeches written by Antiphon to teach students the skills of prosecuting and defending clients in cases. He had also written treatises On Concord and The Statesman. Another Antiphon was the Sophist, who was the author of the treatises On Truth and Interpretation of Dreams. It is however doubtful whether this treatise was written by Antiphon the Sophist or by some third Antiphon. In On Truth, his work supported the views of philosopher Parmenides, who believed that there was a single sole reality and the visible world around was unreal. In On Concord, he criticized the city laws by arguing, if these laws satisfy the natural needs of an individual. Some historians doubted whether the two treatises On Truth and On Concord were written by the same Antiphon. Others claimed that the remaining two treatises, The Statesman and Interpretation of Dreams were not written by Antiphon. Some scholars adopted the view of Caecilius, concluding that at least the orator and sophist Antiphon were the same people. Antiphon was also a part of an anti-democratic movement. According to Thucydides, he was the leader of the movement. He said Antiphon was the one,” who conceived the whole matter and the means by which it was brought to pass.” However, this movement failed, and he was tried and executed for treason. He died in 411 BC. Talking about his contributions, he was the first to solve the problem of squaring a circle. It was believed that like most of the other Greek thinkers, he discussed the formation of the universe and heavenly bodies and he might have related the rising and setting of the sun to changes in the atmosphere of the earth. Also, he used to prefer things that are natural over things that are by convention. A very small amount of his work survived, which we will discuss next.

Contributions in Mathematics

Antiphon made a very significant contribution to mathematics by solving the problem of squaring the circle. He was the first to give upper and lower bounds for pi by inscribing a polygon in a circle and then inscribing the same circle into another polygon so that on doubling the sides of the polygons, the difference in the areas of the polygons becomes exhausted. We know about this work of Antiphon via Aristotle and his commentators. However, Simplicius failed to understand what Antiphon exactly said, and thought that he tried to square a circle. Simplicius wrote,

Antiphon thought that in this way the area of the circle would be used up, and we should some time have a polygon inscribed in the circle the sides of which, owing to their smallness, coincide with the circumference of the circle. And as we can make a square equal to any polygon, we shall be in a position to make a square equal to a circle.’

Heath understood the theory of Antiphon differently and wrote,

Antiphon therefore deserves an honourable place in the history of geometry as having originated the idea of exhausting an area by means of inscribed regular polygons with an ever increasing number of sides, an idea upon which Eudoxus founded his epoch-making method of exhaustion.’

It is believed in modern times, that Antiphon was just making a mistake in geometry by assuming that a polygon with many sides can be approximated with the circumference of a circle. However, this is not an appropriate way to deal with the theory. According to him, an approximation could be attained by this method and he assumed the circle to be a polygon with a very large number of sides.


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