Theaetetus of Athens

c. 417-c. 369 b.c.

Greek Mathematician

Believed to have influenced Euclid's work in books X and XI of the Elements, Theaetetus studied what Pappus described as "The commensurable and the incommensurable, the rational and irrational continuous quantities"—that is, rational and irrational numbers. Student of Theodorus of Cyrene, he was a friend and associate of both Socrates and the latter's pupil Plato.

In fact Plato (427-347 b.c.), who clearly admired him, is the principal source regarding Theaetetus, who became a central figure in two Platonic dialogues, Theaetetus and the Sophist. In the first of these, Plato recorded a discussion between Socrates (c. 470-390 b.c.), Theodorus (465-398 b.c.), and Theaetetus that apparently occurred in 399 b.c.

Plato noted that Theaetetus's father was a wealthy man named Euphronius of Sunium who had left his son a large fortune. Trustees of the will had cheated Theaetetus out of most of his wealth, however, but he remained generous, and Plato described him as the essence of a gentleman. The great philosopher also described Theaetetus's mind as a beautiful one, though apparently his outward appearance—he had a flat nose and bug eyes—did not match his inward one.

As is the case with all too many ancient thinkers, all traces of Theaetetus's work have disappeared, but an echo of them is preserved in the most influential geometry text ever written: the Elements of Euclid (c. 325-c. 250 b.c.). In his introduction to the latter book, Pappus (fl. c. a.d. 320) noted that Euclid's discussion of irrational numbers—those that continue indefinitely without any repeating pattern, and cannot be expressed as fractions—had its origins in Theaetetus's interpretations of Pythagorean ideas.

Theodorus had first shown that the square roots of nonsquare numbers from 3 to 17 were irrational, but as Plato wrote in Theaetetus, it was the student who managed an early generalization of these results. In this version, Theaetetus and Socrates used the term "square" as it is known today, and described numbers with irrational roots as "oblong numbers." (Of course the latter term is meaningless in a geometric sense, because a number with an irrational root can still be represented in theory by a square—with the caveat that the sides are of indefinite or approximate length.)

Sometimes credited for the entirety of Euclid's Book X (though it is almost certain the latter is Euclid's own work, a development of ideas put forth by Theaetetus and others), Theaetetus investigated a number of other mathematical questions. These included a theory of proportion; studies of the octahedron and isocahedron; and work on the medial, binomial, and apotome.

In 369 b.c. Theaetetus fought for Athens in a battle with Corinth. He gained honor on the battlefield, but was wounded and contracted dysentery. He died in Athens, having been brought back to his home.

JUDSON KNIGHT