Xenocrates of Chalcedon

Born: 396 BC in Chalcedon (now Kadiköy, near Istanbul), Bithynia (now Turkey)
Died: 314 BC in Athens, Greece

Xenocrates of Chalcedon was a student of Plato who entered the Academy in Athens in about 376 BC. In about 367 BC Xenocrates accompanied Plato on his journey to Syracuse following the death of Dionysius I. Xenocrates left Athens with Aristotle after Plato's death in 347 BC when they were both invited to Assos. Xenocrates remained for around five years in Assos.

Plato's nephew Speusippus had become head of the Academy on Plato's death, but in 340 BC he sent for Xenocrates to return to Athens to prepare to become his successor. Despite Xenocrates having been chosen to head the Academy by Speusippus, an election took place to find a successor to Speusippus after his death. It was a close battle between Xenocrates, Menedemus of Pyrrha and Heraclides Ponticus but Xenocrates triumphed by just a few votes.

Although Xenocrates had been many years in Athens he had refused to become a citizen of that state since he did not approve of its close relations with Macedonia. In this respect he contrasted strongly with his predecessor Speusippus who had strongly supported the political ties between Athens and Macedonia. It is clear that the Academy at this time was far from what many picture it as, namely an institution where scholars sat thinking, isolated from the world around them. On the contrary, the Academy was highly involved in the politics of the day and different political views strove for supremacy.

In 322 BC Xenocrates found himself in a directly political post when he headed a team negotiating a political settlement with Macedonia. To say 'political settlement' is perhaps rather wide of the mark since effectively they had to negotiate terms for the surrender of Athens. The fact that Xenocrates was not an Athenian citizen became a sore point with the Macedonians and he was deemed to be illegitimate as an ambassador for Athens.

Xenocrates remained head of the Academy in Athens for the rest of his life. A hard working man, Xenocrates is described as [1]:-

... good-natured, gentle and considerate but ... he lacked the graciousness of his teacher Plato.
Xenocrates wrote on philosophy and mathematics. Diogenes Laertius gives the titles of two mathematics books by Xenocrates, namely On numbers and The theory of numbers. All his books are lost and it would appear that there only ever existed a single copy of each in his own hand.

In many ways Xenocrates was not a particularly original thinker. Certainly he saw it his duty as the head of the Academy to promote the views of Plato as exactly as he possibly could. As Dorrie writes in [1]:-

Xenocrates' lifework consisted of producing a kind of codification - and thus of necessity, a transformation - of Plato's philosophy. But it immediately became apparent that others, especially Aristotle, understood Plato in a wholly different way with respect to certain key questions.
Xenocrates believed that matter is composed of indivisible units, so he may be regarded as an early believer in the atomic theory. He agreed with Pythagoras regarding the importance of numbers in philosophy and attributed to Pythagoras an atomic view of acoustics where sound, perceived as a single entity, consists of discrete sounds.

Xenocrates believed in human beings having threefold existence, mind, body and soul. It is not clear whether he was the instigator of this belief. He also believed that people die twice, once on Earth, then for a second time on the Moon when the mind separates from the soul and travels to the Sun.

Plutarch writes about an attempt by Xenocrates to calculate the total number of syllables which could be made from the letters of the alphabet. The result which Xenocrates obtained was, according to Plutarch, 1,002,000,000,000. If true this probably represents the first attempt at solving a combinatorial problem involving permutations.

Article by: J J O'Connor and E F Robertson

April 1999
MacTutor History of Mathematics