1. TMTh:: XENOCRATES OF CHALCEDON xenocrates of chalcedon (fl. 397 314 BC). Life Most of what we know aboutXenocrates comes from a biography by Diogenes Laertius. He was born in Chalcedon http://www.tmth.edu.gr/en/aet/1/98.html

Home Ancient Greek Scientists AGRICULTURALISTS ARCHITECTS ... PHYSICISTS MATHEMATICIAN, ASTRONOMER XENOCRATES OF CHALCEDON (fl. 397 - 314 BC) Life Work Xenocrates' writings are lost except for fragments. He is cited by Stobaeus; and Diogenes Laertius, in his biography, provides the following list of his works: "On logic": 9 books. "On mathematics": 6 books. "On the theory of numbers" "On intervals" "On astrology": 6 books. "On geometry": 5 books. History of geometry. "On the teachings of Pythagoras" "On learning" "On physics": 6 books. Contact the Science Center and Technology Museum

2. Xenocrates xenocrates of chalcedon was a student of Plato who entered the Academy in Athensin about 376 BC. In about 367 BC Xenocrates accompanied Plato on his http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Xenocrates.html

Xenocrates of Chalcedon Born: Died: 314 BC in Athens, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing 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

3. Xenocrates Biography, bibliography and list of related topics. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Xenocrates.html

Xenocrates of Chalcedon Born: Died: 314 BC in Athens, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing 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

4. TMTh XENOCRATES OF CHALCEDON xenocrates of chalcedon (fl. 397 314 BC) Life Most of what we know about Xenocrates comes from a biography by Diogenes Laertius. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Xenocrates. The Columbia Encyclopedia, Sixth Edition. 2001 2001. Xenocrates (z n k r t z) (KEY) , 396314 B.C., Greek philosopher, b. Chalcedon, successor of Speusippus as head of the Academy. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Xenocrates Of Chalcedon xenocrates of chalcedon (396 314 BC) was a Greek philosopher and scholarch orrector of the Academy from 339 to 314 BC. Removing to Athens in early youth, http://www.mlahanas.de/Greeks/Bios/XenocratesOfChalcedon.html

Xenocrates of Chalcedon Xenocrates of Chalcedon (396 - 314 BC) was a Greek philosopher and scholarch or rector of the Academy from 339 to 314 BC. Removing to Athens in early youth, he became the pupil of the Socratic Aeschines , but presently joined himself to Plato , whom he attended to Sicily in 361. Upon his master's death, in company with Aristotle he paid a visit to Hermias at Atarneus. In 339, Aristotle being then in Macedonia, Xenocrates succeeded Speusippus in the presidency of the school, defeating his competitors Menedemus and Heraclides Ponticus by a few votes. On three occasions he was member of an Athenian legation, once to Philip, twice to Antipater. Soon after the death of Demosthenes (fl 322), resenting the Macedonian influence then dominant at Athens, Xenocrates declined the citizenship offered to him at the instance of Phocion , and, being unable to pay the tax levied upon resident aliens, was, it is said, sold, or on the point of being sold, into slavery. He died in 314, and was succeeded as scholarch by Polemon, whom he had reclaimed from a life of profligacy. Besides Polemon, the statesman Phocion , Chaeron (tyrant of Pellene), the Academic Crantor, the Stoic Zeno and Epicurus are said to have frequented his lectures.

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8. Combinatorics And Hipparchus xenocrates of chalcedon, (?e?t ? ?a?d?) who succeeded Speusippus (Spe?s?pp? a?) as head of Plato s Academy (339 BC 314 BC) solved a http://www.mlahanas.de/Greeks/Combinatorics.htm

Hipparchus on Compound Statements Michael Lahanas Griechische Mathematik: Kombinatorik von Hipparchos Technology Museum of Thessaloniki ) Archimedes considered a combinatorial problem of the stomachion n, n ), contain no points above the line y = x , and are composed only of steps (0, 1), (1, 0), and (1, 1). The corresponding s(n) are obtained using: (s(1),s(2),s(3), s(4),...,s(10),...) = (1,1,3,11,...,103049,...) The numbers can be obtained by the following relation: (n+2)s(n+2)-3(2n+1)s(n+1)+(n-1)s(n) = 0, n greater or equal 1., s(1)=s(2)=1. "In pure mathematics [Hipparchus] is said to have considered a problem in permutations and combinations, the problem of finding the number of different possible combinations of 10 axioms or assumptions, which he made to be 103049 (v.l. 101049) or 310952 according as the axioms were affirmed or denied. It seems impossible to make anything of these figures." Heath History of Greek Mathematics According to [ST97] Plutarch (50-120 AD) said: "Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103049 compound statements, and on the negative side 310952.)" Stanley's article points out that 103049 is the number of "bracketings" of a string of 10 letters, as shown by Schröder. This certainly suggests that Hipparchus was calculating something that was somehow related to the number of bracketing of n letters, although as Stanley says:

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13. Xenocrates Xenocrates. Xenocrates (of Chalcedon), according to Diogenes Laertius (D.L.) iv 14, became head of the Academy after Speusippus died, in 339/338 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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