Xenocrates

Xenocrates

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Xenocrates of Chalcedon
Chalcedon
Chalcedon , sometimes transliterated as Chalkedon) was an ancient maritime town of Bithynia, in Asia Minor, almost directly opposite Byzantium, south of Scutari . It is now a district of the city of Istanbul named Kadıköy...

 was a Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

 philosopher, mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, and leader (scholarch
Scholarch
A scholarch is the head of a school. The term was especially used for the heads of schools of philosophy in ancient Athens, such as the Platonic Academy, whose first scholarch was Plato himself...

) of the Platonic Academy
Platonic Academy
The Academy was founded by Plato in ca. 387 BC in Athens. Aristotle studied there for twenty years before founding his own school, the Lyceum. The Academy persisted throughout the Hellenistic period as a skeptical school, until coming to an end after the death of Philo of Larissa in 83 BC...

 from 339/8 to 314/3 BC. His teachings followed those of Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

, which he attempted to define more closely, often with mathematical elements. He distinguished three forms of being, the sensible, the intelligible, and a third compounded of the two, to which correspond respectively, sense
Sense
Senses are physiological capacities of organisms that provide inputs for perception. The senses and their operation, classification, and theory are overlapping topics studied by a variety of fields, most notably neuroscience, cognitive psychology , and philosophy of perception...

, intellect
Intellect
Intellect is a term used in studies of the human mind, and refers to the ability of the mind to come to correct conclusions about what is true or real, and about how to solve problems...

 and opinion
Opinion
In general, an opinion is a subjective belief, and is the result of emotion or interpretation of facts. An opinion may be supported by an argument, although people may draw opposing opinions from the same set of facts. Opinions rarely change without new arguments being presented...

. Unity and duality he considered to be gods which rule the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

, and the soul is a self-moving number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

. God
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

 pervades all things, and there are daemonical
Daemon (mythology)
The words dæmon and daimôn are Latinized spellings of the Greek "δαίμων", a reference to the daemons of Ancient Greek religion and mythology, as well as later Hellenistic religion and philosophy...

 powers, intermediate between the divine
Divinity
Divinity and divine are broadly applied but loosely defined terms, used variously within different faiths and belief systems — and even by different individuals within a given faith — to refer to some transcendent or transcendental power or deity, or its attributes or manifestations in...

 and the mortal
Human
Humans are the only living species in the Homo genus...

, which consist in conditions of the soul. He held that mathematical objects and the Platonic Ideas
Theory of Forms
Plato's theory of Forms or theory of Ideas asserts that non-material abstract forms , and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form is often capitalized...

 are identical, unlike Plato who distinguished them. In Ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...

, he taught that virtue
Virtue
Virtue is moral excellence. A virtue is a positive trait or quality subjectively deemed to be morally excellent and thus is valued as a foundation of principle and good moral being....

 produces happiness
Happiness
Happiness is a mental state of well-being characterized by positive emotions ranging from contentment to intense joy. A variety of biological, psychological, religious, and philosophical approaches have striven to define happiness and identify its sources....

, but that external goods can minister to it and enable it to effect its purpose.

Life


Xenocrates was a native of Chalcedon
Chalcedon
Chalcedon , sometimes transliterated as Chalkedon) was an ancient maritime town of Bithynia, in Asia Minor, almost directly opposite Byzantium, south of Scutari . It is now a district of the city of Istanbul named Kadıköy...

. By the most probable calculation he was born 396/5 BC, and died 314/3 BC at the age of 82. Moving to Athens
Athens
Athens , is the capital and largest city of Greece. Athens dominates the Attica region and is one of the world's oldest cities, as its recorded history spans around 3,400 years. Classical Athens was a powerful city-state...

 in early youth, he became the pupil of Aeschines Socraticus
Aeschines Socraticus
Aeschines of Sphettus or Aeschines Socraticus , son of Lysanias, of the deme Sphettus of Athens was in his youth a follower of Socrates. Historians call him Aeschines Socraticus—"the Socratic Aeschines"—to distinguish him from the more historically influential Athenian orator also named...

, but subsequently joined himself to Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

, whom he accompanied to Sicily
Sicily
Sicily is a region of Italy, and is the largest island in the Mediterranean Sea. Along with the surrounding minor islands, it constitutes an autonomous region of Italy, the Regione Autonoma Siciliana Sicily has a rich and unique culture, especially with regard to the arts, music, literature,...

 in 361. Upon his master's death, he paid a visit with Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

 to Hermias of Atarneus
Hermias of Atarneus
Hermias of Atarneus , who lived in Atarneus, was Aristotle's father-in-law.The first mention of Hermias is as a slave to Eubulus, a Bithynian banker who ruled Atarneus. Hermias eventually won his freedom and inherited the rule of Atarneus...

. In 339/8 BC, Xenocrates succeeded Speusippus
Speusippus
Speusippus was an ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, Speusippus inherited the Academy and remained its head for the next eight years. However, following a stroke, he passed the chair to Xenocrates. Although the successor to Plato...

 in the presidency of the school, defeating his competitors Menedemus of Pyrrha
Menedemus of Pyrrha
Menedemus of Pyrrha, Lesbos, lived c. 350 BC, was a member of Plato's Academy, during the time of Speusippus.Upon the death of Speusippus in 339 BC, an election was held for the next scholarch of the Academy. Menedemus and Heraclides narrowly lost to Xenocrates. Menedemus left the Academy, and set...

 and Heraclides Ponticus
Heraclides Ponticus
Heraclides Ponticus , also known as Herakleides and Heraklides of Pontus, was a Greek philosopher and astronomer who lived and died at Heraclea Pontica, now Karadeniz Ereğli, Turkey. He is best remembered for proposing that the earth rotates on its axis, from west to east, once every 24 hours...

 by a few votes. On three occasions he was member of an Athenian legation, once to Philip
Philip II of Macedon
Philip II of Macedon "friend" + ἵππος "horse" — transliterated ; 382 – 336 BC), was a king of Macedon from 359 BC until his assassination in 336 BC. He was the father of Alexander the Great and Philip III.-Biography:...

, twice to Antipater
Antipater
Antipater was a Macedonian general and a supporter of kings Philip II of Macedon and Alexander the Great. In 320 BC, he became Regent of all of Alexander's Empire. Antipater was one of the sons of a Macedonian nobleman called Iollas or Iolaus and his family were distant collateral relatives to the...

.

Xenocrates resented the Macedon
Macedon
Macedonia or Macedon was an ancient kingdom, centered in the northeastern part of the Greek peninsula, bordered by Epirus to the west, Paeonia to the north, the region of Thrace to the east and Thessaly to the south....

ian influence then dominant at Athens. Soon after the death of Demosthenes
Demosthenes
Demosthenes was a prominent Greek statesman and orator of ancient Athens. His orations constitute a significant expression of contemporary Athenian intellectual prowess and provide an insight into the politics and culture of ancient Greece during the 4th century BC. Demosthenes learned rhetoric by...

 (c. 322 BC), he declined the citizenship offered to him at the instance of Phocion
Phocion
Phocion was an Athenian statesman and strategos, and the subject of one of Plutarch's Parallel Lives....

 as a reward for his services in negotiating peace with Antipater
Antipater
Antipater was a Macedonian general and a supporter of kings Philip II of Macedon and Alexander the Great. In 320 BC, he became Regent of all of Alexander's Empire. Antipater was one of the sons of a Macedonian nobleman called Iollas or Iolaus and his family were distant collateral relatives to the...

 after Athens' unsuccessful rebellion. The settlement was reached "at the price of a constitutional change: thousands of poor Athenians were disenfranchised," and Xenocrates said "that he did not want to become a citizen within a constitution he had struggled to prevent." Being unable to pay the tax levied upon resident aliens, he is said to have been saved only by the courage of the orator Lycurgus
Lycurgus of Athens
Lycurgus was a logographer in Ancient Greece. He was one of the ten Attic orators included in the "Alexandrian Canon" compiled by Aristophanes of Byzantium and Aristarchus of Samothrace in the third century BCE.Lycurgus was born at Athens about 396 BC, and was the son of Lycophron, who belonged...

, or even to have been bought by Demetrius Phalereus
Demetrius Phalereus
Demetrius of Phalerum was an Athenian orator originally from Phalerum, a student of Theophrastus and one of the first Peripatetics...

, and then emancipated. In 314/3, he died from hitting his head, after tripping over a bronze pot in his house.

Xenocrates was succeeded as scholarch by Polemon
Polemon (scholarch)
Polemon of Athens was an eminent Platonist philosopher and Plato's third successor as scholarch or head of the Academy from 314/313 to 270/269 BC...

, whom he had reclaimed from a life of profligacy. Besides Polemon, the statesman Phocion, Chaeron (tyrant of Pellene
Pellene
Pellene was the easternmost ancient Greek city-state in Achaea. The ruins are southwest of Xylokastro. The city sided with Sparta throughout the Peloponnesian War. It was a member of the Achaean League until that League was dissolved by Alexander the Great. In the fourth century it was ruled for...

), the academic Crantor
Crantor
Crantor was a Greek philosopher of the Old Academy, probably born around the middle of the 4th century BC, at Soli in Cilicia.-Life:Crantor moved to Athens in order to study philosophy, where he became a pupil of Xenocrates and a friend of Polemo, and one of the most distinguished supporters of...

, the Stoic Zeno
Zeno of Citium
Zeno of Citium was a Greek philosopher from Citium . Zeno was the founder of the Stoic school of philosophy, which he taught in Athens from about 300 BC. Based on the moral ideas of the Cynics, Stoicism laid great emphasis on goodness and peace of mind gained from living a life of virtue in...

 and Epicurus
Epicurus
Epicurus was an ancient Greek philosopher and the founder of the school of philosophy called Epicureanism.Only a few fragments and letters remain of Epicurus's 300 written works...

 are said to have frequented his lectures.

Wanting in quickness of apprehension and natural grace he compensated by persevering and thorough-going industry, pure benevolence, purity of morals, unselfishness, and a moral earnestness, which compelled esteem and trust even from the Athenians of his own age.

Xenocrates adhered closely to the Platonist doctrine, and he is accounted the typical representative of the Old Academy. In his writings, which were numerous, he seems to have covered nearly the whole of the Academic program; but metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...

 and ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...

 were the subjects which principally engaged his thoughts. He is said to have made more explicit the division of philosophy into the three parts of Physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, Dialectic
Dialectic
Dialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...

 and Ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...

.

Writings


With a comprehensive work on Dialectic
Dialectic
Dialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...

 (τῆς περὶ τὸ διαλέγεσθαι πραγματείας βιϐλία ιδ΄) there were also separate treatises On Knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

, On Knowledgibility (περὶ ἐπιστήμης α΄, περὶ ἐπιστημοσύνης α΄), On Divisions (διαιρέσεις η΄), On Genera and Species (περὶ γενῶν καὶ εἰδῶν α΄), On Ideas (περὶ ἰδεῶν), On the Opposite (περὶ τοῦ ἐναντίου), and others, to which probably the work On Mediate Thought (τῶν περὶ τὴν διάνοιαν η΄) also belonged. Two works by Xenocrates on Physics are mentioned (περὶ φύσεως ϛ΄ - φυσικῆς ἀκροάσεως ϛ΄), as are also books On the Gods (περὶ Θεῶν β΄), On the Existent (περὶ τοῦ ὄντος), On the One (περὶ τοῦ ἑνός), On the Indefinite (περὶ τοῦ ἀορίστου), On the Soul (περὶ ψυχῆς), On the Emotions (περὶ τῶν παθῶν α΄) On Memory
Memory
In psychology, memory is an organism's ability to store, retain, and recall information and experiences. Traditional studies of memory began in the fields of philosophy, including techniques of artificially enhancing memory....

(περὶ μνήμης), etc. In like manner, with the more general Ethical treatises On Happiness
Happiness
Happiness is a mental state of well-being characterized by positive emotions ranging from contentment to intense joy. A variety of biological, psychological, religious, and philosophical approaches have striven to define happiness and identify its sources....

(περὶ εὐδαιμονίας β΄), and On Virtue
Virtue
Virtue is moral excellence. A virtue is a positive trait or quality subjectively deemed to be morally excellent and thus is valued as a foundation of principle and good moral being....

(περὶ ἀρετῆς) there were connected separate books on individual Virtues, on the Voluntary, etc. His four books on Royalty
Royal family
A royal family is the extended family of a king or queen regnant. The term imperial family appropriately describes the extended family of an emperor or empress, while the terms "ducal family", "grand ducal family" or "princely family" are more appropriate to describe the relatives of a reigning...

 he had addressed to Alexander (στοιχεῖα πρὸς Ἀλέξανδρον περὶ βασιλείας δ΄). Besides these he had written treatises On the State (περὶ πολιτείας α΄; πολιτικός α΄), On the Power of Law
Law
Law is a system of rules and guidelines which are enforced through social institutions to govern behavior, wherever possible. It shapes politics, economics and society in numerous ways and serves as a social mediator of relations between people. Contract law regulates everything from buying a bus...

(περὶ δυνάμεως νόμου α΄), etc., as well as upon Geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, Arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, and Astrology
Astrology
Astrology consists of a number of belief systems which hold that there is a relationship between astronomical phenomena and events in the human world...

. Besides philosophical treatises, he wrote poetry (epē) and paraenesis.

Epistemology


Xenocrates made a more definite division between the three departments of philosophy, than Speusippus
Speusippus
Speusippus was an ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, Speusippus inherited the Academy and remained its head for the next eight years. However, following a stroke, he passed the chair to Xenocrates. Although the successor to Plato...

, but at the same time abandoned Plato's heuristic method of conducting through doubts (aporiai), and adopted instead a mode of bringing forward his doctrines in which they were developed dogmatically.

Xenocrates recognized three grades of cognition, each appropriated to a region of its own: knowledge, sensation, and opinion. He referred knowledge (episteme) to that essence which is the object of pure thought, and is not included in the phenomenal world; sensation (aisthesis) to that which passes into the world of phenomena; opinion (doxa) to that essence which is at once the object of sensuous perception, and, mathematically, of pure reason - the essence of heaven
Heaven
Heaven, the Heavens or Seven Heavens, is a common religious cosmological or metaphysical term for the physical or transcendent place from which heavenly beings originate, are enthroned or inhabit...

 or the star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s; so that he conceived of doxa in a higher sense, and endeavoured, more definitely than Plato, to exhibit mathematics as mediating between knowledge and sensuous perception All three modes of apprehension partake of truth; but in what manner scientific perception (epistemonike aisthesis) did so, we unfortunately do not learn. Even here Xenocrates's preference for symbolic modes of sensualising or denoting appears: he connected the above three stages of knowledge with the three Fates
Moirae
The Moirae, Moerae or Moirai , in Greek mythology, were the white-robed incarnations of destiny . Their number became fixed at three...

: Atropos
Atropos
Atropos or Aisa , in Greek mythology, was one of the three Moirae, goddesses of fate and destiny. Her Roman equivalent was Morta.Atropos or Aisa was the oldest of the Three Fates, and was known as the "inflexible" or "inevitable." It was Atropos who chose the mechanism of death and ended the life...

, Clotho
Clotho
Clotho is one of the Three Fates or Moirae, in ancient Greek mythology. Her Roman equivalent is Nona. Clotho was responsible for spinning the thread of human life. She also made major decisions, such as when a person was born, thus in effect controlling people's lives...

, and Lachesis
Lachesis (mythology)
In Greek mythology, Lachesis was the second of the Three Fates, or Moirae, also known as the Triple Moon Goddesses or the Lunar Dieties. Each phase of the moon representing each of the fates - Clotho, Lachesis and Atropos...

. We know nothing further about the mode in which Xenocrates carried out his dialectic
Dialectic
Dialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...

, as it is probable that what was peculiar to Aristotelian
Aristotelianism
Aristotelianism is a tradition of philosophy that takes its defining inspiration from the work of Aristotle. The works of Aristotle were initially defended by the members of the Peripatetic school, and, later on, by the Neoplatonists, who produced many commentaries on Aristotle's writings...

 logic did not remain unnoticed in it, for it can hardly be doubted that the division of the existent into the absolutely existent, and the relatively existent, attributed to Xenocrates, was opposed to the Aristotelian table of categories
Categories (Aristotle)
The Categories is a text from Aristotle's Organon that enumerates all the possible kinds of thing that can be the subject or the predicate of a proposition...

.

Metaphysics


We know from Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...

 that Xenocrates, if he did not explain the Platonic construction of the world-soul as Crantor
Crantor
Crantor was a Greek philosopher of the Old Academy, probably born around the middle of the 4th century BC, at Soli in Cilicia.-Life:Crantor moved to Athens in order to study philosophy, where he became a pupil of Xenocrates and a friend of Polemo, and one of the most distinguished supporters of...

 after him did, nevertheless drew heavily on the Timaeus
Timaeus (dialogue)
Timaeus is one of Plato's dialogues, mostly in the form of a long monologue given by the title character, written circa 360 BC. The work puts forward speculation on the nature of the physical world and human beings. It is followed by the dialogue Critias.Speakers of the dialogue are Socrates,...

; and further that he was at the head of those who, regarding the universe as unoriginated and imperishable, looked upon the chronological succession in the Platonic theory as a form in which to denote the relations of conceptual succession. Plutarch unfortunately, does not give us any further details, and contented himself with describing the well-known assumption of Xenocrates, that the soul is a self-moving number. Probably we should connect with this the statement that Xenocrates called unity and duality (monas and duas) deities, and characterised the former as the first male existence, ruling in heaven, as father and Zeus
Zeus
In the ancient Greek religion, Zeus was the "Father of Gods and men" who ruled the Olympians of Mount Olympus as a father ruled the family. He was the god of sky and thunder in Greek mythology. His Roman counterpart is Jupiter and his Etruscan counterpart is Tinia.Zeus was the child of Cronus...

, as uneven number and spirit; the latter as female, as the mother of the gods, and as the soul of the universe which reigns over the mutable world under heaven, or, as others have it, that he named the Zeus who ever remains like himself, governing in the sphere of the immutable, the highest; the one who rules over the mutable, sublunary world, the last, or outermost.

If, like other Platonists, he designated the material principle as undefined duality, the world-soul was probably described by him as the first defined duality, the conditioning or defining principle of every separate definitude in the sphere of the material and changeable, but not extending beyond it. He appears to have called it in the highest sense the individual soul, in a derivative sense a self-moving number, that is, the first number endowed with motion. To this world-soul Zeus, or the world-spirit, has entrusted - in what degree and in what extent, we do not learn - dominion over that which is liable to motion and change. The divine power of the world-soul is then again represented, in the different spheres of the universe, as infusing soul into the planets, Sun and Moon, - in a purer form, in the shape of Olympic gods
Twelve Olympians
The Twelve Olympians, also known as the Dodekatheon , in Greek mythology, were the principal deities of the Greek pantheon, residing atop Mount Olympus. Zeus, Hera, Poseidon, Demeter, Hestia, and Hades were siblings. Ares, Hermes, Hephaestus, Athena, Apollo, and Artemis were children of Zeus...

. As a sublunary daemonical
Daemon (mythology)
The words dæmon and daimôn are Latinized spellings of the Greek "δαίμων", a reference to the daemons of Ancient Greek religion and mythology, as well as later Hellenistic religion and philosophy...

 power (as Hera
Hera
Hera was the wife and one of three sisters of Zeus in the Olympian pantheon of Greek mythology and religion. Her chief function was as the goddess of women and marriage. Her counterpart in the religion of ancient Rome was Juno. The cow and the peacock were sacred to her...

, Poseidon
Poseidon
Poseidon was the god of the sea, and, as "Earth-Shaker," of the earthquakes in Greek mythology. The name of the sea-god Nethuns in Etruscan was adopted in Latin for Neptune in Roman mythology: both were sea gods analogous to Poseidon...

, Demeter
Demeter
In Greek mythology, Demeter is the goddess of the harvest, who presided over grains, the fertility of the earth, and the seasons . Her common surnames are Sito as the giver of food or corn/grain and Thesmophoros as a mark of the civilized existence of agricultural society...

), it dwells in the elements, and these daemonical natures, midway between gods and men, are related to them as the isosceles triangle is to the equilateral
Equilateral
In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...

 and the scalene
Scalene
Scalene may refer to:* A scalene triangle, one in which all sides are different* A scalene ellipsoid, one in which the lengths of all three semi-principal axes are different* Scalene muscles of the neck...

. The divine world-soul which reigns over the whole domain of sublunary changes he appears to have designated as the last Zeus, the last divine activity.

It is not till we get to the sphere of the separate daemonical powers of nature
Nature
Nature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general...

 that the opposition between good and evil begins, and the daemonical power is appeased by means of a stubbornness which it finds there congenial to it; the good daemonical power makes happy those in whom it takes up its abode, the bad ruins them; for eudaimonia
Eudaimonia
Eudaimonia or eudaemonia , sometimes Anglicized as eudemonia , is a Greek word commonly translated as happiness or welfare; however, "human flourishing" has been proposed as a more accurate translation...

 is the indwelling of a good daemon, the opposite the indwelling of a bad one.

How Xenocrates tried to establish and connect scientifically these assumptions, which appear to be taken chiefly from his books on the nature of the gods, we do not learn, and can only discover the one fundamental idea at the basis of them, that all grades of existence are penetrated by divine power, and that this grows less and less energetic in proportion as it descends to the perishable and individual. Hence he also appears to have maintained that as far as consciousness extends, so far also extends an intuition of that all-ruling divine power, of which he represented even irrational animals as partaking. But neither the thick nor the thin, to the different combinations of which he appears to have tried to refer the various grades of material existence, were regarded by him as in themselves partaking of soul; doubtless because he referred them immediately to the divine activity, and was far from attempting to reconcile the duality of the principia, or to resolve them into an original unity. Hence too he was for proving the incorporeality of the soul by the fact that it is not nourished as the body is.

It is probable, that, after the example of Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

, he designated the divine principium as alone indivisible, and remaining like itself; the material, as the divisible, partaking of multiformity, and different, and that from the union of the two, or from the limitation of the unlimited by the absolute unity, he deduced number, and for that reason called the soul of the universe, like that of individual beings, a self moving number, which, by virtue of its twofold root in the same and the different, shares equally in permanence and motion, and attains to consciousness by means of the reconciliation of this opposition.

Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

, in his Metaphysics
Metaphysics (Aristotle)
Metaphysics is one of the principal works of Aristotle and the first major work of the branch of philosophy with the same name. The principal subject is "being qua being", or being understood as being. It examines what can be asserted about anything that exists just because of its existence and...

, recognized amongst contemporary Platonists three principal views concerning the ideal numbers, and their relation to the ideas and to mathematical number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s:
  1. those who, like Plato, distinguished ideal and mathematical numbers;
  2. those who, like Xenocrates, identified ideal numbers with mathematical numbers
  3. those who, like Speusippus, postulated mathematical numbers only


Aristotle has much to say against the Xenocratean interpretation of the theory, and in particular points out that, if the ideal numbers are made up of arithmetical units, they not only cease to be principles, but also become subject to arithmetical operations.

In the derivation of things according to the series of the numbers he seems to have gone further than any of his predecessors. He approximated to the Pythagoreans in this, that (as is clear from his explanation of the soul) he regarded number as the conditioning principle of consciousness, and consequently of knowledge also; he thought it necessary, however, to supply what was wanting in the Pythagorean assumption by the more accurate definition, borrowed from Plato, that it is only insofar as number reconciles the opposition between the same and the different, and has raised itself to self-motion, that it is soul. We find a similar attempt at the supplementation of the Platonic doctrine in Xenocrates's assumption of indivisible lines. In them he thought he had discovered what, according to Plato, God alone knows, and he among men who is loved by him, namely, the elements or principia of the Platonic triangles. He seems to have described them as first, original lines, and in a similar sense to have spoken of original plain figures and bodies, convinced that the principia of the existent should be sought not in the material, not in the divisible which attains to the condition of a phenomenon, but merely in the ideal definitude of form. He may very well, in accordance with this, have regarded the point as a merely subjectively admissible presupposition, and a passage of Aristotle respecting this assumption should perhaps be referred to him.

Ethics


The information on his Ethics is scanty. He tried to supplement the Platonic doctrine at various points, and at the same time to give it a more direct applicability to life. He distinguished from the good and the bad something which is neither good nor bad. Following the ideas of his Academic predecessors, he viewed the good as that which should be striven after for itself, that is, which has value in itself, while the bad is the opposite of this. Consequently, that which is neither good nor bad is what in itself is neither to be striven after nor to be avoided, but derives value or the opposite according as it serves as means for what is good or bad, or rather, is used by us for that purpose.
While, however, Xenocrates (and with him Speusippus and the other philosophers of the older Academy) would not accept that these intermediate things, such as health
Health
Health is the level of functional or metabolic efficiency of a living being. In humans, it is the general condition of a person's mind, body and spirit, usually meaning to be free from illness, injury or pain...

, beauty
Beauty
Beauty is a characteristic of a person, animal, place, object, or idea that provides a perceptual experience of pleasure, meaning, or satisfaction. Beauty is studied as part of aesthetics, sociology, social psychology, and culture...

, fame
Celebrity
A celebrity, also referred to as a celeb in popular culture, is a person who has a prominent profile and commands a great degree of public fascination and influence in day-to-day media...

, good fortune
Destiny
Destiny or fate refers to a predetermined course of events. It may be conceived as a predetermined future, whether in general or of an individual...

, etc. were valuable in themselves, he did not accept that they were absolutely worthless or indifferent. According, therefore, as what belongs to the intermediate region is adapted to bring about or to hinder the good, Xenocrates appears to have designated it as good or evil, probably with the proviso, that by misuse what is good might become evil, and vice versa, that by virtue, what is evil might become good.

Still he maintained that virtue alone is valuable in itself, and that the value of every thing else is conditional. According to this, happiness should coincide with the consciousness of virtue, though its reference to the relations of human life requires the additional condition, that it is only in the enjoyment of the good things and circumstances originally designed for it by nature that it attains to completion; to these good things, however, sensuous gratification does not belong. In this sense he on the one hand denoted (perfect) happiness as the possession of personal virtue, and the capabilities adapted to it, and therefore reckoned among its constituent elements, besides moral actions conditions and facilities, those movements and relations also without which external good things cannot be attained, and on the other hand did not allow that wisdom, understood as the science of first causes or intelligible essence, or as theoretical understanding, is by itself the true wisdom which should be striven after by people, and therefore seems to have regarded this human wisdom as at the same time exerted in investigating, defining, and applying. How decidedly he insisted not only on the recognition of the unconditional nature of moral excellence, but on morality of thought, is shown by his declaration, that it comes to the same thing whether one casts longing eyes, or sets one's feet upon the property of others. His moral earnestness is also expressed in the warning that the ears of children should be guarded against the poison of immoral speeches.

Mathematics


Xenocrates is known to have written a book On Numbers, and a Theory of Numbers, besides books on geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...

 writes that Xenocrates once attempted to find the total number of syllables that could be made from the letters of the alphabet. According to Plutarch, Xenocrates result was 1,002,000,000,000 (a "myriad
Myriad
Myriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...

-and-twenty times a myriad-myriad"). This possibly represents the first instance that a combinatorial problem involving permutations was attempted. Xenocrates also supported the idea of "indivisible lines" (and magnitudes) in order to counter Zeno's paradoxes
Zeno's paradoxes
Zeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...

.

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