Plato's number is a number enigmatically referred to by
Plato
Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
in his dialogue the ''
Republic
A republic, based on the Latin phrase ''res publica'' ('public affair' or 'people's affair'), is a State (polity), state in which Power (social and political), political power rests with the public (people), typically through their Representat ...
'' (8.546b). The text is notoriously difficult to understand and its corresponding translations do not allow an unambiguous interpretation. There is no real agreement either about the meaning or the value of the number. It also has been called the "geometrical number" or the "nuptial number" (the "number of the bride"). The passage in which Plato introduced the number has been discussed ever since it was written, with no consensus in the debate. As for the number's actual value,
216
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Year 216 ( CCXVI) was a leap year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Sabinus and Anullinus (or, less frequently, year 969 ''Ab urbe condita''). The denomination 216 f ...
is the most frequently proposed value for it, but 3,600 or 12,960,000 are also commonly considered.
An incomplete list
[for more names and references see Dupuis J., ''Le Nombre Geometrique de Platon'', Paris: Hachette, 1885] of authors who mention or discourse about includes the names of
Aristotle
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
,
Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
for antiquity;
Ficino
Marsilio Ficino (; Latin name: ; 19 October 1433 – 1 October 1499) was an Italian scholar and Catholic priest who was one of the most influential humanist philosophers of the early Italian Renaissance. He was an astrologer, a reviver of Neo ...
and
Cardano during the
Renaissance
The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
;
Zeller Zeller, meaning both prisoner and monk in German, may refer to:
Places
*Zeller Ache, a river of Upper Austria
*Zeller Bach (Isar), a river of Bavaria, Germany, tributary of the Isar
*Zeller Bach (Memminger Ach), a river of Bavaria, Germany, tributa ...
,
Friedrich Schleiermacher
Friedrich Daniel Ernst Schleiermacher (; ; 21 November 1768 – 12 February 1834) was a German Reformed Church, Reformed theology, theologian, philosopher, and biblical scholar known for his attempt to reconcile the criticisms of the Age o ...
,
Paul Tannery
Paul Tannery (20 December 1843 – 27 November 1904) was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose ''Notions Mathématiques'' he contributed an historical chapter. Thou ...
and
Friedrich Hultsch
Friedrich Otto Hultsch (22 July 1833, Dresden – 6 April 1906, Dresden) was a German classical philologist and historian of mathematics in antiquity.
Biography
After graduating from the Dresden '' Kreuzschule'', Friedrich Hultsch studied classic ...
in the 19th century and further new names are currently added.
[McNamee K., and Jacovides M., '' Annotations to the Speech of the Muses (Plato "Republic" 546B-C)'', Zeitschrift für Papyrologie und Epigraphik, Bd. 144, (2003), pp. 31-50]
Further in the ''Republic'' (9.587b) another number is mentioned, known as the "Number of the Tyrant".
Plato's text
Great lexical and syntactical differences are easily noted between the many translations of the ''Republic''. Below is a typical text from a relatively recent translation of ''Republic'' 546b–c:
Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable 46cwith one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the
rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births.
The 'entire geometrical number', mentioned shortly before the end of this text, is understood to be Plato's number. The introductory words mention (a period comprehended by) 'a perfect number' which is taken to be a reference to Plato's
perfect year mentioned in his ''
Timaeus'' (39d). The words are presented as uttered by the
muses
In ancient Greek religion and Greek mythology, mythology, the Muses (, ) were the Artistic inspiration, inspirational goddesses of literature, science, and the arts. They were considered the source of the knowledge embodied in the poetry, lyric p ...
, so the whole passage is sometimes called the 'speech of the muses' or something similar.
Indeed,
Philip Melanchthon
Philip Melanchthon (born Philipp Schwartzerdt; 16 February 1497 – 19 April 1560) was a German Lutheran reformer, collaborator with Martin Luther, the first systematic theologian of the Protestant Reformation, an intellectual leader of the L ...
compared it to the proverbial obscurity of the
Sibyl
The sibyls were prophetesses or oracles in Ancient Greece.
The sibyls prophet, prophesied at holy sites.
A sibyl at Delphi has been dated to as early as the eleventh century BC by Pausanias (geographer), PausaniasPausanias 10.12.1 when he desc ...
s.
Cicero
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, orator, writer and Academic skeptic, who tried to uphold optimate principles during the political crises tha ...
famously described it as 'obscure' but others have seen some playfulness in its tone.
Interpretations

Shortly after Plato's time his meaning apparently did not cause puzzlement as Aristotle's casual remark attests. Half a millennium later, however, it was an enigma for the
Neoplatonist
Neoplatonism is a version of Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as a series of thinkers. Among the common id ...
s, who had a somewhat mystic penchant and wrote frequently about it, proposing geometrical and numerical interpretations. Next, for nearly a thousand years, Plato's texts disappeared from Western Europe and it is only in the
Renaissance
The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
that the enigma briefly resurfaced. During the 19th century, when classical scholars restored original texts, the problem reappeared. Schleiermacher interrupted his edition of Plato for a decade while attempting to make sense of the paragraph .
Victor Cousin
Victor Cousin (; ; 28 November 179214 January 1867) was a French philosopher. He was the founder of " eclecticism", a briefly influential school of French philosophy that combined elements of German idealism and Scottish Common Sense Realism. ...
inserted a note that it has to be skipped in his French translation of Plato's works. In the early 20th century, scholarly findings suggested a
Babylon
Babylon ( ) was an ancient city located on the lower Euphrates river in southern Mesopotamia, within modern-day Hillah, Iraq, about south of modern-day Baghdad. Babylon functioned as the main cultural and political centre of the Akkadian-s ...
ian origin for the topic.
Most interpreters argue that the value of Plato's number is 216 because it is the cube of 6, i.e. , which is remarkable for also being the sum of the cubes for the
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...
(3, 4, 5): .
Such considerations tend to ignore the second part of the text where some other numbers and their relations are described. The opinions tend to converge about their values being 480,000 and 270,000 but there is little agreement about the details. It has been noted that 6 yields 1296 and . Instead of multiplication some interpretations consider the sum of these factors: .
Other values that have been proposed include:
*, by Otto Weber (1862).
*, 19 being obtained from and being the number of years in the
Metonic cycle
The Metonic cycle or enneadecaeteris (from , from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The recurrence is not perfect, and by precise obser ...
.
*, a
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
proposed by Cardano. It is known that such numbers can be decomposed into the sum of consecutive odd cubes, so .
*, by Marsilio Ficino (1496).
[Allen M., ''Nuptial Arithmetic:Marsilio Ficino's Commentary on the Fatal Number in Book VIII of Plato's Republic'', UCLA 1994, p75ff.]
*, by
Jacob Friedrich Fries (1823).
See also
*
Euler's sum of powers conjecture
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers ...
References
Further reading
* Donaldson J., "On Plato's Number", ''Proceedings of the Philological Society'', vol.1, iss. 8, p. 81-90, April 7, 1843
*
Adam J., ''The nuptial number of Plato: its solution and significance'', London: C.J. Clay and Sons, 1891.
*Laird, A.G., ''Plato's Geometrical Number and the Comment of Proclus'', The Collegiate Press, George Banta Publishing Company, Menasha, Wisconsin. 1918
*Diès A., ''Le Nombre de Platon: Essai d'exégèse et d'Histoire'', Paris 1936
*Allen M., ''Nuptial Arithmetic: Marsilio Ficino's Commentary on the Fatal Number in Book VIII of Plato's Republic'', UCLA 1994
*Dumbrill R., ''Four Mathematical Texts from the Temple Library of Nippur: a source for Plato's number'', ARANE 1 (2009): 27-3
External links
Five translations of Rep. 8.546 and 9.587* {{MathWorld, title=Plato's Numbers, urlname=PlatosNumbers
*
ttp://mathworld.wolfram.com/DiophantineEquation3rdPowers.html Math world : Diophantine Equation--3rd PowersSum of Consecutive Cubes Equals a Cube
Integers
Republic (Plato)
Greek mathematics
Numerology