Gauss–Jacobi quadrature

Plato’s number is a number enigmatically referred to by Plato in his dialogue the Republic (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 is the most frequently proposed value for it, but 3,600 or 12,960,000 are also commonly considered.

An incomplete list of authors who mention or discourse about includes the names of Aristotle, Proclus for antiquity; Ficino and Cardano during the Renaissance; Zeller, Friedrich Schleiermacher, Paul Tannery and Friedrich Hultsch in the 19th century and further new names are currently added.

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 [546c] with 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, so the whole passage is sometimes called the 'speech of the muses' or something similar. Indeed Philip Melanchthon compared it to the proverbial obscurity of the Sibyls. Cicero 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 Neoplatonists, 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 an it is only in the Renaissance that the enigma briefly resurfaced. During the 19th century, when classical scholars restored original texts, the problem reappeared. Schleiermacher interrupted for a decade his edition of Plato while attempting to make sense of the paragraph. Victor Cousin 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 Babylonian origin for the topic.

Most interpretators argue that the value of Plato's number is 216 because it is the cube of 6 i.e. $6^{3}=216$ which is remarkable for being also a sum of the cubes for the Pythagorean triple 3,4,and 5:

$3^{3}+4^{3}+5^{3}=6^{3}.\,$

Such considerations tend to ignore the second part of the text where some others numbers and their relations are described. The opinions tend to converge about their values being 480000 and 270000 but there is little agreement about the details. It has been noted that 6 raised to fourth power yields 1296 and $48\times 27=36\times 36=1296$. Instead of multiplication some interpretations consider the sum of these factors: 48+27=75.

Other values that have been proposed include:

• $17500=100\times 100+4800+2700$ by Otto Weber(1862)
• $760000=750000+10000=19\times 4\times 10000$, 19 being obtained from $({\tfrac {4}{3}}+5)\times 3$ and it is the number of years in the Metonic cycle
• $8128=2^{6}\cdot (2^{7}-1)$, a perfect number proposed by Cardano. It is known that such numbers can be decomposed in the sum of consecutive odd cubes, so 8128=1³+3³+5³+...+15³
• $1728=12^{3}=8\cdot 12\cdot 18$ by Marsilio Ficino (1496)
• 5040 $=144\times 35=(3+4+5)^{2}\cdot (2^{3}+3^{3})$ by Jacob Friedrich Fries (1823)

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 [1]

External links

• Five translations of Rep. 8.546 and 9.587
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