τετράγωνος: the four-square man and the four-square ψυχή

τετράγωνος (tetrágōnos) is τετρα- (four) + γωνία (angle): four-angled, square. The corpus applies it to a person, to the ψυχή, and to a number. The passages are set out below in chronological order, in the original Greek, transliterated and translated, each anchored to its work and reference.

The four-square man: Simonides, in Plato (5th c. BCE)

Plato's Protagoras quotes a poem of Simonides of Ceos addressed to Scopas of Thessaly. The dialogue then takes the poem as the subject of an extended discussion between Socrates and Protagoras.

ἄνδρ' ἀγαθὸν μὲν ἀλαθέως γενέσθαι χαλεπόν, χερσίν τε καὶ ποσὶ καὶ νόῳ τετράγωνον, ἄνευ ψόγου τετυγμένον.

ándr' agathòn mèn alathéōs genésthai chalepón, chersín te kai posì kai nóō tetrágōnon, áneu psógou tetygménon.

A good man, truly, it is hard to become, four-square in hands and feet and νόος, fashioned without fault.

(Simonides, in Plato, Protagoras, Eulogikon: ffk-al, ref. Prot.338.e)

The word here is τετράγωνον, in agreement with ἄνδρα (man). The three domains named are hands, feet, and νόος (the faculty of νοεῖν).

The four-square man bearing fortune: Aristotle (4th c. BCE)

Aristotle quotes the Simonidean phrase in the Nicomachean Ethics, in the discussion of whether the εὐδαίμων (the one who flourishes) can be touched by changes of fortune (τύχη):

καὶ τὰς τύχας οἴσει κάλλιστα καὶ πάντῃ πάντως ἐμμελῶς ὅ γ' ὡς ἀληθῶς ἀγαθὸς καὶ τετράγωνος ἄνευ ψόγου.

kai tàs týchas oísei kállista kai pántē pántōs emmelōs hó g' hōs alēthōs agathòs kai tetrágōnos áneu psógou.

And he will bear his fortunes most finely and altogether harmoniously, the one who is truly good and four-square without fault.

(Aristotle, Nicomachean Ethics, Eulogikon: hgw-aw, ref. EN 1100b)

The square stone: Aspasius (2nd c. CE)

The Peripatetic commentator Aspasius, in his commentary on the Nicomachean Ethics, glosses the word at this passage. He gives two uses of τετράγωνος, one blamed and one not:

οἱ μὲν γὰρ τετράγωνοι λέγονται οἱ συμμετατιθέμενοι τοῖς ἤθεσι τῶν συγγινομένων… οὗτοι μὲν δὴ ἐπίψογοι· ἄνευ δὲ ψόγου τετράγωνοι οἱ τὰς τύχας ἐμμελῶς φέροντες καὶ μήτε ὑπὸ δυστυχίας καταπίπτοντες μήτε ὑπὸ εὐτυχίας ἐπαιρόμενοι, ἀλλ' ὥσπερ οἱ τετράγωνοι λίθοι ἐφ' ὃ ἂν πέσωσι μέρος ἑστᾶσιν, οὕτως καὶ οὗτοι κατὰ πᾶσαν τύχην ἑστήκασιν ὀρθοί.

hoi mèn gàr tetrágōnoi légontai hoi symmetatithémenoi tois ḗthesi tōn synginoménōn… houtoi mèn dḕ epípsogoi; áneu dè psógou tetrágōnoi hoi tàs týchas emmelōs phérontes kai mḗte hypò dystychías katapíptontes mḗte hypò eutychías epairómenoi, all' hṓsper hoi tetrágōnoi líthoi eph' hò àn pésōsi méros hestasin, hoútōs kai houtoi katà pasan týchēn hestḗkasin orthoí.

Some are called 'four-square' who adjust themselves to the characters of those they keep company with… These are blameworthy. But 'four-square without fault' are those who bear their fortunes harmoniously and neither fall down under misfortune nor are lifted up by good fortune, but just as square stones stand on whatever side they fall, so these too stand upright under every fortune.

(Aspasius, Commentary on Aristotle's Nicomachean Ethics, Eulogikon: pms-aa, ref. 29)

The four-square ψυχή: the Pythagorean tradition (1st–2nd c. CE)

John the Lydian, in On the Months, reports a saying attributed to Pythagoras, in which the word is applied to the ψυχή, with four named supports:

τὰ γὰρ τῆς ψυχῆς στηρίγματα τέσσαρά ἐστι, νοῦς ἐπιστήμη δόξα αἴσθησις. ψυχὰ γὰρ ἀνθρώπου, ὡς Πυθαγόρας ἔφη, ἔστι τετράγωνον ὀρθογώνιον.

tà gàr tēs psychēs stērígmata téssará esti, noûs epistḗmē dóxa aísthēsis. psychà gàr anthrṓpou, hōs Pythagóras éphē, ésti tetrágōnon orthogṓnion.

For the supports of the ψυχή are four: νοῦς, ἐπιστήμη (knowledge), δόξα (opinion), αἴσθησις (perception). For the ψυχή of a human being, as Pythagoras said, is a right-angled square.

(John the Lydian, On the Months, Eulogikon: vhw-ab, ref. Mens 2 9)

The same passage continues with a contrasting report attributed to Archytas, who is said to define the ψυχή by the circle rather than the square, on the ground that the ψυχή is τὸ αὑτὸ κινοῦν, the self-moving, and the first mover is a circle or sphere. The four-square attribution to Pythagoras also appears in Pseudo-Archytas (Eulogikon: Ten Categories, ref. 47) and in the Pythagoras fragments (Eulogikon: Sacred Doctrine, ref. 165).

The tetrad of bodies, shapes, and faculties: Theon of Smyrna (2nd c. CE)

Theon of Smyrna, in On Mathematics Useful for Reading Plato, lists a series of tetrads. The fourth is the four simple bodies; the fifth, their shapes; the eighth, the same four faculties named in the Pythagorean saying:

τετάρτη δὲ τετρακτύς ἐστι τῶν ἁπλῶν σωμάτων, πυρὸς ἀέρος ὕδατος γῆς… πέμπτη δ' ἐστὶ τετρακτὺς ἡ τῶν σχημάτων τῶν ἁπλῶν σωμάτων. ἡ μὲν γὰρ πυραμὶς σχῆμα πυρός, τὸ δὲ ὀκτάεδρον ἀέρος, τὸ δὲ εἰκοσάεδρον ὕδατος, κύβος δὲ γῆς.

tetártē dè tetraktýs esti tōn haplōn sōmátōn, pyròs aéros hýdatos gēs… pémptē d' estì tetraktỳs hē tōn schēmátōn tōn haplōn sōmátōn. hē mèn gàr pyramìs schēma pyrós, tò dè oktáedron aéros, tò dè eikosáedron hýdatos, kýbos dè gēs.

The fourth tetrad is of the simple bodies: fire, air, water, earth… The fifth tetrad is of the shapes of the simple bodies. The pyramid is the shape of fire, the octahedron of air, the icosahedron of water, the cube of earth.

ὀγδόη δὲ τετρακτὺς ἥδε, τούτων κριτικὴ καὶ νοητή τις οὖσα· νοῦς ἐπιστήμη δόξα αἴσθησις.

ogdóē dè tetraktỳs hḗde, toútōn kritikḕ kai noētḗ tis oûsa; noûs epistḗmē dóxa aísthēsis.

The eighth tetrad is this, the one that judges and apprehends these: νοῦς, ἐπιστήμη, δόξα, αἴσθησις.

(Theon of Smyrna, On Mathematics Useful for Reading Plato, Eulogikon: pic-aa, ref. 97)

The square from seven: Proclus (5th c. CE)

Proclus, commenting on the Timaeus and the numbers in the constitution of the ψυχή, names a square among them:

πάντως ἄρα οἱ ἀριθμοὶ πάντες εἰσὶ τῇ οὐσίᾳ τῆς ψυχῆς, μονάς, δυάς, τριάς, τετράς, πεντάς, ἑξάς, ἑπτάς, καὶ ἐπὶ πᾶσιν ὁ ἀπὸ τῆς ἑπτάδος τετράγωνος.

pántōs ára hoi arithmoì pántes eisì tē ousíā tēs psychēs, monás, dyás, triás, tetrás, pentás, hexás, heptás, kai epì pasin ho apò tēs heptádos tetrágōnos.

In every way, then, all the numbers are in the οὐσία (being) of the ψυχή: monad, dyad, triad, tetrad, pentad, hexad, heptad, and above all the square from the heptad.

(Proclus, Commentary on Plato's Timaeus, Eulogikon: uqy-ah, ref. in Ti.2.271)

"The square from the heptad" (ὁ ἀπὸ τῆς ἑπτάδος τετράγωνος) is 7 × 7 = 49.

The phrase as quotation: Julian and Damascius (4th–6th c. CE)

The Simonidean phrase τετράγωνος ἄνευ ψόγου is quoted by later writers with attribution to Simonides. Julian, in The Caesars, applies it to Marcus Aurelius:

δοκεῖ γὰρ εἶναί μοί πως ἀνὴρ κατὰ τὸν Σιμωνίδην «τετράγωνος ἄνευ ψόγου τετυγμένος.»

dokeî gàr eînaí moí pōs anḕr katà tòn Simōnídēn "tetrágōnos áneu psógou tetygménos."

For he seems to me to be a man, in Simonides' phrase, 'four-square, fashioned without fault.'

(Julian, The Caesars, Eulogikon: sno-am, ref. 38)

Damascius, in the Life of Isidore, applies the phrase to the philosopher Isidore, again with attribution to Simonides:

σεμνὸς δὲ ἦν καὶ εὐσχήμων τὰ πάντα καὶ τετράγωνος ἄνευ ψόγου τεταγμένος, ὡς φάναι κατὰ τὸν Σιμωνίδην.

semnòs dè ēn kai euschḗmōn tà pánta kai tetrágōnos áneu psógou tetagménos, hōs phánai katà tòn Simōnídēn.

He was dignified and well-formed in all things, and four-square without fault, as one might say in Simonides' phrase.

(Damascius, Life of Isidore, Eulogikon: uyg-ab, ref. 332)

The phrase is also collected, with attribution to Simonides, in the anthology of Stobaeus (Eulogikon: Anthology, ref. 4.5), and applied to Homer by the 12th-century commentator Eustathius (Eulogikon: Commentary on Homer's Iliad, ref. 1.753).