The Square ψυχή

The earliest attestation: Simonides of Ceos (c. 556–468 BCE)

The metaphor originates in a poem addressed to Scopas of Thessaly, preserved within Plato's Protagoras:

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

"A truly good man it is hard to become, square in hands and feet and νοῦς, fashioned without blame."

(Plato, Protagoras, Eulogikon: ffk-al, ref. Prot.362.a)

Three observations. The square is applied to the ἀνήρ (man), not yet to ψυχή. The domains are threefold: hands (action), feet (movement), νοῦς (intellect), covering the whole person. And the context is a rejection of Pittacus' dictum: Simonides says becoming good is hard, remaining good is for a god alone. The square denotes stable character that withstands fortune.

Plato's Socrates then analyses this poem line by line, making it the centrepiece of a debate on whether ἀρετή can be taught. The square man becomes a philosophical problem.

Aristotle (384–322 BCE): the ethical application

Aristotle takes the Simonidean phrase and applies it to the εὐδαίμων:

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

"And he will bear his fortunes most beautifully and in every way harmoniously, the man who is truly good and square, without blame."

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

Aristotle's innovation: the square is now explicitly a disposition toward τύχη (fortune). The stable man, like a square block, stands firm no matter which side it falls on.

Aspasius (c. 100–150 CE): the first explicit commentary

The Peripatetic commentator Aspasius gives the fullest ancient gloss:

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.

"For some are called 'square' who adapt themselves to the characters of those they associate with… These are blameworthy. But 'square without blame' are those who bear fortunes harmoniously, neither collapsing under misfortune nor being elated by good fortune, but just as square stones stand firm on whatever side they fall, so these men stand upright in every fortune."

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

Aspasius distinguishes two senses: a negative sense of the chameleon-like person (blameworthy), and the positive Aristotelian sense of the man whose character is so stable that fortune cannot topple him. The key image is the τετράγωνος λίθος: the square stone.

The Pythagorean tradition: the ψυχή as a square

A separate strand identifies the ψυχή itself as a square. Attributed to Pythagoras by later doxographers, via John the Lydian:

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

"For the supports of the ψυχή are four: νοῦς, ἐπιστήμη, δόξα, αἴσθησις. 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 attribution appears in Pseudo-Archytas (Eulogikon: mxk-aa, ref. 47) and the Pythagoras fragments (Eulogikon: bgs-ae, ref. 165). The square ψυχή is stable, balanced, and proportioned: a mathematical expression of psychic harmony. Four cognitive supports hold it up the way four sides hold a square.

Proclus (412–485 CE): the Neoplatonic synthesis

Proclus, commenting on Plato's Timaeus, integrates the square into the ψυχή's numerical structure:

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.

"All numbers are therefore in the οὐσία 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 here is 49 (7²): the product of the hebdomad, linked to the World ψυχή's division in the Timaeus. It is the culminating number in the ψυχή's structure, not merely a metaphor for stability.

The four elements and the tetrad: Theon of Smyrna

Theon of Smyrna (c. 70–135 CE) gives the most systematic synthesis of Pythagorean tetrads. The fourth tetrad is the four simple bodies:

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.

"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."

Then the tetrad of the ψυχή's faculties, the eighth:

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

"The eighth tetrad is critical and intelligible: νοῦς, ἐπιστήμη, δόξα, αἴσθησις."

(Theon of Smyrna, On the Usefulness of Mathematics, Eulogikon: pic-aa, ref. 97–98)

The number-four structure runs from the four στοιχεῖα through the four Platonic solids (cube = earth) through the four cognitive faculties. Theon's ninth tetrad then adds the tripartite ψυχή (λογιστικόν, θυμικόν, ἐπιθυμητικόν) plus the σῶμα as a fourth term, making the embodied ψυχή a fourfold structure that maps onto the four-element cosmos.

Late antique usage: the ethical ideal becomes proverbial

The phrase τετράγωνος ἄνευ ψόγου became a standard citation. Julian the Apostate:

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

"For he seems to me to be a man, in Simonides' words, 'square, fashioned without blame.'"

(Julian, Symposium or Kronia, Eulogikon: sno-am, ref. 38)

And Damascius applies it to the philosopher Isidore:

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

"He was dignified and well-formed in all things and ordered square without blame, as one might say in Simonides' words."

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

By the time of Eustathius (12th c. CE), Homer himself receives the epithet: (Eulogikon: xpw-ad, ref. 1.753), the square poet, fashioned without blame.

The architecture of the four στοιχεῖα connection

The link between the square ψυχή and the four elements runs through the architecture of the tetrad:

1. The tetraktys (1 + 2 + 3 + 4 = 10) is the root of all things: the oldest Pythagorean doctrine.
2. The four στοιχεῖα are one application of the tetrad (Theon's fourth tetrad: πῦρ, ἀήρ, ὕδωρ, γῆ).
3. The four cognitive faculties (νοῦς, ἐπιστήμη, δόξα, αἴσθησις) are another (Theon's eighth tetrad).
4. The ψυχή as τετράγωνον is the geometric expression of this tetradic structure.
5. The four Platonic solids (with the cube (κύβος) corresponding to earth) provide the geometrical bridge: just as square faces compose the cube, so the square ψυχή is the principle that structures the physical world.

The cube = γῆ (earth) is the third dimension of the square: the square made solid. The same number runs from the ethical (the man who stands firm under fortune) through the psychological (the four cognitive supports) to the cosmological (the element-structure of the world).

Summary