Shapes, Numbers, Symbols, Tokens


1 Introduction


Proclus and Porphyry

2 Numbers, proportions and harmony


Martianus Capella, Iamblichus, Nicomachus, Proclus, Hermias, Damascius etc.
Proclus, Essay 13 (number symbolism, octave etc.)

3 Two-dimensional shapes

According to Proclus’ commentary On the Republic (vol. 2, p. 48), the circle is appropriate to Kronos, the first intellect, while the triangle belongs to the life-originating goddess, or second intellect (the Mother-of-Gods), from whom issue the souls. In this connection, he cites Xenocrates, who assigns

  • the equilateral triangle to divine souls (gods),
  • the isosceles triangle to daemonic souls (i.e., daemons or Greater Beings),
  • while the scalene triangle “is an image of souls that ascend and descend”, i.e., humans.

(Cf. Plutarch, On the Obsolescence of Oracles 416d.)

However, in another work, Proclus cites the Pythagorean Philolaus as ascribing the angle of the square to Rhea, Demeter and Hestia, all life-originating goddesses in whom the Earth has a share (On Euclid, p. 173), and that of the triangle to Kronos, Hades, Ares and Dionysus, as rulers over the four elemental divisions of the zodiac:

  • Kronos over moist and cold substance, i.e., water,
  • Ares over fire,
  • Hades over earth,
  • and Dionysus over moist and hot, i.e., air.

(Proclus, On Euclid, p. 167.)

Philolaus further ascribes the angle of the dodecagon, or twelve-angled plane figure, to Zeus, as all-encompassing (On Euclid, p. 174).

Plutarch gives us very similar information, saying that “the nature of the triangle belongs to Hades, Dionysus and Ares; that of the square, to Rhea, Aphrodite, Demeter, Hestia and Hera; and that of the 56-angled figure to Typhon, as Eudoxus recorded.”

Damascius includes this material from Philolaus in a general theoretical account: “What is intelligible shape and the kinds of shape? Shape is universally the circumscription of essence, by which it is self-circumscribed. On this account, shape is even prior to Intellect, because Intellect is self-circumscribed, insofar as it converges upon itself, and each form in the same manner, and the round, rectilinear and mixed are in accord with the character (idiótēs) of each of the shape-giving gods.

“Or why did the Pythagoreans consecrate the circle to one deity, the triangle to another, the square to still another, and each of the other rectilinear shapes to different (gods), and likewise the mixed, as they (dedicated) the half-circles to the Dioscuri? And Philolaus, among these wise men, often assigns different ones to a single (deity) in accord with their different characteristics.

“And perhaps, to speak generally, the circular shape is common to all intellective gods insofar as they are intellective, whereas the different rectilinear shapes belong to each of them individually, in accord with the characteristics of numbers, angles and sides – as for instance the triangle belongs to Athena, and the square to Hermes.¹ And beyond that, Philolaus says that, of the square, this angle belongs to Rhea, and this to Hera, and another to another goddess; and all this is the theological definition of shapes.

“[…] As anciently seemed germane to include, a shape at an angle, even if it is not closed, even one observed in just one line, even the unclosed spiral, this shall be a shape for us, as it is in theology. For even (a shape such as) this belongs to some god, as the Egyptians consecrated the angle; and the spiral and many other unclosed shapes are handed down, like the so-called tét² among the Egyptians, which is one straight vertical line and the vertical ones across it, one at the top and two below it; as well as some other among the Heliopolitans,³ and another, belonging to Zeus, among the people of Gaza.⁴

“Why say much more, when even the gods in their Oracles⁵ have handed down a single line ‘drawn in bulging shape’,⁶ and there is much use of linear shape among them?⁷

“And generally, if it belongs to the line to possess a beginning, middle and end,⁸ why not also the shape which is made from it?”

(Damascius, On the Parmenides, pp. 127–128.)

1: These associations are not from Philolaus, but common Greek observance.
2: Egyptian ḏd (Coptic jōt), conventionally called “djed pillar” in English, is a symbol of Ptah and Osiris.
3: If this has been securely identified, I do not know it.
4: Thought to refer to the Aramaic letter mem 𐡌, used as a symbol for Zeus Marnas, but perhaps no longer recognized as a letter at this time due to the development of different scripts for writing Aramaic.
5: The so-called Chaldaic Oracles.
6: Fragment 63 in des Places and Majercik. The phrase “bulging shape” is also found in Proclus.
7: That is, in the Chaldaic corpus. Probably a reference to characters used in ritual.
8: I take it that the encompassing of beginning, middle and end is being positioned as a divine attribute.

Porphyry; Triangle (Plutarch, Isis and Osiris 373f), numbers and shapes (ibid. 381e); square Hermes. Hero, Definitiones 136.11;24;38;50;55, 137.5. Characteres

4 Three-dimensional shapes

cube Hermes

5 Summary of the material

[Work in progress]

Status: work in progress