Unit IV — Timaeus: Geometric Proportion & Harmonic Division of the World-Soul — Structural Synopsis

Argumentative Architecture & Interactive Diagram Guide · Timaeus 31b–36d

Timaeus 31c-36d: Arithmetic, Geometric, and Harmonic Means & Circles of the World-Soul ( O B ): (x - (3))^2 + (y - (0))^2 - (9) = 0 ( O J ): (x - (3))^2 + (y - (0))^2 - (1) = 0 [ J H ]: (-2*sqrt(2)/9)*x + (8/9)*y + (4*sqrt(2)/9) = 0 [ A B ]: (0)*x + (1)*y + (0) = 0 [ J D ]: (1)*x + (0)*y + (-2) = 0 [ O G ]: (-2*sqrt(2))*x + (-1)*y + (6*sqrt(2)) = 0 [ P Q ]: (-3)*x + (3*sqrt(3))*y + (9) = 0 A = (0, 0) B = (6, 0) J = (2, 0) G = (2, 2*sqrt(2)) H = (26/9, 2*sqrt(2)/9) P = (3 - 3*sqrt(3)/2, -3/2) Q = (3*sqrt(3)/2 + 3, 3/2) O = (3, 0) A B J G H P Q O
The Three Platonic Means & Circles of the World-Soul (Timaeus 31c–36d): on octave diameter A J B (2 + 4 = 6), radius O G = 3 is the arithmetic mean, ordinate J G = 2*sqrt(2) is the geometric mean, projection G H = 8/3 is the harmonic mean (giving O G / G H = 9/8), and axis [P, Q] crosses at O like Chi (X).

Architectural Thesis

In Timaeus 31b–36d, Timaeus of Locri presents the grandest proportional and harmonic synthesis of antiquity, constructing both the four-element body of the cosmos and the musical fabric of the World-Soul from exact rational and quadratic invariants:

  1. Single vs. Double Geometric Means (31b–32c): To bind fire and earth together, the Demiurge employs continuous geometric proportion (ἀναλογία), the fairest of bonds. Had the cosmos been a two-dimensional plane without depth, one mean proportional would have sufficed (\(a^2 : ab : b^2\), Elements VIII.11); because the cosmos is a three-dimensional solid, two mean proportionals are required (\(a^3 : a^2 b : a b^2 : b^3\), Elements VIII.12), placing air and water between fire and earth so that fire : air = air : water = water : earth.

  2. Harmonic Division of the World-Soul (35a–36b): Slicing the mixed psychic substance into the geometric tetractys of doubles 1, 2, 4, 8 and triples 1, 3, 9, 27 (spanning four octaves plus a major sixth, \(27:1\)), the Demiurge inserts into every interval the harmonic mean \(H = \frac{2ab}{a+b}\) (G H) and arithmetic mean \(A = \frac{a+b}{2}\) (O G), generating sesquialter fifths (\(3:2\)), sesquitertian fourths (\(4:3\)), and epogdoic whole tones (\(9:8\)), and filling every \(4:3\) fourth with two \(9:8\) tones plus the exact diatonic leimma 256 : 243.

  3. Circles of the Same and the Different (36b–d): Splitting the calibrated harmonic strip lengthwise and bending the two bands into great circles intersecting like the letter Chi (\(\mathrm{X}\), P Q across equator A B on (O, B)), the Demiurge establishes the celestial equator (Same) and the ecliptic subdivided in the seven ratios 1, 2, 3, 4, 8, 9, 27 (Different).

Stephanus Conspectus

Section

Stephanus Span

Mathematical & Philosophical Argument

Tracks

IV.1 Cosmic Analogia

31b–32c

Plane (1 mean) vs. solid (2 means) proportion binding Fire, Air, Water, and Earth.

EN · EL

IV.2 World-Soul Gamut

35a–36d

Doubles & triples, arithmetic/harmonic means, 9:8 tone, 256:243 leimma, and \(\mathrm{X}\)-circles.

EN · EL