Unit IV — Timaeus: Geometric Proportion & Harmonic Division of the World-Soul — Critical Commentary & Notes

Exact Symbolic Reconstruction, Greek Mathematical Philology & Canon Reception · 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).

I. Exact Geometric & Symbolic Reconstruction

In geometor.plato.harmonics and construct_timaeus_proportion_circles (see Demonstrations), both proportions of Timaeus 31b–36d are verified in 100% exact rational and quadratic arithmetic:

  1. Plane vs. Solid Geometric Means (31c–32b): - Between square plane numbers \(p^2\) and \(q^2\), a single mean \(pq\) yields \(p^2 : pq = pq : q^2\) (e.g., \(4 : 6 = 6 : 9\)). - Between cubic solid numbers \(p^3\) and \(q^3\), two means \(p^2 q\) and \(p q^2\) yield \(p^3 : p^2 q = p^2 q : p q^2 = p q^2 : q^3\) (e.g., \(8 : 12 = 12 : 18 = 18 : 27\), Elements VIII.11–12).

  2. Pappus Semicircle Construction of the Three Means (`timaeus_world_soul.svg`): - Joining segments \(AJ = 2\) and \(JB = 4\) along diameter A B of circle (O, B) (center O, radius \(3\)) and erecting ordinate J G with perpendicular projection G H onto radius O G exhibits all three Platonic means simultaneously:

    \[A = |OG| = \frac{2+4}{2} = 3, \quad G = |JG| = \sqrt{2 \cdot 4} = 2\sqrt{2}, \quad H = |GH| = \frac{2 \cdot 2 \cdot 4}{2 + 4} = \frac{8}{3},\]

    with exact algebraic identity \(A \cdot H = G^2\) (\(3 \cdot \frac{8}{3} = 8\)) and whole-tone ratio \(\frac{|OG|}{|GH|} = \frac{3}{8/3} = \frac{9}{8}\)!

  3. Exact Factorization of the 34-Term World-Soul Gamut (35b–36b): - Merging the double (1, 2, 4, 8) and triple (1, 3, 9, 27) chains with their harmonic and arithmetic means (1, 4/3, 3/2, 2, 8/3, 3, 4, 9/2, 16/3, 6, 8, 9, 27/2, 18, 27) and subdividing every \(4:3\) fourth into two \(9:8\) whole tones plus the \(256:243\) leimma (and every \(3:2\) fifth into \(4:3\) plus \(9:8\)) yields 34 unique exact rational pitches containing 24 whole tones and 9 leimmata:

    \[\left(\frac{9}{8}\right)^{24} \cdot \left(\frac{256}{243}\right)^{9} = \frac{3^{48}}{2^{72}} \cdot \frac{2^{72}}{3^{45}} = 3^3 = 27.\]

II. Literal Greek Mathematical Philology (Contra Paraphrasim)

  • 36a3–5 — τὴν μὲν ταὐτῷ μέρει τῶν ἄκρων αὐτῶν ὑπερέχουσαν καὶ ὑπερεχομένην: Jowett translates accurately ‘one mean exceeding and exceeded by equal parts of its extremes’, which defines the harmonic mean \(H\): if \(H = a + \frac{a}{k} = b - \frac{b}{k}\), solving for \(1/k = \frac{b-a}{a+b}\) yields \(H = a\left(1 + \frac{b-a}{a+b}\right) = \frac{2ab}{a+b}\) (for \(a=1, b=2\), \(H = 4/3\) exceeds \(1\) by one-third of \(1\) and falls short of \(2\) by one-third of \(2\)!).

  • 36a6–b5 — ἡμιολίων δὲ διαστάσεων καὶ ἐπιτρίτων καὶ ἐπογδόων… λείπων αὐτῶν ἑκάστου μόριον… ἀριθμοῦ πρὸς ἀριθμὸν ἐχούσης τοὺς ὅρους ἓξ καὶ πεντήκοντα καὶ διακοσίων πρὸς τρία καὶ τετταράκοντα καὶ διακόσια: Notice Plato’s extraordinary numerical precision: Greek superparticular adjectives hēmiolios (\(1 + \frac{1}{2} = 3/2\)), epitritos (\(1 + \frac{1}{3} = 4/3\)), and epogdoos (\(1 + \frac{1}{8} = 9/8\)) culminate in spelling out verbatim in Attic prose 256 to 243 (ἓξ καὶ πεντήκοντα καὶ διακοσίων πρὸς τρία καὶ τετταράκοντα καὶ διακόσια). Note also that two leimmata fall short of a \(9:8\) whole tone by the Pythagorean comma (\(531441 : 524288\)), leaving the apotome \((9/8)/(256/243) = 2187/2048\).

III. Reception in Euclid, Crantor & Proclus

All four sister-corpus citations (plato:timaeus:31c-36d, 32-a-b, 32-a-sqq, 32.a.b across euclid, elements, and proclus) center on Elements VIII.11–12 and Archytas's theorem (Euclid Sectio Canonis Prop. 3) that no superparticular ratio \((n+1):n\) admits a rational geometric mean—explaining why the Demiurge divides octaves and fourths via rational harmonic/arithmetic means (4:3, 3:2, 9:8, 256:243) rather than equal-tempered irrational roots!

IV. Executable Symbolic Verification

In accordance with GEOMETOR’s strict content/code separation for historical editions, executable geometor.model SymPy constructions and Rational Trigonometry verification routines for Unit IV live in src/geometor/plato/ and the companion walkthrough Demonstrations.