Unit IV — The Ten Neo-Pythagorean Means & the Cube’s Most Perfect Proportion — Critical Commentary

Pappus Semicircle Geometry, Robbins’s Textual Corrections on Means 8–9 & Euler–Philolaus Cube Duality · Book II, Chs. 21–29

Nicomachus II.21–28 — The Three Classical Archetypes & Synoptic Decad of Ten Means Left: Semicircle on PQ = 6, QR = 12 proving A ≥ G ≥ H & A · H = G² · Right: Hoche p. 144 Ten Means Semicircle Construction (a = 6, c = 12) P(0) Q(6) O(9) R(18) S F G = 6√2 A = 9 H = 8 A = 9 > G = √72 ≈ 8.49 > H = 8 · A·H = 72 = G² Synoptic Decad Table (Book II.28.11, Hoche p. 144) 1. Arithmetic α, β, γ (1, 2, 3) (z−m):(m−x) = 1 2. Geometric α, β, δ (1, 2, 4) (z−m):(m−x) = z:m 3. Harmonic γ, δ, ϛ (3, 4, 6) (z−m):(m−x) = z:x 4. Subcontr. Harmonic γ, ε, ϛ (3, 5, 6) (m−x):(z−m) = z:x 5. Subcontr. Geom. I β, δ, ε (2, 4, 5) (m−x):(z−m) = m:x 6. Subcontr. Geom. II α, δ, ϛ (1, 4, 6) (m−x):(z−m) = z:m 7. Neo-Pythagorean I ϛ, η, ϑ (6, 8, 9) (z−x):(m−x) = z:x 8. Neo-Pythagorean II ϛ, ζ, ϑ (6, 7, 9) (z−x):(z−m) = z:x 9. Neo-Pythagorean III δ, ϛ, ζ (4, 6, 7) (z−x):(m−x) = m:x 10. Additive / Fibonacci γ, ε, η (3, 5, 8) (z−x):(z−m) = m:x
Exact Rational Trigonometry verification over Q(sqrt(2)) for a=6, c=12: Q(O,S) = 81 (A=9), Q(Q,S) = 72 (G=6*sqrt(2)), and Q(S,F) = 64 (H=8) with A * H = G^2.

I. Exact Rational Trigonometry of the Semicircle of Means (II.21–27 & Pappus III)

In geometor.nicomachus.construct_three_classical_means_model, we model Pappus of Alexandria's celebrated semicircle construction (pappus:synagoge:iii, Synagoge III.12–16) for Nicomachus’s canonical extremes \(a = 6, c = 12\):

  • On diameter \(PR\) of length \(a + c = 18\) with \(P(0, 0), Q(6, 0), R(18, 0)\) bisected at center \(O(9, 0)\), the three radii PO, OR, and OS are strictly equal with exact quadrance \(Q(O, S) = A^2 = 81\) (\(A = 9\)).

  • Erecting the perpendicular chord at \(Q(6, 0)\) meets semicircle PR at \(S(6, 6\sqrt{2})\) over \(\mathbb{Q}(\sqrt{2})\), giving ordinate QS exact quadrance \(Q(Q, S) = G^2 = a \cdot c = 72\) (\(G = 6\sqrt{2}\), Euclid Elements VI.13).

  • Dropping the altitude from \(Q(6, 0)\) onto radius \(OS\) meets \(OS\) at exact foot \(F\bigl(\frac{26}{3}, \frac{2\sqrt{2}}{3}\bigr)\): by right-triangle similarity \(\triangle SQF \sim \triangle OQS\), radial segment SF satisfies \(|SF| = \frac{QS^2}{OS} = \frac{G^2}{A} = \frac{72}{9} = \mathbf{8} = H\) with exact quadrance \(Q(S, F) = H^2 = 64\) and spread \(s(OS, PR) = \frac{72}{81} = \frac{8}{9}\)!

  • Simultaneously, in construct_cube_proportion_model, Euclidean bisection constructs \(T_9\) and complete-quadrilateral harmonic conjugation constructs \(T_8\) with exact projective cross-ratio \((T_6, T_{12}; T_8, Z) = -1\).

II. Textual & Mathematical Critique of Means 8, 9 & 10 (II.28)

As Frank Egleston Robbins and Martin Luther D'Ooge establish in their critical notes (dooge:1926, pp. 282–284 nn.; cf. heath:hgm:1921, Vol. I, pp. 86–89), the manuscript tradition of Book II, Ch. 28.3–4 preserves two instructive slips in Nicomachus’s preliminary summary list (corrected in Ch. 28.9–10 and in Iamblichus / Pappus):

  1. 8th & 9th Means: In the preliminary list at II.28.3 (Hoche p. 140), the manuscripts give \(4, 7, 8\) and \(3, 5, 6\) alongside compressed verbal phrasing (“middle among the differences”), whereas in the detailed exposition at II.28.9–10 (Hoche pp. 142–143) the 8th mean \((c - a) : (c - b) = c : a\) is illustrated properly by 6, 7, 9 (\((9-6):(9-7) = 3:2 = 9:6\)) and the 9th mean \((c - a) : (b - a) = b : a\) is satisfied by 4, 6, 7 (\((7-4):(6-4) = 3:2 = 6:4\)). All ten canonical proportions and closed forms are verified symbolically in geometor.nicomachus.compute_all_ten_means and verify_ten_means_identities.

  2. 10th Mean as the Fibonacci Recurrence: The 10th mean \((c - a) : (c - b) = b : a\) expands to \(a(c - a) = b(c - b) \iff (b - a)(b + a - c) = 0\). Since \(a < b < c\) guarantees \(b \neq a\), this reduces identically to the Fibonacci additive law \(a + b = c\), which is why Nicomachus’s exemplar (3, 5, 8) consists of three consecutive Fibonacci numbers!

III. Why the Cube Embodies 6 : 8 :: 9 : 12 (II.26 & II.29)

Nicomachus’s finale (II.29, echoed by Boethius, De Inst. Arith. II.54 maxima et perfecta symphonia, Luca Pacioli, and Johannes Kepler, verified in verify_cube_proportion_identities) brings together three mathematical worlds in a single proportion:

  • Polyhedral Topology: Every combinatorial cube has \(F = 6\) faces, \(V = 8\) vertices, and \(E = 12\) edges (satisfying Euler’s polyhedral invariant \(V - E + F = 8 - 12 + 6 = 2\)). Because \(\frac{1}{6} + \frac{1}{12} = \frac{2}{8} \iff H(6, 12) = 8\), the vertex count \(V = 8\) is literally the harmonic mean of faces \(F = 6\) and edges \(E = 12\)! (Dually, for the octahedron, \(V = 6, F = 8, E = 12\) exhibits the exact same harmonic triple!)

  • Duality of Arithmetic and Harmonic Means: Inserting the arithmetic mean \(A(6, 12) = 9\) alongside \(H(6, 12) = 8\) satisfies the universal identity \(A \cdot H = G^2 \implies 9 \times 8 = 6 \times 12 = 72\), turning the four terms \(6, 8, 9, 12\) into a disjunct geometric proportion \(6 : 8 :: 9 : 12\) (\(3 : 4\)) and \(6 : 9 :: 8 : 12\) (\(2 : 3\)).

  • Complete Diatonic Scale Scaffolding: Dividing a string of length \(12\) at \(9, 8, 6\) sounds the root, fourth (\(12:9 = 4:3\)), fifth (\(12:8 = 3:2\)), and octave (\(12:6 = 2:1\)), separated centrally by the Pythagorean whole tone (\(9:8\), epogdoos)—the exact harmonic framework upon which Plato constructed the World-Soul in Timaeus 35b–36b (plato:timaeus:27d-36d).