Content of type 'notes'

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 Gold circles = Prime & incomposite (πρῶτοι καὶ ἀσύνθετοι) · Crossed = Secondary & composite (δεύτεροι καὶ σύνθετοι; 9 = 3², 15 = 3×5) I.16 · Perfect Numbers (τέλειοι): Equal to the Sum of Their Proper Aliquot Parts 6 = 2¹(2² − 1) = 1 + 2 + 3 1 2 3 Whole = 6 (first perfect number, in units) 28 = 2²(2³ − 1) = 1 + 2 + 4 + 7 + 14 1+2 4 7 14 Whole = 28 (second perfect number, in tens) Dyadic Rule: P = 2ᵖ⁻¹ · (2ᵖ − 1) p = 2 : 1+2 = 3 (prime) → 2·3 = 6 p = 3 : 1+2+4 = 7 (prime) → 4·7 = 28 p = 5 : 2⁵ − 1 = 31 (prime) → 16·31 = 496 p = 7 : 2⁷ − 1 = 127 (prime)→ 64·127 = 8128 p = 11: 2¹¹ − 1 = 2047 = 23 × 89 Composite! Breaks naive 10⁴ alternation

Unit I — Philosophy, the Quadrivium, the Odd Sieve & Perfect Numbers — Notes

Poson vs. Pēlikon and Aristotle’s Categories: In aristotle:cat:4b20 (Categories 6, 4b20), Aristotle divided quantity (to poson) into discrete (number and speech) and continuous (line, surface, body, time, place). Nicomachus refines the terminology by reserving poson (multitudo) strictly for bounded discrete multitude and pelikon (magnitudo) for bounded continuous magnitude. Five centuries later, Proclus reproduces Nicomachus’s fourfold schema verbatim in his Prologue to Euclid (proclus:in-euclid:35-37, pp. 35.17–36.25 Friedlein), while Boethius renders Nicomachus’s “four methods” (tessares methodoi), “bridges” (gephyrai), and “ladders” (klimakes) by the newly minted Latin compound quadruvium (“fourfold crossroads”) at De Institutione Arithmetica I.1 (boethius:friedlein:1867, p. 7.25).

1 · Multiplex (πολλαπλάσιος) 2 : 1 (duple) m = 2, n = 1 2 · Superparticularis (ἐπιμόριος) 3 : 2 (hemiolios) m = 3, n = 2 3 · Superpartiens (ἐπιμερής) 5 : 3 (epidimoiros) m = 5, n = 3 4 · Mult. superparticularis 5 : 2 (2 + 1/2) m = 5, n = 2 5 · Mult. superpartiens 8 : 3 (2 + 2/3) m = 8, n = 3

Unit II — Five Species of Inequality & the Equality Unfolding Tree — Notes

Nicomachus observes with wonder at the close of Book I (I.23.16, Hoche p. 71; D’Ooge p. 229) that in every triple generated from Equality (1, 1, 1), the first and third terms are squares while the middle term is their geometric mean. Exact algebra reveals why Nicomachus’s three prostagmata (attributed by Iamblichus [iamblichus:pistelli:1894] and Theon of Smyrna [theon:hiller:1878, p. 107] to Eratosthenes and Adrastus) are nothing less than the symmetric square representation \(\mathrm{Sym}^2(\mathbb{Z}^2)\) of the Euclidean unimodular group \(\mathrm{SL}_2(\mathbb{Z}_{\ge 0})\)!

THE OTHER (θάτερον) · Heteromecic H₄ = 4 × 5 = 20 Sum of Evens: 2 + 4 + 6 + 8 = 20 (sides 4 : 5) Book II.19 Alternating Weave: Every Heteromecic Hₙ = n(n+1) is the Geometric Mean of Sₙ = n² and Sₙ₊₁ = (n+1)² S₁=1 H₁=2 S₂=4 H₂=6 S₃=9 H₃=12 S₄=16 H₄=20 S₅=25 Sₙ · Sₙ₊₁ = Hₙ² (e.g. 4 × 9 = 6², 9 × 16 = 12², 16 × 25 = 20²) & Hₙ = 2 · Tₙ

Unit III — Inverse Reduction, Polygonal Gnomons, Same/Other & Sum of Cubes — Notes

In geometor.nicomachus.reduce_to_equality, Book II’s inverse rule \((x, y, z) \mapsto (x, y - x, z - x - 2(y - x))\) applied to any ascending coprime geometric progression \((p^2, pq, q^2)\) (\(0 < p < q\)) evaluates to:

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

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

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 \(a = 6, c = 12\):