Content of type 'summary'

MULTITUDE (τὸ πλῆθος) Discrete · Rooted in Monad, advancing infinitely MAGNITUDE (τὸ μέγεθος) Continuous · Bounded whole, infinitely divisible I · καθ’ ἑαυτό (Per se) ARITHMETIC Ἀριθμητική Even, Odd, Prime, Perfect Sovereign Mother (I.4–5) II · πρὸς ἄλλο (Ad aliud) MUSIC (HARMONICS) Μουσική 2:1 · 3:2 · 4:3 · 9:8 Relative Ratio III · ἐν μονῇ (At rest) GEOMETRY Γεωμετρία Triangle, Square, Cube Stationary Form IV · ἐν κινήσει (In motion) ASTRONOMY Σφαιρική Ordered Revolutions Celestial Harmony Ontological Priority (συναναιρεῖ μὲν, οὐ συναναιρεῖται δέ): Arithmetic precedes Music, Geometry, and Sphaerics

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

Unit I establishes the philosophical charter of the Western mathematical curriculum before classifying odd primes and even perfect numbers:

(1, 1, 1) · α, α, α Equality · 1:1 (1, 2, 4) · α, β, δ 1. Multiplex · 2:1 (1, 3, 9) · α, γ, ϑ Multiplex · 3:1 (4, 6, 9) · δ, ϛ, ϑ 2. Superparticularis · 3:2 (1, 4, 16) Multiplex · 4:1 (9, 12, 16) Epitritos · 4:3 (4, 10, 25) · δ, ι, κε 4. Mult. superpartic. · 5:2 (9, 15, 25) · ϑ, ιε, κε 3. Superpartiens · 5:3 (9, 24, 64) 5. Mult. superpartiens · 8:3

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

Unit II transitions from absolute to relative quantity (pros ti poson), proving that every rational inequality is classified into five exact species and generated deterministically from Equality:

2³ = 8 3³ = 27 4³ = 64 0 T₁=1 T₂=3 T₃=6 T₄=10 Nicomachus's Consecutive Odd-Block Theorem (II.20) k = 1 : 1 = 1 = 1³ k = 2 : 3 + 5 = 8 = 2³ k = 3 : 7 + 9 + 11 = 27 = 3³ k = 4 : 13 + 15 + 17 + 19 = 64 = 4³ k = 5 : 21 + 23 + 25 + 27 + 29 = 125 = 5³ Why Shell k Has Exact Area k³: Tₖ² − Tₖ₋₁² = (Tₖ − Tₖ₋₁) · (Tₖ + Tₖ₋₁) = k · k² = k³ (By II.12, two consecutive triangles Tₖ₋₁ + Tₖ = k²!)

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

Unit III completes Nicomachus’s philosophy of Equality by proving the inverse reduction theorem (II.1–2) and then constructs the entire hierarchy of plane and solid figurate numbers from alpha-pebble gnomons (II.6–20):

6 Faces · 8 Vertices · 12 Edges Harmonic Cube: 8 = 2·6·12 / (6 + 12) 6 = 1×2×3 (scalene) · 8 = 2×2×2 (cube) 9 = 1×3×3 (brick) · 12 = 2×2×3 (beam) Euler: F − E + V = 6 − 12 + 8 = 2 Projective Harmonic Conjugate & Monochord Consonances K (Steiner apex) Z=0 6 (ϛ) 8 (η) 9 (ϑ) 12 (ιβ) 9:8 Tone 4:3 Fourth 12:9 = 4:3 Diapente (Fifth) 9:6 = 12:8 = 3:2 Diapason 12:6 = 2:1 · 6×12 = 8×9 = 72 · CR(6,12;8,0) = −1

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

Unit IV crowns the Introductio Arithmetica by uniting arithmetic, geometry, stereometry, and harmonics in the theory of means: