Unit III — Inverse Reduction, Polygonal Gnomons, Same/Other & Sum of Cubes — Critical Commentary¶
Euclidean Anthyphairesis, Polygonal Identities & Exact Gnomonic Proof of Sum of Cubes · Book II, Chs. 1–2, 6–20
I. Book II, Chs. 1–2 as Quadratic Euclidean Anthyphairesis¶
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:
If \(p \le q - p\), the triple remains ascending and represents \((p,\; q - p)\); if \(q - p < p\), Nicomachus’s instruction to place the lesser extreme first reverses it to \(\bigl((q - p)^2,\; (q - p)p,\; p^2\bigr)\), representing \((q - p,\; p)\). Thus Nicomachus’s three subtractions execute Euclid’s reciprocal subtraction algorithm (Elements VII.1–2, anthyphairesis) simultaneously on the quadratic forms \((p^2, pq, q^2)\) until \(\gcd(p, q) = 1\) terminates at \((1, 1, 1)\)! This provides a rigorous constructive proof of Nicomachus’s philosophical thesis (echoing Plato's Timaeus and Philebus) that Equality is both the generative origin and the terminal element (stoicheion) of all numerical relation.
II. Exact Symbolic Proof of Nicomachus’s Sum-of-Cubes Theorem (II.20)¶
Book II, Ch. 20.5 (hoche:1866, p. 118; dooge:1926, p. 263) contains the oldest extant statement in world mathematics of the odd-block generation of cubes, from which the closed-form sum of cubes follows immediately (explicitly completed by the Roman agrimensores, Abu Bakr al-Karaji, and Nilakantha, and inherent in combining Nicomachus II.9 with II.20; see heath:hgm:1921, Vol. I, pp. 109–110):
Odd-Block Identity: The \(k\)-th block consists of \(k\) consecutive odd integers beginning after the first \(T_{k-1} = \frac{k(k-1)}{2}\) odd numbers:
\[B_k = \bigl\{\, 2(T_{k-1} + j) - 1 \;\big|\; j = 1, \dots, k \,\bigr\} = \bigl\{\, k^2 - k + 1,\; k^2 - k + 3,\; \dots,\; k^2 + k - 1 \,\bigr\}.\]This block has \(k\) terms symmetrically centered around arithmetic mean \(k^2\), so its sum is identically:
\[\sum_{x \in B_k} x = k \cdot k^2 = k^3.\]Square-of-Triangle Gnomon Identity: Together, the first \(n\) blocks \(B_1 \sqcup B_2 \sqcup \dots \sqcup B_n\) consist of the first \(T_n = \frac{n(n+1)}{2}\) odd integers without gap or overlap. By Nicomachus II.9 (\(\sum_{m=1}^{M} (2m - 1) = M^2\) with \(M = T_n\)):
\[\sum_{k=1}^n k^3 = \sum_{m=1}^{T_n} (2m - 1) = T_n^2 = \left(\frac{n(n+1)}{2}\right)^2.\]Geometric Gnomon Realization (`construct_nicomachus_cubes_model`): In the T4 = 10 square, bordering the \(T_{k-1} \times T_{k-1}\) square by strip width \(T_k - T_{k-1} = k\) creates two rectangles of area \(k \cdot T_{k-1}\) plus a \(k \times k\) corner square:
\[T_k^2 - T_{k-1}^2 = 2 k \cdot \frac{k(k-1)}{2} + k^2 = k^2(k - 1) + k^2 = k^3,\]giving each cube 1³, 2³, 3³, 4³ a visible two-dimensional L-gnomon realization!