UNIT V NOTES — SCHOLARLY & SYMBOLIC COMMENTARY¶
Abū Kāmil’s Pentagon Algebra, the Irreducibility of the Flos Cubic, and Golden Convergence
English Edition · Latin (1220 / Boncompagni 1862) · Summary
1. The Quadrans Shadow Square as Pre-Trigonometric Cotangent/Tangent¶
On the medieval quadrans ($12 \times 12$ puncta), umbra recta ($u_r$) and umbra versa ($u_v$) encode exact rational cotangent and tangent scales scaled by $12$:
$$u_r = 12 \cot\theta, \qquad u_v = 12 \tan\theta, \qquad u_r \cdot u_v = 144.$$
By switching between umbra recta for $\theta \ge 45^\circ$ and umbra versa for $\theta \le 45^\circ$, the surveyor always reads an integer scale bounded in $[0, 12]$, avoiding unbounded tangents near the zenith! construct_quadrans_surveying() models this exact $12 \times 12$ lattice and plumb-line intersection in geometor.model.
2. Abū Kāmil’s Kitāb fī al-mukhammas wa-al-muʿashshar in Distinctio VIII¶
Roshdi Rashed and Barnabas Hughes have traced Distinctio VIII’s pentagon and decagon equations directly to Abū Kāmil Shujāʿ ibn Aslam’s tenth-century treatise On the Pentagon and Decagon (fibonacci:pg-8). What makes this chapter historic is the bidirectional isomorphism between Euclid Book XIII and Arabic *al-jabr*:
In Euclid XIII.10, $s_5^2 = r^2 + s_{10}^2$ is proved purely by congruent triangles and arcs without numbers;
In Distinctio VIII, setting $d = 10$ turns Euclid’s apotome ($s_{10}$) and minor ($s_5$) lines into concrete nested radical expressions over $\mathbb{Q}(\sqrt{5})$ solved via census census ($x^4 + 3125 = 125x^2$), verified symbolically in
solve_pentagon_ptolemy_quartic(10).
3. Galois Theory Avant la Lettre in Flos (1225)¶
John of Palermo borrowed the cubic $x^3 + 2x^2 + 10x = 20$ from ʿUmar al-Khayyāmī’s Treatise on Algebra (where it is constructed by intersecting a circle and a rectangular hyperbola). Instead of stopping at conic intersection, Leonardo asks a genuinely number-theoretic question: can the root $x$ belong to any of the thirteen quadratic irrational classes of Euclid Book X?
In modern field theory, $P(x) = x^3 + 2x^2 + 10x - 20$ is irreducible over $\mathbb{Q}$ (by the Rational Root Theorem on divisors of $20$), so $[\mathbb{Q}(x) : \mathbb{Q}] = 3$. Every Euclidean Book X irrational generates an iterated quadratic tower of degree $2^k$; since $3 \nmid 2^k$, $x$ cannot lie in any Euclidean ruler-and-compass field! Leonardo’s elimination of Book X binomials, apotomes, and medials in $1225$ is the earliest known European proof that a cubic algebraic irrational transcends Euclid Book X.
In flos_sexagesimal_cubic(), evaluating $P\!\left(\frac{1{,}596{,}577{,}777}{1{,}166{,}400{,}000}\right)$ yields a residual of $+5.40 \times 10^{-10}$, corresponding to a root error of $+3.18 \times 10^{-11}$ (only the sixth sexagesimal place $40''''''$ differs by $1$ unit from the closest possible rounding $39''''''$!).
4. From Liber Abaci Rabbits to the Divine Proportion¶
Although Liber Abaci (Vol. I, pp. 283–284) presents the rabbit sequence as an exercise in additive recurrence, fibonacci_rabbit_convergents(12) and verify_cassini_identity() reveal why the GEOMETOR ecosystem (geometor/divine, geometor/phyllotaxis, geometor/pacioli) treats Liber Abaci XII and Practica Geometriae VI/VIII as two halves of a single mathematical jewel:
$$\frac{F_{n+1}}{F_n} - \phi = \frac{(-1)^n}{F_n (F_n \phi + F_{n-1})}, \qquad \frac{377}{233} \approx 1.61802575 \quad (\text{error } 8.2 \times 10^{-6}\text{ from } \phi).$$
Every golden chord constructed by compass and straightedge in dodecahedron_golden_chord.svg and pentagon_decagon_circle.svg is the geometric limit of the rational integer ratios generated in Liber Abaci!