UNIT II NOTES — SCHOLARLY & SYMBOLIC COMMENTARY¶
Heron’s Formula, Rational Trigonometry, and Fibonacci’s Small-Number Archimedean Bounds
English Edition · Latin (1220 / Boncompagni 1862) · Summary
1. The Canonical $(13, 14, 15)$ Heronian Triangle¶
Built by juxtaposing the Pythagorean right triangles $(5, 12, 13)$ and $(9, 12, 15)$ back-to-back along their common cathetus $12$, the $(13, 14, 15)$ scalene triangle is the universal benchmark of ancient and medieval mensuration—appearing in Heron’s Metrica I.8 (heron:metrica), Savasorda’s Liber Embadorum (savasorda:embadorum), and throughout Practica Geometriae (Dist. III and Dist. IV).
In geometor.fibonacci.mensuration.construct_heron_incenter_proof(), with $B = (0, 0)$, $G = (14, 0)$, and $A = (5, 12)$:
The angle bisectors meet at the exact rational incenter $X(1) = T = (6, 4)$ (cross-referenced to
geometor/etcX(1));All three perpendicular feet are exact over $\mathbb{Q}$: $H = (6, 0)$, $E = \left(\frac{42}{13}, \frac{100}{13}\right)$, $Z = \left(\frac{46}{5}, \frac{32}{5}\right)$, with common inradius quadrance $Q(T, H) = Q(T, E) = Q(T, Z) = 16$ ($r = 4$);
Extending $[BG]$ to $L = (21, 0)$ realizes $BL = 21 = s$, with $BH = 6 = s - 15$, $HG = 8 = s - 13$, and $GL = 7 = s - 14$.
2. Heron’s Polynomial Identity & Archimedes’ Theorem¶
Leonardo did not read Heron’s Greek Metrica directly (which remained in Constantinople until its rediscovery in $1896$); instead, he drew the incenter proof through the ninth-century Baghdadi treatise of the Banū Mūsā, translated at Toledo by Gerard of Cremona as Verba filiorum Moysi filii Sekir (banu-musa:verba).
In Rational Trigonometry (geometor/wildberger), verify_heron_polynomial_identity() proves over $\mathbb{Z}[x, y, z]$ that Heron’s factored semiperimeter product is identically Archimedes’ quadrea formula in terms of side quadrances $Q_1 = a^2, Q_2 = b^2, Q_3 = c^2$:
$$16\Delta^2 = 16 s(s-a)(s-b)(s-c) = (a+b+c)(-a+b+c)(a-b+c)(a+b-c) = 4 Q_1 Q_2 - (Q_1 + Q_2 - Q_3)^2.$$
3. Refuting the Agrimensor Quadrilateral Fallacy¶
Roman gromatic manuals routinely approximated any four-sided field $(a, b, c, d)$ by $\frac{a+c}{2}\cdot\frac{b+d}{2}$. By Brahmagupta’s and Bretschneider’s identities, for a quadrilateral with semiperimeter $s$ and opposite angles $\alpha, \gamma$:
$$\Delta^2 = (s-a)(s-b)(s-c)(s-d) - abcd \cos^2\!\left(\frac{\alpha + \gamma}{2}\right) \le \left(\frac{a+c}{2}\right)^2 \left(\frac{b+d}{2}\right)^2,$$
with equality if and only if $a = c$, $b = d$, and $\alpha = \gamma = 90^\circ$. Leonardo insists that at least one diagonal or angle must be measured so the quadrilateral is decomposed into rigid Euclidean triangles.
4. Why Fibonacci’s $\frac{864}{275}$ Improves Upon $\frac{22}{7}$¶
In archimedes:circle, Archimedes started his $30^\circ$ rational bounds from $265 < 153\sqrt{3} < 1351/5$, producing four-digit numerators and denominators ($14{,}688$ and $4{,}673\frac{1}{2}$). Leonardo instead fixes the circle diameter uniformly at $d = 1440$ ($r = 720 = 6!$), a highly composite sexagesimal-duodecimal scale in which all four bisections remain transparent.
Comparing absolute errors against $\pi = 3.14159265358979\dots$ (verified in fibonacci_circle_bounds()):
Archimedean upper bound $\frac{22}{7} \approx 3.14285714$: error $+1.264 \times 10^{-3}$;
Fibonacci’s circumscribed upper bound $\frac{7200}{2291} \approx 3.14273243$: error $+1.140 \times 10^{-3}$;
Fibonacci’s inscribed lower bound $\frac{6480}{2063} \approx 3.14105671$: error $-5.359 \times 10^{-4}$;
Fibonacci’s mean ratio $\frac{864}{275} \approx 3.14181818$: error $+2.255 \times 10^{-4}$—more than five times closer to $\pi$ than $\frac{22}{7}$!