UNIT II SUMMARY — DISTINCTIO III: PLANAR MENSURATION & CIRCLE BOUNDS¶
Structural Synopsis of Boncompagni Vol. II, pp. 30–110 (fol. 18v–67v)
English Edition · Latin (1220 / Boncompagni 1862) · GEOMETOR Notes
Overview¶
Distinctio III is the longest mensuration book of Practica Geometriae, advancing systematically from triangles to quadrilaterals, regular and irregular polygons, and finally circular circumference, area, sectors, and segments.
Core Propositions & Numerical Benchmarks¶
Section / Folio |
Geometric Problem |
Exact Result in Practica Geometriae |
|---|---|---|
Scalene Altitude (fol. 24r) |
Casus & cathetus of $\triangle(13, 14, 15)$ |
$bd = 5$, $dg = 9$, $ad = 12$, $\Delta = 84$ |
Heron’s Formula (fol. 24v–25v) |
Area from $(a, b, c)$ via incenter $T$ & semiperimeter $s$ |
$\Delta^2 = r^2 s^2 = s(s-a)(s-b)(s-c) = 7{,}056$ |
Quadrilaterals (fol. 34r–53v) |
Rhombi, trapezoids (caput abscisum), irregular 4-gons |
Triangulation via measured diagonal; refutes false $(a+c)(b+d)/4$ |
Circumscribed 96-gon (fol. 54r) |
4 bisections of $30^\circ$ tangent at $r = 720$ ($d = 1440$) |
$\pi < \frac{1440}{458\frac{1}{5}} = \frac{7200}{2291} \approx 3.142732$ |
Inscribed 96-gon (fol. 55r) |
4 bisections of hexagon chord in semicircle $d = 1440$ |
$\pi > \frac{1440}{458\frac{4}{9}} = \frac{6480}{2063} \approx 3.141057$ |
Fibonacci’s Mean $\pi$ (fol. 56r) |
Intermediate divisor $458\frac{1}{3}$ & reduction to $\frac{22}{7}$ |
$\frac{1440}{458\frac{1}{3}} = \frac{864}{275} = 3\frac{39}{275} \approx 3.141818$ |