UNIT III SUMMARY — DISTINCTIO IV: DIVISIONS OF PLANAR FIGURES¶
Structural Synopsis of Boncompagni Vol. II, pp. 110–148 (fol. 67v–92v)
English Edition · Latin (1220 / Boncompagni 1862) · GEOMETOR Notes
Overview¶
Distinctio IV preserves the fullest Latin witness to Euclid’s lost Greek treatise On Divisions of Figures ($\Pi\varepsilon\rho\grave{\iota}\ \delta\iota\alpha\iota\rho\acute{\varepsilon}\sigma\varepsilon\omega\nu\ \beta\iota\beta\lambda\acute{\iota}\omicron\nu$, euclid:divisions), systematically augmenting every pure ruler-and-compass theorem with numerical coordinates and quadratic solutions for Pisan estate partitioning among consortes.
Catalogue of Partition Constructions¶
Problem Class |
Geometric Construction |
Canonical Example & Invariant |
|---|---|---|
Vertex Partition (fol. 67v) |
Divide opposite base $BG$ in ratio $p : q$, join to $A$ |
$\triangle ABD : \triangle ADG = BD : DG$ (Euclid VI.1) |
Boundary Point $D$ (fol. 68v) |
Draw $EZ \parallel AD$ from midpoint $E$, cut along $DZ$ |
On $(13, 14, 15)$ with $BD=5, DG=9$: $GZ = \frac{35}{3} = 11\frac{2}{3}$ |
Centroid $X(2)$ (fol. 69r, 74r) |
Concurrency of medians $AE, BZ, GT$ in $2 : 1$ ratio at $D$ |
Three equal shares $\triangle DAB = \triangle DBG = \triangle DGA = \frac{1}{3}\Delta$ |
Interior / Exterior $P$ (fol. 69v–72r) |
Application of areas (Euclid VI.28–29) & Book II quadratics |
Line through arbitrary landmark $P$ cutting off $\lambda \Delta$ |
Parallel to Base (fol. 73v) |
Semicircle mean proportional $AE^2 = \lambda\, AB^2$ (Euclid VI.13, 19) |
On $(10, 14, 12)$ for $\lambda = \frac{1}{2}$: $AE^2 = 50, AZ^2 = 72, EZ^2 = 98$ |
Trapezoid Parallel Cut (fol. 78r) |
Weighted root-mean-square of parallel bases $a, b$ |
$c^2 = \frac{q a^2 + p b^2}{p + q}$; for $(7, 17)$, $c = 13$ |
Circles & Segments (fol. 90r–92v) |
Concentric rings ($r_k^2 = \frac{k}{n} R^2$) & chord partitions |
Euclid On Divisions Props. 28–29 |