UNIT V — TRACTATUS GEOMETRIAE: HERON, SURFACE DIVISION & REGULAR SOLIDS¶
Part II, Distinctiones I–VIII: Plane Mensuration, Archimedes’s Circle, De Divisione Superficierum, Inscribed Polyhedra & the 1494 Colophon (fols. 1r–76r)
Euclid Digest & Heron’s (13, 14, 15) Triangle Mensuration (Part II, fol. 9v)
Opening the separately foliated second tome (Tractatus Geometriae, fols. 1r–2r), Fra Luca Pacioli condenses the foundational definitions and constructions of Euclid Books I, II, and VI before turning in Distinctio I, Cap. 8 (fol. 9v) to the crown jewel of practical surveyor’s mensuration inherited from Heron of Alexandria and Leonardo Pisano's Practica Geometriae: determining the exact area, perpendicular cathetus (catetto), and incircle of a scalene triangle from its three sides alone.
Let scalene triangle GBA have sides GA $= 13$, base GB $= 14$, and BA $= 15$ braccia ([I.22]). Sum the three sides ($13 + 14 + 15 = 42$) and take the semiperimeter (mità dela summa) $s = 21$. Subtract each side in turn to obtain the three excesses (differentie):
$$21 - 15 = 6, \qquad 21 - 14 = 7, \qquad 21 - 13 = 8$$
Multiply $6 \times 7 \times 8 = 336$, and multiply $336$ by the semiperimeter $21$ to obtain $7056$, whose exact square root gives the area:
$$\text{Area}(\triangle GBA) = \sqrt{7056} = 84 \text{ square braccia}$$
Dividing $2 \times 84 = 168$ by base GB ($14$) yields the perpendicular cathetus AD $= 12$, splitting $GB$ at D into right-triangle base segments GD $= 5$ (inside $(5, 12, 13)$ right triangle GDA) and DB $= 9$ (inside $(9, 12, 15)$). Dividing area $84$ by semiperimeter $21$ locates the incircle (IT) centered at I $= (6, 4)$ of exact integer inradius IT $= 4$ ([IV.4]).
Archimedes’s Circle (fol. 31r) & De Divisione Superficierum (Dist. IV)
In Distinctio III, Cap. 6 (fol. 31r), Pacioli expounds Archimedes's Dimensio Circuli, bounding the circumference-to-diameter ratio by inscribed and circumscribed $96$-gons between $3\frac{10}{71}$ and $3\frac{1}{7}$ ($22 : 7$). In Distinctio IV (De divisione superficierum), he solves estate-partition problems dividing triangles, quadrilaterals, and circles in prescribed ratios $p : q$ both through vertices ([VI.1]) and parallel to a side ([VI.13], [VI.19]).
In Distinctio V–VI, Pacioli gauges wine casks (botte) and solves algebraic-geometric circle packings via Regola dela Cosa—culminating on fol. 57r in inscribing three mutually tangent equal circles inside the $(10, 10, 12)$ triangle via the Cossic quadratic $5\frac{3}{4}.\text{ce.} \; \bar{p} \; 16.\text{co.} = 48$ ($r = \frac{8\sqrt{85} - 32}{23}$), three centuries before Malfatti (1803).
De Corporibus Regularibus (Dist. VIII) & the November 10, 1494 Venetian Colophon
Closing the entire work in Distinctio VIII (fols. 68v–70v), Pacioli inscribes the five regular Platonic solids ([XIII.13]–[XIII.17]) and dependent Archimedean truncations (following Piero della Francesca's Libellus de quinque corporibus regularibus) inside a sphere (OP) of diameter $D = 6$ (radius $R = 3$):
Tetrahedron (`fol. 69r`): Possanza $e^2 = \frac{2}{3} D^2 = 24$ ($e = \sqrt{24}$); Cube: $e^2 = \frac{1}{3} D^2 = 12$ ($e = \sqrt{12}$); Octahedron (`fol. 68v`): $e^2 = \frac{1}{2} D^2 = 18$ ($e = \sqrt{18}$).
Truncated Tetrahedron (`fol. 69v, prob. 15–16`, 4 triangles + 4 hexagons): Exact squared edge $e^2 = \frac{2}{11} D^2 = \frac{72}{11} = 6\frac{6}{11}$, matching Pacioli’s printed radical $\mathcal{R}.6\frac{6}{11}$!
Cuboctahedron (`fol. 69v–70r, prob. 17–19`, 6 squares + 8 triangles): Its equatorial great-circle section PQRSTU across diameters PS, QT, and RU is a regular hexagon whose side PQ is equal to the sphere radius $R = 3$!
Icosahedron & Dodecahedron (`fol. 70r, prob. 21–22`): Spanned internally by golden rectangle WXYZ (WX : XY $= \phi : 1$). In Problem 21 (fol. 70r), for a sphere whose axis has possanza $48$ ($D = \sqrt{48}$), the inscribed cube has edge $\sqrt{48/3} = 4$, and dividing $4$ in extreme and mean ratio yields the exact dodecahedron edge $\sqrt{20} - 2$ and minor segment $6 - \sqrt{20}$, right beside Problem 20’s hydrostatic method of Archimedes measuring irregular sculptural bodies by displaced water.