Unit IV · Appendix Geometrica: Area Ratios, Ungula, Centroids & Indivisibles (pp. 73–104)

Original GEOMETOR English translation from Giovanni Ceva’s De lineis rectis se invicem secantibus statica constructio (Milan: Ludovico Monza, 1678, Appendix Geometrica, pp. 73–104; Tab. 8–10, Figs. 62–85).

1. Pure Synthetic Area-Ratio Proofs Free of Mechanics

Opening the Appendix Geometrica (pp. 73–76, Tab. 8, Figs. 62–64), Giovanni Ceva fulfills a promise to strict classical geometers: every concurrency and proportion proved via static weights in Libri I–II can be demonstrated purely synthetically without mechanics by substituting sub-triangle areas for weights!

Indeed, if cevian $AD$ passes through an interior (or signed exterior) point $P$ of $\triangle ABC$ and meets side $BC$ at $D$, then by Euclid ([VI.1] — triangles on bases $BD, DC$ sharing the same vertex have areas proportional to those bases):

$$\frac{[\triangle ABD]}{[\triangle ACD]} = \frac{\overline{BD}}{\overline{DC}}, \qquad \frac{[\triangle PBD]}{[\triangle PCD]} = \frac{\overline{BD}}{\overline{DC}}.$$

Subtracting numerator from numerator and denominator from denominator ([V.19]) immediately yields the fundamental barycentric area identity:

$$\frac{[\triangle PAB]}{[\triangle PAC]} = \frac{\overline{BD}}{\overline{DC}}, \qquad \frac{[\triangle PBC]}{[\triangle PAB]} = \frac{\overline{CE}}{\overline{EA}}, \qquad \frac{[\triangle PAC]}{[\triangle PBC]} = \frac{\overline{AF}}{\overline{FB}}.$$

Multiplying these three ratios telescopes the three sub-triangle areas identically to $1$:

$$\frac{\overline{AF}}{\overline{FB}} \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} = \frac{[\triangle PAC]}{[\triangle PBC]} \cdot \frac{[\triangle PAB]}{[\triangle PAC]} \cdot \frac{[\triangle PBC]}{[\triangle PAB]} = 1.$$

Thus Ceva’s static vertex weights $(u : v : w)$ are canonically isomorphic to the signed oriented sub-triangle areas $([\triangle PBC] : [\triangle PCA] : [\triangle PAB])$!

2. Lune Quadrature (Prop. II) & Cylindrical Ungula Cubature (Prop. VII)

Continuing through curvilineal and solid geometry (Tab. 8–9), Ceva establishes a suite of classical and Galilean quadratures and cubatures:

  • Proposition II — Hippocratean Lune Quadrature: Over an isosceles right triangle triABC of legs $R$ and hypotenuse $R\sqrt{2}$, the crescent lune bounded between the semicircle arcHyp on the hypotenuse (radius $R/\sqrt{2}$, area $\frac{1}{4}\pi R^2$) and the quadrant arc arcQuad from the right-angle vertex (sector area $\frac{1}{4}\pi R^2$) equals the rectilinear right triangle triABC itself:

    $$[\text{Lune}] = \frac{1}{4}\pi R^2 - \left(\frac{1}{4}\pi R^2 - \frac{1}{2}R^2\right) = \frac{1}{2}R^2 = [\triangle ABC],$$

    following Hippocrates of Chios via [XII.2] without any transcendental remainder.

  • Proposition VII — Cubature of the Cylindrical Hoof (*Ungula*): Let a right circular cylinder of base radius $R$ and altitude $H$ be cut obliquely by a plane passing through a diameter of the base (Tab. 8–9), detaching a cylindrical ungula ungula inscribed inside a right-triangular prism of base $2R \times R$ and height $H$ ($\text{Vol}_{\text{prism}} = R^2 H$). Slicing parallel to the cylinder’s axis and perpendicular to the diameter produces similar right-triangular cross-sections sliceTri of base $\sqrt{R^2 - x^2}$ and height $\frac{H}{R}\sqrt{R^2 - x^2}$, whose area $\frac{H}{2R}(R^2 - x^2)$ integrates (by Archimedes' parabolic exhaustion) to:

    $$\operatorname{Vol}(\text{Ungula}) = \frac{2}{3} R^2 H = \frac{2}{3} \operatorname{Vol}(\text{Circumscribed Prism}).$$

Lune 1/2 R^2 App. Prop. II: Lune = Triangle [Lune] = [Delta] = 1/2 R^2 App. Prop. VII: Cylindrical Ungula Vol(Ungula) = 2/3 R^2 H = 2/3 Prism V O G (5:3) ds != dz Prop. XIII (5:3 CG) & Indivisibles Paradox Curved Surface Requires Slant Frustum ds, Not dz

3. Solid Centroids: Valerio’s $5:3$ Hemisphere (Prop. XIII) & Torricelli’s $2:1$ Conoid

Turning to solid centers of gravity (Tab. 9–10), Ceva salutes two giants of the Italian school:

  • Proposition XIII — Center of Gravity of the Hemisphere ($5 : 3$): Along the symmetry axis axisVO of a solid hemisphere hemi from pole $V$ ($z = 0$) to base center $O$ ($z = R$), the center of gravity cgH divides the axis in the exact ratio $5 : 3$ ($\bar{z} = \frac{5}{8} R$ from the vertex, $\frac{3}{8} R$ above the equatorial base). Ceva hails Luca Valerio (1604) as “subtilissimus nostri æui Archimedes” (“the most subtle Archimedes of our age”).

  • Proposition XIV & Hyperbolic Solids — Tribute to Torricelli: For a parabolic conoid of revolution of altitude $H$, the center of gravity divides the axis from vertex to base in the exact ratio $2 : 1$ ($\bar{z} = \frac{2}{3} H$), while Evangelista Torricelli's (1644) acute infinite hyperbolic solid possesses a finite volume despite extending infinitely along its asymptote.

4. Closing Caution & Paradox on Cavalieri’s Indivisibles (pp. 100–104)

In the celebrated closing pages of the Appendix Geometrica, Ceva—who had praised Bonaventura Cavalieri in the Prooemium—issues a razor-sharp methodological warning (indivisibilia & slant frustum):

Suppose one slices a hemisphere hemi (or cone) by equidistant parallel planes (chordSlice) of spacing $dz$ and naively equates the curved surface area to the sum of the parallel circular circumferences $2\pi r(z)\,dz$. For a hemisphere $r(z) = \sqrt{R^2 - u^2}$, that naive integral $\int_0^R 2\pi\sqrt{R^2 - u^2}\,du = \frac{\pi^2}{2} R^2$ falsely disagrees with Archimedes’ true surface area $2\pi R^2$ (On the Sphere and Cylinder I.33)! Why? Because on a curved or oblique surface, an infinitesimal band between parallel planes is a conical frustum whose slant generator slantArc has length $ds = \frac{du}{\cos\alpha} = \frac{R}{\sqrt{R^2 - u^2}}\,du$, so the true surface element is:

$$2\pi r(u)\,ds = 2\pi \sqrt{R^2 - u^2} \cdot \frac{R}{\sqrt{R^2 - u^2}}\,du = 2\pi R\,du \implies \int_0^R 2\pi R\,du = 2\pi R^2!$$

Without Archimedean frustum slant correction, Ceva concludes, parallel indivisibles are treacherous on curved surfaces, even where they succeed brilliantly for planar areas and solid volumes!