Unit IV · Executive Geometric Summary

Core Results of Appendix Geometrica (1678, pp. 73–104)

  1. Barycentric Weights = Oriented Sub-Triangle Areas: Eliminating physical mechanics via [VI.1] and [V.19], Ceva proves that for any concurrency point $P = (u : v : w)$,

    $$[\triangle PBC] : [\triangle PCA] : [\triangle PAB] = u : v : w, \qquad \frac{[\triangle PAB]}{[\triangle PAC]} = \frac{\overline{BD}}{\overline{DC}},$$

    verified both symbolically over $\mathbb{Z}[x_A, y_A, x_B, y_B, x_C, y_C, u, v, w]$ (prove_ceva_barycentric_identity()) and rationally for every canonical center (solve_static_barycentrics()).

  2. Exact Quadratures, Cubatures & Solid Centroids (verify_appendix_invariants()):

Verified Appendix Geometrica Invariants (geometor.ceva.polygon_tetrahedron)

1678 Locus

Geometric Object

Exact Symbolic Value

Classical / Modern Link

Appendix Prop. II

Hippocratean Lune on isosceles right $\triangle$

$[\text{Lune}] = \frac{1}{2} R^2 = [\triangle ABC]$

Hippocrates of Chios / Euclid XII.2

Appendix Prop. VII

Cylindrical Hoof (Ungula) in prism $R^2 H$

$\text{Vol} = \frac{2}{3} R^2 H = \frac{2}{3}\,\text{Prism}$

Archimedes The Method (rediscovered 1906!)

Appendix Prop. XIII

Solid Hemisphere centroid along axis $VO$

$\bar{z}_{\text{vertex}} : \bar{z}_{\text{base}} = 5 : 3$

Luca Valerio (De centro gravitatis, 1604)

Appendix Prop. XIV

Parabolic Conoid centroid along axis

$\bar{z}_{\text{vertex}} : \bar{z}_{\text{base}} = 2 : 1$

Archimedes / Commandino (1565) / Torricelli (1644)

  1. Ceva’s Closing Caution on Indivisibles: Summing parallel circles $2\pi r(z)\,dz$ along an axis omits the Jacobian slant factor $\sec\alpha = \sqrt{1 + (dr/dz)^2}$ of inscribed Archimedean conical frusta—explaining why naive surface indivisibles yield $\frac{\pi^2}{2}R^2$ instead of $2\pi R^2$ on a hemisphere.