Giovanni Ceva

Static Barycentrics, Cevian Concurrency, Spatial Equilibrium, and Synthetic Geometry (Milan, 1678)

Substitutisque linearum vice ponderibus… nulla adhibita circulorum, linearumque præuia constructione, quæ ad linearum se se intersecantium symptomata spectant, solis ponderibus staticè enodare. Hinc staticam constructionem libuit appellare.

Having substituted weights in place of straight lines… with no prior construction of circles and auxiliary lines employed, I resolved statically—by weights alone—the properties pertaining to mutually intersecting straight lines. Hence I was pleased to call it a Static Construction.

—Giovanni Ceva, De lineis rectis se invicem secantibus statica constructio, Prooemium (Milan, 1678)

Overview

GEOMETOR Ceva (geometor-ceva) unites a 4-Track Bilingual Digital Scriptorium Book Codex (English, Latin, Summary, Notes) with 100% exact symbolic SymPy / ``geometor.model`` constructions and proofs for Giovanni Ceva's (1647–1734) landmark quarto treatise De lineis rectis se invicem secantibus statica constructio (Milan: Ludovico Monza, 1678):

  1. *Liber Primus* — Static Foundations, Negative Weights (`F - A`), Ceva & Menelaus: Dedicating the work to Ferdinando Carlo Gonzaga and honoring his Pisan mentor Donato Rossetti, Ceva replaces auxiliary Euclidean lines with Archimedes' Law of the Lever ($u \cdot \overline{AF} = v \cdot \overline{FB}$). He proves Menelaus’s Transversal Theorem (Elementum I, Prop. VIII), Ceva’s Concurrency Theorem $\frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA} = 1$ (Elementum II, Prop. VII), introduces algebraic negative weights (“pondus… more analytico, seu algebrico ita scribemus $F - A$”, Elementum III & p. 5 Scholium) 149 years before Möbius, preserves Pietro Paolo Caravaggio Jr.'s pure Euclidean parallel-line proof (Prop. VI), and establishes 3D skew-quadrilateral (Elementum V Coroll.) and tetrahedral 7-line concurrency (Prop. XXIII, “mira quadam concordia”).

  2. *Liber Secundus* — Circumscribed Conics, Isotomic Conjugates & Odd Polygons: Using Serenus and Borelli's cylindrical projection (Lemma I), Ceva proves contact-ratio invariance on circles and ellipses, discovers the concurrency of incircle/inellipse contact cevians (Prop. I, anticipating Gergonne $X(7)$) and its isotomic conjugate (Prop. I Corollarium, anticipating Nagel $X(8)$), and generalizes cevian concurrency to arbitrary odd $(2m + 1)$-gons.

  3. *Appendix Geometrica* — Pure Area Ratios, Ungula Cubature & Indivisibles Caution: Dispensing with mechanics altogether, Ceva re-grounds ratio geometry in pure Euclidean triangle areas ($[\triangle PAB] / [\triangle PAC] = BD / DC$), squares Hippocratean lunes (Prop. II), cubes the cylindrical ungula as $\frac{2}{3}$ of its circumscribed prism (Prop. VII), establishes the $5 : 3$ hemisphere centroid (Prop. XIII, praising Luca Valerio) and $2 : 1$ paraboloid centroid (Prop. XIV, citing Evangelista Torricelli), and closes with his celebrated caution on Cavalieri’s indivisibles on curved surfaces.

Architecture of the Repository

  • Digital Scriptorium Book Codex (De Lineis Rectis Codex (1678)): Four multi-track units (:layout: book) pairing original GEOMETOR English translations with folio-verified 1678 Neo-Latin primary texts, executive geometric summaries, scholarly commentary, 8 interactive .euclid-graphic SVG plates, and linked apparatus registries (Concordance, Bibliography, Persons, Glossary).

  • Exact Symbolic Demonstrations (Symbolic Demonstrations): Pure-symbolic Python scripts (Model(…, auto_intersect=False)) free of floating-point coordinates, verifying canonical centers $X(1), X(2), X(4), X(6), X(7), X(8), J_A$ over $\mathbb{Q}$, harmonic ranges $(B, C; D, D') = -1$, Spread Ceva’s Theorem, pentagonal concurrency, 3D 7-line tetrahedral equilibrium, and Appendix integrals.

  • Python API Reference (Modules): Documented symbolic modules in geometor.ceva (static_barycentrics, polygon_tetrahedron, sphinx_ext, __main__).

  • Project Context & Log (About GEOMETOR Ceva, log): Provenance, editorial conventions, and development log.