Unit I · Executive Geometric Summary¶
Core Theses of the Frontmatter & Liber Primus Opening (1678, pp. 1–7)¶
The Principle of *Statica Constructio*: Rather than introducing clever auxiliary parallels or circles ad hoc for each incidence theorem, Giovanni Ceva places exact non-zero point masses $(u, v, w)$ at the vertices $(A, B, C)$ of a geometric configuration. Because addition of weighted points is commutative and associative and the global center of gravity $P = \frac{uA + vB + wC}{u + v + w}$ is unique (Axioma I), grouping $(B, C)$ first at fulcrum $D$, $(C, A)$ at $E$, or $(A, B)$ at $F$ forces $AD$, $BE$, and $CF$ to concur at $P$.
Signed / Negative Weights (1678 $\to$ 1827): On printed page 5 (Elementum III Scholium), Ceva extends Archimedes’ physical lever to exterior concurrency by writing subtractive weights “more analytico, seu algebrico” as $F - A$ (pondus negativum). Whenever one vertex carries a negative sign (e.g., $(-u, +v, +w)$ with $u + v + w \neq 0$), exactly two side fulcrums fall externally on extended sides, producing two negative directed ratios and one positive ratio whose product remains $(-)(+)(-) = +1$.
3D Non-Planar Skew Quadrilateral Coplanarity: In Elementum V Corollarium (pp. 6–7), Ceva proves that four weights $(m_A, m_B, m_C, m_D)$ on a non-planar 3D quadrilateral determine four edge fulcrums $E_{AB}, E_{BC}, E_{CD}, E_{DA}$ in cyclic product $1$ whose opposite joins $E_{AB}E_{CD}$ and $E_{BC}E_{DA}$ intersect at the spatial center $O = \frac{m_A A + m_B B + m_C C + m_D D}{m_A + m_B + m_C + m_D}$, forcing all four spatial fulcrums to be coplanar.
1678 Locus |
Vertex Weights |
Directed Side Ratios |
Geometric Regime |
|---|---|---|---|
Elementum II (Tab. 1, Fig. 5) |
$(+u, +v, +w)$, same sign |
$(+ , + , +)$, product $= +1$ |
Interior concurrency $P \in \operatorname{int}(\triangle ABC)$ |
Elementum III & p. 5 Scholium |
$(-u, +v, +w)$, $u \neq v, w$ |
$(- , + , -)$, product $= +1$ |
Exterior concurrency (e.g., excenter $J_A$) |
Limit $v + w = 0$ |
$(u, v, -v)$ |
Fulcrum $D \to \infty$ |
Cevian $AD \parallel BC$ (ideal projective point) |
Elementum V Corollarium |
$(m_A, m_B, m_C, m_D)$ in $\mathbb{R}^3$ |
$r_1 r_2 r_3 r_4 = +1$, $\det = 0$ |
Coplanar fulcrums & intersecting joins on 3D skew cycle |