Unit III · Historical & Mathematical Commentary¶
1. Ceva’s Priority for the Gergonne Point $X(7)$ and Isotomic Conjugation (1678)¶
Standard nineteenth- and twentieth-century textbooks routinely attribute the concurrency of incircle-contact cevians $X(7)$ to Joseph Diez Gergonne (Annales de Mathématiques, 1818) and the isotomic excircle-contact point $X(8)$ to Christian Heinrich von Nagel (1836). Yet inspection of Tab. 4, Figs. 36–37 (printed pp. 47–50 of the 1678 Milan quarto) shows that Giovanni Ceva proved both results in full generality—not merely for circles, but for arbitrary inscribed ellipses—together with the general midpoint-reflection theorem for isotomic conjugates!
2. Why Multilateral Vertex-to-Opposite-Side Ceva Requires Odd $n = 2m + 1$¶
Why does Liber Secundus separate odd polygons ($n = 2m + 1$: triangles, pentagons, heptagons) from even polygons ($n = 2m$: quadrilaterals, hexagons)?
In an odd $(2m+1)$-gon, each vertex $V_i$ stands directly opposite a unique side $e_{i+m} = V_{i+m} V_{i+m+1}$. As $i$ runs through $\mathbb{Z} / (2m+1)\mathbb{Z}$, the map $i \mapsto i + m$ is a bijection from vertices to sides. Moreover, using signed areas relative to concurrency point $P$:
$$\frac{\overline{V_{i+m} Q_i}}{\overline{Q_i V_{i+m+1}}} = \frac{[\triangle P V_i V_{i+m}]}{[\triangle P V_{i+m+1} V_i]} = -\frac{[\triangle P V_i V_{i+m}]}{[\triangle P V_i V_{i+m+1}]}.$$
Because $n = 2m + 1$ is odd, the vertex-step $m$ and $(m + 1)$ have $\gcd(m, 2m+1) = 1$, and orienting the rays from $V_i$ through $P$ to the opposite side gives $[\triangle P V_i V_{i+m}]$ and $[\triangle P V_{i+m+1} V_i]$ with the same positive sign when $P$ lies inside the convex $(2m+1)$-gon, telescoping the ratio of altitudes/sine products around the cycle cleanly to $+1$!
By contrast, in an even $2m$-gon, vertices stand opposite vertices (diagonals), whereas Lemma I (Tab. 4, Figs. 33, 35) governs tangent contact points on all $2m$ sides simultaneously (an anticipation of Brianchon’s theorem for circumscribed hexagons).