UNIT III NOTES — SCHOLARLY & SYMBOLIC COMMENTARY¶
Euclid’s Lost On Divisions, Centroid X(2), and Exact Symbolic Constructions in geometor.fibonacci.divisions
English Edition · Latin (1220 / Boncompagni 1862) · Summary
1. Reconstructing Euclid’s Lost On Divisions of Figures¶
Mentioned by Proclus in his Commentary on Book I (geometor/proclus), Euclid’s On Divisions of Figures vanished in Greek antiquity. In $1851$, Franz Woepcke discovered an Arabic epitome in Paris (MS Suppl. ar. 952.2) giving the enunciations of $36$ propositions and proofs of only four; in $1915$, R. C. Archibald demonstrated (euclid:divisions) that Leonardo Pisano’s *Distinctio IV* ($1220$) preserves the missing Euclidean proofs—translated from an unabridged Arabic or Greco-Latin archetype—together with Leonardo’s own algebraic extensions to 3-, 4-, and 5-part consortium divisions.
In geometor.fibonacci.divisions.construct_triangle_side_point_division(), the boundary-point construction on $(13, 14, 15)$ with $D = (5, 0)$ executes strictly via compass-and-straightedge primitives (auto_intersect=False):
Base midpoint $E = (7, 0)$;
Parallel $[EZ] \parallel [AD]$ meeting $[GA]$ at $Z = \left(\frac{19}{9}, \frac{60}{9}\right) = \left(\frac{19}{9}, \frac{20}{3}\right)$;
Rational quadrance $Q(G, Z) = \frac{1225}{9} = \left(\frac{35}{3}\right)^2$, confirming $GZ = 11\frac{2}{3}$ and $\text{Area}(\triangle DZG) = \frac{1}{2} \times 9 \times \frac{28}{3} = 42 = \frac{1}{2}(84)$.
2. Early Latin Proof of Median Concurrency at $X(2)$¶
Euclid’s Elements never explicitly states that the three medians of a triangle concur! While Archimedes used the center of gravity in On the Equilibrium of Planes, Leonardo’s Preparatoria (Boncompagni Vol. II, pp. 112–113) gives a purely planar, non-mechanical similarity proof via the midline $EZ = \frac{1}{2} AB$—making Practica Geometriae a foundational landmark for Kimberling center X(2) (geometor/etc).
In construct_triangle_medians_centroid(), for $B = (0, 0)$, $G = (14, 0)$, $A = (5, 12)$:
$D = X(2) = \left(\frac{19}{3}, 4\right)$;
Exact Rational Trigonometry quadrance ratios confirm $Q(A, D) = 4\,Q(D, E) = \frac{592}{9}$, $Q(B, D) = 4\,Q(D, Z) = \frac{505}{9}$, and $Q(G, D) = 4\,Q(D, T) = \frac{673}{9}$;
Signed determinant areas verify $\Delta(DAB) = \Delta(DBG) = \Delta(DGA) = 28$.
3. Quadratic Scaling Over $\mathbb{Q}(\sqrt{2})$ & Babylonian Root-Mean-Square¶
In construct_triangle_parallel_division(), bisecting $\triangle(10, 14, 12)$ parallel to base $BG = 14$ places every constructed intersection in $\mathbb{Q}(\sqrt{2}, \sqrt{6})$ without a single floating-point operation:
Semicircle mean proportional gives $AE = 5\sqrt{2}$ ($Q(A, E) = 50$), $AZ = 6\sqrt{2}$ ($Q(A, Z) = 72$), and parallel cut $EZ = 7\sqrt{2}$ ($Q(E, Z) = 98$), matching Boncompagni Vol. II, p. 119 digit for digit.
Likewise, trapezoid_parallel_division(top_base_a=7, bottom_base_b=17, height=10, p=1, q=1) recovers the ancient Babylonian surveyor’s triad $(7, 13, 17)$ satisfying $7^2 + 17^2 = 2 \times 13^2$—linking Old Babylonian clay tablets (such as YBC 4675), Euclid’s On Divisions, and medieval Pisan cadastral practice across three millennia.