UNIT V SUMMARY — DISTINCTIO VII–VIII, FLOS & LIBER ABACI¶
Structural Synopsis of Boncompagni Vol. II, pp. 202–247 & Vol. I, pp. 283–284
English Edition · Latin (1220 / Boncompagni 1862) · GEOMETOR Notes
Overview¶
Unit V completes Practica Geometriae with optical trigonometry (Distinctio VII) and algebraic-geometric subtilitates (Distinctio VIII), then bridges directly to Leonardo’s two sister masterpieces—Flos ($1225$) and Liber Abaci ($1202/1228$)—exhibiting the deep unity of Pisan mensuration, Islamic al-jabr, Euclidean irrationals, and the golden ratio $\phi$.
Summary of Propositions & Exact Values¶
Locus |
Problem / Construction |
Exact Symbolic or Sexagesimal Form |
|---|---|---|
Quadrans (Dist. VII) |
Tower elevation from $12 \times 12$ umbra recta / versa |
$H = h_{\text{eye}} + \frac{12}{u_r} D$ (for $u_r = 8, D = 40$: $H = 61$) |
Decagon in $d = 10$ (Dist. VIII) |
Golden section of hexagon side $r = 5$ (Euclid XIII.9) |
$s_{10} = \sqrt{31\frac{1}{4}} - 2\frac{1}{2} = \frac{5}{\phi}$ |
Pentagon in $d = 10$ (Dist. VIII) |
Euclid XIII.10 & census census $x^4 + 3125 = 125x^2$ |
$s_5 = \sqrt{62\frac{1}{2} - \sqrt{781\frac{1}{4}}} = \frac{5\sqrt{10 - 2\sqrt{5}}}{2}$ |
Congruum of $5$ (Flos) |
Three rational squares in arithmetic progression of step $5$ |
$\left(\frac{31}{12}\right)^2,\ \left(\frac{41}{12}\right)^2,\ \left(\frac{49}{12}\right)^2$ |
Imperial Cubic (Flos) |
$x^3 + 2x^2 + 10x = 20$ ($\notin$ Euclid Book X) |
$1; 22, 7, 42, 33, 4, 40_{60} = \frac{1{,}596{,}577{,}777}{1{,}166{,}400{,}000}$ |
Rabbit Sequence (Liber Abaci XII) |
$F_{n+1} = F_n + F_{n-1}$ ($1, 2, 3, \dots, 377$) |
$\lim_{n\to\infty} \frac{F_{n+1}}{F_n} = \phi$, $F_{n+1}F_{n-1} - F_n^2 = (-1)^n$ |