UNIT I — MATHEMATICAL & ARCHITECTURAL SUMMARY

Structural Digest of the Preliminaries & Part I, Distinctio I

1. Bibliographic & Dedicatory Architecture

• Imprint: Venice, Paganino de' Paganini, November 10, 1494 (Part II colophon) and November 20, 1494 (Part I register), folio ([8], 224, 76 leaves).

• Dual Patronage: - Courtly & Humanist: Guidobaldo da Montefeltro, Duke of Urbino ([prelim. fols. 2r–3r]). - Venetian Patrician & Scientific: Marco Sanudo ([prelim. fol. 1v]), whose financial support made the presswork and woodcut marginalia possible.

• Default Source Convention (`Part I, fol. 1v`): Unattributed propositions and algorithms throughout the Summa derive from Leonardo Pisano's Liber Abaci and Practica Geometriae.

2. Boethian Division of Quantity & Quadrivium (fols. 1r–1v)

Pacioli’s Bifurcation of Quantity (La quantità è immediata bimembre)

Genus	Mode of Consideration	Mathematical Science	Primary Ancient Authority
Discrete (Moltitudo)	Absolute (per sé)	Arithmetic	Euclid VII–IX, Boethius
Discrete (Moltitudo)	Relative (ad altro)	Music (proportione)	Boethius, Prosdocimo
Continuous (Magnitudo)	Immobile (immobile)	Geometry ( & Perspective)	Euclid I–VI, XI–XIII, Piero
Continuous (Magnitudo)	Mobile (mobile celeste)	Astronomy (Astrologia)	Ptolemy, Sacrobosco

3. Figurate & Perfect Number Identities (fols. 6r–18v)

• Polygonal Numbers: For $s$-gonal numbers of side rank $k \ge 1$,

  $$P(s, k) = \frac{(s - 2)k^2 - (s - 4)k}{2}, \qquad T_k = P(3, k) = \frac{k(k+1)}{2}$$

• Euclid’s Even Perfect Number Formula (:just:`IX.36`, `fol. 8r`):

  $$M_p = 2^p - 1 \text{ prime} \implies P_p = 2^{p-1}(2^p - 1) \text{ satisfies } \sigma(P_p) = 2 P_p$$

  For $p \in \{2, 3, 5, 7, 13\}$, this yields $6$, $28$, $496$, $8128$, and $33{,}550{,}336$ ($p = 9$ and $p = 11$ fail because $2^9 - 1 = 511 = 7 \cdot 73$ and $2^{11} - 1 = 2047 = 23 \cdot 89$ are composite).