UNIT III — HISTORICAL, PHILOLOGICAL & DEDUCTIVE NOTES

Commentary on Proportione vs. Proportionalità, Piero della Francesca & the 1509 Compendio

1. Renaissance Terminological Precision: Proportione vs. Proportionalità

Throughout Distinctio VI (and in the title Summa de arithmetica, geometria, proportioni et proportionalita itself), Pacioli adheres strictly to Campanus of Novara's Latin rendering of Euclid Book V:

  • Proportione translates Greek logos / Latin habitudo—a single ratio $a : b$ between two homogeneous magnitudes.

  • Proportionalità translates Greek analogia / Latin proportio—a similitude or equality of ratios $a : b :: c : d$.

Only in the seventeenth century did proportio shift in French and English usage to denote the four-term equality while ratio replaced proportio for the two-term relation. Keeping Pacioli’s distinction clear illuminates why fol. 74r calls Euclid’s six rules modi arguendi proportionalitatum (rules of inference transforming one valid proportion into five equivalent proportions).

2. Tribute to Piero della Francesca on fol. 68v

Scan n0155 (Part I, fol. 68v) refutes the legend that Pacioli suppressed his master’s name: in one of the warmest tributes of the Italian Quattrocento, Pacioli hails Maestro Piero della Francesca as “el monarca a li tempi nostri de la pictura” and explicitly commends his treatise De prospectiva pingendi, which Pacioli notes Duke Guidobaldo already possessed in the ducal library at Urbino.

3. Bridge from Summa Dist. VI (1494) to De Divina Proportione (1509)

Every mathematical ingredient of Pacioli’s 1509 Milanese–Venetian treatise *De Divina Proportione* is forged in Distinctio VI of the 1494 Summa:

  • The genealogical tree of rational and irrational ratios printed on fol. 82r–83r directly prefigures the full-page Arbor Proportionis et Proportionalitatis woodcut (Folio 0311) closing the 1509 edition.

  • The canonical numerical archetype dividing $10$ into the apotomes $\sqrt{125}-5$ and $15-\sqrt{125}$ recurs verbatim in *Compendio* Chapters VIII–XXII, elevated in Milan (1496–1498) to a theological and cosmological contemplation illustrated by Leonardo da Vinci.