UNIT I — DEDICATIONS, CONTINENTIA & SPECULATIVE ARITHMETIC¶
Preliminaries (fols. 1r–8v) & Part I, Distinctio I: Continuous and Discrete Quantity, Figurate & Perfect Numbers (fols. 1r–18v)
General Contents (Continentia de tutta lopera) & Venetian Preliminaries
In the opening pages printed by Paganino de' Paganini in Venice (November 10–20, 1494), Fra Luca Pacioli maps out the five universal divisions of the Summa de arithmetica, geometria, proportioni et proportionalita:
Theoretical & Practical Numbers (De numeri e misure): Speculative properties of integers, Hindu–Arabic algorism, fractions, and proportions (Distinctiones I–VI).
The Great Art of Algebra (Arte Maggiore, ditta dal vulgo la Regola dela Cosa): Operations on radicals, binomials, apotomes, and the six canonical equations (Distinctio VIII).
Mercantile Arithmetic (Compagnie, baratti, cambi, leghe): Partnerships, bills of exchange, alloyage, and interrupted games of chance (Distinctio IX, Tractatus I–X).
Venetian Double-Entry Bookkeeping (De computis et scripturis): Ordering every merchant’s Inventario, Memoriale, Giornale, and Quaderno Grande after the custom of Venice (Distinctio IX, Tractatus XI).
Practical & Regular Geometry (Tractatus Geometriae): Plane and solid mensuration, surface division, and the regular and dependent bodies inscribed in the sphere (Part II, Distinctiones I–VIII).
On the verso ([prelim. fol. 1v]), Pacioli salutes the Venetian patrician and mathematician Marco Sanudo (astrologiae, geometriae, arithmeticaeque cultor exquisitissimus), thanking him for financing the typographical enterprise so that secrets hitherto locked in Latin manuscripts might nourish Italian scholars, architects, and merchants alike, followed by a Latin elegiac epigram of Paolo Pompilio and an Italian sonnet by Giorgio Sommaripa.
Dedicatory Epistles to Guidobaldo da Montefeltro, Duke of Urbino
Addressing Guidobaldo da Montefeltro in both vernacular Italian and humanist Latin, Pacioli recalls the mathematical splendor of the Ducal Palace of Urbino erected by Guidobaldo’s father Federico da Montefeltro, argues that military art, perspective, music, cosmography, and architecture all stand or fall with exact measurement, and deliberately defends writing in the Italian mother tongue so that practical artisans and abbacisti share one unified science with university philosophers.
Division of Quantity & Explicit Acknowledgment of Authorities
On fol. 1v, Pacioli states his famous scholarly rule of attribution: he draws his demonstrations chiefly from Euclid, Severinus Boethius, Leonardo Pisano, Jordanus Nemorarius, Biagio Pelacani of Parma, Johannes de Sacrobosco, and Prosdocimo de' Beldomandi—adding explicitly that wherever a passage bears no marginal author’s name, it is to be understood as drawn from Leonardo Pisano.
Figurate Polygonal Numbers & Euclid’s Theorem on Perfect Numbers ([IX.36])
Following Boethius and Euclid ([VII.def.22], [IX.36]), Pacioli classifies integers both geometrically as figurate polygonal numbers—such as the fourth triangular number (tetractys) $T_4 = 1 + 2 + 3 + 4 = 10$ bounded by the equilateral array ABC where AB is equal to BC and CA—and multiplicatively into abundant (superflui), deficient (diminuti), and perfect (numeri perfetti).
Let successive powers of two be set out from the unit in continuous dupla progression:
$$1, \; 2, \; 4, \; 8, \; 16, \; \dots, \; 2^{n-1}$$
Whenever their cumulative sum $S_n = 1 + 2 + 4 + \dots + 2^{n-1} = 2^n - 1$ is a prime incomposite number (numero primo incomposto), the product of that prime sum $S_n$ and the last summed power $2^{n-1}$ forms an exact perfect number:
$$P_n = 2^{n-1}(2^n - 1)$$
In the dyadic rectangle DEFG for $n = 3$, the base DE has length $1 + 2 + 4 = 7$ (prime) and the altitude GD has length $2^2 = 4$ (equal to EF), beginning from the unit strip DHKG of area $1 \cdot 4 = 4$. Their product $7 \times 4 = 28$ equals the exact sum of its proper aliquot divisors:
$$1 + 2 + 4 + 7 + 14 = 28$$
On fol. 8r, Pacioli warns the reader against the widespread medieval error of assuming that $2^n - 1$ is prime for every odd exponent $n$ (pointing out that $1 + 2 + \dots + 256 = 511 = 7 \times 73$ and $2047 = 23 \times 89$ are composite and therefore must be skipped!), notes that every even perfect number terminates alternately in the digits $6$ or $8$, and prints an explicit numerical register extending all the way to a proposed Quartus decimus numerus perfectus.