UNIT IV — PARTICULARIS DE COMPUTIS ET SCRIPTURIS & THE PROBLEM OF POINTS

Part I, Distinctio IX, Tractatus X–XI: Interrupted Games of Chance (fol. 197r–197v) & Venetian Double-Entry Bookkeeping (fols. 198v–210v)

[Folio Part I, fols. 197r–197v]

The Problem of Points (Giuoco della Balla & Balestrare)

Immediately preceding his bookkeeping treatise, at the close of Tractatus X (fols. 197r–197v), Fra Luca Pacioli poses in print for the first time the celebrated Problem of Points (giuoco della balla), destined to launch mathematical probability in the July–October 1654 correspondence between Blaise Pascal and Pierre de Fermat (La règle des partis):

  • Two-Player Ball Game (`fol. 197r`): A company plays a balla to $60$ points (at $10$ points per winning caccia—i.e., first to $6$ rounds) for a stake of $10$ ducats; forced by an accident to break off when one side has $50$ ($5$ rounds) and the other $30$ ($3$ rounds, or $20$), how should the $10$ ducats be divided?

  • Problem 51 — Three Crossbowmen (`fol. 197v`): Three companions compete a balestrare (“whoever first achieves $6$ best shots wins $10$ ducats”); interrupted when the first has $4$ shots, the second $3$, and the third $2$, how much belongs to each?

Treating the contest like a commercial partnership (compagnia), Pacioli divides the stake proportionally to rounds already achieved: $\left(\frac{5}{8}, \frac{3}{8}\right) \times 10 = \left(6\frac{1}{4}, 3\frac{3}{4}\right)$ ducats for $(5, 3)$, and $\left(\frac{4}{9}, \frac{3}{9}, \frac{2}{9}\right) \times 10 = \left(4\frac{4}{9}, 3\frac{1}{3}, 2\frac{2}{9}\right)$ ducats for $(4, 3, 2)$—whereas Pascal and Fermat look forward to the remaining rounds needed ($1$ vs. $3$ in the two-player game), yielding exact fair probabilities $\left(\frac{7}{8}, \frac{1}{8}\right) \implies \left(8\frac{3}{4}, 1\frac{1}{4}\right)$ ducats, and $\left(\frac{451}{729}, \frac{195}{729}, \frac{83}{729}\right) \times 10$ ducats in the three-player crossbow match.

[Folio Part I, fols. 198v–210v]

Particularis de Computis et Scripturis — Double-Entry Bookkeeping al Modo de Vinegia

On fol. 198v, Pacioli opens the world’s first printed treatise on double-entry bookkeeping (Tractatus XI, 36 chapters), declaring that three things above all are required by anyone who wishes to trade with due diligence: (1) ready cash (la pecunia numerata) or solid credit, (2) mastery as a good reckoner (buon ragionieri), and (3) systematic ordering of every affair in Dare and Avere so that he may know at a glance both his debt and his credit.

  1. The Opening Inventory (`Cap. 2–3, fols. 198v–199r`): Drawn up on a single day (exemplified on November 8, 1493), recording all cash, jewels, ginger, real estate, debtors, and creditors.

  2. The Three Principal Books (`Cap. 5–14, fols. 199v–202r`), authenticated and stamped at the Venetian Consoli de’ Mercanti (Cap. 7): - Memoriale (Squarzo / Vacchetta): Day-book recording raw transactions hour-by-hour in whatever foreign or local coin occurred. - Giornale: Private chronological journal converting every item into uniform Venetian money of account via the formula Per <Debitore> // A <Creditore> (Cap. 11, fol. 201r), cross-referenced by a fraction $\frac{\text{Carta Dare}}{\text{Carta Avere}}$. - Quaderno Grande: General ledger with alphabetical Repertorio, placing Dare (debits) on the left man sinistra and Avere (credits) on the facing right man dextra across four columns of Venetian money of account (fol. 201v): lire, soldi, grossi, and piccoli ($1\text{ lira} = 10\text{ ducati} = 20\text{ soldi}$; $1\text{ ducato} = 24\text{ grossi}$; $1\text{ grosso} = 32\text{ piccoli}$).

K (Cavedal) S (Cassa) M (Zenzer) B (Bilancio) P (Pro e Danno) D (Martino) DARE (Per · Sinistra) AVERE (A · Dextra) 1ᵀ · B = 0ᵀ ⇔ Σ Dare = Σ Avere
Fundamental Conservation Law of the Quaderno (`Cap. 14, fol. 202r`): “Never must anything be placed in **Dare* that is not also placed in the same amount in Avere, and conversely never must anything be placed in Avere that is not placed in the same amount in Dare; whence at the closing balance of the book, the grand total of Dare must equal the grand total of Avere.”*
[Euclid I.cn.2 & Skew-Symmetric Conservation]

In the directed ledger network inscribed on circle (OK), each Giornale entry Per u // A v of weight $w_e > 0$ forms an oriented chord between accounts K (Cavedal), S (Cassa), M (Zenzer), D (Martino), P (Pro e Danno), and B (Bilancio). Opening cash KS (Per Cassa // A Cavedal), ginger purchase SM, credit sale MD and gain DP, collection DS, and profit transfer PK trace the closed trading cycle KSMDP. Because every column of the oriented vertex–edge incidence matrix $B \in \{-1, 0, +1\}^{|V| \times |E|}$ contains exactly one $+1$ (Dare) and one $-1$ (Avere), the pairwise flow matrix $F_{ij}$ is skew-symmetric ($F^T = -F$), and:

$$\mathbf{1}^T B = \mathbf{0}^T \implies \sum_{u \in V} (\text{Dare}_u - \text{Avere}_u) = \mathbf{1}^T (B w) = 0 \iff \text{Summa}(\text{Dare}) = \text{Summa}(\text{Avere})$$

Together with the domain invariant of Cap. 12 (fol. 201r) that Cassa can never have a net credit balance ($\text{Dare}_{\text{Cassa}} - \text{Avere}_{\text{Cassa}} \ge 0$, since no merchant can pay out more physical coin than has entered the chest), sweeping nominal accounts through Pro e Danno into Cavedal (Cap. 27) and closing open accounts via SB and KB into Bilancio del libro (Cap. 34) establishes the exact Summa summarum equilibrium of modern accounting.