UNIT IV — CONSERVATION ALGEBRA & PROBABILITY SUMMARY

Symbolic Matrix Formulation of the Quaderno & Exact Problem of Points Shares

1. Venetian Four-Column Currency Radix (fol. 201v)

$$\begin{aligned} 1 \text{ grosso} &= 32 \text{ piccoli}, & 1 \text{ soldo} &= 12 \text{ grossi} = 384 \text{ piccoli}, \\ 1 \text{ ducato} &= 24 \text{ grossi} = 2 \text{ soldi} = 768 \text{ piccoli}, & 1 \text{ lira a oro} &= 20 \text{ soldi} = 10 \text{ ducati} = 7{,}680 \text{ piccoli}. \end{aligned}$$

2. Graph-Theoretic Conservation Laws (geometor.pacioli.ledger_conservation)

Let $V = \{1, \dots, n\}$ be the registered Quaderno accounts and $E = \{1, \dots, m\}$ the chronological Giornale postings $e_k = (u_k, v_k, w_k)$ (Per u_k // A v_k of positive weight $w_k \in \mathbb{Q}_{>0}$):

  • Oriented Incidence Matrix $B \in \{-1, 0, +1\}^{n \times m}$: $B_{u_k, k} = +1$, $B_{v_k, k} = -1$, so $\mathbf{1}^T B = \mathbf{0}^T$.

  • Skew-Symmetric Pairwise Flow Matrix $F \in \mathbb{Q}^{n \times n}$: $F_{ij} = \sum_{e: (i,j)} w_e - \sum_{e: (j,i)} w_e = -F_{ji}$ ($F^T = -F$).

  • Summa Summarum Identity: $\sum_{i=1}^n \text{Dare}_i = \sum_{k=1}^m w_k = \sum_{i=1}^n \text{Avere}_i$, and $\mathbf{1}^T (B w) = 0$.

  • Cap. 12 Cash Non-Negativity: $(B w)_{\text{Cassa}} \ge 0$ after every prefix of postings.

  • Cap. 27 & Cap. 34 Closing Cycle: Sweeping nominal accounts into Pro e Danno, transferring net profit to Cavedal, and sweeping open balances into Bilancio del libro drives $B_{\text{closed}} w_{\text{closed}} = \mathbf{0}$.

3. Pacioli (1494) vs. Pascal–Fermat (1654) on fol. 197r–197v

Exact Stake Divisions for Pot = $10$ Ducats, Target = $6$ Rounds

Interrupted Contest

Pacioli 1494 (Past Scores)

Pascal–Fermat 1654 (Future Paths)

Difference ($\text{PF} - \text{Pac}$)

2-Player Balla $(5, 3)$

$\frac{25}{4}, \; \frac{15}{4}$ ($6.25, \; 3.75$)

$\frac{35}{4}, \; \frac{5}{4}$ ($8.75, \; 1.25$)

$+\frac{5}{2}, \; -\frac{5}{2}$

2-Player Balla $(5, 2)$

$\frac{50}{7}, \; \frac{20}{7}$

$\frac{150}{16} = \frac{75}{8}, \; \frac{5}{8}$

$+\frac{125}{56}, \; -\frac{125}{56}$

3-Player Balestrare $(4, 3, 2)$

$\frac{40}{9}, \; \frac{30}{9}, \; \frac{20}{9}$

$\frac{4510}{729}, \; \frac{1950}{729}, \; \frac{830}{729}$

$+\frac{1270}{729}, \; -\frac{480}{729}, \; -\frac{790}{729}$