UNIT II — ALGORISMUS, EUCLIDEAN SURDS & COSSIC ALGEBRA (REGOLA DELA COSA)¶
Part I, Distinctiones II–V (Practical Algorism) & Distinctio VIII: Arte Maggiore / Algebra e Almucabala (fols. 112r–150r)
Practical Algorism, Lattice Multiplication (Gelosia), Regola del Tre & El Cataym
Across Distinctiones II–V, Fra Luca Pacioli codifies the complete computational repertory of the Italian scuole d’abbaco inherited from Leonardo Pisano's Liber Abaci and Johannes de Sacrobosco's Algorismus:
Finger Reckoning (*Indigitatione*) & Place-Value Numeration: Woodcut hand positions representing units through tens of thousands (fol. 36v).
Eight Modes of Multiplication: Scacchieri (chessboard), castelluccio, colonna, crocetta, quadrilatero, gelosia (Venetian lattice grating), repiego (factor decomposition), and a scapezza (distributive partitioning).
Four Modes of Division: A regolo, per ripiego, a danda (forerunner of modern long division), and a galea (galley scratch-method prized by Venetian shipwrights and merchants).
Rule of Three (*Regola del Tre*) & Double False Position (*El Cataym*): Linear interpolation of unknowns via two trial hypotheses $(x_1, e_1)$ and $(x_2, e_2)$, eliminating affine error prior to invoking full algebraic unknowns.
Syncopated Cossic Notation, Sign Laws & Euclid Book X Surds ([X.36], [X.73])
Entering Distinctio VIII (De l’Arte Maggiore, ditta dal vulgo la Regola dela Cosa), Pacioli standardizes in print the syncopated Renaissance algebraic lexicon:
Gradation of Cossic Powers: Numero ($1$), .co. (cosa) ($x$), .ce. (censo) ($x^2$), .cu. (cubo) ($x^3$), .ce.ce. (censo de censo) ($x^4$), primo relato ($x^5$), and ce.cu. ($x^6$).
Operational & Grouping Symbols: .p̄. (più) and .m̄. (meno) governed by the four sign rules (più via più sempre fa più; meno via meno sempre fa più; più via meno sempre fa meno), together with the compound radical .℞.u. (radice universale).
Six Binomials & Six Apotomes (*Residui*): Before solving quadratic equations, Pacioli translates Euclid's geometric theory of compound irrationals ([X.36], [X.73]) into radical arithmetic. In the left diagram, the binomial segment AB joins rational AM ($3$) and surd $MB = \sqrt{5}$, so its square decomposes by [II.4] into AMPQ ($9$), $b = 5$, and two symmetric gnomon wings MBRP ($2 \cdot 3\sqrt{5}$), totaling $14 + 6\sqrt{5}$. In the right diagram, subtracting $NF = \sqrt{5}$ from EF ($3$) leaves at N the apotome (residuo) $EN = 3 - \sqrt{5}$, whose square ENKL inside EFGH equals $14 - 6\sqrt{5}$ ([II.7]).
The Six Canonical Cossic Equations & Geometric Gnomon Proofs ([II.4]–[II.6])
Following Al-Khwārizmī (Al-jabr wa'l-muqābala) and Leonardo Pisano, since all terms on each side of a classic equation must carry positive coefficients, fol. 145r arranges all linear and quadratic equations into three simple (semplici: $a x^2 = b x$, $a x^2 = c$, $b x = c$) and three compound (capitoli composti), printing Latin hexameter mnemonics for each compound canon:
First Compound Canon (*Primi canonis versus*, `fol. 145r`) — Censo e cose equale a numero ($x^2 + b x = c$): “Si res et census numero coaequantur, a rebus dimidio sumpto censum producere debes, addere quem numero, radicem tollere toti, dimidiumque rei de radice abjicere.” Halve the cose ($b/2$), square that half ($(b/2)^2$), add it to the numero ($c$), extract the square root, and subtract half the cose:
$$x = \sqrt{\left(\frac{b}{2}\right)^2 + c} - \frac{b}{2}$$
Second Compound Canon (*Secundi canonis versus*) — Censo e numero equale a cose ($x^2 + c = b x$): Valid whenever $(b/2)^2 \ge c$, yielding two positive roots $x = \frac{b}{2} \mp \sqrt{(b/2)^2 - c}$ (proved via [II.5]).
Third Compound Canon (*Tertii canonis versus*) — Cose e numero equale a censo ($b x + c = x^2$): Yielding $x = \sqrt{(b/2)^2 + c} + \frac{b}{2}$ (proved via [II.6]).
On fol. 146r, Pacioli proves the First Compound Canon geometrically on the archetype $1.\text{ce.} \; \bar{p} \; 10.\text{co.} = 39$ ($x^2 + 10x = 39$). Let AEFG represent the unknown censo ($x^2$) on side $AE = x$. Halving the $10$ cose into two equal rectangles of width $EB = GD = 5$ attaches the gnomon wings EBHF and GFID, whose combined area with AEFG is given as $39$. Filling in the corner square FHCI of side $5$ and area $5^2 = 25$ completes the great square $ABCD$ whose sides AB and DA are equal to BC and CD along diagonal AC:
$$\text{Area}(ABCD) = 39 + 25 = 64 \implies AB = \sqrt{64} = 8 \implies x = AE = 8 - 5 = 3$$
Higher Cossic Gradations & the Historic Verdict on Cubic Equations (fol. 150r)