UNIT III — TRACTATUS DE PROPORTIONIBUS ET PROPORTIONALITATIBUS¶
Part I, Distinctio VI: Universal Praise of Proportion, the Five Boethian Genera, Classical Means, Six Euclidean Modes & the Golden Ratio (fols. 68r–98r)
Universal Sovereignty of Proportion in Philosophy, Perspective & Architecture
Opening Distinctio VI (fols. 68r–68v), Fra Luca Pacioli proclaims proportion to be the mother and queen of every speculative and practical art: without proportion no syllogism holds in logic, no consonance sounds in music, no physician compounds medicines in due gradus, no painter foreshortens figures in perspective—where Pacioli salutes his fellow citizen Maestro Piero della Francesca (“el monarca a li tempi nostri de la pictura”) and his De prospectiva pingendi—and no architect erects durable temples, invoking Vitruvius (De Architectura III.1) and Leon Battista Alberti (De Re Aedificatoria).
The Five Boethian Genera of Rational Ratio & Musical Compounding
Following Boethius (De Institutione Arithmetica and De Institutione Musica) and Jordanus Nemorarius, Pacioli divides rational ratio (proportione) into equality (aequalitas, $1:1$) and inequality (inaequalitas), subdividing greater inequality ($p > q$) into five exhaustive genera (paired with five sub- genera of lesser inequality):
Multiplex ($n : 1$): dupla ($2:1$, musical diapason/octave), tripla ($3:1$), quadrupla ($4:1$).
Superparticularis ($(n + 1) : n$): sesquialtera ($3:2$, musical diapente/fifth), sesquitertia ($4:3$, diatessaron/fourth), sesquioctava ($9:8$, whole tone).
Superpartiens ($(n + r) : n$, $1 < r < n$, $\gcd(r, n) = 1$): superbipartiens tertias ($5:3$), supertripartiens quartas ($7:4$).
Multiplex superparticularis ($k n + 1 : n$, $k \ge 2$): dupla sesquialtera ($5:2$).
Multiplex superpartiens ($k n + r : n$, $k \ge 2, 1 < r < n$): dupla supertripartiens quartas ($11:4$).
On fol. 83r, Pacioli illustrates compounded ratios (proportioni composte) with arched chord diagrams: compounding sesquialtera ($3:2$) with sesquitertia ($4:3$) produces dupla ($\frac{3}{2} \cdot \frac{4}{3} = \frac{2}{1}$), while subtracting sesquitertia from sesquialtera isolates the Pythagorean whole tone sesquioctava ($\frac{3/2}{4/3} = \frac{9}{8}$).
Semicircle Construction of the Three Classical Means ([VI.13]) & Six Euclidean Modes (fol. 74r)
Let $AP = 9$ and $PB = 4$, so diameter AB $= 13$ and radius MQ $= \frac{9 + 4}{2} = \frac{13}{2} = A$ (the Arithmetic mean). Erecting the perpendicular PQ at P to meet semicircle (MB) at Q, right triangle $\triangle AQB$ ([III.31]) has altitude PQ $= \sqrt{9 \cdot 4} = 6 = G$ (the Geometric mean, [VI.13]). Dropping PN perpendicular to radius MQ at N, right triangle MPQ is similar to PQN ([VI.8]), whence:
$$PQ^2 = MQ \cdot QN \implies QN = \frac{G^2}{A} = \frac{2ab}{a + b} = \frac{72}{13} = 5\frac{7}{13} = H$$
demonstrating visually that $A > G > H$ ($\frac{13}{2} > 6 > \frac{72}{13}$) and $G^2 = A \cdot H = 36$.
On fol. 74r (De sex speciebus sive modis arguendi proportionalitatum), Pacioli codifies the six transformational modes of proportionality from Euclid Book V on $a : b = c : d$: (1) conversa ($b:a = d:c$, [V.7.por]), (2) permutata ($a:c = b:d$, [V.16]), (3) congionta ($(a+b):b = (c+d):d$, [V.18]), (4) disgiunta ($(a-b):b = (c-d):d$, [V.17]), (5) eversa ($a:(a-b) = c:(c-d)$, [V.19.por]), and (6) equa ($a:e = c:f$ when $b:e = d:f$, [V.22]).
Division of $10$ in Extreme and Mean Ratio ([II.11], [VI.30], [XIII.1])
To divide AB $= 10$ in extreme and mean ratio so $AB : AC = AC : CB$: construct the square ABKD of side $10$, bisect $AD$ at E ($AE = 5$), join hypotenuse EB $= \sqrt{10^2 + 5^2} = \sqrt{125}$ ([I.47]), swing arc (EB) to cut $DA$ produced at F so $AF = \sqrt{125} - 5$, and transfer $AF$ via arc (AF) onto AB at C, making AC equal in golden square ACHF to rectangle $AB \cdot CB$:
$$AC = \sqrt{125} - 5, \qquad CB = 15 - \sqrt{125}, \qquad AB \cdot CB = AC^2 = 150 - \sqrt{12500}$$