Technical Glossary

Cossic Algebra, Boethian Proportions, Venetian Double-Entry Bookkeeping & Stereometry

Lexicon of technical Venetian volgare, Latin scholastic, and syncopated Cossic terminology in Pacioli’s 1494 Summa.

giuoco della balla / balestrare — Interrupted Ball or Crossbow Contest (Problem of Points)

Language: Italian

Stake-division problem on Summa Part I, fol. 197r–197v asking how a pot of 10 ducats should be divided fairly when a race to 6 winning rounds is interrupted prematurely.

bilancio del libro — Ledger Closing Balance Sheet

Language: Italian

Closing summary procedure (Cap. 34) transferring open asset, debtor, creditor, and Cavedal balances from an exhausted Quaderno A into a fresh Quaderno B (marked with the Sign of the Cross +).

binomio (binomialis) — Euclidean Binomial Irrational

Language: Italian / Latin

An additive compound irrational a + √b or √a + √b classified into six species following Euclid Book X, Definitions II.1–6 and Propositions X.36–X.72.

misura delle botti — Wine-Barrel Stereometric Gauging

Language: Italian

Practical solid mensuration in Summa Part II, Dist. V modeling curved wine casks as frusta of cones or averaged cylinders from bung and head diameters.

capitoli composti (tre canoni composti) — Three Compound Quadratic Canons

Language: Italian

The three positive-coefficient three-term quadratic forms on fol. 145r: (1) censo + cose = numero, (2) censo + numero = cose, and (3) cose + numero = censo, proved geometrically on fol. 146r by Euclid II.4–II.6.

cassa de contanti — Cash Account

Language: Italian

Real ledger account tracking physical specie on hand; in Cap. 12 (fol. 201r) Pacioli states the domain invariant that Cassa must always be debtor or zero—never creditor.

catetto (cathetus) — Perpendicular Altitude of a Triangle

Language: Italian / Greek-Latin

Perpendicular dropped from a triangle vertex onto the opposite base; in Summa Part II, fol. 9v, dividing twice the Heron area (168) by base 14 yields catetto AD = 12.

cavedal (capitale) — Merchant’s Capital Equity Account

Language: Venetian / Italian

Venetian orthography for capitale (‘tutto il tuo monte corpo di facoltà presente’, Cap. 12, fol. 201r), credited at opening against Inventario assets and credited/debited with net profit or loss from Pro e Danno at closing.

censo (.ce. / census) — Square of the Unknown (x^2)

Language: Italian / Latin

The second Cossic power, equal to cosa multiplied by itself (x^2), abbreviated .ce. from the Latin census translating Arabic māl (‘wealth’ or ‘square estate’).

censo de censo (.ce.ce.) — Fourth Power / Biquadratic (x^4)

Language: Italian

The fourth Cossic power ((x^2)^2 = x^4); Pacioli shows on fol. 149v–150r that biquadratic trinomials in .ce.ce., .ce., and numero reduce directly to the three compound quadratic canons.

cosa (.co. / res / radix) — First Unknown Power (x)

Language: Italian / Latin

The primary algebraic unknown (‘the Thing’) in the Regola dela Cosa (Arte Maggiore, Summa Part I, Dist. VIII), printed with the syncopated abbreviation .co. and corresponding geometrically to the side of a square.

cubo (.cu. / cubus) — Cube of the Unknown (x^3)

Language: Italian / Latin

The third Cossic power (x^3), abbreviated .cu.; on fol. 150r Pacioli famously noted that general equations mixing cubo, censo, cosa, and numero had not yet yielded to a universal rule.

lira di grossi a oro (l. s. g. p.) — Venetian Four-Column Money of Account

Language: Venetian

Standard Venetian Quaderno accounting unit recorded on fol. 201v: 1 lira = 20 soldi = 10 gold ducats; 1 soldo = 12 grossi; 1 ducato = 24 grossi; 1 grosso = 32 piccoli (1 lira = 7,680 piccoli).

giornale — Venetian Journal

Language: Italian

Second principal ledger book converting Memoriale items into uniform money of account using the syntax ‘Per <Debitore> // A <Creditore>’ with marginal Quaderno folio fractions (Cap. 10–12, fol. 201r–201v).

inventario — Opening Merchant Inventory

Language: Italian

Complete dated enumeration on a single day of all cash, jewels, merchandise, real estate, debtors, and creditors owned by a merchant (De computis Cap. 2–3, fols. 198v–199r).

proportione avente el mezzo e doi estremi — Extreme and Mean Ratio (Golden Section)

Language: Italian

Euclidean division of a magnitude AB at C so AB : AC = AC : CB (Euclid II.11, VI.def.3, VI.30, XIII.1), called ‘Divina Proportione’ in Pacioli’s 1509 treatise.

memoriale (squarzo / vacchetta) — Day-Book / Waste-Book Memorandum

Language: Italian / Venetian

First chronological book of the Venetian counting-house in which every transaction is written down immediately as it occurs by any clerk or factor in whatever local coinage was used (Cap. 5–6).

modi arguendi proportionalitatum — Six Euclidean Transformational Modes of Proportion

Language: Latin

The six inference rules expounded on fol. 74r from Euclid Book V: conversa, permutata (alternando), congionta (componendo), disgiunta (dividendo), eversa (convertendo), and equa (ex aequali).

multiplex (moltiplice) — Multiple Genus of Ratio (n : 1)

Language: Latin / Italian

First Boethian genus of greater inequality in which the antecedent contains the consequent an exact integer number n >= 2 times (dupla 2:1, tripla 3:1, quadrupla 4:1).

numero perfetto — Perfect Number

Language: Italian

An integer equal to the sum of its proper aliquot divisors (6, 28, 496, 8128, …), generated via Euclid IX.36 on Summa Part I, fol. 8r whenever the geometric sum 2^n - 1 is prime.

Per // A — Debit // Credit Journal Prepositions

Language: Venetian Italian

Canonical Venetian Giornale markers explained in Cap. 11 (fol. 201r): ‘Per’ designates the debtor account first, separated by two oblique strokes ‘//’ from ‘A’, which designates the creditor account.

più (.p̄.) e meno (.m̄.) — Plus (+) and Minus (−) Operators

Language: Italian

Syncopated operational signs standardized in print across Summa Dist. VIII for additive (.p̄.) and subtractive (.m̄.) terms, accompanied by the four sign laws of multiplication (‘meno via meno fa più’).

possanza (potentia) — Square of a Line Segment (Quadrance)

Language: Italian / Latin

The area of the square on a segment (Euclidean dunamis / Wildberger quadrance): e.g., a sphere whose axis has possanza 48 (length √48) on Part II, fol. 70r.

pro e danno (utile e danno) — Profit and Loss Summary Account

Language: Venetian / Italian

Summary nominal account (Cap. 27) collecting merchandise gains on Avere and losses/expenses on Dare before transferring net income into Cavedal.

proportionalità (proportio / analogia) — Similitude of Ratios (Proportion)

Language: Italian / Latin

Equality or similitude of two or more ratios (e.g., continuous a:b = b:c or discontinuous a:b = c:d), governed by the six Euclidean modi arguendi on fol. 74r.

proportione (ratio / habitudo) — Ratio Between Two Homogeneous Magnitudes

Language: Italian / Latin

In Renaissance Euclidean terminology (fol. 68r–69v), the quantitative relation (habitudo) of one magnitude to another of the same kind—whereas a relation of ratios (a:b = c:d) is called proportionalità.

quaderno grande — General Double-Entry Ledger

Language: Italian

Third principal book indexed by an alphabetical Repertorio (Alfabeto), placing Dare (debits) on the left verso and Avere (credits) on the facing right recto in four currency columns l. s. g. p. (Cap. 13–14).

radice legata / universale (.℞.u.) — Compound Radical Grouping

Language: Italian

Pacioli’s prefix radical operator .℞.u. governing an entire binomial or polynomial radicand (playing the grouping role of the later vinculum over √(…)).

residuo (apotome) — Euclidean Apotome Irrational

Language: Italian / Greek-Latin

A subtractive compound irrational such as √125 − 5 or 15 − √125 (the two segments of 10 divided in extreme and mean ratio), classified into six species following Euclid X.73.

summa summarum — Grand Trial Balance Equality

Language: Latin

Grand total of all Dare postings and grand total of all Avere postings across the entire Quaderno (Cap. 34, 36); their exact equality confirms structural conservation before closing.

superparticularis (superparticolare) — Superparticular Genus ((n + 1) : n)

Language: Latin / Italian

Second Boethian genus in which the greater term contains the lesser once plus a single aliquot part 1/n: sesquialtera (3:2, musical fifth), sesquitertia (4:3, musical fourth), sesquioctava (9:8, whole tone).

superpartiens (superpartiente) — Superpartient Genus ((n + r) : n, 1 < r < n)

Language: Latin / Italian

Third Boethian genus in which the greater term contains the lesser once plus r > 1 proper parts (e.g., 5:3 = superbipartiens tertias).