Glossary of Geometric, Architectural & Latin Terms

Multilingual Lexicon of Renaissance Mathematical & Architectural Terminology

Technical terms in Italian, Latin, Greek, and German across Fra Luca Pacioli’s De Divina Proportione (1509 / 1889).

abaco (abacus) (abaco)

Language: Greek / Latin / Italian | Translation: Abacus of a Capital

The flat slab forming the uppermost member of a column capital immediately beneath the architrave (Architectura Cap. VII, Folio 0213).

Occurrences: Arch. Cap. 4 (EN), Arch. Cap. 4 (IT), Arch. Cap. 7 (IT), Comp. Cap. 57 (IT), Comp. Cap. 70 (IT), Plates: Architectural (IT)

abscisum / absciso / scapezzo (ἀποτετμημένον) (abscisum)

Language: Latin / Italian / Greek | Translation: Truncated Polyhedron

A dependent (Archimedean) polyhedron derived from a regular body by cutting off its vertices at one-third or one-half of each edge (generating the truncated tetrahedron, cuboctahedron, truncated octahedron, truncated icosahedron, and icosidodecahedron).

Occurrences: Comp. Cap. 47 (Notes), Comp. Cap. 48 (EN), Comp. Cap. 48 (Notes), Comp. Cap. 49 (Notes), Comp. Cap. 50 (Notes), Comp. Cap. 51 (Notes), Comp. Cap. 52 (IT), Comp. Cap. 52 (Notes), Comp. Cap. 53 (IT), Comp. Cap. 68 (Notes), Comp. Cap. 69 (EN), Comp. Cap. 69 (IT), Winterberg Intro (EN), Winterberg Anmerkungen (EN), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

acroterio / regolo (acroterio)

Language: Greek / Italian | Translation: Acroterium / Crowning Fillet

Pedestal at the apex and eaves of a pediment, or the uppermost crowning fillet (regolo) at the top of the cornice on Folio 0215 (Architectura Cap. XIV, XVI).

algebra e almucabala (regola de la cosa) (algebra-e-almucabala)

Language: Arabic / Italian | Translation: Algebra and Almucabala (The Rule of the Thing)

The speculative practice of restoration (al-jabr) and balancing (al-muqābala) of equations, termed in the Italian vernacular ‘la regola de la cosa’ (Chapter VIII).

Occurrences: Comp. Cap. 52 (IT)

ambliogonio (triangulo obtusangulo) (ambliogonio)

Language: Greek / Italian | Translation: Obtuse-Angled Triangle (Amblygonion)

A triangle containing one obtuse angle greater than 90° (Compendio Cap. LX).

angulo acuto (angulo-acuto)

Language: Italian / Latin | Translation: Acute Angle (< 90°)

A plane angle smaller than a right angle (Euclid I.def.12; Compendio Cap. LX).

angulo dela iustitia (angolo recto) (angulo-dela-iustitia)

Language: Italian | Translation: Angle of Justice (Right Angle, 90°)

Pacioli’s name in Chapter LIV for the invariant 90° right angle determined by the plummet and square, without which good cannot be distinguished from bad in architecture and measurement.

Occurrences: Comp. Cap. 54 (EN)

angulo obtuso (angulo-obtuso)

Language: Italian / Latin | Translation: Obtuse Angle (> 90°)

A plane angle greater than a right angle (Euclid I.def.11; Compendio Cap. LX).

angulo recto (angulo-recto)

Language: Italian / Latin | Translation: Right Angle (90°)

An angle formed when a straight line standing on another makes the adjacent angles equal (Euclid I.def.10); called by Pacioli the ‘Angle of Justice’ (Compendio Cap. LIV, LX).

angulo solido (angulo-solido)

Language: Italian / Latin | Translation: Solid Angle (Polyhedral Vertex)

An angle formed by the meeting of three or more plane angles at a single point not lying in the same plane; the sum of its plane angles must be strictly less than four right angles (360°, Euclid XI.21).

Occurrences: Comp. Cap. 25 (EN), Comp. Cap. 25 (IT), Comp. Cap. 25 (Notes), Comp. Cap. 27 (EN), Comp. Cap. 28 (EN), Comp. Cap. 29 (EN), Comp. Cap. 30 (EN), Comp. Cap. 34 (EN), Comp. Cap. 35 (EN), Comp. Cap. 36 (IT), Comp. Cap. 37 (EN), Comp. Cap. 38 (EN), Comp. Cap. 40 (EN), Comp. Cap. 43 (EN), Comp. Cap. 45 (EN), Comp. Cap. 46 (EN), Comp. Cap. 48 (EN), Comp. Cap. 50 (IT), Comp. Cap. 50 (Notes), Comp. Cap. 53 (EN), Comp. Cap. 55 (EN), Comp. Cap. 57 (EN)

Arbor Proportio et Proportionalitas (arbor-proportionis)

Language: Latin | Translation: Tree of Ratio and Proportion

The full-page dichotomous classification diagram printed in red and black on Folio 0311, dividing continuous and discrete quantity into rational and irrational ratios and branching into the five species of commensurate ratios (multiplex, superparticularis, superpartiens, multiplex superparticularis, multiplex superpartiens).

Occurrences: Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

archipendolo / perpendiculo (piombo) (archipendolo)

Language: Italian / Latin | Translation: Plumb-Level (Plummet and A-Frame Level)

Instrument using a weighted plumb-line (catheto) to determine true verticality and horizontal leveling in building (Compendio Cap. LIV, LXXI).

arco (arcus) (arco)

Language: Italian / Latin | Translation: Circular Arc

A portion of the circumference of a circle cut off by a chord (Compendio Cap. LXII, LXXI).

arithmetica

Language: Greek / Latin / Italian | Translation: Arithmetic (Science of Discrete Quantity)

The mathematical discipline treating of discrete quantity (number) in itself and its ratios (Compendio Cap. III).

Occurrences: Arch. Prohemio (IT), Comp. Cap. 2 (IT), Comp. Cap. 3 (IT), Comp. Cap. 4 (IT), Comp. Cap. 10 (IT), Comp. Cap. 19 (IT), Comp. Cap. 71 (IT), Plates: Architectural (IT)

astragalo / tondino (astragalo)

Language: Greek / Italian | Translation: Astragal / Bead Molding (Tondino)

Small convex semicircular molding with fillet at the top of a column shaft beneath the capital necking, or dividing fasciae of the architrave (Architectura Cap. VII, Folio 0213, 0215).

astronomia / astrologia (astronomia)

Language: Greek / Latin / Italian | Translation: Astronomy (Science of Celestial Motion)

The mathematical science of mobile continuous magnitude and the revolutions of the celestial spheres (Compendio Cap. III).

Occurrences: Comp. Cap. 1 (IT), Comp. Cap. 3 (IT)

binomio

Language: Italian / Latin | Translation: Binomial Irrational (a + √b)

The sum of two segments commensurable only in square (Euclid X.36, X.48–53), conjugate to the apotome (residuo).

capitello

Language: Italian | Translation: Column Capital

The crowning head of a column shaft, whose form distinguishes the Doric, Ionic (pulvinated), and Corinthian orders (Architectura Cap. IV, VII, Folio 0213).

Occurrences: Arch. Cap. 4 (IT), Arch. Cap. 5 (IT), Arch. Cap. 6 (IT), Arch. Cap. 7 (IT), Arch. Cap. 8 (IT), Arch. Cap. 12 (IT), Comp. Cap. 57 (IT), Plates: Architectural (IT)

catheto / perpendiculare (catheto)

Language: Greek / Italian | Translation: Cathetus / Perpendicular Altitude (κάθετος)

Defined in Chapter LXXI: a plumbed line falling at right angles upon a plane or base line, or the altitude dropped from the apex of a triangle or pyramid to its base.

Occurrences: Arch. Cap. 1 (EN), Arch. Cap. 1 (IT), Arch. Cap. 2 (IT), Arch. Cap. 20 (DE), Arch. Cap. 20 (EN), Arch. Cap. 20 (IT), Comp. Cap. 26 (IT), Comp. Cap. 27 (IT), Comp. Cap. 28 (IT), Comp. Cap. 29 (IT), Comp. Cap. 29 (Notes), Comp. Cap. 30 (IT), Comp. Cap. 31 (IT), Comp. Cap. 33 (IT), Comp. Cap. 71 (IT), Winterberg Anmerkungen (EN)

Ceneraccio (ceneraccio)

Language: Italian | Translation: Ash-Lye Bundle (Annibale Caro’s Epithet for Pacioli’s Prose Style)

Satirical term coined by the humanist Annibale Caro (referring to the coarse cloth bundle of wood-ash used in boiling laundry lye) to describe Fra Luca Pacioli’s unpolished, hybrid Latin-Tuscan-Lombard vernacular prose, beneath which precious mathematical and artistic jewels lie hidden (Winterberg Einleitung).

Occurrences: Winterberg Intro (DE), Winterberg Intro (EN)

censo (ce.) (censo)

Language: Italian | Translation: The Square of the Unknown (x²) in Cossic Algebra

Renaissance abacus term (translating Arabic māl / Latin census) for the square of the unknown quantity.

censo de censo (ce. ce.) (censo-de-censo)

Language: Italian | Translation: Fourth Power of the Unknown (x⁴) in Cossic Algebra

Abacus algebraic term for the square of the square (x⁴), extensively used by Piero della Francesca and Pacioli in the biquadratic problems of the Libellus.

Occurrences: Arch. Cap. 3 (EN), Arch. Cap. 3 (IT), Arch. Cap. 3 (Notes)

centro

Language: Greek / Italian | Translation: Center (κέντρον)

Defined in Chapter LXXI: the middle point inside a circle or sphere from which all straight lines drawn to the circumference are equal.

cimento e copella (cimento-e-copella)

Language: Italian | Translation: Touchstone and Assaying-Cup (Cupel)

Metallurgical metaphor used by Pacioli in Chapters I–II for the mathematical disciplines as the infallible test of intellectual rigor (‘aurum probatur igni et ingenium mathematicis’).

Occurrences: Comp. Cap. 1 (EN), Comp. Cap. 1 (IT)

circino / sesto (compasso) (circino)

Language: Latin / Italian | Translation: Pair of Compasses (Dividers)

Along with the straightedge/ruler (regolo/norma), one of the two universal instruments of geometry and all mechanical arts, generating the circle and transferring proportions (Compendio Cap. LIV; Alphabet Folios 0139–0183).

Occurrences: Arch. Cap. 6 (EN), Arch. Cap. 9 (EN), Arch. Cap. 9 (IT), Arch. Cap. 11 (EN), Arch. Cap. 11 (IT), Arch. Prohemio (EN), Arch. Prohemio (IT)

circonferentia / peripheria (circonferentia)

Language: Latin / Italian / Greek | Translation: Circumference / Periphery of a Circle

The curved boundary line of a circle, whose ratio to its diameter was shown by Archimedes to lie between 3 10/71 and 3 1/7 (approx. 22:7; Compendio Cap. LXII).

Occurrences: Plates: Alphabet (IT)

circulo / cerchio (circulo)

Language: Latin / Italian | Translation: Circle

A plane figure bounded by a single curved line (circumference) such that all straight lines drawn to it from an interior center are equal (Euclid I.def.15).

colonna corinthia (colonna-corinthia)

Language: Italian | Translation: Corinthian Column and Order

The most slender and ornate of the three classical orders, invented by Callimachus at Corinth in imitation of maidenly grace and an acanthus-wreathed basket over a maiden’s tomb, with a capital height equal to an entire lower shaft diameter (Architectura Cap. III–IV, VII; Folio 0215).

colonna dorica (colonna-dorica)

Language: Italian | Translation: Doric Column and Order

The first and most virile of the three classical Greek orders, modeled on the naked proportion of a man (6:1 or 7:1 height-to-diameter), built without a base in ancient Greek temples and crowned by a plain echinus-and-abacus capital beneath a triglyph frieze (Architectura Cap. III–IV, VII, XII).

Columna laterata exagona (Κίων πλευρώδης ἑξάγωνος) (colonna-exagona)

Language: Latin / Greek / Italian | Translation: Hexagonal Prismatic Column (Hexagonal Prism)

Six-sided right prism with parallel regular hexagonal bases and six rectangular lateral faces (Compendio Cap. LXI, Plates XLVII–XLVIII).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

colonna ionica (pulvinata) (colonna-ionica)

Language: Italian | Translation: Ionic Column and Order

The middle classical order, modeled on matronly slenderness (8:1 height-to-diameter), furnished with a base with scotia and tori, fluted shaft, and a voluted (‘pulvinated’) capital mimicking coiled hair (Architectura Cap. III, V, VII; Folio 0213).

colonna laterata (columna laterata / κίων πλευρώδης) (colonna-laterata)

Language: Italian / Latin / Greek | Translation: Prismatic (Polygonal) Column / Prism

An oblong solid whose two opposite bases are congruent rectilinear polygons (triangles, squares, pentagons, hexagons) connected by rectangular lateral faces (Compendio Cap. LVIII–LXI, LXIII).

Occurrences: Arch. Cap. 9 (EN), Comp. Cap. 59 (EN), Comp. Cap. 63 (EN), Comp. Cap. 65 (IT), Comp. Cap. 67 (IT)

Columna laterata pentagona (Κίων πλευρώδης πεντάγωνος) (colonna-pentagona)

Language: Latin / Greek / Italian | Translation: Pentagonal Prismatic Column (Pentagonal Prism)

Five-sided right prism with parallel regular pentagonal bases and five rectangular lateral faces (Compendio Cap. LXI, Plates XLV–XLVI).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Columna laterata quadrilatera (Κίων πλευρώδης τετράπλευρος) (colonna-quadrilatera)

Language: Latin / Greek / Italian | Translation: Quadrangular Prismatic Column (Square Prism)

Four-sided right prism with parallel square bases and four oblong rectangular lateral faces (Compendio Cap. LIX–LXI, Plates XLIII–XLIV).

colonna rotonda (columna rotunda / κίων στρογγύλος) (colonna-rotonda)

Language: Italian / Latin / Greek | Translation: Round Column / Right Circular Cylinder

A cylinder generated by the rotation of a rectangle about one of its stationary vertical sides, bounded by two parallel circular bases (Compendio Cap. LVIII, LXII; Plate LVII).

Occurrences: Arch. Cap. 3 (IT), Arch. Cap. 4 (EN), Arch. Cap. 7 (IT), Arch. Cap. 9 (IT), Arch. Cap. 20 (IT), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

cono dela pyramide (cono-pyramidale)

Language: Italian | Translation: Apex (Vertex) of a Pyramid or Cone

Defined in Chapter LXXI: the supreme summit point of a pyramid where all lateral edges ascending from its base concur.

Occurrences: Comp. Cap. 65 (EN), Comp. Cap. 71 (IT)

corausto (coraustus) (corausto)

Language: Latin / Italian | Translation: Coraustus (Upper Horizontal Chord Joining Erected Perpendiculars)

Defined in Chapter LXXI (from Campanus on Euclid XIII.16): a straight line joining the upper extremities of two equal perpendiculars (catheti) erected upon a base plane, forming the upper horizontal ring of the icosahedron.

Occurrences: Comp. Cap. 26 (Notes), Comp. Cap. 45 (EN), Comp. Cap. 48 (Notes), Comp. Cap. 71 (EN), Comp. Cap. 71 (IT), Comp. index

corda (chorda) (corda)

Language: Italian / Latin | Translation: Chord of a Circle or Polygon

A straight line joining the endpoints of a circular arc or two non-adjacent vertices of a polygon (Compendio Cap. XVIII, LXII, LXXI).

corda pentagonica (corda-pentagonica)

Language: Italian | Translation: Pentagonal Chord (Diagonal of a Regular Pentagon)

Defined in Chapters XVIII and LXXI: a straight line drawn inside a regular pentagon from any vertex to an opposite non-adjacent vertex; two intersecting pentagonal chords divide each other in the Divine Proportion (Euclid XIII.8).

Occurrences: Comp. Cap. 41 (EN), Comp. Cap. 46 (EN), Comp. Cap. 71 (IT)

cornice / cornicione (cornicione)

Language: Italian | Translation: Cornice (Crowning Entablature Molding)

The projecting uppermost assembly of the entablature above the frieze, comprising dentils, corona (dripstone), echinus (ovolo), and acroterium/regolo (Architectura Cap. XIV, XVI, Folio 0215).

Occurrences: Arch. Cap. 16 (EN), Arch. Cap. 16 (IT), Arch. Cap. 20 (IT), Arch. index, Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

corona / gociolatoro (corona)

Language: Latin / Italian | Translation: Corona / Dripstone (Gocciolatoio)

The boldly projecting vertical-faced member of the cornice that throws rainwater clear of the frieze and architrave below (Architectura Cap. XVI, Folio 0215).

Occurrences: Plates: Architectural (IT)

corpi dependenti (corpi-dependenti)

Language: Italian | Translation: Dependent (Archimedean Truncated and Stellated/Elevated) Bodies

Semi-regular and stellated polyhedra generated systematically from the five regular solids by uniform truncation of edges (‘abscisi / scapezzi’) or by erecting pyramids upon every face (‘elevati / pontuti’).

Occurrences: Comp. Cap. 5 (EN), Comp. Cap. 47 (Notes)

corpi regulari (quinque corpora regularia) (corpi-regulari)

Language: Italian / Latin | Translation: The Five Regular (Platonic) Bodies

The five convex polyhedra bounded by congruent equilateral and equiangular polygons meeting in congruent solid angles, each inscribable and circumscribable in a sphere (Euclid XIII.18): tetrahedron, hexahedron (cube), octahedron, icosahedron, and dodecahedron.

Occurrences: Arch. Cap. 6 (IT), Arch. Cap. 9 (IT), Arch. Cap. 12 (IT), Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 19 (IT), Comp. Cap. 23 (IT), Comp. Cap. 24 (IT), Comp. Cap. 25 (IT), Comp. Cap. 26 (IT), Comp. Cap. 32 (IT), Comp. Cap. 46 (IT), Comp. Cap. 47 (IT), Comp. Cap. 55 (IT), Comp. Cap. 57 (IT), Comp. Cap. 69 (IT)

corpo seratile (columna laterata triangula / σχῆμα πριστόν) (corpo-seratile)

Language: Italian / Latin / Greek | Translation: Triangular Prism (Sawn Body)

A five-faced prism with two parallel triangular bases and three rectangular lateral faces, called ‘seratile’ (from Latin serratus, ‘sawn’) because it is obtained by sawing a parallelepiped diagonally in half (Euclid XI.def.13, XII.7; Compendio Cap. LIX, LXVII; Plates XLI–XLII).

Occurrences: Comp. Cap. 41 (IT), Comp. Cap. 59 (EN), Comp. Cap. 59 (IT), Comp. Cap. 59 (Notes), Comp. Cap. 67 (EN), Comp. Cap. 67 (IT), Comp. Cap. 67 (Notes), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

corpo solido (solidum / στερεόν) (corpo-solido)

Language: Italian / Latin / Greek | Translation: Three-Dimensional Solid Body

That which has length, breadth, and depth (altitude), bounded by plane or curved surfaces (Euclid XI.def.1–2).

cosa (co.) (cosa)

Language: Italian | Translation: The Unknown Quantity (x) in Cossic Algebra

Renaissance abacus term (translating Arabic shay’ / Latin res) for the first power of the unknown in algebraic equations.

cosmographia

Language: Greek / Latin / Italian | Translation: Cosmography / Mathematical Geography

The mathematical description and projection of the terrestrial globe in correspondence with the celestial spheres, practiced by Ptolemy, Strabo, and Marinus (Compendio Cap. II–III).

cotozzo (occipite) (cotozzo)

Language: Italian | Translation: Nape / Occiput of the Head

The rear occipital point of the human skull forming the apex of the equilateral triangle that governs the facial profile on Folio 0137 (Architectura Cap. I–II).

Occurrences: Arch. Cap. 2 (EN), Arch. Cap. 2 (IT), Arch. index

crosiera (intersecatione de li diametri) (crosiera)

Language: Italian | Translation: Cross-Intersection of Diagonals and Circle

In Pacioli’s Roman geometric alphabet (Folios 0139–0183), the key construction points where the diagonals of the bounding square intersect one another or the inscribed circle.

Occurrences: Plates: Alphabet (IT)

cubo (cu.) (cubo-algebraico)

Language: Italian | Translation: The Cube of the Unknown (x³) in Cossic Algebra

Abacus algebraic term for the third power of the unknown quantity, produced by multiplying the square (censo) by the root (cosa) (Tabula Folio 0016, Libellus).

decagono equilatero (decagono)

Language: Greek / Italian | Translation: Regular Decagon

A ten-sided equilateral polygon inscribed in a circle; its side is the greater segment when the radius (side of the inscribed hexagon) is divided according to the Divine Proportion (Euclid XIII.9–10; Compendio Cap. XV–XVI, XXIX).

Occurrences: Comp. Cap. 16 (IT), Comp. Cap. 17 (IT), Comp. Cap. 20 (IT), Comp. Cap. 29 (IT), Comp. Cap. 31 (IT), Comp. Cap. 71 (IT)

denticeli / rastro (denticeli)

Language: Latin / Italian | Translation: Dentils (‘Rake’ Molding)

Series of small rectangular tooth-like blocks beneath the corona of an Ionic or Corinthian cornice, called ‘rastro’ (rake) by Renaissance stonecutters (Folio 0215).

Occurrences: Plates: Architectural (IT)

diagonale

Language: Greek / Italian | Translation: Diagonal of a Quadrate or Parallelogram

Defined in Chapter LXXI: the straight line drawn from one angle to the opposite angle of a quadrilateral figure, dividing it into two triangles.

Occurrences: Arch. Cap. 4 (DE), Arch. Cap. 4 (IT), Comp. Cap. 27 (Notes), Comp. Cap. 35 (DE), Comp. Cap. 67 (DE), Comp. Cap. 71 (DE), Winterberg Anmerkungen (DE)

divina proportione (divina-proportione)

Language: Italian / Latin | Translation: Divine Proportion (Golden Ratio, φ = (1 + √5)/2)

Pacioli’s theological and mathematical name for extreme and mean ratio (Euclid VI.def.3, II.11, VI.30, XIII.1–18), possessing five analogies with God: unity, trinity of terms, irrationality/ineffability, self-similarity (invariability), and formal generation of the cosmic dodecahedron and the five Platonic solids.

Occurrences: Arch. Cap. 6 (DE), Arch. Cap. 18 (DE), Arch. Cap. 18 (EN), Arch. Cap. 19 (EN), Arch. Prohemio (DE), Comp. Cap. 1 (DE), Comp. Cap. 5 (DE), Comp. Cap. 5 (Notes), Comp. Cap. 7 (DE), Comp. Cap. 7 (EN), Comp. Cap. 12 (DE), Comp. Cap. 12 (EN), Comp. Cap. 13 (DE), Comp. Cap. 13 (EN), Comp. Cap. 20 (DE), Comp. Cap. 20 (EN), Comp. Cap. 22 (DE), Comp. Cap. 22 (EN), Comp. Cap. 24 (EN), Comp. Cap. 25 (EN), Comp. Cap. 29 (EN), Comp. Cap. 30 (EN), Comp. Cap. 31 (EN), Comp. Cap. 35 (Notes), …and 12 more

Dodecahedron / duodecedron (corpo de dodeci pentagoni) (dodecahedron)

Language: Latin / Greek / Italian | Translation: Regular Dodecahedron (12-Base Pentagonal Body, Figure of the Cosmos)

The fifth and noblest regular solid, bounded by 12 regular pentagons, 30 edges, and 20 three-valent solid angles; constructed in Euclid XIII.17 via the Divine Proportion (Compendio Cap. XXX, Plates XXVII–XXXIV) and assigned by Plato to the Fifth Essence (Heaven).

Occurrences: Comp. Cap. 4 (Notes), Comp. Cap. 5 (DE), Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 5 (Notes), Comp. Cap. 18 (Notes), Comp. Cap. 19 (Notes), Comp. Cap. 22 (DE), Comp. Cap. 22 (EN), Comp. Cap. 22 (IT), Comp. Cap. 24 (EN), Comp. Cap. 24 (Notes), Comp. Cap. 25 (DE), Comp. Cap. 25 (EN), Comp. Cap. 25 (IT), Comp. Cap. 30 (DE), Comp. Cap. 30 (EN), Comp. Cap. 30 (IT), Comp. Cap. 30 (Notes), Comp. Cap. 31 (DE), Comp. Cap. 31 (EN), Comp. Cap. 31 (IT), Comp. Cap. 31 (Notes), Comp. Cap. 32 (DE), …and 64 more

Dodecahedron abscisum (Δωδεκάεδρον ἀποτετμημένον / Icosidodecahedron) (dodecahedron-abscisum)

Language: Latin / Greek | Translation: Icosidodecahedron (32-Base Truncated Dodecahedron)

Archimedean quasi-regular solid obtained by bisecting the thirty edges of a regular dodecahedron, bounded by 12 regular pentagons and 20 equilateral triangles, with 60 edges and 30 vertices (Compendio Cap. LII, Plates XXIX–XXX).

Occurrences: Comp. Cap. 46 (Notes), Comp. Cap. 52 (EN), Comp. Cap. 52 (Notes), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Dodecahedron abscisum elevatum (Δωδεκάεδρον ἀποτετμημένον ἐπηρμένον) (dodecahedron-abscisum-elevatum)

Language: Latin / Greek | Translation: Elevated Icosidodecahedron

Thirty-two-pointed stellated polyhedron formed by erecting 12 pentagonal pyramids and 20 triangular pyramids of equilateral triangles upon the 32 faces of a core icosidodecahedron, bounded externally by 120 equilateral triangles (Compendio Cap. LII, Plates XXXIII–XXXIV).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Dodecahedron elevatum (Δωδεκάεδρον ἐπηρμένον) (dodecahedron-elevatum)

Language: Latin / Greek | Translation: Elevated (Stellated) Dodecahedron

Twelve-pointed stellated polyhedron formed by erecting 12 pentagonal pyramids of equilateral triangles upon the 12 pentagonal faces of a core regular dodecahedron, bounded externally by 60 equilateral triangles and requiring the Divine Proportion at every vertex (Compendio Cap. LII, Plates XXXI–XXXII).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

dyametro

Language: Greek / Italian | Translation: Diameter (διάμετρος)

Defined in Chapter LXXI: a straight line passing through the center of a circle or sphere and terminated at both ends by the circumference, bisecting the figure.

Occurrences: Arch. Cap. 7 (IT), Comp. Cap. 26 (IT), Comp. Cap. 27 (IT), Comp. Cap. 31 (IT), Comp. Cap. 32 (IT), Comp. Cap. 56 (IT), Comp. Cap. 58 (IT), Comp. Cap. 62 (IT), Comp. Cap. 66 (IT), Comp. Cap. 67 (IT), Comp. Cap. 71 (IT)

echino / huovolo (echino)

Language: Greek / Italian | Translation: Echinus / Ovolo (Egg-and-Dart Molding)

Convex quarter-round classical molding, called ‘echino’ by the ancients and ‘huovolo’ (egg-molding) by Renaissance masons (Folio 0215).

Occurrences: Arch. Cap. 6 (EN), Arch. Cap. 6 (IT), Arch. Cap. 14 (EN), Arch. Cap. 14 (IT), Plates: Architectural (IT)

effecto (effetto)

Language: Italian | Translation: Miraculous Mathematical Effect / Theorem

Pacioli’s term in Chapters VII–XXIII for the thirteen passion-like mathematical properties (‘mirabili, ineffabili, inestimabili effecti’) of a line divided according to the Divine Proportion, derived from Euclid XIII.1–6 and XIV.2.

elevatum / elevato / pontuto (ἐπηρμένον) (elevatum)

Language: Latin / Italian / Greek | Translation: Elevated / Stellated (Cumulated) Polyhedron

A stellated polyhedron formed by erecting equilateral triangular pyramids upon every polygonal face of a core convex polyhedron (such as the elevated octahedron, later rediscovered by Kepler as the Stella Octangula).

Occurrences: Comp. Cap. 48 (EN), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

elmuariffe (trapezium) (elmuariffe)

Language: Arabic / Italian | Translation: Irregular Quadrilateral / Trapezium

Campanus’s Arabic-derived term (from al-munḥarif) used by Pacioli in Chapter LX for irregular four-sided plane figures that are neither squares, rectangles, rhombi, nor rhomboids.

Occurrences: Comp. Cap. 60 (DE), Comp. Cap. 60 (EN), Comp. Cap. 60 (IT), Comp. Cap. 60 (Notes), Comp. Cap. 63 (EN)

elmuhaym (rhombus) (elmuhaym)

Language: Arabic / Italian | Translation: Rhombus (Equilateral Non-Right-Angled Quadrilateral)

Campanus’s Arabic-derived term (from al-mu’ayyan) used by Pacioli in Chapter LX for an equilateral four-sided figure whose angles are not right angles.

Occurrences: Comp. Cap. 60 (Notes)

epistilio / architrave (epistilio)

Language: Greek / Latin / Italian | Translation: Epistyle / Architrave

The lowest of the three principal horizontal members of a classical entablature, resting directly upon the capitals of the columns to bind them together; divided into fasciae and crowned by the taenia/cymatium (Architectura Cap. XIV, Folio 0215).

Occurrences: Arch. Cap. 6 (EN), Arch. Cap. 6 (IT), Arch. Cap. 12 (EN), Arch. Cap. 12 (IT), Arch. Cap. 13 (EN), Arch. Cap. 13 (IT), Arch. Cap. 13 (Notes), Arch. Cap. 14 (EN), Arch. Cap. 14 (IT), Arch. Cap. 14 (Notes), Arch. Cap. 15 (EN), Arch. Cap. 15 (IT), Arch. Cap. 16 (EN), Arch. Cap. 16 (IT), Arch. Cap. 17 (IT), Arch. Cap. 20 (IT), Arch. Prohemio (IT), Arch. index, Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

equilatero (triangulo equilatero / ysopleuro) (equilatero)

Language: Italian / Greek | Translation: Equilateral Triangle (Isopleuron)

A three-sided plane figure with three equal sides and three 60° angles (Euclid I.1), serving as the elementary face of the tetrahedron, octahedron, and icosahedron, and as Plato’s geometric atom of fire, air, and water (Compendio Cap. XXIV, XXV, LX).

Occurrences: Comp. Cap. 16 (IT), Comp. Cap. 17 (IT), Comp. Cap. 18 (IT), Comp. Cap. 20 (IT), Comp. Cap. 22 (IT), Comp. Cap. 25 (IT), Comp. Cap. 26 (IT), Comp. Cap. 29 (IT), Comp. Cap. 30 (IT), Comp. Cap. 61 (IT), Plates: Architectural (IT)

exagono equilatero (exagono)

Language: Greek / Italian | Translation: Regular Hexagon

A six-sided equilateral polygon whose side equals the radius (semidiameter) of its circumscribed circle (Euclid IV.15); when the side of the hexagon is divided in the Divine Proportion, its greater segment is the side of the regular decagon in the same circle (Euclid XIII.9; Compendio Cap. XV–XVI).

Occurrences: Comp. Cap. 16 (IT), Comp. Cap. 17 (IT), Comp. Cap. 20 (IT), Comp. Cap. 25 (IT), Comp. Cap. 29 (IT), Comp. Cap. 31 (IT), Comp. Cap. 51 (IT), Comp. Cap. 71 (IT)

Exahedron sive Cubus / exacedron (cubo) (exahedron)

Language: Latin / Greek / Italian | Translation: Regular Hexahedron or Cube (6-Base Square Body, Figure of Earth)

The second regular solid, bounded by 6 square faces, 12 edges, and 8 three-valent solid angles; constructed in Euclid XIII.15 (Compendio Cap. XXVII, Plates VII–XIV) and assigned by Plato to the element of Earth.

Occurrences: Comp. Cap. 5 (DE), Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 25 (EN), Comp. Cap. 27 (EN), Comp. Cap. 30 (EN), Comp. Cap. 33 (DE), Comp. Cap. 33 (IT), Comp. Cap. 36 (DE), Comp. Cap. 36 (EN), Comp. Cap. 36 (IT), Comp. Cap. 37 (DE), Comp. Cap. 37 (EN), Comp. Cap. 37 (IT), Comp. Cap. 38 (EN), Comp. Cap. 41 (EN), Comp. Cap. 44 (EN), Comp. Cap. 46 (EN), Comp. Cap. 49 (DE), Comp. Cap. 49 (EN), Comp. Cap. 49 (IT), Comp. Cap. 49 (Notes), Comp. Cap. 52 (EN), Comp. Cap. 53 (DE), …and 11 more

Exahedron abscisum (Ἑξάεδρον ἀποτετμημένον / Cuboctahedron) (exahedron-abscisum)

Language: Latin / Greek | Translation: Cuboctahedron (14-Base Truncated Cube)

Archimedean quasi-regular solid obtained by bisecting all twelve edges of a cube, bounded by 6 squares and 8 equilateral triangles, with 24 edges and 12 four-valent vertices (Compendio Cap. XLIX, Plates IX–X).

Occurrences: Comp. Cap. 49 (EN), Comp. Cap. 49 (Notes), Comp. Cap. 50 (EN), Comp. Cap. 52 (EN), Comp. Cap. 52 (Notes), Comp. Cap. 54 (Notes), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Exahedron abscisum elevatum (Ἑξάεδρον ἀποτετμημένον ἐπηρμένον) (exahedron-abscisum-elevatum)

Language: Latin / Greek | Translation: Elevated Cuboctahedron

Compound stellated polyhedron formed by erecting 6 square pyramids and 8 triangular pyramids upon the 14 faces of a core cuboctahedron, bounded externally by 48 equilateral triangles (Plates XIII–XIV).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Exahedron elevatum (Ἑξάεδρον ἐπηρμένον) (exahedron-elevatum)

Language: Latin / Greek | Translation: Elevated (Stellated) Hexahedron / Cube

Stellated polyhedron formed by erecting 6 square-based pyramids of equilateral triangles upon the 6 square faces of a core cube, bounded externally by 24 equilateral triangles (Compendio Cap. L, Plates XI–XII).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

proportione avente el mezzo e doi extremi (extrema-et-media-ratio)

Language: Italian / Latin | Translation: Proportion having a mean and two extremes (Extreme and Mean Ratio)

The technical Euclidean definition (Elements VI.def.3) whereby a magnitude AB is divided at C into unequal parts AC > CB such that AB : AC = AC : CB, and AB · CB = AC².

Occurrences: Comp. Cap. 5 (Notes), Comp. Cap. 7 (Notes), Comp. Cap. 8 (Notes), Comp. Cap. 11 (Notes), Comp. Cap. 12 (Notes), Comp. Cap. 13 (Notes), Comp. Cap. 14 (Notes), Comp. Cap. 15 (Notes), Comp. Cap. 16 (Notes), Comp. Cap. 18 (Notes), Comp. Cap. 20 (Notes), Comp. Cap. 21 (Notes), Comp. Cap. 24 (EN), Comp. Cap. 29 (Notes), Comp. Cap. 30 (EN), Comp. Cap. 30 (Notes), Comp. Cap. 44 (EN), Winterberg Intro (EN), Winterberg Anmerkungen (EN)

fastigio / frontespicio (fastigio)

Language: Latin / Italian | Translation: Pediment / Gable (Fastigium)

The triangular crowning gable above the entablature of a temple facade or portal, enclosing the tympanum (Architectura Cap. VI, XVII; Folio 0185).

Occurrences: Arch. Cap. 6 (EN), Arch. Cap. 6 (IT), Arch. Cap. 14 (IT), Arch. Cap. 17 (EN), Arch. Cap. 17 (IT), Comp. Cap. 54 (IT), Plates: Architectural (IT)

gamba grossa (gamba-grossa)

Language: Italian | Translation: Thick Main Stem of a Roman Capital Letter (1/9 of Height)

In Pacioli’s geometric alphabet, the primary vertical or diagonal downstroke of a Roman capital letter, standardized at one-ninth (1:9) of the side of the bounding square, whereas the thin stroke (gamba sutile) is half or one-third as thick.

Occurrences: Plates: Alphabet (IT)

geometria

Language: Greek / Latin / Italian | Translation: Geometry (Science of Continuous Magnitude)

The mathematical discipline treating of stationary continuous magnitude—points, lines, surfaces, and solid bodies—and the sole domain capable of expressing irrational ratios such as the Divine Proportion (Compendio Cap. III, VI).

Occurrences: Arch. Cap. 19 (IT), Arch. Prohemio (IT), Comp. Cap. 2 (IT), Comp. Cap. 3 (IT), Comp. Cap. 4 (IT), Comp. Cap. 26 (IT), Comp. Cap. 54 (IT), Comp. Cap. 63 (Notes), Comp. Cap. 71 (IT), Plates: Architectural (DE), Plates: Architectural (IT)

gnomone (gnomon) (gnomone)

Language: Greek / Italian | Translation: Geometric Gnomon (L-Shaped Carpenter’s Square Figure)

Defined in Euclid II.def.2 and used throughout Compendio Chapters X–XXII ( following Euclid XIII.1–5): the L-shaped figure composed of one of the two parallelograms about the diagonal of a square together with the two complements.

Goldener Schnitt (sectio aurea / stetige Proportion) (goldener-schnitt)

Language: German | Translation: The Golden Section (Continuous / Extreme and Mean Proportion)

The nineteenth-century German mathematical and aesthetic term (‘Die Lehre vom Goldenen Schnitt’) used by Constantin Winterberg and Adolf Zeising for Pacioli’s Divina Proportione (x : a = (a - x) : x, where x = a(√5 - 1)/2).

Occurrences: Comp. Cap. 7 (Notes), Comp. Cap. 22 (Notes), Winterberg Intro (DE), Winterberg Intro (EN)

corpo humano al tondo e al quadro (homo-ad-circulum)

Language: Italian | Translation: Vitruvian Man Inscribed in the Circle and Square

The harmonic canon of Vitruvius III.1 expounded by Pacioli in Architectura Chapters I–III: with arms and legs extended, the navel is the center of the circumscribed circle; with arms stretched horizontally, the span equals the height from sole to crown, forming an exact square.

Icosahedron / ycocedron (corpo de vinti basi triangulari) (icosahedron)

Language: Latin / Greek / Italian | Translation: Regular Icosahedron (20-Base Triangular Body, Figure of Water)

The fourth regular solid, bounded by 20 equilateral triangles, 30 edges, and 12 five-valent solid angles; constructed in Euclid XIII.16 (Compendio Cap. XXIX, Plates XXI–XXVI) and assigned by Plato to the element of Water.

Occurrences: Arch. Cap. 18 (DE), Arch. Cap. 18 (EN), Arch. index, Comp. Cap. 4 (Notes), Comp. Cap. 5 (DE), Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 5 (Notes), Comp. Cap. 19 (Notes), Comp. Cap. 21 (Notes), Comp. Cap. 24 (EN), Comp. Cap. 24 (Notes), Comp. Cap. 25 (DE), Comp. Cap. 25 (EN), Comp. Cap. 25 (IT), Comp. Cap. 28 (Notes), Comp. Cap. 29 (DE), Comp. Cap. 29 (EN), Comp. Cap. 29 (IT), Comp. Cap. 29 (Notes), Comp. Cap. 31 (DE), Comp. Cap. 31 (EN), Comp. Cap. 31 (IT), Comp. Cap. 31 (Notes), …and 59 more

Icosahedron abscisum (Εἰκοσάεδρον ἀποτετμημένον) (icosahedron-abscisum)

Language: Latin / Greek | Translation: Truncated Icosahedron (32-Base Archimedean Solid)

Archimedean semi-regular solid obtained by trisecting the thirty edges of a regular icosahedron, bounded by 20 regular hexagons and 12 regular pentagons, with 90 edges and 60 vertices (Compendio Cap. LII, Plates XXIII–XXIV).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Icosahedron elevatum (Εἰκοσάεδρον ἐπηρμένον) (icosahedron-elevatum)

Language: Latin / Greek | Translation: Elevated (Stellated) Icosahedron

Twenty-pointed stellated polyhedron formed by erecting 20 triangular pyramids of equilateral triangles upon the 20 faces of a core regular icosahedron, bounded externally by 60 equilateral triangles (Compendio Cap. LII, Plates XXV–XXVI).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

imoscapo (grossezza inferiore dela colonna) (imoscapo)

Language: Italian | Translation: Lower Shaft Diameter of a Column (Imoscapus)

The diameter of a column shaft at its lowest part just above the base fillet, serving as the fundamental module for all architectural proportions (Architectura Cap. IV, VI–VIII).

in agibilibus (in-agibilibus)

Language: Scholastic Latin | Translation: In Practical / Moral Matters of Action

Scholastic philosophical phrase (‘in practical affairs of human action’) contrasted with speculative science (in speculabilibus); glossed by Constantin Winterberg in Compendio Chapter LIV, fn. 1.

Occurrences: Arch. Cap. 19 (EN), Arch. Cap. 19 (IT), Winterberg Intro (DE), Winterberg Intro (EN), Plates: Architectural (IT)

irrationale / incommensurabile (irrationale)

Language: Italian / Latin | Translation: Irrational / Incommensurable Magnitude

A magnitude that has no common measure with a given rational unit (Euclid X.def.1–3) and cannot be expressed by a ratio of whole numbers.

Occurrences: Arch. Cap. 1 (IT), Arch. Cap. 2 (DE), Arch. Cap. 2 (IT), Comp. Cap. 3 (IT), Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 6 (IT), Comp. Cap. 6 (Notes), Comp. Cap. 7 (IT), Comp. Cap. 8 (IT), Comp. Cap. 9 (EN), Comp. Cap. 9 (IT), Comp. Cap. 29 (DE), Comp. Cap. 29 (IT), Comp. Cap. 30 (DE), Comp. Cap. 30 (IT), Comp. Cap. 33 (IT), Comp. Cap. 67 (IT)

isoscele (triangulo de doi lati equali) (isoscele)

Language: Greek / Italian | Translation: Isosceles Triangle

A triangle having two equal sides; in particular, the golden isosceles triangle of Euclid IV.10 (with base angles double the vertex angle, 72°–72°–36°), whose legs and base are in the Divine Proportion (Compendio Cap. XVII, LX).

Kanon (κανών / Proportionsschlüssel) (kanon)

Language: Greek / German | Translation: Artistic Canon of Human Proportions

Rule or standard system of modular ratios governing the proportions of the human figure in sculpture and painting, from Polykleitos, Phidias, and Vitruvius to Alberti, Leonardo da Vinci, Pacioli, Carus, and Zeising (Winterberg Einleitung).

Occurrences: Arch. Cap. 12 (DE), Winterberg Intro (DE), Winterberg Intro (EN)

linea (γραμμή) (linea)

Language: Italian / Latin / Greek | Translation: Line (Length without Breadth)

Breadthless length whose extremities are points (Euclid I.def.2–3), divided by Pacioli into the straight line (ruler) and circular curve (compass).

linea minore (minor) (linea-minore)

Language: Italian / Latin | Translation: Minor Irrational Line (Euclid X.76, XIII.16)

The fourth species of irrational line obtained by subtraction in Euclid Book X (X.76), proven in XIII.16 to be the edge of the regular icosahedron inscribed in a sphere of rational diameter.

metopa

Language: Greek / Latin / Italian | Translation: Metope

The square space between two triglyphs in a Doric frieze, left plain or carved with bucrania and paterae (Architectura Cap. XII, XVII).

multiplex (moltiplice) (multiplex)

Language: Latin / Italian | Translation: Multiple Ratio (n : 1)

The first of the five Boethian species of greater inequality in rational ratios, in which the greater term contains the lesser an exact whole number of times (dupla 2:1, tripla 3:1, quadrupla 4:1; Folio 0311).

Occurrences: Comp. Cap. 5 (Notes)

musica / harmonia (musica)

Language: Greek / Latin / Italian | Translation: Music / Harmonic Science (Sonorous Number)

The mathematical science of discrete quantity in relation to sound and harmonic consonance (sesquialtera 3:2, sesquitertia 4:3, dupla 2:1), paired by Pacioli with Perspective in Chapter III.

norma / squadra (norma)

Language: Latin / Italian | Translation: Carpenter’s and Mason’s Try-Square

Instrument defining the invariant 90° right angle (‘angulo dela iustitia’), indispensable in architecture and stonecutting (Compendio Cap. LIV; Architectura Cap. IX–X).

Occurrences: Arch. Cap. 19 (EN), Arch. Cap. 19 (IT), Comp. Cap. 54 (IT), Comp. Cap. 57 (EN), Comp. Cap. 57 (IT)

octagono equilatero (octagono)

Language: Greek / Italian | Translation: Regular Octagon

An eight-sided equilateral polygon, formed as the cross-section and face of the truncated cube and 26-base body (Compendio Cap. LIII, Libellus).

Occurrences: Comp. Cap. 71 (IT)

Octahedron / octocedron (corpo de octo basi triangulari) (octahedron)

Language: Latin / Greek / Italian | Translation: Regular Octahedron (8-Base Triangular Body, Figure of Air)

The third regular solid, bounded by 8 equilateral triangles, 12 edges, and 6 four-valent solid angles; constructed in Euclid XIII.14 (Compendio Cap. XXVIII, Plates XV–XX) and assigned by Plato to the element of Air.

Occurrences: Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 5 (Notes), Comp. Cap. 25 (DE), Comp. Cap. 25 (EN), Comp. Cap. 25 (IT), Comp. Cap. 27 (Notes), Comp. Cap. 28 (DE), Comp. Cap. 28 (EN), Comp. Cap. 28 (IT), Comp. Cap. 28 (Notes), Comp. Cap. 31 (DE), Comp. Cap. 31 (EN), Comp. Cap. 31 (IT), Comp. Cap. 31 (Notes), Comp. Cap. 32 (DE), Comp. Cap. 32 (IT), Comp. Cap. 33 (DE), Comp. Cap. 33 (EN), Comp. Cap. 33 (IT), Comp. Cap. 34 (DE), Comp. Cap. 34 (EN), Comp. Cap. 34 (Notes), Comp. Cap. 35 (EN), …and 48 more

Octahedron abscisum (Ὀκτάεδρον ἀποτετμημένον) (octahedron-abscisum)

Language: Latin / Greek | Translation: Truncated Octahedron (14-Base Archimedean Solid)

Archimedean semi-regular solid obtained by trisecting the twelve edges of a regular octahedron, bounded by 8 regular hexagons and 6 squares, with 36 edges and 24 vertices (Compendio Cap. LI, Plates XVII–XVIII, Libellus Tract. II).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Octahedron elevatum (Ὀκτάεδρον ἐπηρμένον / Stella Octangula) (octahedron-elevatum)

Language: Latin / Greek | Translation: Elevated Octahedron (Stellated Octahedron / Stella Octangula)

Eight-pointed stellated polyhedron formed by erecting 8 equilateral triangular pyramids upon the 8 faces of a core regular octahedron, bounded externally by 24 equilateral triangles (Compendio Cap. L, Plates XIX–XX; later named Stella Octangula by Johannes Kepler in 1609).

Occurrences: Comp. Cap. 36 (Notes), Comp. Cap. 47 (Notes), Comp. Cap. 48 (Notes), Comp. Cap. 49 (Notes), Comp. Cap. 50 (Notes), Comp. Cap. 51 (Notes), Comp. Cap. 52 (Notes), Winterberg Anmerkungen (EN), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

orthogonio (triangulo rectangulo) (orthogonio)

Language: Greek / Italian | Translation: Right-Angled Triangle (Orthogonion)

A triangle containing one 90° right angle, governed by the Pythagorean theorem (Euclid I.47) and producing the right circular cone when rotated about either of its perpendicular legs (Compendio Cap. LX, LXIV).

oxigonio (triangulo acutangulo) (oxigonio)

Language: Greek / Italian | Translation: Acute-Angled Triangle (Oxygonion)

A triangle all three of whose angles are acute (< 90°) (Compendio Cap. LX).

paralellogramo

Language: Greek / Italian | Translation: Parallelogram

Defined in Chapter LXXI: a four-sided plane figure whose opposite sides are parallel and equidistant (Euclid I.34).

Occurrences: Comp. Cap. 5 (Notes), Comp. Cap. 58 (EN), Comp. Cap. 71 (EN), Comp. index, Winterberg Intro (EN)

pentagono equilatero e equiangolo (pentagono)

Language: Greek / Italian | Translation: Regular Pentagon

The five-sided equilateral and equiangular polygon (Euclid IV.11, XIII.7–11), which cannot be constructed without the Divine Proportion and forms the twelve faces of the cosmic dodecahedron (Compendio Cap. V, XVIII–XX, XXX).

Occurrences: Comp. Cap. 18 (IT), Comp. Cap. 20 (IT), Comp. Cap. 22 (IT), Comp. Cap. 25 (IT), Comp. Cap. 29 (IT), Comp. Cap. 30 (IT), Comp. Cap. 31 (IT), Comp. Cap. 52 (IT), Comp. Cap. 61 (IT)

perspectiva (prospectiva) (perspectiva)

Language: Latin / Italian | Translation: Perspective / Mathematical Optics

Argued by Pacioli in Compendio Chapter III to deserve full rank alongside Music in the mathematical Quadrivium (making five mathematical sciences in all), since sight is the first portal of the intellect and visual rays obey strict arithmetic and geometric proportions.

Occurrences: Comp. Cap. 3 (IT), Comp. Cap. 3 (Notes)

pilastro (columna quadrangula) (pilastro)

Language: Italian / Latin | Translation: Square Pillar / Pier (Pilaster)

A four-sided upright structural support of square or rectangular plan, contrasted with the round column (Compendio Cap. LVIII; Architectura Cap. VIII, XI).

Occurrences: Arch. Cap. 5 (IT), Arch. Cap. 6 (IT)

planum / piano (ἐπίπεδον) (planum)

Language: Latin / Italian / Greek | Translation: Plane / Unstellated Polyhedron

Modifier in Leonardo and Pacioli’s systematic polyhedral nomenclature designating a convex polyhedron with flat polygonal faces, prior to stellation (elevation).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

plintho (plinthus / zoccolo) (plintho)

Language: Greek / Latin / Italian | Translation: Plinth (Square Base Block)

The square lowest slab of a column base resting directly on the stylobate or pavement (Architectura Cap. V, VII, Folio 0213, 0215).

Pol und Polare (reciproke Polarität) (pol-und-polare)

Language: German | Translation: Pole and Polar (Projective Reciprocal Polarity)

Projective geometric duality with respect to a concentric sphere invoked by Constantin Winterberg (Einleitung and Sachliche Anmerkung 5) to explain how the vertices and tangent face-planes of a circumscribed polyhedron correspond reciprocally to the face-planes and vertices of its dual inscribed polyhedron.

Occurrences: Winterberg Intro (DE)

Porta Templi Domini dicta Speciosa (porta-speciosa)

Language: Latin / Italian | Translation: The Gate Called Beautiful of Solomon’s Temple

Architectural exemplar cited in Architectura Chapter VI and illustrated on Folio 0185, displaying the proportional synthesis of stylobate, stereobate, column, architrave, frieze, and cornice.

Occurrences: Arch. Cap. 6 (EN), Arch. Cap. 6 (Notes), Arch. Cap. 17 (EN), Arch. Cap. 17 (Notes), Arch. Prohemio (EN), Arch. index, DE DIVINA PROPORTIONE, Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

potentia

Language: Latin / Italian | Translation: Power / Square of a Line (δύναμις)

Defined by Pacioli in Chapter IV (after Euclid I.47 and X.def.2): ‘the power (potentia) of a line is its square; what a line can do in power is its square.’

Occurrences: Comp. Cap. 4 (IT), Comp. Cap. 4 (Notes), Comp. Cap. 6 (IT), Comp. Cap. 6 (Notes), Comp. Cap. 21 (IT), Comp. Cap. 24 (IT), Comp. Cap. 27 (IT), Comp. Cap. 30 (IT), Comp. Cap. 31 (IT)

proportio / proportione (proportio)

Language: Latin / Italian | Translation: Ratio (proportio)

A determinate quantitative relation (habitudo) of two magnitudes of the same genus to one another (Euclid V.def.3).

proportionalitas / proportionalità (proportionalitas)

Language: Latin / Italian | Translation: Proportion / Similarity of Ratios (proportionalitas)

The equality or similitude of two or more ratios (Euclid V.def.6), requiring at least three terms (in continuous proportion) or four terms (in discontinuous proportion).

proportione continua (proportione-continua)

Language: Italian | Translation: Continuous Proportion (a : b = b : c)

A three-term progression in which the middle term serves both as consequent of the first ratio and antecedent of the second.

Occurrences: Comp. Cap. 5 (Notes), Comp. Cap. 30 (DE), Winterberg Intro (DE)

proportione discontinua (proportione-discontinua)

Language: Italian | Translation: Discontinuous (Four-Term) Proportion (a : b = c : d)

A proportion among four distinct terms in which the ratio of the first to the second equals the ratio of the third to the fourth without a shared middle term (Compendio Cap. VI).

colonna pulvinata (ionica) (pulvinata)

Language: Latin / Italian | Translation: Pulvinated (Cushion-Voluted) Ionic Column

Vitruvian term for the Ionic capital whose lateral volutes resemble coiled cushions or matronly curls hanging down on either side (Architectura Cap. IV, VII, Folio 0213).

Occurrences: Arch. Cap. 7 (EN), Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

punto (punctum / σημεῖον) (punto)

Language: Italian / Latin / Greek | Translation: Geometric Point

That which has no part (Euclid I.def.1), the indivisible principle and extremity of lines.

pyramide corta / troncata (pyramide-corta)

Language: Italian | Translation: Truncated Pyramid / Frustum

A pyramid or cone whose apex has been cut off by a plane parallel to the base, leaving two similar parallel bases (Compendio Cap. LXV, LXVIII).

Occurrences: Comp. Cap. 68 (IT), Comp. Cap. 69 (IT)

Pyramis laterata exagona (Πυραμὶς πλευρώδης ἑξάγωνος) (pyramide-exagona)

Language: Latin / Greek / Italian | Translation: Hexagonal Pyramid

Six-sided pyramid rising from a regular hexagonal base with six isosceles triangular lateral faces meeting at the apex (Compendio Cap. LXV, Plates LV–LVI).

pyramide laterata (pyramis laterata / πυραμὶς πλευρώδης) (pyramide-laterata)

Language: Italian / Latin / Greek | Translation: Polygonal Pyramid

A solid bounded by a rectilinear polygonal base and triangular lateral faces meeting at a single apex, whose volume is exactly one-third that of the prism of equal base and altitude (Euclid XII.7; Compendio Cap. LXV–LXVII).

Occurrences: Comp. Cap. 49 (IT), Comp. Cap. 50 (IT), Comp. Cap. 51 (IT), Comp. Cap. 53 (IT), Comp. Cap. 64 (EN), Comp. Cap. 65 (EN), Comp. Cap. 65 (IT)

Pyramis laterata pentagona (Πυραμὶς πλευρώδης πεντάγωνος) (pyramide-pentagona)

Language: Latin / Greek / Italian | Translation: Pentagonal Pyramid

Five-sided pyramid rising from a regular pentagonal base with five isosceles triangular lateral faces meeting at the apex (Compendio Cap. LXV, Plates LIII–LIV).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Pyramis laterata quadrilatera (Πυραμὶς πλευρώδης τετράπλευρος) (pyramide-quadrilatera)

Language: Latin / Greek / Italian | Translation: Quadrangular (Square-Based) Pyramid

Four-sided pyramid rising from a square base with four isosceles triangular lateral faces meeting at the apex (Compendio Cap. LXV, Plates LI–LII).

pyramide rotonda (pyramis rotunda / πυραμὶς στρογγύλη) (pyramide-rotonda)

Language: Italian / Latin / Greek | Translation: Round Pyramid / Right Circular Cone

A cone generated by rotating a right-angled triangle about one of its perpendicular legs, whose volume is one-third that of the cylinder of equal circular base and altitude (Euclid XII.10; Compendio Cap. LXIV, LXVI; Plate LVIII).

Occurrences: Comp. Cap. 64 (EN), Comp. Cap. 64 (IT), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Pyramis laterata triangula (Πυραμὶς πλευρώδης τρίγωνος) (pyramide-triangula)

Language: Latin / Greek / Italian | Translation: Triangular Pyramid

Three-sided pyramid rising from an equilateral triangular base to an elevated apex (Compendio Cap. LXV, Plates XLIX–L).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

quadratura del cerchio (quadratura-del-cerchio)

Language: Italian | Translation: Quadrature (Squaring) of the Circle

Determination of the area of a circle in square units, approximated by Archimedes (Dimensio Circuli) as 11/14 of the square of its diameter (π ≈ 22/7; Compendio Cap. LXII).

Occurrences: Comp. Cap. 66 (IT)

quadrivio / mathematiche scienze (quadrivio)

Language: Italian / Latin | Translation: The Mathematical Quadrivium

The classical Boethian fourfold division of the mathematical sciences—Arithmetic, Geometry, Astronomy, and Music—which Pacioli in Chapter III argues must either be reduced to three (Arithmetic, Geometry, Astronomy) with Music and Perspective subalternate, or expanded to five to include Perspective on an equal footing with Music.

Occurrences: Comp. Cap. 3 (EN), Comp. Cap. 3 (Notes)

quantità continua e discreta (quantita-continua)

Language: Italian | Translation: Continuous and Discrete Quantity (Magnitude and Multitude)

The fundamental Aristotelian and Boethian division of quantity into continuous magnitude (lines, surfaces, solids, studied by geometry and admitting both rational and irrational ratios) and discrete multitude (numbers, studied by arithmetic and admitting only rational ratios).

Occurrences: Arch. Cap. 19 (IT), Comp. Cap. 5 (IT)

quantità discreta (numero) (quantita-discreta)

Language: Italian | Translation: Discrete Quantity (Number / Multitude)

Quantity composed of indivisible units studied by Arithmetic, admitting only rational ratios expressible by whole numbers (Compendio Cap. VI–VII, Folio 0311).

quinta essentia (quinta-essentia)

Language: Italian / Latin | Translation: Fifth Essence (Quintessence / Celestial Aether)

The celestial substance beyond the four sublunary elements (earth, water, air, fire), to which Plato in the Timaeus assigned the regular dodecahedron of twelve pentagonal faces.

Occurrences: Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 5 (Notes), Comp. Cap. 22 (EN), Comp. Cap. 30 (EN), Comp. Cap. 46 (EN), Comp. Cap. 46 (Notes), Comp. Cap. 55 (EN), Winterberg Anmerkungen (EN)

radice (R.) (radice)

Language: Italian | Translation: Square Root (√)

Defined in Chapter IX as a quantity which, multiplied by itself, produces the number or magnitude of which it is called the root; denoted in Pacioli’s notation by R.

radice universale / radice legata (R. V.) (radice-universale)

Language: Italian | Translation: Universal (Bound) Square Root (√(a ± √b))

A nested radical whose root sign governs an entire binomial or apotome expression, introduced in Chapter XXI (Twelfth Effect) and used throughout the Libellus.

Occurrences: Comp. Cap. 21 (Notes)

rationale / commensurabile (rationale)

Language: Italian / Latin | Translation: Rational / Commensurable Magnitude

A magnitude commensurable in length or in square (potentia) with an assigned unit line (Euclid X.def.3).

reciproke Körper (duale Polyeder) (reciproke-koerper)

Language: German | Translation: Reciprocal (Dual) Polyhedra

Polyhedra standing in projective polar reciprocity to one another with respect to a sphere—specifically the self-dual tetrahedron, the reciprocal cube and octahedron, and the reciprocal icosahedron and dodecahedron (Winterberg Einleitung; Compendio Cap. LV–LVII; Sachliche Anmerkungen 1–5).

Occurrences: Comp. Cap. 47 (Notes), Comp. Cap. 48 (Notes), Comp. Cap. 49 (Notes), Comp. Cap. 50 (Notes), Comp. Cap. 51 (Notes), Comp. Cap. 52 (Notes), Winterberg Anmerkungen (DE)

residuo (apotome) (residuo)

Language: Italian / Latin | Translation: Apotome / Binomial Residue (a − √b or √a − b)

An irrational difference of two terms commensurable only in square (Euclid X.73–78, X.85–90), such as the golden-section segments √125 − 5 (first apotome) and 15 − √125 (fourth apotome) of a rational line of length 10, or the edge of the regular dodecahedron (Euclid XIII.17).

Occurrences: Comp. Cap. 2 (IT), Comp. Cap. 8 (Notes), Comp. Cap. 13 (Notes), Comp. Cap. 15 (EN), Comp. Cap. 15 (IT), Comp. Cap. 15 (Notes), Comp. Cap. 18 (Notes), Comp. Cap. 21 (Notes), Comp. Cap. 30 (EN), Comp. Cap. 30 (IT), Comp. Cap. 30 (Notes)

la Rotonda a Roma (Pantheon) (rotonda-pantheon)

Language: Italian / Latin | Translation: The Roman Pantheon (Santa Maria ad Martyres)

Cited by Pacioli in Compendio Chapter LIV as the supreme ancient exemplar of a hemispherical coffered dome whose geometric law derives from the 72-base polyhedral sphere.

Occurrences: Arch. Cap. 8 (DE), Arch. Cap. 8 (EN), Arch. Cap. 8 (IT), Arch. Cap. 11 (Notes), Comp. Cap. 54 (EN), Comp. Cap. 54 (IT), Comp. Cap. 54 (Notes)

saetta (sagitta / frezza) (saetta)

Language: Italian / Latin | Translation: Sagitta / Versine of a Circular Arc

Defined in Chapter LXXI: the perpendicular altitude erected from the midpoint of a chord to the midpoint of its circular arc, like an arrow resting on a bowstring.

Occurrences: Comp. Cap. 71 (EN), Comp. Cap. 71 (IT), Comp. Cap. 71 (Notes), Comp. index

Borgo San Sepolcro (san-sepolcro)

Language: Italian | Translation: Borgo Sansepolcro (Tuscany)

Native town of both Fra Luca Pacioli and his master Piero della Francesca in the upper Tiber valley, celebrated throughout the Compendio, the Architecture dedication, and the Libellus.

Occurrences: Arch. Cap. 8 (EN), Arch. Cap. 19 (EN), Arch. Prohemio (EN), Arch. Prohemio (Notes), Arch. index, Comp. index, DE DIVINA PROPORTIONE, Winterberg Intro (DE), Winterberg Intro (EN)

S. Maria de le Gratie in Milano (santa-maria-delle-grazie)

Language: Italian | Translation: Church and Convent of Santa Maria delle Grazie, Milan

Dominican church in Milan featuring Bramante’s domed tribune (cited in Cap. LIV alongside the 72-base body) and Leonardo da Vinci’s Last Supper in the refectory (cited in Cap. I and XXIII).

Occurrences: Comp. Cap. 1 (EN), Comp. Cap. 1 (Notes), Comp. Cap. 23 (EN), Comp. Cap. 23 (Notes), Comp. Cap. 54 (EN), Comp. Cap. 54 (Notes), Comp. Cap. 69 (EN), Comp. Cap. 69 (Notes), Winterberg Intro (EN)

scaleno (gradato) (scaleno)

Language: Greek / Italian | Translation: Scalene Triangle

A triangle whose three sides are all unequal, such as the right-angled half-equilateral triangle (30°–60°–90°) into which Plato subdivides the faces of the regular solids (Compendio Cap. LIV, LX).

scothia / gola (trochilo) (scothia)

Language: Greek / Italian | Translation: Scotia / Trochilus (Concave Hollow Molding)

Deep concave molding between the tori of an Attic or Ionic column base or in an entablature, called ‘scothia’ or ‘trochilo’ by Vitruvius and ‘gola’ by modern masons (Architectura Cap. V, Folio 0215).

Occurrences: Arch. Cap. 5 (EN), Arch. Cap. 5 (IT), Arch. Cap. 6 (EN), Arch. Cap. 6 (IT), Plates: Architectural (IT)

semicirculo / mezzo cerchio (semicirculo)

Language: Latin / Italian | Translation: Semicircle

The figure bounded by a diameter and the half-circumference cut off by it (Euclid I.def.18); used in Euclid XIII.13–18 to construct the edges of all five regular solids inside the sphere’s diameter (Compendio Cap. XXVI–XXXI) and rotated in XI.def.14 to generate the sphere.

Occurrences: Comp. Cap. 26 (IT), Comp. Cap. 27 (IT), Comp. Cap. 28 (IT), Comp. Cap. 29 (IT), Comp. Cap. 30 (IT), Comp. Cap. 31 (IT), Comp. Cap. 47 (IT), Comp. Cap. 56 (IT)

Septuaginta duarum basium (corpo de 72 basi / Ἑβδομηκονταδισσάεδρον) (septuaginta-duarum-basium)

Language: Latin / Italian / Greek | Translation: 72-Base Body (Faceted Geodesic Sphere / Coffered Dome Model)

Polyhedral sphere bounded by 48 isosceles trapezoids and 24 isosceles triangles arranged in six meridian zones of twelve around two poles (Compendio Cap. LIV, Plates XXXIX–XL), celebrated by Pacioli as the geometric archetype for the coffered dome of the Roman Pantheon (la Rotonda) and Santa Maria delle Grazie.

Occurrences: Comp. Cap. 54 (EN), Comp. index, Plates: Polyhedra (EN), Plates: Polyhedra (LA)

sexquialtera

Language: Latin / Italian | Translation: Sesquialter Ratio (3 : 2)

A superparticular ratio in which the greater term contains the lesser once plus its half (e.g., 6 : 4 = 3 : 2, the musical fifth and the ratio of total surface areas of the octahedron and tetrahedron in the same sphere).

Occurrences: Comp. Cap. 7 (DE), Comp. Cap. 7 (EN), Comp. Cap. 7 (IT), Comp. Cap. 26 (EN), Comp. Cap. 31 (EN), Comp. Cap. 33 (DE)

sexquitertia

Language: Latin / Italian | Translation: Sesquitertian Ratio (4 : 3)

A superparticular ratio in which the greater term contains the lesser once plus its third (e.g., 8 : 6 = 4 : 3, the musical fourth and the ratio of a single face of the tetrahedron to a single face of the octahedron in the same sphere).

Occurrences: Comp. Cap. 31 (EN), Comp. Cap. 33 (DE), Comp. Cap. 33 (Notes)

simetria / commensuratione (symmetria) (simetria)

Language: Greek / Latin / Italian | Translation: Symmetry / Modular Commensurability

Defined after Vitruvius I.2 and III.1 as the harmonious agreement of the members of a work among themselves and of the individual parts to the whole universe of the figure according to a fixed module (Architectura Cap. I–IV).

simulacro de l’ardente desiderio de nostra salute (simulacro)

Language: Italian | Translation: Leonardo da Vinci’s Last Supper (Il Cenacolo)

Pacioli’s name in Chapters I and XXIII for Leonardo da Vinci’s mural of the Last Supper in the refectory of Santa Maria delle Grazie in Milan, where Christ and the twelve apostles embody the sacred number thirteen.

Occurrences: Comp. Cap. 1 (IT), Comp. Cap. 1 (Notes), Comp. Cap. 3 (IT), Comp. Cap. 3 (Notes), Comp. Cap. 23 (Notes), Comp. Cap. 69 (Notes), Winterberg Intro (DE), Winterberg Intro (EN)

solidum / solido (στερεόν) (solidum)

Language: Latin / Italian / Greek | Translation: Solid / Opaque-Faced Polyhedron

Modifier in Leonardo and Pacioli’s plate nomenclature designating a polyhedron rendered as an opaque mass whose front faces conceal its interior and rear faces.

Occurrences: Comp. Cap. 47 (Notes), Comp. Cap. 48 (Notes), Comp. Cap. 49 (Notes), Comp. Cap. 50 (Notes), Comp. Cap. 51 (Notes), Comp. Cap. 52 (Notes), Winterberg Intro (EN), Winterberg Anmerkungen (EN), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

sommoscapo (grossezza superiore dela colonna) (sommoscapo)

Language: Italian | Translation: Upper Shaft Diameter of a Column (Summoscapus)

The contracted upper diameter of a column shaft just below the astragal and capital, diminished relative to the lower diameter according to column height (Architectura Cap. VI, XIV).

spera / sphaera (σφαῖρα) (spera)

Language: Italian / Latin / Greek | Translation: Sphere

The most uniform and capacious of all solid bodies, defined by Euclid (XI.def.14) as the solid generated by the full revolution of a semicircle about its fixed diameter, and by Theodosius/Dionysius as a solid bounded by a single surface equidistant from an interior center (Compendio Cap. LVI–LVII; Plate LIX).

Occurrences: Comp. Cap. 5 (IT), Comp. Cap. 21 (IT), Comp. Cap. 26 (IT), Comp. Cap. 27 (IT), Comp. Cap. 28 (IT), Comp. Cap. 29 (IT), Comp. Cap. 30 (IT), Comp. Cap. 31 (IT), Comp. Cap. 32 (IT), Comp. Cap. 33 (IT), Comp. Cap. 47 (IT), Comp. Cap. 55 (IT), Comp. Cap. 56 (IT), Comp. Cap. 57 (IT), Comp. Cap. 58 (IT), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

admiranda e stupenda equestre statua (statua-equestre)

Language: Italian | Translation: Leonardo da Vinci’s Sforza Equestrian Colossus

Monumental bronze horse designed by Leonardo da Vinci in Milan in memory of Duke Francesco Sforza; Pacioli records its exact metrological dimensions in Chapter I: 12 Milanese braccia in height (37 4/5 times the printed segment ab on Folio 0017) and 200,000 twelve-ounce pounds of bronze.

Occurrences: Comp. Cap. 1 (EN), Comp. Cap. 1 (IT)

steriobata

Language: Greek / Latin / Italian | Translation: Stereobate (Subterranean Foundation)

The solid underground masonry foundation beneath the stylobate and column base, extending down to firm earth (Architectura Cap. VI, XX, Folio 0213).

Occurrences: Arch. Cap. 4 (Notes), Arch. Cap. 6 (EN), Arch. Cap. 6 (IT), Arch. Cap. 17 (EN), Arch. Cap. 20 (IT), Arch. index, Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

stilobata / pilastrello / basamento (stilobata)

Language: Greek / Latin / Italian | Translation: Stylobate / Pedestal (Pilastrello)

The rectangular pedestal supporting a column above ground level, furnished with its own base moldings, die (dado), and cornice (Architectura Cap. IV, VI, Folio 0213).

Occurrences: Arch. Cap. 4 (IT), Arch. Cap. 4 (Notes), Arch. Cap. 5 (IT), Arch. Cap. 6 (EN), Arch. Cap. 6 (IT), Arch. Cap. 7 (IT), Arch. Cap. 16 (EN), Arch. Cap. 16 (IT), Arch. Cap. 17 (EN), Arch. index, Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)

superficie (ἐπιφάνεια) (superficie)

Language: Italian / Latin / Greek | Translation: Surface (Length and Breadth without Depth)

That which has length and breadth only, and whose boundaries are lines (Euclid I.def.5–6).

superparticularis (superparticolare) (superparticularis)

Language: Latin / Italian | Translation: Superparticular Ratio ((n + 1) : n)

The second species of rational ratio of greater inequality, in which the greater term contains the lesser once plus a single aliquot part (1/n), such as sesquialtera (3:2) or sesquitertia (4:3) (Folio 0311).

superpartiens (superpartiente) (superpartiens)

Language: Latin / Italian | Translation: Superpartient Ratio ((n + k) : n, where 1 < k < n)

The third species of rational ratio of greater inequality, in which the greater term contains the lesser once plus several aliquot parts (e.g., superbipartiens tertias, 5:3; Folio 0311).

tetragono / quadrato (tetragono)

Language: Greek / Italian | Translation: Square (Equilateral and Right-Angled Quadrate)

A four-sided equilateral and right-angled plane figure, forming the six faces of the cube (hexahedron) and the eighteen square faces of the 26-base body (Compendio Cap. XXVII, LIII, LX).

Occurrences: Arch. Cap. 2 (IT), Comp. Cap. 67 (IT)

tetragono longo (parte altera longior) (tetragono-longo)

Language: Italian / Latin | Translation: Oblong Rectangle

A right-angled quadrilateral whose length exceeds its breadth (Chapter LX).

Occurrences: Arch. Cap. 1 (EN), Arch. Cap. 1 (IT), Arch. Cap. 2 (IT), Comp. Cap. 71 (EN), Comp. Cap. 71 (IT)

Tetrahedron / tetracedron (corpo de quatro basi triangulari) (tetrahedron)

Language: Latin / Greek / Italian | Translation: Regular Tetrahedron (4-Base Triangular Pyramid, Figure of Fire)

The first regular solid, bounded by 4 equilateral triangles, 6 edges, and 4 three-valent solid angles; constructed in Euclid XIII.13 (Compendio Cap. XXVI, Plates I–VI) and assigned by Plato to the element of Fire.

Occurrences: Comp. Cap. 5 (DE), Comp. Cap. 5 (EN), Comp. Cap. 5 (IT), Comp. Cap. 5 (Notes), Comp. Cap. 25 (DE), Comp. Cap. 25 (EN), Comp. Cap. 25 (IT), Comp. Cap. 26 (DE), Comp. Cap. 26 (EN), Comp. Cap. 26 (IT), Comp. Cap. 26 (Notes), Comp. Cap. 27 (EN), Comp. Cap. 27 (Notes), Comp. Cap. 31 (EN), Comp. Cap. 31 (Notes), Comp. Cap. 32 (DE), Comp. Cap. 32 (IT), Comp. Cap. 33 (DE), Comp. Cap. 33 (EN), Comp. Cap. 33 (IT), Comp. Cap. 34 (DE), Comp. Cap. 34 (EN), Comp. Cap. 34 (IT), Comp. Cap. 34 (Notes), …and 52 more

Tetrahedron abscisum (Τετράεδρον ἀποτετμημένον) (tetrahedron-abscisum)

Language: Latin / Greek | Translation: Truncated Tetrahedron (8-Base Archimedean Solid)

The first dependent body derived from the regular tetrahedron by trisecting its edges and cutting off its four corners, bounded by 4 regular hexagons and 4 equilateral triangles (8 faces, 18 edges, 12 vertices; Compendio Cap. XLVII, Plates III–IV).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Tetrahedron elevatum (Τετράεδρον ἐπηρμένον) (tetrahedron-elevatum)

Language: Latin / Greek | Translation: Elevated (Stellated) Tetrahedron

Stellated polyhedron formed by erecting 4 equilateral triangular pyramids upon the 4 faces of a core regular tetrahedron, bounded externally by 12 equilateral triangles (Compendio Cap. XLVIII, Plates V–VI).

Occurrences: Plates: Polyhedra (EN), Plates: Polyhedra (LA)

timpano (tympanum) (timpano)

Language: Greek / Latin / Italian | Translation: Tympanum of a Pediment

The recessed triangular wall surface enclosed between the horizontal cornice and the raking cornices of a pediment (Architectura Cap. XVII).

Occurrences: Arch. Cap. 7 (IT)

tondo e quadro (circolo e quadrato) (tondo-e-quadro)

Language: Italian | Translation: Circle and Square (Compass and Straightedge Grid)

The two primary geometric figures—representing the curved line (compass/circino) and the straight line (ruler/libella)—from which both the Vitruvian human body and all twenty-three letters (A–Y) of Pacioli’s Roman geometric alphabet (Folios 0139–0183) are generated.

Occurrences: Plates: Alphabet (IT)

toro / bastone (ozzolo) (toro)

Language: Latin / Italian | Translation: Torus (Large Convex Semicircular Base Molding)

Heavy convex cushion molding at the base of a column above the plinth, flanking the concave scotia (Architectura Cap. V, Folio 0213, 0215).

tygraphi / triglyphi (tygraphi)

Language: Greek / Italian | Translation: Triglyphs

Vertically channeled blocks in the Doric frieze representing the ends of timber ceiling beams, alternating with square metopes (Architectura Cap. XII–XIII, XVII).

Occurrences: Arch. Cap. 12 (EN), Arch. Cap. 12 (Notes), Arch. Cap. 13 (EN), Arch. Cap. 13 (Notes), Arch. Cap. 17 (EN), Arch. index

vacuum / vacuo (κενόν) (vacuum)

Language: Latin / Italian / Greek | Translation: Hollow / Skeletal Open-Frame Polyhedron

Leonardo da Vinci’s revolutionary visual invention in which each polyhedral edge is rendered as a slender wooden strut of square cross-section with open fenestrated faces, allowing the viewer to see both front and rear edges simultaneously.

Occurrences: Comp. Cap. 47 (Notes), Comp. Cap. 48 (Notes), Comp. Cap. 49 (Notes), Comp. Cap. 50 (Notes), Comp. Cap. 51 (Notes), Comp. Cap. 52 (Notes), Winterberg Intro (EN), Winterberg Anmerkungen (EN), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Vigintisex basium (corpo de 26 basi / Εἰκοσιεξάεδρον) (vigintisex-basium)

Language: Latin / Italian / Greek | Translation: 26-Base Body (Small Rhombicuboctahedron)

Archimedean semi-regular polyhedron bounded by 18 squares and 8 equilateral triangles, with 48 edges and 24 four-valent vertices, generated by uniform second truncation of the cube (Compendio Cap. LIII, Plates XXXV–XXXVIII).

Occurrences: Comp. Cap. 53 (EN), Comp. Cap. 53 (Notes), Plates: Polyhedra (EN), Plates: Polyhedra (LA)

Vigintisex basium elevatum (Εἰκοσιεξάεδρον ἐπηρμένον) (vigintisex-basium-elevatum)

Language: Latin / Greek | Translation: Elevated 26-Base Body (Stellated Rhombicuboctahedron)

Twenty-six-pointed stellated polyhedron formed by erecting 18 square pyramids and 8 triangular pyramids upon the 26 faces of the small rhombicuboctahedron, bounded externally by 96 equilateral triangles (Compendio Cap. LIII, Plates XXXVII–XXXVIII).

Occurrences: Comp. Cap. 53 (Notes)

ypothesi

Language: Greek / Italian | Translation: Hypothesis / Given Supposition

Defined in Pacioli’s Chapter LXXI lexicon: the initial supposition or given data assumed at the outset of a geometric construction or demonstration.

Occurrences: Comp. Cap. 29 (IT), Comp. Cap. 58 (IT), Comp. Cap. 71 (IT)

ypotumissa (hypotenusa) (ypotumissa)

Language: Greek / Italian | Translation: Hypotenuse / Slant Edge Joining Apex to Base Circle

Defined in Chapter LXXI and used in Chapter XXIX (icosahedron construction): the slant line descending from the top of an erected perpendicular (catheto) to a vertex on the base circle, subtending the right angle.

zophoro / fregio (zophoro)

Language: Greek / Latin / Italian | Translation: Zophorus / Frieze

The middle horizontal member of the entablature resting upon the epistyle (architrave), called ‘zophoros’ (‘animal-bearing’) by the Greeks because it is often sculpted with foliage, scrolls, and figures (Architectura Cap. XIV–XV, Folio 0215).

Occurrences: Arch. Cap. 6 (IT), Arch. Cap. 14 (EN), Arch. Cap. 14 (IT), Arch. Cap. 15 (EN), Arch. Cap. 15 (IT), Arch. Cap. 16 (EN), Arch. Cap. 16 (IT), Arch. Cap. 17 (EN), Arch. Cap. 17 (IT), Arch. Cap. 20 (IT), Comp. Cap. 2 (IT), Plates: Architectural (DE), Plates: Architectural (EN), Plates: Architectural (IT)