Glossary of Cusan Mathematical & Noetic Terms

Latin, Greek & Constructive Lexicon of De Docta Ignorantia & Scripta Mathematica

Systematic lexicon of 26 technical concepts governing Nicholas of Cusa’s docta ignorantia, coincidentia oppositorum, three-step mathematical transsumptio, infinite figures, decentered cosmology, and circle rectification.

docta ignorantia

• Latin: docta ignorantia | Greek: σοφὴ ἀμαθία / εἰδέναι ὅτι οὐκ οἶδεν

• Rendering: Learned ignorance

• Primary Loci: Dedicatio; I.1; I.3; I.12; I.16–17; II.11–12; Epistola auctoris

Cusa’s central epistemological method: rigorous mathematical and dialectical knowledge of why finite comparative reason (ratio) cannot comprehend precise infinite truth, turning that self-knowledge into intellectual vision.

coincidentia oppositorum

• Latin: coincidentia oppositorum / contradictoriorum | Greek: σύμπτωσις τῶν ἐναντίων

• Rendering: Coincidence of opposites

• Primary Loci: I.2; I.4; I.13–16; I.22; Epistola auctoris; De math. perf. §2

The noetic principle—revealed to Cusa at sea returning from Constantinople in 1437/1438—that predicates opposed in the finite domain of ‘more and less’ (maximum/minimum, straight/curved, chord/arc, center/circumference) coincide identically in the infinite Maximum.

maximum absolutum

• Latin: maximum absolutum / maximum simpliciter | Greek: τὸ ἁπλῶς μέγιστον

• Rendering: The Absolute Maximum

• Primary Loci: I.2; I.4; I.12; I.16–23

That than which nothing can be greater (quo nihil maius esse potest), freed from all quantitative contraction and relative opposition, in which all possibility is eternally actual and with which the Absolute Minimum coincides.

minimum

• Latin: minimum | Greek: τὸ ἐλάχιστον

• Rendering: The Minimum

• Primary Loci: I.2; I.4; I.13; II.11; De math. perf. §§3–4

That than which nothing can be lesser; when abstracted from finite quantity, the superlative minimum coincides with the superlative maximum, and in Cusa’s 1458 calculus of chords the unassignable minimal chord coincides with its arc.

proportio / habitudo

• Latin: proportio / comparativa proportio / habitudo | Greek: λόγος / ἀναλογία

• Rendering: Proportion / Comparative ratio

• Primary Loci: I.1; I.3; I.11; De geom. trans. §§4–11; Aurea prop. §§1–7

Simultaneous agreement in one respect and alterity in another, presupposing number; all finite inquiry proceeds by comparative proportion from the known to the unknown, whereas the infinite escapes all finite proportion (infiniti ad finitum proportionem non esse).

transsumptio

• Latin: transsumptio / translative ascendere | Greek: μετάληψις / ἀναγωγή

• Rendering: Mathematical transsumption (three-step ascent)

• Primary Loci: I.12; I.16; I.19; I.21; I.23

Cusa’s three-stage symbolic method in Book I, Chapter 12: (1) inspect finite mathematical figures and their demonstrated invariants; (2) transfer those relations to corresponding infinite figures; (3) transsume them beyond all figure to the simple Infinite.

mathematicalia

• Latin: mathematicalia / signa mathematicalia | Greek: τὰ μαθηματικά

• Rendering: Mathematical objects and signs

• Primary Loci: I.11–12

Abstract intelligible figures free from the fluctuating material instability of sensible bodies while retaining imaginative clarity, giving them ‘incorruptible certainty’ (incorruptibilis certitudo) as symbols of divine truth.

linea infinita

• Latin: linea maxima et infinita | Greek: ἄπειρος γραμμή

• Rendering: The infinite line

• Primary Loci: I.13–19; I.23

Cusa’s primary geometric paradigm: an uncontracted straight line that is actually triangle, circle, and sphere, indivisible yet the exact ratio and measure (adaequatissima mensura) of every finite straight and curved line.

circulus infinitus

• Latin: circulus maximus et infinitus | Greek: ἄπειρος κύκλος

• Rendering: The infinite circle

• Primary Loci: I.12–13; I.15; I.21; Aurea prop. §3

Limit of coaxial circles of increasing radius: as diameter grows without bound, curvature (1/R) vanishes so the circumference is minimally curved and maximally straight, coinciding with its tangent line and identifying center, diameter, and circumference.

triangulus infinitus

• Latin: triangulus maximus et infinitus | Greek: ἄπειρον τρίγωνον

• Rendering: The infinite triangle

• Primary Loci: I.12; I.14–15; I.19

Limit of a triangle whose apex angle opens toward two right angles (180°, spread s_C -> 0) or whose base becomes infinite: without composition of parts, its three sides and three angles are one simple infinite line (unitrinum).

sphaera infinita

• Latin: sphaera maxima et infinita | Greek: ἄπειρος σφαῖρα

• Rendering: The infinite sphere

• Primary Loci: I.12–13; I.15; I.23; II.11–12

Ultimate actuality of the line’s generative unfolding by semicircle revolution, in which length, breadth, and depth concur and center, diameter, and circumference are identical.

polygonia

• Latin: polygonia inscripta / isopleura / isoperimetra | Greek: πολύγωνον

• Rendering: Regular / isoperimetric polygon

• Primary Loci: I.3; De geom. trans. §§4–11; De math. compl. §§4–10

Multi-angled rectilinear figure whose inscribed multiplication of sides approaches circularity monotonically without ever attaining exact algebraic identity with the indivisible circle unless resolved into circularity itself.

curvitas & rectitudo

• Latin: curvitas / rectitudo | Greek: καμπυλότης καὶ εὐθύτης

• Rendering: Curvature and straightness

• Primary Loci: I.12–13; I.16; I.18; De geom. trans. §§2–3

Classic Aristotelian-Euclidean contraries (cf. Parmenides 137e, Aristotle Physics VII.4): Cusa argues that finite curvature admits ‘more and less’ by participating indivisible rectitude, with which minimal curvature coincides.

sagitta

• Latin: sagitta (versed sine) | Greek: βέλος / κάθετος

• Rendering: Sagitta (arc-to-chord rise)

• Primary Loci: I.13; De math. perf. §§3–4

Perpendicular rise h(x, R) = R - sqrt(R^2 - x^2) = x^2 / (R + sqrt(R^2 - x^2)) between chord and arc; vanishes at O(1/R) as R -> oo and at O(theta^2) as theta -> 0.

chorda & arcus

• Latin: chorda / semichorda / arcus / semiarcus | Greek: χορδὴ καὶ περιφέρεια

• Rendering: Chord and circular arc

• Primary Loci: I.13; I.15; De math. perf. §§2–6; Aurea prop. §§1–6

Rectilinear subtense versus curved circular portion; their ratio sin(theta)/theta is approximated to fifth order by Cusa’s harmonic-radial formula 3 / (2 + cos theta).

prima & secunda linea

• Latin: prima linea (inradius) / secunda linea (circumradius) | Greek: κάθετος ἐγγεγραμμένου / ἐκκέντρου

• Rendering: First line (inradius) & second line (circumradius)

• Primary Loci: De math. compl. §§4–10; De math. perf. §§5–6

In De mathematicis complementis (1453), Cusa names the apothem from center to side-midpoint the ‘first line’ (r_n = cos(pi/n)) and the vertex radius the ‘second line’ (R_n); maximally distinct in the equilateral triangle, they coincide in the circle.

proportio sesquiquarta

• Latin: proportio sesquiquarta | Greek: ἐπιτέταρτος λόγος (5:4)

• Rendering: Sesquiquarta ratio (5 : 4)

• Primary Loci: De geom. trans. §§9–10

Superparticular ratio 5:4 by which Cusa extends ray AE (from center A to the quarter-division point E of inscribed equilateral triangle side BC) to AH = 5*sqrt(7)/16, implying pi_Cusa = 24*sqrt(21)/35 ~= 3.142337.

isoperimetra

• Latin: figura isoperimetra | Greek: ἰσοπερίμετρος

• Rendering: Isoperimetric figure (equal perimeter)

• Primary Loci: De geom. trans. §§4–11; De math. compl. §§5–8

Figures sharing an identical perimeter: Cusa invokes Zenodorus’s and Pappus’s isoperimetric hierarchy from the minimum-capacity equilateral triangle up to the maximum-capacity circle.

quadratura circuli & rectilineatio curvae

• Latin: quadratura circuli / rectilineatio curvae | Greek: τετραγωνισμὸς τοῦ κύκλου

• Rendering: Quadrature of the circle & rectification of the curve

• Primary Loci: De geom. trans. §§2–10; De math. compl. §§1–8; Aurea prop. §§1–6

Constructing a square equal in area to a circle or a straight segment equal in length to an arc; via Archimedes’ Dimensio circuli Prop. 1 and Euclid VI.13, Cusa reduces area quadrature directly to perimeter rectification.

ratio vs. intellectus

• Latin: ratio / discursus vs. intellectus / visio intellectualis | Greek: διάνοια vs. νοῦς

• Rendering: Discursive reason vs. intellectual vision

• Primary Loci: I.4; I.12; I.14; II.11; De math. perf. §§3–4

Discursive ratio enforces the Aristotelian principle of non-contradiction among finite magnitudes, whereas higher intellectus beholds the coincidence of opposites prior to contraction.

complicatio & explicatio

• Latin: complicatio / explicatio | Greek: συναίρεσις καὶ ἀνάπτυξις

• Rendering: Enfolding and unfolding

• Primary Loci: I.13–16; I.22–23; II.3; II.11

As unity enfolds all number and the point/infinite line enfolds triangle, circle, and sphere while finite figures unfold that potency successively, God is the complicatio omnium and the cosmos their explicatio.

contractio

• Latin: contractio / maximum contractum | Greek: συστολή / μέθεξις

• Rendering: Ontological contraction

• Primary Loci: I.2; I.4; II.2; II.4; II.11–12

Restriction of infinite unity or rectitude to a determinate finite degree, species, place, and motion; the physical universe is the ‘contracted Maximum’ subsisting solely in plurality.

mensura adaequatissima

• Latin: adaequatissima mensura / metrum et mensura | Greek: μέτρον ἀκριβέστατον

• Rendering: Most adequate / exact measure

• Primary Loci: I.16–18; I.21; I.23

Expanding Aristotle Metaphysics X.1 (1052b18), Cusa demonstrates that only where Maximum and Minimum coincide can one simple essence measure every large and small finite entity without defect or excess.

unitrinum / triunum

• Latin: unitrinum / triunum | Greek: τριὰς ἐν μονάδι

• Rendering: Unitrine / Triune principle

• Primary Loci: I.19; II.11; Aurea prop. §7

Coincidence of unity and trinity prior to numerical counting, symbolized both by the infinite line that is three angles/sides in one and by the three rays of Aurea propositio §7.

machina mundi

• Latin: machina mundi / machina mundana | Greek: κόσμου σύστημα

• Rendering: Fabric / architecture of the universe

• Primary Loci: II.11–12

Cusa’s decentered physical cosmos—neither absolutely infinite nor enclosed within a fixed physical sphere—whose center is everywhere and circumference nowhere (quasi habens undique centrum et nullibi circumferentiam).

coniectura

• Latin: coniectura | Greek: στοχασμός / εἰκασία

• Rendering: Conjecture (participative approximation)

• Primary Loci: II.11; De math. perf. §1

Positive human assertion participating truth in otherness: all astronomical measurements of poles, centers, and orbital periods remain conjectural because no physical body moves around an immobile mathematical point.