Glossary of Platonic Mathematical & Noetic Terms

Polytonic Greek, Scholastic Latin & Mathematical Lexicon of Loci Mathematici et Geometrici

Systematic lexicon of 34 foundational Platonic concepts governing geometric figure (schēma), quadratic powers (dynameis), discursive mathematical thought (dianoia), harmonic proportion (analogia), elementary triangles (stoicheia), and Limit (peras).

σχῆμα (schēma)

  • Greek: σχῆμα | Latin: figura

  • Rendering: Geometric figure

  • Primary Stephanus Loci: Meno 75d–76a; Parmenides 145b; Philebus 51c

Defined dialectically by Socrates at Meno 76a7 as ‘the limit of solid’ (στερεοῦ πέρας), anticipating Euclid Elements I, Defs. 13–14 and XI, Def. 2.

στερεόν (stereon)

  • Greek: στερεόν | Latin: solidum

  • Rendering: Solid / Three-dimensional body

  • Primary Stephanus Loci: Meno 76a; Theaetetus 148b; Republic VII 528a–d; Timaeus 32b; Laws VII 817e

Magnitude possessing depth (βάθος) in addition to length and breadth; requires two geometric means for continuous proportion (Timaeus 32b; Elements VIII.12).

ἐπίπεδον (epipedon)

  • Greek: ἐπίπεδον | Latin: planum / superficies

  • Rendering: Plane surface / Plane number

  • Primary Stephanus Loci: Meno 76a; Theaetetus 148a; Republic VII 528a; Timaeus 32a, 53c

Two-dimensional surface or plane product of two factors; joined in continuous proportion by a single mean proportional (Timaeus 32a; Elements VIII.11).

πέρας (peras)

  • Greek: πέρας | Latin: terminus / finis

  • Rendering: Limit / Determinate ratio

  • Primary Stephanus Loci: Meno 76a; Philebus 16c–26d

In Meno 76a, the bounding extremity of a magnitude; in Philebus 25a–d, the supreme ontological genus of equal, double, and every exact ratio of number to number or measure to measure that brings indeterminate continua into symmetry.

ἄπειρον (apeiron)

  • Greek: ἄπειρον | Latin: infinitum / indeterminatum

  • Rendering: The Unlimited / Indeterminate continuum

  • Primary Stephanus Loci: Parmenides 137d, 145a; Philebus 16c–26d

The domain of ‘the more and less’ (τὸ μᾶλλόν τε καὶ ἧττον) lacking intrinsic numeric or geometric measure prior to the imposition of Limit (peras).

ἀνάμνησις (anamnēsis)

  • Greek: ἀνάμνησις | Latin: reminiscentia

  • Rendering: Noetic recollection

  • Primary Stephanus Loci: Meno 81d–85d

Plato’s epistemological thesis that geometric discovery—exemplified when Meno’s untutored attendant recognizes Q = 4 + 4 = 8 from the diagonal diagram—consists in recovering necessary truths latent within rational soul.

διάμετρος (diametros)

  • Greek: διάμετρος | Latin: diameter / diagonalis

  • Rendering: Diagonal of a square or diameter of a circle

  • Primary Stephanus Loci: Meno 85b; Republic VI 510d

Technical geometric term (‘as the sophists/geometers call it’, Meno 85b4) for the line cutting a square into two equal isosceles right triangles, whose square has double the area of the original square.

ὑπόθεσις (hypothesis)

  • Greek: ὑπόθεσις | Latin: suppositio

  • Rendering: Mathematical hypothesis / Starting assumption

  • Primary Stephanus Loci: Meno 86e–87b; Republic VI 510b–511d; VII 533c

Both the geometer’s analytic reduction of a problem to an auxiliary condition (Meno 86e–87a) and the unexamined postulates of odd, even, figures, and angles from which dianoia reasons downward (Republic VI 510c).

παρατείνειν / ἐλλείπειν (parateinein / elleipein)

  • Greek: παρατείνειν, ἐλλείπειν | Latin: applicatio arearum cum defectu

  • Rendering: Elliptic application of area

  • Primary Stephanus Loci: Meno 87a

To apply (παρατείνειν) a given area along the diameter of a circle so as to fall short (ἐλλείπειν) by a figure similar to the applied area—equivalent to inscribing a triangle of given area in a circle and to Euclid Elements VI.28.

μῆκος (mēkos)

  • Greek: μῆκος | Latin: longitudo

  • Rendering: Rational / Commensurable length

  • Primary Stephanus Loci: Theaetetus 148a–b; Laws VII 819e

In Theaetetus’s classification (148a–b), any line segment commensurable in linear measure with the chosen unit—specifically the side of an equilateral square number (1, 4, 9, 16, …).

δύναμις (dynamis)

  • Greek: δύναμις | Latin: potentia

  • Rendering: Quadratic power / Incommensurable square-root line

  • Primary Stephanus Loci: Theaetetus 147d–148b; Republic VIII 546b; Timaeus 31c

A line segment incommensurable in length (μήκει οὐ σύμμετρος) with the unit, yet commensurable by the plane square area it has the ‘power’ (δύναται) to generate (√2, √3, √5, …, √17; cf. Euclid Elements X, Defs. 1–4 & X.9).

ἴσος ἰσάκις (isos isakis)

  • Greek: ἴσος ἰσάκις | Latin: aequaliter aequalis / numerus quadratus

  • Rendering: Equilateral square number

  • Primary Stephanus Loci: Theaetetus 147e

An integer factorable as equal times equal (n = k · k), likened to a square geometric figure; its side is commensurable in length (mēkos) with the unit (Euclid VII, Def. 18).

προμήκης (promēkēs)

  • Greek: προμήκης ἀριθμός | Latin: numerus oblongus

  • Rendering: Oblong (non-square) number

  • Primary Stephanus Loci: Theaetetus 148a

An integer enclosed by unequal factors (greater times less, e.g., 3 = 3×1, 6 = 3×2), whose equivalent square has a side that is a quadratic surd (dynamis).

εἰκασία (eikasia)

  • Greek: εἰκασία | Latin: imaginatio / conjectura

  • Rendering: Apprehension of shadows and reflections

  • Primary Stephanus Loci: Republic VI 509e–511e; VII 534a

Lowest segment (/ A B /) of the Divided Line, directed toward cast shadows and specular reflections in water and polished surfaces.

πίστις (pistis)

  • Greek: πίστις | Latin: fides

  • Rendering: Sensible belief / Conviction

  • Primary Stephanus Loci: Republic VI 510a, 511e; VII 534a

Upper segment (/ B C /) of the visible realm on the Divided Line, grasping physical organisms and artifacts—the very objects geometers redeploy as visible diagrams in dianoia (/ C D /) with exact proportional equality |BC| = |CD|.

διάνοια (dianoia)

  • Greek: διάνοια | Latin: ratio discursiva / cogitatio

  • Rendering: Discursive mathematical understanding

  • Primary Stephanus Loci: Republic VI 510b–511e; VII 533d–534a

Lower segment (/ C D /) of the intelligible realm: the proper cognitive state of geometry and the sister sciences, reasoning deductively from unexamined hypotheses with the aid of visible diagrams toward immutable intelligible figures.

νόησις / νοῦς (noēsis / nous)

  • Greek: νόησις, νοῦς | Latin: intellectus / intelligentia

  • Rendering: Pure dialectical intuition

  • Primary Stephanus Loci: Republic VI 511b–e; VII 534a

Supreme segment (/ D E /) of the Divided Line, treating hypotheses literally as stepping-stones to ascend to the unhypothetical first principle without sensible images.

ἀνυπόθετον (anhypotheton)

  • Greek: τὸ ἀνυπόθετον | Latin: principium non suppositum

  • Rendering: The unhypothetical first principle

  • Primary Stephanus Loci: Republic VI 510b, 511b; VII 533c

The self-grounding first principle of the whole (ἡ τοῦ παντὸς ἀρχή), identified with the Idea of the Good, from which dialectic validates the axioms of mathematics.

ἀριθμητική (arithmētikē)

  • Greek: ἀριθμητική | Latin: arithmetica

  • Rendering: Pure number theory

  • Primary Stephanus Loci: Republic VII 525a–526c; Philebus 56d–e

First science of the Platonic Quadrivium, contemplating abstract equal units completely indivisible and free of sensible inequality (cf. Euclid Books VII–IX).

λογιστική (logistikē)

  • Greek: λογιστική | Latin: logistica

  • Rendering: Science of numerical ratios and calculation

  • Primary Stephanus Loci: Republic VII 525a–c; Laws VII 817e

Paired with arithmetic as the study of the mutual quantitative relations of numbers; distinguished into vulgar mercantile reckoning vs. philosophical ratio theory.

στερεομετρία / τριχῇ αὔξη (stereometria)

  • Greek: ἡ τῶν κύβων αὔξη / στερεόν | Latin: stereometria

  • Rendering: Solid geometry (three-dimensional increase)

  • Primary Stephanus Loci: Republic VII 528a–d

Third science of the Quadrivium—inserted by Socrates between plane geometry and astronomy—investigating cubes and figures sharing depth; historically realized in Theaetetus’s and Eudoxus’s work culminating in Euclid Elements Books XI–XIII.

διαλεκτική / θριγκός (dialektikē / thrinkos)

  • Greek: διαλεκτική, θριγκός | Latin: dialectica

  • Rendering: Dialectic, coping-stone of the mathematical sciences

  • Primary Stephanus Loci: Republic VII 531d–534e

The synoptic science that grasps the kinship (συγγένεια) and community of the five mathematical disciplines and ascends to the essence (λόγος τῆς οὐσίας) of each thing.

ἀναλογία (analogia)

  • Greek: ἀναλογία | Latin: proportio

  • Rendering: Geometric proportion

  • Primary Stephanus Loci: Timaeus 31c–32c; Republic VI 509d, VII 534a

Continuous equality of ratios (a : b = b : c), revered in Timaeus 31c as the fairest of all bonds because it makes the mean simultaneously first, middle, and last.

δεσμός (desmos)

  • Greek: δεσμός | Latin: vinculum

  • Rendering: Proportional bond

  • Primary Stephanus Loci: Timaeus 31c

The unifying mathematical relation binding disparate elements into an indissoluble cosmos: one mean for planes (a, ab, b²), two means for solids (a³, a²b, ab², b³).

μεσότης (mesotēs)

  • Greek: μεσότης | Latin: medietas

  • Rendering: Mathematical mean (arithmetic, geometric, harmonic)

  • Primary Stephanus Loci: Timaeus 32a, 36a

The three classical Archytean means inscribed in Timaeus 31c–36a: geometric √(ab), harmonic 2ab/(a+b) (‘exceeding and exceeded by the same fraction of the extremes’), and arithmetic (a+b)/2 (‘exceeding and exceeded by an equal number’).

ἐπίτριτος (epitritos)

  • Greek: ἐπίτριτος διάστασις | Latin: proportio sesquitertia

  • Rendering: Sesquitertian ratio / Perfect fourth (4 : 3)

  • Primary Stephanus Loci: Timaeus 36a–b

Superparticular ratio 1 + 1/3 = 4/3 generated between a double-interval baseline a and its harmonic mean 4a/3, as well as between its arithmetic mean 3a/2 and double 2a.

ἡμιόλιος (hēmiolios)

  • Greek: ἡμιόλιος διάστασις | Latin: proportio sesquialtera

  • Rendering: Sesquialter ratio / Perfect fifth (3 : 2)

  • Primary Stephanus Loci: Timaeus 36a

Superparticular ratio 1 + 1/2 = 3/2 generated both in the double intervals (between 1 and 3/2) and throughout the triple intervals (1, 3/2, 2, 3).

ἐπόγδοος (epogdoos)

  • Greek: ἐπόγδοον διάστημα | Latin: proportio sesquioctava / tonus

  • Rendering: Whole-tone ratio (9 : 8)

  • Primary Stephanus Loci: Timaeus 36a–b

Superparticular ratio 1 + 1/8 = 9/8 separating the arithmetic mean (3/2) from the harmonic mean (4/3), used by the Demiurge to measure out every 4:3 fourth.

λεῖμμα (leimma)

  • Greek: λεῖμμα | Latin: limma

  • Rendering: Pythagorean diatonic semitone (256 : 243)

  • Primary Stephanus Loci: Timaeus 36b

The exact rational remainder left when two 9:8 whole tones (81/64) are divided out of a 4:3 fourth: (4/3) / (81/64) = 256/243 (‘number to number, 256 to 243’).

στοιχεῖον (stoicheion)

  • Greek: στοιχεῖον | Latin: elementum

  • Rendering: Geometric element

  • Primary Stephanus Loci: Theaetetus 201e; Timaeus 48b, 53c–55c

Originally a letter/syllabic element; transferred by Plato in Timaeus 48b–53d from fire, air, water, and earth (which are ‘not even syllables’) down to their planar right-triangle constituents—whence Euclid’s title Στοιχεῖα.

ἡμιτρίγωνον (hēmitrigōnon)

  • Greek: τρίγωνον σκαληνές / ἡμιτρίγωνον | Latin: semitriangulum

  • Rendering: Fairest scalene right triangle (1 : √3 : 2)

  • Primary Stephanus Loci: Timaeus 54a–55b

Half of an equilateral triangle bisected by its altitude, having exact side quadrances 1, 3, 4 (‘the greater side triple in power of the lesser’, 54b5) and rational spreads 1/4, 3/4, 1.

ἐπιπρόσθησις (epiprosthēsis)

  • Greek: ἐπίπροσθεν εἶναι | Latin: occultatio optica

  • Rendering: Optical line-of-sight occultation

  • Primary Stephanus Loci: Parmenides 137e

Plato’s celebrated definition of the straight line (εὐθύ): ‘that of which the middle stands in front of (blocks) both extremes’ along a visual ray.

τόρνος καὶ κανών (tornos kai kanōn)

  • Greek: τόρνος, κανών, διαβήτης | Latin: circinus et regula

  • Rendering: Compass/lathe and straightedge

  • Primary Stephanus Loci: Philebus 51c, 56b

Canonical constructive instruments generating pure, non-relative geometric beauty (circles, straight lines, planes, and solids) intrinsic to form itself.

σύμμετρος / ἀσύμμετρος (symmetros / asymmetros)

  • Greek: σύμμετρος, ἀσύμμετρος | Latin: commensurabilis / incommensurabilis

  • Rendering: Commensurable & Incommensurable

  • Primary Stephanus Loci: Theaetetus 147d–148b; Laws VII 819e–820c

Having (or lacking) a common unit measure across lengths, surfaces, and solids; declared in Laws VII 819e–820c indispensable knowledge distinguishing rational human beings from ‘swine-like’ ignorance.