Greek & Neoplatonic Mathematical Glossary

Lexicon of Greek mathematical, logical, and Neoplatonic epistemological terminology in Proclus’s Commentary on Book I, cross-referenced with Euclid’s Elements and Petrus Hoenen’s Scholastic-noetic vocabulary.

apagoge eis adynaton (ἀπαγωγὴ εἰς ἀδύνατον) · Lat. reductio ad absurdum — reduction to the impossible

Indirect proof assuming the contradictory of the sought conclusion to deduce the overthrow of a conceded axiom, hypothesis, or prior theorem. Friedlein loci: 254–256.

aitema (αἴτημα) · Lat. postulatum / petitio — postulate

A constructive geometric demand (Posts. 1–5) whose subject-matter is easy to contrive or grant without prior demonstration. Friedlein loci: 178–193.

analysis (ἀνάλυσις) · Lat. analysis — geometric analysis

Upward path assuming the thing sought as granted and tracing consequences back to an admitted principle; transmitted by Plato to Leodamas. Friedlein loci: 43, 66–67, 211, 255.

anamnesis (ἀνάμνησις) · Lat. reminiscentia — recollection

Platonic awakening of the soul’s substantial mathematical logoi from within, illustrated by the Meno’s doubling of the square. Friedlein loci: 44–46.

antistrophe (ἀντιστροφή) · Lat. conversio — geometric conversion

Interchanging hypothesis and conclusion between leading (proegoumenon) and converse (antistrophon) theorems, either completely or partially. Friedlein loci: 252–254, 407.

apagoge (ἀπαγωγή) · Lat. reductio — geometric reduction

Transition from a harder problem or theorem to another whose solution immediately yields the proposed goal (e.g. Hippocrates on duplicating the cube). Friedlein loci: 212–213.

apeiron (ἄπειρον) · Lat. infinitum — the Unlimited / Indefinite

Complementary cosmic principle supplying inexhaustible progression, continuous divisibility, multiplicity, and infinite extendibility in magnitude and number. Friedlein loci: 5–7, 185, 278.

apodeixis (ἀπόδειξις) · Lat. demonstratio — scientific demonstration (5th of 6 parts)

Syllogistic deduction inferring the sought attribute or construction validity from first principles through essential causes (aitiai). Friedlein loci: 203–207.

axioma / koine ennoia (ἀξίωμα / κοινὴ ἔννοια) · Lat. axioma / communis animi conceptio — axiom / common notion

An indemonstrable, self-evident truth common to multiple sciences (e.g. ‘things equal to the same are equal to one another’). Friedlein loci: 76, 178–184, 193–198.

dianoia (διάνοια) · Lat. ratio discursiva — discursive understanding

The specifically mathematical faculty of the soul, unfolding innate intelligible logoi step-by-step from principles to conclusions. Friedlein loci: 3–4, 10–18, 51–57.

diorismos (διορισμός) · Lat. determinatio — specification / condition of possibility (3rd of 6 parts)

In theorems/problems, restating the sought goal on the set-out figure; in problems like I.22, also the necessary and sufficient condition for solvability. Friedlein loci: 66–67, 203–205, 330–331.

doxa (δόξα) · Lat. opinio — opinion

Cognitive faculty directed toward mutable sensible things, lacking knowledge of Demonstrative Causes (aitiai). Friedlein loci: 3–4, 11.

ekthesis (ἔκθεσις) · Lat. expositio — setting-out (2nd of 6 parts)

Instantiating the given of the enunciation in a concrete lettered figure in phantasia (‘Let AB be the given finite straight line…’). Friedlein loci: 203–207.

elleipsis (ἔλλειψις) · Lat. defectus — falling-short in application of areas

Applying an area along a line so as to fall short of its extremity by a similar figure; origin of the name of the ellipse. Friedlein loci: 419–420.

enstasis (ἔνστασις) · Lat. instantia — logical objection

A counter-argument blocking either a geometric construction or a step of demonstration (e.g. Zeno of Sidon against I.1). Friedlein loci: 212–213, 214.

epharmoge (ἐφαρμογή) · Lat. superpositio / congruentia — superposition / coincidence

Fitting one geometric magnitude onto another in imagination without physical distortion (CN 4, Prop. I.4, I.8). Friedlein loci: 188, 196, 234–240.

kata gnomona (κατὰ γνώμονα) · Lat. ad gnomonem / perpendiculariter — gnomon-wise (perpendicular)

Oenopides of Chios’s archaic astronomical name for the perpendicular (Prop. I.12), drawn from the upright sundial pointer. Friedlein loci: 283.

homoiomeres (ὁμοιομερής) · Lat. uniformis / homoeomeris — homoeomeric / uniform in all parts

A curve every part of which coincides with every other part: proved by Geminus to comprise solely the straight line, circle, and cylindrical helix. Friedlein loci: 105, 112.

horos (ὅρος) · Lat. definitio / terminus — definition / boundary

Both a conceptual definition delimiting an essence (Defs. 1–23) and the geometric boundary enclosing a figure (Def. 13). Friedlein loci: 76, 85, 136–137.

hyle noete (ὕλη νοητή) · Lat. materia intelligibilis — intelligible / imaginative matter

Aristotelian term reinterpreted by Proclus as the pure spatial extension in phantasia individualizing plural congruent circles and triangles; central to Hoenen’s Caput IV. Friedlein loci: 48–54.

hyperbole (ὑπερβολή) · Lat. excessus — exceeding in application of areas

Applying an area along a line so as to overlap beyond its extremity by a similar figure; origin of the name of the hyperbola. Friedlein loci: 419–420.

hypothesis (ὑπόθεσις) · Lat. hypothesis / suppositio — hypothesis

An unproved assumption posited by the teacher and conceded by the learner as the starting point of demonstration. Friedlein loci: 75–77, 178–184.

hysterogenes (ὑστερογενής) · Lat. posterius genitus — later-born / abstracted universal

Concept formed empirically a posteriori from sense-objects; rejected by Proclus for mathematics because sensibles lack exactness and necessity. Friedlein loci: 12–15, 50.

kataskeue (κατασκευή) · Lat. constructio — machinery / construction (4th of 6 parts)

Auxiliary lines and circles drawn via postulates and earlier problems to prepare the figure for demonstration. Friedlein loci: 203–207.

keratoeides gonia (κερατοειδὴς γωνία) · Lat. angulus cornicularis — horn angle

Curvilinear-rectilinear angle between a circle and its tangent, smaller than every acute rectilinear angle yet non-zero—challenging Archimedean magnitude. Friedlein loci: 121–127.

lemma (λῆμμα) · Lat. lemma — auxiliary proposition

A proposition assumed in a proof while requiring independent confirmation elsewhere. Friedlein loci: 73, 211.

metaxy / mesotes (μεταξύ / μεσότης) · Lat. medietas — intermediate status

The middle rank occupied by mathematical reality and dianoia between pure Intellect (nous) and sense-perception/opinion (aisthesis/doxa). Friedlein loci: 3–5, 10–11.

nous (νοῦς) · Lat. intellectus — Intellect

Supreme intuitive intelligence contemplating simple, partless, unextended archetypal Forms prior to discursive unfolding. Friedlein loci: 3–4, 10–18.

ousia mathematike (οὐσία μαθηματική) · Lat. essentia mathematica — mathematical essence / being

The intermediate ontological realm of mathematical objects: unmoved and exact unlike sensibles, yet pluralized and extended unlike pure indivisible Forms. Friedlein loci: 3–5.

parabole (παραβολή) · Lat. applicatio spatiorum — application of areas (exact)

Pythagorean construction applying a given area exactly along a given segment (Prop. I.44); transferred by Apollonius to the parabola. Friedlein loci: 419–420.

peras (πέρας) · Lat. terminus / finis — Limit

First cosmic principle (from Plato’s Philebus) bestowing boundary, definite ratio, symmetry, and intelligible form upon mathematical figures; also geometric extremity. Friedlein loci: 5–7, 136.

phantasia (φαντασία) · Lat. imaginatio / phantasia — mathematical imagination

Formative intellective-imaginative screen (‘passive nous’) onto which dianoia projects universal logoi as extended, divisible, yet exact geometric figures. Friedlein loci: 51–57, 78–79, 141.

porisma (πόρισμα) · Lat. porisma / corollarium — porism / corollary

Either (1) an incidental gain emerging unbidden from a demonstration (as I.15 Porism), or (2) finding an existing geometric entity intermediate between problem and theorem. Friedlein loci: 212, 301–305.

problema (πρόβλημα) · Lat. problema — problem

A proposition proposing the constructive generation, division, or placement of a geometric figure not yet brought to view (ending ‘what it was required to do’). Friedlein loci: 77–81, 200–202, 241–243.

probole (προβολή) · Lat. proiectio — projection

The active projection by which dianoia unfolds its partless internal reason-principles onto the mirror-like screen of phantasia to examine their properties. Friedlein loci: 51–56, 78–79, 141.

protasis (πρότασις) · Lat. propositio — enunciation (1st of 6 parts)

Universal statement declaring what is given and what is sought, prior to any lettered diagram. Friedlein loci: 203–204.

ptosis (πτῶσις) · Lat. casus — geometric case

Positional variation of a construction (e.g. point inside, on, or outside a figure) requiring distinct verification of the same theorem. Friedlein loci: 212, 221.

rhysis (ῥύσις) · Lat. fluxus — generative flux

Unextended motion of a point generating a line, or uniform undeviating flux generating a straight line (cited in Greek by Hoenen at p. 121). Friedlein loci: 97, 100, 185.

stoicheion (στοιχεῖον) · Lat. elementum — element

A foundational proposition whose mastery pervades and unlocks the demonstration of many subsequent theorems, as letters compose speech. Friedlein loci: 71–74.

stoicheiosis (στοιχείωσις) · Lat. elementatio — elementary treatise / axiomatic system

Systematic deductive arrangement ascending from first principles through elementary propositions to the capstone of a science. Friedlein loci: 70–75, 84, 432.

symperasma (συμπέρασμα) · Lat. conclusio — conclusion (6th of 6 parts)

Re-ascending from the particular lettered diagram back to the universal truth of the enunciation, sealed by Q.E.F. or Q.E.D. Friedlein loci: 203–207.

synthesis (σύνθεσις) · Lat. synthesis — geometric synthesis

Downward deductive path proceeding from admitted principles through intermediate steps to the conclusion. Friedlein loci: 43, 211, 255.

theorema (θεώρημα) · Lat. theorema — theorem

A proposition demonstrating an essential attribute belonging per se to a given geometric figure (ending ‘what it was required to prove’). Friedlein loci: 77–81, 200–202, 241–243.

topikon theorema (τοπικὸν θεώρημα) · Lat. theorema locale — locus theorem

Theorem invariant across an infinite locus of positions—inaugurated in the Elements by Prop. I.35 (equal parallelograms between parallel lines). Friedlein loci: 394–396.