Heronian & Rational-Trigonometric Glossary

Lexicon of Greek mensurational, optical, geodetic, and geometric terminology in Hero of Alexandria’s Metrica, Definitiones, Catoptrica, and Dioptra, paired with modern Rational Trigonometry invariants.

anaklasis (ἀνάκλασις) · Lat. reflexio — optical reflection

The bending back of visual rays from polished mirrors at equal angles, demonstrated in Hero’s Catoptrica Prop. 4 from the teleological minimum-path principle that nature acts along shortest lines (διʼ ἐλαχίστων). Loci: Catoptrica Prop. 4 / Greek Fragment (Schmidt pp. 324–329, 368–372).

analogia (ἀναλογία) · Lat. proportio — geometric proportion & distributive justice

Ratio-equality across geometric magnitudes; celebrated in the Preface to Metrica Book III as the sole infallible instrument of proportional justice (‘not exceeding or falling short by a single millet-seed’). Loci: Metrica III Preface (Schöne pp. 140–142), III.1–23.

dioptra (διόπτρα) · Lat. dioptra — surveying transit / theodolite-level

Hero’s precision geared sighting and water-level instrument capable of laying out exact orthogonal sight-lines in horizontal and vertical planes—the ancient ancestor of the modern theodolite. Loci: Dioptra Chs. 1–38 (esp. Ch. 15 mountain tunnel, Schöne pp. 238–240).

dynamodynamis (δυναμοδύναμις) · Lat. quadrato-quadratum — fourth power (square-square, x⁴)

Remarkable algebraic fourth-degree power introduced explicitly by Hero in Metrica I.17 (and implied in I.8’s product of two squares ΓΘ² · ΕΗ²), violating strict three-dimensional homogeneity centuries before Diophantus. Loci: Metrica I.8 (Schöne p. 22), I.17 (Schöne p. 48).

embadon (ἐμβαδόν) · Lat. area — area / planar content

The numerical measure of a plane or curved surface expressed in unit squares (μονάδες τετράγωνοι), standard throughout Hero’s Metrica Book I and Book III. Loci: Metrica I Preface, I.1–39, III.1–19.

ephodikon (τὸ Ἐφοδικόν (Ἔφοδος)) · Lat. Methodus — Archimedes’ Method of Mechanical Theorems

Hero’s title for Archimedes’ heuristic mechanical treatise to Eratosthenes, cited in Metrica I.32, II.14, and II.15 for parabolic areas, the cylindrical ungula (V = d²h/6), and the bicylinder groin vault (V = 2a³/3). Loci: Metrica I.32, II.14–15 (Schöne pp. 80, 130–132).

gnomon (γνώμων) · Lat. gnomon — universal gnomon (similarity-preserving increment)

Generalized by Hero in Definitiones §58 (Hultsch §59) beyond Euclid II.Def.2: ‘Universally, a gnomon is anything which, when added to anything whatever, number or figure, makes the whole similar to that to which it was added.’ Loci: Definitiones §58 [Hultsch §59] (p. 21).

itys (ἴτυς) · Lat. annulus / corona circularis — circular ring / annulus

The planar ring bounded by two concentric circles of diameters D₁ and D₂; Hero avoids measuring two circles separately in Metrica I.26 by factoring D₁² − D₂² = 4w(D₂ + w) where w = (D₁ − D₂)/2 is the rim width. Loci: Metrica I.26 (Schöne pp. 68–70), III.9.

kathetos (κάθετος) · Lat. perpendicularis / altitudo — perpendicular / altitude

The perpendicular dropped from a vertex or apex to the opposite base line or base plane. Metrica I.5–7 compute the altitude via Elements II.12–13, whereas I.8 finds the area directly ‘without an altitude’ (χωρὶς καθέτου). Loci: Metrica I.3–8, II.1–19.

kolouros (κόλουρος πυραμὶς / κόλουρος κῶνος) · Lat. frustum pyramidis / coni — truncated pyramid or cone (frustum)

Literally ‘dock-tailed’ or truncated pyramid or cone cut parallel to its base; in Metrica II.6–10 Hero proves the remarkable decomposition (a + b)/2 squared plus 1/3 of ((a − b)/2)², equivalent to (a² + ab + b²)/3. Loci: Metrica II.6–10 (Schöne pp. 104–120), III.21–22.

perimetros (περίμετρος) · Lat. perimetros / ambitus — perimeter / circumference

Total boundary length of a polygon or circle; in Metrica I.8, half the perimeter (τὸ ἥμισυ τῆς περιμέτρου, s = (a+b+c)/2) governs the area product Δ² = s(s−a)(s−b)(s−c). Loci: Metrica I.8, I.26.

pleura (πλευρά) · Lat. latus / radix quadrata — side / square root

Literally the ‘side’ of a geometric figure or of an arithmetic square; hence ‘take the side of N’ (λαβὲ πλευράν) means extract √N, either exactly when rational (ῥητή) or iteratively by Hero’s method when irrational. Loci: Metrica I.8 (ψκ πλευρά), I.17, II.16–17.

quadrance (δύναμις / τὸ ἀπὸ τῆς εὐθείας) · Lat. potentia / quadratum — squared length Q(A, B) = (x₂−x₁)² + (y₂−y₁)²

The exact algebraic square on a segment (Euclid’s καὶ Ἥρωνος ‘τὸ ἀπὸ ΑΒ’ / ‘δυνάμει δοθεῖσα’), fundamental both to Greek geometric algebra (Metrica III.2: ΑΔ is ‘given in square’, δοθεῖσα δυνάμει) and to GEOMETOR’s exact symbolic verification. Loci: Metrica I.5–8, I.17–18, III.2; geometor/model.

quadrea (ἑκκαιδεκαπλάσιον τοῦ ἀπὸ τοῦ ἐμβαδοῦ) · Lat. quadrea — 16 times squared triangle area, A = 16Δ²

In Rational Trigonometry (Wildberger W.07), the polynomial invariant A(Q₁, Q₂, Q₃) = (Q₁+Q₂+Q₃)² − 2(Q₁²+Q₂²+Q₃²) = 16s(s−a)(s−b)(s−c) = 16Δ², encoding Hero’s formula over any commutative ring without square roots. Loci: Metrica I.8; geometor/wildberger W.07.

rhetos (ῥητὴ πλευρά / ἄλογος) · Lat. rationalis / irrationalis — rational (expressible) / irrational root

A number possessing an exact integer or fractional root has a ‘rational side’ (ῥητὴ πλευρά, e.g., √7056 = 84); when it lacks a rational side (e.g., 720 in Metrica I.8), Hero approximates it to arbitrary rational precision ‘with minimal difference’ (μετὰ διαφόρου ἐλαχίστου). Loci: Metrica I.8 (Schöne p. 18), I.17 (Schöne p. 48).

speira (σπεῖρα) · Lat. spira / torus — torus (solid ring of revolution)

Surface and solid generated when a circle perpendicular to a base plane revolves around a fixed coplanar center; measured in Metrica II.13 via Dionysodorus’s theorem and by unrolling into a right cylinder of length 2πR. Loci: Metrica II.13 (Schöne pp. 126–130).

spread · Lat. extensio angularis — rational angle invariant s(ℓ₁, ℓ₂) = sin²θ

Exact rational measure of inclination between two lines in Rational Trigonometry (s = 0 parallel, s = 1 perpendicular), used in geometor.heron to certify Hero’s angle bisectors, cyclic quadrilaterals, and equal-angle reflection without transcendental functions. Loci: geometor/model & geometor/wildberger.