Greek–Latin–English Technical Glossary¶
Lexicon of Syracusan Doric, Hellenistic Koine, Arabic-Latin (Liber Assumptorum),
and modern geometric terminology wired to the interactive :term: role.
Term Key & Headword |
Greek / Latin |
English Sense |
Mathematical Definition & Loci |
|---|---|---|---|
|
ἄρβηλος / scalprum sutorium |
shoemaker’s knife |
Curvilinear plane region bounded by three mutually tangent semicircles with collinear diameters AB = AC + CB on the same side of line AB; its area equals the circle whose diameter is the common internal tangent perpendicular CD (Book of Lemmas Prop. 4). (Loci: Book of Lemmas Props. 4–6 (Heath 1897, pp. 304–308); Pappus IV.14) |
|
δίδυμοι κύκλοι / circuli aequales in arbelo |
Archimedes’ Twin Circles |
The two circles inscribed on opposite sides of the perpendicular CD dividing the arbelos, each tangent to CD, the outer semicircle AB, and one inner semicircle (AC or CB); Archimedes proves they are strictly equal with radius r = (r1·r2)/(r1 + r2) (Book of Lemmas Prop. 5). (Loci: Book of Lemmas Prop. 5 (Heath 1897, pp. 305–306)) |
|
σάλινον / salinum |
salt-cellar |
Symmetric curvilinear figure formed by four semicircles on collinear segments with AD = EB: an outer semicircle AB and central opposite semicircle DE bridged by two inward equal semicircles AD, EB; its area equals the circle on the axial symmetry segment CF (Book of Lemmas Prop. 14). (Loci: Book of Lemmas Prop. 14 (Heath 1897, pp. 316–317)) |
|
νεῦσις / inclinatio |
verging / marked-straightedge insertion |
Geometric construction inserting a line segment of given length between two given curves or lines such that the extended line ‘verges’ (νεύει) toward a given point; used by Archimedes to trisect an arbitrary angle (Book of Lemmas Prop. 8) and construct spiral tangents (On Spirals Props. 5–9). (Loci: On Spirals Props. 5–9 (pp. 156–161); Book of Lemmas Prop. 8 (p. 309)) |
|
κέντρον τοῦ βάρεος / centrum gravitatis |
center of gravity / centroid |
Unique point of a plane figure or solid body such that if suspended therefrom it remains in equilibrium in any orientation; proved in Equilibrium of Planes I.13–14 to coincide for any triangle with the concurrency of its medians (Kimberling center X(2)). (Loci: Equilibrium I Post. 1–7, Props. 13–14 (pp. 189–202); Method Postulates) |
|
ἰσορροπία / aequilibrium |
static equilibrium |
State in which magnitudes suspended at distances along a lever balance one another about a fulcrum; governed by Archimedes’ seven postulates of plane statics. (Loci: Equilibrium of Planes I.Post.1–7, Props. 1–7 (pp. 189–194)) |
|
ἀντιπεπονθέναι / reciproce proportionalia |
inverse proportionality (Law of the Lever) |
Fundamental law proved for both commensurable (I.6) and incommensurable (I.7) magnitudes: two magnitudes balance at distances reciprocally proportional to their weights (A : B = d_B : d_A). (Loci: Equilibrium of Planes I.6–7 (pp. 192–194); Method Prop. 1) |
|
ἔφοδος / methodus mechanica |
heuristic mechanical method |
Archimedes’ analytic art of investigating unknown areas, volumes, and centroids by decomposing figures into parallel line or plane elements and balancing them on a virtual lever against known comparison solids prior to synthetic proof by exhaustion. (Loci: The Method (Heath 1912, pp. 12–51; Heiberg II, 426–507)) |
|
λῆμμα Ἀρχιμήδους / axioma Archimedis |
Axiom of Archimedes |
Of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with one another (credited by Archimedes to earlier geometers / Eudoxus; formalized as Hilbert Axiom V 1). (Loci: Sphere & Cyl. I Assump. 5 (p. 4); Spirals Pref.; Quadrature Pref. (p. 234)) |
|
ἐπὶ τὰ αὐτὰ κοῖλα / ad easdem partes concava |
convexity / monotonic curvature condition |
Rigorous geometric definition of convexity for curves and surfaces in Sphere and Cylinder I: a curve or surface is concave in the same direction if the segment joining any two points on it lies wholly on one side (or on) the curve/surface, guaranteeing that an included convex perimeter/surface is strictly less than the enveloping one. (Loci: Sphere & Cylinder I Defs. & Assump. 1–4 (Heath 1897, pp. 2–4)) |
|
δι᾽ ἐγγραφομένων καὶ περιγραφομένων / methodus exhaustionis |
method of exhaustion |
Rigorous Greek proof paradigm compressing an unknown curvilinear magnitude X between inscribed and circumscribed rectilinear/polygonal sequences (P_n < X < Q_n with Q_n - P_n < ε or Q_n / P_n < 1 + ε) and refuting both X > M and X < M by double reductio ad absurdum. (Loci: Measurement Prop. 1; Quadrature Props. 20–24; Sphere & Cyl. I.33–34) |
|
ἕλιξ / linea spiralis |
Archimedean spiral |
Plane kinematic locus traced by a point moving uniformly along a straight ray while the ray revolves uniformly in a plane about its fixed extremity O (polar equation r = aθ). (Loci: On Spirals Def. 1 & Props. 12–28 (Heath 1897, pp. 151–188)) |
|
ἡ πρὸς ὀρθὰς τᾷ ἀρχᾷ τᾶς περιφορᾶς / subtangens spiralis |
spiral subtangent |
Segment cut off on the perpendicular to the radius vector at the origin by the tangent to the spiral; in On Spirals Props. 18–20 Archimedes proves that at the end of the n-th turn its length equals n times the circumference of the corresponding circle, achieving the kinematic rectification of the circle. (Loci: On Spirals Props. 18–20 (Heath 1897, pp. 171–176)) |
|
τμῆμα ὀρθογωνίου κώνου τομῆς / segmentum parabolae |
segment of a parabola |
Region comprehended by a straight chord AC and a ‘section of a right-angled cone’ (Archimedes’ pre-Apollonian term for a parabola); its area equals 4/3 of the inscribed triangle of greatest area having the same base AC and vertex B where the tangent is parallel to AC. (Loci: Quadrature Props. 17, 24 (pp. 246, 252); Method Prop. 1 (1912, p. 15)) |
|
ὀρθογώνιον κωνοειδές / conoides rectangulum |
paraboloid of revolution |
Solid generated by revolving a parabola about its axis; Archimedes proves any segment cut off by a plane equals 3/2 of the cone having the same base and axis (half of the enveloping cylinder). (Loci: On Conoids and Spheroids Props. 21–22 (pp. 131–134); Method Prop. 4) |
|
ἀμβλυγώνιον κωνοειδές / conoides obtusangulum |
hyperboloid of revolution |
Solid generated by revolving one branch of a hyperbola (‘section of an obtuse-angled cone’) about its transverse diameter. (Loci: On Conoids and Spheroids Props. 25–26 (pp. 137–140); Method Prop. 5) |
|
σφαιροειδές / sphaeroides |
prolate or oblate spheroid |
Solid generated by revolving an ellipse (‘section of an acute-angled cone’) about its major axis (elongate/prolate spheroid, παραμᾶκες) or minor axis (flat/oblate spheroid, ἐπιπλατύ). (Loci: On Conoids and Spheroids Defs. & Props. 27–32 (pp. 101, 140–150); Method Prop. 3) |
|
τὸ ἐξ ἀμφοτέρων τῶν κυλίνδρων κοινὸν σχῆμα / solidum commune duorum cylindrorum |
Archimedean bicylinder |
Solid common to two right circular cylinders inscribed in a cube with perpendicular intersecting axes; in the concluding theorem of The Method Archimedes discovers and proves that its volume equals 2/3 of the cube (16/3 r³), matching an exclusively planar cross-sectional square construction. (Loci: The Method Preface & Prop. 15 (Heath 1912, pp. 14, 46–51)) |
|
τμῆμα κυλίνδρου / ungula cylindrica |
cylindrical hoof / wedge |
Solid cut from a right circular cylinder inscribed in a rectangular prism by a plane through a diameter of one base and a parallel edge of the opposite face; Archimedes proves both mechanically (Prop. 12–13) and purely geometrically via triangular cross-sections (Prop. 14) that its volume is 1/6 of the prism (2/3 r²h). (Loci: The Method Props. 12–14 (Heath 1912, pp. 38–46)) |
|
τρισκαίδεκα σχήματα ἰσόπλευρα μὲν καὶ ἰσογώνια, οὐχ ὅμοια δέ / corpora semiregularia Archimedis |
Archimedean solids |
The complete finite family of 13 convex non-prismatic, non-Platonic polyhedra bounded by regular polygons of two or three species meeting in congruent vertex configurations, discovered by Archimedes and catalogued in Pappus’s Synagoge V.19. (Loci: Pappus Synagoge V.19; Heath 1897 Ch. V; Pacioli De Divina Proportione) |
|
ἀριθμοὶ πρῶτοι καὶ δεύτεροι / περίοδοι / ordines et periodi numerorum |
Archimedean octad exponential notation |
Positional naming system constructed in The Sand-Reckoner taking 10^8 (a myriad myriads) as the base of successive ‘orders’ up to the (10^8)-th order (First Period = 10^(8·10^8)) and iterating through 10^8 Periods up to 10^(8·10^16), accompanied by the law of exponents A_m · A_n = A_(m+n-1). (Loci: Sand-Reckoner III.1–6 (Heath 1897, pp. 227–230)) |
|
Στομάχιον / Loculus Archimedius |
Archimedes’ Stomachion |
Combinatorial and areal dissection of a 12 × 12 square into 14 rational-area polygonal tiles (areas 3, 6, 9, 12, 14, 21 out of 144) admitting 17,152 (536 distinct modulo dihedral symmetry) square assemblies. (Loci: Heiberg Vol. II (1913, pp. 416–424); Palimpsest fol. 177) |
|
τετραπλάσιον ἐμβαδόν / functio Archimedis |
quadrea A(Q1, Q2, Q3) |
Symmetric quadratic polynomial A(Q1, Q2, Q3) = (Q1 + Q2 + Q3)^2 - 2(Q1^2 + Q2^2 + Q3^2) = 16·Area^2 in the three side quadrances of a triangle (attributed to Archimedes via Arabic Heron tradition and central to Rational Trigonometry in geometor/wildberger). (Loci: geometor/wildberger; Book of Lemmas & Heron Metrica I.8) |