Commentary on the Definitions (I.Def.1–23) — GEOMETOR Commentary & Concordance Notes

GEOMETOR Philosophical & Mathematical Commentary

  1. Numbering of the Definitions (Euclid/Friedlein 1–23 vs. Taylor/Barocius I–XXXV): In modern critical editions of Euclid (Heiberg, Heath in geometor/euclid, and geometor/elements) as well as Friedlein's Greek text, Book I contains 23 Definitions (I.Def.1–I.Def.23). Renaissance editors (Barocius 1560) and Thomas Taylor subdivided compound definitions—such as obtuse/acute angles, the seven species of triangles, and the five species of quadrilaterals—into 35 numbered sub-headings (DEFINITION I–XXXV). Our Codex preserves Taylor’s full text while grouping sections strictly by Euclid’s canonical 23 Definitions (sec-def-01 through sec-def-23) so every :just: link resolves directly to euclid and elements.

  2. Flux of a Point (`Friedlein 97, 6`) & Protagoras on Tangency (`Friedlein 139–140`): Both loci play key roles in geometor/hoenen: - At Caput IV, p. 107 n. 8 (proclus:in-euclid:p97), Hoenen cites Friedlein 97, 6 on the genetic definition of the line as flux of a point (ῥύσις σημείου), pairing it with Geminus's Postulate 1 commentary (Friedlein 185). - At Caput III, p. 67 (proclus:in-euclid:12.14 & 12.14-22), Hoenen cites Friedlein 139, 26–140, 7 where Proclus refutes Protagoras's claim that a circle touches a ruler along a line rather than at a point, demonstrating the chasm between sensory thresholds and exact geometric limits.

  3. Horn Angles (`Friedlein 121–127`) and Non-Archimedean Magnitude: Proclus’s analysis of the horn angle (cited in geometor/euclid Vol. II, p. 40 on [III.16]) foreshadows both medieval debates (Campanus, Peletier, Clavius) and modern non-Archimedean ordered fields.

Friedlein Loci Cited in GEOMETOR (32 Loci in pp. 85–177)

  • Friedlein 95, 21 (text) — Pythagorean definition of the point as a ‘monad having position’ (monas thesin echousa) compared with Euclid’s negative definition. - euclid (proclus:in-euclid:95.21): Heath Vol. I, p. 155 [folio 0169]

  • Friedlein 96, 21–97, 3 (text) — Platonic and Euclidean definitions of a line as ‘breadthless length’ (mekos aplates) and as the boundary of a surface. - euclid (proclus:in-euclid:96.21-97.3): Heath Vol. I, p. 158 [folio 0172]

  • Friedlein 97, 6 (text) — Genetic definition of the line as the ‘flux of a point’ (rhysis semeiou), generating length through unextended motion. - hoenen (proclus:in-euclid:p97): De Noetica Geometriae Caput IV, n. 8, p. 107

  • Friedlein 97, 8–13 (text) — Exposition of the line as the path or flux of a point (rhysis semeiou) and comparison with physical impressions of light. - euclid (proclus:in-euclid:97.8-13): Heath Vol. I, p. 159 [folio 0173]

  • Friedlein 100, 5–19 (text) — How the point generates the line by flowing and how rectilinear vs. circular motion produces simple lines. - euclid (proclus:in-euclid:100.5-19): Heath Vol. I, p. 159 [folio 0173]

  • Friedlein 100, 6 (text) — Terminology of common notions and first concepts in Proclus’s discussion of linear magnitude. - euclid (proclus:in-euclid:100.6): Heath Vol. I, p. 221 [folio 0235]

  • Friedlein 103, 21–107, 10 (text) — Classification of lines; Geminus and Apollonius on homoeomeric lines (straight line, circle, cylindrical helix). - euclid (proclus:in-euclid:103.21-107.10): Heath Vol. I, p. 40 [folio 0054]

  • Friedlein 105, 5 (text) — Proof from Geminus that only three lines are homoeomeric (alike in all parts): the straight line, the circle, and the cylindrical helix. - euclid (proclus:in-euclid:105.5): Heath Vol. I, p. 162 [folio 0176]

  • Friedlein 105, 11 (text) — Why the cylindrical helix, though homoeomeric, is composite rather than simple because it is generated by two dissimilar motions. - euclid (proclus:in-euclid:105.11): Heath Vol. I, p. 162 [folio 0176]

  • Friedlein 110, 18–23 (text) — Famous alternative definitions of a straight line: stretched to the utmost, Plato’s ‘middle covers the ends’ (Parmenides 137E), and Archimedes’ shortest path. - euclid (proclus:in-euclid:110.18-23): Heath Vol. I, p. 168 [folio 0182]

  • Friedlein 111, 9–20 (text) — Geminus’s systematic classification of lines into simple (circular, straight, indefinite) and composite/mixed (in planes vs. on solids). - euclid (proclus:in-euclid:111.9-20): Heath Vol. I, p. 161 [folio 0175]

  • Friedlein 112, 1 (text) — Perseus’s discovery of the three spiric curves (sections of the torus/spira) and quotation of Perseus’s dedicatory epigram. - euclid (proclus:in-euclid:112.1): Heath Vol. I, p. 162 [folio 0176]

  • Friedlein 116, 17 (text) — Terminological distinction between surface in general (epiphaneia) and plane surface (epipedon) in Euclid and older geometers. - euclid (proclus:in-euclid:116.17): Heath Vol. I, p. 169 [folio 0183]

  • Friedlein 117, 8 (text) — Definitions of a plane surface, including the surface to all parts of which a straight line fits (epharmozei). - euclid (proclus:in-euclid:117.8): Heath Vol. I, p. 172 [folio 0186]

  • Friedlein 117, 20 (text) — Archimedean and Platonic characterizations of the plane surface as minimum area with given perimeter or ‘middle overshadowing the ends’. - euclid (proclus:in-euclid:117.20): Heath Vol. I, p. 172 [folio 0186]

  • Friedlein 121, 24–122, 6 (text) — Ancient debate on the ontological category of the angle (relation, quality, or quantity) and the paradox of the horn angle (keratoeides). - euclid (proclus:in-euclid:121-24-122-6): Heath Vol. II, p. 40 [folio 0046]

  • Friedlein 123, 16 (text) — Classification of plane and solid angles and the inclination of planes. - euclid (proclus:in-euclid:123.16): Heath Vol. III, p. 264 [folio 0268]

  • Friedlein 126, 7–127, 16 (text) — Geminus’s exhaustive combinatorics of plane angles formed by straight lines, circles, cissoids, hippopedes, and conchoids (rectilinear, lunular, horn-like, mixed). - euclid (proclus:in-euclid:126.7-127.16): Heath Vol. I, p. 178 [folio 0192]

  • Friedlein 133, 15 (text) — Three species of rectilinear angles (right, obtuse, acute): why the right angle is the determinate measure of equality while obtuse and acute belong to the indeterminate. - euclid (proclus:in-euclid:133.15): Heath Vol. I, p. 182 [folio 0196]

  • Friedlein 136, 8 (text) — Commentary on Def. 13 (Horos / Boundary): why Euclid uses ‘boundary’ rather than ‘extremity’ (peras) to include surfaces and lines bounding figures. - euclid (proclus:in-euclid:136.8): Heath Vol. I, p. 182 [folio 0196]

  • Friedlein 139, 26–140, 7 (text) — Refutation of Protagoras’s empiricist objection that a circle touches a straight ruler across a sensible segment rather than at an indivisible point. - hoenen (proclus:in-euclid:12.14-22): De Noetica Geometriae Caput III, p. 67

  • Friedlein 140, 4 (text) — Exactitude of intelligible tangency at a single point versus the coarse imperfection of sensible rulers and drawn circles. - hoenen (proclus:in-euclid:12.14): De Noetica Geometriae Caput III, p. 67

  • Friedlein 143, 8 (text) — Comparison of Euclid’s definition of Figure (Def. 14) with Posidonius’s definition (‘enclosing limit’), explaining why Euclid includes the contained interior. - euclid (proclus:in-euclid:143.8): Heath Vol. I, p. 183 [folio 0197]

  • Friedlein 153, 3 (text) — Commentary on Def. 16 (Center of a Circle): why Euclid specifies ‘a point lying within the figure’, distinguishing the center from poles outside the plane. - euclid (proclus:in-euclid:153.3): Heath Vol. I, p. 185 [folio 0199]

  • Friedlein 157, 10 (text) — Commentary on Def. 17 (Diameter): Thales’s first demonstration (recorded by Eudemus) that the diameter bisects the circle. - euclid (proclus:in-euclid:157.10): Heath Vol. I, p. 185 [folio 0199]

  • Friedlein 165, 24 (text) — Combinatorial proof that there are seven (not nine) species of triangles when classified simultaneously by sides and by angles. - euclid (proclus:in-euclid:165.24): Heath Vol. I, p. 27 [folio 0041]

  • Friedlein 166, 10 (text) — Terminology and classification of four-sided figures and the Pythagorean significance of the square (tetragonon). - euclid (proclus:in-euclid:166.10): Heath Vol. I, p. 188 [folio 0202]

  • Friedlein 168, 24 (text) — Etymology and geometric properties of equilateral, isosceles, and scalene triangles. - euclid (proclus:in-euclid:168.24): Heath Vol. I, p. 188 [folio 0202]

  • Friedlein 176, 5–17 (text) — Posidonius’s equidistant definition of parallel lines contrasted with Euclid’s non-secant coplanar definition; excerpt from Geminus. - euclid (proclus:in-euclid:176.5-17): Heath Vol. I, p. 40 [folio 0054]

  • Friedlein 176, 6–17 (text) — Critique of defining parallels by equidistance: non-intersecting lines vs. equidistant loci. - euclid (proclus:in-euclid:176.6-17): Heath Vol. I, p. 190 [folio 0204]

  • Friedlein 177, 11 (text) — Geminus’s crucial warning on asymptotic curves (hyperbola and conchoid of Nicomedes): lines can converge continually forever without ever meeting. - euclid (proclus:in-euclid:177.11): Heath Vol. I, p. 191 [folio 0205]

  • Friedlein 177, 21 (text) — Closing attribution of the parallel-line classification to Geminus’s Philokalia / Theory of Mathematics. - euclid (proclus:in-euclid:177.21): Heath Vol. I, p. 27 [folio 0041]