Friedlein Citation Concordance

Complete bidirectional concordance mapping every passage of Proclus’s Commentary on the First Book of Euclid’s Elements (ed. Gottfried Friedlein, Leipzig: B. G. Teubner, 1873) cited across GEOMETOR (euclid and hoenen) — 164 canonical Friedlein loci encompassing 156 distinct euclid citation keys (159 citing occurrences across Heath Vols. I–III) and 10 philosophical-epistemological loci in Hoenen’s De Noetica Geometriae.

Overview by Codex Unit

Codex Unit

Friedlein Pages

Cited Loci

Scope

Prologue I

pp. 3–47

4

Intermediate Being (Metaxy), Limit & Unlimited, Common Mathematical Principles, Critique of Abstraction, and the Ascent to Dialectic

Prologue II

pp. 48–84

27

Mathematical Imagination (Phantasia & Probole), Intelligible Matter, The Eudemian Summary of Greek Geometry, and the Purpose of the Elements

Definitions

pp. 85–177

32

Point, Line & Flux, Straight Line, Surface, Plane & Horn Angles, Figure, Circle, Triangles, Quadrilaterals, and Parallel Lines

Postulates & Axioms

pp. 178–207

19

Epistemology of First Principles, Compass & Straightedge Postulates, Critique of Postulate 5, Common Notions, and the Six Parts of a Proposition

Propositions I.1–I.15

pp. 208–305

42

Equilateral Construction & Zeno’s Critique, Superposition (I.4), Pappus’s Proof of I.5, Converses & Reductio, Oenopides’ Gnomon (I.12), Vertical Angles & Porisms

Propositions I.16–I.32

pp. 305–384

26

Exterior Angle, Epicurean Critique of I.20, Thales’ Ship-Distance Theorem (I.26), Ptolemy & Proclus on Postulate 5 (I.29), and the Pythagorean Angle-Sum Proof (I.32)

Propositions I.33–I.48

pp. 384–436

14

Parallelograms & Locus Theorems (I.35), Pythagorean Application of Areas (Parabole, Hyperbole, Elleipsis, I.44), and Euclid’s Proof of I.47–I.48

Prologue I (Friedlein pp. 3–47)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 12, 14 — Chap. V–VI: Dianoia, Innate Logoî, and Critique of Abstraction

Proclus’s critique of Aristotelian abstraction (hysterogenes): sensible particulars lack the precision and apodictic necessity of mathematical objects.

  • hoenen proclus:in-euclid:12.14 — De Noetica Geometriae Caput III, p. 67 (In Euclid. Elem. ed. Friedlein 12, 14 ; 49, 17 sqq. 140, 4)

Friedlein 12, 14–22 — Chap. V–VI: Dianoia, Innate Logoî, and Critique of Abstraction

Refutation of deriving mathematical forms from sense-objects: where among sensibles is the indivisible point, breadthless line, or exact equality of radii?

  • hoenen proclus:in-euclid:12.14-22 — De Noetica Geometriae Caput III, p. 67 (ibid. 12, 14-22 ; 49, 12-17 ; 139, 26-140, 7 etc.)

Friedlein 22, 15 — Chap. VII–X: Platonic Utility of Mathematics & Reply to Critics

Proclus’s dialectical formula introducing and answering objections against the utility of theoretical geometry.

  • euclid proclus:in-euclid:22.15 — Heath Vol. I, p. 34 (chapter_iv) [folio 0048]

Friedlein 38, 1–42, 8 — Chap. XIII (cont.): Geminus’s Classification of Pure & Mixed Sciences

Geminus’s rival classification of the mathematical sciences into intelligible (arithmetic, geometry) and sensible/mixed sciences (mechanics, astronomy, optics, geodesy, canonics, logistic).

  • euclid proclus:in-euclid:38.1-42.8 — Heath Vol. I, p. 39 (chapter_iv) [folio 0053]

  • euclid proclus:in-euclid:38.1-42.8 — Heath Vol. I, p. 39 (chapter_iv) [folio 0053]

Prologue II (Friedlein pp. 48–84)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 49, 12–17 — Chap. I–II: Intelligible Matter, Sensible Inexactitude & Phantasia

Inexactitude of sensible figures versus the unvarying precision of universal forms projected in mathematical imagination (phantasia).

  • hoenen proclus:in-euclid:12.14-22 — De Noetica Geometriae Caput III (cf. Caput I §1), p. 67 (Friedlein 49, 12–17 (inexactitude of sensible figures vs. phantasia))

Friedlein 49, 17 — Chap. I–II: Intelligible Matter, Sensible Inexactitude & Phantasia

Mathematical being requires a receptacle (phantasia / intelligible matter) where universal logoi are unfolded with indivisible exactitude.

  • hoenen proclus:in-euclid:12.14 — De Noetica Geometriae Caput III, p. 67 (Friedlein 49, 17 sqq. (phantasia as intelligible screen / materia intelligibilis))

Friedlein 60, 7–9 — Chap. III: Projection (Probole) & Cosmic Geometry

Distinction between leading principles and derivative consequences in universal mathematical reasoning.

  • euclid proclus:in-euclid:60.7-9 — Heath Vol. I, p. 137 (chapter_ix) [folio 0151]

Friedlein 64–70 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

The Eudemian Summary: Proclus’s preservation of Eudemus of Rhodes’s History of Geometry from Thales and Pythagoras to Plato’s Academy and Euclid.

  • euclid proclus:in-euclid:64-70 — Heath Vol. I, p. 37 (chapter_iv) [folio 0051]

Friedlein 64, 8 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Transition from cosmic geometry to the historical origin of geometry among the Egyptians and Phoenicians.

  • euclid proclus:in-euclid:64.8 — Heath Vol. I, p. 38 (chapter_iv) [folio 0052]

Friedlein 64, 19 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Origin of geometry in Egyptian land-measurement after the Nile floods and arithmetic among the Phoenicians through commerce.

  • euclid proclus:in-euclid:64.19 — Heath Vol. I, p. 37 (chapter_iv) [folio 0051]

Friedlein 65, 19 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Pythagoras transforms geometry into a liberal philosophical education and investigates its principles from above (anothen).

  • euclid proclus:in-euclid:65.19 — Heath Vol. I, p. 412 (excursus_i) [folio 0426]

Friedlein 65, 20 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Pythagorean discovery of cosmic figures and golden-section pentagon constructions preserved in the Eudemian Summary.

  • euclid proclus:in-euclid:65-20 — Heath Vol. II, p. 97 (Bk. IV, propositions) [folio 0103]

  • euclid proclus:in-euclid:65.20 — Heath Vol. I, p. 413 (excursus_i) [folio 0427]

Friedlein 66, 20 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Leon (pupil of Neoclides) composes improved Elements and invents diorismoi determining when a problem is possible or impossible.

  • euclid proclus:in-euclid:66.20 — Heath Vol. I, p. 116 (chapter_ix) [folio 0130]

Friedlein 67, 6 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Eudoxus’s investigation of ‘the section’ (tome) begun by Plato, applying geometric analysis to golden-ratio division.

  • euclid proclus:in-euclid:67-6 — Heath Vol. II, p. 99 (Bk. IV, propositions) [folio 0105]

  • euclid proclus:in-euclid:67.6 — Heath Vol. I, p. 137 (chapter_ix) [folio 0151]

Friedlein 67, 14 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Theudius of Magnesia arranges the Elements admirably and renders many partial propositions more universal.

  • euclid proclus:in-euclid:67.14 — Heath Vol. I, p. 117 (chapter_ix) [folio 0131]

Friedlein 67, 22 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Hermotimus of Colophon advances Eudoxus’s and Theaetetus’s discoveries, discovers many Elements, and composes treatises on Loci.

  • euclid proclus:in-euclid:67.22 — Heath Vol. I, p. 117 (chapter_ix) [folio 0131]

Friedlein 68, 4–7 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Close of Eudemus’s chronicle with Philippus of Medma, disciple of Plato who directed geometric research toward Platonic philosophy.

  • euclid proclus:in-euclid:68.4-7 — Heath Vol. I, p. 35 (chapter_iv) [folio 0049]

Friedlein 68, 20 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Euclid’s famous reply to King Ptolemy I Soter: ‘There is no royal road to geometry.’

  • euclid proclus:in-euclid:68.20 — Heath Vol. I, p. 2 (chapter_i) [folio 0016]

Friedlein 69, 3 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Survey of Euclid’s other mathematical, optical, astronomical, and musical works alongside the Elements.

  • euclid proclus:in-euclid:69.3 — Heath Vol. I, p. 17 (chapter_ii) [folio 0031]

Friedlein 69, 4 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Characterization of Euclid’s Data, Phenomena, Optics, Catoptrics, and Divisions of Figures.

  • euclid proclus:in-euclid:69.4 — Heath Vol. I, p. 8 (chapter_ii) [folio 0022]

Friedlein 70, 1–18 — Chap. IV: Origin of Geometry & The Eudemian Summary (Thales to Euclid)

Euclid’s Pseudaria (Book of Fallacies): pedagogical treatise training beginners to detect paralogisms alongside true demonstrations.

  • euclid proclus:in-euclid:70.1-18 — Heath Vol. I, p. 7 (chapter_ii) [folio 0021]

Friedlein 70, 19–71, 21 — Chap. V–VII: Purpose of Book I & Meaning of Element (Stoicheion)

Aim (skopos) of Euclid’s Elements both as a whole (construction of the five Platonic cosmic figures) and relative to the learner.

  • euclid proclus:in-euclid:70.19-71.21 — Heath Vol. I, p. 116 (chapter_ix) [folio 0130]

Friedlein 73, 4 — Chap. V–VII: Purpose of Book I & Meaning of Element (Stoicheion)

Menaechmus’s two senses of ‘element’ (stoicheion): that which proves vs. the simplest constituent into which compounds resolve, contrasted with ‘lemma’.

  • euclid proclus:in-euclid:73.4 — Heath Vol. I, p. 133 (chapter_ix) [folio 0147]

Friedlein 74, 23 — Chap. V–VII: Purpose of Book I & Meaning of Element (Stoicheion)

Euclid’s strict economy in the Elements: excluding whatever is alien to elementary stoicheiosis while furnishing irrefragable foundations.

  • euclid proclus:in-euclid:74.23 — Heath Vol. III, p. 10 (Bk. X, introductory_note) [folio 0014]

Friedlein 75, 10–77, 2 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

Why every demonstrative science must proceed from unhypothetical common principles, definitions, postulates, and axioms.

  • euclid proclus:in-euclid:75.10-77.2 — Heath Vol. I, p. 122 (chapter_ix) [folio 0136]

Friedlein 77, 7–12 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

First division of the things following from first principles into Problems (constructions/generations) and Theorems (essential attributes).

  • euclid proclus:in-euclid:77.7-12 — Heath Vol. I, p. 125 (chapter_ix) [folio 0139]

Friedlein 79, 11–80, 5 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

Distinction of theoretical demonstration vs. problematic construction in Book I of the Elements.

  • euclid proclus:in-euclid:79.11-80.5 — Heath Vol. I, p. 126 (chapter_ix) [folio 0140]

Friedlein 80, 15–81, 4 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

Internal architecture of Euclid’s Book I: triangles, parallels, and the theory of parallelogrammic equality of areas.

  • euclid proclus:in-euclid:80.15-81.4 — Heath Vol. I, p. 126 (chapter_ix) [folio 0140]

Friedlein 81, 5–22 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

How Book I ascends from elementary triangle constructions through congruence and parallels to the equality of areas and the square on the hypotenuse.

  • euclid proclus:in-euclid:81.5-22 — Heath Vol. I, p. 127 (chapter_ix) [folio 0141]

Friedlein 84, 8 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

Mention of Aegaeas (or Aenaeas) of Hierapolis and earlier commentators/epitomators on the Elements.

  • euclid proclus:in-euclid:84.8 — Heath Vol. I, p. 19 (chapter_iii) [folio 0033]

Friedlein 84, 13 — Chap. VIII–IX: Organization of First Principles, Problems & Theorems

Proclus’s methodological remark at the close of Prologue II introducing his selective philosophical and mathematical commentary on the Definitions.

  • euclid proclus:in-euclid:84.13 — Heath Vol. I, p. 30 (chapter_iv) [folio 0044]

  • euclid proclus:in-euclid:84.13 — Heath Vol. I, p. 45 (chapter_iv) [folio 0059]

Definitions (Friedlein pp. 85–177)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 95, 21 — Def. 1: Point (Semeion) as Indivisible Limit with Position · Euclid [I.def.1]

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 (Bk. I, first_principles) [folio 0169]

Friedlein 96, 21–97, 3 — Def. 2: Line (Gramme), Flux of a Point & Classification of Curves · Euclid [I.def.2]

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 (Bk. I, first_principles) [folio 0172]

Friedlein 97, 6 — Def. 2: Line (Gramme), Flux of a Point & Classification of Curves · Euclid [I.def.2]

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 (Proclum pag. 97, 6 (line as flux of a point))

Friedlein 97, 8–13 — Def. 2: Line (Gramme), Flux of a Point & Classification of Curves · Euclid [I.def.2]

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 (Bk. I, first_principles) [folio 0173]

Friedlein 100, 5–19 — Def. 2: Line (Gramme), Flux of a Point & Classification of Curves · Euclid [I.def.2]

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 (Bk. I, first_principles) [folio 0173]

Friedlein 100, 6 — Def. 2: Line (Gramme), Flux of a Point & Classification of Curves · Euclid [I.def.2]

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 (Bk. I, first_principles) [folio 0235]

Friedlein 103, 21–107, 10 — Def. 3: Extremities of a Line & Homoeomeric Lines · Euclid [I.def.3]

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 (chapter_iv) [folio 0054]

Friedlein 105, 5 — Def. 3: Extremities of a Line & Homoeomeric Lines · Euclid [I.def.3]

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 (Bk. I, first_principles) [folio 0176]

Friedlein 105, 11 — Def. 3: Extremities of a Line & Homoeomeric Lines · Euclid [I.def.3]

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 (Bk. I, first_principles) [folio 0176]

Friedlein 110, 18–23 — Def. 4: Straight Line (Eutheia) & Spiric/Conic Classifications · Euclid [I.def.4]

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 (Bk. I, first_principles) [folio 0182]

Friedlein 111, 9–20 — Def. 4: Straight Line (Eutheia) & Spiric/Conic Classifications · Euclid [I.def.4]

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 (Bk. I, first_principles) [folio 0175]

Friedlein 112, 1 — Def. 4: Straight Line (Eutheia) & Spiric/Conic Classifications · Euclid [I.def.4]

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 (Bk. I, first_principles) [folio 0176]

Friedlein 116, 17 — Defs. 5–7: Surface (Epiphaneia), Boundaries & Plane Surface · Euclid [I.def.7]

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 (Bk. I, first_principles) [folio 0183]

Friedlein 117, 8 — Defs. 5–7: Surface (Epiphaneia), Boundaries & Plane Surface · Euclid [I.def.7]

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 (Bk. I, first_principles) [folio 0186]

Friedlein 117, 20 — Defs. 5–7: Surface (Epiphaneia), Boundaries & Plane Surface · Euclid [I.def.7]

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 (Bk. I, first_principles) [folio 0186]

Friedlein 121, 24–122, 6 — Defs. 8–9: Plane Angle, Category of Angle & Horn Angles (Keratoeides) · Euclid [I.def.8]

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 (Bk. III, propositions) [folio 0046]

Friedlein 123, 16 — Defs. 8–9: Plane Angle, Category of Angle & Horn Angles (Keratoeides) · Euclid [I.def.8]

Classification of plane and solid angles and the inclination of planes.

  • euclid proclus:in-euclid:123.16 — Heath Vol. III, p. 264 (Bk. XI, definitions) [folio 0268]

Friedlein 126, 7–127, 16 — Defs. 8–9: Plane Angle, Category of Angle & Horn Angles (Keratoeides) · Euclid [I.def.8]

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 (Bk. I, first_principles) [folio 0192]

Friedlein 133, 15 — Defs. 10–12: Right Angle, Perpendicular, Obtuse & Acute Angles · Euclid [I.def.10]

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 (Bk. I, first_principles) [folio 0196]

Friedlein 136, 8 — Defs. 10–12: Right Angle, Perpendicular, Obtuse & Acute Angles · Euclid [I.def.10]

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 (Bk. I, first_principles) [folio 0196]

Friedlein 139, 26–140, 7 — Defs. 13–14: Boundary (Horos), Figure (Schema) & Protagoras on Tangency · Euclid [I.def.14]

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 139, 26–140, 7 (Protagoras’s objection on circle-line tangency))

Friedlein 140, 4 — Defs. 13–14: Boundary (Horos), Figure (Schema) & Protagoras on Tangency · Euclid [I.def.14]

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 140, 4 (indivisible tangency in intelligible geometry))

Friedlein 143, 8 — Defs. 13–14: Boundary (Horos), Figure (Schema) & Protagoras on Tangency · Euclid [I.def.14]

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 (Bk. I, first_principles) [folio 0197]

Friedlein 153, 3 — Defs. 15–18: Circle, Center, Diameter (Thales) & Semicircle · Euclid [I.def.15]

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 (Bk. I, first_principles) [folio 0199]

Friedlein 157, 10 — Defs. 15–18: Circle, Center, Diameter (Thales) & Semicircle · Euclid [I.def.15]

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 (Bk. I, first_principles) [folio 0199]

Friedlein 165, 24 — Defs. 19–21: Rectilinear Figures & Sevenfold Species of Triangles · Euclid [I.def.20]

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 (chapter_iii) [folio 0041]

Friedlein 166, 10 — Defs. 19–21: Rectilinear Figures & Sevenfold Species of Triangles · Euclid [I.def.20]

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 (Bk. I, first_principles) [folio 0202]

Friedlein 168, 24 — Defs. 19–21: Rectilinear Figures & Sevenfold Species of Triangles · Euclid [I.def.20]

Etymology and geometric properties of equilateral, isosceles, and scalene triangles.

  • euclid proclus:in-euclid:168.24 — Heath Vol. I, p. 188 (Bk. I, first_principles) [folio 0202]

Friedlein 176, 5–17 — Def. 23: Parallel Straight Lines (Euclid, Posidonius & Geminus) · Euclid [I.def.23]

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 (chapter_iv) [folio 0054]

Friedlein 176, 6–17 — Def. 23: Parallel Straight Lines (Euclid, Posidonius & Geminus) · Euclid [I.def.23]

Critique of defining parallels by equidistance: non-intersecting lines vs. equidistant loci.

  • euclid proclus:in-euclid:176.6-17 — Heath Vol. I, p. 190 (Bk. I, first_principles) [folio 0204]

Friedlein 177, 11 — Def. 23: Parallel Straight Lines (Euclid, Posidonius & Geminus) · Euclid [I.def.23]

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 (Bk. I, first_principles) [folio 0205]

Friedlein 177, 21 — Def. 23: Parallel Straight Lines (Euclid, Posidonius & Geminus) · Euclid [I.def.23]

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 (chapter_iii) [folio 0041]

Postulates & Axioms (Friedlein pp. 178–207)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 178, 12–179, 8 — Distinction of Hypotheses, Postulates & Axioms

Aristotelian and Stoic/Geminian distinctions among Hypothesis, Postulate (aitema), and Axiom (axioma).

  • euclid proclus:in-euclid:178.12-179.8 — Heath Vol. I, p. 122 (chapter_ix) [folio 0136]

Friedlein 181, 4–11 — Distinction of Hypotheses, Postulates & Axioms

Why drawing a straight line and circle are postulates (simple and easily granted), whereas constructing a single-turn spiral or equilateral triangle requires complex proof.

  • euclid proclus:in-euclid:181.4-11 — Heath Vol. I, p. 123 (chapter_ix) [folio 0137]

Friedlein 181, 16–23 — Distinction of Hypotheses, Postulates & Axioms

Speusippus and Amphinomus on the difference between postulates (constructive generations) and axioms (indemonstrable truths), citing Archimedes’ Equilibria.

  • euclid proclus:in-euclid:181.16-23 — Heath Vol. I, p. 123 (chapter_ix) [folio 0137]

Friedlein 182, 21–183, 13 — Distinction of Hypotheses, Postulates & Axioms

Third distinction between postulate and axiom following Aristotle’s Posterior Analytics (demonstrable/assumed without learner’s prior conviction vs. self-evident).

  • euclid proclus:in-euclid:182.21-183.13 — Heath Vol. I, p. 124 (chapter_ix) [folio 0138]

Friedlein 185 — Postulates 1–3: Straightedge & Compass Constructibility (Geminus) · Euclid [I.post.1]

Commentary on Postulate 1 quoting Geminus in Greek: because the line is the flux of a point and the straight line a uniform undeviating flux, joining two points is postulated.

  • hoenen proclus:in-euclid:p185 — De Noetica Geometriae Caput IV, §4, p. 121 (In Euclid. Comment. ed. Friedlein pag. 185 (quoting Geminus on uniform undeviating flux))

Friedlein 189–190 — Postulate 4: Equality of All Right Angles · Euclid [I.post.4]

Discussion of Postulate 4 and Pappus’s remark that the converse (‘an angle equal to a right angle is not necessarily right’ if curvilinear, e.g. lunular angle) fails.

  • euclid proclus:in-euclid:189.190 — Heath Vol. I, p. 25 (chapter_iii) [folio 0039]

Friedlein 189, 11 — Postulate 4: Equality of All Right Angles · Euclid [I.post.4]

Geminus’s attempted proof of Postulate 4 (that all right angles are equal) from the uniformity of right-angled inclination.

  • euclid proclus:in-euclid:189.11 — Heath Vol. I, p. 201 (Bk. I, first_principles) [folio 0215]

Friedlein 191, 21 — Postulate 5: The Parallel Postulate & Asymptotic Curves · Euclid [I.post.5]

Historical testimony that from the outset geometers attacked Postulate 5 as lacking self-evidence and sought either to prove it or redefine parallels.

  • euclid proclus:in-euclid:191.21 — Heath Vol. I, p. 202 (Bk. I, first_principles) [folio 0216]

  • hoenen prose — De Noetica Geometriae Caput I, §5, p. 15 (teste Proclo: ancient objections and replacement postulates for Postulate V (Friedlein 191))

Friedlein 191, 23 — Postulate 5: The Parallel Postulate & Asymptotic Curves · Euclid [I.post.5]

Reference to Claudius Ptolemy’s dedicated book attempting to demonstrate Postulate 5 from earlier Euclidean principles.

  • euclid proclus:in-euclid:191.23 — Heath Vol. I, p. 43 (chapter_iv) [folio 0057]

Friedlein 194, 8 — Common Notions (Axioms 1–5): Apollonius, Heron & Pappus · Euclid [I.cn.1]

Apollonius’s attempted demonstration of Common Notion 1 (‘things equal to the same are equal to one another’) via a middle space/coincidence, and Proclus’s refutation.

  • euclid proclus:in-euclid:194.8 — Heath Vol. I, p. 221 (Bk. I, first_principles) [folio 0235]

  • euclid proclus:in-euclid:194.8 — Heath Vol. I, p. 121 (chapter_ix) [folio 0135]

Friedlein 194, 10 — Common Notions (Axioms 1–5): Apollonius, Heron & Pappus · Euclid [I.cn.1]

Critique of proving axioms through things less evident than the axioms themselves.

  • euclid proclus:in-euclid:194.10 — Heath Vol. I, p. 62 (chapter_v) [folio 0076]

Friedlein 196, 21 — Common Notions (Axioms 1–5): Apollonius, Heron & Pappus · Euclid [I.cn.1]

Rejection of the interpolated axiom ‘two straight lines do not enclose a space’ as belonging to the special matter of geometry rather than common notions.

  • euclid proclus:in-euclid:196.21 — Heath Vol. I, p. 232 (Bk. I, first_principles) [folio 0246]

Friedlein 197, 6–198, 5 — Common Notions (Axioms 1–5): Apollonius, Heron & Pappus · Euclid [I.cn.1]

Heron’s restriction of authentic Euclidean Common Notions to three, and Pappus’s proposed additional axioms.

  • euclid proclus:in-euclid:197.6-198.5 — Heath Vol. I, p. 224 (Bk. I, first_principles) [folio 0238]

Friedlein 198, 5–15 — Common Notions (Axioms 1–5): Apollonius, Heron & Pappus · Euclid [I.cn.1]

Editorial criterion separating Euclid’s genuine five (or three) Common Notions from later scholiastic interpolations.

  • euclid proclus:in-euclid:198.5-15 — Heath Vol. I, p. 45 (chapter_iv) [folio 0059]

Friedlein 200, 10 — Problems vs. Theorems & The Six Canonical Parts of a Proposition

Speusippus and Amphinomus (school of Plato) vs. Menaechmus on whether all geometric propositions should be called ‘theorems’ or ‘problems’.

  • euclid proclus:in-euclid:200.10 — Heath Vol. I, p. 19 (chapter_iii) [folio 0033]

Friedlein 200, 10–13 — Problems vs. Theorems & The Six Canonical Parts of a Proposition

Philosophical reason why Platonists preferred ‘theorem’: eternal intelligible realities do not undergo temporal generation.

  • euclid proclus:in-euclid:200.10-13 — Heath Vol. I, p. 33 (chapter_iv) [folio 0047]

Friedlein 202, 5 — Problems vs. Theorems & The Six Canonical Parts of a Proposition

Zenodotus (pupil of Oenopides) and Posidonius on how a problem inquires whether something exists while a theorem inquires what attribute belongs to it.

  • euclid proclus:in-euclid:202.5 — Heath Vol. I, p. 243 (Bk. I, propositions) [folio 0257]

Friedlein 205, 13–15 — Problems vs. Theorems & The Six Canonical Parts of a Proposition

The Six Canonical Parts of a Proposition (Protasis, Ekthesis, Diorismos, Kataskeue, Apodeixis, Symperasma): which three parts are indispensable in every proposition.

  • euclid proclus:in-euclid:205.13-15 — Heath Vol. I, p. 132 (chapter_ix) [folio 0146]

Friedlein 207, 4–25 — Problems vs. Theorems & The Six Canonical Parts of a Proposition

When Ekthesis and Kataskeue may be omitted (as in universal arithmetic propositions), and how Symperasma converts the particular proof back to the universal Protasis.

  • euclid proclus:in-euclid:207.4-25 — Heath Vol. I, p. 131 (chapter_ix) [folio 0145]

Propositions I.1–I.15 (Friedlein pp. 208–305)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 208, 21 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Commentary on Prop. I.1: why Euclid opens Book I with the problem of constructing the equilateral triangle.

  • euclid proclus:in-euclid:208.21 — Heath Vol. I, p. 130 (chapter_ix) [folio 0144]

Friedlein 210, 18 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Symbolic and cosmic primacy of the equilateral triangle (Plato’s Timaeus) alongside its deductive role in Book I.

  • euclid proclus:in-euclid:210.18 — Heath Vol. I, p. 31 (chapter_iv) [folio 0045]

Friedlein 211, 1–212, 4 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Definition of Lemma (λαμβανόμενον requiring confirmation) vs. Case (ptosis), Porism, Objection (enstasis), and Reduction (apagoge).

  • euclid proclus:in-euclid:211.1-212.4 — Heath Vol. I, p. 133 (chapter_ix) [folio 0147]

Friedlein 211, 19 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Definition of geometrical Analysis (ascent to admitted principles) and Synthesis, attributed to Plato’s transmission to Leodamas of Thasos.

  • euclid proclus:in-euclid:211.19 — Heath Vol. I, p. 38 (chapter_iv) [folio 0052]

Friedlein 211, 19–23 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Plato hands down the analytical method to Leodamas, through which many discoveries in geometry were made.

  • euclid proclus:in-euclid:211.19-23 — Heath Vol. I, p. 36 (chapter_iv) [folio 0050]

Friedlein 212, 5–11 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Meaning of Case (ptosis): varying configurations of a construction that preserve the same demonstrative force.

  • euclid proclus:in-euclid:212.5-11 — Heath Vol. I, p. 134 (chapter_ix) [folio 0148]

Friedlein 212, 14 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Dual meaning of Porism: a corollary won as a windfall from another demonstration vs. an intermediate inquiry seeking to find an existing thing (as in Euclid’s Porisms).

  • euclid proclus:in-euclid:212.14 — Heath Vol. I, p. 13 (chapter_ii) [folio 0027]

Friedlein 212, 18–23 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Aristotelian Objection (enstasis) in geometry: blocking either the construction or the demonstration.

  • euclid proclus:in-euclid:212.18-23 — Heath Vol. I, p. 135 (chapter_ix) [folio 0149]

Friedlein 212, 24–213, 11 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Geometric Reduction (apagoge), exemplified by Hippocrates of Chios’s reduction of duplicating the cube to finding two mean proportionals.

  • euclid proclus:in-euclid:212.24-213.11 — Heath Vol. I, p. 135 (chapter_ix) [folio 0149]

Friedlein 214, 18 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Zeno of Sidon’s Epicurean critique of Prop. I.1 (what if two straight lines share a common segment?) and Posidonius’s refutation book against Zeno.

  • euclid proclus:in-euclid:214.18 — Heath Vol. I, p. 196 (Bk. I, first_principles) [folio 0210]

Friedlein 215, 24 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Posidonius’s reply to Zeno of Sidon: Postulate 2 implicitly guarantees that a finite straight line can be produced in one unique straight continuation.

  • euclid proclus:in-euclid:215.24 — Heath Vol. I, p. 197 (Bk. I, first_principles) [folio 0211]

Friedlein 217, 10 — Prop. I.1: Equilateral Triangle, Cases & Zeno of Sidon’s Objection · Euclid [I.1]

Critique of attempting to prove uniqueness of straight segments from circle bisection without circularity.

  • euclid proclus:in-euclid:217.10 — Heath Vol. I, p. 198 (Bk. I, first_principles) [folio 0212]

Friedlein 221, 7–11 — Props. I.2–I.3: Transferring & Subtracting Segments · Euclid [I.2]

Commentary on Prop. I.2: enumeration of geometric cases depending on whether the given point lies on the segment, at its extremity, or outside it.

  • euclid proclus:in-euclid:221.7-11 — Heath Vol. I, p. 129 (chapter_ix) [folio 0143]

Friedlein 234, 21 — Prop. I.4: First Congruence Theorem (SAS), Superposition & Carpus · Euclid [I.4]

Commentary on Prop. I.4: why this is the first Theorem in the Elements and why SAS congruence relies on superposition (epharmoge) without prior theorems.

  • euclid proclus:in-euclid:234.21 — Heath Vol. I, p. 128 (chapter_ix) [folio 0142]

Friedlein 241, 19–243, 11 — Prop. I.4: First Congruence Theorem (SAS), Superposition & Carpus · Euclid [I.4]

Carpus of Antioch (in his Astronomical Treatise) vs. Geminus on whether Problems precede Theorems in natural order.

  • euclid proclus:in-euclid:241.19-243.11 — Heath Vol. I, p. 127 (chapter_ix) [folio 0141]

Friedlein 241, 21 — Prop. I.4: First Congruence Theorem (SAS), Superposition & Carpus · Euclid [I.4]

Citation of Carpus the Mechanician (‘Carpus of Antioch’) on the methodological priority of Problems.

  • euclid proclus:in-euclid:241.21 — Heath Vol. I, p. 44 (chapter_iv) [folio 0058]

Friedlein 242, 16 — Prop. I.4: First Congruence Theorem (SAS), Superposition & Carpus · Euclid [I.4]

Proclus’s synthesis resolving Carpus and Geminus: problems predominate in genesis/art, theorems in dignity and contemplative science.

  • euclid proclus:in-euclid:242.16 — Heath Vol. I, p. 140 (chapter_ix) [folio 0154]

Friedlein 248, 22–249, 19 — Prop. I.5: Isosceles Base Angles (Thales & Pappus’s Symmetry Proof) · Euclid [I.5]

Proclus’s alternative proof of Prop. I.5 (isosceles base angles) taking points inside the equal sides rather than producing them.

  • euclid proclus:in-euclid:248.22-249.19 — Heath Vol. I, p. 254 (Bk. I, propositions) [folio 0268]

Friedlein 249, 20–250, 12 — Prop. I.5: Isosceles Base Angles (Thales & Pappus’s Symmetry Proof) · Euclid [I.5]

Pappus of Alexandria’s celebrated zero-construction symmetry proof of Prop. I.5, comparing triangle ABC with itself as ACB.

  • euclid proclus:in-euclid:249.20-250.12 — Heath Vol. I, p. 254 (Bk. I, propositions) [folio 0268]

Friedlein 250, 20 — Prop. I.5: Isosceles Base Angles (Thales & Pappus’s Symmetry Proof) · Euclid [I.5]

Thales of Miletus’s discovery of Prop. I.5, calling the equal base angles ‘similar’ (homoias) in archaic fashion.

  • euclid proclus:in-euclid:250.20 — Heath Vol. I, p. 252 (Bk. I, propositions) [folio 0266]

Friedlein 251, 2–11 — Prop. I.5: Isosceles Base Angles (Thales & Pappus’s Symmetry Proof) · Euclid [I.5]

Geminus’s classification of theorems and introductory transition to the theory of geometric conversion.

  • euclid proclus:in-euclid:251.2-11 — Heath Vol. I, p. 41 (chapter_iv) [folio 0055]

Friedlein 252, 5 — Prop. I.6: Converse of I.5, Conversion Theory & Reductio ad Absurdum · Euclid [I.6]

Commentary on Prop. I.6: exact theory of complete/leading conversion vs. partial conversion, developed by Menaechmus and Amphinomus.

  • euclid proclus:in-euclid:252.5 — Heath Vol. I, p. 256 (Bk. I, propositions) [folio 0270]

Friedlein 254, 22–27 — Prop. I.6: Converse of I.5, Conversion Theory & Reductio ad Absurdum · Euclid [I.6]

Logical structure of Reductio ad Absurdum (apagoge eis adynaton): assuming the contradictory of the conclusion alongside the hypothesis to refute a known truth.

  • euclid proclus:in-euclid:254.22-27 — Heath Vol. I, p. 136 (chapter_ix) [folio 0150]

Friedlein 255, 8–26 — Prop. I.6: Converse of I.5, Conversion Theory & Reductio ad Absurdum · Euclid [I.6]

Correspondence between Reductio ad Absurdum and geometric Analysis, and why difficult converses are characteristically established indirectly.

  • euclid proclus:in-euclid:255.8-26 — Heath Vol. I, p. 137 (chapter_ix) [folio 0151]

Friedlein 269 — Props. I.7–I.8: Rigidity Lemma & SSS Congruence (Philo’s Proof) · Euclid [I.8]

Commentary on Prop. I.8: Philo of Byzantium’s elegant direct three-case proof of SSS congruence without needing Lemma I.7, and terminology of circumference.

  • euclid proclus:in-euclid:269 — Heath Vol. II, p. 111 (Bk. IV, propositions) [folio 0117]

Friedlein 270, 5–24 — Props. I.9–I.10: Angle & Line Bisection, Trisection & Indivisible Lines · Euclid [I.9]

Commentary on Prop. I.9: refutation of objections to Euclid’s angle-bisection construction (ensuring the inner equilateral vertex lies cleanly within the angle).

  • euclid proclus:in-euclid:270.5-24 — Heath Vol. I, p. 45 (chapter_iv) [folio 0059]

Friedlein 272, 3–7 — Props. I.9–I.10: Angle & Line Bisection, Trisection & Indivisible Lines · Euclid [I.9]

Why trisecting a general rectilinear angle transcends straightedge and compass, requiring Nicomedes’ conchoids, Hippias’s quadratrices, or Archimedes’ spirals.

  • euclid proclus:in-euclid:272.3-7 — Heath Vol. I, p. 161 (Bk. I, first_principles) [folio 0175]

Friedlein 272, 10 — Props. I.9–I.10: Angle & Line Bisection, Trisection & Indivisible Lines · Euclid [I.9]

Historical record of angle-trisection curves (Nicomedes, Hippias, Archimedes) in Hellenistic geometry.

  • euclid proclus:in-euclid:272.10 — Heath Vol. I, p. 26 (chapter_iii) [folio 0040]

Friedlein 277, 25–279, 4 — Props. I.9–I.10: Angle & Line Bisection, Trisection & Indivisible Lines · Euclid [I.9]

Commentary on Prop. I.10: refutation of Xenocrates’ doctrine of indivisible lines (atomoi grammai) from the universal bisectability of segments.

  • euclid proclus:in-euclid:277.25-279.4 — Heath Vol. I, p. 268 (Bk. I, propositions) [folio 0282]

Friedlein 279, 16–280, 4 — Props. I.9–I.10: Angle & Line Bisection, Trisection & Indivisible Lines · Euclid [I.9]

Apollonius of Perga’s symmetric two-circle construction for bisecting a finite straight line (Prop. I.10) without first invoking Prop. I.9.

  • euclid proclus:in-euclid:279.16-280.4 — Heath Vol. I, p. 268 (Bk. I, propositions) [folio 0282]

Friedlein 280, 9 — Props. I.11–I.12: Perpendiculars, Apollonius & Oenopides’ Gnomon · Euclid [I.12]

Commentary on Prop. I.11: relation between bisecting a segment and erecting a perpendicular at a given point.

  • euclid proclus:in-euclid:280.9 — Heath Vol. I, p. 32 (chapter_iv) [folio 0046]

Friedlein 282, 8 — Props. I.11–I.12: Perpendiculars, Apollonius & Oenopides’ Gnomon · Euclid [I.12]

Apollonius’s alternative construction for erecting a perpendicular to a straight line from a point on it (Prop. I.11).

  • euclid proclus:in-euclid:282.8 — Heath Vol. I, p. 270 (Bk. I, propositions) [folio 0284]

Friedlein 283, 7–10 — Props. I.11–I.12: Perpendiculars, Apollonius & Oenopides’ Gnomon · Euclid [I.12]

Commentary on Prop. I.12: Oenopides of Chios first investigated dropping a perpendicular for astronomy and called the perpendicular ‘drawn gnomon-wise’ (kata gnomona).

  • euclid proclus:in-euclid:283.7-10 — Heath Vol. I, p. 271 (Bk. I, propositions) [folio 0285]

Friedlein 283, 9 — Props. I.11–I.12: Perpendiculars, Apollonius & Oenopides’ Gnomon · Euclid [I.12]

Archaic Greek terminology of Oenopides: calling the perpendicular kata gnomona because the sundial gnomon stands at right angles to the horizon.

  • euclid proclus:in-euclid:283.9 — Heath Vol. I, p. 181 (Bk. I, first_principles) [folio 0195]

Friedlein 286, 12–289, 6 — Props. I.11–I.12: Perpendiculars, Apollonius & Oenopides’ Gnomon · Euclid [I.12]

Continuity analysis of circle-line intersection in Prop. I.12 and the three-case classification of objections.

  • euclid proclus:in-euclid:286.12-289.6 — Heath Vol. I, p. 272 (Bk. I, propositions) [folio 0286]

Friedlein 297, 1–298, 10 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Commentary on Prop. I.14: Porphyry’s precision regarding the ‘on opposite sides’ condition in proving two adjacent supplementary rays form one straight line.

  • euclid proclus:in-euclid:297.1-298.10 — Heath Vol. I, p. 24 (chapter_iii) [folio 0038]

Friedlein 298, 14–24 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Commentary on Prop. I.15: exact distinction between adjacent angles (ephexes) and vertical angles (kata koryphen).

  • euclid proclus:in-euclid:298.14-24 — Heath Vol. I, p. 278 (Bk. I, propositions) [folio 0292]

Friedlein 299, 3–6 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Eudemus’s testimony that Thales first discovered the equality of vertical angles (Prop. I.15), while Euclid first gave its scientific demonstration.

  • euclid proclus:in-euclid:299.3-6 — Heath Vol. I, p. 278 (Bk. I, propositions) [folio 0292]

Friedlein 303–304 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Excursus on Porisms following Prop. I.15: distinction between Corollary-Porisms and the higher Porisms of Euclid’s lost three-book treatise.

  • euclid proclus:in-euclid:303-4 — Heath Vol. I, p. 82 (chapter_vii) [folio 0096]

Friedlein 303, 6 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Characterization of a Porism as an unintended ‘windfall’ or ‘gain’ (kerdos) emerging directly from a completed demonstration.

  • euclid proclus:in-euclid:303-6 — Heath Vol. II, p. 175 (Bk. V, propositions) [folio 0181]

Friedlein 304–305 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Examples of geometric corollaries and porismatic deductions in the Elements.

  • euclid proclus:in-euclid:304-5 — Heath Vol. II, p. 98 (Bk. IV, propositions) [folio 0104]

Friedlein 304, 2 — Props. I.13–I.15: Adjacent & Vertical Angles (Thales) & Porisms · Euclid [I.15]

Classification of Porisms into geometric vs. arithmetic, following either theorems (as I.15 Porism) or problems (as IV.15 Porism).

  • euclid proclus:in-euclid:304-2 — Heath Vol. II, p. 109 (Bk. IV, propositions) [folio 0115]

Propositions I.16–I.32 (Friedlein pp. 305–384)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 305, 24 — Props. I.16–I.17: Exterior Angle & Sum of Two Interior Angles · Euclid [I.16]

Opening of Prop. I.16 commentary: why omitting the construction misleads critics like Philippus into questioning the exterior-angle enunciation.

  • euclid proclus:in-euclid:305.24 — Heath Vol. I, p. 20 (chapter_iii) [folio 0034]

Friedlein 315, 11–316, 13 — Props. I.18–I.19: Greater Side & Greater Angle (Division Proof) · Euclid [I.18]

Commentary on Prop. I.18: alternative construction cutting off the equal segment from the other extremity (attributed to Porphyry/Heron).

  • euclid proclus:in-euclid:315.11-316.13 — Heath Vol. I, p. 283 (Bk. I, propositions) [folio 0297]

Friedlein 318, 16–23 — Props. I.18–I.19: Greater Side & Greater Angle (Division Proof) · Euclid [I.18]

Commentary on Prop. I.19: proof by exhaustive trichotomy / division (ek diaireseos) eliminating equality and inferiority.

  • euclid proclus:in-euclid:318.16-23 — Heath Vol. I, p. 285 (Bk. I, propositions) [folio 0299]

Friedlein 322 — Props. I.20–I.21: Triangle Inequality (Epicurean Critique, Heron, Porphyry) · Euclid [I.20]

Commentary on Prop. I.20: Epicurean mockery of the triangle inequality as ‘known even to an ass walking straight to its fodder’, and Proclus’s defense of scientific cause (aitia).

  • euclid proclus:in-euclid:322 — Heath Vol. I, p. 287 (Bk. I, propositions) [folio 0301]

Friedlein 323–326 — Props. I.20–I.21: Triangle Inequality (Epicurean Critique, Heron, Porphyry) · Euclid [I.20]

Heron’s and Porphyry’s elegant alternative demonstrations of the triangle inequality (Prop. I.20) without producing a side of the triangle.

  • euclid proclus:in-euclid:323-6 — Heath Vol. I, p. 287 (Bk. I, propositions) [folio 0301]

Friedlein 327, 12 — Props. I.20–I.21: Triangle Inequality (Epicurean Critique, Heron, Porphyry) · Euclid [I.20]

Commentary on Prop. I.21: why the two interior lines must rise specifically from the extremities of the base (counterexample when drawn from interior points).

  • euclid proclus:in-euclid:327.12 — Heath Vol. I, p. 290 (Bk. I, propositions) [folio 0304]

Friedlein 328, 15 — Props. I.20–I.21: Triangle Inequality (Epicurean Critique, Heron, Porphyry) · Euclid [I.20]

Resolution of the geometric paradox in Prop. I.21 (shorter segments containing a greater angle) and connection to Pappus’s generalizes chevrons.

  • euclid proclus:in-euclid:328.15 — Heath Vol. I, p. 27 (chapter_iii) [folio 0041]

Friedlein 330, 19 — Props. I.22–I.23: Triangle Diorismos & Copying an Angle (Oenopides) · Euclid [I.23]

Commentary on Prop. I.22: necessity of the triangle-inequality diorismos when constructing a triangle out of three given straight lines.

  • euclid proclus:in-euclid:330.19 — Heath Vol. I, p. 34 (chapter_iv) [folio 0048]

Friedlein 331, 8 — Props. I.22–I.23: Triangle Diorismos & Copying an Angle (Oenopides) · Euclid [I.23]

Proof (against Zeno-style objections) that the two circles in Prop. I.22 must intersect whenever the triangle-inequality diorismos holds.

  • euclid proclus:in-euclid:331.8 — Heath Vol. I, p. 293 (Bk. I, propositions) [folio 0307]

Friedlein 333, 5 — Props. I.22–I.23: Triangle Diorismos & Copying an Angle (Oenopides) · Euclid [I.23]

Commentary on Prop. I.23: Eudemus records that constructing an angle equal to a given rectilinear angle was ‘rather the discovery of Oenopides’.

  • euclid proclus:in-euclid:333.5 — Heath Vol. I, p. 295 (Bk. I, propositions) [folio 0309]

Friedlein 335, 19 — Props. I.22–I.23: Triangle Diorismos & Copying an Angle (Oenopides) · Euclid [I.23]

Apollonius’s streamlined compass-arc construction for copying a rectilinear angle (Prop. I.23).

  • euclid proclus:in-euclid:335.19 — Heath Vol. I, p. 296 (Bk. I, propositions) [folio 0310]

Friedlein 339, 11 — Props. I.24–I.25: Hinge Theorems (Heron, Porphyry & Menelaus) · Euclid [I.24]

Commentary on Prop. I.24 (Hinge Theorem): proving the missing cases where the transferred side falls on or within the base (Heron/Porphyry).

  • euclid proclus:in-euclid:339.11 — Heath Vol. I, p. 298 (Bk. I, propositions) [folio 0312]

Friedlein 345–347 — Props. I.24–I.25: Hinge Theorems (Heron, Porphyry & Menelaus) · Euclid [I.24]

Commentary on Prop. I.25: direct (non-reductio) alternative demonstrations of the converse Hinge Theorem by Menelaus of Alexandria and Heron.

  • euclid proclus:in-euclid:345-7 — Heath Vol. I, p. 300 (Bk. I, propositions) [folio 0314]

Friedlein 351 — Prop. I.26: ASA/AAS Congruence & Thales’ Ship-Distance Theorem · Euclid [I.26]

Commentary on Prop. I.26: Porphyry’s and Eudemus’s exhaustive combinatorial audit of the six possible three-part triangle congruence hypotheses (why SSA and AAA fail).

  • euclid proclus:in-euclid:351 — Heath Vol. I, p. 306 (Bk. I, propositions) [folio 0320]

Friedlein 352, 14–18 — Prop. I.26: ASA/AAS Congruence & Thales’ Ship-Distance Theorem · Euclid [I.26]

Eudemus’s famous report in his History of Geometry attributing Prop. I.26 to Thales, who must have used it in his method of finding the distance of ships at sea.

  • euclid proclus:in-euclid:352.14-18 — Heath Vol. I, p. 36 (chapter_iv) [folio 0050]

Friedlein 355, 20–356, 16 — Props. I.27–I.28: Book IV Opening & Sufficient Conditions for Parallels · Euclid [I.27]

Opening of Book IV of the Commentary (before Prop. I.27): systematic review of the three defining symptoms of parallel straight lines in a plane.

  • euclid proclus:in-euclid:355.20-356.16 — Heath Vol. I, p. 42 (chapter_iv) [folio 0056]

Friedlein 356, 11 — Props. I.27–I.28: Book IV Opening & Sufficient Conditions for Parallels · Euclid [I.27]

Mention of Hippias’s quadratrix (tetragonizousa) and higher curves in connection with geometric properties.

  • euclid proclus:in-euclid:356.11 — Heath Vol. I, p. 266 (Bk. I, propositions) [folio 0280]

Friedlein 357, 9 — Props. I.27–I.28: Book IV Opening & Sufficient Conditions for Parallels · Euclid [I.27]

Commentary on Prop. I.27: definition and directional analysis of alternate angles (enallax goniai) formed by a transversal.

  • euclid proclus:in-euclid:357.9 — Heath Vol. I, p. 308 (Bk. I, propositions) [folio 0322]

Friedlein 361, 21 — Props. I.27–I.28: Book IV Opening & Sufficient Conditions for Parallels · Euclid [I.27]

Commentary on Prop. I.28: comparison of Euclid’s two criteria with Ptolemy’s symmetric formulation of interior angles on both sides of the transversal.

  • euclid proclus:in-euclid:361.21 — Heath Vol. I, p. 28 (chapter_iii) [folio 0042]

Friedlein 365–373 — Prop. I.29: Ptolemy’s Treatise & Proclus’s Attempted Proof of Postulate 5 · Euclid [I.29]

Grand excursion under Prop. I.29: Claudius Ptolemy’s attempted proof of Postulate 5, Proclus’s refutation of its hidden assumption, and Proclus’s own bounded-divergence proof.

  • hoenen prose — De Noetica Geometriae Caput I, §5, p. 15 (teste Proclo: ancient attempted proofs of Postulate V by Ptolemy and Proclus (Friedlein 365–373))

Friedlein 365, 7–11 — Prop. I.29: Ptolemy’s Treatise & Proclus’s Attempted Proof of Postulate 5 · Euclid [I.29]

Citation of Claudius Ptolemy’s treatise ‘That straight lines produced from angles less than two right angles meet’ and summary of his preliminary lemmas.

  • euclid proclus:in-euclid:365.7-11 — Heath Vol. I, p. 204 (Bk. I, first_principles) [folio 0218]

Friedlein 365, 14–367, 27 — Prop. I.29: Ptolemy’s Treatise & Proclus’s Attempted Proof of Postulate 5 · Euclid [I.29]

Full reproduction of Ptolemy’s attempted symmetry proof of Prop. I.29 and deduction of Euclid’s Fifth Postulate.

  • euclid proclus:in-euclid:365.14-367.27 — Heath Vol. I, p. 205 (Bk. I, first_principles) [folio 0219]

Friedlein 368, 26 — Prop. I.29: Ptolemy’s Treatise & Proclus’s Attempted Proof of Postulate 5 · Euclid [I.29]

Proclus’s critique of Ptolemy’s parallel proof and refutation of a Zeno/Achilles-style infinite-bisection sophism claiming converging lines never meet.

  • euclid proclus:in-euclid:368.26 — Heath Vol. I, p. 206 (Bk. I, first_principles) [folio 0220]

Friedlein 371, 10 — Prop. I.29: Ptolemy’s Treatise & Proclus’s Attempted Proof of Postulate 5 · Euclid [I.29]

Proclus’s own attempted proof of Postulate 5, explicitly invoking Aristotle’s De Caelo I.5 lemma that diverging rays separate beyond any given finite distance.

  • euclid proclus:in-euclid:371.10 — Heath Vol. I, p. 45 (chapter_iv) [folio 0059]

Friedlein 379, 2–5 — Prop. I.32: Triangle Angle-Sum, Pythagorean Proof & Polygon Formula · Euclid [I.32]

Commentary on Prop. I.32: Eudemus’s testimony on the ancient Pythagorean discovery of the universal angle-sum theorem for all triangles.

  • euclid proclus:in-euclid:379.2-5 — Heath Vol. I, p. 317 (Bk. I, propositions) [folio 0331]

Friedlein 379, 2–15 — Prop. I.32: Triangle Angle-Sum, Pythagorean Proof & Polygon Formula · Euclid [I.32]

Preservation of the elegant Pythagorean proof of Prop. I.32 drawing a single parallel to the base through the triangle’s apex.

  • euclid proclus:in-euclid:379.2-15 — Heath Vol. I, p. 320 (Bk. I, propositions) [folio 0334]

Propositions I.33–I.48 (Friedlein pp. 384–436)

Codex tracks: English · Greek (1873) · Summary · Notes

Friedlein 393 — Props. I.33–I.34: Parallelogrammic Areas & Diameter Bisection · Euclid [I.34]

Commentary on Prop. I.34: why Euclid introduces the term ‘parallelogrammic areas’ (parallelogramma choria) and how Prop. I.34 constructs the concept.

  • euclid proclus:in-euclid:393 — Heath Vol. I, p. 325 (Bk. I, propositions) [folio 0339]

Friedlein 394–396 — Prop. I.35: Equal Parallelograms & Birth of Locus Theorems · Euclid [I.35]

Commentary on Prop. I.35: birth of Locus Theorems (topika theoremata) in the Elements and distinction between equality of area and congruence of figure.

  • euclid proclus:in-euclid:394-6 — Heath Vol. I, p. 329 (Bk. I, propositions) [folio 0343]

Friedlein 396–398 — Prop. I.35: Equal Parallelograms & Birth of Locus Theorems · Euclid [I.35]

Analysis of the paradoxical wonder of Prop. I.35 (shearing parallelograms of arbitrarily large perimeter having equal area, anticipating isoperimetric theory).

  • euclid proclus:in-euclid:396-8 — Heath Vol. I, p. 26 (chapter_iii) [folio 0040]

Friedlein 399–400 — Prop. I.35: Equal Parallelograms & Birth of Locus Theorems · Euclid [I.35]

Completion of the remaining two positional cases of Prop. I.35 omitted in Euclid’s concise text.

  • euclid proclus:in-euclid:399-400 — Heath Vol. I, p. 328 (Bk. I, propositions) [folio 0342]

Friedlein 403, 5 — Props. I.36–I.41: Equal Parallelograms & Triangles Between Parallels · Euclid [I.41]

Textual transition between Props. I.36 and I.37 on equal parallelograms and triangles between the same parallels.

  • euclid proclus:in-euclid:403.5 — Heath Vol. I, p. 332 (Bk. I, propositions) [folio 0346]

Friedlein 407, 4–17 — Props. I.36–I.41: Equal Parallelograms & Triangles Between Parallels · Euclid [I.41]

Commentary on Prop. I.39: partial conversion when two hypotheses yield a conclusion, and exclusion of triangles on opposite sides of the base.

  • euclid proclus:in-euclid:407.4-17 — Heath Vol. I, p. 337 (Bk. I, propositions) [folio 0351]

Friedlein 414, 15–415, 16 — Props. I.36–I.41: Equal Parallelograms & Triangles Between Parallels · Euclid [I.41]

Commentary on Prop. I.41: Proclus’s geometric exercise extending triangle-parallelogram area relations to trapezia.

  • euclid proclus:in-euclid:414.15-415.16 — Heath Vol. I, p. 338 (Bk. I, propositions) [folio 0352]

Friedlein 419 — Props. I.44–I.45: Pythagorean Application of Areas (Parabole, Hyperbole, Elleipsis) · Euclid [I.44]

Commentary on Prop. I.44: Eudemus’s testimony that Application of Areas (parabole), Exceeding (hyperbole), and Falling-Short (elleipsis) are ancient discoveries of the Pythagorean muse.

Friedlein 423, 18 — Prop. I.46: Constructing the Square & Posidonius’s Definition · Euclid [I.46]

Commentary on Prop. I.46: philological distinction between ‘constructing’ (systesasthai) a triangle from parts and ‘describing on’ (anagrapsai apo) a square from a single side.

  • euclid proclus:in-euclid:423.18 — Heath Vol. I, p. 348 (Bk. I, propositions) [folio 0362]

Friedlein 424–425 — Prop. I.46: Constructing the Square & Posidonius’s Definition · Euclid [I.46]

Proofs that squares described on equal straight lines are equal, and conversely that equal squares have equal sides.

  • euclid proclus:in-euclid:424-5 — Heath Vol. I, p. 348 (Bk. I, propositions) [folio 0362]

Friedlein 426, 6 — Prop. I.47: The Theorem of Pythagoras, Euclid’s Generalization & Triples · Euclid [I.47]

Commentary on Prop. I.47: critical assessment of the story that Pythagoras sacrificed an ox, and Proclus’s glowing tribute to Euclid’s Book I proof and Book VI generalization.

  • euclid proclus:in-euclid:426.6 — Heath Vol. I, p. 350 (Bk. I, propositions) [folio 0364]

Friedlein 426, 6–9 — Prop. I.47: The Theorem of Pythagoras, Euclid’s Generalization & Triples · Euclid [I.47]

Proclus’s cautious phrasing (‘if we listen to those who wish to recount ancient history…’) regarding the legend of Pythagoras’s sacrifice on discovering I.47.

  • euclid proclus:in-euclid:426.6-9 — Heath Vol. I, p. 319 (Bk. I, propositions) [folio 0333]

Friedlein 429, 9–15 — Prop. I.47: The Theorem of Pythagoras, Euclid’s Generalization & Triples · Euclid [I.47]

Two ancient methods for generating integer right-angled triangles: the Pythagorean odd-number rule and the Platonic even-number rule.

  • euclid proclus:in-euclid:429.9-15 — Heath Vol. I, p. 26 (chapter_iii) [folio 0040]

Friedlein 432, 9 — Prop. I.48: Converse of Pythagoras, Peroration & Manuscript Scholium · Euclid [I.48]

Closing peroration of the Commentary on Book I (Prop. I.48), honoring Syrianus’s oral teaching and inviting successors to expound the remaining Books of Euclid.

  • euclid proclus:in-euclid:432.9 — Heath Vol. I, p. 32 (chapter_iv) [folio 0046]