Chapter III Summary and Outline: Greek Commentators Other Than Proclus¶
Executive Summary¶
Sir Thomas L. Heath investigates the rich tradition of Greek commentaries on Euclid’s Elements produced before or alongside Proclus (5th century A.D.). Although Proclus expresses impatience with the pedantic multiplication of minor cases and lemmas in earlier commentaries, he drew heavily upon several major Hellenistic and late-antique commentators whose works are reconstructed through Proclus and the Arabic tradition:
Heron of Alexandria (established as 3rd century A.D., intermediate between Ptolemy and Pappus):
Textual Transmission: Heron’s systematic commentary on the Elements (reaching at least Book VIII.27) survived in Arabic and is extensively preserved in an-Nairīzī’s (Anaritius) commentary, edited from the Leiden Arabic MS (Codex Leidensis 399, 1) by Besthorn & Heiberg and from Gherard of Cremona’s 12th-century Latin translation by Maximilian Curtze.
Methodological Features:
Axiomatic economy: Heron restricted the fundamental axioms to three.
Exhaustive case distinctions (e.g., adding external contact to III.11 in III.12).
Geometric algebra: Semi-algebraic proofs of Book II (Props. 1–10) “without a figure.”
Meeting the “lack of space” objection (ἔνστασις): Alternative constructions for I.11, I.16, and I.20 avoiding the production of straight lines into hypothetical undefined space.
Direct proofs eliminating reductio ad absurdum (I.19, I.25).
Structural extensions: Demonstrating concurrency of lines $AL, BK, CF$ in the Pythagorean theorem (I.47), and extending III.20 to obtuse angles.
Porphyry (c. 232–304 A.D.):
Neoplatonic philosopher whose commentary focused on precise logical dissection of Euclidean enunciations (e.g., defending the phrase “not on the same side” in I.14, and rigorous conditions in I.26).
Alternative proofs of I.20 (avoiding line production) and I.18.
Pappus of Alexandria (fl. c. 300–320 A.D.):
Extensive commentary on the Elements attested by Marinus, Eutocius, and the Kitāb al-Fihrist.
Surviving Arabic Text: A complete Arabic translation of Pappus’ commentary on Book X by Abū ‘Uthmān survives in Paris MS. 952.
Geometrical Criticisms and Paradoxes:
Rejection of the converse of Postulate 4: pointing out that curvilinear angles between intersecting semicircles can equal a right angle without being rectilineal right angles.
Philosophical axioms on dimension and limits (points dividing lines, lines dividing surfaces).
Celebrated proof of I.5 (angles at the base of an isosceles triangle).
Paradoxes of internal lines (I.21) drawn from the collection of Erycinus.
Application of higher curves (spiral of Archimedes, conchoid, quadratrix) to angle division (I.9).
Simplicius (fl. c. 500 A.D.):
Commentary on Euclid’s introductory definitions, postulates, and axioms preserved in an-Nairīzī.
The “Aganis” Conundrum: Quoting word-for-word an attempt by “Aganis” to demonstrate the Parallel Postulate based on equidistant parallel lines. Heath, endorsing Paul Tannery, refutes the old identification of Aganis with Geminus, showing that Simplicius’ phrase “our associate/master Aganis” proves Aganis was a contemporary 5th/6th-century colleague.
Minor Epitomizers:
Aegaeas (or Aenaeas) of Hierapolis, author of an epitome known only through Proclus.
Detailed Outline¶
§ 1. Proclus’ Attitude Toward Earlier Commentators (pp. 19–20)¶
Proclus’ critique of “interminable diffuseness,” pedantic lemmas, and case-splitting.
The Pythagorean ideal: “a figure and a platform, but not a figure and sixpence” (knowledge as ascent rather than mercenary gain).
Identification of the primary ancient authorities: Heron, Porphyry, and Pappus.
§ 2. Heron of Alexandria (pp. 20–24)¶
Chronological Determination:
Refutation of 1st-century B.C. placement based on Ctesibius.
Terminus post quem: Hipparchus’ table of chords (150 B.C.) and Posidonius of Rhodes (135–51 B.C.).
Vitruvius comparison (14 B.C.): lack of cross-citation; divergence in $\pi$ approximation ($3$ vs $3\frac{1}{7}$).
Decisive terminus post quem: Claudius Ptolemy (100–178 A.D.) on great circle navigation and diving experiments in Peri ropōn.
Upper bound: Pappus (writing under Diocletian, 284–305 A.D.) quotes Heron. Result: 3rd century A.D.
Heron’s Commentary Preserved in Arabic:
The Kitāb al-Fihrist testimonies.
An-Nairīzī’s (Anaritius) commentary: Leiden Arabic MS (al-Ḥajjāj translation, ed. Besthorn-Heiberg) and Gerard of Cremona’s Latin MS (ed. Curtze).
Substantive Characteristics of Heron’s Work:
Restricting axioms to three.
Case-analysis: I.35, I.36; III.7, III.8; external contact in III.12; VI.19; VII.19.
Algebraic proofs “without a figure” for II.1–10.
The enstasis (objection based on spatial boundedness): proofs avoiding line extension (I.11, I.16, I.20, I.48).
Direct demonstrations replacing indirect reduction (I.19, I.25).
Important additions: Concurrency of $AL, BK, CF$ in I.47; obtuse angles in III.20/III.22.
Van Pesch’s attributions: Menelaus’ proof of I.25 and Philo’s proof of I.8 preserved through Heron.
§ 3. Porphyry (pp. 24–25)¶
Neoplatonic philosopher (c. 232–304 A.D.).
Textual evidence in Proclus and the Fihrist.
Critical scrutiny of Euclidean conditions: I.14 (“not on the same side”) and I.26.
Alternative proofs: I.20 (avoiding production), I.18 (meeting pedantic objections regarding segment orientation).
§ 4. Pappus of Alexandria (pp. 25–27)¶
Comprehensive scope of Pappus’ lost commentary on the Elements.
Extant Arabic version of Commentary on Book X (Paris MS 952, tr. Abū ‘Uthmān, studied by Woepcke, Heiberg, and Suter).
Philological identification of Bālos / Bānos with Bābos (Pappus).
Testimonies in Marinus (Preface to Data) and Eutocius (inscribing similar polygons in circles, Book XII).
Specific citations in Proclus:
Critique of the converse of Postulate 4 (curvilinear horn angles and semicircles).
Additional axioms (unequals added to equals; continuous space and limits).
Famous simplified proof of I.5.
Investigation of triangle paradoxes (I.21) drawing on Erycinus.
Use of Archimedean spirals and higher curves for angle division (I.9).
Isoperimetric theorems (Zenodorus’ recension).
§ 5. Simplicius and the “Aganis” Mystery (pp. 27–28)¶
Simplicius’ commentary on the principles preserved in an-Nairīzī.
The parallel postulate demonstration of “Aganis.”
Deconstruction of the “Aganis = Geminus” hypothesis:
Paul Tannery’s demonstration that Simplicius refers to Aganis as “our associate” / “our teacher,” placing him c. 500 A.D.
Linguistic proof: Aganis refers to critics who objected to the postulate “even in ancient times” (iam antiquitus).
Aegaeas of Hierapolis and his lost epitome.