Chapter IV Summary and Outline: Proclus and His Sources¶
Executive Summary¶
Sir Thomas L. Heath undertakes a forensic source-critical analysis of Proclus Diadochus (410–485 A.D.) and his Commentary on Book I of Euclid’s Elements. Along with the Collection of Pappus, Proclus’ commentary constitutes the paramount surviving historical authority for pre-Euclidean Greek geometry.
Drawing primarily on the foundational source-study by J. G. van Pesch (De Procli fontibus, 1900), and evaluating earlier hypotheses by Paul Tannery (La Géométrie grecque, 1887), Heath delineates the intellectual environment, textual practices, primary library, and original contributions of Proclus:
Character and Purpose of Proclus’ Commentary:
Educated in Alexandria (Aristotelian logic under Olympiodorus, mathematics under Heron) and Athens (Plutarch and Syrianus), Proclus was fundamentally a Neoplatonist philosopher rather than an inventive mathematician.
Mathematics served an anagogic function: a propaedeutic (propaideia) clearing the spiritual eye to attain intellectual apprehension of universal forms (νοερὰ γνῶσις) and elucidate Plato’s Timaeus.
Origin: Transcripts of elementary lectures delivered to beginners at the Neoplatonic Academy, subsequently revised, enlarged, and deepened for a scholarly audience.
Textual limit: Despite expressed intentions to continue through Books II, III, and XII, Proclus never completed commentaries beyond Book I.
Quotation Habits and Source Methodology:
Proclus was a compiler of earlier authorities who systematically translated borrowed arguments into his own stylistic idiom; even when purporting to quote verbatim (Plato says, Plotinus says, the geometer says), he rarely transcribed word-for-word.
He seldom cited titles of treatises, omitting names when dealing with well-known commentaries. Consequently, an absence of citation does not imply Proclean originality.
Catalogue of Directly Consulted Works:
Eudemus of Rhodes (History of Geometry, On the Angle): The supreme 4th-century B.C. Peripatetic historical authority, cited directly for Thales, Oenopides, the Pythagoreans, and the geometric application of areas (παραβολή, ὑπερβολή, ἔλλειψις).
Geminus of Rhodes (The Theory of the Mathematics): 1st-century B.C. Stoic encyclopaedia of mathematical philosophy; source for the overarching classification of mathematical sciences, the theory of “uniform lines” (straight line, circle, cylindrical helix), asymptotes vs parallels, and the foundations of axioms and postulates.
Earlier Commentators: Heron of Alexandria, Porphyry, and Pappus of Alexandria.
Apollonius of Perga: A foundational work on elementary geometry (General Treatise, καθόλου πραγματεία) attempting to remodel definitions, prove Axiom 1, and supply alternate proofs for I.10, I.11, and I.23.
Claudius Ptolemy: Monograph on the Parallel Postulate (On the meeting of straight lines produced from angles less than two right angles).
Posidonius: Monograph refuting Zeno of Sidon, the Epicurean sceptic who attacked Euclid’s unstated lemmas (e.g. that straight lines cannot share a common segment).
Carpus mechanicus: Astronomical treatise debating the ontological primacy of problems versus theorems.
Classical and Neoplatonic Literature: Plato, Aristotle, Archimedes, Plotinus, Syrianus, and Pythagorean mystical literature.
The Eudemian Summary Controversy:
The famous historical summary (pp. 64–70) divides into two sections at Euclid’s appearance. Heath refutes Tannery’s attribution of the whole to Geminus, concluding it represents a compilation from a Hellenistic compendium of Eudemus’ history supplemented by Proclus.
Proclus’ Original Mathematical Contributions:
Investigation into the ultimate geometric causes (τὴν αἰτίαν καὶ τὸ διὰ τί) of propositions (I.8, I.16, I.17, I.18, I.32, I.47).
Dialectical refutation of Ptolemy’s parallel proof.
Proclus’ Own Demonstration of the Fifth Postulate: Based on Aristotle’s principle (De Caelo I.5) that straight lines diverging from a point at an angle separate by a distance exceeding any finite magnitude when extended indefinitely.
Detailed Outline¶
§ 1. Proclus’ Biography and Philosophical Outlook (pp. 29–31)¶
Importance of Proclus and Pappus as surviving repositories of lost Greek mathematical history.
Chronology: 410–485 A.D.
Education in Alexandria (Olympiodorus, Heron) and Athens (Plutarch, Syrianus).
Platonist teleology: mathematics subordinate to metaphysics; the “eye of the soul” and Timaeus cosmology.
The elementary student audience (δυσθεωρήτους τοῖς εἰσαγομένοις) and subsequent revision for publication.
§ 2. Scope and Incompleteness of the Commentary (pp. 31–33)¶
Internal evidence of unfulfilled intentions: projected discussions on Books II, III, and XII.
Closing prayer and farewell to the reader (ed. Friedlein, p. 432).
Rejection of the hypothesis of lost commentaries on later books; deconstruction of the Vatican Scholia (Scholia Vaticana).
§ 3. Principles of Source Criticism for Proclus (pp. 33–34)¶
Nature of the commentary as a selective compilation.
Distaste for verbatim citation; loose paraphrasing of Plato, Aristotle, Plotinus, and Euclid.
Rare occurrence of explicit book titles.
Criteria established by van Pesch for attributing anonymous notes to specific authorities.
§ 4. Eudemus of Rhodes as Primary Historical Source (pp. 35–38)¶
Eudemus’ Peripatetic historical trilogy: geometry, arithmetic, astronomy.
Direct availability of Eudemus’ History of Geometry to Proclus, Simplicius, and Eutocius.
Stylistic traits: summary, memorandum style (ὑπομνηματικὸς τρόπος).
Specific citations:
Thales: determination of ship distances (I.26), vertically opposite angles (I.15), bisection of circle by diameter.
Oenopides of Chios: perpendicular construction (I.12) as the “gnomonic line,” angle copy (I.23).
Pythagoreans: triangle angle-sum (I.32), space-filling regular polygons, application of areas (παραβολή, ὑπερβολή, ἔλλειψις).
Plato and Leodamas: the analytical method.
The ox-sacrifice legend (I.47) traced to Apollodorus Logisticus and Plutarch rather than Eudemus.
The Eudemian Historical Summary (pp. 37–38): Analysis of structure, non-Eudemian second half, and refutation of Tannery’s Geminus hypothesis.
§ 5. Geminus of Rhodes and the Encyclopaedic Classification (pp. 38–42)¶
Life, Stoic affiliation, and date (fl. c. 70 B.C.).
The treatise The Theory (or Doctrine) of the Mathematics (τῶν μαθημάτων θεωρία).
Geminus’ systematic classification:
Division of mathematics into pure (arithmetic, geometry) and applied (mechanics, astronomy, optics, geodesy, harmonics, logistics).
Classification of curves: simple vs mixed, uniform (homoeomeric) vs non-uniform. The three uniform curves: straight line, circle, cylindrical helix.
Asymptotes and non-intersecting lines (ἀσύμπτωτοι): hyperbola and conchoid illustrating that convergence does not guarantee intersection.
Philosophical foundations: axioms vs postulates, motion of points, διορισμοί.
The Epicurean critique of I.20 (“obvious even to an ass”).
§ 6. Apollonius of Perga’s Foundational Work (pp. 42–43)¶
The General Treatise (καθόλου πραγματεία) on the elements of geometry.
Empirical definitions of line and surface.
Verbatim proof of Axiom 1 (things equal to the same thing are equal to one another).
Alternate solutions to I.10, I.11, and I.23 criticized by Proclus.
§ 7. Other Direct Sources Consulted by Proclus (pp. 43–44)¶
Claudius Ptolemy: Monograph on the Parallel Postulate.
Posidonius: Polemical work refuting Zeno of Sidon; defense of Euclidean principles against Epicurean scepticism.
Carpus mechanicus: Classification of problems and theorems.
Syrianus: Neoplatonic doctrine of the angle.
Comprehensive bibliography synthesized by van Pesch (p. 44).
§ 8. Proclus’ Original Contributions (pp. 44–45)¶
Search for essential geometric causes (τὸ διὰ τί).
Critical rejoinders to Pappus’ axioms and proofs.
Detailed critique of Ptolemy’s attempt on the Parallel Postulate.
Proclus’ Proof of Postulate 5: Utilizing Aristotle’s theorem on diverging straight lines.