Chapter II Summary and Outline: Euclid’s Other Works¶
Executive Summary¶
Sir Thomas L. Heath provides a critical catalogue and mathematical evaluation of all Euclidean treatises outside the Elements, drawing on ancient testimonies from Proclus, Pappus of Alexandria, Archimedes, Eutocius, and Arabic bio-bibliographies (notably Ibn an-Nadīm’s Kitāb al-Fihrist).
Heath divides these works into lost treatises, extant mathematical and physical treatises, and spurious/apocryphal attributions:
Lost Advanced Geometric Treatises:
The Pseudaria (Ψευδάρια, Book of Fallacies): An introductory pedagogical work designed as a “cathartic and exercise” to train students in recognizing geometric fallacies and paralogisms.
The Porisms (Πορίσματα, 3 Books): A masterpiece of higher geometry contained in Pappus’ Treasury of Analysis (τόπος ἀναλυόμενος), comprising 171 theorems and 38 lemmas. Heath explores the ancient definitions (distinguishing a porism from a theorem and a problem), Simson’s celebrated restoration, Chasles’ link to projective geometry and anharmonic ratios, and Zeuthen’s conic hypothesis.
The Surface-Loci (τόποι πρὸς ἐπιφανείᾳ, 2 Books): Loci consisting of surfaces (cylinders, cones, and surfaces of revolution of the second degree). Preserves Pappus’ lemma containing the first complete ancient proof of the general focus-and-directrix property of conics.
The Conics (4 Books): Euclid’s systematization of conic sections connecting the work of Aristaeus (Solid Loci) with the subsequent monumental synthesis of Apollonius of Perga.
Extant Works:
The Data (Δεδομένα): Supplementary to the planimetric books of the Elements, serving as an elementary manual of geometric analysis and the geometric solution of quadratic equations.
The On Divisions (of Figures) (περὶ διαιρέσεων βιβλίον): Lost in Greek; recovered via Woepcke’s 1851 translation of a Paris Arabic manuscript (emended by Thābit ibn Qurra) and John Dee / Commandino’s 1570 Latin edition of Muḥammad al-Baghdādī; comprehensively reconstructed by R. C. Archibald.
The Phaenomena (Φαινόμενα): Astronomical application of spherical geometry (Sphaerica), resting on Autolycus of Pitane and Eudoxus of Cnidus.
The Optics (Ὀπτικά): Genuine Euclidean treatise on direct vision and perspective (distinguished from the spurious Catoptrica likely due to Theon of Alexandria).
The Elements of Music (αἱ κατὰ μουσικὴν στοιχειώσεις): Containing the mathematical Sectio Canonis (κατατομὴ κανόνος, Pythagorean intervals), while the Harmonic Introduction (εἰσαγωγὴ ἁρμονική) is identified as the work of Aristoxenus’ pupil Cleonides.
Arabic Bibliographical Records:
The Fihrist catalogue analyzed by H. Suter, documenting Arabic knowledge of the Euclidean corpus and rejecting mechanical tracts such as De levi et ponderoso as Hellenistic interpolations postdating Archimedes.
Detailed Outline¶
§ 1. The Pseudaria (Book of Fallacies) (pp. 7–8)¶
Proclus’ characterization (ed. Friedlein, p. 70): cathartic training in the discovery of paralogisms.
Elementary scope and pedagogical function alongside the Elements.
Alexander of Aphrodisias’ reference in his commentary on Aristotle’s Sophistici Elenchi.
§ 2. The Data (pp. 8)¶
Role in the Treasury of Analysis (Pappus VII, p. 638).
Conceptual framework: proving what properties become given (in magnitude, species, or position) under given conditions.
Connection with the geometric solution of quadratic equations.
Modern editions: Heiberg & Menge (Teubner Vol. VI), Marinus’ commentary.
§ 3. On Divisions of Figures (pp. 8–10)¶
Proclus’ notice: dividing figures into like (e.g. triangles into triangles) and unlike figures (e.g. triangle into triangle and quadrilateral).
Transmission history:
John Dee’s discovery, Commandino’s 1570 edition (Machometus Bagdadinus / Abū Bakr Muḥammad ibn ‘Abd al-Bāqī al-Baghdādī).
Franz Woepcke’s discovery of the Paris Arabic MS (1851), containing 36 propositions.
Restoration and commentary by R. C. Archibald (1915) incorporating Leonardo of Pisa’s Practica geometriae (1220).
§ 4. The Porisms (pp. 10–15)¶
Significance and extensive historiography (Robert Simson, Michel Chasles, J. L. Heiberg, Gino Loria).
Complete extract of Pappus’ preface to the Porisms (Collection VII, pp. 648–660):
Nature intermediate between theorems and problems.
Ancient vs. later definitions: “that which falls short of a locus-theorem in respect of its hypothesis.”
Enunciation of the grand ten-proposition porism (the complete quadrangle / collinearity theorem).
Generalization to $n$ straight lines (Simson’s theorem).
List of 29 classes of things sought across 3 books (171 theorems, 38 lemmas).
Proclus’ distinction: corollary (incidental bonus) vs. Euclidean porism (seeking and exhibiting an existing entity, e.g. Prop. III.1).
Modern mathematical interpretations: projective geometry, transversals, anharmonic ratios (Chasles), and Zeuthen’s conic-locus hypothesis.
§ 5. The Surface-Loci (pp. 15–16)¶
Placement in the Treasury of Analysis (2 books).
Conjectured contents: cones, cylinders, and quadric surfaces of revolution (Chasles).
Pappus’ two lemmas:
Lemma 1: Elliptical parallel sections of cylinders and cones (Tannery’s restoration).
Lemma 2: First complete ancient enunciation and demonstration of the focus-and-directrix property of conics.
§ 6. The Conics (pp. 16)¶
Four books of Euclid completed and extended to eight by Apollonius.
Relation to Aristaeus’ Solid Loci.
Archimedes’ references to the “Elements of Conics.”
Early terminology: sections of right-angled, acute-angled, and obtuse-angled cones.
§ 7. The Phaenomena (pp. 16–17)¶
Extant astronomical work on spherical geometry.
Textual recensions: MS Vindobonensis philos. Gr. 103 vs. Gregory’s text; Menge’s edition (Teubner Vol. VIII).
Historical dependence on Autolycus (On the Moving Sphere) and Eudoxus.
§ 8. The Optics and Catoptrica (pp. 17)¶
Euclid’s genuine Optics (Heiberg Teubner Vol. VII) vs Theon’s recension.
Spuriousness of the Catoptrica.
§ 9. The Elements of Music and the Arabic Corpus (pp. 17–18)¶
Sectio Canonis (κατατομὴ κανόνος): Pythagorean interval theory, quoted by Porphyry.
Harmonic Introduction (εἰσαγωγὴ ἁρμονική): attributed correctly to Cleonides.
Analysis of Ibn an-Nadīm’s Fihrist (Suter’s edition):
Data, Phaenomena, Divisions (emended by Thābit), Porisms, Canon.
Spurious mechanical tracts: De levi et ponderoso (Basel 1537).