Chapter V Summary and Outline: The Text¶
Executive Summary¶
Sir Thomas L. Heath provides a definitive account of the textual history, manuscript tradition, and critical recension of Euclid’s Elements. Grounded in the monumental critical edition of J. L. Heiberg (Euclidis Elementa, 5 vols., 1883–1888) and his supplementary studies (Litterargeschichtliche Studien über Euklid, 1882; Paralipomena zu Euklid, 1903), Heath elucidates how modern philology recovered the authentic text of Euclid from the accumulated revisions of antiquity and the Middle Ages.
1. The Theonine Recension and Vatican MS. 190 (P)¶
The Historical Problem: Prior to the 19th century, all printed editions of Euclid (including the 1533 Basel editio princeps, Commandino, Gregory, and Robert Simson in 1756) were based on manuscripts descending from the 4th-century A.D. recension by Theon of Alexandria. These MSS uniformly bear titles describing them as “from the edition of Theon” (ἐκ τῆς Θέωνος ἐκδόσεως) or “from the lectures of Theon” (ἀπὸ συνουσιῶν τοῦ Θέωνος).
The Decisive Clue: In his commentary on Ptolemy’s Syntaxis, Theon explicitly recorded: “that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements at the end of the sixth book.”
Peyrard’s Discovery: When François Peyrard examined Vatican MS. 190 (designated P) in Paris (having been brought from the Vatican under Napoleon), he discovered that it omitted Theon’s programmatic addition to VI. 33 as well as the Theonine colophons. A marginal note by the first hand at XIII. 6 confirmed that its scribe possessed both traditions and consciously adhered to the “old edition” (παλαιὰ ἔκδοσις) over the “new edition” (καινὴ ἔκδοσις).
2. The Primary Manuscript Witnesses¶
Heath surveys the seven primary Greek manuscripts and crucial auxiliary witnesses deployed in Heiberg’s text:
P (Vatican 190): 10th c. 4to. The sole primary surviving representative of the ante-Theonine text.
F (Laurentian XXVIII, 3): 10th c. Beautiful scholarly hand; best witness among the Theonine family, though marred by 16th c. restorations ($\phi$) copied from Laurentian XXVIII, 6 (f).
B (Bodleian D’Orville 301): Written A.D. 888 by Stephen clericus for Arethas of Caesarea; richly annotated in uncial scholia by Arethas himself.
V (Viennese Philos. Gr. 103): 12th c. Complex composite manuscript containing marginal references to “the book of the Ephesian.”
b (Bologna 18–19): 11th–12th c. Contains a unique Byzantine abbreviation of XI. 36 through Book XII.
p (Paris 2466) and q (Paris 2344): 12th c. witnesses preserving distinct sub-branches of the Theonine tradition.
Palimpsests & Papyri: British Museum palimpsest Add. 17211 (7th–8th c. uncials); Papyrus Herculanensis 1061; Oxyrhynchus Papyrus 29 (c. 300 A.D., illustrating II. 5 without lettering on the figure); Fayūm Papyrus (Vienna Pap. Gr. 26011).
3. Critical Method and Stemmatics¶
The Theonine MSS (F, B, V, p, q) descend from a lost common archetype ($\delta$). Because this archetype was already ancient and copied in uncial script, errors common to all Theonine MSS represent corruption in $\delta$.
Where P and the Theonine family agree, the reading is virtually certain to be ancient and genuine.
Where P and the Theonine family diverge, the agreement of ancient papyri (such as Oxyrhynchus and Fayūm) demonstrates that P does not uniquely monopolize original readings; Theon frequently preserved genuine ancient readings where P’s antecedent had suffered accidental corruption or scribal omission (e.g. the accidental loss of I. 40 in the Fayūm papyrus and P).
4. Classification of Theon’s Alterations¶
Heath details Heiberg’s taxonomy of Theon’s editorial interventions:
Removal of Real and Fancied Blots: Harmonizing apparent mathematical anomalies (e.g. inserting the porism to VI. 19; modifying IX. 19; correcting perceived omissions in solid geometry XI. 1, XI. 38, XIII. 17).
Emendations of Form and Diction: Smoothing colloquial expressions, substituting standard technical formulae (e.g. XII. 4, IX. 15), and enforcing grammatical symmetry across enunciations and proofs.
Additions Supplementing Euclid: Supplying additional cases (VI. 27 case 2; VI. 33 second part on sectors; VII. 22), interpolating intermediate mathematical steps, adding corollaries (porisms), and standardizing diagrams.
Omissions: Pruning concise phrases deemed redundant, eliminating the customary closing formulas (ὅπερ ἔδει δεῖξαι / Q.E.D. and ὅπερ ἔδει ποιῆσαι / Q.E.F.).
5. Pre-Theonine Interpolations and Textual Degeneration¶
Textual corruption did not begin with Theon; the Elements had been actively edited and corrupted centuries earlier:
Saccheri and Simson: Simson’s assumption that Euclid was mathematically flawless (“Euclidem ab omni naevo vindicare”) led him to attribute every infelicity to Theon, whereas Euclid himself occasionally made oversights (e.g. unproved criteria for equal solid angles, incomplete steps in XII. 17).
Alternative Proofs & Cases: Ancient teachers and commentators—notably Heron of Alexandria—composed alternate demonstrations (III. 9, III. 10, III. 12, VI. 30, VI. 31, XI. 22) and supplied missing cases (e.g. external vs internal contact of circles) that gradually crept into the text.
Lemmas and Porisms: Post-propositional lemmas (such as those following VI. 22 and XII. 4) are intrinsically suspect as signs of editorial patching; likewise spurious corollaries (I. 15, III. 31, VI. 20 Por. 2).
Scholia Infiltration: Marginal explanations using forms like καλεῖ or ἐκάλεσε (“he calls” / “called”), as well as alternate analyses and syntheses to XIII. 1–5, migrated into the body text.
Book X and Common Notions: The Aristotelian proof of the incommensurability of the diagonal (X. 116) and propositions X. 112–115 represent late additions. Furthermore, Euclid originally recognized only three (or at most five) Common Notions; the axioms concerning unequals, doubles, halves, and the impossibility of two lines enclosing a space are post-Euclidean interpolations.
Historical Verdict: The text suffered its greatest wave of interpolation in the 3rd century A.D. (between Sextus Empiricus’ pristine text and Iamblichus’ corrupt text), before Theon’s recension codified the late antique school tradition.
Detailed Outline¶
§ 1. The Theonine Recension and Vatican MS. 190 (pp. 46–47)¶
Simson’s editorial claims regarding Theon’s corruption of Euclid.
Theonine manuscript titles: ἐκ τῆς Θέωνος ἐκδόσεως and ἀπὸ συνουσιῶν τοῦ Θέωνος.
Theon’s explicit testimony in his Ptolemy Commentary regarding VI. 33 (sectors).
Peyrard’s discovery of Vatican MS. 190 (P) and the marginal note at XIII. 6 distinguishing the παλαιὰ from the καινὴ ἔκδοσις.
§ 2. The Primary Greek Manuscripts (pp. 47–50)¶
P (Vatican 190): Structure, provenance, hands (P m. 1, P m. 2, P m. rec.).
F (Laurentian XXVIII, 3): Date, paleography, and the 16th c. restorations ($\phi$) from f.
B (Bodleian D’Orville 301): Scribe Stephen clericus, patron Arethas of Caesarea (A.D. 888), marginal notes, and uncial scholia.
V (Viennese Philos. Gr. 103): Paleographical analysis, multi-hand layers, and citations of “the Ephesian.”
b (Bologna 18–19): Proem to Geometry, Byzantine condensed recension of XI. 36–Book XII.
p (Paris 2466) and q (Paris 2344): Independent Theonine witnesses.
§ 3. Palimpsests and Papyri (pp. 50–51)¶
London Palimpsest (Brit. Mus. Add. 17211): 7th–8th c. uncial leaves of Books X–XIII.
Papyrus Herculanensis 1061: Ante-Theonine Epicurean fragment citing Book I.
Oxyrhynchus Papyrus 29: c. 300 A.D. diagrammatic fragment of II. 5 without lettered vertices.
Fayūm Papyrus (Vienna Pap. Gr. 26011): Omission of I. 40 and critical implications for P.
§ 4. Heiberg’s Principles of Textual Criticism (pp. 51–53)¶
The archetype $\delta$ of the Theonine family.
Rules governing agreement and disagreement between P and the Theonine MSS.
Revised evaluation of F and the independent authority of P.
The lack of an early bipartite stemma: ancient horizontal contamination and independent collation.
§ 5. Taxonomy of Theon’s Editorial Alterations (pp. 54–57)¶
Class I: Removal of real and fancied blemishes (VI. 19 Por., IX. 19, IX. 11 Por., V. 7 Por., XI. 1, XI. 38, XIII. 17).
Class II: Stylistic and terminological emendations (XII. 4, IX. 15, standard Greek grammatical forms).
Class III: Editorial interpolations and additions (VI. 27 case 2; VI. 33 second part; VII. 22; intermediate demonstration links).
Class IV: Omissions and condensations (pruning Euclid’s exactness, omission of ὅπερ ἔδει δεῖξαι / ποιῆσαι).
General appraisal: Theon as an honest, conservative schoolmaster rather than a reckless corrupter.
§ 6. Pre-Theonine Interpolations (pp. 57–63)¶
Girolamo Saccheri (Euclides vindicatus, 1733) and Robert Simson: Critiquing the myth of Euclidean perfection.
Class I: Spurious alternative proofs and extra cases:
Heron of Alexandria’s contributions (III. 10, III. 11, III. 12 on touching circles).
Alternatives to III. 9, VI. 30, VI. 31, XI. 22, X. 115.
Class II: Spurious lemmas:
Post-propositional lemmas (VI. 22, XII. 2, XII. 4, XIII. 2, XIII. 13, XIII. 18).
Substantive propositions mistaken for lemmas (XI. 38, XIII. 6).
Book X lemmas (X. 10, X. 14, X. 17, X. 19, X. 20, X. 24, X. 29, X. 42, X. 60).
Class III: Corollaries (Porisms) subject to interpolation (I. 15, III. 31, VI. 20 Por. 2, IV. 5, VI. 8).
Class IV: Infiltration of ancient marginal scholia (καλεῖ, ἐκάλεσε; analyses/syntheses to XIII. 1–5).
Class V: Late additions to Book X:
The side-and-diameter proof (X. 116) and advanced treatise on irrationals (X. 112–115).
Class VI: Minor glosses and unnecessary demonstrations (XI. 26, XI. 35).
Class VII: Interpolated definitions and axioms:
Definition of segment in Book I; VI. Def. 2 (reciprocal figures); VII. Def. 10; XI. Def. 11 (solid angle).
Common Notions: Apollonius’ critique; genuine status of the first 3 (or 5) axioms; rejection of “two lines cannot enclose a space.”
Comparative value of external witnesses: Proclus, Heron, Sextus Empiricus, and Iamblichus.