Chapter VI Summary and Outline: The Scholia

Executive Summary

Sir Thomas L. Heath synthesizes the philological and mathematical findings of J. L. Heiberg (Euclidis Elementa, Vol. V, 1888, and Om Scholierne til Euklids Elementer, 1888) regarding the approximately 1,500 ancient and Byzantine Greek marginal notes (scholia) that accompany Euclid’s Elements.

While the majority of scholia are elementary glosses or trivial calculations, critical analysis reveals a rich secondary transmission containing invaluable historical fragments, textual benchmarks, and advanced computational techniques.

1. Historical and Mathematical Value of the Scholia

The scholia preserve irreplaceable historical testimonies concerning early Greek geometry:

  • Pythagorean Discoveries: IV. No. 2 attributes the constructions of Book IV to the Pythagoreans. X. No. 1 records the Pythagorean discovery of incommensurability and its mystical suppression. XIII. No. 1 records that three regular polyhedra (cube, pyramid, dodecahedron) were discovered by the Pythagoreans.

  • Theaetetus of Athens: X. No. 62 attributes Euclid X. 9 (commensurability in square) to Theaetetus; XIII. No. 1 attributes the octahedron and icosahedron to Theaetetus.

  • Eudoxus of Cnidus: V. No. 1 and V. No. 3 explicitly attribute the general theory of proportion (Book V) to Eudoxus, with Euclid responsible for its systematic arrangement.

  • Etymology of the Gnomon (γνώμων): II. Nos. 11 and 13 explain how the gnomon received its mathematical name because it enables one to know (γνωρίζεται) the whole or remainder when added or subtracted.

  • Pappus and Apollonius: X. No. 1 details Apollonius’ classification of irrationals; XII. No. 2 preserves Pappus’ construction for inscribing similar polygons in circles (as attested by Eutocius).

2. The Primary Collections: Schol. Vat. and Schol. Vind.

Heiberg classified the vast body of scholia into two primary corpora:

  1. Scholia Vaticana (Schol. Vat.) (138 core numbers):

    • Primary Witnesses: MSS. P, B, F, Vat. 204 (10th c.), and Vat. 192 (14th c.).

    • Sources: Book I comprises intelligent, recast excerpts from Proclus; Books II–XIII derive primarily from the lost commentary of Pappus of Alexandria (confirmed by Woepcke’s Arabic text).

    • Dating & Origin: Compiled after 500 A.D. and before 888 A.D., likely in the late 6th century during the mathematical revival under Isidore of Miletus. Originally written in the margins of an ante-Theonine MS. of the P family.

    • Text-Critical Decisiveness: Demonstrates that Pappus did not have the Book X lemmas in his text, proving that all Book X lemmas are 4th-century post-Pappus interpolations anterior to Theon.

  2. Scholia Vindobonensia (Schol. Vind.) (~450 numbers):

    • Primary Witnesses: MSS. V (Viennese 103), q (Paris 2344), and l (Laurentian XXVIII, 2).

    • Sources: More literal but less intelligent extracts from Proclus on Book I; a Theonine adaptation of Schol. Vat.; and numerous late Byzantine glosses, trivial numerical examples, and sophistical difficulties (ἀπορίαι).

    • Dating: Compiled in the 11th century, likely associated with the polymath Michael Psellus.

3. Later Byzantine Commentators and Humanist Editions

  • Michael Psellus (11th c.): Associated with the definitions of Book X and scholia on Book I (Nos. 40, 49).

  • Maximus Planudes (late 13th c.): Lectures on the Elements transcribed into MS. l (VI. No. 6, X. No. 223).

  • Ioannes Pediasimus (early 14th c.): Byzantine scholar cited in MS. l.

  • Theodorus Cabasilas (15th c.): Owned both MSS. B and b, copying and collating scholia between them; cited Demetrius Cydonius.

  • Isaak Argyrus (Isaak Monachus) (14th c.): Author of Varia miscellanea and notes on Books I–VI, edited by Conrad Dasypodius (1573, 1579).

  • Barlaam of Seminara (14th c.): Monastic scholar whose arithmetical commentary on Book II was printed by Dasypodius (1564).

  • Renaissance Prints: Basel editio princeps (Simon Grynaeus, 1533), Giorgio Valla (De expetendis et fugiendis rebus, 1501), and Federico Commandino (Scholia antiqua, 1572).

4. Sexagesimal Calculations and Numerical Precision

Heath and Friedrich Hultsch highlight the remarkable computational accuracy in the Byzantine scholia to Book X:

  • Computations carried out to fourth-sixtieths ($1/60^4$).

  • Approximations of irrational roots matching Hellenistic astronomical tables: $$\sqrt{27} \approx 5^\circ\ 11'\ 46''\ 10''' \implies \sqrt{3} \approx 1^\circ\ 43'\ 55''\ 23'''$$ (Hipparchus’ Table of Chords value, correct to seven decimal places). $$\sqrt{8} \approx 2^\circ\ 49'\ 42''\ 20'''\ 10'''' \implies \sqrt{2} \approx 1.41421335$$

Detailed Outline

§ 1. Corpus, Character, and Historical Content (pp. 64–65)

  • Heiberg’s collection of ~1,500 scholia and analytical monograph (Om Scholierne til Euklids Elementer, 1888).

  • General limitations and notable mathematical exceptions (extensions of solid angles XI. 22–23; gap in XII. 17).

  • Historical attributions:

    • Gnomon etymology (II. Nos. 11, 13).

    • Pythagorean origin of Book IV (IV. No. 2).

    • Eudoxian origin of Book V (V. Nos. 1, 3).

    • Incommensurability and Apollonius (X. No. 1).

    • Theaetetus and X. 9 (X. No. 62).

    • Platonic regular solids (XIII. No. 1).

  • Spurious additions distinguished from genuine scholia: marginal notes that infiltrated the text before Theon.

§ 2. Scholia Vaticana (Schol. Vat.) (pp. 65–69)

  • Manuscript witnesses: P, B, F, Vat. 204, Vat. 192, Vᶜ.

  • Contents and structure:

    • Book I: Recast excerpts from Proclus filling lacunae at I. 36 and I. 43.

    • Books II–XIII: Direct derivation from the lost commentary of Pappus of Alexandria (Woepcke’s Arabic text, Eutocius’ citation of XII. 2).

  • Stemmatic significance:

    • Exclusion of Book X lemmas from Pappus’ text; dating lemma interpolations to early 4th century.

    • Concordance with MS. P against Theonine alterations (e.g. IX. 19).

  • Historical synthesis: Origin in the late 6th century School of Isidore of Miletus.

§ 3. Scholia Vindobonensia (Schol. Vind.) (pp. 69–71)

  • Manuscript witnesses: Vᵃ, q, l, Vᵇ, q¹.

  • Contrast with Schol. Vat.: Verbatim excerpts from Proclus, inclusion of trivial scholia, headings, and numerical examples.

  • Association with Theonine text (notes on interpolated VII. 22).

  • Dating and attribution: 11th century; connection to Michael Psellus (Magliabecch. MS. XI, 53).

§ 4. Paleography of the Secondary Hands (pp. 71–73)

  • Distinct scribal layers in MSS. V, q, and P.

  • Introduction of East-Arabian numerals into Greek manuscripts (11th–12th c.).

  • Byzantine scholars:

    • Theodorus Cabasilas (15th c.) and the collation between MSS. B and b.

    • Maximus Planudes (late 13th c.) and notes in MS. l.

    • Ioannes Pediasimus (14th c.).

§ 5. Printed Editions and Renaissance Transmission (pp. 73–74)

  • Greek publications: Basel editio princeps (1533), Oxford edition (1703), August (1829), Wachsmuth, and Knoche.

  • Latin publications:

    • Giorgio Valla (De expetendis et fugiendis rebus, 1501).

    • Federico Commandino (Scholia antiqua, 1572).

    • Conrad Dasypodius: Lexicon mathematicum (1573), Isaaci Monachi Scholia (1579), Barlaam (1564).

  • Friedrich Hultsch’s analysis: Geminus fragments in Book I scholia; Eudoxian tradition in Book V.

§ 6. Sexagesimal Calculations in Book X (p. 74)

  • Mathematical facility of Byzantine commentators.

  • Use of sexagesimal fractions to fourth-sixtieths ($1/60^4$).

  • Square root extractions: $\sqrt{27} \approx 5^\circ\ 11'\ 46''\ 10'''$; $\sqrt{8} \approx 2^\circ\ 49'\ 42''\ 20'''\ 10''''$.

Concordance Table of Commentators, Manuscripts, and Editors Cited

Name / Manuscript

Era / Date

Role & Provenance

Key Contribution & Context

J. L. Heiberg

1883–1888

Copenhagen

Collected and classified ~1,500 scholia in Euclidis Elementa Vol. V; published Om Scholierne til Euklids Elementer.

Scholia Vaticana (Schol. Vat.)

c. 500–600 A.D.

Alexandria / Constantinople

Most authoritative ancient collection; extracts from Proclus (Book I) and Pappus (Books II–XIII).

Scholia Vindobonensia (Schol. Vind.)

11th c.

Constantinople

Second major collection; preserves fuller Proclus excerpts and late Byzantine annotations.

Pappus of Alexandria

fl. 320 A.D.

Alexandria

Ancient commentator whose lost Euclid commentary is the primary source of Schol. Vat. II–XIII.

Proclus Diadochus

410–485 A.D.

Athens

Primary source for Book I scholia in both Schol. Vat. and Schol. Vind.

Franz Woepcke

1856

Paris

Discovered the Arabic translation of Pappus’ commentary on Book X, confirming Pappus as source for Schol. Vat.

Isidore of Miletus

fl. 532 A.D.

Miletus / Constantinople

Architect of Hagia Sophia and mathematician; probable compiler of the Schol. Vat. archetype.

Michael Psellus

1018–1078

Constantinople

Byzantine statesman and polymath; attributed author of scholia on Book X definitions and Schol. Vind. material.

Theodorus Cabasilas

15th c.

Constantinople / Mistra

Byzantine scholar who owned MSS. B and b and transferred scholia between them.

Maximus Planudes

c. 1260–1305

Constantinople

Polymath monk whose lectures on Euclid are preserved in scholia in MS. Laurentian XXVIII, 2 (l).

Ioannes Pediasimus

14th c.

Constantinople / Thessalonica

“Keeper of the Chancellery” (chartophylax); lectured on Euclid; cited in MS. l.

Isaak Argyrus (Isaak Monachus)

14th c.

Constantinople

Byzantine monk whose geometric scholia were published in Latin translation by Dasypodius.

Barlaam of Seminara

c. 1290–1348

Calabria / Constantinople

Monk and scholar who composed an arithmetical commentary on Euclid Book II.

Conrad Dasypodius

1532–1596

Strasbourg

Mathematician and astronomer; published Lexicon mathematicum (1573) and Latin translations of Isaak and Barlaam.

Friedrich Hultsch

1888–1904

Dresden

Analyzed sexagesimal fractional computations in Book X scholia and the Geminus extracts in Pauly-Wissowa.

Giorgio Valla

1501

Venice

Published ~200 Euclidean scholia in Latin translation in De expetendis et fugiendis rebus.

Federico Commandino

1572

Urbino

Translated the majority of Schol. Vat. into Latin under the title Scholia antiqua.

Simon Grynaeus

1533

Basel

Editor of the Greek editio princeps of Euclid; printed the first selection of Greek scholia.

MS. Vat. 204

10th c.

Vatican Library

Preserves Schol. Vat. as an independent collection without Euclid’s text.

MS. Vat. 192

14th c.

Vatican Library

Contains complete Schol. Vat. (I. No. 88 to XIII. No. 44) as an independent appendix.

MS. Vienn. Philos. Gr. 103 (V)

12th c.

Vienna National Library

Primary witness of Schol. Vind.; contains East-Arabian numerals and Byzantine calculations.

MS. Paris. Gr. 2344 (q)

12th c.

Paris

Key witness for Schol. Vind. with annotations by Theodorus Antiochita.

MS. Laurent. XXVIII, 2 (l)

13th–14th c.

Florence

Witness for Schol. Vind. containing lecture notes from Planudes and Pediasimus.

MS. Magliabecch. XI, 53

15th c.

Florence

Compendium explicitly linking Michael Psellus to the Euclidean scholia.