Chapter VIII Summary and Outline: Principal Translations and Editions

Executive Summary

Sir Thomas L. Heath provides an authoritative historical survey and critical bibliography of the translations, commentaries, and printed editions of Euclid’s Elements across two millennia, from the earliest Latin notices in republican Rome to modern 20th-century critical editions.

1. Roman and Early Medieval Latin Tradition

• Early Roman Acquaintance: Theoretical geometry held little appeal for ancient Romans; Cicero contrasted Greek reverence for abstract geometry with Roman restriction to utilitarian surveying and accounting. Roman agrimensores (Balbus De mensuris, Censorinus fragment) preserved only bare definitions and axioms. Martianus Capella (c. 470 A.D.) recorded the philosophical acclaim for Proposition I. 1 in Greek.

• The Verona Palimpsest (4th c.): Earliest surviving physical evidence of Latin Euclid; palimpsest beneath Pope Gregory the Great’s Moralia in Job, containing fragments of Books XII and XIII in a distinct, free Latin arrangement.

• Boethius (c. 480–524) and Pseudo-Boethius: The surviving two-book geometry attributed to Boethius is a 9th–10th century compilation (Pseudo-Boethius) incorporating Euclidean definitions, postulates, and proposition enunciations without proofs, intermingled with surveying treatises. An authentic 11th-century Munich fragment demonstrates that Boethius originally translated the enunciations and proofs of at least Books I–III directly from Greek.

2. The Medieval Arabic-Latin Influx (12th–13th Centuries)

Western Europe recovered the full Elements not from Byzantium, but through Islamic Spain:

• Athelhard of Bath (c. 1120): English scholar who travelled to Spain and translated the complete Elements from Arabic into Latin, surviving in Oxford and Bruges manuscripts. Characterized by compressed, concise demonstrations.

• Gherard of Cremona (1114–1187): Renowned translator of the Toledo School who rendered the Elements and an-Nairīzī’s commentary into Latin directly from Arabic (preserved in Cracow MS. 569, ed. Curtze, and Boulogne MS.).

• Johannes Campanus of Novara (13th c.): Papal chaplain and mathematician who revised Athelhard’s translation against Arabic commentaries, producing an expansive, pedagogically clear Latin text that became the standard textbook of the late Middle Ages (represented in Munich MS. R, copied by Sigboto c. 1250).

• The English Masonry Legend: 14th-century Middle English poem (Constituciones artis gometriae secundum Euclydem, BM MS. Reg. 17 A. I) illustrating the folkloric transmission attributing the invention of geometry and masonry to Euclid in Ptolemaic Egypt.

3. Printed Editions and Humanist Scholarship

Heath classifies the vast printed corpus into distinct categories:

1. Latin Translations Prior to 1533:

   • Erhard Ratdolt (Venice, 1482): The editio princeps of Euclid; printed Campanus’ Latin version with groundbreaking typographic diagrams in the margins.

   • Bartolomeo Zamberti (Venice, 1505): First humanist translation directly from a Greek manuscript, fiercely attacking Campanus’ Arabicisms and errors.

   • Luca Pacioli (Venice, 1509): Emended edition of Campanus based on his public lectures in Venice.

   • Jacques Lefèvre d’Étaples (Paris, Henri Estienne, 1516): Combined Campanus and Zamberti in parallel columns.

2. Editions of the Greek Text:

   • Simon Grynaeus (Basel, Johann Herwagen, 1533): The Greek editio princeps, dedicated by Cuthbert Tonstall; printed Euclid’s Greek text along with Proclus’ commentary on Book I from codices supplied by Lazarus Bayfius and John Claymond.

   • David Gregory (Oxford, 1703): Landmark folio Opera Omnia in Greek and Latin, collated with Bodleian manuscripts by John Hudson.

   • François Peyrard (Paris, 1814–1818): Groundbreaking trilingual (Greek, Latin, French) critical edition that first exploited Vatican MS. 190 (P) to eliminate Theon’s interpolations.

   • J. L. Heiberg (Leipzig, Teubner, 1883–1888): The definitive modern critical edition (Euclidis Elementa, 5 vols.).

3. Major Latin Commentaries After 1533:

   • Federico Commandino (Pesaro, 1572): Rigorous translation and commentary with ancient scholia.

   • Christopher Clavius (Rome, 1574): Vast mathematical encyclopaedia containing exhaustive demonstrations, alternative proofs, and supplementary propositions.

   • Girolamo Saccheri (Milan, 1733): Euclides ab omni naevo vindicatus, foundational to the discovery of non-Euclidean geometry.

   • Robert Simson (Glasgow, 1756): Classic Latin and English edition attempting to restore Euclid’s pristine text.

4. Vernacular Translations:

   • Italian: Niccolò Tartaglia (Venice, 1543 — first vernacular edition printed in Europe); Commandino (1575); Vincenzo Viviani (1674/1690).

   • German: Johann Scheubel (1558); Wilhelm Holtzmann / Xylander (1562); J. F. Lorenz (1773/1781).

   • French: Pierre Forcadel (1564); Didier Henrion (1615); François Peyrard (1804).

   • Dutch: Jan Pieterszoon Dou (1606); Frans van Schooten (1617); Hendrik Coets (1702).

   • English:

     • Henry Billingsley (London, John Daye, 1570): Monumental folio containing John Dee’s celebrated Mathematicall Praeface and innovative 3D pop-up fold-out paper polyhedra in Book XI.

     • Isaac Barrow (1660); Robert Simson (1756); John Playfair (1795); Isaac Todhunter (1862).

   • Other Languages: Spanish (Rodrigo Çamorano 1576); Russian (Astaroff 1739, Vachtchenko-Zakhartchenko 1880); Swedish (Mårten Strömer 1744); Danish (Thyra Eibe 1897–1917).

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Detailed Outline

§ 1. The Roman and Early Medieval Background (pp. 91–96)

• Cicero on the utility of geometry among Romans vs Greek speculative mathematics.

• Roman land-surveyors (gromatici, Balbus De mensuris, Censorinus).

• Martianus Capella (c. 470 A.D.) and Greek philosophical reverence for Euclid.

• The Verona Palimpsest (4th c.): Latin fragments of Books XII–XIII.

• Boethius (c. 480–524) and the Pseudo-Boethian geometric corpus.

• The Munich 11th-century fragment proving an authentic Boethian translation with proofs.

• Early medieval translations from Arabic:

  • Athelhard of Bath (c. 1120).

  • Gherard of Cremona (1114–1187) and Cracow MS. 569.

  • Johannes Campanus of Novara (13th c.) and Munich MS. R (Sigboto).

  • The 14th-century Middle English Constituciones artis gometriae secundum Euclydem.

  • Regiomontanus’ library and plans for printing the Greek text.

§ 2. Section I: Latin Translations Prior to 1533 (pp. 97–100)

• Erhard Ratdolt (Venice, 1482): The editio princeps of Campanus; typographic innovation of marginal diagrams.

• Bartolomeo Zamberti (Venice, 1505): Humanist translation from Greek MSS.; polemics against Campanus.

• Luca Pacioli (Venice, 1509): Revision of Campanus with Italian annotations.

• Jacques Lefèvre d’Étaples (Paris, 1516): Synchronous edition of Campanus and Zamberti.

§ 3. Section II: Editions of the Greek Text (pp. 100–104)

• Simon Grynaeus (Basel, 1533): The Greek editio princeps with Proclus on Book I; Cuthbert Tonstall’s preface.

• Minor 16th-century Greek enunciations (Oronce Fine 1536, Angelo Caiani 1545, Joachim Camerarius 1549, Jean Scheubel 1550).

• David Gregory (Oxford, 1703): Definitive Greek-Latin folio Opera Omnia.

• François Peyrard (Paris, 1814–1818): First post-Theonine critical edition based on Vatican MS. 190.

• Modern 19th-century German editions (Camerer & Hauber 1824, August 1826–1829).

• J. L. Heiberg (Leipzig, Teubner, 1883–1888): Definitive critical edition.

§ 4. Section III: Latin Versions or Commentaries After 1533 (pp. 104–106)

• Petrus Ramus (1545), Jacques Peletier (1557), Jean Borrel / Buteo (1559).

• François de Foix Candalla (1566): Expansion of Book X and Books XIV–XVI.

• Federico Commandino (1572): Standard humanist commentary.

• Christopher Clavius (1574): The monumental Jesuit commentary.

• Sir Henry Savile (1621), André Tacquet (1654), Isaac Barrow (1655).

• Girolamo Saccheri (1733): Euclides ab omni naevo vindicatus.

• Robert Simson (1756): Text-critical reconstruction.

§ 5. Sections IV–XIII: Vernacular Translations (pp. 106–113)

• Italian: Niccolò Tartaglia (1543), Commandino (1575), Cataldi (1613), Viviani (1674/1690).

• German: Scheubel (1558), Xylander (1562), Lorenz (1773/1781).

• French: Forcadel (1564), Henrion (1615), Hérigone (1639), Peyrard (1804).

• Dutch: Dou (1606), van Schooten (1617), Coets (1702).

• English:

  • Henry Billingsley (1570): First English translation; John Dee’s preface; 3D pop-up fold-out diagrams.

  • Barrow (1660), Simson (1756), Playfair (1795), Potts (1845), Todhunter (1862).

• Other Traditions: Spanish (Çamorano 1576), Russian (1739 onwards), Swedish (Strömer 1744), Danish (Thyra Eibe 1897–1917), Modern Greek (1820).

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Concordance Table of Primary Editions and Translators

Year	Editor / Translator	Location / Publisher	Language	Significance & Distinctive Features
c. 4th c.	Unknown	Northern Italy (Verona)	Latin	Verona palimpsest; oldest surviving Latin fragment (Books XII–XIII).
c. 500	Boethius	Rome	Latin	Direct translation from Greek; foundation of early medieval mathematical curriculum.
c. 1120	Athelhard of Bath	England / Spain	Latin	First complete medieval translation from Arabic; concise demonstrations.
c. 1170	Gherard of Cremona	Toledo	Latin	Direct Arabic translation; Cracow MS. 569 containing an-Nairīzī.
c. 1250	Johannes Campanus	Novara / Rome	Latin	Standard scholastic textbook; based on Athelhard and Arabic commentaries.
1482	Erhard Ratdolt	Venice	Latin	Editio princeps of Euclid; first printed mathematical book with marginal geometric diagrams.
1505	Bartolomeo Zamberti	Venice	Latin	First printed humanist translation directly from Greek codices; anti-Campanus polemics.
1509	Luca Pacioli	Venice (Paganini)	Latin	Emended Campanus text based on Pacioli’s Venice lectures.
1516	Jacques Lefèvre d’Étaples	Paris (Estienne)	Latin	Comparative edition printing Campanus and Zamberti in parallel columns.
1533	Simon Grynaeus	Basel (Herwagen)	Greek	Editio princeps of the Greek text; includes Proclus on Book I; preface by Cuthbert Tonstall.
1543	Niccolò Tartaglia	Venice	Italian	First translation into any modern European vernacular.
1570	Henry Billingsley	London (John Daye)	English	First English translation; John Dee’s Praeface; pop-up 3D paper models of polyhedra.
1572	Federico Commandino	Pesaro	Latin	Standard humanist edition with scholia and rigorous mathematical commentary.
1574	Christopher Clavius	Rome	Latin	Exhaustive Jesuit encyclopaedic commentary with extensive additions.
1703	David Gregory	Oxford	Greek/Latin	Complete works folio edition; benchmark Oxford critical text.
1733	Girolamo Saccheri	Milan	Latin	Euclides vindicatus; foundational milestone in non-Euclidean geometry.
1756	Robert Simson	Glasgow (Foulis)	Latin/English	Standard British school recension; attempted elimination of Theonine corruptions.
1814	François Peyrard	Paris	Greek/Latin/Fr.	First critical edition utilizing Vatican MS. 190 (P) to restore ante-Theonine text.
1883	J. L. Heiberg	Leipzig (Teubner)	Greek/Latin	Definitive modern critical edition establishing scientific stemmatics.