Abstract
According to a story told by the Roman antiquarian Aulus Gellius, Euclid of Megara exhibited a singular passion for philosophy. At the outset of the Peloponnesian War, Athens imposed sanctions against the nearby city of Megara and banned its citizens from entering Athens. Euclid, a student of Socrates, was distraught to be kept away from his master’s debates, and came up with a clever subterfuge: each night, under the cover of darkness, he crept into town dressed in women’s clothing and joined his fellow philosophers to hear their teacher’s discourses, leaving the city again before dawn.1
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- 1.
Aulus Gellius, Attic nights VII.10.
- 2.
On this paradox and its association with Euclid of Megara, see Wheeler (1983).
- 3.
See Sorensen (2003, pp. 71–74), on Euclid, the identity paradox and the Theaetetus.
- 4.
Euclid (1505). On Zamberti (c. 1474- after 1539), see Rose (1976, especially pp. 301–302).
- 5.
Ibid., fol. 6v: “Nam nostrum non est de tantorum virorum scriptis ausu temerario iudicare.”
- 6.
Valerius Maximus, Facta et dicta memorabilia, VIII.xii ext. 1: “Platonis quoque eruditissimum pectus haec cogitatio attigit, qui conductores sacrae arae de modo et forma eius secum sermonem conferre conatos ad Eucliden geometren ire iussit scientiae eius cedens, immo professioni.”
- 7.
Plutarch, De genio Socratico, 579B–D. See Euclid (1926, vol. 1, p. 3); Heiberg (1882, p. 23); and Knorr (1986, p. 2).
- 8.
Cited in Heiberg (1882, p. 24); and Euclid (1926, vol. 1, p. 3), where Theodorus is said to be the first person (besides Valerius) to confuse the two Euclids.
- 9.
Euclid, Elements (Venice: Erhard Ratdolt, 1482, fol. 1v: “Euclides igitur Megarensis, serenissime princeps, qui xv libris omnem geometrie rationem consummatissime complexus est.” The similarity of Ratdolt’s phrase “consummatissime complexus est” to Theodorus’s description of Euclid’s activity is quite striking.
- 10.
See Heiberg (1882, p. 24), for a list of editions attributing the Elements to Euclid of Megara; and Thomas-Stanford (1926) for early editions of Euclid in general.
- 11.
Ibid., fol. 7r: “Mirum siquidem fuerit quod is auctor nullam aliorum operum ab Euclide conscriptorum fecerit mentionem.”
- 12.
Ibid.: “Annos postmodum octo et viginti natus Plato Socraticis secum assumptis… ad Euclidem nobilissimum ea tempestate geometram Maegara secessit. Id autem oppidum florentissimum fuerat longe ab Athenis milia passuum viginti distans cuius oriundus erat Euclides Socratis aliquando discipulus. Cui cum aliquandiu studiosissime vacasset, Cyrenem profectus est.”
- 13.
See Sabbadini (1896, p. 136).
- 14.
Plutarch (1514, fol. 366v, obviously not the edition consulted by Zamberti).
- 15.
Ficino, in the life of Plato prefaced to his translation of the philosopher’s works, seems to have been influenced by Guarino. Although Ficino does not mention Euclid, he presents Plato’s trip to Megara as the first step of his travels in search of knowledge. Without mentioning the death of Socrates, Ficino says that, at about the age of 28 (which is how old Plato in fact was when his master died), Plato travelled first to Megara, and thence to visit the places and mathematicians listed by Guarino. See Plato (1491, sig. a2r).
- 16.
Euclid (1505, fol. 2r).
- 17.
Ibid., fol. 4r.
- 18.
See p. 7 above.
- 19.
The third edition of the Oxford Classical Dictionary (1996) states that Euclid’s dates are uncertain – between 325 and 250 B.C. – and that it is only Proclus’s “worthless inferences” that link him to Alexandria and the Hellenistic king Ptolemy I. In fact, “nothing is known of his life.”
- 20.
Proclus (1873, p. 68): “neôteros men oun esti tôn peri Platôna, presbuteros de Eratosthenous kai Arkhimêdous.”
- 21.
Ramus critiques this view at some length, at Ramus (1569, pp. 43–44).
- 22.
Euclid (1505, fols 7r–v): “Non admodum iunior sed aliquanto posterior quam Platonis tempore vixerunt [sic for ’vixerit’]. Sed Eratosthene et Archimede antiquior.”
- 23.
Ibid., fol. 7v: “Si ergo Euclidis tempore primi Ptolemaei Aegyptii regis ex libris Chronicis datur intelligi quot anni ab ipso Euclide usque ad nostra tempora fluxerunt. Ptolemaeus igitur primus Aegypti Rex fuit anno a mundi creatione MMMMDCCCCVIII, ante Salvatoris adventum annis CCLXXXXI. Quibus annis CCLXXXXI si addas Annos MDV qui a salute nostra hucusque fluxerunt fiunt anni MDCCLXXXXVI. Ab ipso igitur Euclide usque ad nostram … aetatem effluxerunt Anni MDCCLXXXXVI. Haec sunt quae de ipso Euclide habere potuimus.”
- 24.
Ibid., fol. 5v. The entire brief biography of Euclid reads: “Euclides vero vir inquam ingenii praestantissimi, qui elementa in unum collegit. Multaque ab Eudoxo, multa a Theaeteto perfecit, et hinc et inde sumpta proclivius et planius quam qui ipsum praecesserunt demonstravit. Vixit Platonis tempestate Socratis auditor, temporibus primi Ptolemaei. Antiquior vero ut inquit Proclus Lycius Eratosthene et Archimede qui uno et eodem tempore vixerunt.”
- 25.
[maffei commentaria 1506] fol. 254r: “Omnium doctrinarum praesertim Geometriae studiosissimus fuit, in qua Euclidem et Architam Tarentinum Theodorumque Cyreneum audivit ex qua sane scientia Secreta quaedam investigasse in libris suis a paucis comprensa quae ad rerum naturam tum divinitatem pertinerent.”
- 26.
Cardano (1663, vol. 4, p. 443): “Fuit, ut in Phaedone apparet, Platonis contemporaneus Euclides, cuius ut vetustissimi, clarissimi extant Elementorum tredecim libri, tum Phaenomena, Optici, Catoptrici.”
- 27.
See p. 13 above for Cardano’s Encomium.
- 28.
Cardano (1663, vol. 4, p. 443): “post Euclidem et Platonem, Cleodamus Thasius …” Cardano also wrote briefly about Euclid in his De subtilitate, stating that the geometer “sprang from Megara” ([cardano subtilitate 1560], p. 1011 (book 16): “Megara fuit oriundus.”)
- 29.
Gesner (1545, fol. 226r): “Euclides Megarensis philosophus, ex Megaris urbe in Isthmo, sectam ab ipso Megaricam introduxit, quam et dialecticam et eristicam, id est contentiosam appelavit. Discipulus fuit Socratis.”
- 30.
Ibid.: “Istis hactenus enumeratis non multo iunior Euclides, elementa conscripsit, quibus multa Eudoxi scripta comprehendit, et multa Theaeteti absoluit.”
- 31.
Bernardino Baldi, in his life of Euclid written in the late 1580s or early 1590s, correctly distinguished between the two Euclids, and identified Gesner and Cardano as the principal sources of the confusion among his contemporaries. See Pace (1993, footnote on pp. 202–204). Gesner used a “cut-and-paste” method (quite literally) to assemble his monumental works, and that may account for the unassimilated juxtaposition of these texts. See Blair (2003, pp. 16–17).
- 32.
For a fuller treatment of this work, see p. 178.
- 33.
Proclus (1560, p. 39): “Qui itaque historias perscripsere, hucusque scientiae huius perfectionem producunt. Non multo autem his iunior Euclides est, qui Elementa collegit…”
- 34.
Proclus (1560, sig. *4r). He says that his path to translating Proclus began with his difficulties in studying “Euclidem Megarensem insignem mathematicum.”
- 35.
Billingsley (1993, p. 3).
- 36.
This point made in Billingsley (1993), passim.
- 37.
The life of Euclid is on pp. 22–24 of the Scholae mathematicae.
- 38.
Ramus (1569, p. 22).
- 39.
Ramus (1569 p. 20): “Atque omnes hi Platonici in Academia unâ conversati, et communibus inter se quaestionibus exercitati, mathematicam philosophiam ad perfectionem deduxerunt, ait Proclus. Ita Procli judicio Platonis Academia mathematum inventrix, vel altrix, certe perfectrix efficitur.”
- 40.
Ibid., p. 21: “Atqui periodus ista est, in qua putat Proclus mathematicam in numeris et figuris inventam et perfectam fuisse.”
- 41.
Proclus (1992, p. 56).
- 42.
Ibid.
- 43.
“Amyclas … et Menechmus … et frater ipsius Dinostratus longe perfectiorem Geometriam reddiderunt.”
- 44.
Ramus (1569, p. 23): “Meminerimus igitur mathematicam adhuc e Procli sententia perfectam in Academia fuisse, cum tamen plurima deinceps a posteris inventa sint. Stoikhei\hat{o}tai quatuor adhuc expositi sunt Hippocrates, Leo, Theudius, Hermotimus. Quintus Euclides deinde a Proclo Platonicis commemoratis paulo iunior efficitur, et dicitur sub Ptolemaeo primo floruisse, eique etiam notus fuisse in Aegypto. Valerius tamen ait libri octavi capite decimo tertio conductores sacrae arae modum et formam eius cum Platone conferre conatos, ad Euclidem Geometram ire iussos, scientiae eius cedente, immo professioni.”
- 45.
Ramus (1569, p. 23): “Sed Proclus Geometriae magister a Theophrasto atque Eudemo praesertim hac de re veritatem edoctus… in hac historia mihi verisimilior est, quam Valerius: nec duplicati cubi quicquam ad Euclidem, sed ad Platonem principem totum referri comperio. Nec Euclides in Elementis quicquam nominatim de duplicando cubo proposuit; alioqui tamen si haec vera essent, non taciturus [this sentence was not in the 1567 Prooemium, added in the 1569 Scholae]. Itaque Euclidem Platone et Platonicis iuniorem a Proclo accipio.”
- 46.
Proclus (1992, p. 56): “All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid …”
- 47.
Ramus (1569, p. 15): “Quarum [sententiarum] prima et ingeniossima est Platonis, cujusque mesographus ad duas medias protinus inveniendum, singularis est: Mathematicae itaque laudis principatus tum penes unum Platonem fuit.”
- 48.
A description of the mechanical solution that Eutocius attributed to Plato can be found in Knorr (1986, pp. 57–61), which shows that the association of such a mechanical contrivance with Plato is very dubious; Eratosthenes was most likely the inventor.
- 49.
See text cited in n. 26 of third chapter above.
- 50.
See text cited at n. 49 of second chapter, and discussion at p. 36 above.
- 51.
See p. 43 above.
- 52.
See discussion at p. 180.
- 53.
Ramus (1569, p. 39): “ut Euclidi praeter inane nomen nihil admodum relinquatur.” See the discussion in third chapter, above.
- 54.
MS Savile 29, fol. 36v.
- 55.
MS Savile 29, fols 41r–44v.
- 56.
There is a long entry for Euclid in Savile’s list of auctores mathematici. The entry begins (at MS Savile 28, fol. *8r) with a summary of Diogenes’s life of Euclid of Megara, followed by details of published works by the mathematician Euclid. At the end of the entry, Savile notes that he flourished under Ptolemy I, that he is mentioned by Aulus Gellius, and that Zamberti has information on him taken from Proclus. Five lines of the entry consist of disjointed notes taken from Ramus’s Prooemium mathematicum. When he compiled this entry (as when he wrote his lecture on Plato), Savile clearly still thought that Euclid of Megara wrote the Elements, despite his marshalling of all the available sources on the biography of Euclid.
- 57.
MS Savile 29, fol. 41r.
- 58.
Ibid.: “Atque hic, auditores, non possum dissimulare sententiam meam; quam profecto proferam non tanquam oraculum vel quod mihi necesse sit postea defendere – sed ita uti sit integrum revocare si eam hoc tempore vobis non probro. Euclidem igitur illum Socratis discipulum, qui sectam megaricorum instituit, cuius toties mentio fit in libris Platonicis, suspicor alterum quendam et aetate superiorem.”
- 59.
MS Savile 29, fol. 41r: “Nam inter bellum Siculum cui Alcibiades Euclidis condiscipulus iam aetate ut par erat provectior, praefuit, et mortem Alexandri cui Ptolemaeus in Aegypto successit, anni intercesserunt 93; et tamen hic noster Euclides, si Proclo credimus, cum Ptolemaeo familiariter versatus est, multis quidem, ut credo, post inaugurationem annis constitutis iam et pacatis negociis.” For Alcibiades’s command of the Sicilian expedition in 416 B.C., see Thucydides, History, VI.8; Diodorus Siculus, Bibliotheca Historica, XII.84; Plutarch, Life of Nicias, 14–15.
- 60.
MS Savile 29, fol. 41r: “Hoc argumentum quam esset inane minimeque necessarium si Socraticum illum intelligat toties a Platone, qui duabus seculis praecessit Archimedem, inductum, quis non videt?”
- 61.
Savile (1621, p. 7): “In hanc sententiam de duplici Euclide, disputatum est a me ante annos quinquaginta et quod excurrit, cum in scholis publicis pro meo modulo interpretarer in ordinariis lectionibus Almagestum Ptolemaei.”
- 62.
In his Praelectiones, Savile drew attention to this fact rather caustically, saying that it was “only fair to believe” that Commandino had been moved by the same arguments as he had. (Savile 1621, p. 7: “In quam opinionem biennio postea Federicum Commandinum Italum iisdem, uti credere par est, permotum argumentis video incidisse.”)
- 63.
Euclid (1572, sig. *5r): “Nemo autem mihi ignotum esse arbitretur, Valerium Maximum scribere Platonem sacrae arae conductores ad Euclidem, tanquam ad primarium mathematicum reiecisse. Sed nos Heronem et Proclum matheseos studio insignes sequimur, vel potius Eudemum ac Theophrastum ex peripateticis post praeceptorem nobilissimos.”
- 64.
The problem of the two Euclids had been raised and resolved a couple of times before in print, but these solutions seem to have passed unnoticed. Caspar Peucer, in an oration delivered in 1557, raised the possibility that there were two Euclids of Megara, an earlier philosopher and a later mathematician (Melanchthon 1834–1860, vol. 12, col. 262). He had probably noticed the impossibility of the chronology; already in the table of mathematicians included in his 1553 Elementa doctrinae he had dated Plato to around 390 B.C., and Euclid of Megara to 292 B.C. – presumably intending by this the author of the Elements (Peucer 1553, sigs A4r–v). It is surprising that Savile, who used Peucer’s book frequently to draw up his list of auctores mathematici, never noticed this feature of his chronology. In 1562, the Sicilian mathematician and humanist Francesco Maurolyco published a collection of sources on Sicilian history. He quoted a letter written by the fifteenth-century Byzantine historian Constantinus Lascaris to the prorex of Sicily, Fernando Acuña, concerning famous ancient Sicilians: “Euclid of Gela, the Platonic philosopher and most famous geometer, is a different man from the one whom Laertius writes about and who composed dialogues. As Proclus says in the second book of his commentary on the first book of Euclid (and as Heron also writes) he lived in the time of Ptolemy I, and was younger than Plato, but older than Eratosthenes and Archimedes. He was from Gela, as one can infer from the words of Laertius. He wrote thirteen books of Elements.” (Maurolyco 1562, fol. 21r). Lascaris’s intention (shared by Maurolyco) was to demonstrate that Euclid the geometer was in fact a native son of Sicily; Diogenes had reported that Alexander Polyhistor said that Euclid had been born in Gela, not Megara, and this was the clue that led Lascaris to consider the existence of two Euclids. See (Heiberg 1882), pp. 22–25 on Lascaris, Maurolyco and the fortuna of the Sicilian Euclid.
- 65.
In his sober Bibliotheca philosophorum classicorum authorum chronologica published 20 years later, Jean-Jacques Frisius has a biographical entry for Euclid of Megara under the year 422 B.C. (Frisius 1592, fol. 13v), and Euclid the geometer has an entry at 320 B.C. In neither entry is there any mention of the fact that the two men were once confused with each other. On Frisius, see sixth chapter.
References
Henry Savile, Praelectiones tresdecim in principium Elementorum Euclidis Oxonii habitae MDCXX, Oxford, 1621.
Dale B. Billingsley, “Authority in Early Editions of Euclid’s Elements,” Fifteenth-Century Studies 20 (1993), pp. 1–14.
Ann Blair, “Reading Strategies for Coping with Information Overload ca. 1550–1700,” Journal of the History of Ideas 64 (2003), pp. 11–28.
J. L. Heiberg, Litterargeschichtliche Studien ber Euklid, Leipzig: B. G. Teubner, 1882.
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Goulding, R. (2010). The Puzzling Lives of Euclid. In: Defending Hypatia. Archimedes, vol 25. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3542-4_5
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