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date: 07 December 2022

Euclidfree

Euclidfree

  • Michalis Sialaros

Summary

Euclid of Alexandria was a Greek geometer whose floruit was c. 300 bce. He is famous as the father of geometry, as his influential Elements of Geometry has been read, edited, praised, or criticized more than any other mathematical book in history. Besides the Elements, Euclid wrote several other treatises which, according to late antique commentators, offer systematic introductions to all mathematical subdisciplines.

Subjects

  • Science, Technology, and Medicine

Updated in this version

Text rewritten to reflect current scholarship.

Life

Euclid (Εὐκλείδης‎) is famous as the author of the Elements of Geometry (Στοιχεῖα‎). Indeed, his magnum opus has been read, edited, praised, or criticized more than any other mathematical book in history. Still, we know almost nothing about his life. The dominant view that Euclid was active in Alexandria (1) around 300 bce is based on a series of hypotheses, most of which rely on later accounts by Proclus (In Eucl. 68.8–20), Pappus (1) (Coll. 7.678.10–12.), and Stobaeus (Flor. 2.31.114).1 These accounts, as well as the (more informative) Arabic biographical tradition, have rightly been approached with scepticism.2 Be that as it may, no 4th-century scholar shows awareness of Euclid (most notably Aristotle), while 3rd-century mathematicians (among whom, Archimedes and Apollonius (2)) assume a part of his work to be known. Besides, papyrological and other documentary remains confirm that at least part of the material included in the Elements was in circulation in the 3rd century bce.3 Regarding Euclid’s education, the idea that he received his training from the pupils of Plato in the Academy must be treated merely as a conjecture.4 An interesting phenomenon concerning Euclid’s biographies appears in the Early Middle Ages, when he was commonly confused with the philosopher Euclides (1) of Megara, a pupil of Socrates. As a result of this misunderstanding, some of these biographies contain elements from the life of both; furthermore, the image of Euclides of Megara—as identified on a Greek coin from the city of Megara—has traditionally been mistakenly regarded as a representation of Euclid the mathematician.5

Works

According to tradition, Euclid wrote several treatises, some of which survive in one form or another. Apart from the Elements of Geometry, the list of the extant works comprises the Data (Δεδομένα‎), the Sectio Canonis (Κατατομὴ Κανόνος‎), the Optics (Ὀπτικά‎), the Phaenomena (Φαινόμενα‎), and the On Divisions (Περὶ Διαιρέσεων‎)—the last survives only in part, in Arabic. The authenticity of two other Arabic treatises that are ascribed to Euclid (the Book of the Balance, and the Book on the Heavy and Light) has been questioned, but not on convincing grounds.6 In addition, four lost treatises are attributed to Euclid: the Conics (Κωνικά‎), the Pseudaria (Ψευδάρια‎), the Porisms (Πορίσματα‎), and the Surface Loci (Τόποι πρὸς ἐπιφανείᾳ‎). Given the enormous size of his work, the apparent diversity of the topics under investigation, and the loss of a substantial part of the corpus, it is not easy to provide a description of Euclid’s project as a whole. If we follow the view of some late antique commentators (Marinus In Dt. 254.13–22; Proclus In Eucl. 68.23–69.19; cf. Pappus, Coll. 7.676.25–678.15), his main aim was to offer a systematic introduction to all mathematical subdisciplines. One must also take into consideration that the practice of writing Elements did not originate with Euclid;7 he must therefore also be understood as a compiler. Still, the remarkably tight structure of the Elements reveals authorial control beyond the limits of a mere editor.8 Finally, despite whether this was Euclid’s original intention or not, his works—the Elements, most notably—played a significant role in mathematical education; thus, Euclid also functioned as a teacher of mathematics.

The Elements

The work upon which Euclid’s fame rests is, undoubtedly, his lengthy Elements of Geometry. The treatise comprises thirteen books in total, dealing with a plethora of topics, such as plane and solid geometry, arithmetic, and irrational magnitudes. Book 1 introduces fundamental geometrical entities (such as points, lines, triangles, and parallelograms) and provides the first extant (and remarkably delicate) proof of the so-called “Pythagoras’s theorem” (I.47–48). Book 2 deals with problems of transformation of areas. Since the 1970s, this book has been at the centre of heated debates among scholars. According to older historiographical approaches, Book 2 should be understood as a kind of algebra textbook in which Euclid expounded the method of “geometrical algebra.”9 Interpretations of this kind have rightly been criticized as ahistorical: the development of algebra, even in its premodern form, only took place centuries after Euclid.10 Books 3–4 deal with fundamental properties of circles, as well as with inscribed and circumscribed polygons. Book 5—one of the highlights of the Elements—exposes what is usually described as the general theory of proportion. It is thought that at least a part of it was the discovery of mathematicians prior to Euclid, most probably Eudoxus (1).11 Book 6 examines geometrical similarity, while Books 7–9 deal with properties of numbers. Among their several highlights is Euclid’s subtle proof that there is an unlimited multitude of prime numbers (9.20). Book 10, the biggest and by far the most complex of the Elements, deals with irrational magnitudes. The final three books of the Elements construct and examine the five regular polyhedra, i.e., the cube, the tetrahedron, the octahedron, the icosahedron, and the dodecahedron. The fact that the Elements concludes with the five regular solids, allowed Proclus (In Eucl. 68.20–21) to propose the questionable idea that Euclid was a Platonist.12

In terms of its content, the mathematical riches of the Elements are invaluable. Its most striking characteristics, however, are its structure and form. Roughly speaking, the Elements is a collection of propositions divided into two groups: those presented without proof and those accompanied by proofs. Following Aristotle’s terminology, scholars usually describe the former group as the “first principles.” Any proposition described as a “definition” (ὅρος, ὁρισμός‎), a “postulate” (αἴτημα‎), or a “common notion” (κοινὴ ἔννοια‎) is a first principle.13 The second group of propositions comprises enunciations accompanied by proofs and diagrams. Their lexicon is surprisingly limited and extremely formularized, and they appear to follow an internal scheme.14 According to whether these propositions bring a new geometrical entity to life or examine a property of an existing geometrical entity, they can be further divided into “problems” and “theorems,” respectively. Euclid himself was aware of this distinction.15 On a more general level, each proposition of the Elements is part of a deductive argument, functioning either as a premise or as a conclusion. The aforementioned type of structure is described as “axiomatic and deductive” or, for the Greeks, “synthetic.”

The text of Euclid’s Elements was the subject of several commentaries in antiquity, for example, by Heron (1), Pappus (1), Proclus, Simplicius, and perhaps Porphyry.16 Only Proclus’s commentary on the first book of the Elements, survives in Greek.17 The critical edition of the Elements was prepared by Johan L. Heiberg, who used Greek manuscripts deriving from a 4th-century re-edition of the Elements by Theon of Alexandria (4), together with a manuscript labelled P, our only apparently pre-Theonian witness. Heiberg received criticism for two of his editorial choices: first, he did not prepare a critical edition of the diagrams;18 and, second, he took into consideration only the Greek tradition of manuscripts.19 Be that as it may, the quest for the original text of the Elements has been the Holy Grail of Euclid scholars for centuries. For a reason aptly explained by Serafina Cuomo, this is an almost impossible enterprise: “We have to keep in mind that the Elements were seen first and foremost as a reservoir of results, not necessarily as a work that should be preserved in its linguistical or methodological integrity.”20

Other Works

The Elements is the flagship of Greek geometrical synthesis. Nonetheless, three of Euclid’s other treatises (the Data, the Porisms and the Surface Loci) were considered essential readings for those who wished to master analysis (Pappus, Coll. 7.636.18–25), a process the ancient writers thought of as reverse of synthesis.21 The Data is a relatively small treatise of ninety-four propositions, all of which deal with plane geometrical objects and ratios. These propositions follow a fixed pattern, namely, they prove that when a certain geometrical object or property is ‘given’ in some respect, then some other geometrical object or property has also been ‘given’ in some respect.22 The Porisms is now lost. On the basis of later descriptions (Pappus, Coll. 7.648.66; Proclus, In Eucl. 212.12–17; 301.21–305.16), we believe that it comprised three books in total and around 200 propositions. A “porism” in Greek mathematics may have two completely different meanings: (1) a corollary, i.e., a proposition the validity of which is established in the proof of a previous proposition; or (2) a third type of proposition—an intermediate between a theorem and a problem—the aim of which is to discover a feature of an existing geometrical entity, for example, to find the centre of a circle. Euclid’s Porisms deals with the latter category of porisms. The Surface Loci, in two books, is also lost. Some hypotheses regarding its contents have been proposed, again based on later descriptions. The treatise probably dealt with cones and cylinders, as well as with other surfaces, among other topics.

Of the remaining treatises, two are included in Pappus’s (1) “Little Astronomy”: the Optics and the Phaenomena. Both works survive in two recensions. The Optics provides an introduction to geometrical optics and discusses fundamental laws of perspective. The Phaenomena is a textbook on elementary spherical geometry which presents some similarity with Autolycus’s On the Moving Sphere. The authorship of two other treatises, the Catoptrics and the Division of the Canon, has been questioned, on unconvincing grounds. The former is a study of mirrors, while the latter provides the foundation of mathematical music theory by means that resemble a Pythagorean approach. The Sectio Canonis is assumed to be the same, or a part of a treatise called the Elements of Music mentioned by ancient authors. The On Divisions has survived only in the Arabic tradition, in thirty-six propositions. It discusses divisions of a geometrical figure by straight lines, so as to fulfil particular specifications. Of the remaining works, the lost Conics comprised four books, and a great part of this material is contained in the first books of Apollonius’s Conics. The (also lost) Pseudaria included, according to Proclus (70.1–18), Euclid’s methods of training beginners in elementary geometry to identify falsehoods.23

Primary Texts

Heiberg, Johan L. and Heinrich Menge, eds. Euclidis Opera Omnia. 8 vols. Leipzig: Teubner, 1883–1916. For the Elements, Stamatis, Evangelos S. (post Johan L. Heiberg), ed. Euclidis Elementa. 5 vols. Leipzig: Teubner, 1969–1977.

Bibliography

  • Acerbi, Fabio, ed. Euclide: Tutte le opere. Milan: Bompiani, 2007.
  • Barbera, André. The Euclidian Division of the Canon: Greek and Latin Sources. Lincoln: University of Nebraska Press, 1991.
  • Barker, Andrew, ed. Greek Musical Writings. 2 vols. Cambridge, UK: Cambridge University Press, 1984–1989.
  • Berggren, Lennart J., and Robert S. D. Thomas, eds. Euclid’s Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy. New York and London: Garland, 1996.
  • Clagett, Marshall. The Science of Mechanics in the Middle Ages. London: University of Wisconsin Press, 1961.
  • Jones, Alexander, ed. Pappus of Alexandria: Book 7 of the Collection. 2 vols. New York: Springer, 1986.
  • Lorch, Richard. “Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages.” Science in Context 14 (2001): 313–331.
  • Vitrac, Bernard, and Maurice Caveing, eds. Euclid: Les Éléments. 4 vols. Paris: Presses Universitaires de France, 1990–2001.

Notes

  • 1. On Euclid’s anecdotes, see Michalis Sialaros, “How Much Does a Theorem Cost?” in Revolutions and Continuity in Greek Mathematics, ed. Michalis Sialaros (Berlin: Walter de Gruyter, 2018), 89–106.

  • 2. On the Arabic biographical tradition, see Bernard Vitrac and Ahmed Djebbar, “Le Livre XIV des Éléments d’ Euclide: Versions grecques et arabes,” SCIAMVS 12 (2011): 77–83.

  • 3. Jürgen Mau and Wolfgang Müller “Mathematische Ostraka aus der Berliner Sammlung,” Archiv für Papyrusforschung und verwandte Gebiete 17 (1960): 1–10. See also David H. Fowler, chap. 6, in The Mathematics of Plato’s Academy, 2nd ed. (Oxford: Clarendon Press, 1999), 199–221.

  • 4. See, for example, the viewpoint of Thomas L. Heath, ed., The Thirteen Books of Euclid’s Elements (New York: Dover, 1956), 2. Cf. Michalis Sialaros, “Euclid of Alexandria: A Child of the Academy?” in Plato’s Academy, ed. Paul Kalligas, Chloe Balla, Effie Baziotopoulou-Valavani, and Vassilis Karasmanis (Cambridge, UK: Cambridge University Press, 2020), 141–152.

  • 5. Cf. Aulus Gellius, NA. 6.10; and Domenico Marolì’s Euclide di Megara si traveste da donna per recarsi ad Atene a seguire le lezioni di Socrate. See also Robert Goulding, Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History (Dordrecht, Germany: Springer, 2010), 116–142.

  • 6. See Wilburn R. Knorr, The Ancient Tradition of Geometric Problems (Boston, MA: Birkhäuser, 1986), 151. NB the arrangement of the letters in the diagrams clearly indicates Greek origin.

  • 7. Proclus (In Eucl. 66.7–67.16) attributes pre-Euclidean Elements to Hippocrates of Chios, Leon, and Theudius.

  • 8. See Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements (Cambridge, MA: MIT Press, 1981).

  • 9. See, for example, Heath, Euclid’s Elements, 372–373.

  • 10. Sabetai Unguru’s notorious 1975 paper (“On the Need to Rewrite the History of Greek Mathematics”), together with the replies to it, is included in Jean Christianidis, ed., Classics in the History of Greek Mathematics (Dordrecht, Germany: Kluwer Academic Publishers, 2004). For Unguru’s latest paper on geometrical algebra, see Sabetai Unguru, “Counter-Revolutions in Mathematics,” in Revolutions and Continuity in Greek Mathematics, ed. Michalis Sialaros (Berlin: Walter de Gruyter, 2018), 17–34. For the distinction between modern and premodern algebra in relation to the debate on geometrical algebra, see Michalis Sialaros and Jean Christianidis, “Situating the Debate on ‘Geometrical Algebra’ within the Framework of Premodern Algebra,” Science in Context 29 (2016): 129–150.

  • 11. See Wilburn R. Knorr, The Evolution of the Euclidean Elements (Dordrecht: Reidel, 1975).

  • 12. Proclus: In Eucl. 68.21–3; 70.19–71.2; 82.25–83.2; cf. Plato, Ti. 53c–55c.

  • 13. There is variation in terminology; see Heron, Df. 137.6.1; Proclus, In Eucl. 57.26–58.2, 75.27–77.6, 178.1–13; cf. Aristotle, [Ath. Pol.] 71b; 74b-76a.

  • 14. See Reviel Netz, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (Cambridge: Cambridge University Press, 1999), 89–167. For the internal scheme of a Greek mathematical proposition, see Proclus, In Eucl. 203–205.11; cf. Heron, Def. 137.1.1. A slightly different scheme appears in the Arabic tradition.

  • 15. In the ending of a theorem he would write “which was required to prove,” while in a problem, “which was required to construct”; cf. Proclus, In Eucl. 81.5, 210.616, 233.112.

  • 16. For Heron and Simplicius, see Anthony Lo Bello, ed., The Commentary of al-Nayrizi on Book I of Euclid’s Elements of Geometry (Boston, MA: Brill, 2003) and The Commentary of al-Nayrizi on Books II–IV of Euclid’s Elements of Geometry (Boston, MA: Brill, 2009); for Proclus, see Glenn R. Morrow, ed., Proclus: A Commentary on the First Book of Euclid’s Elements. (Princeton, MJ: Princeton University Press, 1970); for Pappus, see William Thomson, ed., The Commentary of Pappus on Book X of Euclid’s Elements (Cambridge, MA: Harvard University Press, 1930).

  • 17. For the transmission of the Elements, see Bernard Vitrac, Complete Dictionary of Scientific Biography, s.v. “Euclid.”

  • 18. On this, see Ken Saito’s project GreekMath.Org.

  • 19. On the so-called ‘Heiberg-Klamroth’ debate, see Wilburn Knorr, “The Wrong Text of Euclid: On Heiberg’s Text and Its Alternatives,” Centaurus 38 (1996): 208–276. For the latest research on this topic, see Bernard Vitrac, “The Euclidean Ideal of Proof in the Elements and Philological Uncertainties of Heiberg's Edition of the Text,” in The History of Mathematical Proof in Ancient Traditions, ed. Karine Chemla (Cambridge: Cambridge University Press, 2012), 69–134.

  • 20. Serafina Cuomo, Ancient Mathematics (London: Routledge 2001), 131.

  • 21. For the definition of analysis, see Pappus, Coll. 7.634.11–23; cf. Marinus, In Dt. 252.27–254.1; Proclus, In Eucl. 242.16–17; Aristotle, Eth.Nic. 1112b16–24.

  • 22. Proposition 79, the only exception to this rule, is probably interpolated; see Christian M. Taisbak, Euclid’s Data: ΔΕΔΟΜΕΝΑ (Copenhagen: Museum Tusculanum Press, 2003), 196. For the latest study on how Greek mathematicians employed the concept of ‘given’, see Nathan Sidoli, “The Concept of Given in Greek Mathematics,” Archive for History of Exact Sciences 72 (2018): 353–402. For the linguistical practices surrounding the term, see Fabio Acerbi “The Language of the ‘Givens’: Its Form and Its Use as a Deductive Tool in Greek mathematics,” Archive for History of Exact Sciences 65 (2011): 119–153.

  • 23. For the latest study on Euclid’s Pseudaria, see Fabio Acerbi, “Euclid’s Pseudaria,” Archive for History of Exact Sciences 62 (2008): 511–551.