Euclid (/ˈjuːklɪd/; Greek: Εὐκλείδης; fl. 300 BCE) was an ancient Greek mathematician active as a geometer and logician.[3] Considered the "father of geometry",[4] he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.

Euclid
Εὐκλείδης
Jusepe de Ribera - Euclid - 2001.26 - J. Paul Getty Museum.jpg
Euclid by Jusepe de Ribera, c. 1630–1635[1]
Known for
Scientific career
FieldsMathematics
InfluencesEudoxus, Hippocrates of Chios, Thales and Theaetetus
InfluencedVirtually all subsequent geometry of the Western world and Middle East[2]

Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken for the earlier philosopher Euclid of Megara, causing his biography to be substantially revised. It is generally agreed that he spent his career under Ptolemy I in Alexandria and lived around 300 BCE, after Plato and before Archimedes. There is some speculation that Euclid was a student of the Platonic Academy. Euclid is often regarded as bridging between the earlier Platonic tradition in Athens with the later tradition of Alexandria.

In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements, Euclid wrote a central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena. Euclid's authorship of two other texts—On Divisions of Figures, Catoptrics—has been questioned. He is thought to have written many now lost works.

Life

Traditional narrative

 
A papyrus fragment of Euclid's Elements dated to c. 75–125 CE CE. Found at Oxyrhynchus, the diagram accompanies Book II, Proposition 5.[5]

The English name 'Euclid' is the anglicized version of the Ancient Greek name Εὐκλείδης.[6][a] It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".[8] The word 'Euclid' less commonly also means "a copy of the same",[7] and is sometimes synonymous with 'geometry'.[3]

Like many ancient Greek mathematicians, Euclid's life is mostly unknown.[9] He is accepted as the author of four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, there is nothing known for certain of him.[10][b] The historian Carl Benjamin Boyer has noted irony in that "Considering the fame of the author and of his best seller [the Elements], remarkably little is known of Euclid".[12] The traditional narrative mainly follows the 5th century CE account by Proclus in his Commentary on the First Book of Euclid's Elements, as well as a few anecdotes from Pappus of Alexandria in the early 4th century.[6][c] According to Proclus, Euclid lived after the philosopher Plato (d. 347 BCE) and before the mathematician Archimedes (c. 287 – c. 212 BCE); specifically, Proclus placed Euclid during the rule of Ptolemy I (r. 305/304–282 BCE).[10][9][d] In his Collection, Pappus indicates that Euclid was active in Alexandria, where he founded a mathematical tradition.[10][14] Thus, the traditional outline—described by the historian Michalis Sialaros as the "dominant view"—holds that Euclid lived around 300 BCE in Alexandria while Ptolemy I reigned.[6]

Euclid's birthdate is unknown; some scholars estimate around 330[15][16] or 325 BCE,[3][17] but other sources avoid speculating a date entirely.[18] It is presumed that he was of Greek descent,[15] but his birthplace is unknown.[12][e] Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this.[20] It is unlikely he was contemporary with Plato, so it is often presumed that he was educated by Plato's disciples at the Platonic Academy in Athens.[21] The historian Thomas Heath supported this theory by noting that most capable geometers lived in Athens, which included many of the mathematicians whose work Euclid later built on.[22][23] The accuracy of these assertions has been questioned by Sialaros,[24] who stated that Heath's theory "must be treated merely as a conjecture".[6] Regardless of his actual attendance at the Platonic academy, the contents of his later work certainly suggest he was familiar with the Platonic geometry tradition, though they also demonstrate no observable influence from Aristotle.[15]

Alexander the Great founded Alexandria in 331 BCE, where Euclid would later be active sometime around 300 BCE.[25] The rule of Ptolemy I from 306 BCE onwards gave the city a stability which was relatively unique in the Mediterranean, amid the chaotic wars over dividing Alexander's empire.[26] Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education.[12][f] On the basis of later anecdotes, Euclid is thought to have been among the Musaeum's first scholars and to have founded the Alexandrian school of mathematics there.[25] According to Pappus, the later mathematician Apollonius of Perga was taught there by pupils of Euclid.[22] Euclid's date of death is unknown; it has been estimated that he died c. 270 BCE, presumably in Alexandria.[25]

Identity and historicity

Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara, a pupil of Socrates who was included in the dialogues of Plato.[6][18] Historically, medieval scholars frequently confused the mathematician and philosopher, mistakenly referring to the former in Latin as 'Megarensis' (lit.'of Megara').[28] As a result, biographical information on the mathematician Euclid was long conflated with the lives of both Euclid of Alexandria and Euclid of Megara.[6] The only scholar of antiquity known to have confused the mathematician and philosopher was Valerius Maximus.[29][g] However, this mistaken identification was relayed by many anonymous Byzantine sources and the Renaissance scholars Campanus of Novara and Theodore Metochites, the latter of whom included it in a 1842 translation of the Elements printed by Erhard Ratdolt.[29] After the mathematician Bartolomeo Zamberti [fr] (1473–1539) affirmed this presumption in his 1505 translation, all subsequent publications passed on this identification.[29][h] Later Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.[29]

Arab sources written many centuries after his death give vast amounts of information concerning Euclid's life, but are completely unverifiable.[6] Most scholars consider them of dubious authenticity;[10] Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.[20] There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".[31] The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his Elements, which Euclid replied with "there is no royal road to geometry".[31] This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus.[32] Both the accounts were written in the 5th century CE, neither indicate their source, and neither story appears in ancient Greek literature.[33]

The traditional narrative of Euclid's activity c. 300 is complicated by no mathematicians of the 4th century BCE indicating his existence.[6] Mathematicians of the 3rd century such as Archimedes and Apollonius "assume a part of his work to be known";[6] however, Archimedes strangely uses an older theory of proportions, rather than that of Euclid.[10] The Elements is dated to have been at least partly in circulation by the 3rd century BCE.[6] Some ancient Greek mathematician mention him by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").[34] In the Middle Ages, some scholars contended Euclid was not a not a historical personage and that his name arose from a corruption of Greek mathematical terms.[35]

Works

Elements

Structure of the Elements[36]


Books I–VI: Plane geometry
Books VII–X: Arithmetic
Books XI–XIII: Solid geometry

Euclid is best known for his thirteen-book treatise, the Elements (Greek: Στοιχεῖα; Stoicheia).[4][37] Although many of the results in the Elements originated with earlier Greek mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.[38] Among the mathematicians whose work is featured includes Eudoxus, Hippocrates of Chios, Thales and Theaetetus.[39]

Although best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the Elements of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 CE.[5]

Other works

 
Euclid's construction of a regular dodecahedron.

In addition to the Elements, at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements, with definitions and proved propositions.

  • Catoptrics concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors, though the attribution is sometimes questioned.[40]
  • The Data (Greek: Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems.[40]
  • On Divisions (Greek: Περὶ Διαιρέσεων‎) survives only partially in Arabic translation, and concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It includes thirty-six propositions and is similar to Apollonius' Conics.[40]
  • The Optics (Greek: Ὀπτικά‎) is the earliest surviving Greek treatise on perspective. It includes an introductory discussion of geometrical optics and basic rules of perspective.[40]
  • The Phaenomena (Greek: Φαινόμενα) is a treatise on spherical astronomy, survives in Greek; it is similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BCE.[40]

Lost works

Four other works are credibly attributed to Euclid, but have been lost.[11]

  • The Conics (Greek: Κωνικά‎) was a four-book survey on conic sections, which was later superseded by a Apollonius' more comprehensive treatment of the same name.[41][40] The work's existence is known primarily from Pappus, who asserts that the first four books of Apollonius' Conics are largely based on Euclid's earlier work.[42] Doubt has been cast on this assertion by the historian Alexander Jones [de], owing to sparse evidence and no other corroboration of Pappus' account.[42]
  • The Pseudaria (Greek: Ψευδάρια‎; lit.'Fallacies'), was—according to Proclus in (70.1–18)—a text in geometrical reasoning, written to advise beginners in avoiding common fallacies.[41][40] Very little is known of its specific contents aside from its scope and a few extant lines.[43]
  • The Porisms (Greek: Πορίσματα; lit.'Corollaries') was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 propositions.[41][40] The term 'porism' in this context does not refer to a corollary, but to "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".[40] The mathematician Michel Chasles speculated that these now-lost propositions included content related to the modern theories of transversals and projective geometry.[41][i]
  • The Surface Loci (Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown contents, aside from speculation based on the work's title.[41] Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.[40]

Legacy

 
The cover page of Oliver Byrne's 1847 colored edition of the Elements

Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity.[15] Many commentators cite him as one of the most influential figures in the history of mathematics.[3] The geometrical system established by the Elements long dominated the field; however, today that system is often referred to as 'Euclidean geometry' to distinguish it from other non-Euclidean geometries discovered in the early 19th century.[44] Among Euclid's many namesakes are the European Space Agency's (ESA) Euclid spacecraft,[45] the lunar crater Euclides,[46] and the minor planet 4354 Euclides.[47]

The Elements is often considered after the Bible as the most frequently translated, published, and studied book in the Western World's history.[44] With Aristotle's Metaphysics, the Elements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds.[44]

The first English edition of the Elements was published in 1570 by Henry Billingsley and John Dee.[29] The mathematician Oliver Byrne published a well-known version of the Elements in 1847 entitled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, which included colored diagrams intended to increase its pedagogical effect.[48] David Hilbert authored a modern axiomatization of the Elements.[49]

References

Notes

  1. ^ In modern English, 'Euclid' is pronounced as /ˈjuːklɪd/ in British English and /ˈjuˌklɪd/ in American English.[7]
  2. ^ Euclid's oeuvre also includes the treatise On Divisions, which survives fragmented in a later Arabic source.[11] He authored numerous lost works as well.
  3. ^ Some of the information from Pappus of Alexandria on Euclid is now lost and was preserved in Proclus's Commentary on the First Book of Euclid's Elements.[13]
  4. ^ See Heath 1981, p. 354 for an English translation on Proclus's account of Euclid's life.
  5. ^ Later Arab sources state he was a Greek born in modern-day Tyre, Lebanon, though these accounts are considered dubious and speculative.[10][6] See Heath 1981, p. 355 for an English translation of the Arab account. He was long held to have been born in Megara, but by the Renaissance it was concluded that he had been confused with the philosopher Euclid of Megara,[19] see §Identity and historicity
  6. ^ The Musaeum would later include the famous Library of Alexandria, but it was likely founded later, during the reign of Ptolemy II Philadelphus (285–246 BCE).[27]
  7. ^ The historian Robert Goulding notes that the "common conflation of Euclid of Megara and Euclid the mathematician in Byzantine sources" suggests that doing so was a "more extensive tradition" than just the account of Valerius.[29]
  8. ^ This misidentification also appeared in Art; the 17th-century painting Euclide di Megara si traveste da donna per recarsi ad Atene a seguire le lezioni di Socrate [Euclid of Megara Dressing as a Woman to Hear Socrates Teach in Athens] by Domenico Maroli portrays the philosopher Euclid of Megara but includes mathematical objects on his desk, under the false impression that he is also Euclid of Alexandria.[30]
  9. ^ See Jones 1986, pp. 547–572 for further information on the Porisms

Citations

  1. ^ Getty.
  2. ^ Asper 2010, § para. 7.
  3. ^ a b c d Bruno 2003, p. 125.
  4. ^ a b Sialaros 2021, § "Summary".
  5. ^ a b Fowler 1999, pp. 210–211.
  6. ^ a b c d e f g h i j k Sialaros 2021, § "Life".
  7. ^ a b OEDa.
  8. ^ OEDb.
  9. ^ a b Heath 1981, p. 354.
  10. ^ a b c d e f Asper 2010, § para. 1.
  11. ^ a b Sialaros 2021, § "Works".
  12. ^ a b c Boyer 1991, p. 100.
  13. ^ Heath 1911, p. 741.
  14. ^ Sialaros 2020, p. 142.
  15. ^ a b c d Ball 1960, p. 52.
  16. ^ Sialaros 2020, p. 141.
  17. ^ Goulding 2010, p. 125.
  18. ^ a b Smorynski 2008, p. 2.
  19. ^ Goulding 2010, p. 118.
  20. ^ a b Heath 1981, p. 355.
  21. ^ Goulding 2010, p. 126.
  22. ^ a b Heath 1908, p. 2.
  23. ^ Sialaros 2020, p. 147.
  24. ^ Sialaros 2020, pp. 147–148.
  25. ^ a b c Bruno 2003, p. 126.
  26. ^ Ball 1960, p. 51.
  27. ^ Tracy 2000, pp. 343–344.
  28. ^ Taisbak & Waerden 2021, § "Life".
  29. ^ a b c d e f Goulding 2010, p. 120.
  30. ^ Sialaros 2021, § "Life" and Note 5.
  31. ^ a b Boyer 1991, p. 101.
  32. ^ Boyer 1991, p. 96.
  33. ^ Sialaros 2018, p. 90.
  34. ^ Heath 1981, p. 357.
  35. ^ Ball 1960, pp. 52–53.
  36. ^ Artmann 2012, p. 3.
  37. ^ Asper 2010, § para. 2.
  38. ^ Struik 1967, p. 51, "their logical structure has influenced scientific thinking perhaps more than any other text in the world".
  39. ^ Asper 2010, § para. 6.
  40. ^ a b c d e f g h i j Sialaros 2021, § "Other Works".
  41. ^ a b c d e Taisbak & Waerden 2021, § "Other writings".
  42. ^ a b Jones 1986, pp. 399–400.
  43. ^ Acerbi 2008, p. 511.
  44. ^ a b c Taisbak & Waerden 2021, § "Legacy".
  45. ^ "NASA Delivers Detectors for ESA's Euclid Spacecraft". Jet Propulsion Laboratory. 9 May 2017.
  46. ^ "Gazetteer of Planetary Nomenclature | Euclides". usgs.gov. International Astronomical Union. Retrieved 3 September 2017.
  47. ^ "4354 Euclides (2142 P-L)". Minor Planet Center. Retrieved 27 May 2018.
  48. ^ Hawes, Susan M.; Kolpas, Sid. "Oliver Byrne: The Matisse of Mathematics - Biography 1810-1829". Mathematical Association of America. Retrieved 10 August 2022.
  49. ^ Hähl & Peters 2022, § para. 1.

Sources

Books and chapters
Journal and encyclopedia articles
Online

External links

Works
The Elements
  • PDF copy, with the original Greek and an English translation on facing pages, University of Texas.
  • All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.