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]

Euclid (/ˈjuːklɪd/; Greek: Εὐκλείδης; fl. 300 BC) 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.

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 BC, after Plato and before Archimedes. There is some speculation that Euclid was a student of the Platonic Academy and later taught at the Musaeum. Euclid is often regarded as bridging 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

Detail of Raphael's impression of Euclid, teaching students in the The School of Athens (1509–1511)
Detail of Raphael's impression of Euclid, teaching students in the The School of Athens (1509–1511)

The English name 'Euclid' is the anglicized version of the Ancient Greek name Εὐκλείδης.[5][a] It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".[7] The word 'Euclid' less commonly also means "a copy of the same",[6] and is sometimes synonymous with 'geometry'.[3]

Like many ancient Greek mathematicians, Euclid's life is mostly unknown.[8] 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.[9][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".[11] The traditional narrative mainly follows the 5th century AD 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.[5][c] According to Proclus, Euclid lived after the philosopher Plato (d. 347 BC) and before the mathematician Archimedes (c. 287 – c. 212 BC); specifically, Proclus placed Euclid during the rule of Ptolemy I (r. 305/304–282 BC).[9][8][d] In his Collection, Pappus indicates that Euclid was active in Alexandria, where he founded a mathematical tradition.[9][13] Thus, the traditional outline—described by the historian Michalis Sialaros as the "dominant view"—holds that Euclid lived around 300 BC in Alexandria while Ptolemy I reigned.[5]

Euclid's birthdate is unknown; some scholars estimate around 330[14][15] or 325 BC,[3][16] but other sources avoid speculating a date entirely.[17] It is presumed that he was of Greek descent,[14] but his birthplace is unknown.[11][e] Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this.[19] 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.[20] 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.[21][22] The accuracy of these assertions has been questioned by Sialaros,[23] who stated that Heath's theory "must be treated merely as a conjecture".[5] 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.[14]

Alexander the Great founded Alexandria in 331 BC, where Euclid would later be active sometime around 300 BC.[24] The rule of Ptolemy I from 306 BC onwards gave the city a stability which was relatively unique in the Mediterranean, amid the chaotic wars over dividing Alexander's empire.[25] Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education.[11][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.[24] According to Pappus, the later mathematician Apollonius of Perga was taught there by pupils of Euclid.[21] Euclid's date of death is unknown; it has been estimated that he died c. 270 BC, presumably in Alexandria.[24]

Identity and historicity

A painting from the 1650s by Domenico Maroli, depicting the philosopher Euclid of Megara. At the time, Euclid the philosopher and Euclid the mathematician were wrongly considered the same person, so this painting includes mathematical objects on the table.
A painting from the 1650s by Domenico Maroli, depicting the philosopher Euclid of Megara. At the time, Euclid the philosopher and Euclid the mathematician were wrongly considered the same person, so this painting includes mathematical objects on the table.

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.[5][17] Historically, medieval scholars frequently confused the mathematician and philosopher, mistakenly referring to the former in Latin as 'Megarensis' (lit.'of Megara').[27] As a result, biographical information on the mathematician Euclid was long conflated with the lives of both Euclid of Alexandria and Euclid of Megara.[5] The only scholar of antiquity known to have confused the mathematician and philosopher was Valerius Maximus.[28][g] However, this mistaken identification was relayed by many anonymous Byzantine sources and the Renaissance scholars Campanus of Novara and Theodore Metochites, which was included in a of 1482 translation of the latter by Erhard Ratdolt.[28] After the mathematician Bartolomeo Zamberti [fr] (1473–1539) affirmed this presumption in his 1505 translation, all subsequent publications passed on this identification.[28][h] Later Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.[28]

Arab sources written many centuries after his death give vast amounts of information concerning Euclid's life, but are completely unverifiable.[5] Most scholars consider them of dubious authenticity;[9] Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.[19] There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".[30] 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".[30] This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus.[31] Both the accounts were written in the 5th century AD, neither indicate their source, and neither story appears in ancient Greek literature.[32]

The traditional narrative of Euclid's activity c. 300 is complicated by no mathematicians of the 4th century BC indicating his existence.[5] Mathematicians of the 3rd century such as Archimedes and Apollonius "assume a part of his work to be known";[5] however, Archimedes strangely uses an older theory of proportions, rather than that of Euclid.[9] The Elements is dated to have been at least partly in circulation by the 3rd century BC.[5] Some ancient Greek mathematician mention him by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").[33] 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.[34]

Works

Elements

Main article: Euclid's Elements

A papyrus fragment of Euclid's Elements dated to c. 75–125 AD. Found at Oxyrhynchus, the diagram accompanies Book II, Proposition 5.[35]
A papyrus fragment of Euclid's Elements dated to c. 75–125 AD. Found at Oxyrhynchus, the diagram accompanies Book II, Proposition 5.[35]

Euclid is best known for his thirteen-book treatise, the Elements (Greek: Στοιχεῖα; Stoicheia), considered his magnum opus.[4][36] Much of its content originates from earlier mathematicians, including Eudoxus (books 10, 12), Hippocrates of Chios (3.14), Thales (1.26) and Theaetetus (10.9), while other theorems are mentioned by Plato and Aristotle.[37] It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics.[38][i] The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps".[37] Despite this, Sialaros furthers that "the remarkably tight structure of the Elements reveals authorial control beyond the limits of a mere editor".[10] The mathematician Serafina Cuomo described it as a "reservoir of results".[39]

The Elements does not exclusively discuss geometry as is sometimes believed.[38] It is traditionally divided into three topics: plane geometry (books 1–6), basic arithmetic (books 7–10:) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this scheme.[40][41] The heart of the text is the theorems scattered throughout.[36] Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles".[42] The first group includes statements labeled as a "definition" (Greek: ὅρος or Greek: ὁρισμός), "postulate" (Greek: αἴτημα‎), or a "common notion" (Greek: κοινὴ ἔννοια);[42][43] only the first book includes postulates—later known as axioms—and common notions.[38][j] The second group consists of propositions, presented alongside mathematical proofs and diagrams.[42][k] It is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit.[10] As a whole, the authorial voice remains general and impersonal.[37]

Contents

Books 1–6
Euclid's postulates and common notions[45]
No. Postulates
Let the following be postulated:
1 To draw a straight line from any point to any point[l]
2 To produce a finite straight line continuously in a straight line
3 To described a circle with any centre and distance
4 That all right angles are equal to one another
5 That, if a straight line falling on two straight lines make the
interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side
on which are the angles less than the two right angles
No. Common notions
1 Things which are equal to the same thing are also equal to one another
2 If equals be added to equals, the wholes are equal
3 If equals be subtracted from equals, the remainders are equal
4 Things which coincide with one another are equal to one another
5 The whole is greater than the part

See also: Foundations of geometry

Book 1 of the Elements is foundational for the entire text.[38] It begins with a series of 20 definitions for lines, angles and different regular polygons among other basic concepts.[46] Euclid then presets 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.[47][m] These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system.[48][n] The common notions exclusively concern the comparison of magnitudes.[50] While postulates 1 through 4 are relatively straight forward,[o] the 5th is known as the parallel postulate and particularly famous.[50][p] Book 1 also includes 48 propositions, which can be loosely divided into those concerning basics theorems of plane geometry (1–26); theories on parallel lines (27–32); theories on parallelograms (33–45); and the Pythagorean theorem (46–48).[50] The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".[42]

Book 2 is traditionally understood as concerning geometric algebra, though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.[42] The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes.[38][50] Book 3 focuses on circles, while the 4th discusses regular polygons, especially the pentagon.[38][52] Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion".[53][q] Book 6 utilizes the "theory of ratios" in the context of plane geometry.[38] It is built almost entirely of its first proposition:[54] "Triangles and parallelograms which are under the same height are to one another as their bases".[55]

Books 7–10

From Book 7 onwards, the mathematician Benno Artmann [de] notes that "Euclid starts afresh. Nothing from the preceding books is used".[56] Number theory is covered by books 7 to 10, the former beginning with a set of 22 definitions for parity, prime numbers and other arithmetic-related concepts.[38] Book 7 includes the Euclidean algorithm, a method for finding the greatest common divisor of two numbers.[56] The 8th book discusses geometric progressions, while book 9 includes a proof that there are an infinite amount of prime numbers.[38]

Of the Elements, book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.[42]

Books 11–13

Books 11 through 13 primarily discuss solid geometry.[40]

Other works

Euclid's construction of a regular dodecahedron.
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.

Lost works

Four other works are credibly attributed to Euclid, but have been lost.[10]

Legacy

See also: List of things named after Euclid

The cover page of Oliver Byrne's 1847 colored edition of the Elements
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.[14] 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.[61] Among Euclid's many namesakes are the European Space Agency's (ESA) Euclid spacecraft,[62] the lunar crater Euclides,[63] and the minor planet 4354 Euclides.[64]

The Elements is often considered after the Bible as the most frequently translated, published, and studied book in the Western World's history.[61] 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.[61]

The first English edition of the Elements was published in 1570 by Henry Billingsley and John Dee.[28] 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.[65] David Hilbert authored a modern axiomatization of the Elements.[66]

References

Notes

  1. ^ In modern English, 'Euclid' is pronounced as /ˈjuːklɪd/ in British English and /ˈjuˌklɪd/ in American English.[6]
  2. ^ Euclid's oeuvre also includes the treatise On Divisions, which survives fragmented in a later Arabic source.[10] 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.[12]
  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.[9][5] 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,[18] 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 BC).[26]
  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.[28]
  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.[29]
  9. ^ The Elements version available today also includes "post-Euclidian" mathematics, probably added later by later editors such as the mathematician Theon of Alexandria in the 4th century.[37]
  10. ^ The use of the term "axiom" instead of "postulate" derives from the choice of Proclus to do so in his highly influential commentary on the Elements. Proclus also substituted the term "hypothesis" instead of "common notion", though preserved "postulate".[43]
  11. ^ Euclid includes Q.E.D. (quod erat demonstrandum; lit.'what was to be demonstrated') at the end of each proof, which has since become a long-standing tradition in the presentation of mathematical proofs.[44]
  12. ^ See also: Euclidean relation
  13. ^ The distinction between these categories is not immediately clear; postulates may simply refer to geometry specifically, while common notions are more general in scope.[47]
  14. ^ The mathematician Gerard Venema notes that this axiomatic system is not complete: "Euclid assumed more than just what he stated in the postulates".[49]
  15. ^ See Heath 1908, pp. 195–201 for a detailed overview of postulates 1 through 4
  16. ^ Since antiquity, enormous amounts of scholarship have been written about the 5th postulate, usually from mathematicians attempting to prove the postulate—which would make it different than the other, unprovable, four postulates.[51]
  17. ^ Much of Book 5 was probably ascertained from earlier mathematicians, perhaps Eudoxus.[42]
  18. ^ 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 c d e f g h i j k Sialaros 2021, § "Life".
  6. ^ a b OEDa.
  7. ^ OEDb.
  8. ^ a b Heath 1981, p. 354.
  9. ^ a b c d e f Asper 2010, § para. 1.
  10. ^ a b c d Sialaros 2021, § "Works".
  11. ^ a b c Boyer 1991, p. 100.
  12. ^ Heath 1911, p. 741.
  13. ^ Sialaros 2020, p. 142.
  14. ^ a b c d Ball 1960, p. 52.
  15. ^ Sialaros 2020, p. 141.
  16. ^ Goulding 2010, p. 125.
  17. ^ a b Smorynski 2008, p. 2.
  18. ^ Goulding 2010, p. 118.
  19. ^ a b Heath 1981, p. 355.
  20. ^ Goulding 2010, p. 126.
  21. ^ a b Heath 1908, p. 2.
  22. ^ Sialaros 2020, p. 147.
  23. ^ Sialaros 2020, pp. 147–148.
  24. ^ a b c Bruno 2003, p. 126.
  25. ^ Ball 1960, p. 51.
  26. ^ Tracy 2000, pp. 343–344.
  27. ^ Taisbak & Waerden 2021, § "Life".
  28. ^ a b c d e f Goulding 2010, p. 120.
  29. ^ Sialaros 2021, § "Life" and Note 5.
  30. ^ a b Boyer 1991, p. 101.
  31. ^ Boyer 1991, p. 96.
  32. ^ Sialaros 2018, p. 90.
  33. ^ Heath 1981, p. 357.
  34. ^ Ball 1960, pp. 52–53.
  35. ^ Fowler 1999, pp. 210–211.
  36. ^ a b Asper 2010, § para. 2.
  37. ^ a b c d Asper 2010, § para. 6.
  38. ^ a b c d e f g h i Taisbak & Waerden 2021, § "Sources and contents of the Elements".
  39. ^ Cuomo 2005, p. 131.
  40. ^ a b Artmann 2012, p. 3.
  41. ^ Asper 2010, § para. 4.
  42. ^ a b c d e f g Sialaros 2021, § "The Elements".
  43. ^ a b Jahnke 2010, p. 18.
  44. ^ Asper 2010, § para. 5.
  45. ^ Heath 1908, pp. 154–155.
  46. ^ Artmann 2012, p. 3–4.
  47. ^ a b Wolfe 1945, p. 4.
  48. ^ Pickover 2009, p. 56.
  49. ^ Venema 2006, p. 10.
  50. ^ a b c d Artmann 2012, p. 4.
  51. ^ Heath 1908, p. 202.
  52. ^ Artmann 2012, p. 5.
  53. ^ Artmann 2012, pp. 5–6.
  54. ^ Artmann 2012, p. 6.
  55. ^ Heath 1908b, p. 191.
  56. ^ a b Artmann 2012, p. 7.
  57. ^ a b c d e f g h i j Sialaros 2021, § "Other Works".
  58. ^ a b c d e Taisbak & Waerden 2021, § "Other writings".
  59. ^ a b Jones 1986, pp. 399–400.
  60. ^ Acerbi 2008, p. 511.
  61. ^ a b c Taisbak & Waerden 2021, § "Legacy".
  62. ^ "NASA Delivers Detectors for ESA's Euclid Spacecraft". Jet Propulsion Laboratory. 9 May 2017.
  63. ^ "Gazetteer of Planetary Nomenclature | Euclides". usgs.gov. International Astronomical Union. Retrieved 3 September 2017.
  64. ^ "4354 Euclides (2142 P-L)". Minor Planet Center. Retrieved 27 May 2018.
  65. ^ Hawes, Susan M.; Kolpas, Sid. "Oliver Byrne: The Matisse of Mathematics - Biography 1810-1829". Mathematical Association of America. Retrieved 10 August 2022.
  66. ^ Hähl & Peters 2022, § para. 1.

Sources

Books and chapters
Journal and encyclopedia articles
Online
Works
The Elements