Euclid
Euclid | |
---|---|
Εὐκλείδης | |
Years active | fl. 300 BC |
Known for | |
Scientific career | |
Fields | Mathematics (Geometry) |
Euclid (
Very little is known of Euclid's life, and most information comes from the scholars
In the Elements, Euclid deduced the
Life
Traditional narrative
The English name 'Euclid' is the anglicized version of the Ancient Greek name Eukleídes (Εὐκλείδης).[4][a] It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".[6] In English, by metonymy, 'Euclid' can mean his most well-known work, Euclid's Elements, or a copy thereof,[5] and is sometimes synonymous with 'geometry'.[2]
As with many ancient Greek mathematicians, the details of Euclid's life are mostly unknown.[7] 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.[8][b] 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.[4][c]
According to Proclus, Euclid lived shortly after several of Plato's (d. 347 BC) followers and before the mathematician Archimedes (c. 287 – c. 212 BC);[d] specifically, Proclus placed Euclid during the rule of Ptolemy I (r. 305/304–282 BC).[8][7][e] Euclid's birthdate is unknown; some scholars estimate around 330[11][12] or 325 BC,[2][13] but others refrain from speculating.[14] It is presumed that he was of Greek descent,[11] but his birthplace is unknown.[15][f] Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this.[17] 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.[18] Historian Thomas Heath supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on;[19] Sialaros considers this a mere conjecture.[20][4] In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition.[11]
In his Collection, Pappus mentions that
Identity and historicity
Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher
Medieval Arabic sources give vast amounts of information concerning Euclid's life, but are completely unverifiable.[4] Most scholars consider them of dubious authenticity;[8] Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.[17] There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".[29] 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".[29] This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus.[30] Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature.[31]
Any firm dating of Euclid's activity c. 300 BC is called into question by a lack of contemporary references.
Works
Elements
Euclid is best known for his thirteen-book treatise, the Elements (
The Elements does not exclusively discuss geometry as is sometimes believed.
Contents
No. | Postulates |
---|---|
Let the following be postulated: | |
1 | To draw a straight line from any point to any point[k] |
2 | To produce a finite straight line continuously in a straight line |
3 | To describe 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 |
Book 1 of the Elements is foundational for the entire text.[37] It begins with a series of 20 definitions for basic geometric concepts such as lines, angles and various regular polygons.[45] Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.[46][l] These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system.[47][m] The common notions exclusively concern the comparison of magnitudes.[49] While postulates 1 through 4 are relatively straightforward,[n] the 5th is known as the parallel postulate and particularly famous.[49][o]
Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry and
Book 2 is traditionally understood as concerning "
Book 3 focuses on circles, while the 4th discusses
From Book 7 onwards, the mathematician Benno Artmann notes that "Euclid starts afresh. Nothing from the preceding books is used".[55] 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.[37] Book 7 includes the Euclidean algorithm, a method for finding the greatest common divisor of two numbers.[55] The 8th book discusses geometric progressions, while book 9 includes the proposition, now called Euclid's theorem, that there are infinitely many prime numbers.[37]
Of the Elements, book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.[41]
The final three books (11–13) primarily discuss solid geometry.[39] By introducing a list of 37 definitions, Book 11 contextualizes the next two.[56] Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates.[56] The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37).[56]
Other works
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.[57]
- The Data (Greek: Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems.[57]
- On Divisions (
- 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.[57]
- 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 BC.[57]
Lost works
Four other works are credibly attributed to Euclid, but have been lost.[9]
- The Conics (Greek: Κωνικά) was a four-book survey on conic sections, which was later superseded by Apollonius' more comprehensive treatment of the same name.[58][57] 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.[59] Doubt has been cast on this assertion by the historian Alexander Jones , owing to sparse evidence and no other corroboration of Pappus' account.[59]
- The Pseudaria (
- The Porisms (Greek: Πορίσματα; lit. 'Corollaries') was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 propositions.[58][57] 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".[57] The mathematician Michel Chasles speculated that these now-lost propositions included content related to the modern theories of transversals and projective geometry.[58][q]
- The Surface Loci (Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown contents, aside from speculation based on the work's title.[58] Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.[57]
Legacy
Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity.[11] Many commentators cite him as one of the most influential figures in the history of mathematics.[2] 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
The first English edition of the Elements was published in 1570 by Henry Billingsley and John Dee.[27] 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
- ^ In modern English, 'Euclid' is pronounced as /ˈjuːklɪd/.[5]
- ^ Euclid's oeuvre also includes the treatise On Divisions, which survives fragmented in a later Arabic source.[9] He authored numerous lost works as well.
- ^ 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.[10]
- ^ Proclus was likely working from (now-lost) 4th-century BC histories of mathematics written by Theophrastus and Eudemus of Rhodes. Proclus explicitly mentions Amyclas of Heracleia, Menaechmus and his brother Dinostratus, Theudius of Magnesia, Athenaeus of Cyzicus, Hermotimus of Colophon, and Philippus of Mende, and says that Euclid came "not long after" these men.
- ^ See Heath 1981, p. 354 for an English translation on Proclus's account of Euclid's life.
- ^ Later Arab sources state he was a Greek born in modern-day Tyre, Lebanon, though these accounts are considered dubious and speculative.[8][4] 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,[16] see §Identity and historicity
- 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).[24]
- ^ The Elements version available today also includes "post-Euclidean" mathematics, probably added later by later editors such as the mathematician Theon of Alexandria in the 4th century.[36]
- ^ 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".[42]
- ^ 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.[43]
- ^ See also: Euclidean relation
- ^ The distinction between these categories is not immediately clear; postulates may simply refer to geometry specifically, while common notions are more general in scope.[46]
- ^ The mathematician Gerard Venema notes that this axiomatic system is not complete: "Euclid assumed more than just what he stated in the postulates".[48]
- ^ See Heath 1908, pp. 195–201 for a detailed overview of postulates 1 through 4
- prove the postulate—which would make it different from the other, unprovable, four postulates.[50]
- ^ Much of Book 5 was probably ascertained from earlier mathematicians, perhaps Eudoxus.[41]
- ^ See Jones 1986, pp. 547–572 for further information on the Porisms
Citations
- ^ Getty.
- ^ a b c d Bruno 2003, p. 125.
- ^ a b Sialaros 2021, § "Summary".
- ^ a b c d e f g h i Sialaros 2021, § "Life".
- ^ a b OEDa.
- ^ OEDb.
- ^ a b Heath 1981, p. 354.
- ^ a b c d e f Asper 2010, § para. 1.
- ^ a b c d Sialaros 2021, § "Works".
- ^ Heath 1911, p. 741.
- ^ a b c d Ball 1960, p. 52.
- ^ Sialaros 2020, p. 141.
- ^ Goulding 2010, p. 125.
- ^ a b Smorynski 2008, p. 2.
- ^ a b Boyer 1991, p. 100.
- ^ Goulding 2010, p. 118.
- ^ a b Heath 1981, p. 355.
- ^ Goulding 2010, p. 126.
- ^ a b Heath 1908, p. 2.
- ^ Sialaros 2020, pp. 147–148.
- ^ Sialaros 2020, p. 142.
- ^ a b c Bruno 2003, p. 126.
- ^ Ball 1960, p. 51.
- ^ Tracy 2000, pp. 343–344.
- ^ Sialaros 2021, § "Life" and Note 5.
- ^ a b c Jones 2005.
- ^ a b c d e Goulding 2010, p. 120.
- ^ Taisbak & Van der Waerden 2021, § "Life".
- ^ a b Boyer 1991, p. 101.
- ^ Boyer 1991, p. 96.
- ^ Sialaros 2018, p. 90.
- ^ Heath 1981, p. 357.
- ^ Ball 1960, pp. 52–53.
- ^ Fowler 1999, pp. 210–211.
- ^ a b Asper 2010, § para. 2.
- ^ a b c d Asper 2010, § para. 6.
- ^ a b c d e f g h i Taisbak & Van der Waerden 2021, § "Sources and contents of the Elements".
- ^ Cuomo 2005, p. 131.
- ^ a b Artmann 2012, p. 3.
- ^ Asper 2010, § para. 4.
- ^ a b c d e f g Sialaros 2021, § "The Elements".
- ^ a b Jahnke 2010, p. 18.
- ^ Asper 2010, § para. 5.
- ^ Heath 1908, pp. 154–155.
- ^ Artmann 2012, p. 3–4.
- ^ a b Wolfe 1945, p. 4.
- ^ Pickover 2009, p. 56.
- ^ Venema 2006, p. 10.
- ^ a b c d Artmann 2012, p. 4.
- ^ Heath 1908, p. 202.
- ^ Artmann 2012, p. 5.
- ^ Artmann 2012, pp. 5–6.
- ^ Artmann 2012, p. 6.
- ^ Heath 1908b, p. 191.
- ^ a b Artmann 2012, p. 7.
- ^ a b c Artmann 2012, p. 9.
- ^ a b c d e f g h i j Sialaros 2021, § "Other Works".
- ^ a b c d e Taisbak & Van der Waerden 2021, § "Other writings".
- ^ a b Jones 1986, pp. 399–400.
- ^ Acerbi 2008, p. 511.
- ^ a b c Taisbak & Van der Waerden 2021, § "Legacy".
- ^ "NASA Delivers Detectors for ESA's Euclid Spacecraft". Jet Propulsion Laboratory. 9 May 2017.
- ^ "Gazetteer of Planetary Nomenclature | Euclides". usgs.gov. International Astronomical Union. Retrieved 3 September 2017.
- ^ "4354 Euclides (2142 P-L)". Minor Planet Center. Retrieved 27 May 2018.
- ^ Hawes & Kolpas 2015.
- ^ Hähl & Peters 2022, § para. 1.
Sources
- Books
- Artmann, Benno (2012) [1999]. Euclid: The Creation of Mathematics. New York: ISBN 978-1-4612-1412-0.
- ISBN 978-0-486-20630-1.
- Bruno, Leonard C. (2003) [1999]. Math and Mathematicians: The History of Math Discoveries Around the World. Baker, Lawrence W. Detroit: U X L. OCLC 41497065.
- ISBN 978-0-471-54397-8.
- ISBN 978-1-134-71019-5.
- ISBN 978-0-19-850258-6.
- Goulding, Robert (2010). Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History. Dordrecht: Springer Netherlands. ISBN 978-90-481-3542-4.
- Heath, Thomas, ed. (1908). The Thirteen Books of Euclid's Elements. Vol. 1. New York: ISBN 978-0-486-60088-8.
- Heath, Thomas, ed. (1908b). The Thirteen Books of Euclid's Elements. Vol. 2. New York: Dover Publications.
- ISBN 0-486-24073-8, 0-486-24074-6
- Jahnke, Hans Niels (2010). "The Conjoint Origin of Proof and Theoretical Physics". In ISBN 978-1-4419-0576-5.
- Jones, Alexander, ed. (1986). Pappus of Alexandria: Book 7 of the Collection. Vol. Part 2: Commentary, Index, and Figures. New York: ISBN 978-3-540-96257-1.
- ISBN 978-1-4027-5796-9.
- Sialaros, Michalis (2018). "How Much Does a Theorem Cost?". In Sialaros, Michalis (ed.). Revolutions and Continuity in Greek Mathematics. Berlin: ISBN 978-3-11-056595-9.
- Sialaros, Michalis (2020). "Euclid of Alexandria: A Child of the Academy?". In Kalligas, Paul; Balla, Vassilis; Baziotopoulou-Valavani, Chloe; Karasmanis, Effie (eds.). Plato's Academy. Cambridge: ISBN 978-1-108-42644-2.
- Smorynski, Craig (2008). History of Mathematics: A Supplement. New York: ISBN 978-0-387-75480-2.
- Tracy, Stephen V (2000). "Demetrius of Phalerum: Who was He and Who was He Not?". In Fortenbaugh, William W.; Schütrumpf, Eckhart (eds.). Demetrius of Phalerum: Text, Translation and Discussion. Rutgers University Studies in Classical Humanities. Vol. IX. New Brunswick and London: ISBN 978-1-3513-2690-2.
- Venema, Gerard (2006). The Foundations of Geometry. Hoboken: ISBN 978-0-13-143700-5.
- Wolfe, Harold E. (1945). Introduction To Non-Euclidean Geometry. New York: Dryden Press.
- Articles
- Acerbi, Fabio (September 2008). "Euclid's Pseudaria". Archive for History of Exact Sciences. 62 (5): 511–551. S2CID 120860272.
- Jones, Alexander (2005). "Euclid, the Elusive Geometer" (PDF). Euclid and His Heritage Meeting, Clay Mathematics Institute, Oxford, 7–8 October 2005.
- Asper, Markus (2010). "Euclid". In Gagarin, Michael (ed.). The Oxford Encyclopedia of Ancient Greece and Rome. Oxford: ISBN 978-0-19-517072-6.
- Hähl, Hermann; Peters, Hanna (10 June 2022). "A Variation of Hilbert's Axioms for Euclidean Geometry". Mathematische Semesterberichte. 69 (2): 253–258. S2CID 249581871.
- Hawes, Susan M.; Kolpas, Sid (August 2015). "Oliver Byrne: The Matisse of Mathematics – Biography 1810–1829". Mathematical Association of America. Retrieved 10 August 2022.
- Heath, Thomas Little (1911). . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 20 (11th ed.). Cambridge University Press. pp. 470–471.
- Sialaros, Michalis (2021) [2015]. "Euclid". ISBN 978-0-19-938113-5.
- Taisbak, Christian Marinus; van der Waerden, Bartel Leendert (5 January 2021). "Euclid". Encyclopædia Britannica. Chicago: Encyclopædia Britannica, Inc.
- Online
- "Euclid". J. Paul Getty Museum. Retrieved 11 August 2022.
- "Euclid, n". OED Online. Oxford: Oxford University Press. Retrieved 10 August 2022. (subscription required)
- "Euclidean (adj.)". Online Etymology Dictionary. Retrieved 18 March 2015.
External links
- Works
- Works by Euclid at Project Gutenberg
- Works by or about Euclid at Internet Archive
- Works by Euclid at LibriVox (public domain audiobooks)
- Euclid Collection at University College London (c.500 editions of works by Euclid), available online through the Stavros Niarchos Foundation Digital Library.
- Scans of Johan Heiberg's edition of Euclid at wilbourhall.org
- 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.