Editing Euclid

{{Short description|Ancient Greek mathematician (fl. 300 BC)}}
{{For-multi|the philosopher|Euclid of Megara|other uses}}
{{protection padlock|small=yes}}
{{Use dmy dates|date=June 2021}}
{{Infobox scientist
| name              = Euclid
| native_name       = Εὐκλείδης
| image             = Jusepe de Ribera - Euclid - 2001.26 - J. Paul Getty Museum.jpg
| caption           = Euclid by [[Jusepe de Ribera]], {{circa|1630–1635}}{{sfn|Getty}}
| years_active      = {{fl.|300}} BC
| known_for         = {{hlist|The [[Euclid's Elements|''Elements'']]|''[[Euclid's Optics|Optics]]''|''[[Euclid's Data|Data]]''}}
{{collapsible list|title={{nobold|Various concepts}}|{{plainlist|
* [[Euclidean geometry]]
* [[Euclidean algorithm]]
* [[Euclid's theorem]]
* [[Euclidean relation]]
* [[Pythagorean triple#Generating a triple|Euclid's formula]]
* [[List of things named after Euclid|Numerous other namesakes]]
}}}}
| field             = [[Mathematics]] ([[Geometry]])
}}

'''Euclid''' ({{IPAc-en|ˈ|j|uː|k|l|ɪ|d}}; {{langx|grc|Εὐκλείδης}}; {{fl.|300}} BC) was an [[Ancient Greece|ancient Greek]] [[mathematician]] active as a [[geometer]] and [[logician]].{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/125 125]}} Considered the "father of geometry",{{sfn|Sialaros|2021|loc=§ "Summary"}} he is chiefly known for the ''[[Euclid's Elements|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 innovations in combination with a synthesis of theories from earlier Greek mathematicians, including [[Eudoxus of Cnidus]], [[Hippocrates of Chios]], [[Thales]] and [[Theaetetus (mathematician)|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 scholars [[Proclus]] and [[Pappus of Alexandria]] many centuries later. [[Mathematics in the medieval Islamic world|Medieval Islamic mathematicians]] invented a fanciful biography, and medieval [[Byzantine Empire|Byzantine]] and early [[Renaissance]] scholars mistook him for the earlier philosopher [[Euclid of Megara]]. It is now generally accepted that he spent his career in [[Alexandria]] and lived around 300 BC, after [[Plato]]'s students and before Archimedes. There is some speculation that Euclid studied at the [[Platonic Academy]] and later taught at the [[Musaeum]]; he is regarded as bridging the earlier Platonic tradition in [[Athens]] with the later tradition of Alexandria.

In the ''Elements'', Euclid deduced the [[theorem]]s from a small set of [[axiom]]s. He also wrote works on [[Perspective (visual)|perspective]], [[conic section]]s, [[spherical geometry]], [[number theory]], and [[Rigour#Mathematical proof|mathematical rigour]]. In addition to the ''Elements'', Euclid wrote a central early text in the [[optics]] field, ''[[Euclid's Optics|Optics]]'', and lesser-known works including ''[[Euclid's Data|Data]]'' and ''Phaenomena''. Euclid's authorship of ''On Divisions of Figures'' and ''Catoptrics'' has been questioned. He is thought to have written many [[#Lost works|lost works]].
{{TOC limit|3}}

==Life==
===Traditional narrative===
[[File:0 Chambre de Raphaël - École d'Athènes - Musées du Vatican.JPG|thumb|upright=1.4|Detail of [[Raphael]]'s impression of Euclid, teaching students in ''[[The School of Athens]]'' (1509–1511)]]

The English name 'Euclid' is the anglicized version of the [[Ancient Greek]] name {{Transliteration|grc|Eukleídes}} ({{lang|grc|Εὐκλείδης}}).{{sfn|Sialaros|2021|loc=§ "Life"}}{{efn|In modern English, 'Euclid' is pronounced as {{IPAc-en|ˈ|j|uː|k|l|ɪ|d}}.{{sfn|''OED''a}}}} It is derived from '[[Wikt:eu-#English|eu-]]' ([[Wikt:εὖ#Ancient Greek|εὖ]]; 'well') and 'klês' ([[Wikt:#-κλῆς|-κλῆς]]; 'fame'), meaning "renowned, glorious".{{sfn|''OED''b}} In English, by [[metonymy]], 'Euclid' can mean his most well-known work, [[Euclid's Elements|Euclid's ''Elements'']], or a copy thereof,{{sfn|''OED''a}} and is sometimes synonymous with 'geometry'.{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/125 125]}}

As with many [[ancient Greek mathematicians]], the details of Euclid's life are mostly unknown.{{sfn|Heath|1981|p=354}} He is accepted as the author of four mostly extant treatises—the ''Elements'', ''[[Euclid's Optics|Optics]]'', ''[[Euclid's Data|Data]]'', ''[[Euclid's Phaenomena|Phaenomena]]''—but besides this, there is nothing known for certain of him.{{sfn|Asper|2010|loc=§ para. 1}}{{efn|Euclid's ''oeuvre'' also includes the treatise ''On Divisions'', which survives fragmented in a later Arabic source.{{sfn|Sialaros|2021|loc=§ "Works"}} He authored numerous [[#Lost works|lost works]] as well.{{sfn|Sialaros|2021|loc=§ "Works"}}}} 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.{{sfn|Sialaros|2021|loc=§ "Life"}}{{efn|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''.{{sfn|Heath|1911|p=741}}}}

<!--Early life-->
According to Proclus, Euclid lived shortly after several of [[Plato]]'s ({{Died in|347}} BC) followers and before the mathematician [[Archimedes]] ({{circa|287|212}} BC);{{efn|Proclus was likely working from (now-lost) 4th-century BC histories of mathematics written by [[Theophrastus]] and [[Eudemus of Rhodes]]. [[Proclus#Commentary on Euclid's Elements|Proclus explicitly mentions]] Amyclas of Heracleia, [[Menaechmus]] and his brother [[Dinostratus]], [[Theudius|Theudius of Magnesia]], [[Athenaeus of Cyzicus]], [[Hermotimus of Colophon]], and [[Philip of Opus|Philippus of Mende]], and says that Euclid came "not long after" these men.}} specifically, Proclus placed Euclid during the rule of [[Ptolemy I Soter|Ptolemy I]] ({{reign|305/304|282}} BC).{{sfn|Heath|1981|p=354}}{{sfn|Asper|2010|loc=§ para. 1}}{{efn|See {{harvnb|Heath|1981|p=354}} for an English translation on Proclus's account of Euclid's life.}} Euclid's birthdate is unknown; some scholars estimate around 330{{sfn|Ball|1960|p=[https://archive.org/details/shortaccountofhi0000ball/page/52/mode/2up 52]}}{{sfn|Sialaros|2020|p=141}} or 325 BC,{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/125 125]}}{{sfn|Goulding|2010|p=125}} but others refrain from speculating.{{sfn|Smorynski|2008|p=2}} It is presumed that he was of Greek descent,{{sfn|Ball|1960|p=[https://archive.org/details/shortaccountofhi0000ball/page/52/mode/2up 52]}} but his birthplace is unknown.{{sfn|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/100/mode/2up?view=theater 100]}}{{efn|Later Arab sources state he was a Greek born in modern-day [[Tyre, Lebanon]], though these accounts are considered dubious and speculative.{{sfn|Asper|2010|loc=§ para. 1}}{{sfn|Sialaros|2021|loc=§ "Life"}} See {{harvnb|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]],{{sfn|Goulding|2010|p=118}} see [[#Identity and historicity|§Identity and historicity]]}} Proclus held that Euclid followed the [[Platonism|Platonic tradition]], but there is no definitive confirmation for this.{{sfn|Heath|1981|p=355}} It is unlikely he was a contemporary of Plato, so it is often presumed that he was educated by Plato's disciples at the [[Platonic Academy]] in Athens.{{sfn|Goulding|2010|p=126}} Historian [[Thomas Heath (classicist)|Thomas Heath]] supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on;{{sfn|Heath|1908|p=[https://archive.org/details/bub_gb_UhgPAAAAIAAJ/page/n13/mode/2up 2]}} historian Michalis Sialaros considers this a mere conjecture.{{sfn|Sialaros|2021|loc=§ "Life"}}{{sfn|Sialaros|2020|pp=147–148}} In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition.{{sfn|Ball|1960|p=[https://archive.org/details/shortaccountofhi0000ball/page/52/mode/2up 52]}}

<!--Later life-->
In his ''Collection'', Pappus mentions that [[Apollonius of Perga|Apollonius]] studied with Euclid's students in [[Alexandria]], and this has been taken to imply that Euclid worked and founded a [[school of thought|mathematical tradition]] there.{{sfn|Asper|2010|loc=§ para. 1}}{{sfn|Sialaros|2020|p=142}}{{sfn|Heath|1908|p=[https://archive.org/details/bub_gb_UhgPAAAAIAAJ/page/n13/mode/2up 2]}} The city was founded by [[Alexander the Great]] in 331 BC,{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/126 126]}} and the rule of Ptolemy I from 306 BC onwards gave it a stability which was relatively unique amid the chaotic [[Wars of the Diadochi|wars over dividing Alexander's empire]].{{sfn|Ball|1960|p=[https://archive.org/details/shortaccountofhi0000ball/page/50/mode/2up 51]}} Ptolemy began a process of [[hellenization]] and commissioned numerous constructions, building the massive [[Musaeum]] institution, which was a leading center of education.{{sfn|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/100/mode/2up?view=theater 100]}}{{efn|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).{{sfn|Tracy|2000|pp=343–344}}}} Euclid is speculated to have been among the Musaeum's first scholars.{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/126 126]}} Euclid's date of death is unknown; it has been speculated that he died {{circa|270 BC}}.{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/126 126]}}

===Identity and historicity===
[[File:Domenico Marolì - Euclid of Megara.jpg|thumb|upright=1.5|[[Domenico Maroli]]'s 1650s painting {{lang|it|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'']. 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.{{sfn|Sialaros|2021|loc=§ "Life" and Note 5}}]]

Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher [[Euclid of Megara]], a pupil of [[Socrates]] included in [[List of speakers in Plato's dialogues|dialogues of Plato]] with whom he was historically conflated.{{sfn|Sialaros|2021|loc=§ "Life"}}{{sfn|Smorynski|2008|p=2}} [[Valerius Maximus]], the 1st century AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for [[Eudoxus of Cnidus|Eudoxus]] (4th century BC) as the mathematician to whom Plato sent those asking how to [[Doubling the cube|double the cube]].{{sfn|Jones|2005}} Perhaps on the basis of this mention of a mathematical Euclid roughly a century early, Euclid became mixed up with Euclid of Megara in medieval [[Byzantine Empire|Byzantine]] sources (now lost),{{sfn|Goulding|2010|p=120}} eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described as {{lang|la|Megarensis}} ({{lit|of Megara}}).{{sfn|Sialaros|2021|loc=§ "Life"}}{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Life"}} The Byzantine scholar [[Theodore Metochites]] ({{c.|1300}}) explicitly conflated the two Euclids, as did printer [[Erhard Ratdolt]]'s 1482 {{lang|la|[[editio princeps]]}} of [[Campanus of Novara]]'s Latin translation of the ''Elements''.{{sfn|Goulding|2010|p=120}} After the mathematician {{ill|Bartolomeo Zamberti|fr||de}} appended most of the extant biographical fragments about either Euclid to the preface of his 1505 translation of the ''Elements'', subsequent publications passed on this identification.{{sfn|Goulding|2010|p=120}} A further confusion, which gives the birthplace of Euclid as [[Gela|Gela, Sicily]], arises from the fact that [[Euclid of Megara]] is sometimes said to have been born in Gela.<ref name=":0">{{Cite web |title=Euclid, Elements, volume 1, page 4 |url=http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0086:volume=1:page=4 |access-date=2025-04-15 |website=www.perseus.tufts.edu}}</ref> Later Renaissance scholars, particularly [[Peter Ramus]], reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.{{sfn|Goulding|2010|p=120}}

[[Mathematics in the medieval Islamic world|Medieval Arabic sources]] give vast amounts of information concerning Euclid's life, but are completely unverifiable.{{sfn|Sialaros|2021|loc=§ "Life"}} Euclid, who was supposedly a [[Tyre, Lebanon|Tyre]]-born Greek domiciled at [[Damascus]], was claimed to have been the son of Naucrates.<ref name=":0" /> Most scholars consider them of dubious authenticity.{{sfn|Asper|2010|loc=§ para. 1}} Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.{{sfn|Heath|1981|p=355}} There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".{{sfn|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/100/mode/2up?view=theater 101]}} 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".{{sfn|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/100/mode/2up?view=theater 101]}} This anecdote is questionable since a very similar interaction between [[Menaechmus]] and Alexander the Great is recorded from [[Stobaeus]].{{sfn|Boyer|1991|p=96}} Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature.{{sfn|Sialaros|2018|p=90}}

Any firm dating of Euclid's activity {{c.|300 BC}} is called into question by a lack of contemporary references.{{sfn|Sialaros|2021|loc=§ "Life"}} The earliest original reference to Euclid is in Apollonius' [[prefatory]] letter to the ''[[Conics]]'' (early 2nd century BC): "The third book of the ''Conics'' contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions of solid [[locus (mathematics)|loci]]. Most of these, and the finest of them, are novel. And when we discovered them we realized that Euclid had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that was not felicitously done."{{sfn|Jones|2005}} The ''Elements'' is speculated to have been at least partly in circulation by the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted;{{sfn|Sialaros|2021|loc=§ "Life"}} however, Archimedes employs an older variant of the [[theory of proportions]] than the one found in the ''Elements''.{{sfn|Asper|2010|loc=§ para. 1}} The oldest physical copies of material included in the ''Elements'', dating from roughly 100 AD, can be found on [[Oxyrhynchus Papyri|papyrus fragments unearthed]] in an ancient rubbish heap from [[Oxyrhynchus]], [[Roman Egypt]]. The oldest extant direct citations to the ''Elements'' in works whose dates are firmly known are not until the 2nd century AD, by [[Galen]] and [[Alexander of Aphrodisias]]; by this time it was a standard school text.{{sfn|Jones|2005}} Some ancient Greek mathematicians mention Euclid by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of ''Elements''").{{sfn|Heath|1981|p=357}} In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from a [[Language change|corruption]] of Greek mathematical terms.{{sfn|Ball|1960|pp=[https://archive.org/details/shortaccountofhi0000ball/page/52/mode/2up 52–53]}}

==Works==
=== ''Elements'' ===
{{anchor|The ''Elements''|Elements}}{{main|Euclid's Elements}}
[[File:Oxyrhynchus papyrus with Euclid's Elements.jpg|thumb|upright=1.6|A [[Papyrus Oxyrhynchus 29|papyrus fragment]] of Euclid's ''Elements'' dated to {{circa|75–125 AD}}. Found at [[Oxyrhynchus]], the diagram accompanies Book II, Proposition 5.{{sfn|Fowler|1999|pp=210–211}}]]

Euclid is best known for his thirteen-book treatise, the ''Elements'' ({{langx|grc|[[Wikt:στοιχεία|Στοιχεῖα]]}}; {{transliteration|grc|Stoicheia}}), considered his ''[[magnum opus]]''.{{sfn|Sialaros|2021|loc=§ "Summary"}}{{sfn|Asper|2010|loc=§ para. 2}} Much of its content originates from earlier mathematicians, including [[Eudoxus of Cnidus|Eudoxus]], [[Hippocrates of Chios]], [[Thales]] and [[Theaetetus (mathematician)|Theaetetus]], while other theorems are mentioned by Plato and Aristotle.{{sfn|Asper|2010|loc=§ para. 6}} 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.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{efn|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.{{sfn|Asper|2010|loc=§ para. 6}}}} 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" and the historian [[Serafina Cuomo]] described it as a "reservoir of results".{{sfn|Cuomo|2005|p=131}}{{sfn|Asper|2010|loc=§ para. 6}} Despite this, Sialaros furthers that "the remarkably tight structure of the ''Elements'' reveals authorial control beyond the limits of a mere editor".{{sfn|Sialaros|2021|loc=§ "Works"}}

The ''Elements'' does not exclusively discuss geometry as is sometimes believed.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} It is traditionally divided into three topics: [[plane geometry]] (books 1–6), basic [[number theory]] (books 7–10) and [[solid geometry]] (books 11–13)—though book 5 (on proportions) and 10 (on [[Irrational number|irrational]] lines) do not exactly fit this scheme.{{sfn|Artmann|2012|p=3}}{{sfn|Asper|2010|loc=§ para. 4}} The heart of the text is the [[theorem]]s scattered throughout.{{sfn|Asper|2010|loc=§ para. 2}} Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles".{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}} The first group includes statements labeled as a "definition" ({{langx|grc|ὅρος}} or {{lang|grc|ὁρισμός}}), "postulate" ({{lang|grc|αἴτημα}}), or a "common notion" ({{lang|grc|κοινὴ ἔννοια}});{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}}{{sfn|Jahnke|2010|p=18}} only the first book includes postulates—later known as [[axiom]]s—and common notions.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{efn|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".{{sfn|Jahnke|2010|p=18}}}} The second group consists of propositions, presented alongside [[mathematical proof]]s and diagrams.{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}} It is unknown if Euclid intended the ''Elements'' as a textbook, but its method of presentation makes it a natural fit.{{sfn|Sialaros|2021|loc=§ "Works"}} As a whole, the [[authorial voice]] remains general and impersonal.{{sfn|Asper|2010|loc=§ para. 6}}

====Contents====
{| class="wikitable plainrowheaders floatright" style="font-size:90%"
|-
|+ Euclid's postulates and common notions{{sfn|Heath|1908|pp=154–155}}
|-
! scope="col" | {{abbr|No.|Number}}
! scope="col" | Postulates
|-
| colspan="2" | Let the following be postulated:
|-
| 1
| To draw a straight line from any point to any point{{efn|See also: [[Euclidean relation]]}}
|-
| 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<br/> interior angles on the same side less than two right angles,<br/> the two straight lines, if produced indefinitely, meet on that side<br/> on which are the angles less than the two right angles
|-
! scope="col" | {{abbr|No.|Number}}
! scope="col" | 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.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} It begins with a series of 20 definitions for basic geometric concepts such as [[Line (geometry)|line]]s, [[angle]]s and various [[regular polygon]]s.{{sfn|Artmann|2012|p=3–4}} Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.{{sfn|Wolfe|1945|p=4}}{{efn|The distinction between these categories is not immediately clear; postulates may simply refer to geometry specifically, while common notions are more general in scope.{{sfn|Wolfe|1945|p=4}}}} These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an [[axiomatic system]].{{sfn|Pickover|2009|p=56}}{{efn|The mathematician Gerard Venema notes that this [[axiomatic system]] is not complete: "Euclid assumed more than just what he stated in the postulates".{{sfn|Venema|2006|p=10}}}} The common notions exclusively concern the comparison of [[Magnitude (mathematics)|magnitude]]s.{{sfn|Artmann|2012|p=4}} While postulates 1 through 4 are relatively straightforward,{{efn|See {{harvnb|Heath|1908|pp=195–201}} for a detailed overview of postulates 1 through 4}} the 5th is known as the [[parallel postulate]] and particularly famous.{{sfn|Artmann|2012|p=4}}{{efn|Since antiquity, enormous amounts of scholarship have been written about the 5th postulate, usually from mathematicians attempting to [[Proving (math)|prove]] the postulate—which would make it different from the other, unprovable, four postulates.{{sfn|Heath|1908|p=202}}}} Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry and [[triangle congruence]] (1–26); [[parallel line]]s (27–34); the [[area]] of [[triangle]]s and [[parallelogram]]s (35–45); and the [[Pythagorean theorem]] (46–48).{{sfn|Artmann|2012|p=4}} The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}}

Book 2 is traditionally understood as concerning "[[Greek geometric algebra|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.{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}} The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{sfn|Artmann|2012|p=4}} It focuses on the area of [[rectangle]]s and [[square]]s (see [[Quadrature (geometry)|Quadrature]]), and leads up to a geometric precursor of the [[law of cosines]].{{sfn|Katz|Michalowicz|2020|p=59}} Book 3 focuses on circles, while the 4th discusses [[regular polygons]], especially the [[pentagon]].{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{sfn|Artmann|2012|p=5}} Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion".{{sfn|Artmann|2012|pp=5–6}}{{efn|Much of Book 5 was probably ascertained from earlier mathematicians, perhaps Eudoxus.{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}}}} Book 6 utilizes the "theory of [[ratio]]s" in the context of plane geometry.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} It is built almost entirely of its first proposition:{{sfn|Artmann|2012|p=6}} "Triangles and parallelograms which are under the same height are to one another as their bases".{{sfn|Heath|1908b|p=191}}

[[File:Platonic Solids Transparent.svg|thumb|The five [[Platonic solids]], foundational components of [[solid geometry]] which feature in Books 11–13]]

From Book 7 onwards, the mathematician {{ill|Benno Artmann|de}} notes that "Euclid starts afresh. Nothing from the preceding books is used".{{sfn|Artmann|2012|p=7}} [[Number theory]] is covered by books 7 to 10, the former beginning with a set of 22 definitions for [[Parity (mathematics)|parity]], [[prime number]]s and other arithmetic-related concepts.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} Book 7 includes the [[Euclidean algorithm]], a method for finding the [[greatest common divisor]] of two numbers.{{sfn|Artmann|2012|p=7}} The 8th book discusses [[geometric progression]]s, while book 9 includes the proposition, now called [[Euclid's theorem]], that there are infinitely many [[prime number]]s.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} Of the ''Elements'', book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}}

The final three books (11–13) primarily discuss [[solid geometry]].{{sfn|Artmann|2012|p=3}} By introducing a list of 37 definitions, Book 11 contextualizes the next two.{{sfn|Artmann|2012|p=9}} Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates.{{sfn|Artmann|2012|p=9}} The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and [[parallelepiped]]al solids (24–37).{{sfn|Artmann|2012|p=9}}

=== Other works ===
<!--Linked from [[Optics]]-->
[[File:Euclid Dodecahedron 1.svg|thumb|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.{{sfn|Sialaros|2021|loc=§ "Other Works"}}
* The ''[[Euclid's Data|Data]]'' ({{langx|grc|Δεδομένα}}), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems.{{sfn|Sialaros|2021|loc=§ "Other Works"}}
* ''On Divisions'' ({{langx|grc|Περὶ Διαιρέσεων}}) survives only partially in [[Arabic language|Arabic]] translation, and concerns the division of geometrical figures into two or more equal parts or into parts in given [[ratio]]s. It includes thirty-six propositions and is similar to Apollonius' ''Conics''.{{sfn|Sialaros|2021|loc=§ "Other Works"}}
* The ''[[Euclid's Optics|Optics]]'' ({{langx|grc|Ὀπτικά}}) is the earliest surviving Greek treatise on perspective. It includes an introductory discussion of [[geometrical optics]] and basic rules of [[Perspective (graphical)|perspective]].{{sfn|Sialaros|2021|loc=§ "Other Works"}}
* The ''[[Euclid's Phaenomena|Phaenomena]]'' ({{langx|grc|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.{{sfn|Sialaros|2021|loc=§ "Other Works"}}

=== Lost works ===
Four other works are credibly attributed to Euclid, but have been lost.{{sfn|Sialaros|2021|loc=§ "Works"}}
* The ''Conics'' ({{langx|grc|Κωνικά}}) was a four-book survey on [[conic section]]s, which was later superseded by Apollonius' more comprehensive treatment of the same name.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Other writings"}}{{sfn|Sialaros|2021|loc=§ "Other Works"}} 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.{{sfn|Jones|1986|pp=399–400}} Doubt has been cast on this assertion by the historian {{ill|Alexander Jones (classicist)|de|Alexander Jones (Wissenschaftshistoriker)|lt=Alexander Jones}}, owing to sparse evidence and no other corroboration of Pappus' account.{{sfn|Jones|1986|pp=399–400}}
* The ''Pseudaria'' ({{langx|grc|Ψευδάρια}}; {{lit|Fallacies}}), was—according to Proclus in (70.1–18)—a text in geometrical [[reasoning]], written to advise beginners in avoiding common fallacies.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Other writings"}}{{sfn|Sialaros|2021|loc=§ "Other Works"}} Very little is known of its specific contents aside from its scope and a few extant lines.{{sfn|Acerbi|2008|p=511}}
* The ''Porisms'' ({{langx|grc|Πορίσματα}}; {{lit|Corollaries}}) was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 propositions.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Other writings"}}{{sfn|Sialaros|2021|loc=§ "Other Works"}} 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".{{sfn|Sialaros|2021|loc=§ "Other Works"}} The mathematician [[Michel Chasles]] speculated that these now-lost propositions included content related to the modern theories of [[Transversal (geometry)|transversal]]s and [[projective geometry]].{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Other writings"}}{{efn|See {{harvnb|Jones|1986|pp=547–572}} for further information on the ''Porisms''}}
* The ''Surface Loci'' ({{langx|grc|Τόποι πρὸς ἐπιφανείᾳ}}) is of virtually unknown contents, aside from speculation based on the work's title.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Other writings"}} Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.{{sfn|Sialaros|2021|loc=§ "Other Works"}}

==Legacy==
{{see also|List of things named after Euclid}}
[[File:Byrne1.png|thumb|upright|The cover page of [[Oliver Byrne (mathematician)|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.{{sfn|Ball|1960|p=[https://archive.org/details/shortaccountofhi0000ball/page/52/mode/2up 52]}} Many commentators cite him as one of the most influential figures in the [[history of mathematics]].{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/125 125]}} 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 geometry|non-Euclidean geometries]] discovered in the early 19th century.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Legacy"}} Among Euclid's [[List of things named after Euclid|many namesakes]] are the [[European Space Agency]]'s (ESA) [[Euclid (spacecraft)|Euclid]] spacecraft,<ref>{{cite news |date=9 May 2017 |title=NASA Delivers Detectors for ESA's Euclid Spacecraft |publisher=[[Jet Propulsion Laboratory]] |url=https://www.jpl.nasa.gov/news/news.php?feature=6840}}</ref> the lunar crater [[Euclides (crater)|Euclides]],<ref>{{cite web |title=Gazetteer of Planetary Nomenclature {{!}} Euclides |url=http://planetarynames.wr.usgs.gov/Feature/1860 |access-date=September 3, 2017 |website=usgs.gov |publisher=[[International Astronomical Union]]}}</ref> and the minor planet [[4354 Euclides]].<ref>{{cite web |title=4354 Euclides (2142 P-L) |url=https://www.minorplanetcenter.net/db_search/show_object?object_id=4354 |access-date=27 May 2018 |publisher=Minor Planet Center}}</ref>

The ''Elements'' is often considered after the [[Bible]] as the most frequently translated, published, and studied book in the [[Western World]]'s history.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Legacy"}} With Aristotle's ''[[Metaphysics (Aristotle)|Metaphysics]]'', the ''Elements'' is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Legacy"}}

The first English edition of the ''Elements'' was published in 1570 by [[Henry Billingsley]] and [[John Dee]].{{sfn|Goulding|2010|p=120}} The mathematician [[Oliver Byrne (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.{{sfn|Hawes|Kolpas|2015}} [[David Hilbert]] authored a [[Hilbert's axioms|modern axiomatization]] of the ''Elements''.{{sfn|Hähl|Peters|2022|loc=§ para. 1}} [[Edna St. Vincent Millay]] wrote that "Euclid alone has looked on Beauty bare."<ref>{{cite book| last=Millay| first=Edna St. Vincent| author-link=Edna St. Vincent Millay| title=Euclid alone has looked on Beauty bare| url=https://www.poetryfoundation.org/poems/148566/euclid-alone-has-looked-on-beauty-bare}}</ref>

== References ==
===Notes===
{{notelist|30em}}

===Citations===
{{reflist|20em}}

===Sources===
:'''Books'''
{{refbegin|30em}}
* {{cite book |last=Artmann |first=Benno |year=2012 |orig-year=1999 |title=Euclid: The Creation of Mathematics |publisher=[[Springer Publishing]] |location=New York |isbn=978-1-4612-1412-0 |url={{google books|plainurl=y|id=F8XgBwAAQBAJ}} }}
* {{cite book |last=Ball |first=W.W. Rouse |author-link=W. W. Rouse Ball |year=1960 |orig-year=1908 |title=A Short Account of the History of Mathematics |edition=4th |publisher=[[Dover Publications]] |location=Mineola |isbn=978-0-486-20630-1 |url=https://archive.org/details/shortaccountofhi0000ball }}
* {{Cite book |last=Bruno |first=Leonard C. |others=Baker, Lawrence W.|date=2003 |orig-year=1999 |title=Math and Mathematicians: The History of Math Discoveries Around the World |publisher=U X L |location=Detroit |isbn=978-0-7876-3813-9 |oclc=41497065 |url=https://archive.org/details/mathmathematicia00brun }}
* {{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |year=1991 |orig-year=1968 |title=A History of Mathematics |edition=2nd|publisher=John Wiley & Sons, Inc. |isbn=978-0-471-54397-8 |url=https://archive.org/details/historyofmathema00boye |url-access=limited }}
* {{cite book |last=Cuomo |first=Serafina |author-link=Serafina Cuomo |year=2005 |orig-year=2001 |title=Ancient Mathematics |publisher=[[Routledge]] |location=London and New York |isbn=978-1-134-71019-5 |url={{google books|plainurl=y|id=KXuFAgAAQBAJ}} }}
* {{cite book |last=Fowler |first=David |author-link=David Fowler (mathematician) |year=1999 |title=The Mathematics of Plato's Academy |edition=2nd |publisher=[[Clarendon Press]] |location=Oxford |isbn=978-0-19-850258-6 |url={{google books|plainurl=y|id=HuwwIdk-xL8C}} }}
* {{cite book |last=Goulding |first=Robert |year=2010 |title=Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History |publisher=Springer Netherlands |location=Dordrecht |isbn=978-90-481-3542-4 }}
* {{cite book|author-link=Thomas Heath (classicist)|editor-last=Heath|editor-first=Thomas |year=1908|title=The Thirteen Books of Euclid's Elements|volume=1 |publisher=[[Dover Publications]] |location=New York|isbn=978-0-486-60088-8|url=https://archive.org/details/bub_gb_UhgPAAAAIAAJ/mode/2up}}
* {{cite book |editor-last=Heath |editor-first=Thomas |editor-link=Thomas Heath (classicist) |year=1908b |title=The Thirteen Books of Euclid's Elements |volume=2 |publisher=[[Dover Publications]] |location=New York }}
* {{cite book |last=Heath |first=Thomas L. |author-link=Thomas Heath (classicist) |year=1981 |orig-year=1921 |title=[[A History of Greek Mathematics]] |volume=2 |publisher=[[Dover Publications]] |location=New York }} {{ISBN|0-486-24073-8|0-486-24074-6}}
* {{cite book |last=Jahnke |first=Hans Niels |editor-last1=Hanna |editor-first1=Gila |editor-link1=Gila Hanna |editor-last2=Jahnke |editor-first2=Hans Niels |editor-last3=Pulte |editor-first3=Helmut |year=2010 |title=Explanation and Proof in Mathematics: Philosophical and Educational Perspectives |chapter=The Conjoint Origin of Proof and Theoretical Physics |publisher=[[Springer US]] |location=Berlin |isbn=978-1-4419-0576-5 |url={{google books|plainurl=y|id=3bLHye8kSAwC}} }}
* {{cite book |editor-last=Jones |editor-first=Alexander |year=1986 |title=Pappus of Alexandria: Book 7 of the Collection |volume=Part 2: Commentary, Index, and Figures |publisher=[[Springer Science+Business Media]] |location=New York |isbn=978-3-540-96257-1 |url=https://archive.org/details/book7ofcollectio0000papp/mode/2up }}
* {{cite book |last1=Katz |first1=Victor J. |author-link=Victor J. Katz |last2=Michalowicz |first2=Karen Dee |year=2020 |orig-year=2005 |title=Historical Modules for the Teaching and Learning of Mathematics |publisher=[[Mathematical Association of America]] |location=Washington D.C. |isbn=978-1-4704-5711-2 }}
* {{cite book |last=Pickover |first=Clifford A. |author-link=Clifford A. Pickover |year=2009 |title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |publisher=[[Sterling Publishing]] |location=New York |isbn=978-1-4027-5796-9 }}
* {{cite book |last=Sialaros |first=Michalis |editor-last=Sialaros |editor-first=Michalis |year=2018 |title=Revolutions and Continuity in Greek Mathematics |chapter=How Much Does a Theorem Cost? |publisher=[[De Gruyter]] |location=Berlin |pages=89–106 |isbn=978-3-11-056595-9 |chapter-url={{google books|plainurl=y|id=DZRdDwAAQBAJ}} }}
* {{cite book |last=Sialaros |first=Michalis |editor-last1=Kalligas |editor-first1=Paul |editor-last2=Balla |editor-first3=Chloe |editor-last3=Baziotopoulou-Valavani |editor-first4=Effie |editor-last4=Karasmanis |editor-first2=Vassilis |year=2020 |title=Plato's Academy |chapter=Euclid of Alexandria: A Child of the Academy? |publisher=[[Cambridge University Press]] |location=Cambridge |pages=141–152 |isbn=978-1-108-42644-2 }}
* {{cite book |last=Smorynski |first=Craig |year=2008 |title=History of Mathematics: A Supplement |publisher=[[Springer Publishing]] |location=New York |isbn=978-0-387-75480-2 |url={{google books|plainurl=y|id=_zliInaOM8UC}} }}
* {{cite book |last=Tracy |first=Stephen V |editor1-last=Fortenbaugh |editor1-first=William W. |editor2-last=Schütrumpf |editor2-first=Eckhart |year=2000 |title=Demetrius of Phalerum: Text, Translation and Discussion |chapter=Demetrius of Phalerum: Who was He and Who was He Not? |series=Rutgers University Studies in Classical Humanities |volume=IX |publisher=[[Transaction Publishers]] |location=New Brunswick and London |isbn=978-1-3513-2690-2 |url={{google books|plainurl=y|id=IvdKDwAAQBAJ}} }}
* {{cite book |last=Venema |first=Gerard |year=2006 |title=The Foundations of Geometry |publisher=[[Pearson Prentice Hall]] |location=Hoboken |isbn=978-0-13-143700-5 }}
* {{cite book |last=Wolfe |first=Harold E. |year=1945 |title=Introduction To Non-Euclidean Geometry |publisher=[[Dryden Press]] |location=New York |oclc= |url=https://archive.org/details/introductiontono031680mbp }}
{{refend}}

:'''Articles'''
{{refbegin|30em}}
* {{cite journal |last=Acerbi |first=Fabio |date=September 2008 |title=Euclid's Pseudaria |journal=Archive for History of Exact Sciences |volume=62 |issue=5 |pages=511–551 |doi=10.1007/s00407-007-0017-3 |jstor=41134289 |s2cid=120860272 }}
* {{cite conference |last=Jones |first=Alexander |year=2005 |title=''"Euclid, the Elusive Geometer"'' |conference=Euclid and His Heritage Meeting, Clay Mathematics Institute, Oxford, 7–8 October 2005 |url=https://archive.nyu.edu/bitstream/2451/63988/2/Jones%202005%20Euclid%20the%20Elusive%20Geometer%20Oxford.pdf }}
* {{cite encyclopedia |last=Asper |first=Markus |editor-last=Gagarin |editor-first=Michael |year=2010 |encyclopedia=The Oxford Encyclopedia of Ancient Greece and Rome |title=Euclid |publisher=[[Oxford University Press]] |location=Oxford |url=https://www.oxfordreference.com/view/10.1093/acref/9780195170726.001.0001/acref-9780195170726-e-455 |isbn=978-0-19-517072-6 }}
* {{cite journal |last1=Hähl |first1=Hermann |last2=Peters |first2=Hanna |date=10 June 2022 |title=A Variation of Hilbert's Axioms for Euclidean Geometry |journal=Mathematische Semesterberichte |volume=69 |issue=2 |pages=253–258 |doi=10.1007/s00591-022-00320-3 |s2cid=249581871 |doi-access=free }}
* {{cite web |last1=Hawes |first1=Susan M. |last2=Kolpas |first2=Sid |date=August 2015 |title=Oliver Byrne: The Matisse of Mathematics – Biography 1810–1829 |publisher=[[Mathematical Association of America]] |url=https://www.maa.org/press/periodicals/convergence/oliver-byrne-the-matisse-of-mathematics-biography-1810-1829 |access-date=10 August 2022 }}
* {{Cite EB1911 |last=Heath |first=Thomas Little |author-link=Thomas Heath (classicist) |wstitle=Pappus of Alexandria |volume=20 |pages=470–471}}
* {{cite encyclopedia |last=Sialaros |first=Michalis |year=2021 |orig-year=2015 |encyclopedia=[[Oxford Classical Dictionary]] |title=Euclid |publisher=[[Oxford University Press]] |location=Oxford |url=https://oxfordre.com/classics/view/10.1093/acrefore/9780199381135.001.0001/acrefore-9780199381135-e-2521 |doi=10.1093/acrefore/9780199381135.013.2521 |isbn=978-0-19-938113-5 }}
* {{cite encyclopedia |last1=Taisbak |first1=Christian Marinus |last2=Van der Waerden |first2=Bartel Leendert |author-link2=Bartel Leendert van der Waerden |date=5 January 2021 |title=Euclid |encyclopedia=[[Encyclopædia Britannica]] |publisher=[[Encyclopædia Britannica, Inc.]] |location=Chicago |url=https://www.britannica.com/EBchecked/topic/194880/Euclid }}
{{refend}}

:'''Online'''
{{refbegin|30em}}
* {{cite web |title=Euclid |publisher=[[J. Paul Getty Museum]] |url=https://www.getty.edu/art/collection/object/108GV4 |access-date=11 August 2022 |ref={{sfnRef|Getty}} }}
* {{cite encyclopedia |encyclopedia=[[OED Online]] |title=Euclid, n |publisher=[[Oxford University Press]] |location=Oxford |url-access=subscription |url=https://www.oed.com/view/Entry/64929 |access-date=10 August 2022 |ref={{sfnRef|''OED''a}} }} {{subscription required}}
* {{cite encyclopedia |title=Euclidean (adj.) |encyclopedia=[[Online Etymology Dictionary]] |url=http://www.etymonline.com/index.php?term=Euclidean |access-date=18 March 2015 |ref={{sfnRef|''OED''b}} }}
{{refend}}

== External links ==
{{Spoken Wikipedia|En-Euclid-article.ogg|date=2020-09-29}}

; Works
* {{Gutenberg author | id=9719| name=Euclid}}
* {{Internet Archive author}}
* {{Librivox author |id=2304}}
* [https://www.ucl.ac.uk/library/special-collections/a-z/euclid Euclid Collection] at [[University College London]] (c.500 editions of works by Euclid), available online through the [https://ucl.primo.exlibrisgroup.com/discovery/collectionDiscovery?vid=44UCL_INST:UCL_VU2&collectionId=81354471580004761 Stavros Niarchos Foundation Digital Library].
* Scans of [[Johan Ludvig Heiberg (historian)|Johan Heiberg]]'s [https://wilbourhall.org/index.html#euclid edition of Euclid] at wilbourhall.org

; The ''Elements''
* [https://farside.ph.utexas.edu/books/Euclid/Euclid.html PDF copy], with the original Greek and an English translation on facing pages, [[University of Texas]].
* [https://euclides.org All thirteen books], in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.

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