Editing Euclid (section)

==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to differentiate the work of Euclid from that of his predecessors, especially because the ''Elements'' essentially superseded much earlier and now-lost Greek mathematics.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{efn|The ''Elements'' version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"}}