View source for Euclid's Elements

{{Short description|Mathematical treatise by Euclid}}
{{DISPLAYTITLE:Euclid's ''Elements''}}
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{{Infobox book
| name              = Elements
| author            = [[Euclid]]
| language          = [[Ancient Greek]]
| country           = 
| genre             = Mathematics
| publisher         = 
| isbn              = 
| italic title      = Elements
| image             = P.Penn. Museum inv. E02748 Euclid's Elements 2.5.jpg
| caption           = [[Papyrus Oxyrhynchus 29]], a fragment of Euclid's ''Elements'' dated to {{circa|3rd&ndash;4th century CE}}.<br>Found at [[Oxyrhynchus]], the diagram accompanies [[Euclid's Elements#Book II|Book II]], {{nowrap|Proposition 5.}}
| title_orig        = 
| translator        = 
| illustrator       = 
| cover_artist      = 
| series            = 
| subject           = [[Euclidean geometry]], [[number theory]], [[commensurability (mathematics)|incommensurability]]
| publisher2        = 
| pub_date          = {{circa}} 300 BC
| english_pub_date  = 
| media_type        = 
| pages             = 13 books
| awards            = 
| oclc              = 
| dewey             = 
| congress          = 
| preceded_by       = 
| followed_by       = 
}}
The '''''Elements''''' ({{langx|grc|Στοιχεῖα}} {{transliteration|grc|Stoikheîa}}) is a mathematical [[treatise]] written {{circa}} 300 BC by the Ancient Greek mathematician [[Euclid]].

The ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as [[Hippocrates of Chios]], [[Eudoxus of Cnidus]], and [[Theaetetus (mathematician)|Theaetetus]], the ''Elements'' is a collection in 13 books of definitions, [[postulates]], [[Compass-and-straightedge construction|geometric constructions]], and [[theorem]]s with their [[mathematical proof|proofs]] that covers plane and solid [[Euclidean geometry]], elementary [[number theory]], and [[Commensurability (mathematics)|incommensurability]]. These include the [[Pythagorean theorem]], [[Thales' theorem]], the [[Euclidean algorithm]] for [[greatest common divisor]]s, [[Euclid's theorem]] that there are infinitely many prime numbers, and the construction of [[regular polygons]] and [[Regular polyhedra|polyhedra]].

Often referred to as the most successful [[textbook]] ever written, the ''Elements'' has continued to be used for introductory geometry. It was translated into Arabic and Latin in the medieval period, where it exerted a great deal of influence on [[mathematics in the medieval Islamic world]] and in Western Europe, and has proven instrumental in the development of [[logic]] and modern [[science]], where its logical rigor was not [[Set theory|surpassed]] until the 19th century.

== Background ==
Euclid's ''Elements'' is the oldest extant large-scale deductive treatment of mathematics.{{sfn|Grant|2002}} [[Proclus]], a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the ''Elements'': "Euclid, who put together the ''Elements'', collecting many of [[Eudoxus of Cnidus|Eudoxus]]' theorems, perfecting many of [[Theaetetus (mathematician)|Theaetetus]]', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".{{efn|name=translation}} Scholars believe that the ''Elements'' is largely a compilation of propositions based on books by earlier Greek mathematicians,{{sfn|Van der Waerden|1975|p=197}} including [[Eudoxus of Cnidus|Eudoxus]], [[Hippocrates of Chios]],{{efn|Hippocrates of Chios should not be confused with his contemporary [[Hippocrates|Hippocrates of Kos]], famous as the "father of medicine".}} [[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''"}} 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, historian Michalis Sialaros opines that its "remarkably tight structure" suggests that Euclid wrote the ''Elements'' himself rather than merely editing together the works of others.{{sfn|Sialaros|2021|loc=§ "Works"}}

The detailed attribution of parts of the ''Elements'' to specific mathematicians is still the subject of scholarly debate. According to [[W. W. Rouse Ball]], [[Pythagoras]] was probably the source for most of books I and II, Hippocrates of Chios for book III, and [[Eudoxus of Cnidus]] for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.{{sfn|Rouse Ball|1915|p=15}} The ''Elements'' may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.{{sfn|Rouse Ball|1915|p=38}} [[Wilbur Knorr]] ascribes the origin of the material in Books I, III, and VI of the ''Elements'' to the time of Hippocrates of Chios, and of the material in books II, IV, X, and XIII to the later period of [[Theodorus of Cyrene]], Theaetetus, and Eudoxos. However, this suggested history has been criticized by [[Bartel Leendert van der Waerden|van der Waerden]], who believed that books I through IV were largely due to the much earlier [[Pythagoras|Pythagorean]] school.<ref name="vdw">Review of ''The Evolution of the Euclidean Elements'' by [[Bartel Leendert van der Waerden]] (1976), ''[[Historia Mathematica]]'' '''3''' (4): 497–499, {{doi|10.1016/0315-0860(76)90092-6}}.</ref>

Other similar works are also reported to have been written by Hippocrates of Chios, [[Theudius of Magnesia]], and [[Leon (mathematician)|Leon]], but are now lost.{{sfn|Unguru|1985}}{{sfn|Merzbach|Boyer|2011|p=93}}

== Contents ==
{| class="wikitable"
|+ Summary Contents of Euclid's ''Elements'' (Heath edition)
! Book
! I
! II
! III
! IV
! V
! VI
! VII
! VIII
! IX
! X
! XI
! XII
! XIII
! Totals
|-
! Definitions
| 23 || 2 || 11 || 7 || 18 || 4 || 22 || – || – || 16 || 28 || – || – || 131
|-
! Postulates
| 5 || – || – || – || – || – || – || – || – || – || – || – || – || 5 
|-
! Common Notions
| 5 || – || – || – || – || – || – || – || – || – || – || – || – || 5 
|-
! Propositions
| 48 || 14 || 37 || 16 || 25 || 33 || 39 || 27 || 36 || 115 || 39 || 18 || 18 || 465
|}
The ''Elements'' does not exclusively discuss geometry as is sometimes believed.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{sfn|Merzbach|Boyer|2011|pp=93–94}} It is traditionally divided into three topics: [[plane geometry]] (books I–VI), basic [[number theory]] (books VII–X) and [[solid geometry]] (books XI–XIII)—though book V (on proportions) and X (on [[Commensurability (mathematics)|incommensurability]]) do not exactly fit this scheme.{{sfn|Artmann|2012|p=3}}{{sfn|Asper|2010|loc=§ para. 4}} The heart of the text is the theorems 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}} The postulates (that is, [[axiom]]s) and common notions occur only in book I.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} Close study of [[Proclus]] suggests that older versions of the ''Elements'' may have followed the same distinctions but with different terminology, instead calling each definition a "hypothesis" ({{lang|grc|ύπόΘεςιζ}}) and the common notions "axioms" ({{lang|grc|άξιώμα}}).{{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 whether Euclid intended the ''Elements'' as a textbook,{{sfn|Sialaros|2021|loc=§ "Works"}} despite its wide subsequent use as one.{{sfn|Merzbach|Boyer|2011|p=90}} As a whole, the [[authorial voice]] remains general and impersonal.{{sfn|Asper|2010|loc=§ para. 6}}
{| class="wikitable plainrowheaders floatright" style="font-size:90%"
|-
|+ Euclid's postulates and {{vanchor|common notion}}s{{sfn|Heath|1908|loc=Vol. I|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.
|-
| 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{{efn|The converse of this postulate is: if the interior angles sum to two right angles, then the lines do not intersect. It is proved in the Elements, using a common notion or postulate not listed here: "two straight lines do not enclose any region", which is equivalent to "two straight lines have at most one intersection point". It is this property that allows distinguishing Euclidean geometry from [[spherical geometry]]}}
|-
! 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
|}

=== Books I to VI: Plane geometry ===

==== Book I ====
[[File:Parallel postulate.svg|thumb|Euclid's 5th postulate: Line {{mvar|g}} falls on the two lines {{mvar|h}} and {{mvar|k}}, making interior angles {{mvar|&alpha;}} and {{mvar|&beta;}} that sum to less than 180°, so lines {{mvar|h}} and {{mvar|k}} must meet at some point {{mvar|S}} on the same side of {{mvar|g}} as the angles.]]
[[File:Byrne 82 diagram 1.svg|thumb|upright=0.8|The [[Bride's Chair]] from the proof of the [[Pythagorean theorem]], in the colored version used by Byrne's 1847 edition. The proof shows that the black and yellow areas are equal, as are the red and blue areas.]]
Book I 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 [[Point (geometry)|point]]s, [[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 and five common notions.{{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}} The common notions exclusively concern the comparison of [[Magnitude (mathematics)|magnitude]]s, the sizes of geometric objects.{{sfn|Artmann|2012|p=4}} In modern mathematics these magnitudes would be treated as [[real number]]s measuring [[arc length]], [[angle]], or [[area]], and compared numerically, but Euclid instead found ways of comparing the magnitude of shapes using geometric operations, without interpreting these magnitudes as numbers.{{sfn|Corry|2021|p=12}} While the first four postulates are relatively straightforward, the fifth is not. It is known as the [[parallel postulate]], and the question of its independence from the other four postulates became the focus of a long line of research leading to the development of [[non-Euclidean geometry]].{{sfn|Artmann|2012|p=4}}

Book I also includes 48 propositions, which can be loosely divided into:  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]] and its converse (46–48).{{sfn|Artmann|2012|p=4}}

Proposition 5, that the base angles of an [[isosceles triangle]] are equal, became known in the [[Middle Ages]] as the [[pons asinorum]], or bridge of asses, separating the mathematicians who could prove it from the fools who could not.{{sfn|Reid|1963|p=20}} [[Papyrus Oxyrhynchus 29]], a 3rd-century CE papyrus, contains fragments of propositions 8–11 and 14–25.{{efn|{{Cite web |date=2022-09-26 |title=P.Oxy. LXXXII 5299. Euclid, Elements 1.4 (Diagram), 8–11, 14–25 (without Proofs) |url=https://portal.sds.ox.ac.uk/articles/online_resource/P_Oxy_LXXXII_5299_Euclid_Elements_1_4_Diagram_8_11_14_25_without_Proofs_/21186181/2 |access-date=2025-07-03 |publisher=University of Oxford |language=en}} For the discovery and content of this fragment see {{harvnb|Henry|2016}} and {{harvnb|Dorandi|2018}}.}} The last two propositions of Book I comprise the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}} The figure for the Pythagorean theorem has itself become well known under multiple names: the [[Bride's Chair]], the windmill, or the peacock's tail.{{sfn|Merzbach|Boyer|2011|p=97}}

==== Book II ====
[[File:EuclidBk2Prop11Heath+Arc.jpg|thumb|upright=0.6|Euclid's subdivision of line segment {{mvar|AB}} into the [[golden ratio]] from Book II Proposition 11, with an arc added to the traditional diagram. The construction finds the midpoint {{mvar|E}} of side {{mvar|AC}} of square {{mvar|ABCD}}, intersects line {{mvar|AC}} at {{mvar|F}} with a circle of radius {{mvar|EB}}, and constructs a second square {{mvar|AFGH}}, whose vertex {{mvar|H}} is the subdivision point.]]
The second book focuses on [[area]], measured through [[quadrature (mathematics)|quadrature]], meaning the construction of a [[square]] of equal area to a given figure. It includes a geometric precursor of the [[law of cosines]], and culminates in the quadrature of arbitrary [[rectangle]]s.{{sfn|Artmann|2012|p=4}} In the late 19th and 20th centuries, Book II was interpreted by some mathematical historians to establish a "[[Greek geometric algebra|geometric algebra]]", an expression of algebraic manipulation of linear and quadratic equations in terms of geometric concepts of length and area,<ref>{{cite journal |last=Høyrup |first=Jens |author-link=Jens Høyrup |year=2017 |title=What is 'geometric algebra', and what has it been in historiography? |journal=AIMS Mathematics |volume=2 |number=1 |pages=128–160 |doi=10.3934/Math.2017.1.128 |doi-access=free }}</ref>{{sfn|Corry|2013}} centered on the quadratic case of the [[binomial theorem]].{{sfn|Artmann|2012|p=4}} This interpretation has been heavily debated since the 1970s;{{sfn|Corry|2013}} critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}} Nevertheless, taken as statements about geometry, many of the propositions in this book are superfluous to modern mathematics, as they can be subsumed by the use of algebra.{{sfn|Merzbach|Boyer|2011|p=98}}

Proposition 11 of Book II subdivides a given line segment into extreme and mean proportions, now called the [[golden ratio]]. It is the first of several propositions involving this ratio: It is later used in Book IV to construct a [[Golden triangle (mathematics)|golden triangle]] and [[regular pentagon]] and in Book XIII to construct the [[regular dodecahedron]] and [[regular icosahedron]], and studied as a ratio in Book VI Proposition 30.{{sfn|Fowler|1980}}{{sfn|Herz-Fischler|1998|pp=1–3}}

==== Book III ====
Book III begins with a list of 11 definitions, and follows with 37 propositions that deal with [[circle]]s and their properties. Proposition 1 is on finding the center of a circle. Propositions 2 through 15 concern [[Chord (geometry)|chord]]s, and intersecting and [[tangent circles]]. [[Tangent lines to circles]] are the subjects of propositions 16 through 19. Next are propositions on [[inscribed angle]]s (20 through 22), and on chords, arcs, and angles (23 through 30), including the [[inscribed angle theorem]] relating inscribed to central angles as proposition 20. Propositions 31 through 34 concern angles in circles, including [[Thales's theorem]] that an angle inscribed in a [[semicircle]] is a [[right angle]] (part of proposition 31). The remaining propositions, 35 through 37, concern intersecting chords and tangents; proposition 35 is the [[intersecting chords theorem]], and proposition 36 is the [[tangent–secant theorem]].{{sfn|Artmann|2012|p=79}}

==== Book IV ====
Book IV treats four problems systematically for different polygons: inscribing a polygon within a circle, circumscribing a polygon about a circle, [[incircle|inscribing a circle]] within a polygon, and [[circumcircle|circumscribing a circle]] about a polygon.{{sfn|Artmann|2012|p=5}} These problems are solved in sequence for triangles and then for [[Constructible polygon|constructible regular polygon]]s (i.e., those that have a [[straightedge and compass construction]]) with 4, 5, 6, and 15 sides.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}

==== Book V ====
Book V, which is independent of the previous four books, concerns [[ratio]]s of [[magnitude (mathematics)|magnitudes]] (intuitively, how much bigger or smaller one shape is relative to another) and the comparison of ratios.{{sfn|Artmann|2012|pp=5–6}} Heath and other translators have formulated its first six propositions in symbolic algebra, as forms of the [[distributive law]] of multiplication over division and the [[associative law]] for multiplication. However, [[Leo Corry]] argues that this is anachronistic and misleading, because Euclid did not treat magnitudes as numbers, nor taking a ratio as a binary operation from numbers to numbers.{{sfn|Corry|2021}}

Much of Book V was probably ascertained from earlier mathematicians, perhaps Eudoxus,{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}} although certain propositions, such as V.16, dealing with "alternation" (if ''a'' : ''b'' :: ''c'' : ''d'', then ''a'' : ''c'' :: ''b'' : ''d'') likely predate Eudoxus.{{sfn|Artmann|2012|p=5-6}}

[[Christopher Zeeman]] has argued that Book V's focus on the behavior of ratios under the addition of magnitudes, and its consequent failure to define ratios of ratios, was a flaw that prevented the Greeks from finding certain important concepts such as the [[cross ratio]] (central to [[projective geometry]]).{{sfn|Zeeman|2008}}

==== Book VI ====
Book VI uses the theory of ratios from Book V in the context of plane geometry,{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}  especially the construction and recognition of [[Similarity (geometry)|similar]] figures. 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". That is, if two triangles have the same height, the ratio of their areas is the same as the ratio of lengths of their two base segments (and analogously for two parallelograms of the same height). This proposition provides a connection between ratios of lengths and ratios of areas.{{sfn|Artmann|2012|p=136}} Proposition 25 constructs, from any two [[polygon]]s, a third polygon similar to the first and with the same area as the second. [[Plutarch]] attributes this construction to Pythagoras, calling it "more subtle and more scientific" than the Pythagorean theorem. The famous ancient Greek problem of [[doubling the cube]], now known impossible with compass and straightedge, is a special case of the analogous 3d problem of constructing a figure with a specified shape and volume.{{sfn|Artmann|2012|p=148}} The book ends as it begins, by connecting two types of ratios: ratios of angles, and ratios of circular arc lengths, in proposition 33.{{sfn|Fletcher|1938}}

=== Books VII to X: Number theory ===
[[Number theory]], the theory of the arithmetic of [[natural number]]s, is covered by books VII to X. Book VII begins with a set of 22 definitions for [[Parity (mathematics)|parity]] (whether a number is even or odd), [[prime number]]s, and other arithmetic-related concepts.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}} The first of these definitions is for the unit (in modern terms, the number one), while the second states that "a number is a multitude composed of units";{{sfn|Heath|1908|loc=Vol. II, [https://archive.org/details/bwb_S0-AHZ-704_2/page/276 p. 277]}} this is generally interpreted to mean that, for Euclid, one is not a number, and the natural numbers begin at two.{{sfn|Caldwell|Reddick|Xiong|Keller|2012}}

==== Book VII ====
Book VII deals with elementary number theory, and includes 39 propositions, which can be loosely divided into: the [[Euclidean algorithm]], a method for determining whether numbers are [[relatively prime]] and for finding the [[greatest common divisor]] (1–4), fractions (5–10), the theory of proportions for numbers (11–19), prime and relatively prime numbers and the theory of greatest common divisors, (20–32), and [[least common multiple]]s (33–39).{{sfn|Artmann|2012|p=7}}

==== Book VIII ====
The topic of Book VIII is [[geometric progression]]s.{{sfn|Artmann|2012|p=7}} For Euclid, these were defined by the property of being in continued proportion (each two consecutive magnitudes have the same ratio) rather than, as in modern treatments, by [[exponentiation]] (the <math>i</math>th term of the progression has the form <math>ax^i</math> for constants <math>a</math> and <math>x</math>). This allowed Euclid to avoid multiplication of more than two values, but led to some awkward proofs of facts that exponential notation would make obvious.{{sfn|Unguru|2018|pp=[https://books.google.com/books?id=DZRdDwAAQBAJ&pg=PA26 26–27]}}

The first part of Book VIII (propositions 1 through 10) deals with the construction and existence of geometric progressions of integers in general, and the [[divisibility]] of members of a geometric progression by each other.
Propositions 11 to 27 deal with [[square number]]s and [[cube number]]s in geometric progressions, and the relation between these special progressions and the elements two or three steps apart in an arbitrary geometric progression.{{sfn|Artmann|2012|p=7}}

==== Book IX ====
After continuing the investigations of Book VIII on squares and cubes in geometric progressions,{{sfn|Artmann|2012|p=7}} Book IX applies the results of the preceding two books and gives the [[infinitude of prime numbers]] (Euclid's theorem, proposition 20), the formula for the sum of a [[finite geometric series]] (proposition 35) and a construction using this sum for even [[perfect number]]s (proposition 36). Here, a number is perfect if it equals the sum of its [[proper divisor]]s, as for instance 28 = 1 + 2 + 4 + 7 + 14.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Sources and contents of the ''Elements''"}}{{sfn|Merzbach|Boyer|2011|pp=103–104}} [[Alhazen]] conjectured c. 1000, and in the 18th century [[Leonhard Euler]] proved, that this construction generates ''all'' even perfect numbers. This result is the [[Euclid–Euler theorem]].{{sfn|O'Connor|Robertson|1999}}

==== Book X ====
Of the ''Elements'', book X is by far the largest and most complex, dealing with (in modern terms) [[irrational number]]s in the context of magnitudes.{{sfn|Sialaros|2021|loc=§ "The ''Elements''"}}{{sfn|Artmann|2012|p=223}} Proposition 9 (as restated in modern terms) proves the irrationality of the square roots of all non-square integers such as <math>\sqrt2</math>, the [[square root of 2]].{{sfn|Heath|1908|loc=Vol. III|pp=28–31}} A lemma to Proposition 29 gives [[Euclid's formula]] for producing all fundamental [[Pythagorean triples]].{{sfn|Knorr|1975|p=[https://books.google.com/books?id=_1H6BwAAQBAJ&pg=PA157 157]}} Additionally, this book classifies irrational lengths into thirteen disjoint categories, related to their construction by various combinations of other lengths that are integers and their square roots.{{sfn|Roskam|2009}} However, [[Wilbur Knorr]] warns that "The student who approaches Euclid's Book X in the hope that its length and obscurity conceal mathematical treasures is likely to be disappointed. ... the mathematical ideas are few."{{sfn|Knorr|1983|p=60}}

Rather than treating magnitudes as [[real number]]s and asking whether these are [[rational number]]s, Euclid handles this material in terms of the [[commensurability (mathematics)|commensurability]] of lengths or areas: whether two line segments or two rectangles can both be measured by an integer number of copies of a common subunit.{{sfn|Artmann|2012|p=223}} His classification of lengths as rational or irrational differs from the modern meaning: for Euclid, a line segment is rational when the square on its side has a rational area. That is, for Euclid, a length such as <math>\sqrt2</math> that is the square root of a rational area is itself rational.<ref>See Book X, Definition 4, {{harvnb|Heath|1908|loc=Vol. III|pp=10, 12–13}}.</ref>

This book is connected to a short passage in [[Plato]]'s dialogue ''[[Theaetetus (dialogue)|Theaetetus]]'' among [[Socrates]], [[Theodorus of Cyrene]], and [[Theaetetus (mathematician)|Theaetetus]], a younger mathematician. This passage discusses a proof by Theodorus that the non-square integers from 3 to 17 have irrational square roots (after the much earlier discovery of the irrationality of <math>\sqrt2</math>), the generalization of this result to all non-square integers by Theaetetus, and a partial classification of the irrational numbers (with fewer than 13 classes).{{sfn|Dyde|1899|pp=[https://books.google.com/books?id=wt29k-Jz8pIC&pg=PA86 86–87]}}{{sfn|Knorr|1983|p=42}}

=== Books XI to XIII: Solid geometry ===
[[File:Platonic Solids Transparent.svg|thumb|The five [[Platonic solids]], foundational components of [[solid geometry]] which feature in Books 11–13]]
The final three books primarily discuss [[solid geometry]].{{sfn|Artmann|2012|p=3}} By introducing a list of 37 definitions, Book XI contextualizes the next two.{{sfn|Artmann|2012|p=9}} Although its foundational character resembles Book I, unlike Book I it features no axiomatic system or postulates.{{sfn|Artmann|2012|p=9}}

==== Book XI ====
Book XI generalizes the results of book VI to solid figures: perpendicularity, parallelism, volumes, and similarity of [[parallelepiped]]s (polyhedra with three pairs of parallel faces). The three sections of Book XI include content on: solid geometry (1–19), solid angles (20–23), and parallelepipeds (24–37).{{sfn|Artmann|2012|p=9}}

==== Book XII ====
Book XII studies the volumes of [[cone (geometry)|cones]], [[Pyramid (geometry)|pyramids]], and [[cylinder (geometry)|cylinders]] in detail by using the [[method of exhaustion]], a precursor to [[integral|integration]],{{sfn|Artmann|2012|p=9}} and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder.{{sfn|Edwards|1979|p=23}} It concludes by showing that the volume of a [[sphere]] is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.{{sfn|Edwards|1979|pp=24–25}}

==== Book XIII ====
Book XIII constructs the five [[Platonic solid]]s (regular polyhedra) inscribed in a sphere, compares the ratios of their edges to the radius of the sphere,{{sfn|Artmann|2012|p=10}} and concludes the ''Elements'' by proving that these are the only regular polyhedra.{{sfn|Merzbach|Boyer|2011|p=106}}

=== Apocryphal books ===
Two additional books, that were not written by Euclid, Books XIV and XV, have been transmitted in the manuscripts of the ''Elements'':{{sfn|Merzbach|Boyer|2011|loc=Apocrypha|p=107}}
* Book XIV was likely written by [[Hypsicles]], following a treatise by [[Apollonius of Perga]]. It continues the study in Book XIII of the Platonic solids and their circumscribed spheres. It concludes that, for a dodecahedron and icosahedron inscribed in a common sphere, the ratio of their surface areas and the ratio of their volumes are equal, both being{{sfn|Merzbach|Boyer|2011|loc=Apocrypha|p=107}}<math display=block>\sqrt{\frac{10}{3(5-\sqrt{5})}} = \sqrt{\frac{5+\sqrt{5}}{6}}.</math>
* Book XV may have been written by a student of [[Isidore of Miletus]]. It also studies the Platonic solids; it inscribes some of them within each other, counts their edges and vertices (without however finding [[Euler's formula]] <math>V-E+F=2</math> relating these counts to each other), and computes the [[dihedral angle]]s between their faces.{{sfn|Merzbach|Boyer|2011|loc=Apocrypha|p=107}}
The practice of adding to the works of famous authors, exemplified by these books, was not unusual in ancient Greek mathematics.{{sfn|Merzbach|Boyer|2011|loc=Apocrypha|p=107}}

==Euclid's method and style of presentation==
{{Quote box
 |quote = 
• To draw a straight line from any point to any point.<br>
• To describe a circle with any center and distance.
 |author = Euclid
 |source = ''Elements'', Book I, Postulates 1 & 3.{{Sfn|Hartshorne|2000|p=18}}
}}
[[File:HexagonConstructionAni.gif|thumb|upright=1|An animation showing how Euclid constructed a hexagon (Book IV, Proposition 15). Every two-dimensional figure in the ''Elements'' can be constructed using only a compass and straightedge.{{Sfn|Hartshorne|2000|p=18}}]]
Euclid's [[Axiomatic system#Axiomatic method|axiomatic approach]] and [[Euclidean geometry#Methods of proof|constructive methods]] were widely influential.{{sfn|Mueller|1969}}{{sfn|Andrade-Molina|Ravn|2016}}

Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to [[Compass-and-straightedge construction|construct]] the object using a [[Compass (drawing tool)|compass]] (circle-drawing tool) and [[straightedge]] (unmarked ruler). His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier.{{Sfn|Hartshorne|2000|pp=18–20}}

The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation.{{sfn|Heath|1931|p= 216}}

No indication is given of the method of reasoning that led to the result, although a different book by Euclid, ''[[Data (Euclid)|Data]]'', does provide instruction about how to approach the types of problems encountered in the first four books of the ''Elements''.{{sfn|Rouse Ball|1915|p=54}} For proofs involving [[Proof by exhaustion|case analysis]], the ''Elements'' often includes details only of the most difficult case; some of these case analyses have been filled out by later editors such as Theon.{{sfn|Heath|1908|loc=Vol. I|p=59}}

Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,{{sfn|Rouse Ball|1915|p=55}} the number 1 was sometimes treated separately from other positive integers, and, as multiplication was treated geometrically, as the area of a rectangle with given side lengths, he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward [[Greek numerals|Alexandrian system of numerals]],{{sfn|Rouse Ball|1915|pp=54,58,127}} an [[alphabetic numeral system]] in which each Greek letter represented a single-digit multiple of a power of ten.{{sfn|Heath|1931|pp=15–18}}

==Reception==
Euclid's ''Elements'' has been referred to as the most successful [[textbook]] ever written.{{sfn|Merzbach|Boyer|2011|p=90}}{{sfn|Hartshorne|2000|p=1}} The ''Elements'' is often considered after the [[Bible]] as the most frequently translated, published, and studied book in 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 Islamic world and Western Europe.{{sfn|Taisbak|Van der Waerden|2021|loc=§ "Legacy"}}{{sfn|Hartshorne|2000|p=1}} It was one of the very earliest mathematical works to be printed after the [[movable type|invention of the printing press]] and has been estimated to be second only to the [[Bible]] in the number of editions published since the first printing in 1482,{{sfn|Merzbach|Boyer|2011|p=108}}{{sfn|Bunt|Jones|Bedient |1988|p=142}} the number reaching well over one thousand.{{sfn|Bunt|Jones|Bedient |1988|p=142}}

=== Classical antiquity ===
{{see also|Ancient Greek mathematics}}
The oldest extant evidence for Euclid's Elements are a set of six [[Ostracon|ostraca]] (clay fragments with writing scratched onto them) found among the [[Elephantine papyri and ostraca]], from the 3rd century BC, that deal with propositions XIII.10 and XIII.16, on the construction of a dodecahedron.{{sfn|Fowler|1999|pp=209-210}} A papyrus recovered from [[Herculaneum]]<ref>P. Herc. 1061</ref> contains an essay by the Epicurean philosopher [[Demetrius Lacon]] on Euclid's ''Elements''.{{sfn|Fowler|1999|pp=209-210}} The earliest extant papyrus containing the actual text of the ''Elements'' is [[Papyrus Oxyrhynchus 29]], a fragment containing the text of Book II, Proposition 5 and an accompanying diagram, dated to {{circa|75–125 AD}}.{{sfn|Fowler|1999|pp=210–211}}

[[File:Euclid Vat ms no 190 I prop 47.jpg|thumb|The [[Pythagorean theorem]] in MS. Vat.gr.190]]
Copies of the Greek text still exist, some of which can be found in the [[Vatican Library]] and the [[Bodleian Library]] in Oxford.{{efn|name=vat190|[https://digi.vatlib.it/mss/detail/Vat.gr.190.pt.1 MS. Vat.gr.190.pt.1] and [https://digi.vatlib.it/mss/detail/Vat.gr.190.pt.2 MS. Vat.gr.190.pt.2]. ''Digital Vatican Library''. Retrieved 2025-09-11.}}{{efn|{{cite web|title=MS. D'Orville 301|work=Bodleian Library|url=https://digital.bodleian.ox.ac.uk/objects/d4a23501-0b98-4aff-acd6-fe06fe9b62e3/|publisher=University of Oxford|access-date=2025-09-11}}}} The manuscripts available are of variable quality, and often incomplete.{{sfn|Heath|1908|loc=Vol. I|pp=47–51}} By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text.{{sfn|Heath|1908|loc=Vol. I|pp=51–53}} Also of importance are the [[scholia]], or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study.{{sfn|Heath|1908|loc=Vol. I|pp=64–74}}

In the 4th century AD, [[Theon of Alexandria]] produced an edition of Euclid which was so widely used that it became the only surviving Greek-language source (in multiple manuscripts) until [[François Peyrard]]'s 1808 discovery at the [[Vatican Library|Vatican]] of a manuscript not derived from Theon's.{{sfn|Knorr|1996}} This manuscript, MS. Vat.gr.190,{{efn|name=vat190}} was transcribed in the 10th century. It does not include text identifying itself as edited by Theon, and is missing a corollary to Book VI Proposition 33 claimed by Theon to be his own addition. Both Greek versions include many explanations beyond the propositions and their proofs that are missing from the Arabic translations of the ''Elements''. This sparked a 19th-century academic debate between M. Klamroth and [[Johan Ludvig Heiberg (historian)|J. L. Heiberg]] over whether the differences between the various versions reflected abridgements or additions to Euclid's text. Revisiting this issue, [[Wilbur Knorr]] sides with Klamroth in suggesting that the Arabic sources were closer to the original, but concludes that "We have never had a 'genuine' text of Euclid, and we never will have one."{{sfn|Knorr|1996}}

Although Euclid was known to [[Cicero]], for instance, no record exists of the text having been translated into Latin prior to [[Boethius]] in the fifth or sixth century.{{sfn|Russell|2013|p=177}}

=== Medieval era ===
{{see also|Manuscript culture}}
[[Image:Woman teaching geometry.jpg|thumb|A woman teaches geometry, from a manuscript ({{circa}} 1309–1316) of [[Adelard of Bath]]'s 12th century translation of the ''Elements'' from Arabic into Latin.{{sfn|Russell|2013|p=177}}]]
From classical antiquity until the western invention of printing, texts such as the ''Elements'' were preserved and duplicated through the process of copying [[manuscript]]s. This was laborious and expensive so manuscripts were often confined to the collections of the wealthy or to institutions such as the [[House of Wisdom]] in the [[Mathematics in the medieval Islamic world|medieval Islamic world]] or the [[Monastery|monasteries]] and early universities of medieval Europe.{{sfn|Schubring|2022}}

The Islamic world received the ''Elements'' from the [[Byzantine Empire]] around 760. According to sources from that milieu, this version was translated into [[Arabic]] under [[Harun al-Rashid]] ({{circa}} 800),{{sfn|Russell|2013|p=177}} in two versions by [[Al-Ḥajjāj ibn Yūsuf ibn Maṭar]]. Another Arabic translation was made later in the 9th century by [[Ishaq ibn Hunayn]] and revised by [[Thābit ibn Qurra]].{{sfn|Elior|2024}}{{sfn|Brentjes|2018}} Although most Arabic manuscripts have been attributed to one or another of these translations, some mix material from both,{{sfn|Elior|2024}} and their attributions are not always in accord with the evidence from textual similarities in surviving manuscripts.{{sfn|Brentjes|2018}} This mixture was also passed down into medieval translations into [[Hebrew language|Hebrew]] from the Arabic.{{sfn|Elior|2024}}

The Byzantine scholar [[Arethas of Caesarea|Arethas]] commissioned the copying of one of the Greek manuscripts of Euclid in the late ninth century;{{sfn|Reynolds|Wilson|1991|p=57}} it and another Byzantine manuscript are the two oldest surviving copies of the Greek text.{{sfn|Wardhaugh|2023|p=46}} Although known in Byzantium, the ''Elements'' was lost to Western Europe until about 1120,{{sfn|Murdoch|1967}} except through fragments of a translation into Latin by [[Boethius]] (circa 500), quoted in other works.{{sfn|Folkerts|1989}}{{efn|It was once thought that a complete Latin translation by Boethius remained available in England before Adelard, and was annotated by [[Alfred the Great]]. However, there is no evidence of any full translation by Boethius surviving (if it ever existed) and a manuscript found in the [[Biblioteca Marciana]] in Venice, holding a marginal note that it was annotated by Alfred, is one of many post-Adelard versions. See {{harvnb|Busard|2005|p=3}} and {{harvnb|Clagett|1954}}.}} In about 1120, the English monk [[Adelard of Bath]] translated the ''Elements'' into Latin from an Arabic translation.{{sfn|Rouse Ball|1915|p=165}} A relatively recent discovery was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the [[Almagest]] to Latin. The Euclid manuscript is extant and quite complete.{{sfn|Murdoch|1967}}

After Adelard's translation (which became known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include [[Herman of Carinthia]] who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), [[John of Tynemouth (geometer)|John of Tynemouth]]{{sfn|Knorr|1990}} (late 12th century; his manuscripts are referred to collectively as Adelard III), and [[Gerard of Cremona]] (sometime after 1120 but before 1187). The detailed transmission history of these translations is still an active area of research.{{sfn|Busard|2005|loc=Survey of the Arabic–Latin and Greek–Latin translations|pp=1-40}} [[Campanus of Novara]] relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today.{{sfn|Folkerts|1989}}{{efn|{{cite web |author=Campanus |title=Manuscript – Pal.lat.1348 |url=https://digi.vatlib.it/mss/detail/Pal.lat.1348 |work=Digital Vatican Library |access-date=20 November 2023}}}} After its availability in Europe, the first books of the ''Elements'' became standard in medieval universities as part of the [[quadrivium]], the second stage of instruction after the [[trivium]] of grammar, logic, and rhetoric.{{sfn|Schubring|2022}}

=== Renaissance and early modern period ===
{{see also|Printing revolution}}
[[Image:Ricci Guangqi 2.jpg|thumb|The Italian [[Jesuit]] [[Matteo Ricci]] (left) and the Chinese mathematician [[Xu Guangqi]] (right) published the first [[Chinese language|Chinese]] edition of ''Euclid's Elements'' ({{Transliteration|zh|Jīhé yuánběn}} {{lang|zh|幾何原本}}) in 1607.]]
The first printed edition of the ''Elements'' was published by Erhard Ratdolt in 1482, based on Campanus's version,{{sfn|Furlong|2015}} and since then it has been translated into many languages and published in over a thousand different editions.{{sfn|Bunt|Jones|Bedient |1988|p=142}} A manuscript descended from Theon's Greek version was recovered and a Latin translation was published in Venice in 1505 by {{ill|Bartolomeo Zamberti|de}}.{{sfn|Mueller|1970|p=[https://books.google.com/books?id=JZEHj2fEmqAC&pg=PR47 xlvii]}} The Greek text itself was [[list of editiones principes in Greek|published in 1533]].{{sfn|Swetz|2021}} The first to translate the ''Elements'' into a modern European language was [[Nicolo Tartaglia]], who published an Italian edition in 1543.{{sfn|Elior|2021}}

In 1570, [[John Dee (mathematician)|John Dee]] provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by [[Henry Billingsley]].{{sfn|Alexanderson|Greenwalt|2012}}{{sfn|Goulding|2009|p=120}}{{sfn|Simpkins|1966}} In 1607, The Italian Jesuit [[Matteo Ricci]] and the Chinese mathematician [[Xu Guangqi]] published the first Chinese edition of Euclid's Elements.{{sfn|Siu|2011}}

The Renaissance also saw the creation of new works about [[polyhedron|polyhedra]], illustrated with [[perspective drawing]]s, including [[Piero della Francesca]]'s ''[[De quinque corporibus regularibus]]'' (late 1400s), its [[plagiarism]] by [[Luca Pacioli]] as ''[[Divina proportione]]'' (1498, illustrated by [[Leonardo da Vinci]]), and [[Wenzel Jamnitzer]]'s ''[[Perspectiva corporum regularium]]'' (1568).{{sfn|Wardhaugh|2021|pp=212–226}} The ''Elements'' were the main inspiration behind della Francesca's initial work in this direction.{{sfn|Field|1997|p=246}}{{sfn|Davis|1977|pp=18–19}} Pacioli lectured in Venice on Euclid, and his commentary was included in a 1509 edition of the ''Elements''.{{sfn|Damiani|2024}} Jamnitzer, likewise, credits the ''Elements'' in the subtitles of his book.{{sfn|Rothstein|2013|p=101}}

Although this period also saw an explosion in newly published textbooks, teachers often stuck to the classics: a list of recommended readings by 16th century Dutch humanist [[Joachim Sterck van Ringelbergh]], for instance, lists the ''Elements'' as its only mathematics book.{{sfn|Grafton|2008}} Even after printed versions existed, a university might expect its students to copy by hand material from the university's copy of the ''Elements''.{{sfn|Wardhaugh|2021}}

=== Modern mathematics ===
In the 19th century the ''Elements'' fell out of favor as a geometry textbook, in part supplanted by newer textbooks such as one by [[Adrien-Marie Legendre]],{{sfn|Moktefi|2005}}{{sfn|Toumasis|1990}}{{sfn|Sinclair|2008|loc=The geometry curriculum in the United States in the early nineteenth century|pages=[https://books.google.com/books?id=G_onDwAAQBAJ&pg=PA14 14–18]}} in part because of the rise of other forms of geometry including [[non-Euclidean geometry]], [[analytic geometry]], and [[descriptive geometry]],{{sfn|Brock|1975}}{{sfn|Sinclair|2008|loc=1860s: A mathematical challenge to the dominance of Euclidean geometry|pages=[https://books.google.com/books?id=G_onDwAAQBAJ&pg=PA23 23–25]}} and in part out of pressure for an approach to mathematics education with more emphasis on intuition and less on memorization.{{sfn|Brock|1975}}{{sfn|Sinclair|2008|loc=1850s: Early pedagogical influences on the geometry curriculum|pages=[https://books.google.com/books?id=G_onDwAAQBAJ&pg=PA19 19–22]}} [[Lewis Carroll|Charles Dodgson]] (better known as Lewis Carroll), in particular, railed against this replacement of Euclid in his book ''Euclid and His Modern Rivals'' (1879).{{sfn|Moktefi|2005}} Another defender of the ''Elements'', mathematician and historian [[W. W. Rouse Ball]], remarked that "the fact that for two thousand years [the ''Elements''] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."{{sfn|Rouse Ball|1915|p=55}} Despite falling out of wide use in education, the ''Elements'' is still occasionally used as a textbook in experimental education projects.<ref>{{harvnb|Jameson|2019}}; {{harvnb|Hoppen|2023}}</ref>

The ''Elements'' remains an object of scholarly study for the [[history of mathematics]], and it has had significant influence on two areas of modern mathematics, the development of [[non-Euclidean geometry]]{{sfn|Bardi|2008}}{{sfn|Wardhaugh|2023|pp=267–268}} and of the [[axiomatic method]].{{sfn|Majer|2006}}{{sfn|Beeson|2022}}

==== Non-Euclidean geometry ====
[[File:Noneuclid.svg|thumb|upright=1.2|The different versions of the parallel postulate result in different geometries.]]
{{main|Non-Euclidean geometry}}
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"}}

One of the most notable influences of Euclid on modern mathematics and, beyond mathematics, modern physics and the discovery of [[general relativity]], is the discussion of the [[parallel postulate]].{{sfn|Bardi|2008}}{{sfn|Wardhaugh|2023|pp=267–268}} In Book I, Euclid lists five postulates, the fifth of which stipulates{{sfn|Heath|1908|loc=Vol. I|p=155}}

{{blockquote|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.}}

This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician [[Nikolai Lobachevsky]] published a description of acute geometry (or [[hyperbolic geometry]]), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate ([[elliptic geometry]]). If one takes the fifth postulate as a given, the result is [[Euclidean geometry]].{{sfn|Laubenbacher|Pengelley|1999}}

==== Axiomatics ====
{{see also|Axiomatic method}}
The axiomatic reasoning of Euclid's ''Elements'' was long considered to set the standard for mathematical rigor,{{sfn|Artmann|2012|p=v}} but the issues of the soundness and completeness of Euclid's axioms came to the foreground in the late 19th century, when gaps were found in his reasoning{{sfn|Jahnke|2003|p=[https://books.google.com/books?id=c8AjDwAAQBAJ&pg=PA156 156]}} and when [[David Hilbert]] began seeking "to revive Euclid's axiomatic point of view", to develop improved axiom systems through which all mathematical and physical questions could be answered by simple calculations.{{sfn|Majer|2006}} Hilbert's hopes were dashed in the [[Foundations of mathematics|foundational crisis]] of the early 20th century, in which [[Kurt Gödel]] and others discovered that any sound axiom system for [[set theory]] must necessarily be incomplete.{{sfn|Robič|2015}} In the 21st century, a new standard for rigor arose, [[computer-assisted proof]]s, and the propositions of the ''Elements'' (with some updates to their proofs) have withstood computer checking.{{sfn|Beeson|2022}}{{sfn|Beeson|Narboux|Wiedijk|2019}}

[[File:Dyad.svg|thumb|upright=0.8|Two circles sharing a radius cross each other, forming a [[vesica piscis]]. This is the first step of Book I Proposition 1, the construction of an [[equilateral triangle]] from the radius and one of the crossing points.]]
Some of the foundational proofs of the ''Elements'' use assumptions that Euclid did not state explicitly as axioms. For example, in the first construction of Book 1, of an [[equilateral triangle]], Euclid used a premise that was neither postulated nor proved: that two circles sharing the same line segment as a radius will cross each other in two points, rather than somehow not crossing.{{sfn|Heath|1908|loc=Vol. I|p= 242}}{{sfn|Toussaint|1993}} This example depends only on topological properties of its diagram, which remain evident even if the diagram is drawn inaccurately.{{sfn|Panza|2012}} However, in other cases, Euclid did not prove that certain objects were distinct or separated from each other, and the possibility that they might coincide (a type of [[degeneracy (mathematics)|degeneracy]]) might not be evident from a single diagram. An example occurs in Euclid's [[bisection]] of an angle, by constructing an isosceles triangle on the given angle and an equilateral triangle with the same base, and connecting by a line the apexes of the two triangles. This breaks down when the initial angle is 60° and the two apexes coincide.{{sfn|Beeson|Narboux|Wiedijk|2019}}

Later editors of the ''Elements'' have included these implicit axiomatic assumptions, such as [[Pasch's axiom]],{{sfn|Szmielew|1974}}{{sfn|Cederberg|2001|pp=41–43}} in their editions' lists of formal axioms.{{sfn|Heath|1908|loc=Vol. I|p= 62}} Early attempts to construct a more complete set of axioms include [[Hilbert's axioms|Hilbert's geometry axioms]]{{sfn|Hilbert|1902}}{{sfn|Baldwin|2018}} and [[Tarski's axioms|Tarski's]].{{sfn|Szmielew|1974}}{{sfn|Tarski|1959}} In 2017, Michael Beeson et al. used computer [[proof assistant]]s to create and check a set of axioms similar to Euclid's. Beeson et al. chose Tarski's system as their starting point, instead of Hilbert's, because it is closer to Euclid's, and uses only points as the variables in its formulas. They provided computer-verified proofs of all propositions in Book I, using these axioms, and they also proved (using a separate logical formalization of the [[real number]]s) that all of their axioms are valid for the points of the [[Cartesian coordinate system]].{{sfn|Beeson|Narboux|Wiedijk|2019}}

==Selected editions==
Over one thousand editions of Euclid's ''Elements'' have been published,{{sfn|Bunt|Jones|Bedient |1988|p=142}} in Greek, Latin, English, and other languages. Some of the more significant of these include:
* {{cite book|year=1482|location=Venice|publisher=[[Erhard Ratdolt]]|title=Preclarissimus liber elementorum Euclidis perspicacissimi in artem geometriam incipit quam foelicissime|url=https://archive.org/details/preclarissimusl00eucl}} The ''[[editio princeps]]'' (in Latin), based on the 13th century translation and commentary of Campanus.{{sfn|Baldasso|2013}}
*{{cite book|editor-first=Bartolomeo|editor-last=Zamberti|location=Venice|year=1505|title=Euclidis megarẽsis philosophi platonici|url=https://www.digitale-sammlungen.de/en/details/bsb11199807}} First full-text Latin translation directly from the Greek.{{sfn|Mueller|1970|p=[https://books.google.com/books?id=JZEHj2fEmqAC&pg=PR47 xlvii]}} The confusion in the title between Euclid of Alexandria, the author of the ''Elements'', and [[Euclid of Megara]], a philosopher from almost a century earlier, was a common mistake in the Middle Ages and Renaissance.{{sfn|Goulding|2009}}
*{{cite book|title=Euclidis Megarensis Geometricorum elementorum liber XV|year=1516|editor-last=Lefèvre d'Étaples|editor-first=Jacques|editor-link=Jacques Lefèvre d'Étaples|location=Paris|publisher=[[Henri Estienne (elder)|Henri Estienne]]|url=https://archive.org/details/hin-wel-all-00000536-001}} Based on Campanus and Zamberti, but without some of Zamberti's commentary. The first edition published in France.{{sfn|Wardhaugh|2024}}
*{{cite book|title=Ευκλείδου Στοιχεῖον|editor-link=Simon Grynaeus|editor-first=Simon|editor-last=Grynaeus|year=1533|location=Basel|publisher=Johann Herwagen}} ''Editio princeps'' of the Greek text. Based on two Greek manuscripts, Paris gr. 2343 and Venetus Marcianus 301,{{sfn|Heath|1908|loc=Vol. I|p=101}} both "very inferior ones".{{sfn|Mueller|1970|p=[https://books.google.com/books?id=JZEHj2fEmqAC&pg=PR47 xlvii]}}
*{{cite book|title=Euclide megarense philosopho solo introduttore delle scientie mathematice|url=https://archive.org/details/bub_gb_ymWnEXab_7UC|year=1543|editor-first=Nicolo|editor-last=Tartalea|editor-link=Nicolo Tartaglia|location=Venice|publisher=Rossinelli}} Tartaglia's translation of the ''Elements'' into Italian was the first into any modern European language, and was based on both Campanus and Zamberti. It was revised in 1565 and reprinted in 1569 and 1586.{{sfn|Elior|2021}}
*{{cite book|year=1557|title=Euclidis Elementorum libri XV grœce et latine|editor1-first=Jean|editor1-last=Magnien|location=Paris|publisher=Cavellat}} Greek with Latin translation. Revised posthumously by Stephanus Gracilis; incorporates a translation of Book X by Pierre de Montdoré. Most proofs omitted.{{sfn|Pantin|2006}}
* {{cite book|title=The Elements of Geometrie|editor-first=H.|editor-last=Billingsley|editor-link=Henry Billingsley|year=1570|url=https://archive.org/details/bim_early-english-books-1641-1700_the-elements-of-geometri_euclid_1570|publisher=[[John Day (printer)|John Daye]]|location=London|others=Preface by [[John Dee]]}} The first published English-language edition.{{sfn|Simpkins|1966}}
*{{cite book|year=1572|title=Euclidis Elementorum Libri XV|editor-first=Federico|editor-last=Commandino|editor-link=Federico Commandino|location=Pisauri [Pesaro, Italy]|publisher=Apud Camillum Francischinum|url=https://library.si.edu/digital-library/book/euclidiselement00eucl}} In Latin. "The reference edition for the scholarly community up until the early nineteenth century"; Italian translation published in 1575 by Commandino's son-in-law, Valerio Spaccioli.{{sfn|Gavagna|2014}}
*{{cite book|year=1574|title=Euclidis elementorum libri XV|url=https://archive.org/details/euclidiselemento01eucl|editor-first=Christopher|editor-last=Clavius|editor-link=Christopher Clavius|location=Rome|publisher=Apud Vincentium Accoltum}} Latin, with added commentary by Clavius. Published in an expanded 2nd edition in 1584. Based on multiple sources, but most closely related to the version of Magnien and Gracilis.{{sfn|Wardhaugh|2024}}
* {{cite book|author-link=Nasir al-Din al-Tusi|author=Nasir al-Din al-Tusi|url=http://pds.lib.harvard.edu/pds/view/13079270 |trans-title=The Recension of Euclid's "Elements"|title=Kitāb taḥrīr uṣūl li-Uqlīdus|language=ar|date=1594|location=Rome|publisher=Typographia Medicea}} Based on [[Laurentian Library]] MS. Or. 20, a copy of MS. Or. 50.{{sfn|De Young|2012}}
*{{cite book|year=1607|script-title=zh:幾何原本|title=Jī hé yuán běn|trans-title=Source of quantity|language=zh|editor1-first=Matteo|editor1-last=Ricci|editor1-link=Matteo Ricci|editor2-last=Xu|editor2-first=Guangqi|editor2-link=Xu Guangqi|location=Beijing}} Translated from the Latin edition of Clavius, but including only books I-VI.{{sfn|Siu|2011}} This translation, together with a later translation of the remaining books, can be found on [[:s:zh:幾何原本]].
* {{cite book|year=1620|editor-link=Henry Briggs (mathematician)|editor-first=Henry|editor-last=Briggs|title=Eukleidou Stoicheiōn biblia 13 / Elementorum Euclidis libri tredecim|location=London|publisher=William Jones}} Despite the title this includes only the first six books, with parallel columns of Greek from Grynaeus 1533 and Latin corrected from Commandino 1572. The first edition in either language published in England.{{sfn|Simpkins|1966}}
* [[Isaac Barrow|Barrow, Isaac]], ed. (1659). ''[https://archive.org/details/b30337094/ Euclidis Elementorum]'' [in Latin]. ''[https://books.google.com/books?id=oTcDAAAAQAAJ&pg=PP5 Euclide's Elements]'' (1660) [in English]. Many later editions.{{sfn|Simpkins|1966}}
* {{cite book|year=1806|editor-last=Simson|editor-first=Robert|editor-link=Robert Simson|title=The elements of Euclid, viz. the first six books, together with the eleventh and twelfth|location=London|publisher=F. Wingrave|url=https://archive.org/details/elementsofeuclid00eucluoft}}{{sfn|Ackerberg-Hastings|2023}} Revised in 1862 by [[Isaac Todhunter]] and republished in 1933 by E. P. Dutton, Everyman's Library 891.{{sfn|Cairns|1934}}
* {{cite book |last=Byrne |first=Oliver |author-link=Oliver Byrne (mathematician)  |year=1847 |title=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 |publisher=William Pickering |place=London |url=https://archive.org/details/firstsixbooksofe00byrn }}{{sfn|Hawes|Kolpas|2015}}{{sfn|Moriarty|2023}} Facsimile edited by Oechslin, Werner (2010). Taschen. {{isbn| 3836517752}}.{{sfn|Moriarty|2023}} [https://www.kroneckerwallis.com/product/euclids-elements-completing-oliver-byrnes-work/ ''Euclid's Elements: Completing Oliver Byrne's work''], a modern redrawing extended to the rest of the ''Elements'', was published by Kronecker Wallis, 2019.{{sfn|Campbell|2017}}
*{{cite book|year=1882|title=The First Six Books of the Elements of Euclid with Copious Annotations and Numerous Exercises|editor-first=John|editor-last=Casey|editor-link=John Casey (mathematician)|url=https://books.google.com/books?id=ycNbaJ1VcQIC|location=Dublin|publisher=Hodges, Figgis, & Co.}}{{sfn|Henrici|1884}} Casey produced many subsequent editions; [https://gutenberg.org/ebooks/21076 the third edition] was republished in a free online edition by [[Project Gutenberg]]. Casey's version of the ''Elements'' was likely "Casey's frost book of page torn on dirty" of the geometry section of [[James Joyce]]'s ''[[Finnegans Wake]]''.{{sfn|McCreedy|2011}} Casey also wrote ''[https://archive.org/details/asequeltofirsts02casegoog A Sequel to the First Six Books of the Elements of Euclid]'' (Dublin: Hodges, Figgis, & Co., 1881).
*[[Johan Ludvig Heiberg (historian)|Heiberg, Johan Ludvig]], ed. (1883–1888) ''Euclidis Opera omnia'' [Euclid's complete works, in Greek]. Leibzig: Teubner. Volumes 1–5 comprise the ''Elements''. [https://archive.org/details/euclidisoperaomn01eucl/ Vol.{{nbsp}}I], [https://archive.org/details/euclidisoperaomn02eucl/ Vol.{{nbsp}}II], [https://archive.org/details/euclidisoperaomn03eucl/ Vol.{{nbsp}}III], [https://archive.org/details/euclidisoperaomn04eucl/ Vol.{{nbsp}}IV], [https://archive.org/details/euclidisoperaomn05eucl/ Vol.{{nbsp}}V]. Heiberg consulted multiple Greek manuscripts for his work, taking the position that the single version not edited by Theon, MS. Vat.gr.190, was the most authentic, but following the others at points where he suspected his primary text to be faulty.{{sfn|Knorr|1996}}
* {{wikicite | ref = {{harvid|Heath|1908}} |reference = {{cite book | ref = none
 | editor-last = Heath | editor-first = Thomas | editor-link = Thomas Heath (classicist)
 | year = 1908
 | title = The Thirteen Books of Euclid's Elements
 | publisher = Cambridge University Press }} 2nd ed., 1926. In three volumes: [https://archive.org/details/bwb_S0-AHZ-704_1/ Vol. I], [https://archive.org/details/bwb_S0-AHZ-704_2/page/n7/mode/2up Vol. II], [https://archive.org/details/bwb_S0-AHZ-704_3/ Vol. III]. Reprints include Dover, 1956; Green Lion Press, 2002, {{isbn|1-888009-18-7}} (single volume, without Heath's commentary);{{sfn|Wardhaugh|2016}} Barnes & Noble, 2006, {{isbn|0-7607-6312-7}} (single volume).}}

==References==
===Notes===
{{notelist|refs=

{{efn|name=translation|This translation is from {{harvnb|Heath|1908}}, p. 1; for another translation, see {{cite book
 | author = Proclus | author-link = Proclus
 | page = [https://archive.org/details/morrow-commentary-on-the-first-book-of-euclids-elements-en-1992/page/67 56]
 | publisher = Princeton University Press
 | title = A commentary on the first book of Euclid's ''Elements''
 | translator-first = Glenn Raymond | translator-last = Morrow
 | url = https://archive.org/details/morrow-commentary-on-the-first-book-of-euclids-elements-en-1992
 | year = 1970}} }}
 
}}

===Citations===
{{Reflist|15em}}

===Sources===
{{Refbegin|indent=yes}}
*{{cite book
 | last = Ackerberg-Hastings | first = Amy
 | editor1-last = Zack | editor1-first = Maria
 | editor2-last = Waszek | editor2-first = David
 | contribution = Analysis and synthesis in Robert Simson's ''The Elements of Euclid''
 | doi = 10.1007/978-3-031-21494-3_8
 | isbn = 978-3-031-21493-6
 | location = Cham
 | mr = 4633160
 | pages = 133–147
 | publisher = Birkhäuser/Springer
 | series = Annals of the Canadian Society for History and Philosophy of Mathematics
 | title = Research in history and philosophy of mathematics—the CSHPM 2021 volume
 | year = 2023}}
* {{cite journal
 | last1 = Andrade-Molina | first1 = Melissa
 | last2 = Ravn | first2 = Ole
 | id = {{EBSCOhost|118987364|dbcode=ehh}} 
 | journal = [[Philosophy of Mathematics Education Journal]]
 | pages = 1–10
 | title = The Euclidean tradition as a paradigm for scientific thinking
 | volume = 30
 | year = 2016}}
* {{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=Alexanderson|first1=Gerald L.|author1-link=Gerald L. Alexanderson|last2=Greenwalt|first2=William S.| title=About the cover: Billingsley's Euclid in English|journal=[[Bulletin of the American Mathematical Society]] |series=New Series|volume=49|issue=1|year=2012|pages=163–167|doi=10.1090/S0273-0979-2011-01365-9|doi-access=free}}
*{{cite book |last=Artmann |first=Benno |year=2012 |orig-year=1999 |title=Euclid: The Creation of Mathematics |publisher=Springer |location=New York |isbn=978-1-4612-1412-0 |url={{google books|plainurl=y|id=F8XgBwAAQBAJ}} |url-access=limited |doi=10.1007/978-1-4612-1412-0 }}
*{{cite journal
 | last = Baldasso | first = Renzo
 | issue = 3
 | journal = La Bibliofilía
 | jstor = 26202240
 | pages = 525–552
 | title = Printing for the Doge: On the first quire of the first edition of the ''Liber elementorum Euclidis''
 | volume = 115
 | year = 2013
 }}
*{{cite journal
 | last = Baldwin | first = John T.
 | doi = 10.1093/philmat/nkx030
 | issue = 3
 | journal = Philosophia Mathematica
 | mr = 3867369
 | pages = 346–374
 | series = Series III
 | title = Axiomatizing changing conceptions of the geometric continuum I: Euclid–Hilbert
 | url = https://homepages.math.uic.edu/~jbaldwin/pub/axconIFeb1717.pdf
 | volume = 26
 | year = 2018}}
*{{cite book
 | last = Rouse Ball | first = Walter William | author-link = W. W. Rouse Ball
 | edition = 6th
 | publisher = MacMillan
 | title = A Short Account of the History of Mathematics
 | url = https://archive.org/details/shortaccountofhi00ballrich/
 | year = 1915}} 1st ed., 1888.
*{{cite book|title=The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe|first=Jason Socrates|last=Bardi|publisher=Wiley|year=2008|isbn=9780470467367}}
*{{cite journal
 | last = Beeson | first = Michael
 | doi = 10.1080/00029890.2022.2069985
 | issue = 7
 | journal = [[The American Mathematical Monthly]]
 | mr = 4457736
 | pages = 623–646
 | title = Euclid after computer proof-checking
 | volume = 129
 | year = 2022
 | arxiv = 2103.09623
 }}
*{{cite journal
 | last1 = Beeson | first1 = Michael
 | last2 = Narboux | first2 = Julien
 | last3 = Wiedijk | first3 = Freek
 | arxiv = 1710.00787
 | doi = 10.1007/s10472-018-9606-x
 | issue = 2–4
 | journal = Annals of Mathematics and Artificial Intelligence
 | mr = 3914603
 | pages = 213–257
 | title = Proof-checking Euclid
 | volume = 85
 | year = 2019}}
*{{cite book|contribution=Who translated Euclid’s ''Elements'' into Arabic?|first=Sonja|last=Brentjes|author-link=Sonja Brentjes|pages=21–54|title=Translation and Transmission: Collection of articles|editor1-first=Jaakko|editor1-last=Hämeen-Anttila|editor2-first=Ilkka|editor2-last=Lindstedt|location=Münster|publisher=Ugarit-Verlag|year=2018|series=The Intellectual Heritage of the Ancient and Mediaeval Near East|volume=3}}
*{{cite journal
 | last = Brock | first = W. H.
 | date = January 1975
 | doi = 10.1080/0046760750040203
 | issue = 2
 | journal = History of Education
 | pages = 21–35
 | title = Geometry and the universities: Euclid and his modern rivals 1860–1901
 | volume = 4}}
*{{cite book
 | last1 = Bunt | first1 = Lucas Nicolaas Hendrik
 | last2 = Jones | first2 = Phillip S.
 | last3 = Bedient | first3 = Jack D.
 | publisher = Dover
 | title = The Historical Roots of Elementary Mathematics
 | year = 1988}}
*{{cite book
 | last = Busard | first = H. L. L.
 | contribution = Introduction to the text
 | contribution-url = https://books.google.com/books?id=tiHUN6jjm6MC&pg=PA1
 | isbn = 978-3-515-08645-5
 | location = Stuttgart
 | publisher = Franz Steiner Verlag
 | title = Campanus of Novara and Euclid's Elements
 | year = 2005}}
*{{cite journal
 | last = Cairns | first = W. D.
 | date = June–July 1934
 | issue = 6
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2301562
 | page = 383
 | title = Review of ''The Elements of Euclid'' by Isaac Todhunter
 | volume = 41
 | doi = 10.2307/2301562
 }}
*{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Reddick | first2 = Angela | last3 = Xiong | first3 = Yeng | last4 = Keller | first4 = Wilfrid | issue = 9 | journal = [[Journal of Integer Sequences]] | mr = 3005523 | page = Article 12.9.8 | title = The history of the primality of one: a selection of sources | url = https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html | volume = 15 | year = 2012 | archive-date = 2018-04-12 | access-date = 2018-01-15 | archive-url = https://web.archive.org/web/20180412081118/https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html | url-status = live }}
*{{cite journal
 | last = Campbell | first = Paul J.
 | date = December 2017
 | doi = 10.4169/math.mag.90.5.395
 | issue = 5
 | journal = [[Mathematics Magazine]]
 | jstor = 10.4169/math.mag.90.5.395
 | pages = 395–396
 | title = Reviews: Babylonian trig; crowd-sourcing Euclid; gerrymandering
 | volume = 90}}
*{{cite book
 | last = Cederberg | first = Judith N.
 | doi = 10.1007/978-1-4757-3490-4
 | edition = 2nd
 | isbn = 0-387-98972-2
 | location = New York
 | mr = 1798736
 | publisher = Springer-Verlag
 | series = Undergraduate Texts in Mathematics
 | title = A Course in Modern Geometries
 | year = 2001}}
*{{cite journal
 | last = Clagett | first = Marshall
 | date = September 1954
 | doi = 10.1086/348338
 | issue = 3
 | journal = [[Isis (journal)|Isis]]
 | jstor = 226713
 | pages = 269–277
 | publisher = University of Chicago Press
 | title = King Alfred and the ''Elements'' of Euclid
 | volume = 45}}
*{{cite journal
 | last = Corry | first = Leo | author-link = Leo Corry
 | date = July 2013
 | doi = 10.1007/s00407-013-0121-5
 | issue = 6
 | journal = Archive for History of Exact Sciences
 | pages = 637–705
 | title = Geometry and arithmetic in the medieval traditions of Euclid's ''Elements'': a view from Book II
 | volume = 67}} See especially p. 637.
*{{cite book
 | last = Corry | first = Leo | author-link = Leo Corry
 | contribution = Distributivity-like results in Euclid's ''Elements''
 | doi = 10.1007/978-3-030-79679-2_2
 | isbn = 9783030796792
 | pages = 5–25
 | publisher = Springer International Publishing
 | title = Distributivity-like Results in the Medieval Traditions of Euclid's Elements
 | series = SpringerBriefs in History of Science and Technology | year = 2021}}
*{{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 journal
 | last = Damiani | first = Giacomo
 | date = June 2024
 | doi = 10.1163/15733823-20240106
 | issue = 3
 | journal = Early Science and Medicine
 | pages = 230–270
 | title = Form and matter of regular geometrical bodies in Luca Pacioli's ''Summa'' (1494) and ''Compendium de divina proportione'' (1498)
 | volume = 29}}
*{{cite book
 | last = Davis | first = Margaret Daly
 | publisher = Longo Editore
 | title = Piero Della Francesca's Mathematical Treatises: The Trattato D'abaco and Libellus de Quinque Corporibus Regularibus
 | year = 1977}}
*{{cite journal
 | last = De Young | first = Gregg
 | date = April 2012
 | doi = 10.1007/s00407-012-0094-9
 | issue = 3
 | journal = Archive for History of Exact Sciences
 | jstor = 41472233
 | pages = 265–294
 | publisher = Springer Science and Business Media LLC
 | title = Further adventures of the Rome 1594 Arabic redaction of Euclid's Elements
 | volume = 66}}
*{{cite journal
 | last = Dorandi | first = Tiziano
 | journal = Zeitschrift für Papyrologie und Epigraphik
 | jstor = 26603972
 | language = it
 | pages = 92–95
 | title = L'Euclide di P.Oxy. 5299
 | volume = 205
 | year = 2018}}
*{{cite book
 |editor-last=Dyde|editor-first=Samuel Walters
 |title=The Theaetetus of Plato
 |publisher=J. Maclehose
 |year=1899}}
*{{cite book
 | last = Edwards | first = C. H.
 | doi = 10.1007/978-1-4612-6230-5
 | isbn = 9781461262305
 | publisher = Springer | location = New York
 | title = The Historical Development of the Calculus
 | year = 1979}}
*{{cite journal
 | last = Elior | first = Ofer
 | doi = 10.2979/aleph.21.1.0123
 | issue = 1
 | journal = Aleph: Historical Studies in Science and Judaism
 | jstor = 10.2979/aleph.21.1.0123
 | pages = 123–148
 | publisher = Indiana University Press
 | title = Niccolò Tartaglia's 1543 edition of Euclid's ''Elements'' and the sources of an early Modern Hebrew version of the ''Elements''
 | volume = 21
 | year = 2021}}
*{{cite journal
 | last = Elior | first = Ofer
 | doi = 10.1016/j.hm.2024.02.005
 | journal = [[Historia Mathematica]]
 | mr = 4799290
 | pages = 1–21
 | title = The Hebrew translation of Euclid's ''Elements'' ascribed to Rabbi Jacob: a new analysis following the 'discovery' of the Arabic version in MS Paris, BULAC ARA. 606
 | volume = 68
 | year = 2024}}
*{{cite journal
 | last = Field | first = J. V. | authorlink = Judith V. Field
 | doi = 10.1007/BF00374595
 | issue = 3–4
 | journal = [[Archive for History of Exact Sciences]]
 | jstor = 41134110
 | mr = 1457069
 | pages = 241–289
 | s2cid = 118516740
 | title = Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler
 | volume = 50
 | year = 1997}}
*{{cite journal
 | last = Fletcher | first = W. C.
 | date = February 1938
 | issue = 248
 | journal = [[The Mathematical Gazette]]
 | jstor = 3607447
 | pages = 58–65
 | title = Euclid
 | volume = 22
 | doi = 10.2307/3607447
 }}
*{{cite book |first1=Menso |last1=Folkerts | author1-link=Menso Folkerts |title=Euclid in Medieval Europe |date=1989 |publisher=The Benjamin Catalogue for History of Science|url=https://math.berkeley.edu/~wodzicki/160/Euclid_in_Middle_Ages.pdf}} Collected in {{cite book
 | last = Folkerts | first = Menso
 | isbn = 0-86078-957-8
 | mr = 2229235
 | publisher = Ashgate Publishing Limited | location = Aldershot
 | series = Variorum Collected Studies Series
 | title = The Development of Mathematics in Medieval Europe
 | volume = 811
 | year = 2006}}
*{{cite journal
 | last = Fowler | first = D. H. | author-link = David Fowler (mathematician)
 | doi = 10.1007/bf00327868
 | issue = 1–2
 | journal = Archive for History of Exact Sciences
 | jstor = 41133572
 | pages = 5–36
 | title = Book II of Euclid's Elements and a pre-Eudoxan theory of ratio
 | volume = 22
 | year = 1980}}
* {{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=Furlong |first=Gillian |title=Treasures from UCL |publisher=UCL Press |chapter-url=https://books.google.com/books?id=ySE6DwAAQBAJ&pg=PA15 |chapter=First printed edition of Euclid's ''Elements'' |year=2015 |page=15 |isbn=9781910634363}}
*{{cite book|first=Veronica|last=Gavagna|contribution=The Euclidean tradition at the Renaissance courts: the case of Federico Commandino|pages=291–297|hdl=2158/1037064|title=Scientific Cosmopolitanism and Local Cultures: Religions, Ideologies, Societes (Proceedings of 5th International Conference of the ESHS, Athens 2012)|location=Athens|year=2014|editor-first=Gianna|editor-last=Katsiampoura|publisher=National Hellenic Research Foundation|isbn= 978-960-98199-3-0|contribution-url=https://flore.unifi.it/bitstream/2158/1037064/1/2014_ESHS_Atene.pdf}}
*{{cite book
 | last = Goulding | first = Robert
 | contribution = The puzzling lives of Euclid
 | doi = 10.1007/978-90-481-3542-4_5
 | isbn = 978-90-481-3542-4
 | pages = 117–142
 | publisher = Springer Netherlands
 | title = Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History
 | series = Archimedes
 | year = 2009
 | volume = 25
 }}
*{{cite book
 | last = Grafton | first = Anthony T.
 | editor1-last = Campi | editor1-first = Emidio
 | editor2-last = Angelis | editor2-first = Simone De
 | editor3-last = Goeing | editor3-first = Anja-Silvia
 | editor4-last = Grafton | editor4-first = Anthony T.
 | contribution = Textbooks and the disciplines
 | pages = 11–38
 | publisher = Librairie Droz
 | title = Scholarly Knowledge: Textbooks in Early Modern Europe
 | year = 2008}}
*{{cite journal
 | last = Grant | first = Hardy
 | date = May 2002
 | issue = 1
 | journal = Cubo Matemática Educacional
 | title = Euclid's ''Elements'' in cultural context
 | url = https://cubo.ufro.cl/index.php/cubo/article/view/1719
 | volume = 4}}
*{{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 book
 | last = Hartshorne | first = Robin | author-link = Robin Hartshorne
 | edition = 2nd
 | isbn = 9780387986500
 | location = New York
 | publisher = Springer]
 | title = Geometry: Euclid and Beyond
 | year = 2000}}
*{{cite journal
 | last1 = Hawes | first1 = Susan M.
 | last2 = Kolpas | first2 = Sid
 | date = August 2015
 | journal = Convergence
 | publisher = [[Mathematical Association of America]]
 | title = Oliver Byrne: The Matisse of mathematics – biography 1810–1829
 | url = https://old.maa.org/press/periodicals/convergence/oliver-byrne-the-matisse-of-mathematics-biography-1810-1829}}
*{{cite book
 | last = Heath | first = Thomas L. | author-link = Thomas Heath (classicist)
 | publisher = Clarendon Press
 | title = A Manual of Greek Mathematics
 | year = 1931}} Reprinted 1963. Dover. {{isbn|978-0-486-43231-1}}.
*{{cite journal
 | last = Henrici | first = O. | author-link = Olaus Henrici
 | date = March 1884
 | doi = 10.1038/029453b0
 | issue = 750
 | journal = Nature
 | pages = 453–454
 | title = The axioms of geometry
 | volume = 29 | bibcode = 1884Natur..29..453H }}
*{{Cite book |last=Henry |first=W. B. |url=https://archive.org/details/p.-oxy.-16/Oxyrhynchus%20Papyri/P.%20Oxy.%2082/page/22/mode/2up?view=theater |title=Oxyrhynchus Papyri Collection |date=2016 |publisher=The Egypt Exploration Society |location=London |pages=23 |language=English}}
*{{cite book
 | last = Herz-Fischler | first = Roger
 | isbn = 9780486152325
 | publisher = Courier Corporation
 | series = Dover Books on Mathematics
 | title = A Mathematical History of the Golden Number
 | year = 1998}}
*{{cite book|first=David|last=Hilbert|author-link=David Hilbert|url=https://www.gutenberg.org/files/17384/17384-pdf.pdf|translator-first=E. J.|translator-last=Townsend|year=1902|publisher=Open Court|title=Foundations of Geometry}}
*{{cite thesis|title=Euclid's Elements for High School Classrooms|first=Faith K.|last=Hoppen|year=2023|publisher=University of Nebraska|id={{ProQuest|2829362062}}}}
*{{cite book
 | last = Jahnke | first = Hans Niels
 | isbn = 9780821826232
 | publisher = American Mathematical Society
 | series = History of mathematics
 | title = A History of Analysis
 | volume = 24
 | year = 2003}}
* {{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 thesis|title=Euclid in the Modern Classroom|first=Tom|last=Jameson|year=2019|publisher=Harvard University|id={{ProQuest|2511164716}}}}
*{{cite book
 | last = Knorr | first = Wilbur R. | author-link = Wilbur Knorr
 | title = The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry
 | series = Synthese Historical Library
 | year = 1975
 | publisher = D. Reidel Publishing Co.
 | location = Dordrecht, Netherlands}}
*{{citation
 | last = Knorr | first = Wilbur
 | doi = 10.1090/S0273-0979-1983-15155-8
 | issue = 1
 | journal = American Mathematical Society
 | mr = 699316
 | pages = 41–69
 | series = New Series
 | title = 'La croix des mathématiciens': the Euclidean theory of irrational lines
 | volume = 9
 | year = 1983| doi-access = free
 }}
*{{cite journal
 | last = Knorr | first = Wilbur R. | author-link = Wilbur Knorr
 | doi = 10.1017/S0007087400044009
 | issue = 3
 | journal = [[The British Journal for the History of Science]]
 | jstor = 4026757
 | pages = 293–330
 | title = John of Tynemouth alias John of London: Emerging portrait of a singular medieval mathematician
 | volume = 23
 | year = 1990}}
*{{cite journal
 | last = Knorr | first = Wilbur R. | author-link = Wilbur Knorr
 | doi = 10.1111/j.1600-0498.1996.tb00610.x
 | issue = 2–3
 | journal = Centaurus
 | mr = 1384401
 | pages = 208–276
 | title = The wrong text of Euclid: on Heiberg's text and its alternatives
 | volume = 38
 | year = 1996}}
*{{cite book
 | last1 = Laubenbacher | first1 = Reinhard
 | last2 = Pengelley | first2 = David
 | contribution = Chapter 1: Geometry: The parallel postulate
 | doi = 10.1007/978-1-4612-0523-4
 | isbn = 9781461205234
 | pages = 1–53
 | publisher = Springer New York
 | series = Undergraduate Texts in Mathematics
 | title = Mathematical Expeditions
 | year = 1999}}
*{{cite book
 | last = Majer | first = Ulrich
 | editor1-last = Hendricks | editor1-first = Vincent F.
 | editor2-last = Jørgensen | editor2-first = Klaus Frovin
 | editor3-last = Lützen | editor3-first = Jesper
 | editor4-last = Pedersen | editor4-first = Stig Andur
 | contribution = Hilbert's axiomatic approach to the foundations of science—A failed research program?
 | doi = 10.1007/978-1-4020-5195-1_5
 | isbn = 9781402051944
 | pages = 155–184
 | publisher = Springer Netherlands
 | title = Interactions: Mathematics, Physics and Philosophy, 1860–1930
 | series = Boston Studies in the Philosophy of Science
 | year = 2006
 | volume = 251
 }}
*{{cite journal
 | last = McCreedy | first = Jonathan
 | date = Spring 2011
 | journal = Genetic Joyce Studies
 | pages = 1–24
 | title = 'Ocone! Ocone!' ALP's 3D Siglum and Dolph's 'Night (''Sex-Education'') Lessons'
 | url = https://www.geneticjoycestudies.org/static/issues/GJS11/GJS11_McCreedy.pdf
 | volume = 11}}
*{{cite book |first1=Uta B.|last1=Merzbach|author1-link=Uta Merzbach| first2=Carl B. | last2=Boyer | author2-link=Carl Benjamin Boyer | title=A History of Mathematics |chapter=Chapter 5: Euclid of Alexandria
 | edition=Third | publisher=John Wiley & Sons | year=2011|isbn=978-0470525487|pages=90–108}}
*{{cite journal
 | last = Moktefi | first = Amirouche
 | date = November 2005
 | issue = 516
 | journal = [[The Mathematical Gazette]]
 | jstor = 3621985
 | pages = 556–558
 | title = Review of ''Euclid and His Modern Rivals'' by Charles L. Dodgson
 | volume = 89
 | doi = 10.1017/S0025557200178775
 }}
*{{cite journal
 | last = Moriarty | first = Clare Marie
 | doi = 10.1162/leon_a_02242
 | issue = 2
 | journal = Leonardo
 | pages = 151–158
 | title = 'Tint and form': The geometric philosophy underlying Oliver Byrne's ''Elements''
 | url = https://muse.jhu.edu/article/887050
 | volume = 56
 | year = 2023}}
*{{cite book|contribution=Preface|first=Ian|last=Mueller|author-link=Ian Mueller|editor=Glenn R. Morrow|title=Proclus: A Commentary on the First Book of Euclid's Elements|publisher=Princeton University Press|year=1970}}
*{{cite journal
 | last = Mueller | first = Ian
 | date = December 1969
 | doi = 10.1093/bjps/20.4.289
 | issue = 4
 | journal = The British Journal for the Philosophy of Science
 | jstor = 686258
 | pages = 289–309
 | publisher = University of Chicago Press
 | title = Euclid's Elements and the axiomatic method
 | volume = 20}}
*{{cite journal
 | last = Murdoch | first = John E. | author-link = John E. Murdoch
 | doi = 10.2307/310767
 | journal = [[Harvard Studies in Classical Philology]]
 | jstor = 310767
 | pages = 249–302
 | title = Euclides Graeco-Latinus: A hitherto unknown medieval Latin translation of the ''Elements'' made directly from the Greek
 | volume = 71
 | year = 1967}}
*{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham|date = 1999}}{{sfn whitelist|CITEREFO&#39;ConnorRobertson1999}}
*{{cite journal
 | last = Pantin | first = Isabelle | author-link = Isabelle Pantin
 | doi = 10.1007/s11191-005-1794-1
 | issue = 2–4
 | journal = Science & Education
 | pages = 189–207
 | title = Teaching mathematics and astronomy in France: The ''Collège Royal'' (1550–1650)
 | volume = 15
 | year = 2006 | bibcode = 2006Sc&Ed..15..189P }}
*{{cite journal
 | last = Panza | first = Marco
 | doi = 10.1007/s11229-012-0074-2
 | issue = 1
 | journal = [[Synthese]]
 | jstor = 41494923
 | mr = 2935330
 | pages = 55–102
 | title = The twofold role of diagrams in Euclid's plane geometry
 | volume = 186
 | year = 2012
 | url = https://hal.science/hal-00192165
 }}
* {{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=Reid|first=Constance|author-link=Constance Reid|title=A Long Way from Euclid: The intelligent reader's guide to modern mathematics|year=1963|publisher=Thomas Y. Crowell Co.|location=New York|url=https://archive.org/details/longwayfromeucli005621mbp}} Reprinted by Courier Dover, 2004, {{isbn|9780486436135}}.
* {{cite book|last1=Reynolds|first1=Leighton Durham|author1-link=L. D. Reynolds |last2=Wilson|first2=Nigel Guy|author2-link=Nigel Wilson (classicist) |title=Scribes and scholars: a guide to the transmission of Greek and Latin literature|url=https://books.google.com/books?id=7iSlvWFf1FQC|edition=2nd|date=9 May 1991|publisher=Clarendon Press|location=Oxford|isbn=978-0-19-872145-1}}
*{{cite book
 | last = Robič | first = Borut
 | contribution = The foundational crisis of mathematics
 | doi = 10.1007/978-3-662-44808-3_2
 | isbn = 9783662448083
 | pages = 9–30
 | publisher = Springer
 | title = The Foundations of Computability Theory
 | year = 2015}}
*{{cite journal
 | last = Roskam | first = Jade
 | date = January 2009
 | doi = 10.54870/1551-3440.1151
 | issue = 1–2
 | journal = The Mathematics Enthusiast
 | pages = 277–294
 | title = Book X of ''The Elements'': Ordering irrationals
 | volume = 6
 | url = https://scholarworks.umt.edu/tme/vol6/iss1/23
 }}
*{{cite journal
 | last = Rothstein | first = Bret
 | doi = 10.1353/jem.2013.0000
 | issue = 1
 | journal = Journal for Early Modern Cultural Studies
 | pages = 96–129
 | publisher = Project MUSE
 | title = Making Trouble: Strange Wooden Objects and the Early Modern Pursuit of Difficulty
 | volume = 13
 | year = 2013}}
* {{cite book|last=Russell|first=Bertrand |author-link=Bertrand Russell|title=History of Western Philosophy: Collectors Edition|url=https://books.google.com/books?id=Gm_cCZBiOhQC&pg=PA177|year=2013|publisher=Routledge|isbn=978-1-135-69284-1}}
*{{cite book
 | last = Schubring | first = Gert
 | contribution = Textbooks before the invention of the printing press: orality and teaching
 | doi = 10.1007/978-3-031-17670-8_2
 | isbn = 9783031176708
 | pages = 15–53
 | publisher = Springer International Publishing
 | title = Analysing Historical Mathematics Textbooks
 | series = International Studies in the History of Mathematics and its Teaching
 | year = 2022}}
* {{cite encyclopedia |last=Sialaros |first=Michalis |editor-first1=Michalis |editor-last1=Sialaros |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|archive-url=https://web.archive.org/web/20210814034203/https://oxfordre.com/classics/view/10.1093/acrefore/9780199381135.001.0001/acrefore-9780199381135-e-2521|archive-date=2021-08-14 |url-access=subscription|url-status=live}}
*{{cite journal
 | last = Simpkins | first = Diana M.
 | date = December 1966
 | doi = 10.1080/00033796600203155
 | issue = 4
 | journal = [[Annals of Science]]
 | pages = 225–249
 | title = Early editions of Euclid in England
 | volume = 22}}
*{{cite book|year=2008|title=The History of the Geometry Curriculum in the United States|first=Nathalie|last=Sinclair|publisher=Information Age Publishing|isbn=9781607527305}}
*{{cite book
 | last = Siu | first = Man-Keung
 | contribution = 1607, a year of (some) significance: Translation of the first European text in mathematics{{snd}}''Elements''{{snd}}into Chinese
 | hdl = 10722/153396 | url = https://hub.hku.hk/handle/10722/153396 <!-- handle broken? -->
 | pages = 573–589
 | publisher = Holzhausen Verlag
 | title = History and Epistemology in Mathematics Education: Proceedings of the 6th European Summer University, Wien, Österreich, 19-23 July 2010
 | via = Hong Kong University Scholars Hub
 | year = 2011}}
*{{cite journal|last=Swetz|first=Frank J.|title=Mathematical Treasures{{snd}}Greek edition of Euclid's ''Elements''|publisher=Mathematical Association of America |url=https://old.maa.org/press/periodicals/convergence/mathematical-treasures-greek-edition-of-euclids-elements|journal=Convergence|date=January 2021}}
* {{cite conference
 | last = Szmielew | first = Wanda | author-link = Wanda Szmielew
 | editor1-last = Henkin | editor1-first = Leon
 | editor2-last = Addison | editor2-first = John
 | editor3-last = Chang | editor3-first = Chen Chung
 | editor4-last = Craig | editor4-first = William
 | editor5-last = Scott | editor5-first = Dana
 | editor6-last = Vaught | editor6-first = Robert
 | contribution = The role of the Pasch axiom in the foundations of Euclidean geometry
 | contribution-url = https://books.google.com/books?id=SAoDCAAAQBAJ&pg=PA123
 | doi = 10.1090/pspum/025/0373872
 | location = Providence, Rhode Island
 | mr = 373872
 | pages = 123–132
 | publisher = American Mathematical Society
 | series = Proceedings of Symposia in Pure Mathematics
 | title = Proceedings of the Tarski Symposium: An international symposium held at the University of California, Berkeley, June 23–30, 1971, to honor Alfred Tarski on the occasion of his seventieth birthday
 | volume = 25
 | year = 1974 | isbn = 978-0-8218-1425-3 }}
* {{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 }}
* {{cite conference
 | last = Tarski | first = Alfred | author-link = Alfred Tarski
 | editor1-last = Henkin | editor1-first = L.
 | editor2-last = Suppes | editor2-first = P.
 | contribution = What is elementary geometry? The axiomatic method, With special reference to geometry and physics
 | mr = 106185
 | pages = 16–29
 | publisher = North-Holland, Amsterdam
 | series = Studies in Logic and the Foundations of Mathematics
 | title = Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958
 | year = 1959}}
*{{cite journal
 | last = Toumasis | first = Charalampos
 | date = December 1990
 | issue = 6
 | journal = Educational Studies in Mathematics
 | jstor = 3482388
 | pages = 491–508
 | title = The epos of Euclidean Geometry in Greek secondary education (1836-1985): Pressure for change and resistance
 | volume = 21
 | doi = 10.1007/BF00315941
 }}
*{{cite journal
 | last = Toussaint | first = Godfried | author-link = Godfried Toussaint
 | doi = 10.1007/BF03024252
 | issue = 3
 | journal = [[The Mathematical Intelligencer]]
 | pages = 12–24
 | title = A new look at Euclid's second proposition
 | url = https://cgm.cs.mcgill.ca/~godfried/teaching/dm-reading-assignments/Euclid-Second-Proposition-Math-Intell.pdf
 | volume = 15
 | year = 1993}}
*{{cite journal
 | last = Unguru | first = Sabetai | author-link = Sabetai Unguru
 | date = May 1985
 | doi = 10.1016/0315-0860(85)90013-8
 | issue = 2
 | journal = [[Historia Mathematica]]
 | pages = 176–184
 | title = Digging for structure into the ''Elements'': Euclid, Hilbert, and Mueller
 | volume = 12| doi-access = free
 }}
*{{cite book
 | last = Unguru | first = Sabetai
 | editor-last = Sialaros | editor-first = Michalis
 | contribution = Counter-revolutions in mathematics
 | isbn = 978-3-11-056365-8
 | mr = 3822179
 | pages = 17–34
 | publisher = De Gruyter, Berlin
 | series = Science, Technology, and Medicine in Ancient Cultures
 | title = Revolutions and continuity in Greek mathematics
 | volume = 8
 | year = 2018}}
*{{cite book|last=Van der Waerden|first=Bartel Leendert|author-link=Bartel Leendert van der Waerden|title=Science awakening|url=https://books.google.com/books?id=djUPAQAAMAAJ|year=1975|publisher=Noordhoff International|isbn=978-90-01-93102-5}}
*{{cite book
 | last = Wardhaugh | first = Benjamin
 | editor1-last = Remmert | editor1-first = Volker R.
 | editor2-last = Schneider | editor2-first = Martina R.
 | editor3-last = Sørensen | editor3-first = Henrik Kragh
 | contribution = Greek Mathematics in English: The Work of Sir Thomas L. Heath (1861–1940)
 | contribution-url = https://ora.ox.ac.uk/objects/uuid:784375d7-3969-4b9d-93f9-c85c55eb19c7
 | doi = 10.1007/978-3-319-39649-1_6
 | isbn = 9783319396491
 | pages = 109–122
 | publisher = Springer International Publishing
 | title = Historiography of Mathematics in the 19th and 20th Centuries
 | series = Trends in the History of Science
 | year = 2016}}
*{{cite book
 | last = Wardhaugh | first = Benjamin
 | editor1-last = Goeing | editor1-first = Anja-Silvia
 | editor2-last = Parry | editor2-first = Glyn
 | editor3-last = Feingold | editor3-first = Mordechai
 | contribution = Defacing Euclid: Reading and Annotating the Elements of Geometry in Early Modern Britain
 | doi = 10.1163/9789004444058_015
 | isbn = 9789004442412
 | publisher = BRILL
 | title = Early Modern Universities: Networks of Higher Learning
 | year = 2021}}
*{{cite book|title=Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World|first=Benjamin|last=Wardhaugh|publisher=Princeton University Press|year=2023|isbn=9780691235769}}
*{{cite journal|last=Wardhaugh|first=Benjamin|title=Euclid's Elements in Latin, 1482–1703: Vocabulary and Classiﬁcation|journal=Revue d'Histoire des Mathématiques|publisher= Société Mathématique de France|year= 2024|url=https://ora.ox.ac.uk/objects/uuid:f27b083c-0f32-4f9f-9110-12f95bc9bfbd}}
*{{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 }}
*{{cite journal
 | last = Zeeman | first = E. C. | author-link = Christopher Zeeman
 | doi = 10.1112/blms/bdm104
 | issue = 1
 | journal = [[Bulletin of the London Mathematical Society]]
 | mr = 2409172
 | pages = 1–17
 | title = What's wrong with Euclid Book V
 | url = https://www.lms.ac.uk/sites/lms.ac.uk/files/2008%20What's%20Wrong%20with%20Euclid%20Book%20V.pdf
 | archive-url = https://web.archive.org/web/20170809033625/https://www.lms.ac.uk/sites/lms.ac.uk/files/2008%20What's%20Wrong%20with%20Euclid%20Book%20V.pdf
 | archive-date = 2017-08-09
 | url-status = dead
 | volume = 40
 | year = 2008}}
{{Refend}}

== External links ==
{{Wikiquote}}
{{Wikisource|The Elements of Euclid}}
{{Commons category|Elements of Euclid}}
*[https://elements.ratherthanpaper.com/ ''Elements'' with highlights] by ratherthanpaper
*[https://www2.hf.uio.no/polyglotta/index.php?page=volume&vid=67 Multilingual edition of ''Elementa'' in the Bibliotheca Polyglotta]
*{{cite web | url = http://aleph0.clarku.edu/~djoyce/java/elements/toc.html | title = Euclid's Elements | editor = David E. Joyce | editor-link = David E. Joyce |year =1997
 }} In HTML with Java-based interactive figures.

{{Euclid}}
{{Greek mathematics}}
{{Authority control}}

[[Category:3rd-century BC books]]
[[Category:Euclidean geometry|*]]
[[Category:Mathematics textbooks]]
[[Category:Works by Euclid]]
[[Category:History of geometry]]
[[Category:Foundations of geometry]]
[[Category:Geometry education]]