View source for Euclid's Elements

{{Short description|Mathematical treatise by Euclid}}
{{DISPLAYTITLE:Euclid's ''Elements''}}
{{Infobox book
| name             = Elements
| author           = [[Euclid]]
| language         = [[Ancient Greek]]
| country          = 
| genre            = Mathematics
| publisher        = 
| isbn             = 
| italic title     = Elements
| image            = Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg
| caption          = Title page of Sir Henry Billingsley's first English version of Euclid's ''Elements'', 1570. During the [[Renaissance]], Euclid was [[Euclid#Identity and historicity|commonly conflated with]] the philosopher [[Euclid of Megara]].
| title_orig       = 
| translator       = 
| illustrator      = 
| cover_artist     = 
| series           = 
| subject          = [[Euclidean geometry|plane]] and [[solid geometry]], [[number theory]], [[commensurability (mathematics)|incommensurable]] lines
| 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 [[mathematics|mathematical]] [[treatise]] consisting of 13 books attributed to the ancient [[Greek mathematics|Greek mathematician]] [[Euclid]] {{circa}} 300 BC. It is a collection of definitions, [[postulates]], [[propositions]] ([[theorem]]s and [[Compass and straightedge constructions|constructions]]), and [[mathematical proof]]s of the propositions. The books cover plane and solid [[Euclidean geometry]], elementary [[number theory]], and [[Commensurability (mathematics)|incommensurable]] lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of [[logic]] and modern [[science]], and its logical rigor was not [[Set theory|surpassed]] until the 19th century.

Euclid's ''Elements'' has been referred to as the most successful{{efn|{{harvnb|Wilson|2006|p=278}} states, "Euclid's Elements subsequently became the basis of all mathematical education, not only in the Roman and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written."}}{{efn|{{harvnb|Boyer|1991|p=100}} notes, "As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written – the ''Elements'' (''Stoichia'') of Euclid".}} and influential{{efn|{{harvnb|Boyer|1991|p=119}} notes, "The ''Elements'' of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [...]The first printed versions of the ''Elements'' appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's ''Elements''".}} [[textbook]] ever written. 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|Boyer|1991|p=100}} the number reaching well over one thousand.{{efn|{{harvnb|Bunt|Jones|Bedient |1988|p=142}} state, "the ''Elements'' became known to Western Europe via the Arabs and the Moors. There, the ''Elements'' became the foundation of mathematical education. More than 1000 editions of the ''Elements'' are known. In all probability, it is, next to the ''Bible'', the most widely spread book in the civilization of the Western world."}} For centuries, when the [[quadrivium]] was included in the curriculum of all university students, knowledge of at least part of Euclid's ''Elements'' was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.{{Citation needed|reason=Sweeping claim|date=April 2023}}

== History ==
[[File:P. Oxy. I 29.jpg|thumb|A fragment of Euclid's ''Elements'' on part of the [[Oxyrhynchus papyri]] ]]
[[File:Illustrated Opening. Arabic Translation of Euclid's Elementa (CBL Ar 3035, ff.105b-106a).jpg|thumb|Double-page from the [[Ishaq ibn Hunayn|Ishaq ibn Hunayn's]] Arabic translation of the ''Elements''. [[Iraq]], 1270. [[Chester Beatty Library]]]]

===Basis in earlier work===
[[Image:Woman teaching geometry.jpg|thumb|An illumination from a manuscript based on [[Adelard of Bath]]'s translation of the ''Elements'', {{circa}} 1309–1316; Adelard's is the oldest surviving translation of the ''Elements'' into Latin, done in the 12th-century work and translated from Arabic.{{sfn|Russell|2013|p=177}}]]
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}}

[[Proclus]] (412–485 AD), 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".

[[Pythagoras]] ({{circa}} 570–495 BC) was probably the source for most of books I and II, [[Hippocrates of Chios]] ({{circa}} 470–410 BC, not the better known [[Hippocrates|Hippocrates of Kos]]) for book III, and [[Eudoxus of Cnidus]] ({{circa}} 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.{{sfn|Ball|1915|loc=[https://archive.org/details/shortaccountofhi00ballrich/page/54/ {{p.|54}}]}} 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|Ball|1915|loc=[https://archive.org/details/shortaccountofhi00ballrich/page/38/ {{p.|38}}]}} Other similar works are also reported to have been written by [[Theudius of Magnesia]], [[Leon (mathematician)|Leon]], and Hermotimus of Colophon.<ref>Unguru, S. (1985). [https://core.ac.uk/download/pdf/82804799.pdf Digging for Structure into the Elements: Euclid, Hilbert, and Mueller]. Historia Mathematica 12, 176</ref><ref>Zhmud, L. (1998). [https://philarchive.org/archive/ZHMPAA Plato as "Architect of Science"]. Phonesis 43, 211</ref>

===Transmission of the text===
In the 4th century AD, [[Theon of Alexandria]] produced an edition of Euclid which was so widely used that it became the only surviving source until [[François Peyrard]]'s 1808 discovery at the [[Vatican Library|Vatican]] of a manuscript not derived from Theon's. This manuscript, the [[Johan Ludvig Heiberg (historian)|Heiberg]] manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions.<ref>[<!-- http://www.historyofscience.com/G2I/timeline/index.php?id=2749 -->http://historyofinformation.com/expanded.php?id=2749 The Earliest Surviving Manuscript Closest to Euclid's Original Text (Circa 850)]; an [http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html image] {{webarchive|url=https://web.archive.org/web/20091220042513/http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html |date=2009-12-20 }} of one page</ref> [[Papyrus Oxyrhynchus 29]] is a tiny fragment of an even older manuscript, but only contains the statement of one proposition.

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}} The Arabs received the ''Elements'' from the Byzantines around 760; this version was translated into [[Arabic]] under [[Harun al-Rashid]] ({{circa}} 800).{{sfn|Russell|2013|p=177}} The Byzantine scholar [[Arethas of Caesarea|Arethas]] commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century.{{sfn|Reynolds|Wilson|1991|p=57}} Although known in Byzantium, the ''Elements'' was lost to Western Europe until about 1120, when the English monk [[Adelard of Bath]] translated it into Latin from an Arabic translation.{{efn|One older work claims Adelard disguised himself as a Muslim student to obtain a copy in Muslim Córdoba.{{sfn|Ball|1915|loc=[https://archive.org/details/shortaccountofhi00ballrich/page/165/ {{p.|165}}]}} However, more recent biographical work has turned up no clear documentation that Adelard ever went to [[Al-Andalus|Muslim-ruled Spain]], although he spent time in Norman-ruled Sicily and Crusader-ruled Antioch, both of which had Arabic-speaking populations. Charles Burnett, ''Adelard of Bath: Conversations with his Nephew'' (Cambridge, 1999); Charles Burnett, ''Adelard of Bath'' (University of London, 1987).}} 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.<ref>{{cite journal |last1=Murdoch |first1=John E. |title=Euclides Graeco-Latinus: A Hitherto Unknown Medieval Latin Translation of the Elements Made Directly from the Greek |journal=Harvard Studies in Classical Philology |date=1967 |volume=71 |pages=249–302 |doi=10.2307/310767|jstor=310767 }}</ref>

After the translation by Adelard of Bath (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), Johannes de Tinemue,<ref>{{cite journal |last1=Knorr |first1=Wilbur R. |title=John of Tynemouth alias John of London: Emerging Portrait of a Singular Medieval Mathematician |journal=The British Journal for the History of Science |date=1990 |volume=23 |issue=3 |pages=293–330 |doi=10.1017/S0007087400044009 |jstor=4026757 |s2cid=144172844 |url=https://www.jstor.org/stable/4026757 |issn=0007-0874}}</ref> possibly also known as [[John of Tynemouth (geometer)|John of Tynemouth]] (his manuscripts are referred to collectively as Adelard III), late 12th century, and [[Gerard of Cremona]] (sometime after 1120 but before 1187). The exact details concerning these translations is still an active area of research.{{sfn|Busard|2005|p=}}{{pn|date=November 2023}} [[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.<ref>{{cite book |last1=Menso |first1=Folkerts |title=Euclid in Medieval Europe |date=1989 |publisher=Benjamin catalogue |url=https://math.berkeley.edu/~wodzicki/160/Euclid_in_Middle_Ages.pdf}}</ref><ref>{{cite web |last1=Campanus`` |first1=Pal.lat.1348 |title=DigiVatLib |url=https://digi.vatlib.it/mss/detail/Pal.lat.1348 |website=digi.vatlib.it |access-date=20 November 2023}}</ref>

[[File:Euclid's Elements 1573 Edition.JPG|thumb|Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in  8:350, (2)pp. THOMAS–STANFORD, Early Editions of Euclid's ''Elements'', n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.]] The first printed edition appeared in 1482 (based on Campanus's translation),{{sfn|Busard|2005|p=1}} and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered and [[list of editiones principes in Greek|published in 1533]]<ref>{{cite web |title=Mathematical Treasures - Greek Edition of Euclid's Elements {{!}} Mathematical Association of America |url=https://maa.org/press/periodicals/convergence/mathematical-treasures-greek-edition-of-euclids-elements |website=maa.org}}</ref> based on Paris gr. 2343 and Venetus Marcianus 301.<ref>{{cite book |last1=Thomas |first1=Heath |title=The thirteen books of Euclid's Elements. Vol. 1: Introduction and books I, II |date=1956 |publisher=Dover Publications |location=New York |isbn=978-0-486-60088-8 |edition=Second revised with additions}}</ref> 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]].

Copies of the Greek text still exist, some of which can be found in the [[Vatican Library]] and the [[Bodleian Library]] in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available).

Ancient texts which refer to the ''Elements'' itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by [[Johan Ludvig Heiberg (historian)|J. L. Heiberg]] and Sir [[Thomas Little Heath]] in their editions of the text.

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.

== Contents ==
* Book 1 contains 5 postulates and 5 common notions, and covers important topics of plane geometry such as the [[Pythagorean theorem]], equality of angles and [[area]]s, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures.
* Book 2 contains a number of [[Lemma (mathematics)|lemmas]] concerning the equality of rectangles and squares, sometimes referred to as "[[Greek geometric algebra|geometric algebra]]", and concludes with a construction of the [[golden ratio]] and a way of constructing a square equal in area to any rectilineal plane figure.
* Book 3 deals with circles and their properties: finding the center, [[inscribe]]d angles, [[tangent]]s, the power of a point, [[Thales' theorem]].
* Book 4 constructs the [[incircle]] and [[circumcircle]] of a triangle, as well as [[regular polygon]]s with 4, 5, 6, and 15 sides.
* Book 5, on proportions of [[magnitude (mathematics)|magnitudes]], gives the highly sophisticated theory of proportion probably developed by [[Eudoxus of Cnidus|Eudoxus]], and proves properties such as "alternation" (if ''a'' : ''b'' :: ''c'' : ''d'', then ''a'' : ''c'' :: ''b'' : ''d'').
* Book 6 applies proportions to plane geometry, especially the construction and recognition of [[Similarity (geometry)|similar]] figures.
* Book 7 deals with elementary number theory: [[divisibility]], [[prime number]]s and their relation to [[composite number]]s, [[Euclidean algorithm|Euclid's algorithm]] for finding the [[greatest common divisor]], finding the [[least common multiple]].
* Book 8 deals with the construction and existence of [[geometric progression|geometric sequences]] of integers.
* Book 9 applies the results of the preceding two books and gives the [[infinitude of prime numbers]] and the construction of all even [[perfect number]]s.
* Book 10 proves the irrationality of the square roots of non-square integers (e.g. <math>\sqrt{2}</math>) and classifies the square roots of [[commensurability (mathematics)|incommensurable]] lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of [[irrational number]]s. He also gives a [[Euclid's formula|formula]] to produce [[Pythagorean triples]].<ref name="Joyce">{{citation |date=June 1997 |last=Joyce |first=D. E. |chapter-url= https://mathcs.clarku.edu/~djoyce/java/elements/bookX/propX29.html |title=Euclid's Elements |chapter=Book X, Proposition XXIX |publisher=Clark University}}</ref>
* Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of [[parallelepiped]]s.
* Book 12 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]], and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. 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.
* Book 13 constructs the five regular [[Platonic solid]]s inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.

{| class="wikitable"
|+ Summary Contents of Euclid's ''Elements''
! 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
|}

==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}}]]
[[File:Euclid Vat ms no 190 I prop 47.jpg|thumb|Scan of pages demonstrating [[Pythagorean theorem]] from manuscript held in the [[Vatican Library]]]]
Euclid's [[Axiomatic system#Axiomatic method|axiomatic approach]] and [[Euclidean geometry#Methods of proof|constructive methods]] were widely influential.

Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a [[Compass-and-straightedge construction|compass and straightedge]]. 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}}

As was common in ancient mathematical texts, when a proposition needed [[Mathematical proof|proof]] in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as [[Theon of Alexandria|Theon]] often interpolated their own proofs of these cases.

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|Ball|1915|loc=[https://archive.org/details/shortaccountofhi00ballrich/page/55/ {{p.|55}}]}} the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically 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|Ball|1915|loc={{nobr|[https://archive.org/details/shortaccountofhi00ballrich/page/54/ pp. 54], [https://archive.org/details/shortaccountofhi00ballrich/page/58/ 58], [https://archive.org/details/shortaccountofhi00ballrich/page/127/ 127]}}}}

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|1963|p= 216}}

No indication is given of the method of reasoning that led to the result, although the ''[[Data (Euclid)|Data]]'' does provide instruction about how to approach the types of problems encountered in the first four books of the ''Elements''.{{sfn|Ball|1915|loc=[https://archive.org/details/shortaccountofhi00ballrich/page/54/ {{p.|54}}]}} Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.{{sfn| Toussaint|1993| pp= 12–23}} 

== Influence ==
[[File:Euclid's_Elements,_1482.jpg|thumb|A page with marginalia from the first printed edition of ''Elements'', printed by [[Erhard Ratdolt]] in 1482]]
The ''Elements'' is still considered a masterpiece in the application of [[logic]] to [[mathematics]]. In historical context, it has proven enormously influential in many areas of [[science]]. Scientists [[Nicolaus Copernicus]], [[Johannes Kepler]], [[Galileo Galilei]], [[Albert Einstein]] and Sir [[Isaac Newton]] were all influenced by the ''Elements'', and applied their knowledge of it to their work.<ref>{{cite news |last1=Andrew. |first1=Liptak |date=2 September 2017 |title=One of the world's most influential math texts is getting a beautiful, minimalist edition |url=https://www.theverge.com/2017/9/2/16247282/euclids-elements-kroncker-wallis-math-text-beautiful-minimalist-kickstarter |newspaper=The Verge}}</ref><ref>{{cite news |last1=Grabiner. |first1=Judith |title=How Euclid once ruled the world |url=https://plus.maths.org/content/how-euclid-once-rules-world |newspaper=Plus Magazine}}</ref> Mathematicians and philosophers, such as [[Thomas Hobbes]], [[Baruch Spinoza]], [[Alfred North Whitehead]], and [[Bertrand Russell]], have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.

The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. [[Abraham Lincoln]] kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in [[Springfield, Illinois|Springfield]], went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight".{{sfn|Ketcham|1901|p=}}<ref>[https://nautil.us/issue/41/selection/euclid-as-founding-father Euclid as Founding Father]</ref> [[Edna St. Vincent Millay]] wrote in her sonnet "Euclid alone has looked on Beauty bare", "O blinding hour, O holy, terrible day, / When first the shaft into his vision shone / Of light anatomized!". Albert Einstein recalled a copy of the ''Elements'' and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".<ref>{{cite web |last=Herschbach |first=Dudley |author-link=Dudley R. Herschbach |title=Einstein as a Student |url=http://www.chem.harvard.edu/herschbach/Einstein_Student.pdf |archive-url=https://web.archive.org/web/20090226002142/http://www.chem.harvard.edu/herschbach/Einstein_Student.pdf |archive-date=2009-02-26 |publisher=Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA |page=3}}: about Max Talmud visited on Thursdays for six years.</ref><ref>{{cite web |last=Prindle |first=Joseph |title=Albert Einstein – Young Einstein |url=http://www.alberteinsteinsite.com/einsteinyoung.html |url-status=live |archive-url=https://web.archive.org/web/20170610041942/http://www.alberteinsteinsite.com/einsteinyoung.html |archive-date=10 June 2017 |access-date=29 April 2018 |website=www.alberteinsteinsite.com}}</ref>

The success of the ''Elements'' is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the ''Elements'', encouraged its use as a textbook for about 2,000 years. The ''Elements'' still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain the cornerstone of mathematics.

=== In modern mathematics ===
One of the most notable influences of Euclid on modern mathematics is the discussion of the [[parallel postulate]]. In Book I, Euclid lists five postulates, the fifth of which stipulates

{{quote|If a [[line segment]] intersects two straight [[Line (mathematics)|lines]] forming two interior angles on the same side that sum to less than two [[right angle]]s, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.}}
[[File:Noneuclid.svg|thumb|263x263px|The different versions of the parallel postulate result in different geometries.]]
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]].{{Citation needed|date=January 2022}}

== Criticism ==
Euclid's ''Elements'' contains errors.  Some of the foundational theorems are proved using axioms that Euclid did not state explicitly.  A few proofs have errors, by relying on assumptions that are intuitive but not explicitly proven.  Mathematician and historian [[W. W. Rouse Ball]] put the criticisms in perspective, remarking 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|Ball|1915|loc=[https://archive.org/details/shortaccountofhi00ballrich/page/55/ {{p.|55}}]}}

Later editors have added Euclid's implicit axiomatic assumptions in their list of formal axioms.{{sfn|Heath|1956a|p= 62}}  For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.{{sfn|Heath|1956a|p= 242}} 

Known errors in Euclid date to at least 1882, when [[Pasch's axiom|Pasch published his missing axiom]].

Early attempts to find all the errors include [[Hilbert's axioms|Hilbert's geometry axioms]] and [[Tarski's axioms|Tarski's]].  In 2017, Michael Beeson et al. used computer [[Proof assistant|proof assistants]] to create a new set of axioms similar to Euclid's and generate proofs that were valid with those axioms.<ref>{{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}}</ref>  

Beeson et al. checked only Book I and found these errors: missing axioms, superfluous axioms, gaps in logic (such as failing to prove points were colinear), missing theorems (such as an angle cannot be less than itself), and outright bad proofs.  The bad proofs were in Book I, Proof 7 and Book I, Proposition 9.

== Apocrypha ==
It was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means that the [[apocryphal]] books XIV and XV of the ''Elements'' were sometimes included in the collection.{{sfn|Boyer|1991|pp=118–119}} The spurious Book XIV was probably written by [[Hypsicles]] on the basis of a treatise by [[Apollonius of Perga|Apollonius]]. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the [[dodecahedron]] and [[icosahedron]] inscribed in the same sphere is the same as the ratio of their volumes, the ratio being
<math display=block>\sqrt{\frac{10}{3(5-\sqrt{5})}} = \sqrt{\frac{5+\sqrt{5}}{6}}.</math>

The spurious Book XV was probably written, at least in part, by [[Isidore of Miletus]]. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.{{efn|{{harvnb|Boyer|1991|pp=118–119}} writes, "In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's ''Elements'' include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, <math>\sqrt{10/[3(5-\sqrt{5})]}</math>. It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. [...] The spurious Book XV, which is inferior, is thought to have been (at least in part) the work of Isidore of Miletus (fl. ca. A.D. 532), architect of the cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also deals with the regular solids, counting the number of edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge.}}

==Editions==
[[Image:Ricci Guangqi 2.jpg|thumb|The [[Italy|Italian]] [[Jesuit]] [[Matteo Ricci]] (left) and the Chinese mathematician [[Xu Guangqi]] (right) published the first [[Chinese language|Chinese]] edition of ''Euclid's Elements'' ({{transl|zh|Jīhé yuánběn}} {{lang|zh|幾何原本}}) in 1607.]]
* 4th century, [[Theon of Alexandria]], 888 AD manuscript extant.
* 9th century, Pre-Theon [[François Peyrard]] Vaticanus Graecus 190
* Many medieval editions, pre 1482
*1460s, [[Regiomontanus]] (incomplete)
*1482, [[Erhard Ratdolt]] (Venice), ''[[editio princeps]]'' (in Latin)<ref>{{harvnb|Alexanderson|Greenwalt|2012|loc= p. 163}}</ref><ref>{{Cite web |title=Editio Princeps of Euclid's Elements, the Most Famous Textbook Ever Published : History of Information |url=https://www.historyofinformation.com/detail.php?entryid=382 |access-date=2023-07-28 |website=www.historyofinformation.com}}</ref>
*1533, ''[[editio princeps]]'' of the Greek text by [[Simon Grynaeus|Simon Grynäus]]<ref>{{Cite web |title=The First Printed Edition of the Greek Text of Euclid is also the First Edition to Include the Diagrams within the Text : History of Information |url=https://historyofinformation.com/detail.php?entryid=2748 |access-date=2023-07-28 |website=historyofinformation.com}}</ref>
*1557, by Jean Magnien and {{Interlanguage link multi|Pierre de Montdoré|fr|3=Pierre de Montdoré}}, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)
*1572, [[Federico Commandino|Commandinus]] Latin edition
*1574, [[Christoph Clavius]]
* 1883–1888, [[Johan Ludvig Heiberg (historian)|Johan Ludvig Heiberg]]

===Translations===
====English====
#1570, [[Henry Billingsley]]
#1651, [[Thomas Rudd]]
#1660, [[Isaac Barrow]]
#1661, John Leeke and Geo. Serle
#1685, [[William Hallifax]]
#1705, [[Charles Scarborough]]
#1708, [[John Keill]]
#1714, W. Whiston
#1756, [[Robert Simson]]
#1781, 1788 James Williamson
#1781, William Austin
#1795, [[John Playfair]]
#1826, George Phillips
#1828, [[Dionysius Lardner]]
#1833, [[Thomas Perronet Thompson]]
#1862, [[Isaac Todhunter]]
#1908, [[Thomas Little Heath]] (revised in 1926) from [[Johan Ludvig Heiberg (historian)|Johan Ludvig Heiberg]]'s edition
#1939, [[R. Catesby Taliaferro]]

====Other languages====
*1505, {{Interlanguage link multi|Bartolomeo Zamberti|de|3=Bartolomeo Zamberti}} (Latin)
*1543, [[Nicolo Tartaglia]] (Italian)
*1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin)
*1558, [[Johann Scheubel]] (German)
*1562, Jacob Kündig (German)
*1562, Wilhelm Holtzmann (German)
*1564–1566, {{Interlanguage link multi|Pierre Forcadel|fr|3=Pierre Forcadel}} de Béziers (French)
*1572, [[Federico Commandino|Commandinus]] (Latin)
*1575, Commandinus (Italian)
*1576, [[Rodrigo de Zamorano]] (Spanish)
*1594, [[Typographia Medicea]] (edition of the Arabic translation of ''The Recension of Euclid's "Elements"''{{sfn|Nasir al-Din al-Tusi|1594}})
*1604, [[Jean Errard]] de Bar-le-Duc (French)
*1606, Jan Pieterszoon Dou (Dutch)
*1607, [[Matteo Ricci]], [[Xu Guangqi]] (Chinese)
*1613, [[Pietro Cataldi]] (Italian)
*1615, [[Denis Henrion]] (French)
*1617, Frans van Schooten (Dutch)
*1637, L. Carduchi (Spanish)
*1639, [[Pierre Hérigone]] (French)
*1651, Heinrich Hoffmann (German)
*1663, Domenico Magni (Italian from Latin)
*1672, [[Claude François Milliet Dechales]] (French)
*1680, Vitale Giordano (Italian)
*1689, Jacob Knesa (Spanish)
*1690, Vincenzo Viviani (Italian)
*1694, Ant. Ernst Burkh v. Pirckenstein (German)
*1695, Claes Jansz Vooght (Dutch)
*1697, [[Samuel Reyher]] (German)
*1702, Hendrik Coets (Dutch)
*1714, Chr. Schessler (German)
*1720s, [[Jagannatha Samrat]] (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi){{sfn|Sarma|1997|pp=460–461}}
*1731, Guido Grandi (abbreviation to Italian)
*1738, Ivan Satarov (Russian from French)
*1744, [[Mårten Strömer]] (Swedish)
*1749, Dechales (Italian)
*1749, Methodios Anthrakitis (Μεθόδιος Ανθρακίτης) (Greek)
*1745, Ernest Gottlieb Ziegenbalg (Danish)
*1752, Leonardo Ximenes (Italian)
*1763, Pibo Steenstra (Dutch)
*1768, Angelo Brunelli (Portuguese)
*1773, 1781, J. F. Lorenz (German)
*1780, [[Baruch Schick of Shklov]] (Hebrew)<ref>{{cite web|url=http://aleph500.huji.ac.il/nnl/dig/books/bk001139706.html|archive-url=https://archive.today/20090622080437/http://aleph500.huji.ac.il/nnl/dig/books/bk001139706.html|url-status=dead|title=JNUL Digitized Book Repository|date=22 June 2009|archive-date=22 June 2009|website=huji.ac.il|access-date=29 April 2018}}</ref>
*1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek)
*1803, H.C. Linderup (Danish)
*1804, [[François Peyrard]] (French). Peyrard discovered in 1808 the ''Vaticanus Graecus 190'', which enables him to provide a first definitive version in 1814–1818
*1807, Józef Czech (Polish based on Greek, Latin and English editions)
*1807, J. K. F. Hauff (German)
*1818, [[Vincenzo Flauti]] (Italian)
*1820, Benjamin of Lesbos (Modern Greek)
*1828, Joh. Josh and Ign. Hoffmann (German)
*1833, E. S. Unger (German)
*1836, H. Falk (Swedish)
*1844, 1845, 1859, P. R. Bråkenhjelm (Swedish)
*1850, F. A. A. Lundgren (Swedish)
*1850, H. A. Witt and M. E. Areskong (Swedish)
*1865, [[Sámuel Brassai]] (Hungarian)
*1873, Masakuni Yamada (Japanese)
*1880, [[Mykhailo Vaschenko-Zakharchenko|Vachtchenko-Zakhartchenko]] (Russian)
*1897, Thyra Eibe (Danish)
*1901, Max Simon (German)
*1907, František Servít (Czech){{sfn|Servít|1907}}
*1953, 1958, 1975, Evangelos Stamatis (Ευάγγελος Σταμάτης) (Modern Greek) 
*1999, Maja Hudoletnjak Grgić (Book I-VI) (Croatian){{sfn|Euklid|1999}}
*2009, Irineu Bicudo ([[Portuguese language|Portuguese]])
*2019, Ali Sinan Sertöz (Turkish){{sfn|Sertöz|2019|p=}}
*2022, Ján Čižmár (Slovak)

=== Book I Editions ===

*1886, Euclid Book I Hall & Stevens (English)
*1891,1896, The Harpur Euclid by Edward Langley and Seys Phillips (English)
*1949, Henry Regnery Company (English)

===Selected editions currently in print===
*''Euclid's Elements – All thirteen books complete in one volume'', Based on Heath's translation, edited by Dana Densmore, et al. Green Lion Press {{isbn|1-888009-18-7}}.
*''The Elements: Books I–XIII – Complete and Unabridged'' (2006), Translated by Sir Thomas Heath, Barnes & Noble {{isbn|0-7607-6312-7}}.
*''The Thirteen Books of Euclid's Elements'', translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. {{isbn|0-486-60088-2}} (vol. 1), {{isbn|0-486-60089-0}} (vol. 2), {{isbn|0-486-60090-4}} (vol. 3)
*''Plane Geometry (Euclid's elements Redux) Books I–VI'', based on John Casey's translation, edited by Daniel Callahan, {{isbn|978-1977730039}}

=== Selected editions based on Oliver Byrne's edition ===

* ''The first six books of the Elements of Euclid'', edited by Werner Oechslin, Taschen, 2010, {{isbn| 3836517752}}, a facsimile of Byrne (1847).
*''Oliver Byrne's Elements of Euclid'', Art Meets Science, 2022, {{isbn|978-1528770439}}, a facsimile of Byrne (1847).
* [https://www.kroneckerwallis.com/product/euclids-elements-completing-oliver-byrnes-work/ ''Euclid’s Elements: Completing Oliver Byrne's work''], Kronecker Wallis, 2019, a modern redrawing extended to the rest of the ''Elements'', originally launched on [[Kickstarter]].

=== Free versions ===
*''Euclid's Elements Redux, Volume 1'', contains books I–III, based on John Casey's translation.{{sfn|Callahan |Casey|2015}}
*''Euclid's Elements Redux, Volume 2'', contains books IV–VIII, based on John Casey's translation.{{sfn|Callahan |Casey|2015}}

==See also==
* [[Bride's Chair]]

==References==
=== Notes ===
{{Notelist}}

===Citations===
{{Reflist|30em}}

===Sources===
{{Refbegin|2|indent=yes}}
*{{citation|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}}
*[[Benno Artmann|Artmann, Benno]]: ''Euclid – The Creation of Mathematics.'' New York, Berlin, Heidelberg: Springer 1999, {{ISBN|0-387-98423-2}}
*{{cite book|last=Ball|first=Walter William Rouse|author-link = W. W. Rouse Ball |title=A Short Account of the History of Mathematics|url=https://archive.org/details/shortaccountofhi00ballrich/ |edition=6th |year=1915 |orig-year=1st ed. 1888 |publisher=MacMillan }}
*{{cite book | first=Carl B. | last=Boyer | author-link=Carl Benjamin Boyer | title=A History of Mathematics| url=https://archive.org/details/historyofmathema00boye | url-access=registration |chapter=Euclid of Alexandria
 | edition=Second | publisher=John Wiley & Sons | year=1991 | isbn=0-471-54397-7}}
*{{cite book|title=The Historical Roots of Elementary Mathematics|first1= Lucas Nicolaas Hendrik |last1=Bunt|first2= Phillip S. |last2=Jones|first3= Jack D. |last3=Bedient |date=1988|publisher= Dover }}
*{{cite book|last=Busard|first=H.L.L.|title=Campanus of Novara and Euclid's Elements|url=https://books.google.com/books?id=tiHUN6jjm6MC&pg=PA1|year=2005|publisher=Franz Steiner Verlag|location=Stuttgart|isbn=978-3-515-08645-5|chapter=Introduction to the Text}}
*{{Cite book|title=Euclid's "Elements" Redux |date=2015 |last1=Callahan |first1=Daniel |last2=Casey |first2=John |url= http://gutenberg.org/ebooks/21076 }}
*{{cite book|last1=Dodgson|first1=Charles L. |author-link1=Lewis Carroll|first2=Amit |last2=Hagar|title=Euclid and His Modern Rivals|url=https://books.google.com/books?id=6Uk42tX9P10C|year=2009|publisher=Cambridge University Press|isbn=978-1-108-00100-7|chapter=Introduction}}
*{{cite book
 |last1=Hartshorne
 |first1=Robin
 |title=Geometry: Euclid and Beyond
 |date=2000
 |publisher=[[Springer Publishing|Springer]]
 |location=[[New York City|New York, NY]]
 |isbn=9780387986500
 |edition=2nd
 }}
*{{cite book
 | last = Heath
 | first = Thomas L.
 | author-link = T. L. Heath
 | title = The Thirteen Books of Euclid's Elements
 | edition = 2nd 
 | year = 1956a|volume=1. Books I and II
 | publisher = Dover Publications
 | location = New York|ol=22193354M
 }}
*{{cite book
 | last = Heath
 | first = Thomas L.
 | author-link = T. L. Heath
 | title = The Thirteen Books of Euclid's Elements
 | edition = 2nd 
 | year = 1956b|volume=2. Books III to IX
 | publisher = Dover Publications
 | location = New York|ol=7650092M
 }}
*{{cite book
 | last = Heath
 | first = Thomas L.
 | author-link = T. L. Heath
 | title = The Thirteen Books of Euclid's Elements
 | edition = 2nd 
 | year = 1956c|volume=3. Books X to XIII and Appendix
 | publisher = Dover Publications
 | location = New York|oclc=929205858
 }} Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.
* {{cite book| last = Heath
 | first = Thomas L.
 | author-link = T. L. Heath
 | title=A Manual of Greek Mathematics|url=https://books.google.com/books?id=_HZNr_mGFzQC|year=1963|publisher=Dover Publications|isbn=978-0-486-43231-1}}
*{{cite book|last=Ketcham|first=Henry |title=The Life of Abraham Lincoln|url=https://archive.org/details/lifeofabrahaml1154ketc/page/n6|year=1901|publisher=Perkins Book Company|location=New York}}
*{{cite book|author-link=Nasir al-Din al-Tusi|author=Nasir al-Din al-Tusi|url=https://www.wdl.org/en/item/10666/ |trans-title=The Recension of Euclid's "Elements"|title=Kitāb taḥrīr uṣūl li-Uqlīdus|language=ar|date=1594}}
*{{cite book|last1=Reynolds|first1=Leighton Durham |last2=Wilson|first2=Nigel Guy |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=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| year=1997 | title = Encyclopaedia of the history of science, technology, and medicine in non-western cultures | author-link=K. V. Sarma|first=K.V.|last= Sarma | editor-first=Helaine | editor-last=Selin | editor-link=Helaine Selin | publisher=Springer | isbn=978-0-7923-4066-9| url=https://books.google.com/books?id=yoiXDTXSHi4C&q=rekhaganita&pg=PA460}}
*{{cite book|url=https://commons.wikimedia.org/wiki/File:Eukleides_Servit.pdf |title=Eukleidovy Zaklady (Elementa)|language=cs|trans-title=Euclid's Elements| date=1907|first=František |last=Servít}}
*{{cite book|last=Sertöz|first=Ali Sinan |title=Öklidin Elemanlari: Ciltli|url=https://books.google.com/books?id=X_eIxgEACAAJ|year=2019|publisher=Tübitak|isbn=978-605-312-329-3|language=tr|trans-title=Euclid's Elements}}
*{{cite journal|last=Toussaint|first=Godfried|author-link =Godfried Toussaint|title=A new look at euclid's second proposition|journal=The Mathematical Intelligencer|volume=15|issue=3|year=1993|pages=12–24|issn=0343-6993|doi=10.1007/BF03024252|s2cid=26811463}}
*{{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|title=Encyclopedia of Ancient Greece |date=2006|first= Nigel Guy|last= Wilson|publisher= Routledge}} 
*{{cite book|title=Elementi I-VI|date=1999|last=Euklid|publisher=KruZak|isbn=953-96477-6-2|translator-last=Hudoletnjak Grgić|translator-first=Maja}}
{{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
 | last = Euclid | title = Elements | orig-year = c. 300 BC
 | url = http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
 | access-date = 2006-08-30 | editor = David E. Joyce | editor-link = David E. Joyce (mathematician) |year =1997
 }} In HTML with Java-based interactive figures.
*[http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf Richard Fitzpatrick's bilingual edition] (freely downloadable PDF, typeset in a two-column format with the original Greek beside a modern English translation; also available in print as {{isbn|979-8589564587}})
* [<!-- http://old.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+1 -->https://www.perseus.tufts.edu/hopper/text?doc=Euc.+1 Heath's English translation] (HTML, without the figures, public domain) (accessed February 4, 2010)
** Heath's English translation and commentary, with the figures (Google Books): [https://books.google.com/books?id=UhgPAAAAIAAJ vol. 1], [https://books.google.com/books?id=lxkPAAAAIAAJ vol. 2], [https://books.google.com/books?id=xhkPAAAAIAAJ vol. 3], [https://books.google.com/books?id=KHMDAAAAYAAJ vol. 3 c. 2]
* [http://www.math.ubc.ca/~cass/Euclid/byrne.html Oliver Byrne's 1847 edition] (also hosted at [https://archive.org/details/firstsixbooksofe00eucl archive.org])– an unusual version by [[Oliver Byrne (mathematician)|Oliver Byrne]] who used color rather than labels such as ABC (scanned page images, public domain)
* [https://www.c82.net/euclid/#books Web adapted version of Byrne’s Euclid] designed by Nicholas Rougeux 
*[https://www.youtube.com/c/SandyBultena/playlists Video adaptation], animated and explained by Sandy Bultena, contains books I-VII. 
* [http://gutenberg.org/ebooks/21076 The First Six Books of the ''Elements''] by John Casey and Euclid scanned by [[Project Gutenberg]].
* [<!-- http://www.du.edu/~etuttle/classics/nugreek/contents.htm -->http://mysite.du.edu/~etuttle/classics/nugreek/contents.htm Reading Euclid] – a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
*[[Thomas More|Sir Thomas More]]'s [https://web.archive.org/web/20040807095930/http://www.columbia.edu/acis/textarchive/rare/24.html manuscript]
*[https://web.archive.org/web/20040807101503/http://www.columbia.edu/acis/textarchive/rare/6.html Latin translation] by [[Aethelhard of Bath]]
*[<!-- http://www.physics.ntua.gr/Faculty/mourmouras/euclid/index.html -->http://www.physics.ntua.gr/~mourmouras/euclid/index.html Euclid ''Elements'' – The original Greek text] Greek HTML
*[[Clay Mathematics Institute]] Historical Archive – [http://www.claymath.org/euclids-elements The thirteen books of Euclid's ''Elements''] copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD
*[http://pds.lib.harvard.edu/pds/view/13079270 Kitāb Taḥrīr uṣūl li-Ūqlīdis] Arabic translation of the thirteen books of Euclid's ''Elements'' by Nasīr al-Dīn al-Ṭūsī. Published by Medici Oriental Press(also, Typographia Medicea). Facsimile hosted by [http://ocp.hul.harvard.edu/ihp/ Islamic Heritage Project].
*[https://archive.org/details/euclid-elements-redux_201809 Euclid's ''Elements'' Redux], an open textbook based on the ''Elements''
*[https://archive.org/search.php?query=%E5%B9%BE%E4%BD%95%E5%8E%9F%E6%9C%AC 1607 Chinese translations] reprinted as part of the ''[[Complete Library of the Four Treasuries]]'', or ''Siku Quanshu''.

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

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