Euclid's Elements

Elements

Papyrus Oxyrhynchus 29, a fragment of Euclid's Elements dated to c. 3rd-4th century CE. Found at Oxyrhynchus, the diagram accompanies Book II, Proposition 5.
Author	Euclid
Language	Ancient Greek
Subject	Euclidean geometry, number theory, incommensurability
Genre	Mathematics
Publication date	c. 300 BC
Pages	13 books

The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.

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, the Elements is a collection in 13 books of definitions, postulates, geometric constructions, and theorems with their proofs that covers plane and solid Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the construction of regular polygons and polyhedra.

Often referred to as the most successful textbook ever written, the Elements has continued to be used for introductory geometry from the time it was written up through the present day. 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 surpassed until the 19th century.

Euclid's Elements is the oldest extant large-scale deductive treatment of mathematics.^[1] 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' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".^[a] Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians,^[2] including Eudoxus, Hippocrates of Chios, Thales, and Theaetetus, while other theorems are mentioned by Plato and Aristotle.^[3] 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to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics.^[4] The Elements version available today also includes "post-Euclidean" mathematics, probably added later by later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.^[3] 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".^[5][3] Despite this, Sialaros furthers that "the remarkably tight structure of the Elements reveals authorial control beyond the limits of a mere editor".^[6]

Pythagoras (c. 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios (c. 470–410 BC, not to be confused with the contemporaneous physician Hippocrates of Kos) for book III, and Eudoxus of Cnidus (c. 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.^[7] 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.^[8]

Other similar works are also reported to have been written by Hippocrates of Chios, Theudius of Magnesia, and Leon, but are now lost.^[9][10]

The Elements does not exclusively discuss geometry as is sometimes believed.^[4][11] 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 incommensurability) do not exactly fit this scheme.^[12][13] The heart of the text is the theorems scattered throughout.^[14] Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles".^[15] The first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια).^[15][16] The postulates (that is, axioms) and common notions occur only in book I.^[4] Close study of Proclus suggests that older versions of the Elements may have followed the same distinctions but with different terminology, instead calling the definitions hypotheses and the common notions axioms.^[16] The second group consists of propositions, presented alongside mathematical proofs and diagrams.^[15] It is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit.^[6] As a whole, the authorial voice remains general and impersonal.^[3]

Book I of the Elements is foundational for the entire text.^[4] It begins with a series of 20 definitions for basic geometric concepts such as lines, angles and various regular polygons.^[18] Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.^[19] These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system.^[20] The common notions exclusively concern the comparison of magnitudes.^[21] While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate. 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.^[21]

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 lines (27-34), the area of triangles and parallelograms (35–45), and the Pythagorean theorem and its converse (46–48).^[21]

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.^[22] Papyrus Oxyrhynchus 29, a 3rd-century CE papyrus, contains fragments of propositions 8-11 and 14-25.^[b] The last two propositions of Book I comprise the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".^[15] 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.^[23]

The second book focuses on area, measured through 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 rectangles.^[21] Book II is traditionally understood as concerning "geometric algebra", a formulation of algebra in geometric terminology,^[24] centered on the quadratic case of the binomial theorem.^[21] This interpretation has been heavily debated since the 1970s;^[24] critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.^[15] 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.^[25]

Book III begins with a list of 11 definitions, and follows with 37 propositions that deal with circles and their properties. Proposition 1 is on finding the center of a circle. Propositions 2 through 15 concern chords, and intersecting and tangent circles. Tangent lines to circles are the subjects of propositions 16 through 19. Next are propositions on inscribed angles (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.^[26]

Book IV treats four problems systematically for different polygons: inscribing a polygon within a circle, circumscribing a polygon about a circle, inscribing a circle within a polygon, and circumscribing a circle about a polygon.^[27] These problems are solved in sequence for triangles, as well as constructible regular polygons with 4, 5, 6, and 15 sides.^[4]

Book V, which is independent of the previous four books, concerns ratios of magnitudes and the comparison of ratios.^[28] 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 taking a ratio as a binary operation from numbers to numbers.^[29]

Much of Book V was probably ascertained from earlier mathematicians, perhaps Eudoxus, ^[15] although certain propositions, such as V.16, dealing with "alternation" (if a : b :: c : d, then a : c :: b : d) likely predate Eudoxus.^[30]

Book VI utilizes the theory of proportions from Book V in the context of plane geometry,^[4] especially the construction and recognition of similar figures. It is built almost entirely of its first proposition:^[31] "Triangles and parallelograms which are under the same height are to one another as their bases".^[32]

Number theory is covered by books VII to X, the former beginning with a set of 22 definitions for parity, prime numbers, and other arithmetic-related concepts.^[4]

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 multiples (33-39).^[33]

The first part of Book VIII (propositions 1 through 10) deals with the construction and existence of geometric sequences of integers in general, and the divisibility of members of a geometric sequence by each other. Propositions 11 to 27 deal with square numbers and cube numbers in geometric progressions, and the relation between these special progressions and the elements two or three steps apart in an arbitrary geometric progression.^[33]

After continuing the investigations of Book VIII on squares and cubes in geometric progressions,^[33] 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 numbers (proposition 36).^[4][34] 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.^[35]

Of the Elements, book X is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.^[15] Book X proves the irrationality of the square roots of non-square integers such as ${\displaystyle {\sqrt {2}}}$ and classifies the square roots of incommensurable lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. He also gives a formula to produce Pythagorean triples.^[36]

The final three books primarily discuss solid geometry.^[12] By introducing a list of 37 definitions, Book XI contextualizes the next two.^[37] Although its foundational character resembles Book I, unlike the latter it features no axiomatic system or postulates.^[37]

Book XI generalizes the results of book VI to solid figures: perpendicularity, parallelism, volumes, and similarity of parallelepipeds. The three sections of Book XI include content on: solid geometry (1-19), solid angles (20-23), and parallelepipeds (24-37).^[37]

Book XII studies the volumes of cones, pyramids, and cylinders in detail by using the method of exhaustion,^[37] a precursor to 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 XIII constructs the five Platonic solids (regular polyhedra) inscribed in a sphere, compares the ratios of their edges to the radius of the sphere,^[38] and concludes the Elements by proving that these are the only regular polyhedra.^[39]

Two additional books, that were not written by Euclid, Books XIV and XV, have been transmitted in the manuscripts of the Elements:^[40]

• 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^[40]$${\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.}$$
• 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), and computes the dihedral angles between their faces.^[40]

The practice of adding to the works of famous authors, exemplified by these books, was not unusual in ancient Greek mathematics.^[40]

Euclid's axiomatic approach and constructive methods were widely influential.^[42]

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. 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.^[43]

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.^[44]

No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach the types of problems encountered in the first four books of the Elements.^[45] As was common in ancient mathematical texts, when a proposition needed 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 often interpolated their own proofs of these cases.^[46]

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,^[47] 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 Alexandrian system of numerals.^[48]

Euclid's Elements has been referred to as the most successful textbook ever written.^[49] The Elements is often considered after the Bible as the most frequently translated, published, and studied book in history.^[50] With Aristotle's 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.^[50] In historical context, it has proven enormously influential in many areas of science. It was one of the very earliest mathematical works to be printed after the 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,^[51] the number reaching well over one thousand.^[52]

The oldest extant evidence for Euclid's Elements are a set of six ostraca 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.^[53] A papyrus recovered from Herculaneum^[54] contains an essay by the Epicurean philosopher Demetrius Lacon on Euclid's Elements.^[53] 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 c. 75–125 AD.^[55]

Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford.^[c][d] The manuscripts available are of variable quality, and often incomplete.^[56] By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text.^[57] 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.^[58]

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 of a manuscript not derived from Theon's.^[59] This manuscript, MS. Vat.gr.190,^[c] 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 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."^[59]

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.^[60]

The Arabs received the Elements from the Byzantines around 760; this version was translated into Arabic under Harun al-Rashid (c. 800),^[60] 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. Although most Arabic manuscripts have been attributed to one or another of these translations, some mix material from both, and this mixture was also passed down into medieval translations into Hebrew from the Arabic.^[61]

The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century.^[62] 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.^[63] 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.^[64]

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), Johannes de Tinemue,^[65] possibly also known as 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.^[66] 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.^[67][e]

The first printed edition appeared in 1482 (based on Campanus's translation),^[68] 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 translated into Latin in an edition published in Venice in 1505 by Bartolomeo Zamberti.^[69] The Greek text itself was published in 1533.^[70] The first to translate the Elements into a modern European language was Nicolo Tartaglia, who published an Italian edition in 1543.^[71]

In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley.^[72][73][74] In 1607, The Italian Jesuit Matteo Ricci and the Chinese mathematician Xu Guangqi published the first Chinese edition of Euclid's Elements.^[75]

The Elements is still considered a masterpiece in the application of logic to mathematics. The mathematician Oliver Byrne published a well-known version of the Elements in 1847 entitled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, which included colored diagrams intended to increase its pedagogical effect.^[76] David Hilbert authored a modern axiomatization of the Elements.^[77]

The geometrical system established by the Elements long dominated the field; however, today that system is often referred to as 'Euclidean geometry' to distinguish it from other non-Euclidean geometries discovered in the early 19th century.^[50]

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

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than 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.^[78]

Some of the foundational theorems are proved using 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 with centers at the distance of their radius will intersect in two points.^[79][80] Although this construction depends only on topological properties of its diagram, which remain evident even if the diagram is drawn inaccurately,^[81] other constructions from Euclid can fail in degenerate cases. 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.^[82]

Later editors of the Elements have included these implicit axiomatic assumptions, such as Pasch's axiom,^[83][84] in their editions' lists of formal axioms.^[85] Early attempts to construct a more complete set of axioms include Hilbert's geometry axioms^[86][87] and Tarski's.^[83][88] In 2017, Michael Beeson et al. used computer proof assistants 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, is a first-order theory, 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 numbers) that all of their axioms are valid for the points of the Cartesian coordinate system.^[82]

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.^[89]

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."^[47]

Many editions of Euclid's Elements have been published in Greek, Latin, English, and other languages. They include:

• Preclarissimus liber elementorum Euclidis perspicacissimi in artem geometriam incipit quam foelicissime. Venice: Erhard Ratdolt. 1482. The editio princeps (in Latin), based on the 13th century translation and commentary of Campanus.^[90]
• Zamberti, Bartolomeo, ed. (1505). Euclidis megarẽsis philosophi platonici. Venice. First full-text Latin translation directly from the Greek.^[69] 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.^[91]
• Lefèvre d'Étaples, Jacques, ed. (1516). Euclidis Megarensis Geometricorum elementorum liber XV. Paris: Henri Estienne. Based on Campanus and Zamberti, but without some of Zamberti's commentary. The first edition published in France.^[92]
• Grynaeus, Simon, ed. (1533). Ευκλείδου Στοιχεῖον. Basel: Johann Herwagen. Editio princeps of the Greek text. Based on two Greek manuscripts, Paris gr. 2343 and Venetus Marcianus 301,^[93] both "very inferior ones".^[69]
• Tartalea, Nicolo, ed. (1543). Euclide megarense philosopho solo introduttore delle scientie mathematice. Venice: 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.^[71]
• Magnien, Jean, ed. (1557). Euclidis Elementorum libri XV grœce et latine. Paris: Cavellat. Greek with Latin translation. Revised posthumously by Stephanus Gracilis; incorporates a translation of Book X by Pierre de Montdoré. Most proofs omitted.^[94]
• Billingsley, H., ed. (1570). The Elements of Geometrie. Preface by John Dee. London: John Daye. The first published English-language edition.^[74]
• Commandino, Federico, ed. (1572). Euclidis Elementorum Libri XV. Pisauri [Pesaro, Italy]: Apud Camillum Francischinum. 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.^[95]
• Clavius, Christopher, ed. (1574). Euclidis elementorum libri XV. Rome: 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.^[92]
• Nasir al-Din al-Tusi (1594). Kitāb taḥrīr uṣūl li-Uqlīdus [The Recension of Euclid's "Elements"] (in Arabic). Rome: Typographia Medicea. Based on Laurentian Library MS. Or. 20, a copy of MS. Or. 50.^[96]
• Ricci, Matteo; Xu, Guangqi, eds. (1607). 幾何原本 Jī hé yuán běn [Source of quantity] (in Chinese). Beijing. Translated from the Latin edition of Clavius, but including only books I-VI.^[75] This translation, together with a later translation of the remaining books, can be found on s:zh:幾何原本.
• Briggs, Henry, ed. (1620). Eukleidou Stoicheiōn biblia 13 / Elementorum Euclidis libri tredecim. London: 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.^[74]
• Barrow, Isaac, ed. (1659). Euclidis Elementorum [in Latin]. Euclide's Elements (1660) [in English]. Many later editions.^[74]
• Simson, Robert, ed. (1806). The elements of Euclid, viz. the first six books, together with the eleventh and twelfth. London: F. Wingrave.^[97] Revised in 1862 by Isaac Todhunter and republished in 1933 by E. P. Dutton, Everyman's Library 891.^[98]
• Byrne, Oliver (1847). 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. London: William Pickering.^[76][99] Facsimile edited by Oechslin, Werner (2010). Taschen. ISBN 3836517752.^[99] Euclid’s Elements: Completing Oliver Byrne's work, a modern redrawing extended to the rest of the Elements, was published by Kronecker Wallis, 2019.^[100]
• Casey, John, ed. (1882). The First Six Books of the Elements of Euclid with Copious Annotations and Numerous Exercises. Dublin: Hodges, Figgis, & Co.^[101] Casey produced many subsequent editions; 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.^[102] Casey also wrote A Sequel to the First Six Books of the Elements of Euclid (Dublin: Hodges, Figgis, & Co., 1881).
• Heiberg, Johan Ludvig, ed. (1883–1888) Euclidis Opera omnia [Euclid's complete works, in Greek]. Leibzig: Teubner. Volumes 1–5 comprise the Elements. Vol. I, Vol. II, Vol. III, Vol. IV, Vol. 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.^[59]
• Heath, Thomas, ed. (1908). The Thirteen Books of Euclid's Elements. Cambridge University Press. 2nd ed., 1926. In three volumes: Vol. I, Vol. II, Vol. III. Reprints include Dover, 1956; Green Lion Press, 2002, ISBN 1-888009-18-7 (single volume, without Heath's commentary);^[103] Barnes & Noble, 2006, ISBN 0-7607-6312-7 (single volume).

Wikiquote has quotations related to Euclid's Elements.

English Wikisource has original text related to this article:

The Elements of Euclid

Wikimedia Commons has media related to Elements of Euclid.

• Elements with highlights by ratherthanpaper
• Multilingual edition of Elementa in the Bibliotheca Polyglotta
• David E. Joyce, ed. (1997). "Euclid's Elements". In HTML with Java-based interactive figures.