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The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical
treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject and its conclusions." Tr ...
consisting of 13 books attributed to the ancient
Greek mathematician Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
in
Alexandria Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
, Ptolemaic Egypt 300 BC. It is a collection of definitions,
postulates An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
,
propositions In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
(
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...
s and constructions), and
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
s of the propositions. The books cover plane and solid
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, elementary number theory, and incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and modern
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textbook ...
ever written. 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 The Bible (from Koine Greek , , 'the books') is a collection of religious texts or scriptures that are held to be sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of texts of ...
in the number of editions published since the first printing in 1482, the number reaching well over one thousand. For centuries, when the
quadrivium From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts— arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in t ...
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.
Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the
Victorian period In the history of the United Kingdom and the British Empire, the Victorian era was the period of Queen Victoria's reign, from 20 June 1837 until her death on 22 January 1901. The era followed the Georgian period and preceded the Edward ...
it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. The standard textbook for this purpose was none other than Euclid's ''The Elements''.

# History

## Basis in earlier work

Scholars believe that the ''Elements'' is largely a compilation of propositions based on books by earlier Greek mathematicians. 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".
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politica ...
( 570–495 BC) was probably the source for most of books I and II,
Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
( 470–410 BC, not the better known Hippocrates of Kos) for book III, and
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...
( 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. 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. Other similar works are also reported to have been written by Theudius of Magnesia, Leon, and Hermotimus of Colophon.

## Transmission of the text

Theon of Alexandria Theon of Alexandria (; grc, Θέων ὁ Ἀλεξανδρεύς;  335 – c. 405) was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's '' Elements'' and wrote commentaries on wor ...
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 Vatican may refer to: Vatican City, the city-state ruled by the pope in Rome, including St. Peter's Basilica, Sistine Chapel, Vatican Museum The Holy See * The Holy See, the governing body of the Catholic Church and sovereign entity recognized ...
of a manuscript not derived from Theon's. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. 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 Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the estab ...
, for instance, no record exists of the text having been translated into Latin prior to
Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tran ...
in the fifth or sixth century. The Arabs received the ''Elements'' from the Byzantines around 760; this version was translated into
Arabic Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter ...
under Harun al Rashid ( 800). The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. 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. 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. The first printed edition appeared in 1482 (based on Campanus of Novara's 1260 edition), and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered in 1533. 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. Copies of the Greek text still exist, some of which can be found in the
Vatican Library The Vatican Apostolic Library ( la, Bibliotheca Apostolica Vaticana, it, Biblioteca Apostolica Vaticana), more commonly known as the Vatican Library or informally as the Vat, is the library of the Holy See, located in Vatican City. Formally es ...
and the
Bodleian Library The Bodleian Library () is the main research library of the University of Oxford, and is one of the oldest libraries in Europe. It derives its name from its founder, Sir Thomas Bodley. With over 13 million printed items, it is the secon ...
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 J. L. Heiberg and Sir Thomas Little Heath in their editions of the text. Also of importance are the
scholia Scholia (singular scholium or scholion, from grc, σχόλιον, "comment, interpretation") are grammatical, critical, or explanatory comments – original or copied from prior commentaries – which are inserted in the margin of th ...
, 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.

# Influence

The ''Elements'' is still considered a masterpiece in the application of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
to mathematics. In historical context, it has proven enormously influential in many areas of
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
. Scientists
Nicolaus Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulated ...
,
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
,
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
and Sir Isaac Newton were all influenced by the ''Elements'', and applied their knowledge of it to their work. Mathematicians and philosophers, such as
Thomas Hobbes Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book ''Leviathan'', in which he expounds an influent ...
,
Baruch Spinoza Baruch (de) Spinoza (born Bento de Espinosa; later as an author and a correspondent ''Benedictus de Spinoza'', anglicized to ''Benedict de Spinoza''; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Portuguese-Jewish origin, b ...
,
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applic ...
, and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
, 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 Abraham Lincoln ( ; February 12, 1809 – April 15, 1865) was an American lawyer, politician, and statesman who served as the 16th president of the United States from 1861 until his assassination in 1865. Lincoln led the nation throu ...
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, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight". 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". 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 geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
. In Book I, Euclid lists five postulates, the fifth of which stipulates
''If a line segment intersects two straight
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
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 Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, k ...
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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
.

# Contents

* Book 1 contains 5 postulates (including the infamous
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
) and 5 common notions, and covers important topics of plane geometry such as the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures. * Book 2 contains a number of
lemmas Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), ...
concerning the equality of rectangles and squares, sometimes referred to as " geometric algebra", and concludes with a construction of the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
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, inscribed angles,
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s, the power of a point,
Thales' theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and prove ...
. * Book 4 constructs the
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incente ...
and
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
of a triangle, as well as
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s with 4, 5, 6, and 15 sides. * Book 5, on proportions of magnitudes, gives the highly sophisticated theory of proportion probably developed by 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 similar figures. * Book 7 deals with elementary number theory:
divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
, prime numbers and their relation to
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, o ...
s, Euclid's algorithm for finding the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
, finding the least common multiple. * Book 8 deals with the construction and existence of
geometric sequences In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
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 numbers. * Book 10 proves the irrationality of the square roots of non-square integers (e.g. $\sqrt$) 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 In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a '' chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
to produce Pythagorean triples. * Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of parallelepipeds. * Book 12 studies the volumes of
cones A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines conn ...
,
pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
, and
cylinders A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...
in detail by using the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...
, a precursor to
integration Integration may refer to: Biology *Multisensory integration * Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technolog ...
, 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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
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 solids inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.

# Euclid's method and style of presentation

Euclid's axiomatic approach and 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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
. 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. 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. 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, 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. 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. 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''. 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.

# Criticism

Euclid's list of axioms in the ''Elements'' was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms. 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. Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.2 and I.3 can be proved trivially by using superposition. Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years he ''Elements''was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."

# 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 Apocrypha are works, usually written, of unknown authorship or of doubtful origin. The word ''apocryphal'' (ἀπόκρυφος) was first applied to writings which were kept secret because they were the vehicles of esoteric knowledge considered ...
books XIV and XV of the ''Elements'' were sometimes included in the collection. The spurious Book XIV was probably written by
Hypsicles Hypsicles ( grc-gre, Ὑψικλῆς; c. 190 – c. 120 BCE) was an ancient Greek mathematician and astronomer known for authoring ''On Ascensions'' (Ἀναφορικός) and the Book XIV of Euclid's ''Elements''. Hypsicles lived in Alexandria. ...
on the basis of a treatise by 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 $\sqrt = \sqrt.$ 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.

# Editions

*1460s, Regiomontanus (incomplete) *1482,
Erhard Ratdolt Erhard Ratdolt (1442–1528) was an early German printer from Augsburg. He was active as a printer in Venice from 1476 to 1486, and afterwards in Augsburg. From 1475 to 1478 he was in partnership with two other German printers. The first book ...
(Venice), first printed edition *1533, '' editio princeps'' by Simon Grynäus *1557, by Jean Magnien and , reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation) *1572, Commandinus Latin edition *1574, Christoph Clavius

## Translations

*1505, (Latin) *1543, Niccolò 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, de Béziers (French) *1570, Henry Billingsley (English) *1572, 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"'' *1604, de Bar-le-Duc (French) *1606, Jan Pieterszoon Dou (Dutch) *1607, Matteo Ricci, Xu Guangqi (Chinese) *1613,
Pietro Cataldi Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of contin ...
(Italian) *1615, Denis Henrion (French) *1617, Frans van Schooten (Dutch) *1637, L. Carduchi (Spanish) *1639, Pierre Hérigone (French) *1651, Heinrich Hoffmann (German) *1651,
Thomas Rudd Thomas Rudd (1583?–1656) was an English military engineer and mathematician. Life The eldest son of Thomas Rudd of Higham Ferrers, Northamptonshire, he was born in 1583 or 1584. He served during his earlier years as a military engineer in ...
(English) *1660, Isaac Barrow (English) *1661, John Leeke and Geo. Serle (English) *1663, Domenico Magni (Italian from Latin) *1672, Claude François Milliet Dechales (French) *1680, Vitale Giordano (Italian) *1685, William Halifax (English) *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) *1705, Charles Scarborough (English) *1708, John Keill (English) *1714, Chr. Schessler (German) *1714, W. Whiston (English) *1720s, Jagannatha Samrat (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi) *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) *1756, Robert Simson (English) *1763, Pibo Steenstra (Dutch) *1768, Angelo Brunelli (Portuguese) *1773, 1781, J. F. Lorenz (German) *1780, Baruch Schick of Shklov (Hebrew) *1781, 1788 James Williamson (English) *1781, William Austin (English) *1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek) *1795,
John Playfair John Playfair FRSE, FRS (10 March 1748 – 20 July 1819) was a Church of Scotland minister, remembered as a scientist and mathematician, and a professor of natural philosophy at the University of Edinburgh. He is best known for his book ''Illu ...
(English) *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) *1826, George Phillips (English) *1828, Joh. Josh and Ign. Hoffmann (German) *1828, Dionysius Lardner (English) *1833, E. S. Unger (German) *1833, Thomas Perronet Thompson (English) *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) *1862, Isaac Todhunter (English) *1865, Sámuel Brassai (Hungarian) *1873, Masakuni Yamada (Japanese) *1880, Vachtchenko-Zakhartchenko (Russian) *1897, Thyra Eibe (Danish) *1901, Max Simon (German) *1907, František Servít (Czech) *1908, Thomas Little Heath (English) *1939, R. Catesby Taliaferro (English) *1953, 1958, 1975, Evangelos Stamatis (Ευάγγελος Σταµάτης) (Modern Greek) *1999, Maja Hudoletnjak Grgić (Book I-VI) (Croatian) *2009, Irineu Bicudo ( Portuguese) *2019, Ali Sinan Sertöz (Turkish) *2022, Ján Čižmár (Slovak)

## Currently in print

*''Euclid's Elements – All thirteen books complete in one volume'', Based on Heath's translation, Green Lion Press . *''The Elements: Books I–XIII – Complete and Unabridged,'' (2006) Translated by Sir Thomas Heath, Barnes & Noble . *''The Thirteen Books of Euclid's Elements'', translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. (vol. 1), (vol. 2), (vol. 3)

## Free versions

*''Euclid's Elements Redux, Volume 1'', contains books I–III, based on John Casey's translation. *''Euclid's Elements Redux, Volume 2'', contains books IV–VIII, based on John Casey's translation.

# References

## Sources

* * Artmann, Benno: ''Euclid – The Creation of Mathematics.'' New York, Berlin, Heidelberg: Springer 1999, * * * * * * * * * * Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text. * * * * * * * * * * * *

Clark University Euclid's elementsMultilingual edition of ''Elementa'' in the Bibliotheca Polyglotta
* In HTML with Java-based interactive figures.
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 )
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)
vol. 1vol. 2vol. 3vol. 3 c. 2

(also hosted a
archive.org
– an unusual version by Oliver Byrne who used color rather than labels such as ABC (scanned page images, public domain)
Web adapted version of Byrne’s Euclid
designed by Nicholas Rougeux
animated and explained by Sandy Bultena, contains books I-VII.
The First Six Books of the ''Elements''
by John Casey and Euclid scanned by Project Gutenberg.
– a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures) *
Sir Thomas More Sir Thomas More (7 February 1478 – 6 July 1535), venerated in the Catholic Church as Saint Thomas More, was an English lawyer, judge, social philosopher, author, statesman, and noted Renaissance humanist. He also served Henry VIII as Lord ...
'
manuscript
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 textGreek HTML *
Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sci ...
Historical Archive
The thirteen books of Euclid's ''Elements''
copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD
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 b
Islamic Heritage Project

Euclid's ''Elements'' Redux
an open textbook based on the ''Elements''
1607 Chinese translations
reprinted as part of Siku Quanshu, or "Complete Library of the Four Treasuries."
Elements with Highlights
Interactive rendering of all 13 books {{Authority control 3rd-century BC books * Mathematics textbooks Ancient Greek mathematical works Works by Euclid History of geometry Foundations of geometry Geometry education