The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a

mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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." Tre ...

consisting of 13 books attributed to the ancient Greek mathematician 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, Alexandri ...

, 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 ( theorems and constructions), and mathematical proofs 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 axioms ...

, elementary number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

, 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 premise ...

and modern science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...

, 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 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 ...

in the number of editions published since the first printing in 1482, the number reaching well over one thousand. 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.
Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the Victorian period 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 politi ...

( 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios ( 470–410 BC, not the better known Hippocrates of Kos
Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kôios; ), also known as Hippocrates II, was a Greek physician of the classical period who is considered one of the most outstanding figures in the history of ...

) for book III, and Eudoxus of Cnidus ( 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

In the 4th century AD,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 esta ...

, for instance, no record exists of the text having been translated into Latin prior to Boethius 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 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 J. L. Heiberg and Sir Thomas Little Heath
Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translat ...

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.
Influence

The ''Elements'' is still considered a masterpiece in the application oflogic
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 premise ...

to mathematics. In historical context, it has proven enormously influential in many areas of science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...

. 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, 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 wa ...

, 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
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...

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 influ ...

, Baruch Spinoza, Alfred North Whitehead, 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, ...

, 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 thro ...

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 theparallel 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 segmen ...

. 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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...

), 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
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...

). 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 axioms ...

.
Contents

* Book 1 contains 5 postulates (including the infamousparallel 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 segmen ...

) and 5 common notions, and covers important topics of plane geometry such as the Pythagorean theorem, equality of angles and area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...

s, 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, inscribe
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figur ...

d 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. Mo ...

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 proved ...

.
* 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 incenter. ...

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 polyg ...

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 number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s 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, ...

s, Euclid's algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ef ...

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 number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
...

s.
* Book 10 proves the irrationality of the square roots of non-square integers (e.g. $\backslash sqrt$) 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 number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s. He also gives a formula to produce Pythagorean triples
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

.
* 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, and cylinders 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 are ...

, 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
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 th ...

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. 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
In the pursuit of knowledge, data (; ) is a collection of discrete Value_(semiotics), values that convey information, describing quantity, qualitative property, quality, fact, statistics, other basic units of meaning, or simply sequences of sy ...

'' 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 arecongruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...

; 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
Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...

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 books XIV and XV of the ''Elements'' were sometimes included in the collection. The spurious Book XIV was probably written byHypsicles 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
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being
$$\backslash sqrt\; =\; \backslash sqrt.$$
The spurious Book XV was probably written, at least in part, by Isidore of Miletus
Isidore of Miletus ( el, Ἰσίδωρος ὁ Μιλήσιος; Medieval Greek pronunciation: ; la, Isidorus Miletus) was one of the two main Byzantine Greek architects (Anthemius of Tralles was the other) that Emperor Justinian I commissioned ...

. 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
Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrument ...

(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
Xu Guangqi or Hsü Kuang-ch'i (April 24, 1562– November 8, 1633), also known by his baptismal name Paul, was a Chinese agronomist, astronomer, mathematician, politician, and writer during the Ming dynasty. Xu was a colleague and collaborato ...

(Chinese)
*1613, Pietro Cataldi (Italian)
*1615, Denis Henrion (French)
*1617, Frans van Schooten (Dutch)
*1637, L. Carduchi (Spanish)
*1639, Pierre Hérigone
Pierre Hérigone (Latinized as Petrus Herigonius) (1580–1643) was a French mathematician and astronomer.
Of Basque people, Basque origin, Hérigone taught in Paris for most of his life.
Works
Only one work by Hérigone is known to exist: ''Cur ...

(French)
*1651, Heinrich Hoffmann (German)
*1651, Thomas Rudd (English)
*1660, Isaac Barrow (English)
*1661, John Leeke and Geo. Serle (English)
*1663, Domenico Magni (Italian from Latin)
*1672, Claude François Milliet Dechales
Claude François Milliet Dechales (1621 – 28 March 1678) was a French Jesuit priest and mathematician. He published a treatise on mathematics and a translation of the works of Euclid, though of lesser quality than that of Gilles Personne de Rob ...

(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
Sir Charles Scarborough or Scarburgh MP FRS FRCP (29 December 1615 – 26 February 1694) was an English physician and mathematician.Robert L. Martensen, "Scarburgh, Sir Charles (1615–1694)", ''Oxford Dictionary of National Biography'' (Oxfor ...

(English)
*1708, John Keill
John Keill FRS (1 December 1671 – 31 August 1721) was a Scottish mathematician, natural philosopher, and cryptographer who was an important defender of Isaac Newton.
Biography
Keill was born in Edinburgh, Scotland on 1 December 1671. His f ...

(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
Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.Baruch Schick of Shklov (Hebrew)
*1781, 1788 James Williamson (English)
*1781, William Austin (English)
*1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek)
*1795,

Clark University Euclid's elements

Multilingual 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. 1

vol. 2

vol. 3

vol. 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

Video adaptation

animated and explained by Sandy Bultena, contains books I-VII.

The First Six Books of the ''Elements''

by John Casey and Euclid scanned by

Reading Euclid

– a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures) * Sir Thomas More'

manuscript

by Aethelhard of Bath * http://www.physics.ntua.gr/~mourmouras/euclid/index.html Euclid ''Elements'' – The original Greek text">!-- 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 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

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
Professor Dionysius Lardner FRS FRSE (3 April 179329 April 1859) was an Irish scientific writer who popularised science and technology, and edited the 133-volume '' Cabinet Cyclopædia''.
Early life in Dublin
He was born in Dublin on 3 Apr ...

(English)
*1833, E. S. Unger (German)
*1833, Thomas Perronet Thompson
Thomas Perronet Thompson (1783–1869) was a British Parliamentarian, a governor of Sierra Leone and a radical reformer. He became prominent in 1830s and 1840s as a leading activist in the Anti-Corn Law League. He specialized in the grass-root ...

(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
Isaac Todhunter FRS (23 November 1820 – 1 March 1884), was an English mathematician who is best known today for the books he wrote on mathematics and its history.
Life and work
The son of George Todhunter, a Nonconformist minister, a ...

(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
Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translat ...

(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
Portuguese may refer to:
* anything of, from, or related to the country and nation of Portugal
** Portuguese cuisine, traditional foods
** Portuguese language, a Romance language
*** Portuguese dialects, variants of the Portuguese language
** Portu ...

)
*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

Notes

Citations

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. * * * * * * * * * * * *External links

Clark University Euclid's elements

Multilingual 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. 1

vol. 2

vol. 3

vol. 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

Video adaptation

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
Project Gutenberg (PG) is a volunteer effort to digitize and archive cultural works, as well as to "encourage the creation and distribution of eBooks."
It was founded in 1971 by American writer Michael S. Hart and is the oldest digital libr ...

.
Reading Euclid

– a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures) * Sir Thomas More'

manuscript

by Aethelhard of Bath * http://www.physics.ntua.gr/~mourmouras/euclid/index.html Euclid ''Elements'' – The original Greek text">!-- 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 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