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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570

Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position A treatise is a formal lengthy systematic Discourse on some subject The Greeks ( Greek: Έλληνες) are a Nation and Ethnic group native to Greece, Cyprus and neighbouring regions A mathematician is a person whose primary area of study and research is the field of Mathematics. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια Circa (often abbreviated c, ca, ca or cca and sometimes Italicized to show it is Latin) means "about" Events By place Egypt Pyrrhus, the King of Epirus, is taken as a hostage to Egypt after the Battle of Ipsus It comprises a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises[1] and it is the oldest extant axiomatic deductive treatment of mathematics. Autolycus of Pitane (c 360 BC–c 290 BC was a Greek Astronomer, Mathematician, and Geographer. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and [2] It has proven instrumental in the development of logic and modern science. Logic is the study of the principles of valid demonstration and Inference. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding

Euclid's Elements is the most successful[3][4] and influential[5] textbook ever written. Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and is second only to the Bible in the number of editions published,[5] with the number reaching well over one thousand. Venice ( Italian: Venezia, Venetian: Venesia or Venexia) is a city in Northern Italy, the capital of the A printing press is a mechanical device for applying pressure to an inked surface resting upon a medium (such as paper or cloth thereby transferring an image Etymology According to the Online Etymology Dictionary, the word bible is from Latin biblia, traced from the same word through Medieval Latin and Late Latin [6] It was used as the basic text on geometry throughout the Western world for about 2,000 years. 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. The quadrivium comprised the four subjects or arts taught in Medieval universities after the trivium. Not until the 20th century, by which time its content was universally taught through school books, did it cease to be considered something all educated people had read. [7]

Contents

History

The frontispiece of an Adelard of Bath Latin translation of Euclid's Elements, c. 1309–1316; the oldest surviving Latin translation of the Elements is a 12th century work by Adelard, which translates to Latin from the Arabic.
The frontispiece of an Adelard of Bath Latin translation of Euclid's Elements, c. Adelard of Bath ( Latin: Adelardus Bathensis) (c 1080 &ndash c 1309–1316; the oldest surviving Latin translation of the Elements is a 12th century work by Adelard, which translates to Latin from the Arabic. [8]

Euclid was a Greek mathematician who wrote Elements in Alexandria during the Hellenistic period (around 300 BC). Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry The Greeks ( Greek: Έλληνες) are a Nation and Ethnic group native to Greece, Cyprus and neighbouring regions This article focuses on the historical aspects of the Hellenistic age for the cultural aspects see Hellenistic civilisation. Scholars believe that the Elements is largely a collection of theorems proved by other mathematicians as well as containing some original work. Proclus, a Greek mathematician who lived several centuries after Euclid, wrote in his commentary of 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". Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos

Although known to, for instance, Cicero, there is no extant record of the text having been translated into Latin prior to Boethius in the fifth or sixth century. Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman Anicius Manlius Severinus Boethius (480&ndash524 or 525 was a Christian philosopher of the 6th century [8] The Arabs received the Elements from the Byzantines in approximately 760; this version, by a pupil of Euclid called Proclo, was translated into Arabic under Harun al Rashid circa 800 AD. Proclo was a later pupil of the Greek Geometer Euclid whose version of Euclid's Elements was translated into Arabic. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language Hārūn al-Rashīd (and Persian: هارون الرشيد) also spelled Harun ar-Rashid; English: Aaron the Upright, Aaron the [8] The first printed edition appeared in 1482 (based on Giovanni Campano's 1260 edition), and since then it has been translated into many languages and published in about a thousand different editions. Johannes Campanus (in Italian, Giovanni Campano; also known as Campanus of Novara or similar (1220-1296 was an Italian Astrologer 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. John Dee (13 July 1527 – 1608 or 1609 was a noted English mathematician, astronomer, astrologer, geographer, occultist Sir Henry Billingsley (died November 22, 1606) was Lord Mayor of London and the first translator of Euclid into English

Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford. The Vatican Library ( Latin: Bibliotheca Apostolica Vaticana) is the Library of the Holy See, currently located in Vatican City. The Bodleian Library ( the main Research library of the University of Oxford, is one of the oldest libraries in Europe, and in England The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been drawn 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. Johan Ludvig Heiberg (1854&ndash1928 was a Danish Philologist and Historian. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical

Also of importance are the scholia, or annotations to the text. A scholium, plural scholia (σχόλιον "comment" "lecture" is a grammatical, critical or explanatory comment either original or extracted 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 elucidation. Some of these are useful and add to the text, but many are not.

A difficult text

Although we now consider the Elements to be an elementary text on geometry, that was not always the case. It is said that King Ptolemy asked for a way in geometry that was shorter than the Elements. For the astronomer see Ptolemy; for others named "Ptolemy" or "Ptolemaeus" see Ptolemy (disambiguation. Euclid answered that "there is no royal road to geometry. The Persian Royal Road was an ancient highway reorganized and rebuilt by the Persian king Darius I of Achaemenid Empire in the 5th Century BC " [9] More recently, Sir Thomas Little Heath wrote, in his introduction to the 1932 Everyman's Library volume of Euclid:

The simple truth is that it was not written for schoolboys or schoolgirls, but for the grown man who would have the necessary knowledge and judgment to appreciate the highly contentious matters which have to be grappled with in any attempt to set out the essentials of Euclidean geometry as a strictly logical system. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist . . " [10].

The first difficult passage of Book I is referred to as the pons asinorum, which is Latin for "Bridge of Asses" (traditionally, it is hard to get asses to cross a bridge). Pons Asinorum ( Latin for "Bridge of Asses" is the name given to Euclid 's fifth proposition in Book 1 of his Elements of The donkey or ass, Equus asinus, is a member of the Equidae or horse family and an odd-toed ungulate. [11]

Outline of the Elements

A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.
A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

The Elements is still considered a masterpiece in the application of logic to mathematics. Logic is the study of the principles of valid demonstration and Inference. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In historical context, it has proven enormously influential in many areas of science. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work. Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Mathematicians and philosophers, such as Bertrand Russell, Alfred North Whitehead, and Baruch Spinoza, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Baruch or Benedict de Spinoza (ברוך שפינוזה Bento de Espinosa Benedictus de Spinoza ( November 24, 1632 – February 21,

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. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics.

Although Elements is primarily a geometric work, it also includes results that today would be classified as number theory. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Euclid probably chose to describe results in number theory in terms of geometry because he couldn't develop a constructible approach to arithmetic. A construction used in any of Euclid's proofs required a proof that it is actually possible. This avoids the problems the Pythagoreans encountered with irrationals, since their fallacious proofs usually required a statement such as "Find the greatest common measure of . . . "[12]

First principles

Euclid's Book 1 begins with 23 definitions — such as point, line, and surface — followed by five postulates and five "common notions" (both of which are today called axioms). In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject These are the foundation of all that follows.

Postulates:

  1. A straight line segment can be drawn by joining any two points.
  2. A straight line segment can be extended indefinitely in a straight line.
  3. Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.
  4. All right angles are equal.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Common notions:

  1. Things which equal the same thing are equal to one another. (Euclidean property of equality)
  2. If equals are added to equals, then the sums are equal. In Mathematics, a Binary relation R over a set X is euclidean if it holds for all a, b, and c in Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric (Addition property of equality)
  3. If equals are subtracted from equals, then the remainders are equal. (Subtraction property of equality)
  4. Things which coincide with one another are equal to one another. (Reflexive property of equality)
  5. The whole is greater than the part. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity.

These basic principles reflect the interest of Euclid, along with his contemporary Greek and Hellenistic mathematicians, in constructive geometry. The first three postulates basically describe the constructions one can carry out with a compass and an unmarked straightedge. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as A straightedge is a tool with an accurately straight edge used for drawing or cutting straight lines or checking the straightness of lines A marked ruler, used in neusis construction, is forbidden in Euclid construction, probably because Euclid could not prove that verging lines meet. A ruler, or rule, is an instrument used in Geometry, Technical drawing and engineering/building to measure distances and/or to rule straight The neusis is a geometric construction method that was used in antiquity by Greek mathematicians

Parallel postulate

Main article: Parallel postulate
If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect.
If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive

The last of Euclid's five postulates warrants special mention. The so-called parallel postulate always seemed less obvious than the others. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Euclid himself used it only sparingly throughout the rest of the Elements. Many geometers suspected that it might be provable from the other postulates, but all attempts to do this failed.

By the mid-19th century, it was shown that no such proof exists, because one can construct non-Euclidean geometries where the parallel postulate is false, while the other postulates remain true. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry For this reason, mathematicians say that the parallel postulate is independent of the other postulates. In Mathematical logic, a sentence &sigma is called independent of a given first-order theory T if T neither proves nor

Two alternatives to the parallel postulate are possible in non-Euclidean geometries: either an infinite number of parallel lines can be drawn through a point not on a straight line in a hyperbolic geometry (also called Lobachevskian geometry), or none can in an elliptic geometry (also called Riemannian geometry). In Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point Elliptic geometry is also sometimes called Riemannian geometry. That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Albert Einstein's theory of general relativity shows that the real space in which we live is non-Euclidean. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

Contents of the books

A fragment of Euclid's elements found at Oxyrhynchus, which is dated to circa 100 AD. The diagram accompanies Proposition 5 of Book II of the Elements.
A fragment of Euclid's elements found at Oxyrhynchus, which is dated to circa 100 AD. Oxyrhynchus (Ὀξύρρυγχος "sharp-nosed" ancient Egyptian Pr-Medjed; Coptic Pemdje; modern Egyptian Arabic The diagram accompanies Proposition 5 of Book II of the Elements.

Books 1 through 4 deal with plane geometry:

Books 5 through 10 introduce ratios and proportions:

Books 11 through 13 deal with spatial geometry:

Criticism

Despite its universal acceptance and success, the Elements has been criticised as having insufficient proofs and definitions. 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 the movement of triangles to prove that if two sides and their angles are equal, then they are congruent; however, he did not postulate or even define movement.

In the 19th century, non-Euclidean geometries attracted the attention of contemporary mathematicians. Leading mathematicians, including Richard Dedekind and David Hilbert, attempted to reformulate the axioms of the Elements, such as by adding an axiom of continuity and an axiom of congruence, to make Euclidean geometry more complete. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most

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. Walter William Rouse Ball ( 14 August 1850 – 4 April 1925) was a British Mathematician, Lawyer and a fellow "[13]

Apocrypha

It was not uncommon in ancient time 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. [14] The spurious Book XIV was likely written by Hypsicles on the basis of a treatise by Apollonius. This article is about Hypsicles of Alexandria For the historian see Hyspicrates (historian. 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{\tfrac{10}{3(5-\sqrt{5})}}. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ

The spurious Book XV was likely written, at least in part, by Isidore of Miletus. Isidore of Miletus (Ισίδωρος ο Μιλήσιοςin Greek) was one of the two Greek Architects (the other being Anthemius This inferior 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. [14]

Editions

The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) published the Chinese edition of Euclid's Elements (幾何原本) in 1607.
The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) published the Chinese edition of Euclid's Elements (幾何原本) in 1607. Italy (Italia officially the Italian Republic, (Repubblica Italiana is located on the Italian Peninsula in Southern Europe, and on the two largest The Society of Jesus ( Latin: Societas Iesu, SJ and SI or SJ, SI) is a Catholic religious order Matteo Ricci SJ ( October 6 1552 &ndash May 11 1610;; Courtesy name: 西泰 Xītài was an Italian Jesuit priest Mathematics in China emerged independently by the 11th century BC Xu Guangqi ( 1562–1633 Courtesy name Zixian (子先 was a Chinese bureaucrat agricultural scientist astronomer and mathematician in the Ming Dynasty

Translations

Currently in print

"Euclid's Elements - All thirteen books in one volume" Green Lion Press. Johannes Müller von Königsberg ( June 6, 1436 &ndash July 6, 1476) known by his Latin Pseudonym Regiomontanus In Classical scholarship, editio princeps is a Term of art. It means roughly the first printed edition of a work that previously had existed only in Christopher Clavius, ( March 25, 1538 &ndash February 12, 1612) was a German Jesuit Mathematician and For the soul singer see Johnny Daye John Day or Daye ( c 1522 &ndash 23 July 1584 was an English Matteo Ricci SJ ( October 6 1552 &ndash May 11 1610;; Courtesy name: 西泰 Xītài was an Italian Jesuit priest Xu Guangqi ( 1562–1633 Courtesy name Zixian (子先 was a Chinese bureaucrat agricultural scientist astronomer and mathematician in the Ming Dynasty Isaac Barrow (October 1630 &ndash May 4, 1677) was an English scholar and Mathematician who is generally given credit for his early role ISBN 1-888009-18-7 Based on Heath's translation.

Notes

  1. ^ Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Euclid of Alexandria", , 101.  “With the exception of the Sphere of Autolycus, surviving work by Euclid are the oldest Greek mathematical treatises extant; yet of what Euclid wrote more than half has been lost,” 
  2. ^ Ball (1960).
  3. ^ Encyclopedia of Ancient Greece (2006) by Nigel Guy Wilson, page 278. Published by Routledge Taylor and Francis Group. Quote:"Euclid's Elements subsequently became the basis of all mathematical education, not only in the Romand and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written. "
  4. ^ Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Euclid of Alexandria", , 100.  “As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written - the Elements (Stoichia) of Euclid. ” 
  5. ^ a b Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Euclid of Alexandria", , 119.  “The Elements of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [. . . ]The first printed versions of the Elements appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's Elements. ” 
  6. ^ The Historical Roots of Elementary Mathematics by Lucas Nicolaas Hendrik Bunt, Phillip S. Jones, Jack D. Bedient (1988), page 142. Dover publications. Quote:"the Elements became known to Western Europe via the Arabs and the Moors. There the Elements became the foundation of mathematical education. More than 1000 editions of the Elements are known. In all probability it is, next to the Bible, the most widely spread book in the civilization of the Western world. "
  7. ^ Ball (1960).
  8. ^ a b c Russell, Bertrand. A History of Western Philosophy. p. 212.
  9. ^ http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+1
  10. ^ Heath: Everyman's Library Euclid Introduction
  11. ^ Oxford Philosophy Dictionary, http://www.answers.com/topic/pons-asinorum?cat=technology
  12. ^ Daniel Shanks (2002). Solved and Unsolved Problems in Number Theory. American Mathematical Society.  
  13. ^ Ball (1960) p. 55.
  14. ^ a b Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Euclid of Alexandria", , 118-119.  “In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's Elements include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, \sqrt{10/[3(5-\sqrt{5})]}. It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. [. . . ] The spurious Book XV, which is inferior, is thought to have been (at least in part) the work of Isidore of Miletus (fl. ca. A. D. 532), architect of the cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also deals with the regular solids, counting the number o edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge. ” 

References

External links

Complete and fragmentary manuscripts of versions of Euclid's Elements :

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