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

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.[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. Not until the 20th century did it cease to be considered something all educated people had read.[7] For other uses, see Venice (disambiguation). ... Events Portuguese fortify Fort Elmina on the Gold Coast Tizoc rules the Aztecs Diogo Cão, a Portuguese navigator, becomes the first European to sail up the Congo. ... The printing press is a mechanical device for printing many copies of a text on rectangular sheets of paper. ... This Gutenberg Bible is displayed by the United States Library. ... The quadrivium comprised the four subjects taught in medieval universities after the trivium. ...

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.[8]

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.[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.[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. 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. For other uses, see Cicero (disambiguation). ... There are several persons called Bo thius: Philosophers: Anicius Manlius Severinus thius - to many scholars this is the Bo thius, a late-Roman writer best known for his works in philosophy and theology. ... Proclo was a later pupil of the Greek geometer Euclid whose version of Euclids Elements was translated into Arabic. ... Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ... Harun al-Rashid (Arabic &#1607;&#1575;&#1585;&#1608;&#1606; &#1575;&#1604;&#1585;&#1588;&#1610;&#1583; also spelled Harun ar-Rashid, Haroun al-Rashid or Haroon al Rasheed; English: Aaron the Upright; ca. ... Events Portuguese fortify Fort Elmina on the Gold Coast Tizoc rules the Aztecs Diogo Cão, a Portuguese navigator, becomes the first European to sail up the Congo. ... For other people named Johannes Campanus, see Campanus. ... The magnificent Cathedral of Chartres was dedicated in 1260. ... Events January 23 - The assassination of regent James Stewart, Earl of Moray throws Scotland into civil war February 25 - Pope Pius V excommunicates Queen Elizabeth I of England with the bull Regnans in Excelsis May 20 - Abraham Ortelius issues the first modern atlas. ... For the American college basketball coach, see John Dee (basketball coach). ... Sir Henry Billingsley (d. ...

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 drawn about the contents of the original text (copies of which are no longer available). The Vatican Library (Latin: Bibliotheca Apostolica Vaticana) is the library of the Holy See, located in Vatican City. ... Entrance to the Library, with the coats-of-arms of several Oxford colleges The Bodleian Library, the main research library of the University of Oxford, is one of the oldest libraries in Europe, and in England is second in size only to the British Library. ...

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. Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ...

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 elucidation. Some of these are useful and add to the text, but many are not. Scholium (tr~bXtoe), the name given to a grammatical, critical and explanatory note, extracted from existing commentaries and inserted on the margin of the manuscript of an ancient author. ...

## 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.

The Elements is still considered a masterpiece in the application of logic to mathematics. In historical context, it has proven enormously influential in many areas of science. Scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work. 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. Image File history File links Euclid-proof. ... Image File history File links Euclid-proof. ... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Part of a scientific laboratory at the University of Cologne. ... â€œCopernicusâ€ redirects here. ... â€œKeplerâ€ redirects here. ... Galileo redirects here. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England â€“ December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... Baruch de Spinoza (â€Ž, Portuguese: , Latin: ) (November 24, 1632 â€“ February 21, 1677) was a Dutch philosopher of Portuguese Jewish origin. ...

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. 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 ..."[9] 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 of problems that arise from their study. ...

### 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). These are the foundation of all that follows. A spatial point is an entity with a location in space but no extent (volume, area or length). ... â€œLineâ€ redirects here. ... An open surface with X-, Y-, and Z-contours shown. ... This article or section does not adequately cite its references or sources. ... This article is about a logical statement. ...

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 congruent.
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: An example of congruence. ...

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. (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.

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. A marked ruler, used in neusis construction, is forbidden in Euclid construction, probably because Euclid could not prove that verging lines meet. This article does not cite any references or sources. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... A straightedge is a tool similar to a ruler, but without markings. ... A variety of rulers A 2 metre carpenters rule Retractable flexible rule A ruler or rule is an instrument used in geometry, technical drawing and engineering/building to measure distances and/or to rule straight lines. ... Neusis construction 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.

The last of Euclid's five postulates warrants special mention. The so-called parallel postulate always seemed less obvious than the others. 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. a and b are parallel, the transversal t produces congruent angles. ... Image File history File links Download high-resolution version (1920x1237, 60 KB) Diagram for the converse of Euclids parallel postulate. ... Image File history File links Download high-resolution version (1920x1237, 60 KB) Diagram for the converse of Euclids parallel postulate. ... a and b are parallel, the transversal t produces congruent angles. ...

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. For this reason, mathematicians say that the parallel postulate is independent of the other postulates. Alternative meaning: Nineteenth Century (periodical) (18th century &#8212; 19th century &#8212; 20th century &#8212; more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ...

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). 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. Lines through a given point P and asymptotic to line l. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (&#1053;&#1080;&#1082;&#1086;&#1083;&#1072;&#769;&#1081; &#1048;&#1074;&#1072;&#769;&#1085;&#1086;&#1074;&#1080;&#1095; &#1051;&#1086;&#1073;&#1072;&#1095;&#1077;&#769;&#1074;&#1089;&#1082;&#1080;&#1081;) (December 1, 1792 - February 24, 1856) was a Russian mathematician. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... â€œEinsteinâ€ redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

### 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.

Books 1 through 4 deal with plane geometry: Image File history File links Size of this preview: 800 Ã— 487 pixelsFull resolution (1694 Ã— 1032 pixel, file size: 320 KB, MIME type: image/jpeg) This is [o]ne of the oldest and most complete diagrams from Euclids Elements of Geometry. It is a fragment of papyrus found among the... Image File history File links Size of this preview: 800 Ã— 487 pixelsFull resolution (1694 Ã— 1032 pixel, file size: 320 KB, MIME type: image/jpeg) This is [o]ne of the oldest and most complete diagrams from Euclids Elements of Geometry. It is a fragment of papyrus found among the... Oxyrhynchus (Greek: ÎŸÎ¾ÏÏÏ…Î³Ï‡Î¿Ï‚; sharp-nosed; ancient Egyptian Per-Medjed; modern Egyptian Arabic el-Bahnasa) is an archaeological site in Egypt, considered one of the most important ever discovered. ... Pliny the Younger advances to consulship. ...

• Book 1 contains the basic propositions of geometry: the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
• Book 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted in terms of algebra.
• Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.
• Book 4 is concerned with inscribing and circumscribing triangles and regular polygons.

Books 11 through 13 deal with spatial geometry: The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... In geometry, Thales theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In arithmetic and number theory, the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... Commensurability in general Generally, two quantities are commensurable if both can be measured in the same units. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... 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. ... This article is about the concept of integrals in calculus. ...

## Criticism

Despite its universal acceptance and success, the Elements has been the subject of substantial criticism. Euclid's parallel postulate, treated above, has been a primary target of critics.[citation needed]

Other criticisms abound. For example, the definitions are not sufficient to describe fully the terms that are defined. 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. Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...

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."[10] Walter William Rouse Ball (1850 August 14&#8211;1925 April 4) was a Brtish mathematician, and a fellow at Trinity College, Cambridge from 1878 to 1905. ...

## 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.[11] The spurious Book XIV was likely written by Hypsicles 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 $sqrt{tfrac{10}{3(5-sqrt{5})}}$. In Judeo-Christian theologies, apocrypha refers to religious Sacred text that have questionable authenticity or are otherwise disputed. ... Hypsicles (ca. ... Apollonius may be: Historical people: Apollonius (philosopher), Greek philosopher is Apollonius of Tyana listed below. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

The spurious Book XV was likely written, at least in part, by Isidore of Miletus. 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.[11] Isidore of Miletus was the architect who together with Anthemius of Tralles designed Hagia Sophia in modern day Istanbul The Emperor Justinian I decided to rebuild the 4th century basilica in Constantinople which was destroyed during the Nika riots of 532. ...

## Editions

The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) in the Chinese edition of Euclid's Elements (幾何原本), printed in 1607.

This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... The Society of Jesus (Latin: Societas Iesu), commonly known as the Jesuits, is a Roman Catholic religious order. ... Matteo Ricci. ... Knowledge of Chinese mathematics before 100 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. ... Xu Guangqi (Simplified Chinese: å¾å…‰å¯; Traditional Chinese: å¾å…‰å•Ÿ; Pinyin: XÃº GuÄngqÇ) (1562â€“1633) was a Chinese agricultural scientist and mathematician born in Shanghai. ... Centuries: 14th century - 15th century - 16th century Decades: 1410s 1420s 1430s 1440s 1450s - 1460s - 1470s 1480s 1490s 1500s 1510s Years: 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 Events and Trends Sonni Ali, first Songhai king, conquers many of his African neighbors. ... Johannes MÃ¼ller von KÃ¶nigsberg (June 6, 1436 â€“ July 6, 1476), known by his Latin pseudonym Regiomontanus, was an important German mathematician, astronomer and astrologer. ... Events January 25 - King Henry VIII of England marries Anne Boleyn, his second Queen consort. ... January 16 - Thomas Howard, 4th Duke of Norfolk is tried for treason for his part in the Ridolfi plot to restore Catholicism in England. ... Year 1574 was a common year starting on Friday (link will display the full calendar) of the Julian calendar. ... Christopher Clavius, born Christoph Clau, (1538 &#8211; February 12, 1612) was a German mathematician and astronomer who was the main architect of the modern Gregorian calendar. ...

### Translations

1505 was a common year starting on Sunday (see link for calendar) of the Gregorian calendar. ... // Events February 21 - Battle of Wayna Daga - A combined army of Ethiopian and Portuguese troops defeat the armies of Adal led by Ahmed Gragn. ... Events Russia breaks 60 year old truce with Sweden by attacking Finland February 2 - Diet of Augsburg begins February 4 - John Rogers becomes first Protestant martyr in England February 9 - Bishop of Gloucester John Hooper is burned at the stake May 23 - Paul IV becomes Pope. ... Year 1562 was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. ... Events March 27 â€” Naples bans kissing in public under the penalty of death June 22 â€” Fort Caroline, the first French attempt at colonizing the New World September 10 â€” The Battle of Kawanakajima Ottoman Turks invade Malta Modern pencil becomes common in England Conquistadors crossed the Pacific Spanish founded a colony... Events January 23 - The assassination of regent James Stewart, Earl of Moray throws Scotland into civil war February 25 - Pope Pius V excommunicates Queen Elizabeth I of England with the bull Regnans in Excelsis May 20 - Abraham Ortelius issues the first modern atlas. ... Woodcut of John Day included in the 1563 and subsequent editions of Actes and Monuments. ... Events May 5 - Peace of Beaulieu or Peace of Monsieur (after Monsieur, the Duc dAnjou, brother of the King, who negotiated it). ... Events February 27 - Henry IV is crowned King of France at Rheims. ... Tusi couple from Vat. ... Year 1607 (MDCVII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Thursday of the 10-day slower Julian calendar). ... Matteo Ricci. ... Xu Guangqi (Simplified Chinese: å¾å…‰å¯; Traditional Chinese: å¾å…‰å•Ÿ; Pinyin: XÃº GuÄngqÇ) (1562â€“1633) was a Chinese agricultural scientist and mathematician born in Shanghai. ... // Events January 1 - Colonel George Monck with his regiment crosses from Scotland to England at the village of Coldstream and begins advance towards London in support of English Restoration. ... Isaac Barrow (October 1630 - May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the...

### Currently in print

"Euclid's Elements - All thirteen books in one volume" Green Lion Press. ISBN 1-888009-18-7 Based on Heath's translation. www.greenlion.com

## Notes

1. ^ Boyer (1991). "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). "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). "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. ^ Daniel Shanks (2002). Solved and Unsolved Problems in Number Theory. American Mathematical Society.
10. ^ Ball (1960) p. 55.
11. ^ a b Boyer (1991). "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.”

Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

## References

• Ball, W.W. Rouse (1960). A Short Account of the History of Mathematics, 4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908], New York: Dover Publications, pp. 50–62. ISBN 0-486-20630-0.
• Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (3 vols.), 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925], New York: Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3).  Heath's authoritative translation plus his extensive historical research and detailed commentary throughout the text.
• Boyer, Carl B. (1991). A History of Mathematics, Second Edition, John Wiley & Sons, Inc.. ISBN 0471543977.

Walter William Rouse Ball (1850 August 14&#8211;1925 April 4) was a Brtish mathematician, and a fellow at Trinity College, Cambridge from 1878 to 1905. ... Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

Results from FactBites:

 Geometry - LoveToKnow 1911 (0 words) Between Euclid and Apollonius there flourished the illustrious Archimedes, whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane. Euclid does not suppose,that the bases of the two pyramids to be compared are equal, and hence he proves that the volumes are as the bases.
 Euclid's Elements, Introduction (416 words) Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages. This edition of Euclid's Elements uses a Java applet called the Geometry Applet to illustrate the diagrams.
More results at FactBites »

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