Euclid ( /jukld/ EWK-lid; Ancient Greek: Eukleids), fl

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ewk-lid;Ancient Greek:Eukleids),fl.300 BC, also known asEuclid of Alexandria, was aGreek mathematician, often referred to as the "Father of Geometry". He was active inAlexandriaduring the reign ofPtolemy I(323283 BC). HisElementsis one of the most influential works in thehistory of mathematics, serving as the maintextbookfor teachingmathematics(especiallygeometry) from the time of its publication until the late 19th or early 20th century.[1]

HYPERLINK "http://en.wikipedia.org/wiki/Euclid" \l "cite_note-1" [2]

HYPERLINK "http://en.wikipedia.org/wiki/Euclid" \l "cite_note-2" [3]In theElements, Euclid deduced the principles of what is now calledEuclidean geometryfrom a small set ofaxioms. Euclid also wrote works onperspective,conic sections,spherical geometry,number theoryandrigor.

"Euclid" is the anglicized version of theGreekname ( Eukleds), meaning "Good Glory".

Contents

[hide] 1Life 2Elements 3Other works 4See also 5Notes 6References 7Further Reading 8External links

LifeLittle is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination.

The few historical references to Euclid were written centuries after he lived, byProclusandPappus of Alexandria.[4]Proclus introduces Euclid only briefly in his fifth-centuryCommentary on the Elements, as the author ofElements, that he was mentioned byArchimedes, and that whenKing Ptolemyasked if there was a shorter path to learning geometry than Euclid'sElements, "Euclid replied there is no royal road to geometry."[5]Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.[6]

HYPERLINK "http://en.wikipedia.org/wiki/Euclid" \l "cite_note-6" [7]In addition, the "royal road" anecdote is questionable since it is similar to a story told aboutMenaechmusandAlexander the Great.[8]In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought."[9]It is further believed that Euclid may have studied atPlato's AcademyinAthens.

Elements

One of the oldest surviving fragments of Euclid'sElements, found atOxyrhynchusand dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[10]Main article:Euclid's ElementsAlthough many of the results inElementsoriginated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.[11]There is no mention of Euclid in the earliest remaining copies of theElements, and most of the copies say they are "from the edition ofTheon" or the "lectures of Theon",[12]while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written theElementswas from Proclus, who briefly in hisCommentary on the Elementsascribes Euclid as its author.

Although best-known for its geometric results, theElementsalso includesnumber theory. It considers the connection betweenperfect numbersandMersenne primes, the infinitude ofprime numbers,Euclid's lemmaon factorization (which leads to thefundamental theorem of arithmeticon uniqueness ofprime factorizations), and theEuclidean algorithmfor finding thegreatest common divisorof two numbers.

The geometrical system described in theElementswas long known simply asgeometry, and was considered to be the only geometry possible. Today, however, that system is often referred to asEuclidean geometryto distinguish it from other so-callednon-Euclidean geometriesthat mathematicians discovered in the 19th century.

Other works

Euclid's construction of a regular dodecahedron

Construction of a dodecahedron basing on a cube

In addition to theElements, at least five works of Euclid have survived to the present day. They follow the same logical structure asElements, with definitions and proved propositions.

Datadeals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of theElements.

On Divisions of Figures, which survives only partially inArabictranslation, concerns the division of geometrical figures into two or more equal parts or into parts in givenratios. It is similar to a third century AD work byHeron of Alexandria.

Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution to Euclid is doubtful. Its author may have beenTheon of Alexandria.

Phaenomena, a treatise onspherical astronomy, survives in Greek; it is quite similar toOn the Moving SpherebyAutolycus of Pitane, who flourished around 310 BC.

Statue of Euclid in theOxford University Museum of Natural History.

Opticsis the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal.Pappusbelieved these results to be important in astronomy and included Euclid'sOptics, along with hisPhaenomena, in theLittle Astronomy, a compendium of smaller works to be studied before theSyntaxis(Almagest) ofClaudius Ptolemy.

Other works are credibly attributed to Euclid, but have been lost.

Conicswas a work onconic sectionsthat was later extended byApollonius of Pergainto his famous work on the subject. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.

Porismsmight have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.

Pseudaria, orBook of Fallacies, was an elementary text about errors inreasoning.

Surface Lociconcerned eitherloci(sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt withquadric surfaces.

Several works onmechanicsare attributed to Euclid by Arabic sources.On the Heavy and the Lightcontains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity.On the Balancetreats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.

See also Axiomatic method Euclidean geometry Euclid's orchard Euclidean relation Euclidean algorithm Extended Euclidean algorithm List of topics named after Euclid