name = Euclid

image_width =
caption =
birth_date = fl. 300 BC
residence = Alexandria, Egypt
ethnicity = Greek
field = Mathematics
known_for = Euclid's "Elements"

Euclid (Greek: . polytonic|Εὐκλείδης — Eukleidēs), fl. 300 BC, also known as Euclid of Alexandria, is often referred to as the Father of Geometry. He was a Greek mathematician, and is believed to have been active in Alexandria during the reign of Ptolemy I (323 BC–283 BC). His "Elements" is the most successful textbook in the history of mathematics. In it, the principles of what is now called Euclidean geometry are deduced from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, and rigor.

Biographical knowledge

Little is known about Euclid other than his writings. What little biographical information we do have comes largely from commentaries by Proclus and Pappus of Alexandria. Euclid was active at the great Library of Alexandria and may have studied at Plato's Academy in Greece. The date and place of Euclid's birth and the date and circumstances of his death are unknown.

Some writers in the Middle Ages confused him with Euclid of Megara, a Greek Socratic philosopher who lived approximately one century earlier. [Heath (1956) vol. I, p. 4]

The "Elements"

and dated to circa AD 100. The diagram accompanies Book II, Proposition 5. [cite web
title=One of the Oldest Extant Diagrams from Euclid
author=Bill Casselman
publisher=University of British Columbia
] ] Although many of the results in "Elements" originated 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 laterFact|date=July 2008.

Although best-known for its geometric results, the "Elements" also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the "Elements" was long known simply as "geometry", and was considered to be the only geometry possible. Today, however, that system is often referred to as "Euclidean geometry" to distinguish it from other so-called "Non-Euclidean geometries" that mathematicians discovered in the 19th century.

Other works

In addition to the "Elements", at least five works of Euclid have survived to the present day.

* "Data" deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the "Elements".
* "On Divisions of Figures", which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a third century AD work by Heron 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 been Theon of Alexandria.
* "Phaenomena" is a treatise on spherical Astronomy, it survives in Greek and is quite similar to "On the Moving Sphere", by Autolycus of Pitane, who flourished around 310 BC.
* "Optics" is 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. Pappus believed these results to be important in astronomy and included Euclid's "Optics", along with his "Phaenomena", in the "Little Astronomy", a compendium of smaller works to be studied before the "Syntaxis" ("Almagest") of Claudius Ptolemy.All of these works follow the basic logical structure of the "Elements", containing definitions and proved propositions.

There are also works credibly attributed to Euclid which have been lost.
* "Conics" was a work on conic sections that was later extended by Apollonius of Perga into 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.
* "Porisms" might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
* "Pseudaria", or "Book of Fallacies", was an elementary text about errors in reasoning.
* "Surface Loci" concerned either loci (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 with quadric surfaces.
* Several works on mechanics are attributed to Euclid by Arabic sources. "On the Heavy and the Light" contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. "On the Balance" treats 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.

ee also

* Axiomatic method




*citeweb|title=Euclid (Greek mathematician)|url=|accessdate=2008-04-18|year=2008|publisher=Encyclopædia Britannica, Inc
*Artmann, Benno (1999). "Euclid: The Creation of Mathematics". New York: Springer. ISBN 0387984232.
*citebook|last=Ball|first=W.W. Rouse|authorlink=W. W. Rouse Ball|title = A Short Account of the History of Mathematics|origyear=1908|url=|edition=4th ed.|year=1960|publisher= Dover Publications
isbn=0486206300|pages =p.50–62

*cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=2d ed.| publisher=John Wiley & Sons, Inc.| year=1991|isbn=0471543977
*citebook|authorlink=T. L. Heath|last=Heath|first=Thomas|year=1956|title=The Thirteen Books of Euclid's Elements|volume=vol.1|origyear=1908|publisher=Dover Publications|isbn=0486600882
*Heath, Thomas L. (1981). "A History of Greek Mathematics", 2 Vols. New York: Dover Publications. ISBN 0486240738 / ISBN 0486240746.
*Kline, Morris (1980). "Mathematics: The Loss of Certainty". Oxford: Oxford University Press. ISBN 019502754X.
*MacTutor Biography|id=Euclid

External links

* [ A short list of Mathematicians born in Egypt]
* [ Euclid's elements] , All thirteen books, with interactive diagrams using Java. Clark University
* [ Euclid's elements] , with the original Greek and an English translation on facing pages (includes PDF version for printing). University of Texas.
* [ Euclid's elements] , All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese .
* [ "Elementa Geometriae"] 1482, Venice. From Rare Book Room.
* [ "Elementa"] 888 AD, Byzantine. From Rare Book Room.
* [ Euclid biography by Charlene Douglass] With extensive bibliography.
* [ Texts on Ancient Mathematics and Mathematical Astronomy] PDF scans (Note: many are very large files). Includes editions and translations of Euclid's "Elements", "Data", and "Optica", Proclus's "Commentary on Euclid", and other historical sources.

ALTERNATIVE NAMES=Euclid of Alexandria; Εὐκλείδης (Greek)
SHORT DESCRIPTION=Greek mathematician

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