Euclid - Revision history

Irlandos: /* ''The Elements'' */

2006-04-03T20:31:03Z

''The Elements''

Irlandos at 20:30, April 3, 2006

2006-04-03T20:30:44Z

Irlandos at 20:28, April 3, 2006

2006-04-03T20:28:52Z

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'''Euclid of Alexandria ''' ([[Greek language|Greek]]: {{polytonic|Ευκλείδης}}) (ca. [[325 BC]]–[[265 BC]]) was a [[Hellenistic]] [[mathematician]] who lived in [[Alexandria]], Egypt almost certainly during the reign of [[Ptolemy I]] ([[323 BC]]–[[283 BC]]). Often considered as the "father of geometry", his most popular work is ''[[Euclid's Elements|Elements]]'', which is often considered to be one of the most successful textbooks in the history of mathematics. Within it, the properties of geometrical objects and integers are deduced from a small set of axioms, thereby anticipating (and partly inspiring) the [[axiomatic method]] of modern mathematics.

Euclid also wrote works on perspective, conic sections, spherical geometry, and possibly [[Quadric|quadric surfaces]]. Neither the year nor place of his birth have been established, nor the circumstances of his death.

== ''The Elements'' ==
{{main|Euclid's Elements}}

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. In addition to providing some missing proofs, Euclid's text also includes sections on number theory and three-dimensional geometry. In particular, Euclid's proof of the infinitude of prime numbers is in Book IX, Proposition 20.

The geometrical system described in ''Elements'' was long known simply as "the" geometry. Today, however, it is often referred to as [[Euclidean geometry]] to distinguish it from other so-called ''non-Euclidean'' geometries which were discovered in the [[19th century]]. These new geometries grew out of more than two [[millennium|millennia]] of investigation into Euclid's [[Parallel postulate|fifth postulate]], one of the most-studied [[axiom]]s in all of mathematics. Most of these investigations involved attempts to prove the relatively complex and presumably non-intuitive fifth postulate using the other four (a feat which, if successful, would have shown the postulate to be in fact a theorem).

==Other works==
In addition to the ''Elements'', four works of Euclid have survived to the present day.
* ''[[Data (Euclid)|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]], except Euclid's work characteristically lacks any numerical calculations.
* ''Phaenomena'' concerns the application of spherical geometry to problems of astronomy.
* ''Optics'', the earliest surviving [[Greek language|Greek]] treatise on perspective, contains propositions on the apparent sizes and shapes of objects viewed from different distances and angles.

All of these works follow the basic logical structure of the ''Elements'', containing definitions and proved propositions.

There are four works credibly attributed to Euclid which have been lost.
* ''Conics'' was a work on [[conic section]]s that was later extended by [[Apollonius of Perga]] into his famous work on the subject.
* ''[[Porism]]s'' 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.

==Biographical sources==
Almost nothing is known about Euclid outside of what is presented in ''Elements'' and his few other surviving books. What little biographical information we do have comes largely from commentaries by [[Proclus]] and [[Pappus of Alexandria]]: he was active at the [[Library of Alexandria|great library in Alexandria]] and may have studied at [[Plato]]'s [[Academe]] in [[Greece]], but his exact lifespan and place of birth are unknown.

In the [[Middle Ages]], writers sometimes referred to him as ''[[Euclid of Megara]]'', confusing him with a Greek [[Socrates|Socratic]] [[philosopher]] who lived approximately one century earlier.

== References ==
* Bulmer-Thomas, Ivor (1971). "Euclid". ''Dictionary of Scientific Biography.''
* Heath, Thomas L. (1956). ''The Thirteen Books of Euclid's Elements'', Vol. 1 (2nd ed.). New York: Dover Publications. ISBN 0-486-60088-2.
* Heath, Thomas L. (1981). ''A History of Greek Mathematics'', 2 Vols. New York: Dover Publications. ISBN 0-486-24073-8 / ISBN 0-486-24074-6.
* Kline, Morris (1980). ''Mathematics: The Loss of Certainty''. Oxford: Oxford University Press. ISBN 0-19-502754-X.

== External links ==
* [http://farside.ph.utexas.edu/euclid.html Euclid's elements], with the original Greek and an English translation on facing pages
* [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html Euclid entry] at the [http://www-groups.dcs.st-and.ac.uk/~history/index.html MacTutor History of Mathematics archive]
* [http://www.worldcatlibraries.org/wcpa/ow/e8ebf8aa9507bdc9.html Library search at WorldCat] for ''The Medieval Latin translation of the Data of Euclid'' by Shuntaro Ito
* [http://www.eucliduniversity.org Euclid University] for ''The only accredited university actually named after Euclid''
* [http://fermatslasttheorem.blogspot.com/2006/04/euclid-of-alexandria.html Biography of Euclid]

{{Credit wikipedia}}
[[Category:365 BC births]]
[[Category:275 BC deaths]]
[[Category:Ancient mathematicians]]

@@ Line 4: / Line 4: @@
 == ''The Elements'' ==
-{{main|Euclid's Elements}}
 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. In addition to providing some missing proofs, Euclid's text also includes sections on number theory and three-dimensional geometry.  In particular, Euclid's proof of the infinitude of prime numbers is in Book IX, Proposition 20.

@@ Line 1: / Line 1: @@
-'''Euclid of Alexandria ''' ([[Greek language|Greek]]: {{polytonic|Ευκλείδης}}) (ca. [[325 BC]]&ndash;[[265 BC]]) was a [[Hellenistic]] [[mathematician]] who lived in [[Alexandria]], Egypt almost certainly during the reign of [[Ptolemy I]] ([[323 BC]]&ndash;[[283 BC]]). Often considered as the "father of geometry", his most popular work is ''[[Euclid's Elements|Elements]]'', which is often considered to be one of the most successful textbooks in the history of mathematics.  Within it, the properties of geometrical objects and integers are deduced from a small set of axioms, thereby anticipating (and partly inspiring) the [[axiomatic method]] of modern mathematics.
+'''Euclid of Alexandria ''' ([[Greek language|Greek]]: Ευκλείδης) (ca. [[325 BC]]&ndash;[[265 BC]]) was a [[Hellenistic]] mathematician who lived in [[Alexandria]], Egypt almost certainly during the reign of [[Ptolemy I]] ([[323 BC]]&ndash;[[283 BC]]). Often considered as the "father of geometry", his most popular work is ''[[Euclid's Elements|Elements]]'', which is often considered to be one of the most successful textbooks in the history of mathematics.  Within it, the properties of geometrical objects and integers are deduced from a small set of axioms, thereby anticipating (and partly inspiring) the axiomatic method of modern mathematics.
-Euclid also wrote works on perspective, conic sections, spherical geometry, and possibly [[Quadric|quadric surfaces]]. Neither the year nor place of his birth have been established, nor the circumstances of his death.
+Euclid also wrote works on perspective, conic sections, spherical geometry, and possibly quadric surfaces. Neither the year nor place of his birth have been established, nor the circumstances of his death.
 == ''The Elements'' ==
@@ Line 8: / Line 8: @@
 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. In addition to providing some missing proofs, Euclid's text also includes sections on number theory and three-dimensional geometry.  In particular, Euclid's proof of the infinitude of prime numbers is in Book IX, Proposition 20.
-The geometrical system described in ''Elements'' was long known simply as "the" geometry. Today, however, it is often referred to as [[Euclidean geometry]] to distinguish it from other so-called ''non-Euclidean'' geometries which were discovered in the [[19th century]]. These new geometries grew out of more than two [[millennium|millennia]] of investigation into Euclid's [[Parallel postulate|fifth postulate]], one of the most-studied [[axiom]]s in all of mathematics. Most of these investigations involved attempts to prove the relatively complex and presumably non-intuitive fifth postulate using the other four (a feat which, if successful, would have shown the postulate to be in fact a theorem).
+The geometrical system described in ''Elements'' was long known simply as "the" geometry. Today, however, it is often referred to as [[Euclidean geometry]] to distinguish it from other so-called ''non-Euclidean'' geometries which were discovered in the [[19th century]]. These new geometries grew out of more than two millennia of investigation into Euclid's fifth postulate, one of the most-studied axioms in all of mathematics. Most of these investigations involved attempts to prove the relatively complex and presumably non-intuitive fifth postulate using the other four (a feat which, if successful, would have shown the postulate to be in fact a theorem).
 ==Other works==
@@ Line 21: / Line 21: @@
 There are four works credibly attributed to Euclid which have been lost.
 * ''Conics'' was a work on [[conic section]]s that was later extended by [[Apollonius of Perga]] into his famous work on the subject.
-* ''[[Porism]]s'' might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
+* ''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]].
+* ''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.
@@ Line 29: / Line 29: @@
 Almost nothing is known about Euclid outside of what is presented in ''Elements'' and his few other surviving books. What little biographical information we do have comes largely from commentaries by [[Proclus]] and [[Pappus of Alexandria]]: he was active at the [[Library of Alexandria|great library in Alexandria]] and may have studied at [[Plato]]'s [[Academe]] in [[Greece]], but his exact lifespan and place of birth are unknown.
-In the [[Middle Ages]], writers sometimes referred to him as ''[[Euclid of Megara]]'', confusing him with a Greek [[Socrates|Socratic]] [[philosopher]] who lived approximately one century earlier.
+In the Middle Ages, writers sometimes referred to him as ''[[Euclid of Megara]]'', confusing him with a Greek [[Socrates|Socratic]] philosopher who lived approximately one century earlier.
 == References ==