Euclid
Euclid , a
Greek mathematician, who lived in
Alexandria,
Hellenistic Egypt, almost certainly during the reign of
Ptolemy I , is often considered to be the "father of
geometry". His most popular work,
Elements, is thought to be one of the most successful
textbooks in the
history of mathematics. Within it, the properties of
geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of
mathematics.
Although best-known for its geometric results, the
Elements also includes various results in number theory, such as the connection between perfect numbers and
Mersenne primes, the proof of the infinitude of prime numbers, Euclid's lemma on factorization , and the
Euclidean algorithm
Encyclopedia
Euclid , a
Greek mathematician, who lived in
Alexandria,
Hellenistic Egypt, almost certainly during the reign of
Ptolemy I , is often considered to be the "father of
geometry". His most popular work,
Elements, is thought to be one of the most successful
textbooks in the
history of mathematics. Within it, the properties of
geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of
mathematics.
Although best-known for its geometric results, the
Elements also includes various results in number theory, such as the connection between perfect numbers and
Mersenne primes, the proof of the infinitude of prime numbers, Euclid's lemma on factorization , and the
Euclidean algorithm for finding the greatest common divisor of two numbers.
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
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 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 .
Tributes
- 4354 Euclides is an asteriod named after him
- Lunar Crater Euclid is named after him
- He is the subject of Euclid: The Creation of Mathematics, 1999, by Benno Artmann
Other works
In addition to the
Elements, four 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 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 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 sections that was later extended by Apollonius of Perga into his famous work on the subject.
- 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 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
Little is known about Euclid outside of what is presented in
Elements and his other surviving books. 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. Euclid's exact lifespan and place of birth are unknown. Some writers in the
Middle Ages erroneously confused him with Euclid of Megara, a Greek
Socratic philosopher who lived approximately one century earlier.
References
- Bulmer-Thomas, Ivor . "Euclid". Dictionary of Scientific Biography.
- Heath, Thomas L. . The Thirteen Books of Euclid's Elements, Vol. 1 . New York: Dover Publications. ISBN 0-486-60088-2.
- Heath, Thomas L. . A History of Greek Mathematics, 2 Vols. New York: Dover Publications. ISBN 0-486-24073-8 / ISBN 0-486-24074-6.
- Kline, Morris . Mathematics: The Loss of Certainty. Oxford: Oxford University Press. ISBN 0-19-502754-X.
External links
- , All thirteen books, with interactive diagrams using Java.
- , with the original Greek and an English translation on facing pages .
- for The Medieval Latin translation of the Data of Euclid by Shuntaro Ito
