# Euclid

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Related to Euclid: Euclid elements

## Euclid

(yo͞o`klĭd), fl. 300 B.C., Greek mathematician. Little is known of his life other than the fact that he taught at Alexandria, being associated with the school that grew up there in the late 4th cent. B.C. He is famous for his Elements, a presentation in thirteen books of the geometry and other mathematics known in his day. The first six books cover elementary plane geometry and have served since as the basis for most beginning courses on this subject. The other books of the Elements treat the theory of numbers and certain problems in arithmetic (on a geometric basis) and solid geometry, including the five regular polyhedra, or Platonic solids. A few modern historians have questioned Euclid's authorship of the Elements, but he is definitely known to have written other works, most notably the Optics.

The great contribution of Euclid was his use of a deductive system for the presentation of mathematics. Primary terms, such as point and line, are defined; unproved assumptions, or postulates, regarding these terms are stated; and a series of statements are then deduced logically from the definitions and postulates. Although Euclid's system no longer satisfies modern requirements of logical rigor, its importance in influencing the direction and method of the development of mathematics is undisputed.

One consequence of the critical examination of Euclid's system was the discovery in the early 19th cent. that his fifth postulate, equivalent to the statement that one and only one line parallel to a given line can be drawn through a point external to the line, can not be proved from the other postulates; on the contrary, by substituting a different postulate for this parallel postulate two different self-consistent forms of non-Euclidean geometrynon-Euclidean geometry,
branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
were deduced, one by Nikolai I. LobachevskyLobachevsky, Nikolai Ivanovich
, 1793–1856, Russian mathematician. A pioneer in non-Euclidean geometry, he challenged Euclid's fifth postulate that one and only one line parallel to a given line can be drawn through a fixed point external to the line; he developed,
(1826) and independently by János BolyaiBolyai
, family of Hungarian mathematicians. The father, Farkas, or Wolfgang, Bolyai, 1775–1856, b. Bolya, Transylvania, was educated in Nagyszeben from 1781 to 1796 and studied in Germany during the next three years at Jena
(1832) and another by Bernhard RiemannRiemann, Bernhard
(Georg Friedrich Bernhard Riemann) , 1826–66, German mathematician. He studied at the universities of Göttingen and Berlin and was professor at Göttingen from 1859.
(1854).

### Bibliography

See D. Berlinski, The King of Infinite Space: Euclid and His Elements (2013).

## Euclid,

city (1990 pop. 54,875), Cuyahoga co., NE Ohio, a suburb adjoining Cleveland, on Lake Erie; settled 1798, inc. 1848. Named for the famous Greek mathematician, the industrial city manufactures metal goods, electrical supplies and equipment, airplane and automobile parts, and machinery. The National American Shrine of Our Lady of Lourdes is there.

## Euclid

1. 3rd century bc, Greek mathematician of Alexandria; author of Elements, which sets out the principles of geometry and remained a text until the 19th century at least
2. the works of Euclid, esp his system of geometry

## Euclid

(language)
(Named after the Greek geometer, fl ca 300 BC.) A Pascal descendant for development of verifiable system software. No goto, no side effects, no global assignments, no functional arguments, no nested procedures, no floats, no enumeration types. Pointers are treated as indices of special arrays called collections. To prevent aliasing, Euclid forbids any overlap in the list of actual parameters of a procedure. Each procedure gives an imports list, and the compiler determines the identifiers that are implicitly imported. Iterators.

Ottawa Euclid is a variant.

["Report on the Programming Language Euclid", B.W. Lampson et al, SIGPLAN Notices 12(2):1-79, Feb 1977].
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Case Parameter Evaluation 1 5 Strategy MAE 0.1351 0.0761 DE/best/1 Euclid 0.1772 0.1218 NCLS 0.8879 0.8153 MAE 0.0920 0.0828 DE/rand/1 Euclid 0.15402 0.1342 NCLS 0.9025 0.7965 MAE 0.0920 0.0828 DE/rand-to-best/1 Euclid 0.1540 0.1342 NCLS 0.9025 0.7965 MAE 0.0889 0.0842 DE/best/2 Euclid 0.1337 0.1364 NCLS 0.9154 0.7854 MAE 17.0368 9.6299 DE/rand/2 Euclid 27.0270 14.6791 NCLS -16.1006 -21.0442 Case Parameter Evaluation 6 7 Strategy MAE 0.0494 0.1308 DE/best/1 Euclid 0.0840 0.1931 NCLS 0.8895 0.7627 MAE 0.0392 0.0653 DE/rand/1 Euclid 0.0593 0.0956 NCLS 0.9220 0.8826 MAE 0.0392 0.0653 DE/rand-to-best/1 Euclid 0.0593 0.0956 NCLS 0.9220 0.8826 MAE 0.0825 0.0966 DE/best/2 Euclid 0.1259 0.1400 NCLS 0.8344 0.8280 MAE 19.0175 2.8862 DE/rand/2 Euclid 28.9367 4.2267 NCLS -37.1143 -4.1928 Note.
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It is at that time that Euclid presented the King with the manuscript we report below.

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