# Euclid

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## 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

(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

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].
References in periodicals archive ?
optimal] = 2 is a reasonable optimal Minkowski exponent and mar perceptual space is considered as Euclidian.
Each of the 113 countries is positioned on the map on the basis of the smallest Euclidian distance between its input data vector and each of the 108 estimated vectors.
By measuring the position of the probe relative to the phantom in two perpendicular camera directions the total Euclidian error can be computed for both the target and entry point.
j]) is the Euclidian distance between a new observation x and the nearest neighbors of reference defined by y.
Clavius believed that Euclidian geometry held the secret for restoring order to society and re-establishing the absolute authority of the church.
According to the HCA results, the common bean cultivars were divided into 4 groups when a Euclidian distance at 1.
Performances of the gradient descent and the coordinate dictionary algorithms are compared in Table V in terms of Euclidian distance error and time consumed while finding the best solution for the objects.
LBP based technique has shown minimum deviation against Euclidian distance, mean squared error and peak signal to noise ratio.
Yet, there is a positive reason for sticking to competition as an explanatory model: it often provides the best model to describe what happens in the real world, just as a Euclidian line with zero thickness is useful to measure real-world lines.
A cluster analysis was also performed based on dissimilarity measures of Euclidian distance and using two-step clustering (Maroco, 2007).
For the Euclidian ball of the radio r > 0 the first eigenvalue is v = 1/r with its own space generated by the coordinate functions.
Quantitative data were submitted to multivariate analysis through Euclidian distance average, and the relative contribution of the characters for genetic divergence by Singh method (1981).

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