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 ...

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Euclid, Euclid - The Elements, Euclid - Other works, Euclid - Biographical sources

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Euclid, Euclid - The Elements, Euclid - Other works, Euclid - Biographical sources

Read more here: » Euclid: Encyclopedia II - Euclid - Other works

Euclid is located at 41°35'44" North, 81°31'9" West (41.595563, -81.519176)GR1. According to the United States Census Bureau, the city has a total area of 30.0 km² (11.6 mi²). 27.7 km² (10.7 mi²) of it is land and 2.2 km² (0.9 mi²) of it is water. The total area is 7.43% water. It is directly on Lake Erie, so it contains an area of beach front and beach front property along the North. ...

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Euclid Ohio, Euclid Ohio - Geography, Euclid Ohio - Demographics, Euclid Ohio - Notable Attractions/History, Euclid Ohio - Ethnic Groups

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As of the censusGR2 of 2000, there are 52,717 people, 24,353 households, and 13,491 families residing in the city. The population density is 1,900.5/km² (4,923.2/mi²). There are 26,123 housing units at an average density of 941.8/km² (2,439.6/mi²). The racial makeup of the city is 66.36% White, 30.57% African American, 0.12% Native American, 0.94% Asian, 0.02% Pacific Islander, 0.35% from other races, and 1.64% from two or more races. 1.15% of the po ...

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Euclid Ohio, Euclid Ohio - Geography, Euclid Ohio - Demographics, Euclid Ohio - Notable Attractions/History, Euclid Ohio - Ethnic Groups

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By around 1915, Euclid Beach Park would only admit African Americans on certain days, as part of its efforts to promote what its management saw as a family-friendly atmosphere. On the other days, the park's special police enforcement team would eject blacks entirely. On the days when blacks were allowed into the park, they were not permitted to interact with white park-goers. A series of protests occurred at Euclid Beach Park in 1946. On July 21, 1946 twenty people from American Youth for Democracy, United Negroes and Allied Veterans ...

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Euclid Beach Park, Euclid Beach Park - Rides and attractions, Euclid Beach Park - The Euclid Beach Carousel, Euclid Beach Park - Roller coasters, Euclid Beach Park - The Euclid Beach Park Riot, Euclid Beach Park - Decline and closure of Euclid Beach Park

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The South Euclid-Lyndhurst City School District is comprised of 7 elementary schools (Adrian, Lowden, Ridgebury, Rowland, Sunview, Southlyn and Greenview); 1 junior high school (Memorial); and 1 high school (Charles F. Brush). There are two elementary level parochial schools in South Euclid: St. Gregory the Great (Roman Catholic), and St. John's (Lutheran). Regina High School is a parochial high school, also located in South Euclid. Their women's basketball team has been the Ohio state champions in 2000, 2001, 2002, 2003 and 2005. South Euclid has ...

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South Euclid Ohio, South Euclid Ohio - Geography, South Euclid Ohio - Demographics, South Euclid Ohio - Schools, South Euclid Ohio - Famous Natives

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Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathemat ...

Including:

Geometry - The earliest geometry Geometry - The Greek period c. 600 B.C. – 600 A.D.
Geometry - Thales and Pythagoras Geometry - Plato Geometry - Euclid Geometry - Archimedes Geometry - After Archimedes
Geometry - The Middle Ages Renaissance and Reformation Geometry - The 17th and early 18th centuries Geometry - The late 18th and 19th centuries
Geometry - Non-Euclidean geometry Geometry - Introduction of mathematical rigor Geometry - Analysis situs or topology
Geometry - The 20th century

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South Euclid is located at 41°31'19" North, 81°31'40" West (41.522071, -81.527677)GR1. It is bordered by Cleveland, East Cleveland, Cleveland Heights, University Heights, Beachwood, Lyndhurst, Richmond Heights, and Euclid. According to the United States Census Bureau, the city has a total area of 12.1 km² (4.7 mi²). 12.1 km² (4.7 mi²) of it is land and none of the area is covered with water. ...

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South Euclid Ohio, South Euclid Ohio - Geography, South Euclid Ohio - Demographics, South Euclid Ohio - Schools, South Euclid Ohio - Famous Natives

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In a formal logical system, that is, a set of propositions that are consistent with one another, it is probable that some of the statements can be deduced from one another. For example, in the syllogism, "All men are mortal; Socrates is a man; Socrates is mortal" the last claim can be deduced from the former two. A first principle is one that cannot be deduced from any other. The classic example is that of Euclid's geometry ...

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Euclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of which are today called axioms). Postulates in Book I: A straight line segment can be drawn by joining any two points. A straight line segment can be extended indefinitely in a straight line. Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center. All right angles are congruent. If two lines are drawn w ...

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Euclid's Elements, Euclid's Elements - First principles, Euclid's Elements - Parallel postulate, Euclid's Elements - History

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This page lists some links to ancient philosophy. In Europe, the spread of Christianity through the Roman world marked the end of Hellenistic philosophy and ushered in the beginnings of Medieval philosophy. Ancient philosophy - Classical. Ancient philosophy - Greek. Pericles (495-429) Aspasia (469-406) Socrates(469-399) Euclid of Megara (450-380) Antisthenes (445-360) Aristippus (435-356) Plato (429-347)Including:

Ancient philosophy - Classical
Ancient philosophy - Greek Ancient philosophy - Roman
Ancient philosophy - Schools of thought in the Hellenistic period Ancient philosophy - Vedic philosophy Ancient philosophy - Classical Indian philosophy Ancient philosophy - Old Iranian philosophy Ancient philosophy - Chinese philosophy

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Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC 297 BC 296 BC 295 BC Births Deaths of children Events End of Jomon era and beginning of Yayoi era in Japan. Euclid's Elements written. Antioch was founded by ...

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Euclid Beach Park was built on a beach, which was part of the attraction, and for a time, most of it. Another early facet of the park was its dance hall. After the Humphreys acquired the park, many more attractions became a part of Euclid Beach. Euclid Beach Park - The Euclid Beach Carousel. The first carousel that Euclid Beach Park used was the Philadelphia Toboggan Company's number nine design. It debuted in 1905; in 1910, it was replaced by carousel number nineteen. This carousel had forty-four horses a ...

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Euclid Beach Park, Euclid Beach Park - Rides and attractions, Euclid Beach Park - The Euclid Beach Carousel, Euclid Beach Park - Roller coasters, Euclid Beach Park - The Euclid Beach Park Riot, Euclid Beach Park - Decline and closure of Euclid Beach Park

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The history of the theorem called Pythagorean can be divided into three parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, and proofs of the theorem. Circa 2500 BC, Megalithic monuments on the British Isles incorporate right triangles with integer sides. B.L. van der Waerden conjectures that these Pythagorean triples were discovered algebraically. Written between 2000 - 1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem, the s ...

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Pythagorean theorem, Pythagorean theorem - History, Pythagorean theorem - Proofs, Pythagorean theorem - Geometrical proof, Pythagorean theorem - A visual proof, Pythagorean theorem - Converse of the theorem, Pythagorean theorem - Algebraic Proof, Pythagorean theorem - Pythagorean triples, Pythagorean theorem - Generalizations, Pythagorean theorem - The Pythagorean theorem in non-Euclidean geometry, Pythagorean theorem - Other facts, Pythagorean theorem - Notes

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As of the censusGR2 of 2000, there are 23,537 people, 9,542 households, and 6,421 families residing in the city. The population density is 1,937.7/km² (5,019.2/mi²). There are 9,854 housing units at an average density of 811.2/km² (2,101.3/mi²). The racial makeup of the city is 75.24% White, 21.38% African American, 0.10% Native American, 1.50% Asian, 0.01% Pacific Islander, 0.42% from other races, and 1.35% from two or more races. 1.02% of ...

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South Euclid Ohio, South Euclid Ohio - Geography, South Euclid Ohio - Demographics, South Euclid Ohio - Schools, South Euclid Ohio - Famous Natives

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Say p is a prime factor of ab, but also state that it is not a factor of a. Therefore, rp = ab, where r is the other corresponding factor to produce ab. As p is prime, and also because it is not a factor of a, a and p must be coprime. This means that two integers x and y can be found so that 1 = px + ay (Bézout's identity). Multiply b on both sid ...

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Euclid's lemma, Euclid's lemma - Proof of Proposition 30

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Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proven in a system where the parallel postulate is true, and that if they are assumed as axioms, then the parallel postulate can be proven. Strictly speaking, some of these are actually equivalent to the conjunction of Euclid's parallel postulate and its converse, and thus can be used to distinguish Euclidean geometry from both elliptic geometry and hyperbolic geometry simultaneously. One of the most important of these p ...

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Parallel postulate, Parallel postulate - Converse of Euclid's parallel postulate, Parallel postulate - Logically equivalent properties, Parallel postulate - History

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The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies “eternal forms”, or abstractions, of whi ...

See also:

Geometry, Geometry - The earliest geometry, Geometry - The Greek period c. 600 B.C. – 600 A.D., Geometry - Thales and Pythagoras, Geometry - Plato, Geometry - Euclid, Geometry - Archimedes, Geometry - After Archimedes, Geometry - The Middle Ages Renaissance and Reformation, Geometry - The 17th and early 18th centuries, Geometry - The late 18th and 19th centuries, Geometry - Non-Euclidean geometry, Geometry - Introduction of mathematical rigor, Geometry - Analysis situs or topology, Geometry - The 20th century

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List of publications in mathematics - Euclid's Elements. Euclid Publication data: c. 300 BC Online version: Interactive Java version Description: This is probably not only the most important work in geometry but the most important work in mathematics. It contains many important results in geometry, number theory and the first algorithm as well. The Elements is still a valuable resource and a good introduction to algorithm. More tha ...

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List of publications in mathematics, List of publications in mathematics - Geometry, List of publications in mathematics - Euclid's Elements, List of publications in mathematics - La Géométrie, List of publications in mathematics - Logic, List of publications in mathematics - Begriffsschrift, List of publications in mathematics - Formulario mathematico, List of publications in mathematics - Principia Mathematica, List of publications in mathematics - Gödel's incompleteness theorem, List of publications in mathematics - Information theory, List of publications in mathematics - Number theory, List of publications in mathematics - Disquisitiones Arithmeticae, List of publications in mathematics - On the Number of Primes Less Than a Given Magnitude, List of publications in mathematics - Vorlesungen über Zahlentheorie, List of publications in mathematics - Number Theory An approach through history from Hammurapi to Legendre, List of publications in mathematics - Calculus, List of publications in mathematics - Philosophiae Naturalis Principia Mathematica, List of publications in mathematics - Newton's Principia for the Common Reader, List of publications in mathematics - Calculus and Calculus on Manifolds, List of publications in mathematics - Numerical analysis, List of publications in mathematics - Method of Fluxions, List of publications in mathematics - Game theory, List of publications in mathematics - Evolution and the Theory of Games, List of publications in mathematics - Theory of Games and Economic Behavior, List of publications in mathematics - On Numbers and Games, List of publications in mathematics - Winning Ways for your Mathematical Plays, List of publications in mathematics - Fractals, List of publications in mathematics - How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, List of publications in mathematics - Early manuscripts, List of publications in mathematics - Rhind Mathematical Papyrus, List of publications in mathematics - The Nine Chapters on the Mathematical Art, List of publications in mathematics - Archimedes Palimpsest, List of publications in mathematics - The Sand Reckoner, List of publications in mathematics - Textbooks, List of publications in mathematics - Course of Pure Mathematics, List of publications in mathematics - Art of Problem Solving, List of publications in mathematics - Metalogic: an Introduction to the Metatheory of Standard First Order Logic, List of publications in mathematics - Popular writing, List of publications in mathematics - Gödel Escher Bach, List of publications in mathematics - The World of Mathematics, List of publications in mathematics - Arithmetic, List of publications in mathematics - Arithmetick: or The Grounde of Arts, List of publications in mathematics - The Schoolmaster's Assistant Being a Compendium of Arithmetic both Practical and Theoretical, List of publications in mathematics - Abstract algebra, List of publications in mathematics - Moderne Algebra, List of publications in mathematics - Linear algebra, List of publications in mathematics - Algebraic geometry, List of publications in mathematics - Faisceaux Algébriques Cohérents, List of publications in mathematics - Géométrie Algébrique et Géométrie Analytique, List of publications in mathematics - Éléments de géométrie algébrique, List of publications in mathematics - Séminaire de géométrie algébrique, List of publications in mathematics - Algebraic Geometry, List of publications in mathematics - Universal algebra, List of publications in mathematics - Universal algebra, List of publications in mathematics - Group theory, List of publications in mathematics - Monoid, List of publications in mathematics - Topology, List of publications in mathematics - Topologie, List of publications in mathematics - Topology, List of publications in mathematics - General Topology, List of publications in mathematics - Graph theory, List of publications in mathematics - Category theory, List of publications in mathematics - Categories for the Working Mathematician, List of publications in mathematics - Category Theory for Computing Science, List of publications in mathematics - Order theory, List of publications in mathematics - Trigonometry, List of publications in mathematics - Differential geometry, List of publications in mathematics - Differential topology, List of publications in mathematics - Topology from the Differentiable Viewpoint, List of publications in mathematics - Algebraic topology, List of publications in mathematics - Algebraic Topology, List of publications in mathematics - Fractal geometry, List of publications in mathematics - Discrete mathematics, List of publications in mathematics - Combinatorics, List of publications in mathematics - Set theory, List of publications in mathematics - Grundzüge der Mengenlehre, List of publications in mathematics - Naive Set Theory, List of publications in mathematics - Cardinal and Ordinal Numbers, List of publications in mathematics - The Consistency of the Continuum Hypothesis, List of publications in mathematics - Set Theory and the Continuum Hypothesis, List of publications in mathematics - Set Theory: An Introduction to Independence Proofs, List of publications in mathematics - Optimization, List of publications in mathematics - The New Variational Method, List of publications in mathematics - Decomposition Principle for Linear Programs., List of publications in mathematics - Network Flows and General Matchings, List of publications in mathematics - Paths trees and Flowers, List of publications in mathematics - The complexity of theorem proving procedures, List of publications in mathematics - Reducibility among combinatorial problems, List of publications in mathematics - How good is the simplex algorithm?, List of publications in mathematics - Linear Programming and Polynomial time algorithms, List of publications in mathematics - New polynomial-time algorithm for linear programming, List of publications in mathematics - Interior Point Polynomial Algorithms in Convex Programming

Read more here: » List of publications in mathematics: Encyclopedia II - List of publications in mathematics - Geometry

While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of non-Euclidean geometries began almost as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions (five axioms and five postulates) and sought to prove all the other results (propositions) in the work. The most notorious of t ...

See also:

Non-Euclidean geometry, Non-Euclidean geometry - History, Non-Euclidean geometry - Reference

Read more here: » Non-Euclidean geometry: Encyclopedia II - Non-Euclidean geometry - History