Euclid

Euclid (/ˈjuːklɪd/; Ancient Greek: ΕὐκλείδηςEukleídēs, pronounced [eu̯.kleː.dɛːs]; fl. 300 BC), sometimes given the name Euclid of Alexandria[1] to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry"[1] or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.[2][3][4] In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.

Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious".[5]

Euclid
Euklid-von-Alexandria 1
Eukleides of Alexandria
BornMid-4th century BC
DiedMid-3rd century BC
ResidenceAlexandria, Hellenistic Egypt
Known for
Scientific career
FieldsMathematics
Scuola di atene 23
Bramante as Euclid or Archimedes in the School of Athens.

Life

Very few original references to Euclid survive, so little is known about his life. He was likely born c. 325 BC, although the place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes (c. 287 BC – c. 212 BC) onward, and is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").[6] The few historical references to Euclid were written centuries after he lived, namely by Proclus c. 450 AD.[7]

A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be fictitious.[8] If he came from Alexandria, he would have known the Serapeum of Alexandria, and the Library of Alexandria, and may have worked there during his time. Euclid's arrival in Alexandria came about ten years after its founding by Alexander the Great, which means he arrived c. 322 BC.[9]

Proclus introduces Euclid only briefly in his Commentary on the Elements. According to Proclus, Euclid supposedly belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato (particularly Theaetetus and Philip of Opus.) Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I (c. 367 BC – 282 BC) because he was mentioned by Archimedes. Although the apparent citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before Archimedes wrote his.[10] Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry."[11] This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.[12]

Euclidis quae supersunt omnia
Euclidis quae supersunt omnia (1704)

Euclid died c. 270 BC, presumably in Alexandria.[9] In the only other key reference to Euclid, Pappus of Alexandria (c. 320 AD) briefly mentioned that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought" c. 247–222 BC.[13][14]

Because the lack of biographical information is unusual for the period (extensive biographies being available for most significant Greek mathematicians several centuries before and after Euclid), some researchers have proposed that Euclid was not a historical personage, and that his works were written by a team of mathematicians who took the name Euclid from Euclid of Megara (à la Bourbaki). However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.[15]

Elements

P. Oxy. I 29
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.[16]

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, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.[17]

There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon",[18] while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

Although best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

Fragments

The Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the Elements of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 AD.[19]

The classic translation of T. L. Heath, reads:[20]

If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

Other works

Euclid Dodecahedron 1
Euclid's construction of a regular dodecahedron.
Cube to dodecahedron 240px
Construction of a dodecahedron by placing faces on the edges of a cube.

In addition to the Elements, at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements, with definitions and proved propositions.

  • 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 first-century AD work by Heron of Alexandria.
  • Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J O'Connor and E F Robertson who name Theon of Alexandria as a more likely author.[21]
  • Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.
EuclidStatueOxford
19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History
  • Optics is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Claudius Ptolemy.

Lost works

Other works are credibly attributed to Euclid, but 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. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.
  • 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 (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.
  • Several works on mechanics are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. On the Balance treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.

Legacy

The European Space Agency's (ESA) Euclid spacecraft was named in his honor.[22]

See also

References

  1. ^ a b Bruno, Leonard C. (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 125. ISBN 978-0-7876-3813-9. OCLC 41497065.
  2. ^ Ball, pp. 50–62.
  3. ^ Boyer, pp. 100–19.
  4. ^ Macardle, et al. (2008). Scientists: Extraordinary People Who Altered the Course of History. New York: Metro Books. g. 12.
  5. ^ Harper, Douglas. "Euclidean (adj.)". Online Etymology Dictionary. Retrieved March 18, 2015.
  6. ^ Heath (1981), p. 357
  7. ^ Joyce, David. Euclid. Clark University Department of Mathematics and Computer Science. [1]
  8. ^ O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Heath 1956, p. 4; Heath 1981, p. 355.
  9. ^ a b Bruno, Leonard C. (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 126. ISBN 978-0-7876-3813-9. OCLC 41497065.
  10. ^ Proclus, p. XXX; O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"
  11. ^ Proclus, p. 57
  12. ^ Boyer, p. 96.
  13. ^ Heath (1956), p. 2.
  14. ^ "Conic Sections in Ancient Greece".
  15. ^ O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Jean Itard (1962). Les livres arithmétiques d'Euclide.
  16. ^ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26.
  17. ^ Struik p. 51 ("their logical structure has influenced scientific thinking perhaps more than any other text in the world").
  18. ^ Heath (1981), p. 360.
  19. ^ Fowler, David (1999). The Mathematics of Plato's Academy (Second ed.). Oxford: Clarendon Press. ISBN 978-0-19-850258-6.
  20. ^ Bill Casselman, One of the oldest extant diagrams from Euclid
  21. ^ O'Connor, John J.; Robertson, Edmund F., "Theon of Alexandria"
  22. ^ "NASA Delivers Detectors for ESA's Euclid Spacecraft". NASA. 2017.

Works cited

Further reading

  • DeLacy, Estelle Allen (1963). Euclid and Geometry. New York: Franklin Watts.
  • Knorr, Wilbur Richard (1975). The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry. Dordrecht, Holland: D. Reidel. ISBN 978-90-277-0509-9.
  • Mueller, Ian (1981). Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge, MA: MIT Press. ISBN 978-0-262-13163-6.
  • Reid, Constance (1963). A Long Way from Euclid. New York: Crowell.
  • Szabó, Árpád (1978). The Beginnings of Greek Mathematics. A.M. Ungar, trans. Dordrecht, Holland: D. Reidel. ISBN 978-90-277-0819-9.

External links

! (The Dismemberment Plan album)

! is the debut studio album by American indie rock band The Dismemberment Plan. It was released on October 2, 1995 on DeSoto Records. The band's original drummer Steve Cummings played on the album but left shortly after its release.

A (New York City Subway service)

The A Eighth Avenue Express is a rapid transit service in the B Division of the New York City Subway. Its route emblem, or "bullet", is colored blue since it uses the IND Eighth Avenue Line in Manhattan.The A operates at all times. Daytime service operates between 207th Street in Inwood, Manhattan and Far Rockaway or Lefferts Boulevard in Richmond Hill, Queens, making express stops in Manhattan and Brooklyn and local stops in Queens. Limited rush hour service also operates to or from Beach 116th Street in Rockaway Park, Queens. Late night service operates only between 207th Street and Far Rockaway, making local stops along its entire route; during this time, a shuttle train (the Lefferts Boulevard Shuttle) operates between Euclid Avenue and Lefferts Boulevard.The A provides the longest one-seat ride in the system—at 32.39 miles (52.13 km), between Inwood and Far Rockaway—and has a weekday ridership of 600,000.

Cleveland Clinic

The Cleveland Clinic is an American academic medical center based in Cleveland, Ohio. Owned and operated by the Cleveland Clinic Foundation, an Ohio nonprofit corporation established in 1921, it runs a 170-acre campus in Cleveland, as well as 10 regional hospitals and 19 family health centers in north-east Ohio, and hospitals in Florida and Nevada. Tomislav Mihaljevic is the president and CEO.Outside the United States, there is a Cleveland Clinic in Toronto, Canada, and a Cleveland Clinic Abu Dhabi in the United Arab Emirates. In the UK, Cleveland Clinic London is scheduled to open in 2021.Cleveland Clinic's operating revenue in 2017 was $8.4 billion and its operating income $330 million. That year it recorded 7.6 million patient visits and 229,132 admissions. As of 2018, it has over 52,000 employees, a figure that includes over 11,800 nurses and over 3,600 physicians and scientists. It publishes the peer-reviewed journal Cleveland Clinic Journal of Medicine.

Dweezil Zappa

Dweezil Zappa (born Ian Donald Calvin Euclid Zappa; September 5, 1969) is an American rock guitarist and occasional actor. He is the son of musical composer and performer Frank Zappa. Exposed to the music industry from an early age, Dweezil developed a strong affinity for playing the guitar and producing music. Able to learn directly from guitarists such as Steve Vai and Eddie Van Halen, Dweezil released his first single (produced by Eddie Van Halen) at the age of 12.In addition to writing and recording his own music, Dweezil has carried on the legacy of his father's music by touring with the group Zappa Plays Zappa. The band features renditions of Zappa's original material and the lineup has often included Zappa alumni such as Napoleon Murphy Brock, Steve Vai, Terry Bozzio and others.

Euclid's Elements

The Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century.

Euclid's Elements has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482, with the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.

Euclid, Ohio

Euclid is a city in Cuyahoga County, Ohio, United States. It is an inner ring suburb of Cleveland. As of the 2010 census, the city had a total population of 48,920. In 2009, Euclid celebrated its bicentennial.

Euclid (spacecraft)

Euclid is a visible to near-infrared space telescope currently under development by the European Space Agency (ESA) and the Euclid Consortium. The objective of the Euclid mission is to better understand dark energy and dark matter by accurately measuring the acceleration of the universe. To achieve this, the Korsch-type telescope will measure the shapes of galaxies at varying distances from Earth and investigate the relationship between distance and redshift. Dark energy is generally accepted as contributing to the increased acceleration of the expanding universe, so understanding this relationship will help to refine how physicists and astrophysicists understand it. Euclid's mission advances and complements ESA's Planck telescope. The mission is named after the ancient Greek mathematician Euclid of Alexandria.

Euclid is a medium-class ("M-class") mission and is part of ESA's "Cosmic Vision" (2015–2025) scientific program. This class of missions have an ESA budget cap at around €500 million. Euclid was chosen in October 2011 together with Solar Orbiter, out of several competing missions. The launch date is planned for June 2022.

Euclid Avenue (Cleveland)

Euclid Avenue is a major street in Cleveland, Ohio. It runs northeasterly from Public Square in Downtown Cleveland, through the cities of East Cleveland, Euclid and Wickliffe, to the suburb of Willoughby as a part of U.S. Route 20 and U.S. Route 6. The street passes Playhouse Square, University Circle, Cleveland State University, the Cleveland Clinic, Severance Hall, Case Western Reserve University’s Maltz Performing Arts Center (formerly The Temple Tifereth Israel ), Case Western Reserve University and University Hospitals Case Medical Center. The HealthLine bus rapid transit line runs in designated bus lanes in the median of Euclid Avenue from Public Square to Louis Stokes Station at Windermere in East Cleveland.

It received nationwide attention from the 1860s to the 1920s for its beauty and wealth, including a string of mansions that came to be known as Millionaires' Row. There are several theaters, banks, and churches along Euclid, as well as Cleveland's oldest extant building, the Dunham Tavern. A large reconstruction project, which brought the HealthLine to the street, was completed in 2008.

Euclid Avenue (IND Fulton Street Line)

Euclid Avenue is an express station on the IND Fulton Street Line of the New York City Subway, located at the intersection of Euclid and Pitkin Avenues in East New York, Brooklyn. It is served by the A train at all times and is the southern terminal for the C train at all times except nights. During nights, this is the northern terminal for the Lefferts Boulevard shuttle train from Ozone Park, Queens.

Construction on the Euclid Avenue station started in 1938, but this part of the Fulton Street Line did not open until 1948. The Fulton Street Line was extended to the east in 1956, connecting to the Fulton Street Elevated via a branch line that runs through the Grant Avenue station. Elevators were installed at Euclid Avenue circa 2000.

The station has four tracks and two island platforms. In terms of railroad directions, this is the southernmost station on the Fulton Street Line. The line was originally planned to extend further east as a four-track underground line; however, the four-track extension was never built. East of the station, there are connections to the Pitkin Yard as well as to the Fulton Street Elevated. The tracks themselves dead-end after the Fulton Street elevated spur diverges.

Euclid Trucks

The Euclid Company of Ohio was a company specialized in heavy equipment for earthmoving, namely dump trucks and wheel tractor-scrapers, that operated from the United States of America from the 1920s to the 1950s, then it was purchased and converted into a section of General Motors and later on by Hitachi Construction Machinery.

Euclid number

In mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.

Euclid of Megara

Euclid of Megara (; also Euclides, Eucleides; Greek: Εὐκλείδης ὁ Μεγαρεύς; c. 435 – c. 365 BC) was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BC, and was present at his death. He held the supreme good to be one, eternal and unchangeable, and denied the existence of anything contrary to the good. Editors and translators in the Middle Ages often confused him with Euclid of Alexandria when discussing the latter's Elements.

Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school (High School) as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas.

Geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, and curves, as well as the more advanced notions of manifolds and topology or metric.Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics.

Kent Smith (politician)

Kent Smith (born September 4, 1966) is the Representative of the 8th district of the Ohio House of Representatives. Smith is a resident of Euclid, Ohio, and served on the Euclid School Board for 12 years. He also served as head of the Democratic Party for Euclid, and has a degree from the Maxine Goodman Levin College of Urban Affairs at Cleveland State University. In 2014, Smith opted to run for the Ohio House of Representatives to replace Armond Budish, who was term-limited and sought election instead as Cuyahoga County Executive. He faced Republican Mikhail Alterman and Independent Jocelyn Conwell, and won with 71.47% of the vote. He would go on to be re-elected in 2016 and 2018. His district includes the Cuyahoga County communities of Beachwood, East Cleveland, Euclid, Richmond Heights, South Euclid, Woodmere, and parts of Cleveland.

Papyrus Oxyrhynchus 29

Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the Elements of Euclid in Greek. It was discovered by Grenfell and Hunt in 1897 in Oxyrhynchus. The fragment was originally dated to the end of the third century or the beginning of the fourth century, although more recent scholarship suggests a date of 75–125 CE. It is housed in the library of the University of Pennsylvania (in a University Museum, E 2748). The text was published by Grenfell and Hunt in 1898.

Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

This postulate does not specifically talk about parallel lines; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates.Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or, in other places known as neutral geometry).

Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

Although it is often argued that knowledge of the theorem predates him, the theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC) as it is he who, by tradition, is credited with its first proof, although no evidence of it exists. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.

The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

South Euclid, Ohio

South Euclid is a city in Cuyahoga County, Ohio, United States. It is an inner-ring suburb of Cleveland located on the city's east side. As of the 2010 census the population was 22,295.

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