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Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus (mathematician), Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's Elements, his ''Elements'' (c. 300 BC). It is an example of an ''algorithm'', a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce Fraction (mathematics), fractions to their Irreducible fraction, simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Εὐκλείδης
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken for the earlier philosopher Euclid of Megara, causing his biography to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. 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 " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Relation
In mathematics, Euclidean relations are a class of binary relations that formalize " Axiom 1" in Euclid's ''Elements'': "Magnitudes which are equal to the same are equal to each other." Definition A binary relation ''R'' on a set ''X'' is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every ''a'', ''b'', ''c'' in ''X'', if ''a'' is related to ''b'' and ''c'', then ''b'' is related to ''c''.. To write this in predicate logic: :\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c). Dually, a relation ''R'' on ''X'' is left Euclidean if for every ''a'', ''b'', ''c'' in ''X'', if ''b'' is related to ''a'' and ''c'' is related to ''a'', then ''b'' is related to ''c'': :\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c). Properties # Due to the commutativity of ∧ in the definition's antecedent, ''aRb'' ∧ ''aRc'' even implies ''bRc'' ∧ ''cRb'' when ''R'' is right Euclidean. Similarly, ''bRa'' ∧ ''cRa'' implies ''bRc' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Things Named After Euclid
This is a list of topics named after the Greek mathematician Euclid. Mathematics Number theory * Euclidean algorithm ** Extended Euclidean algorithm * Euclidean division * Euclid–Euler theorem * Euclid number * Euclid's lemma * Euclid's orchard * Euclid–Mullin sequence * Euclid's theorem Algebra * Euclidean domain * Euclidean field Geometry * Euclidean group * Euclidean geometry **Non-Euclidean geometry * Euclid's formula * Euclidean distance **Euclidean distance matrix * Euclidean space **Pseudo-Euclidean space * Euclidean vector * Euclidean relation * Euclidean topology * Euclid's fifth postulate Other * Euclid's Elements * Euclid's Optics * Euclid (spacecraft) * Euclid, Ohio Euclid, Minnesota * Euclidean rhythm a term coined by Godfried Toussaint in his 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms" * Euclid (computer program) * Euclid (programming language) * Euclid, a supercomputer built by the fictional character Maximillian Cohen in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometer
A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * Manava (c. 750 BC–690 BC) – Euclidean geometry * Thales of Miletus (c. 624 BC – c. 546 BC) – Euclidean geometry * Pythagoras (c. 570 BC – c. 495 BC) – Euclidean geometry, Pythagorean theorem * Zeno of Elea (c. 490 BC – c. 430 BC) – Euclidean geometry * Hippocrates of Chios (born c. 470 – 410 BC) – first systematically organized '' Stoicheia – Elements'' (geometry textbook) * Mozi (c. 468 BC – c. 391 BC) * Plato (427–347 BC) * Theaetetus (c. 417 BC – 369 BC) * Autolycus of Pitane (360–c. 290 BC) – astronomy, spherical geometry * Euclid (fl. 300 BC) – '' Elements'', Euclidean geometry (sometimes called the "father of geometry") * Apollonius of Perga (c. 262 BC – c. 190 BC) – Euclidean geometry, conic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars. The earliest mathematical texts available are from Mesopotamia and Egypt – '' Plimpton 322'' ( Babylonian c. 2000 – 1900 BC), the ''Rhind Mathematical Papyrus'' ( Egyptian c. 1800 BC) and the '' Moscow Mathematical Papyrus'' (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most anci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid Of Megara
Euclid of Megara (; grc-gre, Εὐκλείδης ; 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''. Life Euclid was born in Megara. In Athens he became a follower of Socrates: so eager was he to hear the teaching and discourse of Socrates, that when, for a time, Athens had a ban on any citizen of Megara entering the city, Euclid would sneak into Athens after nightfall disguised as a woman, to hear him speak. He is represented in the preface of Plato's '' Theaetetus'' as being responsible for writing down the conversation between Socrates and the young Theaetetus many years earlier. Socrates is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
