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Geometry

Geometry

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 mathematics and algebraic geometry.)

The earliest geometry

The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. Chinese culture at this same time period was equally advanced, so it is likely that they had an equally advanced mathematics, but no artifacts have survived from which we could learn about it. This may be partly due to their early use of paper, rather than clay tablets or stone, to record their achievements.

The Greek period (c. 600 B.C. – 600 A.D.)

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 which physical objects are only approximations; and they developed the idea of an “axiomatic theory”, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, “Let none enter here who are ignorant of geometry.” Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. # Any two points can be joined by a straight line. # Any finite straight line can be extended in a straight line. # A circle can be drawn with any center and any radius. # All right angles are equal to each other. # If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks. The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The Middle Ages, Renaissance, and Reformation

The great library of Alexandria was burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. The Islamic ascendency in the Middle East, north Africa, and Spain began about 640 A.D. Original Arab mathematics during this period was primarily algebraic rather than geometric, though there were important commentaries on geometry. Omar Khayyám, for example, was a geometer as well as a poet. Scholarship in Europe declined until even the great works of antiquity were lost to them, and survived only in the Islamic centers of learning. When Europe started to emerge from the intellectual darkness of the Middle Ages, the writers of Ancient Greece and Rome were rediscovered in Islamic libraries and translated from Arabic into Latin. Euclid’s Elements of Geometry was recovered, and the rigorous deductive methods of geometry were relearned. Development of geometry in the style of Euclid resumed, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

The 17th and early 18th centuries

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591-1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Greek geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The late 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. It remained to prove mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The 20th century

External links

- [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at cut-the-knot
- [http://www.elvenkids.com/tools/geometria/Geometria.php Geometria] An online tool to compute lines, surfaces and volumes of the main plane and solid figures, through direct and indirect formulas.
- [http://www.geogebra.at/ Geogebra] A free dynamic geometry tool, useful for exploring geometry.
- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
- [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
- Stanford Encyclopedia of Philosophy:
- [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
- [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
- [http://www.egwald.com/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens Category:Geometry ko:기하학 ja:幾何学 simple:Geometry zh-min-nan:Kí-hô-ha̍k

Areas of mathematics

The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of Wikipedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — most of the traditional material would be divided amongst topics under analysis. This illustrates, in part, the difficulty of communicating the principles of any large-scale organisation. The research on most calculus topics was carried out in the eighteenth century, and has long been assimilated. The story of why fields exist as specialties involves in most cases quite a long intellectual history (and sometimes institutional history). The American Mathematical Society's [http://www.ams.org/msc/ Mathematics Subject Classification (2000 edition)] has been used as a starting point to ensure all areas are covered, and related areas are close together. However, the MSC aims to classify mathematical papers, not mathematics itself, so additional categories have been used. See also the list of lists of mathematical topics (not to be confused with the far longer list of mathematical topics).

Foundations / general

- 00: General
- 01: History and biography
- 03: Mathematical logic and foundations
- 97: Mathematics education
- 00: Recreational mathematics

Algebra

The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces is studied in linear algebra. ; Combinatorics (MSC 05) : Studies finite collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). It includes graph theory, used to describe inter-connected objects (a graph in this sense is a collection of connected points). See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. ; Order theory (MSC 06) : With any set of real numbers, it is possible to write them out in ascending order. Order Theory extends this idea to sets in general. It includes notions like lattices and ordered algebraic structures. See also the order theory glossary and the list of order topics. ; General algebraic systems (MSC 08) : Given a set, ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems. ; Number theory (MSC 11) : Number theory is traditionally concerned with the properties of integers. More recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. It can be divided into elementary number theory (where the integers are studied without use of techniques from other mathematical fields); analytic number theory (where calculus and complex analysis are used as tools); algebraic number theory (which studies the algebraic numbers - the roots of polynomials with integer coefficients); Geometric number theory; combinatorial number theory and computational number theory. See also the list of number theory topics
- 12: Field theory and polynomials
- 13: Commutative rings and algebras
- 15: Linear and multilinear algebra; matrix theory
- 16: Associative rings and associative algebras
- 17: Non-associative rings and non-associative algebras
- 18: Category theory; homological algebra
- 19: K-theory
- 20: Group theory and generalizations
- 22: Topological groups, Lie groups, and analysis upon them (Also transformation groups, abstract harmonic analysis)

Analysis

Analysis is primarily concerned with change. Rates of change, accumulated change, multiple things changing relative to (or independently of) one another, etc.
- 26: Real functions, including derivatives and integrals
- 28: Measure and integration
- 30: Complex functions, including approximation theory in the complex domain
- 31: Potential theory
- 32: Several complex variables and analytic spaces
- 33: Special functions
- 34: Ordinary differential equations
- 35: Partial differential equations
- 37: Dynamical systems and ergodic theory
- 39: Difference equations and functional equations
- 40: Sequences, series, summability
- 41: Approximations and expansions
- 42: Fourier analysis, including Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions
- 43: Abstract harmonic analysis
- 44: Integral transforms, operational calculus
- 45: Integral equations
- 46: Functional analysis, including infinite-dimensional holomorphy, integral transforms in distribution spaces
- 47: Operator theory
- 49: Calculus of variations and optimal control; optimization (including geometric integration theory)
- 58: Global analysis, analysis on manifolds (including infinite-dimensional holomorphy) (Also: probabilistic potential theory, numerical approximation, representation theory, analysis on manifolds)

Geometry

Geometry (MSC 51) deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. ;Convex geometry (MSC 52) ;Discrete or combinatorial geometry (MSC 52): may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation. It includes the study of shapes such as the Platonic solids and the notion of tessellation. ;Differential geometry (MSC 53): is the study of geometry using calculus, and is very closely related to differential topology. Covers such areas as Riemannian geometry, curvature and differential geometry of curves. See also the glossary of differential geometry and topology. ;Algebraic geometry (MSC 14): Given a polynomial of two real variables, then the points on a plane where that function is zero will form a curve. An algebraic curve extends this notion to polynomials over a field in a given number of variables. Algebraic geometry may be viewed the study of these curves. See also the list of algebraic geometry topics and list of algebraic surfaces. ;Topology: Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below. ;General topology (MSC 54): Also called point set topology. Properties of topological spaces. Includes such notions as open and closed sets, compact spaces, continuous functions, convergence, separation axioms, metric spaces, dimension theory. See also the glossary of general topology and the list of general topology topics. ;Algebraic topology (MSC 55): Properties of algebraic objects associated with a topological space and how these algebraic objects capture properties of such spaces. Contains areas like homology theory, cohomology theory, homotopy theory, and homological algebra. The latter deals with homotopy groups (including the fundamental group) as well as simplicial complexes and CW complexes (also called cell complexes). See also the list of algebraic topology topics. ;Manifolds (MSC 57): A manifold can be thought of as an n-dimensional generalization of a surface in the usual 3-dimensional Euclidean space. The study of manifolds includes differential topology, which looks at the properties of differentiable functions defined over a manifold. See also complex manifolds.

Applied mathematics

Probability and statistics

;Probability theory (MSC 60) : the study of how likely a given event is to occur.
- 60G/H: Stochastic processes (including probabilistic potential theory) ;Statistics (MSC 62): Analysis of data, and how representative it is. See also the list of statistical topics.

Computational sciences

- 65: Numerical analysis, including numerical methods
- 68: Computer science

Physical sciences

;Mechanics: addresses what happens when a real physical object is subjected to forces. This divides naturally into the study of rigid solids, deformable solids, and fluids, detailed below. ;Particle mechanics (MSC 70): In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics — the study of the motion of celestial objects. ;Mechanics of deformable solids (MSC 74) : Most real-world objects are not point-like nor perfectly rigid. More importantly, objects change shape when subjected to forces. This subject has a very strong overlap with continuum mechanics, which is concerned with continuous matter. It deals with such notions as stress, strain and elasticity. See also continuum mechanics. ;Fluid mechanics (MSC 76): Fluids in this sense includes not just liquids, but flowing gases, and even solids under certain situations. (For example, dry sand can behave like a fluid). It includes such notions as viscosity, turbulent flow and laminar flow (its opposite). See also fluid dynamics.
- 78: Optics, electromagnetic theory
- 80: Classical thermodynamics, heat transfer
- 81: Quantum theory, including quantum optics
- 82: Statistical mechanics, structure of matter
- 83: Relativity and gravitational theory, including relativistic mechanics
- 85: Astronomy and astrophysics
- 86: Geophysics

Non-physical sciences

- 90: Operations research, mathematical programming
- 91: Game theory, economics, social and behavioral sciences
- 92: Biology (see also mathematical biology) and other natural sciences
- 93: Systems theory; control, including optimal control
- 94: Information and communication, circuits
- 97: Mathematics education Category:Mathematics th:สาขาของคณิตศาสตร์

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry between the three sides of a right-angled triangle. The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

Where c is the length of the hypotenuse and a and b are the lengths of the other two sides, the theorem can be expressed as the following equation: :

a^2 + b^2 = c^2. \,

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third. A paraphrase of the Pythagorean theorem is : : In the diagram, the sum of the areas of the blue and red squares is equal to the area of the purple square. This works for any right triangle laid out on a flat plane.

A visual proof

This theorem may have a greater variety of known proofs than any other (though the law of quadratic reciprocity may also be a contender for that distinction). Image:Pythagorean_proof.png This illustration depicts one of them. The area of each large square is (a + b)². In both, the area of four identical triangles is removed. The remaining areas, a² + b² and c², are equal. Q.E.D. NB: This proof is very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry. In particular, while it is quite easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see proof 6 in the external link).

History

Euclidean geometry 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 BCE, 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 BCE, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem, the solution to which is a Pythagorean triple.
- Written between 1790 - 1750 BCE, during the reign of Hammurabi, the Mesopotamian tablet Plimpton 322 contains a large number of entries closely related to Pythagorean triples.
- Written sometime between 800 - 500 BCE in India, the Sulba Sutras contain a statement of the Pythagorean theorem and a list of Pythagorean triples discovered algebraically. The Apastamba Sulba Sutra (c. 600 BC) also contains a numerical proof of the theorem, using an area computation. (Numerical proof is a proof that uses specific numbers but in such a way that it can be generalized.) Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Bŭrk, this is the original proof of the theorem, and Pythagoras copied it. Many scholars find Bŭrk's claim unsubstantiated, however.
- Pythagoras, whose dates are commonly given as 569 - 475 BCE, used algebraic methods to construct Pythagorean triples, according to Proklos' commentary on Euclid. Proklos, however, wrote between 410 - 485 CE. According to Sir Thomas L. Heath, there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero did attribute the theorem to Pythagoras, they did so in such a way as to suggests that the attribution was widely known and undoubted.
- Circa 400 BCE, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry.
- Circa 300 BCE, in Euclid's "Elements" we find the oldest extant abstract proof of the theorem (that is, a proof that does not use specific numbers).
- Written during the Han dynasty, 200 BCE - 200 CE the Chinese text "Chou Pei Suan Ching" gives a numerical proof of the Pythagorean theorem, using the (3, 4, 5) right triangle. From the same period, Pythagorean triples appear in "Nine Chapters on the Mathematical Art", together with a mention of right triangles. (Some authorities place the "Chou Pei Suan Ching" as much as three hundred years earlier than the Han dynasty.) There has been much debate on whether the Pythagorean theorem was discovered once or many times. B.L. van der Waerden asserts a single discovery, by someone in Neolithic Britain, knowledge of which then spread to Mesopotamia and Egypt circa 2000 BCE, and from there to India, China, and Greece circa 600 BCE. Most authorities, however, favor independent discovery. In the West, the theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras. In China, the theorem goes by the name "Gougu Theorem" (勾股定理), based on the numerical proof in the Chou Pei Suan Ching (周髀算经) (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, variously dated between 500 BCE -200 CE, see image above). James Garfield, who later became a President of the United States, devised an original proof of the Pythagorean theorem in 1876. (See the external links below for a sampling of the many different proofs of the Pythagorean theorem.) In heraldry, the Pythagorean theorem appears as a charge in the arms of Seissenegger.

Other facts

The converse of the Pythagorean theorem is also true: : For any three positive numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. This converse also appears in Euclid's Elements. This can be proven using the law of cosines (see below under Generalizations). It can also be proven by reductio ad absurdum. Suppose there exists a triangle for which a² + b² = c² but the angle between sides a and b is not a right angle. Then we can construct another triangle with a right angle between sides of lengths a and b. It follows that the hypotenuse of this triangle also has length c. Thus we have two triangles with the side lengths a, b and c but different angles between the a and b sides. But triangles with the same side lengths are congruent, and so we have a contradiction. The theorem is referenced in an episode of The Simpsons. After finding a pair of glasses at the Nuclear Power Plant, Homer puts them on and in an attempt to sound smart, comments "the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (This was a homage to The Wizard of Oz. When the Scarecrow receives his diploma from the Wizard, he recites the Pythagorean theorem incorrectly).

Generalizations

- The Pythagorean theorem was generalised by Euclid in his Elements: : If one erects similar figures (see Euclidean geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.
- The law of cosines (or cosine rule) is a version of the Pythagorean theorem that applies to all (Euclidean) triangles, not just right-angled ones. It states that: ::

a^2+b^2-2ab\cos=c^2, \,

:where θ is the angle between sides a and b. :When θ is 90 degrees, then cos(θ) = 0, so the formula reduces to the usual Pythagorean theorem.
- The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane -- if (x₀, y₀) and (x₁, y₁) are points in the plane, then the distance between them is given by ::

\sqrt.

- More generally, given two vectors v and w in a complex inner product space, the Pythagorean theorem takes the following form: ::

\|\mathbf+\mathbf\|^2 = \|\mathbf\|^2 + \|\mathbf\|^2 + 2\,\mbox\,\langle\mathbf,\mathbf\rangle

:In particular, ||v + w||² = ||v||² + ||w||² if v and w are orthogonal, and these two statements are equivalent in any real inner product space.
- Using mathematical induction, the previous result can be extended to any finite number of pairwise orthogonal vectors. Let v₁, v₂,..., v_n be vectors in an inner product space such that i, v_j> = 0 for 1 ≤ i < j ≤ n. Then ::

\left\|\,\sum_^\mathbf_k\,\right\|^2 = \sum_^ \|\mathbf_k\|^2

:The generalisation of this result to infinite-dimensional inner product spaces is known as Parseval's identity.
- The Pythagorean theorem also generalises to higher-dimensional simplexes. If a tetrahedron has a right angle corner (a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This is called de Gua's theorem.
- Edsger Dijkstra discovered this related proposition, not actually a generalisation: ::sgn(α + β − γ) = sgn(a² + b² − c²) :where α is the angle opposite to side a, β is the angle opposite to side b and γ is the angle opposite to side c. :This formula holds in all triangles, not just the right triangles. If γ is a right angle (γ equals

\pi/2

radians or 90°), then sgn(α + β − γ) = 0 since the sum of the angles of a triangle is

\pi

radians (or 180°). Thus, a² + b² − c² = 0. :In a triangle with three acute angles, α + β > γ holds. Therefore, a² + b² > c² holds. :In a triangle with an obtuse angle, α + β < γ holds. Therefore, a² + b² < c² holds.

The Pythagorean theorem in non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to

\pi/2

; this violates the Euclidean Pythagorean theorem because

(\pi/2)^2+(\pi/2)^2\neq (\pi/2)^2

. This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean theorem. There are two cases to consider -- spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines:
- For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form ::

\cos \left(\frac\right)=\cos \left(\frac\right)\,\cos \left(\frac\right).

:By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.
- For any triangle in the hyperbolic plane (with Gaussian curvature −1), the Pythagorean theorem takes the form ::

\cosh c=\cosh a\,\cosh b

:where cosh is the hyperbolic cosine. By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.

References

- Euclid, The Elements, Translated with an introduction and commentary by Sir Thomas L. Heath, Dover, (3 vols.), 2nd edition, 1956.
- B. L. van der Waerden, Geometry and Algebra in Ancient Civilizations, Springer, 1983.

External links

- [http://www.cut-the-knot.org/pythagoras/index.html Over 50 proofs of the Pythagorean theorem] at cut-the-knot
- [http://buster2058.netfirms.com/octagon/octagon.htm Using the Pythagorean theorem and a Microsoft Windows calculator to calculate the sides of an octagon]
- [http://www.cs.utexas.edu/users/EWD/ewd09xx/EWD975.PDF Dijkstra's generalization]
- [http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml The Pythagorean Theorem is Equivalent to the Parallel Postulate] at cut-the-knot
- [http://maths.dur.ac.uk/~dma0rcj/custer.html The true origin]
- [http://www.mathsisfun.com/pythagoras.html Some animated proofs of the Pythagorean theorem]
- [http://www.gosai.com/chaitanya/saranagati/html/vishnu_mjs/math/math.html#Vedic-Mathematics Vedic Mathematics]
- [http://mathworld.wolfram.com/PythagoreanTheorem.html Pythagorean Theorem -- From MathWorld] Category:Euclidean plane geometry Category:Mathematical theorems Category:Triangles Category:Angle ko:피타고라스의 정리 ja:ピタゴラスの定理 simple:Pythagorean theorem th:ทฤษฎีบทพีทาโกรัส

Forms

Plato spoke of forms (sometimes capitalized: The Forms) in formulating his solution to the problem of universals. The forms, according to Plato, are roughly speaking archetypes or abstract representations of the many types and properties (that is, of universals) of things we see all around us. There are, therefore, on Plato's view, forms of dogs, of human beings, of mountains, as well as of the color red, of courage, of love, and of goodness. Indeed, for Plato, God is identical to the Form of the Good. For Plato, the forms exist in what is known as a "Platonic heaven," and when human beings die, their souls achieve reunion with the forms. Plato makes clear that souls originate in this "Platonic heaven" and have recollection of it even in life. Form and idea are terms used to translate the Greek word eidos (plural eide). "Idea" is a misleading translation, because for Plato, the eide do not exist in the mind. Several of Plato's dialogues make use of the Forms, including Plato's Parmenides, which outline several of Plato's own objections to his Theory of Forms. A serious problem for the Theory of Forms is similar to that for Cartesian Dualism, if these Forms exist in 'another world', or are not of the same order of reality as matter, how can they interact with matter to 'inform' it. One standard response was to argue that Forms are part of the same Cosmos and so can interact with matter. But in this case we have the so-called Third Man Problem. If all things (men for instance) have Forms, and things are components of the Cosmos, then Forms are things too and so must have their own Forms ad infinitum... This is not a total refutation of the theory, but its remedy is difficult. For more information about Plato's theory of universals (forms, ideas), see Platonic realism. See also the divided line of Plato. It is interesting to note that al-Farabi, an excellent student of Plato and Aristotle, didn't even mention the Forms. (cf. "The Philosophy of Plato and Aristotle" by al-Farabi.) Plato's concept of the Forms found visual representation in the work of Conceptual artist Joseph Kosuth in his work "One and Three Chairs" and other similar works. Category:Platonism

Thales

:For the French electronics and defence contractor, see Thales Group Thales Group Thales (in Greek: Θαλής) of Miletus (ca. 635 BC-543 BC), also known as Thales the Milesian, was a pre-Socratic Greek philosopher and one of the Seven Sages of Greece. Many regard him as the first philosopher in the Greek tradition as well as the father of science.

Life

Thales lived in the city of Miletus, in Ionia, now western Turkey. According to Herodotus, he was of Phoenician descent. It was said that Thales had no children but adopted his nephew as his son. The well-traveled Ionians had many dealings with Egypt and Babylon, and Thales may have studied in Egypt as a young man. In any event, Thales almost certainly had exposure to Egyptian mythology, astronomy, and mathematics, as well as to other traditions alien to the Homeric traditions of Greece. Perhaps because of this his inquiries into the nature of things took him beyond traditional mythology. Several anecdotes suggest that Thales was not solely a thinker; he was involved in business and politics. One story recounts that he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Another version of this same story states that he bought the presses to demonstrate to his fellow Milesians that he could use his intelligence to enrich himself. However, looking at his way of thinking, getting rich was not his intent; merely to show people that by being a philosopher it was easy to enrich himself without it being the point of the exercise. Herodotus recorded that Thales advised the city-states of Ionia to form a federation. Thales is said to have died in his seat, while watching an athletic contest.

Theories and influence

Before Thales, the Greeks explained the origin and nature of the world through myths of anthropomorphic gods and heroes. Phenomena like lightning or earthquakes were attributed to actions of the gods. By contrast, Thales attempted to find naturalistic explanations of the world, without reference to the supernatural. He explained earthquakes by imagining that the Earth floats on water, and that earthquakes occur when the Earth is rocked by waves. Herodotus cites him as having predicted the solar eclipse of 585 BC that put an end to fighting between the Lydians and the Medes. Thales' most famous belief was his cosmological doctrine, which held that the world originated from water. Aristotle considered this belief roughly equivalent to the later ideas of Anaximenes, who held that everything in the world was composed of air. Thus it is sometimes assumed that Thales considered everything to be made from water. According to Lloyd, however, it is likely that while Thales saw water as an origin, he never pondered whether water continued to be the substance of the world. Thales had a profound influence on other Greek thinkers and therefore on Western history. Some believe Anaximander was a pupil of Thales. Early sources report that one of Anaximander's more famous pupils, Pythagoras, visited Thales as a young man, and that Thales advised him to travel to Egypt to further his philosophical and mathematical studies. Many philosophers followed Thales' lead in searching for explanations in nature rather than in the supernatural; others returned to supernatural explanations, but couched them in the language of philosophy rather than myth or religion. When you specifically look at the influence Thales had in the pre-Socrates era, he was one of the first thinkers who thought more in the way of logos than mythos. The difference between these two more profound ways of seeing the world is that mythos is concentrated around the stories of holy origin, while logos is concentrated around the argumentation. When the mythical man wants to explain the world the way he sees it, he explains it based on gods and powers. The mythical thought does not differ between things and persons and furthermore it does not differ between nature and culture. The way a logos thinker would present the view on the world is radically different than the mythical thinker. In its concrete form, logos is a way of thinking not only about individualism, but also the abstract. Furthermore, it focuses on sensible and continuous argumentation. This lays the foundation of philosophy and it's way of explaining the world in terms of abstract argumentation, and not in the way of gods and mythical stories. Thales is credited with first popularizing geometry in ancient Greek culture, mainly that of spatial relationships. He is the first one who separated trigonometry as an independent group from Mathematics, to be one of the four basic "elements" of geometry. The other three elements of geometry are about long, square and cube of an object.

Sources

Most of our sources for information on the Miletian philosophers (Thales, Anaximander, and Anaximenes) are the works of much later writers. The primary source for Thales' philosophy is Aristotle, who credited him with the first inquiry into the causes of things. Thales may or may not have written books. It is certain, however, that Aristotle did not have access to any work of Thales, and was writing from secondary sources of his own. While Thales' historical importance is unquestioned, this introduces a good deal of uncertainty into our understanding of him.

Interpretations

Nietzsche, in his Philosophy in the Tragic Age of the Greeks, § 3, wrote: "Greek philosophy seems to begin with an absurd notion, with the proposition that water is the primal origin and the womb of all things. Is it really necessary for us to take serious notice of this proposition? It is, and for three reasons. First, because it tells us something about the primal origin of all things; second, because it does so in language devoid of image or fable, and finally, because contained in it, if only embryonically, is the thought, 'all things are one.'"

Trivia

- In the A&E television rendition of Nero Wolfe, one of the antagonists, a mathematician, uses the name "Milton Thales" as a pseudonym, a reference to Thales of Miletus.

References

G.E.R. Lloyd, Early Greek Science: Thales to Aristotle

External links

- [http://www.iep.utm.edu/t/thales.htm Thales of Miletus from The Internet Encyclopedia of Philosophy]
- [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html Thales of Miletus] from the MacTutor History of Mathematics archive
- [http://www.livius.org Livius], [http://www.livius.org/th/thales/thales.html Thales of Miletus] by Jona Lendering Category:635 BC births Category:543 BC deaths Category:Ancient Greek mathematicians Category:Ancient philosophers Category:Presocratic philosophers ko:탈레스 ja:タレス

Ionia

:This article is about the region of western Anatolia. For the group of islands west of Greece, see Ionian Islands. Ionian Islands Ionia (Greek Ιωνία; see also List of traditional Greek place names) was an ancient region of southwestern coastal Anatolia (now in Turkey) on the Aegean Sea. It comprised a narrow coastal strip from Phocaea in the north near the mouth of the river Hermus (now the Gediz), to Miletus in the south near the mouth of the river Maeander, and included the islands of Chios and Samos. It was bounded by Aeolia to the north, Lydia to the east and Caria to the south. According to the universal Greek tradition, the cities of Ionia were founded by migrants from the other side of the Aegean and their settlement was connected with the legendary history of the Ionic race in Attica, by the statement that the colonists were led by Neleus and Androclus, sons of Codrus, the last king of Athens. In accordance with this view the "Ionic migration", as it was called by later chronologers, was dated by them one hundred and forty years after the Trojan war, or sixty years after the return of the Heraclidae into the Peloponnese. Without assigning any definite date, we may say that recent research [as of 1910] has tended to support the popular Greek idea that Ionia received its main Greek element rather late - after the descent of the Dorians, and, therefore, after any part of the Aegean period. The only Aegean objects yet found (1910) in or near Ionia are some shards of the very late Minoan age at Miletus. It is improbable that all the Greek colonists were of the not numerous Ionian race. Herodotus tells us that the settlers were from many different tribes and cities of Greece (a fact indicated also by the local traditions of the cities), and that they intermarried with the native races. In Asia, Greeks were named with derivations of "Ionian", such as Yona in Pali. Josephus relates the Ionians to the biblical character Javan son of Japheth: "but from Javan, Ionia, and all the Grecians, are derived" (Antiquities of the Jews I:6). In Greek mythology, Ion, regarded as the founder of the Ionian tribe, was the son of Creusa (daughter of Erechtheus); his father was either Creusa's husband Xuthus (according to Hesiod's Eoiae) or Apollo (according to Euripides).

Geography

The cities called Ionian in historical times were twelve in number, an arrangement copied as it was supposed from the constitution of the Ionian cities in Greece which had originally occupied the territory in the north of the Peloponnese subsequently held by the Achaeans. These were (from south to north) Miletus, Myus, Priene, Ephesus, Colophon, Lebedus, Teos, Erythrae, Clazomenae and Phocaea, together with Samos and Chios. Smyrna, originally an Aeolic colony, was afterwards occupied by Ionians from Colophon, and became an Ionian city — an event which had taken place before the time of Herodotus. But at what period it was admitted as a member of the league is unknown. The Ionian cities formed a religious and cultural (as opposed to a political or military) confederacy (see Ionian League), of which participation in the Panionic festival (Panionia) was a distinguishing characteristic. This festival took place on the north slope of Mt. Mycale in a shrine called the Panionium. In addition to the Panionic festival at Mycale, which was celebrated mainly by the Asian Ionians, both European and Asian coast Ionians convened on Delos Island each summer to worship at the temple of the Delian Apollo. But like the Amphictyonic league in Greece, the Ionic was rather of a sacred than a political character; every city enjoyed absolute autonomy, and, though common interests often united them for a common political object, they never formed a real confederacy like that of the Achaeans or Boeotians. The advice of Thales of Miletus to combine in a political union was rejected. Ionia was of small extent, not exceeding 90 geographical miles in length from north to south, with a breadth varying from 20 to 30 miles, but to this must be added the peninsula of Mimas, together with the two large islands. So intricate is the coastline that the voyage along its shores was estimated at nearly four times the direct distance. A great part of this area was, moreover, occupied by mountains. Of these the most lofty and striking were Mimas and Corycus, in the peninsula which stands out to the west, facing the island of Chios; Sipylus, to the north of Smyrna; Corax, extending to the south-west from the Gulf of Smyrna, and descending to the sea between Lebedus and Teos; and the strongly marked range of Mycale, a continuation of Messogis in the interior, which forms the bold headland of Trogilium or Mycale, opposite Samos. None of these mountains attains a height of more than 4000 feet The district comprised three extremely fertile valleys formed by the outflow of three rivers, among the most considerable in Asia Minor: the Hermus in the north, flowing into the Gulf of Smyrna, though at some distance from the city of that name; the Caster, which flowed under the walls of Ephesus; and the Maeander, which in ancient times discharged its waters into the deep gulf that once bathed the walls of Miletus, but which has been gradually filled up by this river's deposits. With the advantage of a peculiarly fine climate, for which this part of Asia Minor has been famous in all ages, Ionia enjoyed the reputation in ancient times of being the most fertile of all the rich provinces of Asia Minor; and even in modern times, though very imperfectly cultivated, it produces abundance of fruit of all kinds, and the raisins and figs of Smyrna supply almost all the markets of Europe. The colonies naturally became prosperous. Miletus especially was at an early period one of the most important commercial cities of Greece; and in its turn became the parent of numerous other colonies, which extended all around the shores of the Euxine Sea and the Propontis from Abydus and Cyzicus to Trapezus and Panticapaeum. Phocaea was one of the first Greek cities whose mariners explored the shores of the western Mediterranean. Ephesus, though it did not send out any colonies of importance, from an early period became a flourishing city and attained to a position corresponding in some measure to that of Smyrna at the present day.

History

The first event in the history of Ionia of which we have any trustworthy account is the inroad of the Cimmerii, who ravaged a great part of Asia Minor, including Lydia, and sacked Magnesia on the Maeander, but were foiled in their attack upon Ephesus. This event may be referred to the middle of the 7th century BC. About 700 BC Gyges, first Mermnad king of Lydia, invaded the territories of Smyrna and Miletus, and is said to have taken Colophon as his son Ardys did Priene. But it was not till the reign of Croesus (560–545 BC) that the cities of Ionia successively fell under Lydian rule. The defeat of Croesus by Cyrus was followed by the conquest of all the Ionian cities. These became subject to the Persian monarchy with the other Greek cities of Asia. In this position they enjoyed a considerable amount of autonomy, but were for the most part subject to local despots, most of whom were creatures of the Persian king. It was at the instigation of one of these despots, Histiaeus of Miletus, that in about 500 BC the principal cities ignited the Ionian Revolt against Persia. They were at first assisted by the Athenians, with whose aid they penetrated into the interior and burnt Sardis, an event which ultimately led to the Persian invasion of Greece. But the fleet of the Ionians was defeated off the island of Lade, and the destruction of Miletus after a protracted siege was followed by the reconquest of all the Asiatic Greeks, insular as well as continental. The victories of the Greeks during the great Persian war had the effect of enfranchizing their kinsmen on the other side of the Aegean; and the battle of Mycale (479 BC), in which the defeat of the Persians was in great measure owing to the Ionians, secured their emancipation. They henceforth became the dependent allies of Athens (see Delian League), though still retaining their autonomy, which they preserved until the peace of Antalcidas in 387 BC once more placed them as well as the other Greek cities in Asia under the nominal dominion of Persia. They appear, however, to have retained a considerable amount of freedom until the invasion of Asia Minor by Alexander the Great. After the battle of the Granicus most of the Ionian cities submitted to the conqueror. Miletus, which alone held out, was reduced after a long siege (334 BC). From this time they passed under the dominion of the successive Macedonian rulers of Asia, but continued, with the exception of Miletus, to enjoy great prosperity both under these Greek dynasties and after they became part of the Roman province of Asia.

Legacy

Ionia has laid the world under its debt not only by giving birth to a long roll of distinguished men of letters and science (see Ionian School of Philosophy), but also by originating the distinct school of art which prepared the way for the brilliant artistic development of Athens in the 5th century BC. This school flourished between 700 and 500 BC, and is distinguished by the fineness of workmanship and minuteness of detail with which it treated subjects, inspired always to some extent by non-Greek models. Naturalism is progressively obvious in its treatment, e.g. of the human figure, but to the end it is still subservient to convention. It has been thought that the Ionian migration from Greece carried with it some part of a population which retained the artistic traditions of the Mycenaean civilization, and so caused the birth of the Ionic school; but whether this was so or not, it is certain that from the 8th century BC onwards we find the true spirit of Hellenic art, stimulated by commercial intercourse with eastern civilizations, working out its development chiefly in Ionia and its neighbouring isles. The great names of this school are Theodorus and Rhoecus of Samos; Bathycles of Magnesia on the Maeander; Glaucus, Melas, Micciades, Archermus, Bupalus and Athenis of Chios. Notable works of the school still extant are the famous archaic female statues found on the Athenian Acropolis in 1885–1887, the seated statues of Branchidae, the Nike of Archermus found at Delos, and the objects in ivory and electrum found by D.G. Hogarth in the lower strata of the Artemision at Ephesus. The Arabic, Turkish & Persian name for Greece is Younan (یونان), a corruption of "Ionia." The same is true for the Hebrew word, "Yavan" (יוון). The Ionians were the first Greek-speaking people that Semitic and Persian language speakers encountered, and the name spread throughout the Near East and Central Asia. This entry was originally from the 1911 Encyclopædia Britannica. Category:Anatolia Category:Ancient Greece

Pythagoras

Pythagoras (approximately 569 BCE – 475 BCE, Greek: Πυθαγόρας) was an Ionian mathematician and philosopher, known best for the Pythagorean theorem which bears his name. Pythagorean theorem Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. Pythagoras and his students believed that everything was related to mathematics, and thought that everything could be predicted and measured in rhythmic patterns or cycles.

Biography

Pythagoras was born on the island of Samos, off the coast of Asia Minor. He was born to Pythais (a native of Samos) and Mnesarchus (a merchant from Tyre). As a young man he left his native city for Crotona in Southern Italy, to escape the tyrannical government of Polycrates. Many writers credit him with visits to the sages of Egypt and Babylon before going west; but such visits feature stereotypically in the biographies of many Greek wise men, and are likely more legend than fact. Upon his migration from Samos to Crotona, Pythagoras established a secret religious society very similar to, and possibly influenced by, the earlier Orphic cult. Pythagoras undertook a reform of the cultural life of Crotona, urging the citizens to follow virtue and form an elite circle of followers around himself. Very strict rules of conduct governed this cultural center. He opened his school to men and women students alike. They called themselves the Mathematikoi. According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory. The Pythagorean theorem that bears his name was known earlier in Mesopotamia, Egypt and India. For a chronology of the theorem and its proofs, see the article on the Pythagorean theorem. Whether Pythagoras himself proved this theorem is not known, as it was common in the ancient world to credit to a famous teacher the discoveries of his students. The earliest known mention of Pythogoras's name in connection with the theorem came five centuries after his death, in the writings of Cicero and Plutarch. right

Pythagoreans

Main articles: Pythagoreans, Pythagoreanism Pythagoras' followers were commonly called "Pythagoreans." For the most part we remember them as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements. The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed ascetic, dietary and moral rules which they believed would enable their soul to achieve a higher rank among the gods. Consequently, they expected they would be set free from the wheel of birth. The Pythagoreans also believed that the sexes are equal. Pythagoras started a school, together with his wife Theano, built on this principle. After he died, his wife and daughters ran and taught at the school. Theano herself discovered a formula to derive the golden rectangle. All slaves were treated humanely, and animals were respected as creatures with souls. The highest purification of the soul was "philosophy", and Pythagoras has been credited with the first use of the term. The Jains in India follow similar beliefs and practices, which leads some neo-Pythagoreans and neo-Platonic authors to believe that Pythagoras had visited India, and studied under the Jains. It was the Pythagoreans who discovered that the relationship between musical notes could be expressed in numerical ratios of small whole numbers. The Pythagoreans elaborated on a theory of numbers the exact meaning of which is still debated among scholars. They taught that all things were numbers, that the essence of everything is a number, and that all relationships can be expressed numerically.

Literary works

No texts by Pythagoras survive, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. And ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses , where Ovid has him expound upon his philosophical viewpoints.

Scientific contributions

Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales, just before the death of the latter. No account exists of the specifics of the meeting, other than the report that Thales recommended that Pythagoras travel to Egypt in order to further his philosophical and mathematical training. Egypt In astronomy, the Pythagoreans were well aware of the periodic numerical relations of the planets, moon, and sun. The celestial spheres of the planets were thought to produce a harmony called the music of the spheres. These ideas, as well as the ideas of the Platonic solids, would later be used by Johannes Kepler in his attempt to formulate a model of the solar system in his work The Harmony of the Worlds. Pythagoreans also believed that the earth itself was in motion and that the laws of nature could be derived from pure mathematics. They may have coined the term cosmos, a term implying a universe with orderly movements and events. It is sometimes difficult to determine which ideas Pythagoras taught originally, as opposed to the ideas his followers later added. While he clearly attached great importance to geometry, classical Greek writers tended to cite Thales as the great pioneer of this science rather than Pythagoras. The later tradition of Pythagoras as the inventor of mathematics stems largely from the Roman period. Whether or not we attribute the Pythagorean theorem to Pythagoras, it seems fairly certain that he had the pioneering insight into the numerical ratios which determine the musical scale, since this plays a key role in many other areas of the Pythagorean tradition, and since no evidence remains of earlier Greek or Egyptian musical theories. Another important discovery of this school -- which upset Greek mathematics, as well as the Pythagoreans' own belief that whole numbers and their ratios could account for everything in nature -- was the incommensurability of the diagonal of a square with its side. This result showed the existence of irrational numbers. The influence of Pythagoras has transcended the field of mathematics, and the Hippocratic Oath — with its central commitment to First do no harm — has its roots in the oath of the Pythagorean Brotherhood [http://www.nlm.nih.gov/hmd/greek/greek_oath.html].

References

Primary sources:

Only a few relevant source texts deal with Pythagoras and the Pythagoreans, most are available in different translations. Other texts usually build solely on information from these four books.
- Diogenes Laertius, Vitae philosophorum VIII (Lives of Eminent Philosophers, which in turn reference the lost work Successions of Philosophers by Alexander Polyhistor) — [http://classicpersuasion.org/pw/diogenes/dlpythagoras.htm Pythagoras, Translation by C.D. Yonge]
- Porphyry, Vita Pythagorae (Life of Pythagoras)
- Iamblichus, De Vita Pythagorica (On the Pythagorean Life)
- Apuleius also writes about Pythagoras in Apologia, including a story of him being taught by Babylonian disciples of Zoroaster

Secondary sources:

- Eric Temple Bell, The Magic of Numbers, Dover, New York, 1991 ISBN 0486267881
- Walter Burkert, Lore and Science in Ancient Pythagoreanism, Harvard University Press (June 1, 1972), ISBN 0674539184
- K. L., Guthrie (Ed.), The Pythagorean Sourcebook and Library, Phanes, Grand Rapids, 1987 ISBN 0-933999-51-8
- Dominic J. O'Meara, Pythagoras Revived, Clarendon Press, Oxford, 1989, Paperback ISBN 0198239130, Hardcover ISBN 0198244851

External links

- [http://users.ucom.net/~vegan Pythagoreanism Web Site]
- [http://www.utm.edu/research/iep/p/pythagor.htm Pythagoras, Internet Encyclopedia of Philosophy]
- [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pythagoras.html Pythagoras of Samos, The MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland]
- [http://history.hanover.edu/texts/presoc/pythagor.htm Pythagoras and the Pythagoreans, Fragments and Commentary, Arthur Fairbanks Hanover Historical Texts Project, Hanover College Department of History]
- [http://www.completepythagoras.net/ The Complete Pythagoras], an on-line book containing all survived biographies and Pythagorean fragments.
- [http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html Pythagoras and the Pythagoreans, Department of Mathematics, Texas A&M University]
- [http://www.newadvent.org/cathen/12587b.htm Pythagoras and Pythagoreanism, The Catholic Encyclopedia]
- [http://cyberspacei.com/jesusi/inlight/philosophy/western/Pythagoreanism.htm Pythagoreanism Web Article]
- [http://groups.yahoo.com/group/Pythagorean-L Pythagoreanism Discussion Group]
- [http://www.geocities.com/go_darkness/god-pythagorean-pentacle.html Occult conception of Pythagoreanism]
- [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html Pythagoras of Samos]
- [http://plato.stanford.edu/entries/pythagoras/ Stanford Encyclopedia of Philosophy entry]
- [http://www.organelle.org/organelle/tetra/tetraktys.html Tetraktys] Category:569 BC births Category:475 BC deaths Category:Ancient Greek mathematicians Category:Pythagoreans Category:Music theorists Category:Vegetarians ko:피타고라스 ja:ピュタゴラス simple:Pythagoras th:พีทาโกรัส

Irrational number

In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a ratio of two integers, i.e., it is not of the form :

\frac

where a and b are integers and b is not zero. It can readily be shown that the irrational numbers are precisely those numbers whose expansion in any given base (decimal, binary, etc) never ends and never enters a periodic pattern, but no mathematician takes that to be a definition. Almost all real numbers are irrational, in a sense which is defined more precisely below. Some irrational numbers are algebraic numbers, such as √2, the square root of two, and ³√5, the cube root of 5, and the golden ratio, symbolized by the Greek letter

\varphi

(phi) or less commonly by

\tau

(tau); others are transcendental numbers such as π and e. When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.

History

The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction (proof below). However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning. The sixteenth century saw the final acceptance of negative, integral and fractional numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. The next hundred years saw the imaginary become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. For the nineteenth century it remained to complete the theory of complex numbers, to separate irrationals into algebraic and transcendental, to prove the existence of transcendental numbers, and to make a scientific study of a subject which had remained almost dormant since Euclid, the theory of irrationals. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray. Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. Transcendental numbers were first distinguished from algebraic irrationals by Kronecker. Lambert proved (1761) that π cannot be rational, and that eⁿ is irrational if n is rational (unless n = 0), a proof, however, which left much to be desired. Legendre (1794) completed Lambert's proof, and showed that π is not the square root of a rational number. Joseph Liouville (1840) showed that neither e nor e² can be a root of an integral quadratic equation. But the existence of transcendental numbers was first established by Liouville (1844, 1851), the proof being subsequently displaced by Georg Cantor (1873). Charles Hermite (1873) first proved

e

transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Hurwitz and Paul Albert Gordan.

The square root of 2

One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the negation and showing that that leads to a contradiction, which means that the proposition must be true. # Assume that √2 is a rational number. This would mean that there exist integers a and b such that a / b = √2. # Then √2 can be written as an irreducible fraction (the fraction is shortened as much as possible) a / b such that a and b are coprime integers and (a / b)² = 2. # It follows that a² / b² = 2 and a² = 2 b². # Therefore a² is even because it is equal to 2 b² which is obviously even. # It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.) # Because a is even, there exists a k that fulfills: a = 2k. # We insert the last equation of (3) in (6): 2b² = (2k)² is equivalent to 2b² = 4k² is equivalent to b² = 2k². # Because 2k² is even it follows that b² is also even which means that b is even because only even numbers have even squares. # By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2). Since we have found a contradiction the assumption (1) that √2 is a rational number must be false. The opposite is proven. √2 is irrational. This proof can be generalized to show that any root of any natural number is either a natural number or irrational.

Another proof

Another reductio ad absurdum showing that √2 is irrational is less well-known and has sufficient charm that it is worth including here. It proceeds by observing that if √2 = m/n then √2 = (2n − m)/(m − n), so that a fraction in lowest terms is reduced to yet lower terms. That is a contradiction if n and m are positive integers, so the assumption that √2 is rational must be false. It is possible to construct from an isosceles right triangle whose leg and hypotenuse have respective lengths n and m, by a classic ruler-and-compass construction, a smaller isosceles right triangle whose leg and hypotenuse have respective lengths m − n and 2n − m. That construction proves the irrationality of √2 by the kind of method that was employed by ancient