A representation of Euclid from The School of Athens by Raphael. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry The School of Athens, or it Scuola di Atene in Italian, is one of the most famous Paintings by the Italian Renaissance artist Raphael Sanzio, usually known by his first name alone (in Italian Raffaello) (April 6 or March 28 1483 – April 6 1520 was an Italian painter and

Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. The Greeks ( Greek: Έλληνες) are a Nation and Ethnic group native to Greece, Cyprus and neighbouring regions A mathematician is a person whose primary area of study and research is the field of Mathematics. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια Euclid's text Elements is the earliest known systematic discussion of geometry. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist

The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. In Mathematics, plane geometry may mean geometry of a plane, geometry of the Euclidean plane, or sometimes Secondary school is a term used to describe an educational Institution where the final stage of compulsory schooling known as Secondary education, takes In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. In Mathematics, solid geometry was the traditional name for the Geometry of three-dimensional Euclidean space &mdash for practical purposes the kind of In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Much of the Elements states results of what is now called number theory, proved using geometrical methods. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes

For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitation is a natural Phenomenon by which objects with Mass attract one another

Axiomatic approach

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a finite number of axioms. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms):

1. Any two points can be joined by a straight line. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume
2. Any straight line segment can be extended indefinitely in a straight line. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers
4. All right angles are congruent. In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i
5. Parallel postulate. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

These axioms invoke the following concepts: point, straight line segment and line, side of a line, circle with radius and center, right angle, congruence, inner and right angles, sum. The following verbs appear: join, extend, draw, intersect. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere.

A proof from Euclid's elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane:

Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive

Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.

Strictly speaking, the constructs of lines on paper etc are models of the objects defined within the formal system, rather than instances of those objects. The Movement for Democracy in Liberia (MODEL was a rebel group in Liberia that became active in March 2003, launching attacks from Côte d'Ivoire. For example a Euclidean straight line has no width, but any real drawn line will.

The Elements also include the following five "common notions":

1. Things that equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things that coincide with one another equal one another.
5. The whole is greater than the part.

Euclid also invoked other properties pertaining to magnitudes. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering 1 is the only part of the underlying logic that Euclid explicitly articulated. 2 and 3 are "arithmetical" principles; note that the meanings of "add" and "subtract" in this purely geometric context are taken as given. 1 through 4 operationally define equality, which can also be taken as part of the underlying logic or as an equivalence relation requiring, like "coincide," careful prior definition. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" 5 is a principle of mereology. In Mathematical logic, mereology is a collection of axiomatic First-order theories dealing with parts and their respective wholes "Whole", "part", and "remainder" beg for precise definitions.

In the 19th century, it was realized that Euclid's ten axioms and common notions do not suffice to prove all of theorems stated in the Elements. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore needs to be an axiom itself. The very first geometric proof in the Elements, shown in the figure on the right, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex. His axioms, however, do not guarantee that the circles actually intersect, because they are consistent with discrete, rather than continuous, space. Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski. Moritz Pasch (8 November 1843 Breslau, Germany (now Wrocław, Poland) --20 September 1930 Bad Homburg, Germany) was a Hilbert's axioms are a set of 20 assumptions (originally 21 David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. In 1932 G D Birkhoff created a set of four Postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary" that is formulable in

To be fair to Euclid, the first formal logic capable of supporting his geometry was that of Frege's 1879 Begriffsschrift, little read until the 1950s. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Begriffsschrift is the title of a short book on Logic by Gottlob Frege, published in 1879, and is also the name of the Formal system We now see that Euclidean geometry should be embedded in first-order logic with identity, a formal system first set out in Hilbert and Wilhelm Ackermann's 1928 Principles of Theoretical Logic. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Wilhelm Friedrich Ackermann ( March 29, 1896, Herscheid municipality Germany &ndash December 24, 1962 Lüdenscheid Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert 's and Wilhelm Ackermann 's classic text Formal mereology began only in 1916, with the work of Lesniewski and A. N. Whitehead. In Mathematical logic, mereology is a collection of axiomatic First-order theories dealing with parts and their respective wholes Stanisław Leśniewski ( March 30 1886 – May 13 1939) was a Polish Mathematician, Philosopher and Logician Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Tarski and his students did major work on the foundations of elementary geometry as recently as between 1959 and his death in 1983. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary" that is formulable in

The parallel postulate

Main article: Parallel postulate

To the ancients, the parallel postulate seemed less obvious than the others; verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive ^[1] Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: the first 28 propositions he presents are those that can be proved without it.

Many geometers tried in vain to prove the fifth postulate from the first four. By 1763 at least 28 different proofs had been published, but all were found to be incorrect. ^[2] In fact the parallel postulate cannot be proved from the other four: this was shown in the 19th century by the construction of alternative (non-Euclidean) systems of geometry where the other axioms are still true but the parallel postulate is replaced by a conflicting axiom. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry One distinguishing aspect of these systems is that the three angles of a triangle do not add to 180°: in hyperbolic geometry the sum of the three angles is always less than 180° and can approach zero, while in elliptic geometry it is greater than 180°. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point If the parallel postulate is dropped from the list of axioms without replacement, the result is the more general geometry called absolute geometry. Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives

Treatment using analytic geometry

The development of analytic geometry provided an alternative method for formalizing geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In this approach, a point is represented by its Cartesian (x,y) coordinates, a line is represented by its equation, and so on. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In the 20th century, this fit into David Hilbert's program of reducing all of mathematics to arithmetic, and then proving the consistency of arithmetic using finitistic reasoning. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered to be theorems. The equation

defining the distance between two points P = (p,q) and Q = (r,s) is then known as the Euclidean metric, and other metrics define non-Euclidean geometries. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry

As a description of physical reality

A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar eclipse. An eclipse is an astronomical event that occurs when one Celestial object moves into the shadow of another The rays of starlight were bent by the Sun's gravity on their way to the earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.

Euclid believed that his axioms were self-evident statements about physical reality.

This led to deep philosophical difficulties in reconciling the status of knowledge from observation as opposed to knowledge gained by the action of thought and reasoning. A major investigation of this area was conducted by Immanuel Kant in The Critique of Pure Reason. Immanuel Kant (ɪmanuəl kant 22 April 1724 12 February 1804 was an 18th-century German Philosopher from the Prussian city of Königsberg The Critique of Pure Reason (Kritik der reinen Vernunft by Immanuel Kant, first published in 1781, second edition 1787, is one

However, Einstein's theory of general relativity shows that the true geometry of spacetime is non-Euclidean geometry. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting the deviations from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the observation of the slight bending of starlight by the Sun during a solar eclipse in 1919, and non-Euclidean geometry is now, for example, an integral part of the software that runs the GPS system. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth It is possible to object to the non-Euclidean interpretation of general relativity on the grounds that light rays might be improper physical models of Euclid's lines, or that relativity could be rephrased so as to avoid the geometrical interpretations. However, one of the consequences of Einstein's theory is that there is no possible physical test that can do any better than a beam of light as a model of geometry. Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, which can then be imagined as a formal system without any intrinsic real-world meaning. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry

Because of the incompatibility of the Standard Model with general relativity, and because of some recent empirical evidence against the former, both theories are now under increased scrutiny, and many theories have been proposed to replace or extend the former and, in many cases, the latter as well. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The disagreements between the two theories come from their claims about space-time, and it is now accepted that physical geometry must describe space-time rather than merely space. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS While Euclidean geometry, the Standard Model and general relativity are all in principle compatible with any number of spatial dimensions and any specification as to which of these if any are compactified (see string theory), and while all but Euclidean geometry (which does not distinguish space from time) insist on exactly one temporal dimension, proposed alternatives, none of which are yet part of scientific consensus, differ significantly in their predictions or lack thereof as to these details of space-time. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Scientific consensus is the collective judgement position and Opinion of the community of Scientists in a particular field of Science at a particular The disagreements between the conventional physical theories concern whether space-time is Euclidean (since quantum field theory in the standard model is built on the assumption that it is) and on whether it is quantized. In quantum field theory (QFT the forces between particles are mediated by other particles Few if any proposed alternatives deny that space-time is quantized, with the quanta of length and time are respectively the Planck length and the Planck time. The Planck length, denoted by \scriptstyle\ell_P \, is the unit of Length approximately 1 In Physics, the Planck time ( tP) is the unit of Time in the system of Natural units known as Planck units. However, which geometry to use - Euclidean, Riemannian, de Stitter, anti de Stitter and some others - is a major point of demarcation between them. Elliptic geometry is also sometimes called Riemannian geometry. Many physicists expect some Euclidean string theory to eventually become the Theory Of Everything, but their view is by no means unanimous, and in any case the future of this issue is unpredictable. A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena Regarding how if at all Euclidean geometry will be involved in future physics, what is uncontroversial is that the definition of straight lines will still be in terms of the path in a vacuum of electromagnetic radiation (including light) until gravity is explained with mathematical consistency in terms of a phenomenon other than space-time curvature, and that the test of geometrical postulates (Euclidean or otherwise) will lie in studying how these paths are affected by phenomena. For now, gravity is the only known relevant phenomenon, and its effect is uncontroversial (see gravitational lensing). A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of

Conic sections and gravitational theory

Apollonius and other Ancient Greek geometers made an extensive study of the conic sections — curves created by intersecting a cone and a plane. The (nondegenerate) ones are the ellipse, the parabola and the hyperbola, distinguished by having zero, one, or two intersections with infinity. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions This turned out to facilitate the work of Galileo, Kepler and Newton in the 17th Century, as these curves accurately modeled the movement of bodies under the influence of gravity. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Using Newton's law of universal gravitation, the orbit of a comet around the Sun is

• an ellipse, if it is moving too slowly for its position (below escape velocity), in which case it will eventually return;
• a parabola, if it is moving with exact escape velocity (unlikely), and will never return because the curve reaches to infinity; or
• a hyperbola, if it is moving fast enough (above escape velocity), and likewise will never return. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass A comet is a small Solar System body that orbits the Sun and when close enough to the Sun exhibits a visible coma (atmosphere or a tail — The Sun (Sol is the Star at the center of the Solar System. In Physics, escape velocity is the speed where the Kinetic energy of an object is equal to the magnitude of its Gravitational potential energy

In each case the Sun will be at one focus of the conic, and the motion will sweep out equal areas in equal times. In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle

Galileo experimented with objects falling small distances at the surface of the Earth, and empirically determined that the distance travelled was proportional to the square of the time. Given his timing and measuring apparatus, this was an excellent approximation. Over such small distances that the acceleration of gravity can be considered constant, and ignoring the effects of air (as on a falling feather) and the rotation of the Earth, the trajectory of a projectile will be a parabolic path. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example External ballistics is the part of the science of Ballistics that deals with the behaviour of a non-powered projectile in flight

Later calculations of these paths for bodies moving under gravity would be performed using the techniques of analytical geometry (using coordinates and algebra) and differential calculus, which provide straightforward proofs. Of course these techniques had not been invented at the time that Galileo investigated the movement of falling bodies. Once he found that bodies fall to the earth with constant acceleration (within the accuracy of his methods), he proved that projectiles will move in a parabolic path using the procedures of Euclidean geometry.

Similarly, Newton used quasi–Euclidean proofs to demonstrate the derivation of Keplerian orbital movements from his laws of motion and gravitation.

Centuries later, one of the first experimental measurements to support Einstein's general theory of relativity, which postulated a non-Euclidean geometry for space, was the orbit of the planet Mercury. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Kepler described the orbit as a perfect ellipse. Newtonian theory predicted that the gravitational influence of other bodies would give a more complicated orbit. But eventually all such Newtonian corrections fell short of experimental results; a small perturbation remained. Einstein postulated that the bending of space would precisely account for that perturbation.

Logical status

Euclidean geometry is a first-order theory. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science That is, it allows statements such as those that begin as "for all triangles . . . ", but it is incapable of forming statements such as "for all sets of triangles . . . ". Statements of the latter type are deemed to be outside the scope of the theory.

We owe much of our present understanding of the properties of the logical and metamathematical properties of Euclidean geometry to the work of Alfred Tarski and his students, beginning in the 1920s. In general metamathematics or meta-mathematics is a scientific reflection and Knowledge about mathematics seen as an entity/ object in Human Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Tarski proved his axiomatic formulation of Euclidean geometry to be complete in a certain sense: there is an algorithm which, for every proposition, can show it to be either true or false. Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary" that is formulable in In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas Gödel's incompleteness theorems showed the futility of Hilbert's program of proving the consistency of all of mathematics using finitistic reasoning. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most Tarski's findings do not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural ^[3]

Although complete in the formal sense used in modern logic, there are things that Euclidean geometry cannot accomplish. In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. The problem of trisecting the angle is a classic problem of Compass and straightedge constructions of ancient Greek mathematics. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Pierre Laurent Wantzel ( June 5, 1814 in Paris – May 21, 1848 in Paris was a French Mathematician who proved

Absolute geometry, first identified by Bolyai, is Euclidean geometry weakened by omission of the fifth postulate, that parallel lines do not meet. Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean Of strength intermediate between absolute geometry and Euclidean are geometries derived from Euclid's by alterations of the parallel postulate that can be shown to be consistent by exhibiting models of them. For example, geometry on the surface of a sphere is a model of elliptical geometry. Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point Another weakening of Euclidean geometry is affine geometry, first identified by Euler, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint). Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths

Classical theorems

• Ceva's theorem
• Heron's formula
• Nine-point circle
• Pythagorean theorem
• Tartaglia's formula
• Menelaus's theorem
• Angle bisector theorem
• The butterfly theorem
• Parallel Postulate

See also

• Interactive geometry software
• Non-Euclidean geometry
• Ordered geometry
• Incidence geometry
• Birkhoff's axioms
• Hilbert's axioms
• Tarski's axioms
• Parallel postulate
• Schopenhauer's criticism of the proofs of the Parallel Postulate

Notes

References

• Ball, W.W. Rouse (1960). Walter William Rouse Ball ( 14 August 1850 – 4 April 1925) was a British Mathematician, Lawyer and a fellow A Short Account of the History of Mathematics, 4th ed. [Reprint. Original publication: London: Macmillan & Co. , 1908], New York: Dover Publications, pp. 50–62. ISBN 0-486-20630-0.  
• Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he A History of Mathematics, Second Edition, John Wiley & Sons, Inc. . ISBN 0471543977.  
• Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters.  
• Heath, Thomas L. (1956). Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical The Thirteen Books of Euclid's Elements (3 vols. ), 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925], New York: Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3).   Heath's authoritative translation of Euclid's Elements plus his extensive historical research and detailed commentary throughout the text.
• Hofstadter, Douglas R. (1979). Douglas Richard Hofstadter (born February 15 1945 in New York New York) is an American academic whose research focuses on consciousness thinking and creativity Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books.  
• Nagel, E. and Newman, J. R. (1958). Gödel's Proof. New York University Press.  
• Alfred Tarski (1951) A Decision Method for Elementary Algebra and Geometry. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Univ. of California Press.

External links

• Kiran Kedlaya, Geometry Unbound (a treatment using analytic geometry; PDF format, GFDL licensed)
• Geometry at cut-the-knot (a collection of geometry problems, HTML with Java applets)
• Geometry Step by Step from the Land of the Incas by Antonio Gutierrez (flash required)
• Geometry lessons in PowerPoint

The Elements online

• A bilingual edition (typset in PDF format, with the original Greek and an English translation on facing pages; free in PDF form, available in print)
• In English (HTML, with the figures in the form of Java applets that the user can manipulate)
• Heath's translation (HTML, without the figures, public domain)
• In ancient Greek (typeset in PDF format, public domain)
• Oliver Byrne's 1847 edition - an unusual version using color rather than labels such as ABC (scanned page images, public domain)
• Reading Euclid - a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
• Mind Map of The Elements Interactive. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.