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Encyclopedia > Euclidean geometry
A representation of Euclid from The School of Athens by Raphael.

The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods. In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ... Secondary school is a term used to describe an institution where the final stage of compulsory schooling, known as secondary education, takes place. ... 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 that some mathematical statement is necessarily true, within the accepted standards of the field. ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space â€” for practical purposes the kind of space we live in. ... For other uses, see Dimension (disambiguation). ... For other uses, see Dimension (disambiguation). ... 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 of problems that arise from their study. ...

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. 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. Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... Einstein redirects here. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... Gravity is a force of attraction that acts between bodies that have mass. ...

## Axiomatic approach GA_googleFillSlot("encyclopedia_square");

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a finite number of axioms. Near the beginning of the first book of the Elements, Euclid gives five postulates (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, theory is used informally to refer to a body of knowledge about mathematics. ... This article or section does not adequately cite its references or sources. ...

1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Parallel postulate. 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.

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: Image File history File links Euclid-proof. ... Image File history File links Euclid-proof. ... a and b are parallel, the transversal t produces congruent angles. ...

Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

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. 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. Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

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. For example a Euclidean straight line has no width, but any real drawn line will. Model may refer to more than one thing : For models in society, art, fashion, and cosmetics, see; role model model (person) supermodel figure drawing modeling section In science and technology, a model (abstract) is understood as an abstract or theoretical representation of a phenomenon,see; geologic modeling model (economics) model...

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. 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. 5 is a principle of mereology. "Whole", "part", and "remainder" beg for precise definitions. 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 of the class of objects to which it belongs. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ...

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 Wroclaw, Poland) -- 20 September 1930 Bad Homburg, Germany) was a German mathematician specializing in the foundations of geometry. ... Hilberts 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 Birkhoffs axioms. ... Tarskis axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called elementary, that is formulable in first order logic with identity, and requiring no set theory. ...

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. 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. Formal mereology began only in 1916, with the work of Lesniewski and A. N. Whitehead. Tarski and his students did major work on the foundations of elementary geometry as recently as between 1959 and his death in 1983. Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... 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 set out in that book. ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany â€“ December 24, 1962 LÃ¼denscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text GrundzÃ¼ge der theoretischen Logik, on elementary mathematical logic. ... Mereology is a collection of axiomatic formal systems 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 Alfred North Whitehead (February 15, 1861 _ December 30, 1947) was a British philosopher and mathematician who worked in logic, mathematics, philosophy of science and metaphysics. ... Alfred Tarski, original name Alfred Teitelbaum (b. ... Tarskis axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called elementary, that is formulable in first order logic with identity, and requiring no set theory. ...

### 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.[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. a and b are parallel, the transversal t produces congruent angles. ...

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. 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°. If the parallel postulate is dropped from the list of axioms without replacement, the result is the more general geometry called absolute geometry. Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... For other uses, see Triangle (disambiguation). ... Lines through a given point P and asymptotic to line l. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Absolute geometry is a geometry 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. In this approach, a point is represented by its Cartesian (x,y) coordinates, a line is represented by its equation, and so on. 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. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. 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 Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Fig. ... | name = David Hilbert | image = Hilbert1912. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

$|PQ|=sqrt{(p-r)^2+(q-s)^2}$

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 between elements of the set is defined. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes 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. 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.

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. Kant redirects here. ... The Critique of Pure Reason is widely regarded as the philosopher Immanuel Kants major work, first published in 1781, with a second edition in 1787. ...

However, Einstein's theory of general relativity shows that the true geometry of spacetime is non-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. 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. â€œEinsteinâ€ redirects here. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... GPS redirects here. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes 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 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. 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. 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. 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. However, which geometry to use - Euclidean, Riemannian, de Stitter, anti de Stitter and some others - is a major point of demarcation between them. 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. 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). The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... This box:      String theory is a still developing mathematical approach to theoretical physics, whose original building blocks are one-dimensional extended objects called strings. ... Scientific consensus is the collective judgment, position, and opinion of the community of scientists in a particular field of science at a particular time. ... Quantum field theory (QFT) is the quantum theory of fields. ... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ... In physics quanta is the plural of quantum. ... The Planck length, denoted by , 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. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... This page discusses Theories of Everything in physics. ... 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 massive galaxy) between the source object and the observer. ...

## 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. 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. Using Newton's law of universal gravitation, the orbit of a comet around the Sun is Apollonius of Perga [Pergaeus] (ca. ... Elliptical redirects here. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... Galileo redirects here. ... Kepler redirects here. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ... Comet Hale-Bopp Comet West For other uses, see Comet (disambiguation). ... Sol redirects here. ...

• 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.

In each case the Sun will be at one focus of the conic, and the motion will sweep out equal areas in equal times. For other senses of this term, see escape velocity (disambiguation). ... In geometry, the focus (pl. ...

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. Look up air in Wiktionary, the free dictionary. ... This article is about Earth as a planet. ... Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... 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. 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. Einstein redirects here. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... This article is about the planet. ...

## Logical status

Euclidean geometry is a first-order theory. 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. First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ...

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. 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. Gödel's incompleteness theorems showed the futility of Hilbert's program of proving the consistency of all of mathematics using finitistic reasoning. Tarski's findings do not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.[3] In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland â€“ October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... Tarskis axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called elementary, that is formulable in first order logic with identity, and requiring no set theory. ... A logical system or theory is decidable if the set of all well-formed formulas valid in the system is decidable. ... In mathematical logic, GÃ¶dels incompleteness theorems, proved by Kurt GÃ¶del in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...

Although complete in the formal sense used in modern logic, there are things that Euclidean geometry cannot accomplish. 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. 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. A logical system or theory is decidable if the set of all well-formed formulas valid in the system is decidable. ... A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... Pierre Wantzel (1814â€“1848) was a French mathematician. ...

Absolute geometry, first identified by Bolyai, is Euclidean geometry weakened by omission of the fifth postulate, that parallel lines do not meet. 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. 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). Absolute geometry is a geometry that does not assume the parallel postulate or any of its alternatives. ... János Bolyai (December 15, 1802–January 27, 1860) was a Hungarian mathematician. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... Euler redirects here. ...

## Classical theorems

Cevas Theorem (pronounced Cheva) is a very popular theorem in elementary geometry. ... A triangle with sides a, b, and c. ... In geometry, the nine-point circle is a circle that can be constructed for any given triangle. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Niccolo Fontana Tartaglia. ... In this diagram, BD:DC = AB:AC. In geometry, the angle bisector theorem relates the length of the side opposite one angle of a triangle to the lengths of the other two sides of the triangle. ... a and b are parallel, the transversal t produces congruent angles. ...

Interactive geometry software (IGS, also called dynamic geometry environments, DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... Incidence geometry is a mathematical structure composed of objects of various types and an incidence relation between them. ... In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoffs axioms. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... Tarskis axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called elementary, that is formulable in first order logic with identity, and requiring no set theory. ... a and b are parallel, the transversal t produces congruent angles. ... Schopenhauers criticism of the proofs of the Parallel Postulate Schopenhauer criticized mathematicians attempts to prove Euclids Parallel Postulate because they try to prove from indirect concepts that which is directly evident from perception. ...

## Notes

1. ^ For the assertion that this was the historical reason for the ancients considering the parallel postulate less obvious than the others, see Nagel and Newman 1958, p. 9.
2. ^ Hofstadter 1979, p. 91.
3. ^ Franzén 2005.

## References

• Ball, W.W. Rouse (1960). 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). 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). 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). 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. Univ. of California Press.

Walter William Rouse Ball (1850 August 14–1925 April 4) was a Brtish mathematician, and a fellow at Trinity College, Cambridge from 1878 to 1905. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ... Douglas Richard Hofstadter (born February 15, 1945) is an American academic. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland â€“ October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ...

Results from FactBites:

 PlanetMath: non-Euclidean geometry (459 words) A non-Euclidean geometry is a geometry in which at least one of the axioms from Euclidean geometry fails. (This sum is not constant as in Euclidean geometry; it depends on the area of the triangle. Note also that, in spherical geometry, two distinct points do not necessarily determine a unique line; however, two distinct points that are not antipodal always determine a unique line.
 Euclidean geometry - Wikipedia, the free encyclopedia (2331 words) Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. 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 probably cannot be constructed within the theory. Absolute geometry, formed by removing the parallel postulate, is also a consistent theory, as is non-Euclidean geometry, formed by alterations of the parallel postulate.
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