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Topic: Parallel Postulate

Related Topics

Elliptic geometry

Euclidean geometry

Finite geometry

Euclid

Projective geometry

Giovanni Gerolamo Saccheri

Axiom

Hyperbolic geometry

Eugenio Beltrami

Affine geometry

Algebraic equation

Line segment

Axiomatic method

Taxicab geometry

Euclidean space

	Parallel postulate - Wikipedia, the free encyclopedia
		A geometry where the parallel postulate cannot hold is known as a non-euclidean geometry.
		Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry.
		For two thousand years the parallel postulate was suspected by some mathematicians to be a theorem which could be proved using Euclid's first four postulates.
	en.wikipedia.org /wiki/Parallel_postulate (1043 words)

	Parallel (geometry) - Wikipedia, the free encyclopedia
		Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these.
		When embedded in Euclidean space a dimension higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center.
		The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space.
	en.wikipedia.org /wiki/Parallel_(geometry) (1050 words)

	parallel postulate - Article and Reference from OnPedia.com
		A geometry where the parallel postulate is violated is known as a non-Euclidean geometry.
		Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proved in a system where the parallel postulate is true, and that if they are assumed as axioms, then the parallel postulate can be proved.
		The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami.
	www.onpedia.com /encyclopedia/parallel-postulate (399 words)

	Euclidean geometry
		The parallel postulate was a subject of deep contention among modern mathematicians in the middle ages.
		They tried to prove this fact by "not assuming" the postulate to hold true, and try and arrive at this postulate as a theorem by means of inference rules alone.
		In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist.
	www.guajara.com /wiki/en/wikipedia/e/eu/euclidean_geometry.html (693 words)

	inleiding
		Parallel lines are straight lines that are in the same plane and that in both directions extended infinitely do not meet in either direction.
		Mathematicians investigated whether the parallel postulate is a proposition that can be deduced from the other axioms or can be replaced by another, simpler postulate that would result in the same mathematics.
		This is because this proposition is equivalent to the parallel postulate.
	members.home.nl /gulikgulikers/NE_inleiding.htm (2309 words)

	Euclid's Fifth Postulate
		Postulates 1 and 3 set up the "ruler and compass" framework that was a standard for geometric constructions until the middle of the 19th century.
		We may think of the fourth postulate as having been justified by the everyday experience acquired by man in the finite, inhabited portion of the universe which is our world and extrapolated (much as the Postulate 2) to that part of the world whose existence (and infinite expense) we sense and believe in.
		He wrote, "This postulate ought even to be struck out of Postulates altogether; for it is a theorem..." Now we know that it is impossible to derive the Parallel Postulate from the first four.
	www.cut-the-knot.org /triangle/pythpar/Fifth.shtml (776 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		This postulate, one of the most controversial topics in the history of mathematics, is one that geometers have tried to eliminate for more than two thousand years.
		It is observed by Pogorelov that the parallel straight lines this proof relies on are not explicitly contained in the other postulates or axioms and therefore cannot be deduced from them.
		In fact, the original postulate that he based the proof on was logically equivalent to Euclid's fifth postulate.
	www.math.rutgers.edu /~cherlin/History/Papers2000/eder.html (2053 words)

	Parallel postulate
		The parallel postulate, also called Euclid's fifth postulate on account of it being the fifth postulate in Euclid's Elements.
		If a line segment intersects two straight lines forming two interior angles on the same side sum to less than two right angles then the two lines segments, if extended indefinitely, meet on that side on which are the angles less than the two right angles.
		The parallel postulate is the only postulate of Euclidean geometry which fails for non-Euclidean geometry.
	www.guajara.com /wiki/en/wikipedia/p/pa/parallel_postulate_1.html (352 words)

	The Parallel Postulate
		Postulates 1 and 3 are based on the geometric construction.
		Postulate 2 illustrates a common belief that straight lines may not terminate.
		So, Postulate 4 is justified by our belief that no matter where two perpendicular lines are drawn, the angle they form is one and the same and is called right.
	pegasus.cc.ucf.edu /~xli/non-euclid.htm (945 words)

	Parallel postulate 1 - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		Start the Parallel postulate 1 article or add a request for it.
		Look for Parallel postulate 1 in Wiktionary, our sister dictionary project.
		Look for Parallel postulate 1 in the Commons, our repository for free images, music, sound, and video.
	www.sciencedaily.com /encyclopedia/parallel_postulate_1 (155 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		A postulate is a statement about a geometrical world that is accepted without justification and makes the frame work for the geometrical world.
		So now to Euclid's fourth and most troublesome postulate: For every line, l, and point, P, which is not on that line, there exists a unique line, m, through P that is parallel to l.
		The disagreement came when mathematicians tried to figure out whether or not the parallel postulate had to be taken as a postulate, or whether it followed from the other four.
	mathforum.org /library/drmath/view/54750.html (543 words)

	Math 123 Course Information (Site not responding. Last check: 2007-11-06)
		The purpose of Math 123 is to study the axiom sets and models for various geometries, with particular attention paid to Euclid's parallel postulate and to models for geometries that violate the parallel postulate (noneuclidean geometries).
		It seemed as if the parallel postulate should be a consequence of the other postulates, that is, it should be a theorem rather than a postulate.
		In the nineteenth century mathematicians began to consider the possibility that there might be geometries for which the parallel postulate fails, and they began to study the properties of such geometries.
	www.math.ucla.edu /undergrad/courses/math123 (402 words)

	Non-Euclidean geometry (Site not responding. Last check: 2007-11-06)
		Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found.
		Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate.
		In fact Beltrami's model was incomplete but it certainly gave a final decision on the fifth postulate of Euclid since the model provided a setting in which Euclid's first four postulates held but the fifth did not hold.
	www-groups.dcs.st-andrews.ac.uk /~history/HistTopics/Non-Euclidean_geometry.html (1843 words)

	SparkNotes: Constructions: Parallel Lines
		Although parallel lines are usually thought of in pairs, an infinite number of lines can be parallel to one another.
		The Parallel Postulate states that there exists one line through C which is parallel to line AB.
		Whenever you encounter three lines, and only two of them are parallel, the third line, known as a transversal, will intersect with each of the parallel lines.
	www.sparknotes.com /math/geometry1/constructions/section3.rhtml (509 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Postulate 3 If two points of a line are in a given plane, then the line is in the plane.
		Postulate 5 Space is determined by at least four points not all in the same plane.
		Theorem 15 The sum of the measures of the angles of a triangle is 180 degrees.
	users.adelphia.net /~mathhomeworkhelp/geometry.html (1527 words)

	Postulate and Theorems (Site not responding. Last check: 2007-11-06)
		If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.
		If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.
		If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary.
	www.wmjh.mcdowell.k12.nc.us /mathportal/chapter_3_postualtes_and_theorems.htm (267 words)

	Body (Site not responding. Last check: 2007-11-06)
		Let the following be postulated: that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
		No version of a parallel postulate has been necessary either on the plane or on a sphere or on a hyperbolic plane.
		Above we noted that Playfair's Postulate is not true on a sphere or a hyperbolic space, and in Problem 10.1 you should have decided whether or not your postulate is true on spheres or on hyperbolic spaces.
	www.math.cornell.edu /~dwh/books/eg99/Ch10/Ch10.html (2091 words)

	Euclid's Elements, Book I, Proposition 30
		A number of the propositions in the Elements are equivalent to the parallel postulate Post.5 in the sense that if the rest of the postulates are assumed and any one of these propositions is assumed, then the parallel postulate can be proved as a proposition.
		Two advantages of Playfair's axiom over Euclid's parallel postulate are that it is a simpler statement, and it emphasizes the distinction between Euclidean and hyperbolic geometry.
		Two disadvantages are that it does not have the historical importance of Euclid's parallel postulate, and the proof of the parallel postulate from Playfair's axiom is nonconstructive.
	aleph0.clarku.edu /~djoyce/java/elements/bookI/propI30.html (643 words)

	Mathmatics - Postulates and Theorems
		If two parallel lines are cut by a transvesal, then the pairs of alternate interior angles are congruent.
		If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
		If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
	www.rio.k12.wi.us /MATH/geo1.html (872 words)

	The Pythagorean Theorem is Equivalent to the Parallel Postulate
		The Pythagorean Theorem is Equivalent to the Parallel Postulate.
		Then r is parallel (that is, doesn’t intersect) with line l, for a triangle with two right angles contradicts Lemma 1.
		In that context, the Parallel Postulate emerges as a paraphrase of the algebraic conditions for two simultaneous linear equations to have no solution, as opposed to either a single solution, or an infinite set of solutions.
	www.cut-the-knot.org /triangle/pythpar/PTimpliesPP.shtml (2205 words)

	Hyperbolic Geometry
		In other words, Neutral geometry is Euclidean Geometry without the parallel postulate or any subsequent axioms derived by using the parallel postulate.
		Euclid's Postulate II: For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE.
		Hilbert's Parallel Postulate: For every line l and for every point P that does not lie on l, there is at most one line m through P such that m is parallel to l.
	www.willamette.edu /~zizza/Courses/SeniorSeminar/G1.4/hyperbolic.html (530 words)

	Foundations and Structure of Mathematics 1 "The Pythagorean Theorem" webpage
		The Parallel Postulate says that "Through a given point not on a given line can be drawn only one line parallel to the given line" (see problem 30 of section 4.6 of The Heart of Mathematics).
		From the Parallel Postulate it follows that: "The sum of the measures of the angles of a triangle is 180
		The Parallel Postulate is neither true nor false - it is merely an assumption of Euclidean geometry.
	www.etsu.edu /math/gardner/5025/pythagorean.htm (1118 words)

	The Parallel Postulate
		Postulate states that lines will always intersect at some point unless they are parallel.
		Even before Beltrami proved the independence of the Parallel Postulate, mathematicians were still able to work on Projective Geometry.
		Century, Kepler suggested the notion of ‘points at infinity’ where parallel lines would intersect; meanwhile Desargues and Pascal began to study Geometry using only intersections.
	www.math.umd.edu /~lidador/Affine/intro2.htm (344 words)

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