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Topic: Fifth postulate

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	Parallel postulate - Wikipedia, the free encyclopedia
		In geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry.
		A geometry where the parallel postulate is violated is known as a non-Euclidean geometry.
		Janos Bolyai (and probably Carl Friedrich Gauss before him) realized that the negation of the fifth postulate leads to logically consistent geometries which were later developed by Lobachevsky, Riemann and PoincarÃ© into hyperbolic geometry and spherical geometry.
	en.wikipedia.org /wiki/Parallel_postulate (619 words)

	International Catholic University 7.6
		Postulate one declares that a straight line can be drawn from any point to any point, whereas the definition says only that a straight line is a line that lies evenly with the points on itself.
		Postulate four asserts that the right angle is a determinate magnitude and so can serve as a standard or criterion by which to measure other angles.
		Hence by means of his fifth postulate Euclid in effect specifies that his is a geometry of a space that is flat, or of zero curvature.
	home.comcast.net /~icuweb/c00706.htm (6029 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.com /triangle/pythpar/Fifth.shtml (768 words)

	Euclid mathematician picture math gift shirt mathematicians famous
		Postulate four asserts that all right angles are equal -- a concept that assumes a commonality to space, with geometrical constructs existing independent of the specific space or location they occupy.
		The parallel postulate states that one, and only one, line can be drawn through a point parallel to a given line -- and it is from this postulate, and on this basis, that what has come to be known as " Euclidean geometry " proceeds.
		It was not until the 19th century that Euclid's fifth postulate -- the "parallel postulate" was rigorously and successfully challenged.
	www.mathematicianspictures.com /Mathematicians/Euclid.htm (456 words)

	Fifth postulate (Site not responding. Last check: 2007-11-04)
		Einstein's Equivalence Postulate and Spacelike Waves James Constant argues that Einstein's equivalence postulate must be abandoned for the study of spacelike wave phenomena and gravitation.
		Proof of Bertrand's Postulate International Mathematics Olympiad tutorial proving the theorem of Chebyshef that there is a prime between n and 2n for all positive integers n > 1.
		A New Concept of Time Travel A bipolar type of discontinuity in spacetime is postulated that would enable time travel, teleportation, antigravity, and warp drive.
	www.serebella.com /encyclopedia/article-Fifth_postulate.html (230 words)

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		Euclid’s fifth postulate is unlike any of the other previous postulates in that not only is not self-evident, but it is also a lot longer and more complex to understand.
		In the eighteenth century it was hoped that by assuming the negation of the fifth postulate, a contradiction could be arrived at, thus finally proving the truth of the postulate (by reductio ad absurdum).
		When Saccheri postulated the obtuse angle hypothesis he arrived at a contradiction, and when he postulated the acute angle hypothesis his results were absurd from the viewpoint of Euclidean geometry.
	www.geocities.com /griseborough/14.txt (2764 words)

	Euclidean geometry: Definition and links by Encyclopedian.com Information about Euclidean geometry
		Euclid's text Elements is an early systematic treatment of this kind of geometry, based on axioms (or postulates).
		Euclidean geometry is distinguished from other geometries by the parallel postulate, which is usually phrased as follows: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
		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.encyclopedian.com /fi/Fifth-Postulate.html (360 words)

	Reference.com/Web Search/postulate
		John Wallis proposed a new axiom that implied the parallel postulate and...
		The Pythagorean Theorem is Equivalent to the Parallel Postulate
		A proof that the Fifth Postulate is equivalen to Pythgoras' Theorem.
	www.reference.com /search?q=postulate (215 words)

	Non-Euclidean geometry
		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.
		The importance of Saccheri's work was that he assumed the fifth postulate false and attempted to derive a contradiction.
		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.meta-religion.com /Mathematics/Articles/non-euclidean_geometry.htm (1899 words)

	The Left Hand of the Electron -Euclid's Fifth
		The fifth postulate talks about 'the interior angles on the same side,' that is, 1 and 4 on one side and 2 and 3 on the other.
		The fifth postulate now states that if the lines GH and CD are extended, they will intersect on the side where the interior angles with a sum less than two right angles are located.
		The fifth postulate is just not contained in the other axioms or in any list of axioms useful in geometry and simpler than itself.
	www.fortunecity.com /emachines/e11/86/l-hand10.html (3876 words)

	Euclid's Fifth Postulate
		By the end of the last century, it was also shown that the fifth postulate is independent of the remaining axioms, i.e., all the attempts at proving it had been doomed from the outset.
		This appears to be the first attempt to prove the postulate by deriving a contradiction from the assumption that the fifth postulate is wrong.
		Adrien-Marie Legendre (1752-1833) was preoccupied with the fifth postulate for decades.
	www.cut-the-knot.org /triangle/pythpar/Attempts.shtml (874 words)

	Euclid’s Fifth Postulate and the Advent of Non-Euclidean Geometries (Site not responding. Last check: 2007-11-04)
		Euclid’s Fifth Postulate and the Advent of Non-Euclidean Geometries
		That particular postulate was postulate #5, which is more commonly known as the parallel postulate.
		The history of Euclid’s fifth postulate and the efforts of mathematicians to prove it is a very colorful story.
	mywebpages.comcast.net /lcrosswell/paper.html (3311 words)

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		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 because of this that Euclid, seeing the necessity of the postulate, selected what he apparently found to be the simplest form of it as his fifth postulate (Sarton, page 40).
		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)

	Fifth postulate (Site not responding. Last check: 2007-11-04)
		Several properties of Euclidean geometry are logicallyequivalent 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 finallydemonstrated by Eugenio Beltrami.
		Some of the other statements that are equivalent to the parallel postulate appear at first to be unrelated to parallelism.Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proved the parallel postulate from Euclid's otherpostulates.
	www.therfcc.org /fifth-postulate-42970.html (381 words)

	The Parallel Postulate
		Postulates 1 and 3 are based on the geometric construction.
		It is conjectured that Euclid himself had mixed feelings about the fifth postulate as he avoided using it until Proposition I.29 in his Elements.
		Now, we know that the fifth postulate is independent of the other postulates and it cannot be derived from the other postulates.
	pegasus.cc.ucf.edu /~xli/non-euclid.htm (945 words)

	The origins of proof
		In addition, in the nineteenth century, Legendre went on to show that postulate 5 is equivalent to the postulate that "the sum of the angles of a triangle is equal to two right angles").
		Many attempts to prove the fifth postulate in this manner were made, and often a putative proof would be accepted for a long period before being shown to be flawed.
		One of the consequences of the failure of the 5th postulate is that it is no longer true that the sum of the angles of a triangle is always 180 degrees.
	pass.maths.org.uk /issue7/features/proof1 (2381 words)

	Outline for Chapter XXVI. (Site not responding. Last check: 2007-11-04)
		Later, Georg Riemann slightly modified the fourth postulate of Euclid to eliminate Sacceri's contradiction with (5c), then chose (5c) to be the fifth axiom, and came up with yet another geometry.
		The geometry with the statement (5b) as its fifth postulate is called the hyperbolic geometry (Gauss geometry, Lobachevsky geometry).
		The geometry with the modified fourth postulate and statement (5c) as its fifth postulate is called the elliptic geometry.
	www.math.psu.edu /elkin/math/035-Su02/outline26.html (632 words)

	Fifth Postulate Problem
		A theorem that is derived from Euclid's postulate.
		Riemann changed the postulate and created a different set of theorems that were nonetheless self-consistent.
		He changed the c+v postulate to (c+v)/(1+v/c) but didn't use it, he said "But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c-v..." and that is not consistent.
	www.pych-one.com /new-6442669-4395.html (1241 words)

	Postulates and Principles
		However, we now understand that Euclid 's fifth postulate is logically independent of the rest of Euclid 's logical structure.
		Einstein formally adopted this conjecture as a postulate, but on a more fundamental level it serves as a principle, since it entails the decision to organize our knowledge in terms of coordinate systems with respect to which the equations of mechanics hold good, i.e., inertial coordinate systems.
		In view of this, some have wondered why he did not simply dispense with his "second postulate” and assert that the "laws of electrodynamics and optics" in the statement of the first principle are none other than Maxwell's equations.
	www.mathpages.com /rr/s3-01/3-01.htm (1824 words)

	What is non-Euclidean geometry?
		Of these postulates, all were considered self-evident except for the fifth postulate.
		The fifth postulate asserted that two lines are parallel (i.e.
		Mathematicians stumbled with ways to prove the validity of the fifth postulate from the first four postulates, which we now call the postulates of absolute geometry.
	ks.essortment.com /noneuclideange_risc.htm (963 words)

	Parallel postulate (Site not responding. Last check: 2007-11-04)
		Geometry that is independent of Euclid's fifth postulate (i.e.
		only assumes the first four postulates) is known as affine geometry.
		The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami.
	www.sciencedaily.com /encyclopedia/parallel_postulate_1 (450 words)

	Hyperbolic Geometry: History
		It was widely believed for many years that this fifth postulate could actually be proven by the other four.
		Yet the attempts to prove the fifth postulate were not totally a waste of time.
		John Playfair reworded Euclid's fifth postulate to the form it is currently known by.
	www.geocities.com /CollegePark/Residence/1492/euclid.html (488 words)

	Parallel postulate (Site not responding. Last check: 2007-11-04)
		Many attempts were made to prove the postulate in terms of Euclid's first four Instead this led to the invention of hyperbolic geometry.
		The independence of the parallel postulate from Euclid's axioms was finally demonstrated by Eugenio Beltrami.
		Some of the other statements that are to the parallel postulate appear at first be unrelated to parallelism.
	www.freeglossary.com /Playfair's_axiom (618 words)

	ISTE \| September (No. 1) (Site not responding. Last check: 2007-11-04)
		However, many believed that the fifth postulate, because of its technical nature, could be established from the first four postulates.
		Because it is equivalent to Euclid’s fifth postulate, the EUC is often called the Euclidean parallel postulate.
		Euclid’s fifth postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
	www.iste.org /inhouse/publications/ll/28/1/32c/index.cfm?Section=LL_28_1 (2731 words)

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