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Topic: Postulate

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	Lake Zurich Middle School South - Gifted Education - Geometry
		Postulate 3: If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, then the two triangles are congruent.
		Postulate 4: If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, then the two triangles are congruent.
		Postulate 6: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, then the two triangles are congruent.
	www.freewebs.com /lzmssgeometry/postulates (529 words)

	Parallel postulate - Wikipedia, the free encyclopedia
		A geometry where the parallel postulate is violated is known as a non-Euclidean geometry.
		Euclid did not postulate the converse of his fifth postulate, which is necessary 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 (980 words)

	§ 9. axiom / postulate / theorem / corollary / hypothesis. 4. Science Terms. The American Heritage Book of English ...
		axiom / postulate / theorem / corollary / hypothesis.
		In fact, this statement defines an important relationship between toves and momes; it is an axiom or postulate of Jabbermetry.
		Thus, axioms and postulates form the roots of a particular deductive system; theorems and corollaries are the logical consequences that fill out the deductive system; hypotheses drive theoretical development forward.
	www.bartleby.com /64/C004/009.html (455 words)

	PlanetMath: axiom (Site not responding. Last check: 2007-10-08)
		Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge.
		A great lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions.
		The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.
	planetmath.org /encyclopedia/Axiom.html (1184 words)

	Postulational Terminology (Site not responding. Last check: 2007-10-08)
		POSTULATING IS ONE OF THE fundamental attributes of a Being, the other attribute is perceiving, of course.
		A postulate and a matching counter-postulate together constitute a Problem; and this can be one of the components of a Goals Problems Mass, which is just a string of Problems deriving from a Goal (the Mass is just the mental mass of a bunch of problems, and accreted locks).
		Postulate 2 is a decision which is so indefinite as to verge on being a mere pious hope.
	freezoneamerica.org /ivy/24/postulational.html (840 words)

	Euclid's Fifth Postulate
		This appears to be the first attempt to prove the postulate by deriving a contradiction from the assumption that the fifth postulate is wrong.
		Assuming that Euclid's second postulate (A piece of straight line may be extended indefinitely.) requires straight lines to be infinitely long, he showed that (B) indeed leads to a contradiction.
		The Fifth Postulate is Equivalent to the Pythagorean Theorem
	www.cut-the-knot.com /triangle/pythpar/Attempts.shtml (873 words)

	Euclid's Elements, Book I, Postulate 1
		This first postulate says that given any two points such as A and B, there is a line AB which has them as endpoints.
		This is one of the constructions that may be done with a straightedge (the other being described in the next postulate).
		Since Euclid uses this postulate as if it includes the uniqueness as part of it, he really ought to have stated the uniqueness explicitly.
	aleph0.clarku.edu /~djoyce/java/elements/bookI/post1.html (165 words)

	POSTULATE - Definition
		The distinction between a postulate and an axiom lies in this, -- that the latter is admitted to be self-evident, while the former may be agreed upon between two reasoners, and admitted by both, but not as proposition which it would be impossible to deny.
		To beg, or assume without proof; as, to postulate conclusions.
		postulated a sort of paramount supremacy over this nation.
	www.hyperdictionary.com /dictionary/postulate (245 words)

	Parallel postulate - Open Encyclopedia (Site not responding. Last check: 2007-10-08)
		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.
		Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proved the parallel postulate from Euclid's other postulates.
	open-encyclopedia.com /Fifth_postulate (384 words)

	Parallel postulate -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A geometry where the parallel postulate is violated is known as a (Geometry based on axioms different from Euclid's) non-Euclidean geometry.
		Instead this led to the invention of (A non-Euclidean geometry in which it is assumed that through any point there are two or more parallel lines that do not intersect a given line in the plane) hyperbolic geometry.
		The sum of the (The space between two lines or planes that intersect; the inclination of one line to another; measured in degrees or radians) angles in a (A three-sided polygon) triangle is 180°.
	www.absoluteastronomy.com /encyclopedia/p/pa/parallel_postulate.htm (455 words)

	Postulate 1 (Site not responding. Last check: 2007-10-08)
		Postulate 2 can be shown to follow from Postulate 1.As Einstein chose to do it, SR is a 2 line plus a one page derivation theory.
		Postulate 1 reduces it to one and a transformation.
		Postulate 1 was not, and still is not, a significant part of modern physics theory.
	home.att.net /~j.neergaard/SR/Postulate1.html (1030 words)

	Einstein's light postulate of Antonio Ferrigno
		We have therefore investigated whether the 2nd postulate of Einstein is also a matter of convention and whether it is possible to find transformations linking physical quantities from one "convention" to another.
		In particular we suppose that a "Galilean observer" would observe the light propagating in accordance with the Ritz's postulate, when the particles are not subjected to forces (according to Ritz [8] the speed of the light is c in the frame where the source was at rest at the instant of emission).
		Non-Lorentz transformations exist which are also in accordance with the postulates of SRT; the existence of these alternative transformations can also be considered a paradox, since it is then not clear how to select the "true" transformations among the various possibilities.
	www.geocities.com /antonioferrigno/npameating.html (2319 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)

	C&EN: SCIENCE & TECHNOLOGY - HAMMOND POSTULATE
		Hammond's postulate built upon seminal work by several organic chemists, including the late Ronald P. Bell of Oxford University, as well as the late Meredith G. Evans and the late Michael Polanyi of Manchester University, in England.
		But unlike earlier work, the postulate allowed organic chemists to discuss the outcome of complex reactions in terms of the structure of their transition states.
		Hammond postulated that in highly exothermic reactions (left) the transition state (Ts) is structurally similar to the reactant (R), but that in highly endothermic reactions (right) the product (P) is a better model of the transition state.
	pubs.acs.org /cen/science/8120/8120sci2.html (737 words)

	Twenty Postulates
		A twentieth postulate, which pertains to strengthening and sustaining teachers, was added in September 2000.
		Programs for the education of the nation's educators must be viewed by institutions offering them as a major responsibility to society and be adequately supported and promoted and vigorously advanced by the institution's top leadership.
		Programs for the education of educators must compete in an arena that rewards efforts to continuously improve on the conditions embedded in all of the postulates and tolerates no shortcuts intended to ensure a supply of teachers.
	depts.washington.edu /cedren/Publications/postulates.htm (771 words)

	Chapter 20. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity. ...
		The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity.
		We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity.
		We must note carefully that the possibility of this mode of interpretation rests on the fundamental property of the gravitational field of giving all bodies the same acceleration, or, what comes to the same thing, on the law of the equality of inertial and gravitational mass.
	www.bartleby.com /173/20.html (1221 words)

	The Pythagorean Theorem is Equivalent to the Parallel Postulate
		The Pythagorean Theorem is Equivalent to the Parallel Postulate.
		As we have already observed that I implies II, we thus establish that the Pythagorean Theorem itself is yet another proposition which may be taken as an equivalent alternative to the Parallel Postulate.
		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 (2165 words)

	Encyclopedia: Parallel postulate (Site not responding. Last check: 2007-10-08)
		In geometry, an internal angle is an angle that 2 sides of a polygon form by touching.
		A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions.
		Spherical geometry is the geometry of the two-dimensional surface of a sphere.
	www.nationmaster.com /encyclopedia/Parallel-postulate (1913 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-groups.dcs.st-and.ac.uk /~history/HistTopics/Non-Euclidean_geometry.html (1843 words)

	How and Why Hyperbolic Geometry Came to Be
		Paul Kelly and Gordon Matthews argue that if a fifth postulate is "logically implied" by the previous four postulates, then the fifth postulate is not a postulate at all, but rather a theorem (Kelly, 2).
		Making the fifth postulate a theorem does not change the system at all because the remaining propositions can be proved using the four postulates and the first theorem.
		Kelly and Matthews imply that the worst situation would be to assume the fifth postulate is a theorem and be unable to prove it, which is the case of Euclid’s Fifth Postulate (Kelly, 2).
	filebox.vt.edu /users/jtoffene/HowandWhyHyperbolicGeometryCametoBe.htm (1887 words)

	Euclid's Elements, Book I, Postulate 5
		Of course, this is a postulate for plane geometry.
		This postulate is usually called the "parallel postulate" since it can be used to prove properties of parallel lines.
		Geometers throughout the ages have tried to show that it could be proved from the remaining postulates so that it wasn't necessary to assume it.
	aleph0.clarku.edu /~djoyce/java/elements/bookI/post5.html (349 words)

	The Postulates of Practical Reason by Kant
		It follows that the postulate of the possibility of the highest derived good (the best world) is likewise the postulate of the reality of a highest original good, that is to say, of the existence of God.
		Moreover, it is not meant by this that it is necessary to suppose the existence of God as a basis of all obligation in general (for this rests, as has been sufficiently proved, simply on the autonomy of reason itself).
		No doubt they were so far right that they established the principle of morals of itself independently of this postulate, from the relation of reason only to the will, and consequently made it the supreme practical condition of the summum bonum; but it was not therefore the whole condition of its possibility.
	www.baylor.edu /~Scott_Moore/God_immortality.html (1832 words)

	SparkNotes: Constructions: Parallel Lines
		The Parallel Postulate states that there exists one line through C which is parallel to line AB.
		In the problem section we'll see how to use the parallel postulate to find the measures of unknown angles.
		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 (507 words)

	Von Neumann's Fifth Postulate (Site not responding. Last check: 2007-10-08)
		To understand von Neumann's fifth postulate, notice that although the conventional interpretation does not uniquely determine the outcome of a particular measurement for a given state, it does predict a unique 'expected value' for that measurement.
		He assumed that, for any hidden variable theory, just as in the conventional interpretation, the averages of X+Y, X and Y evaluated over a set of identical systems are additive.
		Thus, as Bell remarked, "[von Neumann's] 'very general and plausible postulate' is absurd".
	mathpages.com /rr/s9-06/9-06.htm (1639 words)

	The Second Postulate has been blown!
		The Original Intention or First Postulate was to create a game with individuals where everyone wins.
		The Counter intention or Second Postulate was that "because change has occurred and static has moved, only one may win all others must sink into the mud".
		From there came all the "to win, you have to be better than everyone else.", "Form alliances and then break them when they no longer suit your purpose to win", "the Haves and the Have Nots" and all that other garbage we could have done without.
	radio.weblogs.com /0114013/stories/2002/11/22/theSecondPostulateHasBeenBlown.html (1063 words)

	Alkene Reactivity
		This Hammond Postulate states that a transition state will be structurally and energetically similar to the species (reactant, intermediate or product) nearest to it on the reaction path.
		The exceptional stability of allyl and benzyl cations is the result of charge delocalization, and the stabilizing influence of alkyl substituents, although less pronounced, has been interpreted in a similar fashion.
		When an unsymmetrically substituted double bond is protonated, we expect the more stable carbocation intermediate to be formed faster than the less stable alternative, because the activation energy of the path to the former is the lower of the two possibilities.
	www.cem.msu.edu /%7Ereusch/VirtualText/addene1.htm (3521 words)

	Third Postulate - the Shadowing Effect (Site not responding. Last check: 2007-10-08)
		That means that if the experiment is repeated many times, then there will be particles that were not registered while close to the counter; they moved away, and they will never be registered in the future.
		To answer this question it is not enough to use the two postulates above.
		Our third postulate reads: prior to any event, and independently of whether any event will happen or not, the state of the system is described by the vector
	quantumfuture.net /quantum_future/papers/time/node5.html (316 words)

	Einstein's Equivalence Postulate and Instantaneous Gravitational Action
		During the end of the 19th century, theoretical physics was based on three areas of knowledge: Newton's equations, Maxwell's equations, and the Galilean transformation.
		However, the question is whether there are some other phenomena which do not obey EEP and can be described by another postulate.
		Here, I begin by noting that Newton's equations are invariant under a GT and while they can be generalized to obey EEP, and perhaps provide new insights for phenomena, I see no reason why they should be exclusively made Lorentz invariant.
	www.coolissues.com /gravitation/gwaves.htm (755 words)

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