Talk:Parallel postulate

1145:

difference between motion and transformation in this context is splitting hairs, so I didn't. The real problems were in the previous sentence where a claim of a "first" was not supported by the citation given and the citation itself was nothing more than a student paper. I toned down the statement to something that could be verified by a legitimate source. There are a couple of other "firsts" that need to be removed which I will take care of in a few minutes. I don't consider the Encyclopedia of the History of Arabic Sciences to be a reliable source. It seems prone to hyperbole and making unfounded assertions. I note that in the quote you gave there is a statement about the proofs of Thales theorems. This is pure BS ... to my knowledge we have no direct evidence of how Thales proved anything. All the sources I have seen make statements about how Thales might have proved this or that, or that someone said that Thales had proved something this way. To say that Thales did something a certain way is pure speculation. I am going to revert your cut because I do think the statement is appropriate, but I might come back to it and cut it out again if when taken as a whole this section on Khayyam is out of proportion to the rest of the history section.

1441:

at least one. As you say, this converse is true in neutral geometry ... parallels exist in neutral geometry. Logically unrelated, yes, but most students think of Playfair's axiom as an if and only if statement (and many authors present it that way). Making this statement explicit is a good way to combat that. As to distinguishing Euclidean geometry from elliptic ... it is the converse which does this. Euclidean geometry, being a neutral geometry, has parallels while elliptic geometry does not and so can not be a neutral geometry. Saying that there is at most one parallel in this Playfair form does not distinguish these two geometries (but it does distinguish Euclidean from hyperbolic, both of which are neutral geometries and so have parallels.) As to changes in other axioms ... elliptic geometry is non-Euclidean (by definition), but it is not a neutral geometry, so changes to axioms other than the fifth must occur if you want to get elliptic geometry.

1317:

predjudices are just bred into research mathematicians). On the other hand, I can't think of a quality math journal that would publish either paper (on the grounds that the content would not be suitable for their readership) and after glancing through them quickly I didn't see anything out of the ordinary, so checking the arguments wouldn't be that difficult. At this point I am more likely to believe the result but I still have that little nagging doubt in the back of my head - it won't go away until I go through one of these proofs myself. So, until then, I will just live with this nag (by the way, your citation, as far as I can tell, just has the one sentence relating the two ... that's just not going to do it for me, but thanks for the effort.)

1673:

but if you restrict to the realm of Euclidean and hyperbolic geometry, then the above argument is valid and you get the equivalence (in context). I don't usually restrict my geometric viewpoint in this way and that caused me to overlook this fact. The Euclidean setting is the natural one for this question, but I now find the following question to be more interesting. What is the largest class of geometries in which the two statements are equivalent? Is this just a Euclidean geometry property or can the axiom set be loosened to permit more general settings for which the equivalence is true? These are not questions for Knowledge, but I did want to point out why I might have fallen into this error. --

1335:: Julien Narboux: I also doubt that pythagorean theorem can be proved equivalent to the parallel postulate. First, how do you state the pythagorean theoreom without parallel postulate ? to state the theorem you need to define the multiplication geometrically, the usual construction of multiplication and addition following Descartes and Hilbert and the proof that it forms a field use the parallel postulate. Nov 2014. Update: Feb 2015: Millman and Parker, Geometry a metric approach with models page 226 contains a proof of the equivalence between pythagoras theorem and euclid 5, it use an axiom system which assume continuity and the reals numbers from the beginning (jnarboux). 1220:

endpoint of the segment. This is a very explicit use of motion to define something geometric. Euclid on the other hand is very circumspect about the concept and introduces superposition in order to talk about motion without talking about motion. Euclid's use is implicit and I believe that Katz is trying to say that Alhazen is the first to explicitly use motion in a geometric definition. Its not that the Greeks didn't discuss or use motion, they certainly did, but because Euclid set the standards, motion was not considered dependable in geometry. This was so ingrained that everyone seemed to jump all over Alhazen for even trying to use motion in the way he did.

1260:

found to be all that reliable). The second citation is to Pruss who wrote a philosophy (not mathematical) book in which this statement is made as an example of what he means by logical equivalence. There is no proof there ... I am sure he just made this up so that he didn't have to say "If X can be proved assuming Y, and Y can be proved assuming X, then X and Y are logically equivalent." These citations make me think that this claim is bogus. I could be wrong. If someone has a real reference I'd like to see it. If someone would like to go through Brodie's paper and verify it, that would also be nice - but I am not willing to spend the time to do it myself.

516:

reads: Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ εὐθείας ἐκβαλεῖν, which Fitzpatrick translates as "And to produce a finite straight-line continuously in a straight-line," Heath as "To produce a finite straight line continuously in a straight line." Note that it really doesn't say "indefinitely," it says "continously," συνεχὲς. Only in the fifth postulate does he use the phrase "if produced indefinitely." I don't think that's a coincidence; one main reason the ancients didn't like the parallel postulate is that they thought it was suspect to have to talk about a potentially infinite process. (See the footnote at "One reason that the ancients..." in

816:

just rambling, but I remember from my logic class, that given a statement, that the inverse or the converse are not always necessarily true. Please correct me if I am wrong. I realize that this may be the base for some of the non-Euclidean geometries, that given this postulate it must be proven otherwise that the inverse (if the sum of the interior angles are not less than two right angles, then the lines will not meet if extended to infinity) is also true, and cannot simply be taken as an assumption. I'm no expert, and I have a lot to study yet about mathematics, but I was wondering if anyone else has some thoughts on this.

525:, you just can't always determine unambiguously what they would mean in a context Euclid didn't anticipate. A similar point comes up in postulate 3. Heath (pp. 199-200) interprets it as meaning that space is infinite, since you can draw a circle with *any* radius, with no limit on how big it can be. Realistically, Euclid's original expression of his axioms is just ambiguous, and also inconvenient if you want to do noneuclidean geometry. From a modern point of view, you'd clearly want more of a strict separation between the postulates that definitely describe absolute geometry and the ones that are definitely not absolute.-- 634:), which I think is logically equivalent to the question of whether two distinct lines can have a common segment (since the segment joining the two points is implicitly unique according to postulate 1). Heath gives a long discussion of this topic, pp. 196-199. His interpretation is that all the constructions referred to in the postulates are implicitly unique, and therefore postulate 2 (extending a line) does imply that distinct lines can't have a common segment. However, "This assumption is not appealed to by Euclid until XI.I.")-- 1039:(2) Every conceivable example, or manifestation, of Postulate 5 may be understood in terms of finite distances (since the lines must intersect somewhere). However, the case of parallel lines cannot be understood solely in terms of finite distances (or even 'indefinite' distances, by which Euclid means no more than 'any and all distances you can imagine'). So PP imports something new which was not present in P5. This shifting of the logical goalposts is something else which seems to have slipped under most commentators' radar. 95: 708:

than one point in common and that space is limitless in extent, in the sense that figures can be scaled up arbitrarily. This isn't consistent with elliptic geometry. Therefore I don't think it's really meaningful to try to prove definitively whether Playfair is exactly equivalent to the parallel postulate. Their equivalence or lack of equivalence is something that can only really be meaningfully determined in an axiomatic system hasn't got the assumption of flatness of space sprinkled all over it.--

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equivalences should be more than just a listing and at least one of those items should be double checked for accuracy. Also, a clear definition of "parallel lines" needs to be given and should not be confused with various properties of such lines which depend upon the geometry in which they live. The criticism section should be either moved into the history section or expanded to include other philosophical issues.

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contradictions are not necessarily contradictions. One author may be focusing on how the axioms were historically interpreted in a particular period. Another may be interested in explaining noneuclidean geometry to the general reader without getting bogged down in technicalities. Yet another author may actually have in mind a particular modern set of axioms that's similar to, but not identical to, Euclid's.--

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other fact that I need to bring in for this proof. It's not on my original list, but it's obvious that it's true. I just didn't list it before because it's so obvious." Re the distinction between the parallel postulate and its converse, this all depends on the question of whether two distinct lines can have two points in common (

1399:"The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan pointed out, this is logically equivalent to (Book I, Proposition 16)." 454:. Almost everyone seems to be in agreement that elliptical geometry is not an absolute geometry, but they rarely say which postulates they're assuming for absolute geometry. Indeed, many of the other axiom systems contain incidence axioms or plane separation axioms that can be used to prove Proposition 16 (see 1440:

This pretty much depends on what you think the fifth postulate is saying. Euclid's formulation is a bit awkward to work with. In the Playfair version, the fifth postulate is saying that there is at most one parallel to a given line through a non-incident point. The converse statement is that there is

1003:

Traveling Matt is absolutely correct. Euclid's Postulate 5 is not synonymous with the "postulate of the parallels"; the latter was a separate inference. Postulate 5 deals explicitly, specifically and literally with internal angles which add up to LESS THAN two right angles. This clearly rules out

815:

I remember a lot from my high school geometry class, and I really loved that class, but looking at the definition now, it seems strange to me that this should be called the parallel postulate, as it doesn't really define parallel lines, but rather defines non-parallel lines. Also, I'm very sorry for

515:

Euclid uses the terms the terms "πεπερασμένος" and "άπειρος," which are usually translated as "finite" and "infinite." I'm not certain, but I believe the root of both words is the same as the root of "perimeter," i.e., they literally mean something like "bounded" and "unbounded." The second postulate

506:

Another central question is whether Euclid's second postulate excludes elliptic geometry. (The other three of the first four apparently don't.) This depends on how you interpret "can be extended indefinitely in a straight line". One might need to understand the Greek original in order to tell whether

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It's failed the google test so I am going to go ahead and remove it. As far as I can see it's just a bit of promotion for an otherwise unnotable minor project. Whoever wrote it up could come back with some references if they wish to put it back up, but it'll take a lot of convincing to show that this

1578:

I too am interested in this ... see my remarks above (Pythagorean theorem ???). The wolfram claim is based on Bodie's article at cut-the-knot, so it isn't an independent source. I would hope to hear more from the IP editor about who the "genuine geometer" is and what was the uncovered error. Even if

1672:

Hmmm. My bad, and I have struck out the offending comment above. In any real discussion of the equivalence of theorems, the axiomatic context has to be taken into account. These two statements are not logically equivalent since there are examples of geometries where one holds and the other doesn't,

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citations – that the Pythagorean theorem is equivalent to the parallel postulate. Having never heard this before I decided to check the references. The Weisstein reference seems to be based on an unpublished, unrefereed paper by Brodie which appears on the cut-the-knot site (not a place that I have

1219:

I am a little annoyed at Katz for not making his point a bit clearer, but I do think I know what he meant. Alhazen actually constructs a parallel line by taking a line segment perpendicular to a given line and "moving" it so that it remains perpendicular. The parallel line is the locus of the other

1007:

In Euclid's time, mathematicians were fully apprised of the special philosophical difficulties surrounding the concept of parallelism. Euclid appreciated that geometry as he understood it depended upon a "postulate of the parallels", but he wanted to short-circuit the philosophical difficulties as

707:

Summarizing some of the points I made above, I think the upshot of all this is that you can't really get absolute geometry from postulates 1-4 as stated by Euclid, simply omitting 5. Euclid's intended interpretation of postulates 2 and 3 is quite strong, implying that distinct lines can't have more

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If, as mentionned in the text, spherical, projective or elliptic geometry is allowed by Euclid 5th postulate then the theorems contained in Euclid's elements are not only valid for usual euclidean geometry but also for projective geometry. So for instance, no pythagoras theorem or anything of this

794:

Yes, I am sorry to say that you are wrong. You've most likely let some subtle assumption equivalent to the parallel postulate creep in unawares into your argument. Don't feel too bad about this, there is a long line of quite brilliant mathematicians who have done the same thing ... you are in good

1477:

Afraid not. This can be tricky business. If you are assuming SAS similarity as an axiom, then it is equivalent to the parallel postulate. If you are proving SAS similarity, you need the parallel postulate to do so. In hyperbolic geometry, where the parallel postulate is not true, if two triangles

1693:

This could be better formulated. A postulate is, after all, an obvious, or even a counter-intuitive (such as that upon which non-Euclidean geometry is based), statement for which there is no proof. There is a much better discussion of this in Heath, who lays out the history of both ancient and

1616:

The student's work, showing that the pythagorean theorem does not hold in hyperbolic geometry (and I haven't looked at it carefully to even guess as to whether or not it is valid) would not give the equivalence she claims even if the argument is valid (you would need to show that the pythagorean

629:

When you say Euclid uses unfounded assumptions, one thing you have to watch out for is that he never intended the list of five postulates to be exhaustive. They were more like a "greatest hits" list or a FAQ. There are places later on in the Elements where he essentially says, "Okay, here's this

1144:

The quote from Katz is verbatim and in context (it appears about 3/4 of the way down the first paragraph in section 7.4.2 The Parallel Postulate). For this reason I left it in. I toyed with the idea of removing the wikilink to transformation since that was not mentioned in Katz's quote, but the

760:

I have proven the existence of a triangle whose interior angle sun is equal to two right angles, or 180 degrees, from first principles. I did so by first (as a lemma) constructing a parallelogram from first principles and showing that its opposite sides are equal as well as its opposite angles

745:

Has anyone questioned the stated equivalence with the Pythagorean Theorem? The references given are highly suspect. One is a philisophical work on implication and seems to use this as an example of what equivalence means (in form only, not content). The other, in turn, refers to a Cut-the-Knot

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for the source for this statement.) Re whether postulate 2 "excludes great circles on a sphere that are finite but without ends," it's a question that has bugged me as well. I think you have to stop for a moment and and consider that this case is one that occurs naturally to us, today, because

683:

Sorry, I didn't answer immediately. Elliptic geometry is what you get from spherical geometry when the antipodal points get identified. In this way, the "other point" in the text is in fact the same point --there can't be two distict straight line through two distinct points (and in spherical

559:

It might be useful as a starting point to look for whether there's a consensus on the net, but that's only going to get you so far. One problem is that sources on the net are typically not as reliable as print sources. Statements like these are also likely to be context-dependent, so seeming

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This article is in need of some serious work. The section on the converse is factually inaccurate and should be removed. The history section needs much weeding and removal of material that is not supported by the references given and where the references are of dubious value. The section on

1316:

After I wrote the above I did some searching myself. I found two fairly recently published research papers that contained the result. Unfortunately they were both published in education journals and I am worried about the quality of the refereeing (I know how snobbish that sounds, but some

934:

I propose that the Criticism section be removed. It is now universally acknowledged that the parallel postulate is independent of the other postulates, and thus can be added or negated as a consistent additional axiom. The argument from direct perception is blatantly false in a world where

412:. The latter says that Playfair's axiom is equivalent to Euclid's parallel postulate, and this article says that's not true. I did a lot of googling and reading to clarify this, but I'm not sure whether I've missed something and I'd like to discuss this here first before making changes. 856:. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. Please see: 1046:

I refrain from making any alteration to the first sentence of the main article, even though I consider it flawed, because I am not a mathematician by training. I strongly believe that only those with the appropriate level of academic expertise should make such alterations.

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Afterthought: it is plausible that the statement is equivalent to the parallel axiom. For 1. in spherical geometry by two antipodal points there are infinitely many lines. 2. In the projective plane the postulate is ambiguous because the complement of a line is connected.

1364:

The parallel postulate is given here, loosely, as "if not right angles, then not parallel" The converse is "if not parallel, then not right angles." The statement offered in the section titles converse is not that one, and it's killing me. Somebody please fix this.

1461:

The article claims that this is equivalent to the parallel postulate. "5.There exists a pair of similar, but not congruent, triangles." This axiom is easily proven by extending two sides of a given triangle away from the point at which they meet, and then using SAS

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much as possible. So he asserted as much as he dared about parallels in Definition 23, and in Postulate 5, said only as much as would enable the reader to INFER a Postulate of Parallels. This was philosophically dishonest, but what was the guy supposed to do???

1036:(1) If it is thus equivalent, then Definition 23 is redundant. Euclid would have realised that (he was quite a bright chap); that he chose to leave Definition 23 in place is, prima facie, evidence that he himself did NOT regard P5 and PP as logically equivalent. 555:) -- but it's not clear what this equivalence is relative to. It is certain that the equivalence holds given some of the modern axiom sets used in studying absolute geometry, but is unclear whether it holds given the first four of Euclid's original postulates. 390:

kind. I never read the elements but I will be quite surprised this to be true. Can you please be precise on that point. I also checked on internet most of the things which are written are imprecise or simply wrong, as you mentionned below. MM November 2008

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Incidentally, in the context of classical Greek mathematics, there is a world of difference between a "postulate" and an "axiom". Euclid's postulates are not axioms (as he understood the word). In fact there are, strictly speaking, no "axioms" in Euclid.

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I think this article would be more precise if it would distinguish the equivalences which are true in neutral geometry with/or without the Archimedian axiom. For example, in Robin Harthshorne book; Geometry: Euclid and beyond, the distinction is made.

592:

Given Euclid's first four postulates as a base, Playfair's axiom is equivalent to the conjunction of Euclid's parallel postulate and its converse. I think those that claim that Playfair's axiom is equivalent to Euclid's parallel postulate are either:

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company. You can look up the theorem that a single triangle with angle sum equal to 180° implies that all triangles have angle sum equal to 180° in Faber (Theorem IV-5, pg. 132) - I have just now added this reference in the further reading section.

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I am not denying, obviously, that P5 irresistibly suggests PP. But that's just a matter of overwhelming psychological plausibility which, as we both know, dear reader, is the truth-seeker's worst enemy. Unless, of course, it's an axiom.

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This article is substantially duplicated by a piece in an external publication. Since the external publication copied Knowledge rather than the reverse, please do not flag this article as a copyright violation of the following source:

1597:

that points to page 128 of this book that claims an equivalence of the 5th postulate to Hilbert's parallel postulate: Greenburg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. W.H. Freeman and Company.

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refers to: Wanda Szmielew. Elementary hyperbolic geometry. In P. Suppes L. Henkin and A. Tarski, editors, The axiomatic Method, with special reference to Geometry and Physics, pages 30–52, Amsterdam, 1959. North-Holland.

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geometry those two points are antipodal). By the way, when you identify a point with its antipodal, the quotient manifold that you get is not orientable and so the very formulation of the parallel axiom makes no sense.

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modern mathematicians who attempted to prove this postulate, and an acknowledgment of Euclid's genius in recognizing that it could not be proved. I think a short discussion of this history would improve the article.

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article and I have found several inaccuracies at that site in general. I don't believe that this equivalence is valid, but at the moment I am not willing to spend the time to locate the errors in the supposed proof.

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triangle exists whose angle sum is 180 degrees, and that this actually implies that all triangles have angle sum 180 is somewhat surprising to me. Any ideas for a proof of this previous step mentioned? Thanks!

890:

15. Proclus' axiom, which states "if a line intersects one of two parallel lines, both of which are coplanar with the original line, then it must intersect the other also", is also equivalent to the parallel

651:

Two remarks: 1. Playfair's axiom is a proposition already considered by Proclus 2. the discussion about elliptic geometry is based on a misconception -it confuses spherical geometry with elliptic geometry.

764:

I'm probably wrong though since probably a million man-hours have been spent on this problem, and I am not sure if my argument is a fully general one. But the fact that I seem to only need to show that a

1087:{{Harvnb|Katz|1998|p=269}}: {{quote|In effect, this method characterized parallel lines as lines always equidistant from one another and also introduced the concept of motion into geometry.}}</ref: --> 503:, which assumes only the first four postulates, mentions only hyperbolic geometry, not elliptic geometry, as an example, and states that Euclids first 28 propositions are valid in absolute geometry. 1478:

have the same angle measures, then they are congruent, i.e., the corresponding sides have the same lengths. In that geometry the only way that two triangles can be similar is if they are congruent.

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noneuclidean geometry has already been developed, but it's not one that would have occurred to Euclid. Because the Elements aren't written in a formally defined mathematical language like, e.g.,

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It's a bit of a nitpick, but theorems aren't "proved". They're "proven". Saying that a theorem is "proved" rather than "proven" is like saying that a toaster is "broke" rather than "broken".--

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defines absolute geometry); however, due to the recognition that there are some problems in Euclid's approach, various other axiom systems have been developed for absolute geometry -- see

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claims it is equivalent and because an IP editor should at first raise such an issue in the talk page. I will let those more qualified than I consider this editor's suggestion.----

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way to distinguish between Euclidean, Elliptic, and Hyperbolic geometry. It also mentions changing other axioms, which then gets into not Euclidean geometries which are neither

860:. That's an old and archived RfC. The point is still valid though, and his contribs need to be doublechecked. I searched the page history, and found 5 edits by Jagged 85 in 726:) is incorrect. If the intersecting line is not coplanar with the two parallel lines, then it need not, and in fact cannot, intersect both of them. If the intersecting line 853: 1727: 849: 202: 206: 1742: 440:. The general idea is that absolute geometry is the geometry that results from Euclid's first four postulates without assuming the fifth (in fact this is how 35: 1752: 141: 339:

I looked on google & aol search and couldn't find 1 reference for that film, so i'm not to sure it even exists.No references for the director either

1004:

parallelism - a point which (if I may echo Russell), with the exception of Matt, has proved too subtle for the philosophers and mathematicians to grasp.

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I don't know how this page got on my watchlist; I am interested in the parallel postulate but am not an expert. I just reverted revision 766900271 by

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The "history" section of this article, which is substantially duplicated in that publication, has been evolving here since 2005, having been seeded

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Axiom 11 is not even true in three dimensions, whereas 15 is a fair representation of the correct axiom. this section is muddled in that 11 refers

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On consideration, I can't resist adding another couple of arguments to show that Postulate 5 is not equivalent to a 'Postulate of the Parallels'.

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only talk about these geometries violating the parallel postulate; they make no statement about the other postulates, but it seems implicit that

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Greenburg only shows that the parallel postulate implies the pythagorean theorem (a fact known to Euclid!) and does not give an equivalence.

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M. Sivasubramanian Department of Mathematics, Dr. Mahalingam College of Engineering and Technology, Polachi, Tamilnadu-642003, India (2009),

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The problem of considering mechanical movement in geometry which we encountered while discussing the proofs of postulate V in the works of

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The question is whether it follows from the other four postulates that there is at least one parallel. This articles denies this, whereas

1409:"Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry" 709: 635: 561: 526: 1555: 1372: 779: 426:, the Exterior Angle Theorem, plays a central role, since it is used in proving that there is at least one parallel. Cut the Knot has 1346: 1287: 1187: 1093: 693: 653: 108: 69: 611:

That said, Euclid also uses unfounded assumptions in propositions 1 and 4; I'm not sure OTTOMH how much of it depends on these. --

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any statement true in euclidean geometry but false in hyperbolic geometry can serve as replacement of the parallel postulate see

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To summarize, there's a strong consensus on the Net that Playfair's axiom is equivalent to Euclid's parallel postulate (see e.g.

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I tagged the statement by Archimedes due to concerns about its accuracy. The full list of his treatises is given in the article

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coplanar with the parallel lines, then this example is a rephrasing of Proclus' axiom, which is discussed separately below.

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I strongly agree with you: see my comments below. So, mentionning spherical geometry here is a misconception. Alain Gen

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Bodie's proof is wrong, that does not mean the result is false, but it would strengthen my feelings on the matter.--

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Cut the Knot is the only source I found that explicitly says that elliptic geometry is an absolute geometry -- see

507:"indefinitely" is meant here in a way that excludes great circles on a sphere that are finite but without ends. ( 1603: 1567: 1467: 900: 821: 1530: 458: 445: 294:

I'm removing the image, because it displays the Corresponding Angles Postulate, not the Parallel Postulate.--

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to two dimensions whereas 15 would be true in n-dimensional euclidean space. This needs to be made clearer.

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The section regarding the converse of the given parallel postulate does not make mention of the converse.

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Perhaps I was a little too vague in the previous section, so let me be more precise. It is stated – with

it is the usual meaning of "equivalent" (implicitely, equivalent when the other axioms are assumed).

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Clarification of the status of Proposition 16 is complicated by the ambiguity in the use of the term

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The following sentence is not true. It's true for hyperbolic geometry, but not elliptic geometry.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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not a great source for history but ok for the point) Could he be talking about Euclid him self?

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I don't believe that Katz, Victor J. (History of Mathematics: An Introduction, Addison-Wesley,

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11. Given two parallel lines, any line that intersects one of them also intersects the other.

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What I'm trying to say is it doesn't belong here and it is probably not in context anyway.

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film/director which so far as I can tell, is basically unknown is 'popular' or notable. --

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Technically a toaster is broke if it is an insolvent person. Now that is another nitpick!

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Given two parallel lines, any line that intersects one of them also intersects the other

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of Proposition 16 and the implicit assumption that Euclid made in proving it. (See also

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during whose lifetime the Euclidean axioms and postulates were not yet formulated. The

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using as a base Euclid's first four postulates plus his unstated assumption (see the

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Last edited at 19:43, 2 October 2008 (UTC). Substituted at 02:26, 5 May 2016 (UTC)

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This is completely unrelated to the parallel postulate. Both of these are true in

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I reverted an IP edit that "felt" wrong to me. An expert should verify.

1171:(in geometry) predates Alhazen so I don't understand what Katz means by 606:

misreading the "if" in Euclid's parallel postulate as "if and only if".

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incorporating content copyrighted by various Knowledge contributors. --

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takes the view that Proposition 16 is based on the second postulate.)

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where he introduced the concept of ] and ] into geometry.<ref: -->

761:(assuming only the first four postulates and Propositions I.1-I.28). 883:

In section "Logically equivalent properties", these two statements:

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Bill its mentioned here as a consequence of the parallel postulate

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should literally mean the same. I suggest one should be removed. --

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Confuses spherical geometry with elliptic geometry in what way? --

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confusing Euclid's parallel postulate with some other formulation

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postulate I postulate II postulate III postulate IV postulate V

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Application of Sivasubramanian Kalimuthu Hypothesis to Triangles

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One of the examples under "Logically equivalent properties" (

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In fact the opposite, the WP stuff has been copied into the

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Parallel_postulate#Converse_of_Euclid.27s_parallel_postulate

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There's a contradiction between this article and the one on

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The History of the Geometry Curriculum in the United States

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https://en.wikipedia.org/Saccheri%E2%80%93Legendre_theorem

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Laubenbacher, Reinhard; Pengelley, David (1998-12-04).

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takes the view that Proposition 16 is not a theorem of

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Encyclopedia of the History of Arabic Science - Vol. 2

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Mathematical Expeditions: Chronicles by the Explorers

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The 5th postulate is 954:Factual accuracy of the "History" section 858:Knowledge:Requests for comment/Jagged 85 1728:Knowledge vital articles in Mathematics 1593:Snooping around the internet I found a 1393:Converse of Euclid's parallel postulate 1387:Converse of Euclid's parallel postulate 854:misused sources here over several years 60: 19: 1552:2A01:E35:2E78:9EA0:1493:7C48:B7F1:D585 958:This appears to have been lifted from 194: 1743:C-Class vital articles in Mathematics 1343:2A01:E35:2E78:9EA0:C7F:E067:78A4:F0AE 7: 1391:What is the purpose of the section " 1243:https://en.wikipedia.org/Dehn_planes 575:Comments would be much appreciated. 496:the parallel postulate is violated. 106:This article is within the scope of 385:More questions on the 5th postulate 230:. See signs of natural development 49:It is of interest to the following 1753:High-priority mathematics articles 1386: 14: 1506:several discussions in past years 1395:"? 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Dvinin (evddev) 1381:03:46, 10 March 2012 (UTC) 1110:GEOMETRIC TRANSFORMATIONS 850:198 in the number of edits 826:23:50, 16 March 2010 (UTC) 805:22:31, 28 April 2012 (UTC) 788:01:42, 28 April 2012 (UTC) 480:The Knowledge articles on 457:("Elliptic Geometry") and 1649:answer by Julien Narboux 1513: 1451:03:47, 16 July 2014 (UTC) 1435:01:12, 16 July 2014 (UTC) 874:19:49, 15 June 2010 (UTC) 740:22:11, 1 March 2010 (UTC) 299:01:01, 24 June 2006 (UTC) 145: 78: 57: 1617:theorem did not hold in 355:05:34, 8 June 2007 (UTC) 344:17:39, 5 June 2007 (UTC) 330:20:03, 17 May 2012 (UTC) 269:20:21 11 Jun 2003 (UTC) 152:project's priority scale 1251:Pythagorean Theorem???? 879:Literally mean the same 424:Euclid's Proposition 16 422:As the article states, 419:explicitly affirms it. 109:WikiProject Mathematics 1718:C-Class vital articles 1127: 482:non-Euclidean geometry 288:non-Euclidean geometry 1108: 999:Another Pedantic Note 368:, and none is called 36:level-5 vital article 1595:student's term paper 1514:Please see the site 1457:Already proven axiom 1357:Misuse of "converse" 132:mathematics articles 1560:matworld.wofram.com 490:hyperbolic geometry 452:elliptical geometry 217:Additional comments 1494:Assessment comment 1183:. 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