Talk:Playfair's axiom

Circular reasoning
The part argument about Playfair's axiom only needing "at most" is rather sketchy and not rigorous at all. The place where it falls dreadfully short is the claim that the other axioms can prove the existence of a parallel line through a point. It is not at all clear why this would be the case. The argument claims to use the Triangle Angle sums theorem; however, the proof of this theorem already assumes the ability to create a parallel line. Euclid's 5th postulate also was not written as a biconditional statement. So, its claim is about if the angles are less than 2 right angles on one side, then they meet on that side does NOT imply that if they do NOT equal 2 right angles then they do NOT meet on that side. The case where the angles equal 2 right angles is outside of the scope of that statement logically, as it is written. That is probably why "exactly" is commonly used.

As it is, it seems like the argument is using the assumption of parallel lines (in the form of the triangle angle sums) to argue the existence of parallel lines--a circular argument. A more clear explanation is needed to convince as to why "at most" is sufficient. — Preceding unsigned comment added by 168.94.245.50 (talk) 02:44, 23 December 2022 (UTC)

The situation is complicated. See nonEuclidean geometry, elliptic geometry and absolute geometry. Elliptic, and hyperbolic geometry, both discovered in the 19th century, violate the Playfair's postulate in different ways, replacing "exactly one" by "no" and "more than one". In the simplest example of elliptic geometry, the geometry of great circles on the surface of the sphere, parallel lines don't exist because any two "lines" (i.e. great circles) intersect at two antipodal points. But this spherical geometry violates other Euclidean axioms, e.g. there is not a unique line joining any two points, and circles of arbitrarily large size cannot be drawn.

Almost nonsense
Not only are sections of this article poorly written, but there is also a knotty problem with the section on the relationship to Euclid's 5th postulate. The article repeats the near-nonsense that can be found in Henderson's text - and that's the rub, this is sourced material. The argument goes – they are not equivalent since one holds in spherical geometry and the other doesn't. Leaving aside Henderson's deliberate confounding of spherical and elliptic geometry (there is a viewpoint from which this makes sense, but that is a very abstract advanced topic which does not belong at this level of treatment), the only thing that this argument shows is that the two concepts are not tautologically related. This is a straw-man argument, no one to my knowledge has ever claimed that the statements were logically equivalent. The equivalence must be established within the context of the geometry in which the statements make sense. Without that context (that is, the axioms which define the geometry) one can not provide a proof that either statement implies the other. What can be said in this situation is that the two statements are equivalent in the context of Euclidean geometry, while they are not equivalent when interpreted in the spherical model. Henderson's fixation on the spherical model distorts the way he emphasizes his statements and it is this distortion which is being repeated in this article. Conclusions about which statement is "stronger" just don't make any sense. It is not clear to me how I should fix this without just wiping it all away. Bill Cherowitzo (talk) 18:03, 9 October 2013 (UTC)

It appears that Henderson has come out with a new edition and has picked up a co-author. Circumstantial evidence indicates that the language has been toned down and is now in line with what I have written above (but I haven't seen a copy of this edition). I think I can now rewrite the offending section appropriately. Bill Cherowitzo (talk) 22:44, 9 October 2013 (UTC)

Secondary source
It is stated policy WP:Secondary source that the ideal references are secondary sources such as the survey by Morris Kline that has just been re-instated. There are not many articles that have extended comments on Playfair’s axiom but Kline’s survey of geometry in 1964 is one. The article first describes the consequences of Playfair’s axiom in the plane before venturing into conditions denied by Playfair: multiple parallels or no parallels. The article has recently become generally available and gives a reader a sense of the utility of this particular phrasing of a geometric condition. The reference is: Discussion can proceed here.Rgdboer (talk) 02:15, 11 January 2015 (UTC)
 * Morris Kline (1964) Geometry, Scientific American 211(3):60–9, Playfair's axiom p. 66
 * This is one of your more ridiculous inclusions. You have placed a totally non-notable statement in the history section for no reason that I can discern. Kline does not say anything more about Playfair's axiom than anyone else who has written about the subject, in a nutshell, it is equivalent to Euclid's parallel postulate and is easier to understand. In what sense do you see this as an extended comment? What is the historical import of this survey article? And, to be perfectly clear, a secondary source uses and details primary sources, Kline is not doing this in this survey, this is not a secondary source! Bill Cherowitzo (talk) 04:37, 11 January 2015 (UTC)

Should be more explained Kavinaya vk (talk) 13:23, 11 June 2018 (UTC)

uniformising citations to Dr Playfair's 'Elements of Geometry'
hi guys,

i noticed that different versions of Dr Playfair's 'Elements of Geometry' are being used as citations. given that google books now requires flash & signing, i was hoping we could use the 1846 version that's freely available on the archives. no wikipedia source should require users to log in to google in order to inspect the supplied citation.

the edition i'm proposing we use for uniformising citations is given here: https://archive.org/details/elementsgeometr05playgoog

this page could really benefit from consolidation of sources and clean up. some cites seem a bit excessive but i think we need to have a healthy discussion as to how to tackle this.

best. g 174.3.155.181 (talk) 23:49, 26 March 2016 (UTC)

--- The page number in the current citation does not appear to correlate with the quote given. — Preceding unsigned comment added by Trachten (talk • contribs) 21:27, 22 May 2016 (UTC)

Can it be explained more precisely?
Needing some more examples Kavinaya vk (talk) 13:23, 11 June 2018 (UTC)