Talk:Euclidean geometry

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Former good articleEuclidean geometry was one of the Mathematics good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
Article milestones
July 19, 2006Good article nomineeListed
October 6, 2007Good article reassessmentDelisted
February 24, 2009Good article nomineeNot listed
Current status: Delisted good article

There are no axioms or postulates in Euclid's Elements[edit]

This article is factually wrong. There are no axioms or postulates in Euclid's Elements.

There are no axioms or postulates in Euclidean geometry. Every "requirement" is well defined from the first, that is, a straight line can be drawn between any two points.

Read my article:

Play with the dynamic applets: — Preceding unsigned comment added by (talk) 15:53, 14 April 2017 (UTC)[]

New Video: — Preceding unsigned comment added by (talk) 00:58, 2 March 2018 (UTC)[]

GA Review[edit]

This review is transcluded from Talk:Euclidean geometry/GA1. The edit link for this section can be used to add comments to the review.

Hello. I am going to have to fail this article's GA nomination, due mainly to referencing issues, but also to a few other things. Here is a list of the major issues that need to be addressed:

  • The main problem is that the article is severely under-referenced. There are many sections that completely lack references, and most sections have at least one paragraph that is unreferenced.
Added a bunch of references.-- (talk) 01:39, 26 February 2009 (UTC)[]
  • Web references need to be formatted with a publisher and access date at the very least.
Done.-- (talk) 01:39, 26 February 2009 (UTC)[]
  • The references section has two books listed in the bullet point part that aren't present in the in-line references. Books that aren't used for in-line references should be removed altogether or moved to a "Further reading" section.
Fixed.-- (talk) 22:14, 25 February 2009 (UTC)[]
  • There are a lot of really short paragraphs and really short sections. These need to be either expanded or combined. As they are, they make the article look very choppy and harder to read.
  • The Euclid proof image in the Axiomatic treatment section needs some tweaks made to its licensing.
Can you explain what the necessary tweaks are? It's not obvious to me what the problem is.-- (talk) 22:14, 25 February 2009 (UTC)[]
If you go to the image page, the license says "Note: This tag should not be used anymore!", and then gives a list of licensing options to choose from to replace the deprecated tag. Dana boomer (talk) 00:13, 26 February 2009 (UTC)[]
I see, thanks. Fixed.-- (talk) 01:39, 26 February 2009 (UTC)[]
  • The first two images in the gallery in the Some important or well known results section need descriptions added to their image pages.
Fixed.-- (talk) 22:14, 25 February 2009 (UTC)[]
  • The formatting of the image gallery in the Applications section needs to be tweaked - the last image falls off the page.
Changed formatting. This will of course depend on your browser and screen.-- (talk) 22:14, 25 February 2009 (UTC)[]

Once these issues are taken care of and a full copyedit is completed, please feel free to renominate this article at GAN. Please let me know if you have any questions or comments. Dana boomer (talk) 18:43, 24 February 2009 (UTC)[]

Thanks for the comments! Some of these I'll fix, others I'll leave to other people.-- (talk) 22:06, 25 February 2009 (UTC)[]

Thanks, Dana boomer, for the helpful comments on the article. I think most of them are now pretty much taken care of, with the possible exception of references (not sure if they're sufficient now), and the definite exception of the choppiness, short paragraphs and short sections. I assume the latter was mostly referring to the section on applications, and I think that's a symptom of the fact that the application section is thin, and lacks any kind of narrative thread, transitions between topics, etc. That's not something that can be taken care of with a quick fix.-- (talk) 07:21, 27 February 2009 (UTC)[]

Using Heath's translation of axioms[edit]

I've replaced the statement of the axioms with Heath's translation. Although rewording them may help in making them a little easier to read and a little more consistent with the typical terminology used in high school geometry courses (e.g., "line segment" rather than "finite line"), I think we run into all kinds of problems with homebrewed formulations of them. E.g., there was some discussion a while back about whether Playfair's axiom was equivalent to Euclid's fifth postulate, and this hinged on the exact interpretation of what the postulates said -- but the discussion was being based on the formulation in the WP article, which differed in subtle ways from Euclid's.-- (talk) 19:19, 27 February 2009 (UTC)[]

Yes, there is a problem with the common assumption that Postulate 5 is in some sense "equivalent" to the Parallel Postulate, and regardless of the form of words used. The difficulty is that a negation of the Parallel Postulate does not entail a negation of Postulate 5. To amplify a little: if one were to assert, as an axiom, "there is NO parallel line in Euclidean Geometry" (ie all lines either converge or diverge, and there is no line separating the two classes, such that it neither converges nor diverges), Postulate 5 would continue to be true. And since it remains equally true, whether there is or is not such a thing as a parallel line, the two postulates are therefore logically independent and can have no necessary connection with each other. Alan1000 (talk) 03:39, 21 March 2016 (UTC)[]

Euclid and physical space[edit]

In the section "The 20th century and general relativity," there is an image that has a caption which contains the words: "A disproof of Euclidean geometry as a description of physical space." Is there any passage in Euclid's books where he claimed that his plane geometry is a description of physical space? I was under the impression that Euclid's geometry purported to be a pure, a priori exercise in the mathematics of forms. How can a book that starts by saying "A point is that which has no parts" be considered to be a book about physical space? Lestrade (talk) 18:40, 28 April 2009 (UTC)Lestrade[]

I think you're imposing modern mathematical attitudes on the ancients. It's a very modern idea that mathematics doesn't have to describe anything about the real world.
What!!!! You're joking, right? Ever heard of a guy called Plato?
Alan1000 (talk) 15:38, 21 March 2016 (UTC)[]

The ancients didn't conceive of the postulates as arbitrarily chosen statements for building an arbitrary logical system, they conceived of them as statements that were obviously true, so that they could be used to prove other things that were equally true in an absolute sense, but not as obvious.-- (talk) 18:51, 29 May 2009 (UTC)[]

You are absolutely right; the conception that postulates can be arbitrarily chosen to build an arbitrary logical system is an entirely modern conception.
Alan1000 (talk) 15:38, 21 March 2016 (UTC)[]
Euclid would have been horrified by the suggestion that his geometry was a description of physical space. One must understand him in his philosophical context. His geometry is a blend of Platonism and Atomism, neither of which had any connection with the sensual perception of space. I would only quibble with 'the mathematics of forms'; I would prefer, 'the Forms of mathematics'!
Alan1000 (talk) 15:38, 21 March 2016 (UTC)[]

The following statement: "A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar eclipse. The rays of starlight were bent by the Sun's gravity on their way to the earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry." is not true.

I have read - although I cannot quote a reference - that the 1919 experimental results were later found to be within the predictable margin of error for the experiment, and thus proved nothing at all. Can anyone verify this?
Alan1000 (talk) 15:38, 21 March 2016 (UTC)[]

Mathematically the Euclidean geometry is true, as well as the other 2 plane geometries are true. Mathematics itself is not about the physical world. One cannot prove or disprove mathematical theorems with experiments. What is more, here we have an axiom, and to prove an axiom is a nonsense. For this specific photography a text for a disproof of the Newton's theory of classical physics is more appropriate. But then we have to talk about physics, not about mathematics

How about a line having an infinite number of points as constituents ? this lead Zenon to postulate that movement is impossible , because to traverse an infinity of points towards your destination would take an infinity of time .Only this single paradox is sufficient to dismiss Euclidean mathematics from the true world.It does make it easier for children up to 10 years old to learn mathematics , but once they can throw a curve ball , i guess the veil is broken. —Preceding unsigned comment added by Pef333 (talkcontribs) 04:19, 7 January 2010 (UTC)[]

OK, so you're saying that when your wife alters the hem on a dress, she uses the Theory of General Relativity? No, I thought not. The mathematical fact is that Euclidean Geometry is both true and valid where the gravitational field is zero, or where the effects of gravity can safely be ignored. Alan1000 (talk) 14:37, 21 March 2016 (UTC)[]

Also in the second paragraph of the article it is stated that there are many geometries. In fact, apart from the Euclidean geometry, there are only 2 other plane geometries, which is not many.

Stefan Stoyanov, May 11, 2009 —Preceding unsigned comment added by (talk) 21:43, 10 May 2009 (UTC)[]

To prove or disprove an axiom as a model of something physical is not nonsense. Re "In fact, apart from the Euclidean geometry, there are only 2 other plane geometries, which is not many," you're incorrect. For example, see: Euclidean geometry, hyperbolic geometry, elliptic geometry, Dehn plane, Taxicab geometry, Discrete geometry, affine geometry, projective geometry.-- (talk) 18:51, 29 May 2009 (UTC)[]
This is a mere quibble over terminology. There are indeed many different geometries, but they can all be classified under the general headings of "Open" or "Closed". Euclidean geometry is merely a special case of a geometry in the unique space between open and closed geometries (which means that, statistically speaking, it will be vanishingly rare in the Universe; but nonetheless valid for all that).

A final thought: suppose you leap from the fiftieth storey of a building. By the time you reach terminal velocity, Euclidean geometry will be true. Follow that to the bus stop, baby!

Euclidean geometry was not the only geometry "for 2000 years"[edit]

The statement "For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived." is obviously incorrect, since mariners have been using spherical trigonometry, which is a non-Euclidean geometry, for a considerable length of time. The thing is that I'm not entirely sure when navigation by means of spherical trigonometry was introduced as part of the standard tool-set of naval and merchant navy officers. Certainly by the middle of the 18th century, but when precisely? —Preceding unsigned comment added by Recoloniser (talkcontribs) 18:06, 16 December 2009 (UTC)[]

That's an interesting point. However: (1) 2000 years after Euclid would be ca. 1700, which is earlier than you're talking about; (2) it's not so clear that people using spherical geometry understood it to be a logically independent system, or merely a system defined on the surface of a sphere embedded in a surrounding Euclidean space. For example, they might have conceived of a spherical angle as a subsidiary concept, rather than as a notion of angle that was complete in and of itself. Did anyone formalize spherical geometry as an independent axiomatic system during this period?-- (talk) 02:09, 18 December 2009 (UTC)[]

Section: Logical Basis[edit]

This article needs attention from an expert on the subject.[edit]

I have tried to makes some sense of this section, which consisted mainly of a paragraph on Tarski that originally seemed disconnected from the subject. It appears that the logic behind Euclidean geometry has shifted focus over the years, but still remains the motivator for deep thought about what is math.

It would be great if someone versed in this area could flesh out the issues and provide a clearer notion of background and developing issues. Brews ohare (talk) 20:01, 28 June 2010 (UTC)[]

Hi- I'm the person who wrote the original version "Logical basis" section, which has now been greatly expanded on by other people. You put the expert-subject template both at the top of the article and in that section, but I think it only belongs in that section, so I've removed
- on whose authority? Mount your argument first, and remove later, if no adequate counter-argument is forthcoming. Nobody appointed you Commissar of Correct Thinking.

Alan1000 (talk) 16:23, 21 March 2016 (UTC)[]

the one at the top of the article. Usually the expert-subject template is used in cases where the article appears to contain wrong information, -
who defines "wrong information?"

- and isn't going to be substantially correct unless an expert works on it; is this your opinion about the current version of the "Logical basis" section? I would consider myself at least somewhat of an "expert" on Euclidean geometry. (I have an undergraduate degree in math and physics, PhD in physics.)-- (talk) 00:46, 2 July 2010 (UTC)[]

The appeal to institutional authority is the badge of fascism. Try addressing the arguments instead.
My view of this section is that it could use some mature perspective from someone versed in the logical developments and historical events involved in the progression of geometry from Euclid, through Gauss, through Hilbert, Tarski and through the more recent developments in constructive type theory. I have put together some treatment of these matters, but it is an amateur effort. Brews ohare (talk) 01:18, 2 July 2010 (UTC)[]
Presenting Tarski's formalism for Euclidean geometry is ill-motivated in this context. Focusing on the Hilbert/Brikhoff program,and the development of neutral geometry is more in the spirit of what this section seems to be for. That is to say how Hilbert and others developed neutral geometry and essentially worked until they couldn't do anything with invoking the parallel pos in some form. The text "Roads to Geometry" by Wallace, and West presents this approach. There is also a wealth of finite geometries whose study was prompted by the question of independence of the parallel pos. DifferentiableF (talk) Jul 10, 2010, 3:13 AM —Preceding undated comment added 07:14, 10 July 2010 (UTC).[]
Wiki, say something to convince me that you are not an academic joke. I've gathered that you forbid original thinking, and you forbid quoting from original sources. Thus far your anti-intellectual credentials are well established. Does this explain why Wikipedia is never cited on any reputable scientific site?

Alan1000 (talk) 16:23, 21 March 2016 (UTC)[]


Perhaps the pronunciation of 'Euclidean' should be added to the article - I had to look this up elsewhere (<>). -- (talk) 19:13, 29 September 2010 (UTC)[]

Euclidean geometry is an invention of historians?!? pls review[edit]

Euclidean geometry is an invention by young acient historinas in the 70s. "They teach Davenportian geometry in high schools now, though of course they call it Euclidean", accordingt to Professor Gene Haddlebury (and redisributed by the University teacher mr. Blank (M.A) at the Univerity in Mainz.) Source:,18209/) —Preceding unsigned comment added by (talk) 08:19, 31 January 2011 (UTC)[]

Please stop adding this joke to the article, as produces satire. DVdm (talk) 08:22, 31 January 2011 (UTC)[]

Earlier textbooks existed and were 'systematic'[edit]

According to Heath in [1] vol. 1 p. 117 Aristole mentions as examples in his text numerous geometric propositions and definitions which Heath belives come from a textbook used by the Platonic Academy written by Theudius of Magnesia who studied closely at the Academy along Eudoxus.

  • [1] The Thirteen Books of Euclid's Elements, translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3)

Mm32pc (talk) 15:19, 26 July 2011 (UTC)[]

But the article doesn't claim that no earlier, systematic textbook existed...?-- (talk) 01:48, 5 January 2012 (UTC)[]

"Lost Girl" writers use Wikipedia for script?[edit]

HA! I noticed something while watching Lost Girl... at 30min, 45seconds in on Season 2, Episode 16, Kenzie WORD FOR WORD repeats part of this article! Not while looking at Wikipedia or using any computer either, she repeats this part:"Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense" part. Just popping in to pass on that discovery. TinyEdit (talk) 01:15, 9 February 2012 (UTC)[]

Curious, but not really something we can include. It doesn't tell us anything about the theorem (or at least no more than however many textbooks on it that we also don't mention). It's just someone using WP, as people are quite legally allowed to do, and which happens more and more. I from time to time use images from WP in my blog. Other people mirror WP wholesale, or copy and paste it into books they then sell as their own work.--JohnBlackburnewordsdeeds 01:35, 9 February 2012 (UTC)[]
No, I'm just mentioning it here, like trivia, since there's no place for Wikipedia trivia, is there? I've never seen this happen before so I thought it was kinda interesting. TinyEdit (talk) 01:43, 9 February 2012 (UTC)[]

First 28 propositions[edit]

The article claims that "the first 28 propositions [Euclid] presents are those that can be proved without [the 5th postulate]." I take this to mean that they can be proved from the first four postulates. However as pointed out at this page, "The first 15 propositions in Book I hold in elliptic geometry, but not [Proposition 16]." (Consider a triangle whose interior angles are each greater than a right angle, in which case its exterior angles are each less than a right angle.) Unless Postulates 1-4 somehow rule out elliptical geometry I don't see how Proposition 16 could follow from them. If 1-4 do achieve this I'd be fascinated to see a reputable source with a proof to that effect.

Absent such a source it seems to me that this claim about the first 28 propositions is unsupportable. --Vaughan Pratt (talk) 04:27, 17 August 2013 (UTC)[]

Absolutely. Such a claim appears to me to be fantastical. To begin with, the first 23(?) propositions are offered as definitions (or stipulations), and are therefore in no need of proof (at least within the context of Euclidean geometry).

Incidentally there are restatements of Postulates 1 and 2 that rule out elliptical geometry. The requirement of uniqueness for Postulate 1 would achieve this, as would the requirement for Postulate 2 that no extension ever self-intersect, but that's not how they're worded. --Vaughan Pratt (talk) 04:38, 17 August 2013 (UTC)[]

You do not need to restate the postulates. Postulate 2 as originally stated rules out elliptic geometry since it can be used to prove that there exist parallel lines -
This would astonish me greatly. Please develop your arguments?

- (you only need postulate 5 to provide uniqueness in the Playfair formulation) -

i'm afraid you will need to explain what this means. You seem to be coming dangerously close to asserting that on the basis of some axioms, it is possible to assert additional facts about other axioms. This is a logical minefield. Since the definition of "axiom" precludes logical relationships with other axioms, you will need to develop your arguments.

- and elliptic geometry has no parallel lines. Postulate 3 also needs to be modified before it can be used in the elliptic geometry setting. Part of the problem here is that Euclid's postulates are not sufficient to give Euclidean geometry (this has been known for over a century) and one must work with a complete set of axioms rather than attempting to repair Euclid's. Neutral geometry (in which the first 28 propositions of Euclid are valid) is obtained by removing the parallel postulate from a complete set of axioms for Euclidean geometry, such as Hilbert's axioms. Bill Cherowitzo (talk) 18:20, 17 August 2013 (UTC)[]

You seem to contradict yourself. On the one hand you say the postulates don't need to be restated. On the other you say that you need to start from neutral geometry. How is that not a restatement?
Since elliptical geometry is a model of the five postulates as originally stated, it is not possible to prove the existence part of Playfair's formulation from them. The original statement of Postulate 2 merely promises the ability to produce a finite straight line continuously in a straight line, which clearly holds on the sphere unless Postulate 2 is strengthened in some way. Furthermore nothing in Euclid's proof of Proposition 16 uses anything to do with such a strengthening, so it's clear Euclid simply made a mistake in his proof, thereby invalidating all subsequent propositions that depend on Proposition 16. --Vaughan Pratt (talk) 16:29, 24 August 2013 (UTC)[]
As I've said, elliptical geometry is NOT a model for Euclid's five postulates. This has been proved over and over again, dating back to Omar Khayyám, Girolamo Saccheri and J. H. Lambert. The main problem is Postulate 2, which as stated is ambiguous in meaning. The two interpretations are 1) straight lines are infinite in extent (length) or 2) straight lines are boundary-less. Of these, only interpretation 2 works on a sphere, -
Please expound; meaning not entirely clear.

- while Euclid implicitly uses interpretation 1 in his proof of Prop. I, 16. (See, Faber, Foundations of Euclidean and Non-Euclidean Geometry, 1983, Marcel Dekker, p.113) It was Riemann in 1854 who clarified these distinctions and showed that they are not equivalent. Another problem that arises is that you can not make Postulate 3 work on a sphere. Bill Cherowitzo (talk) 18:00, 24 August 2013 (UTC)[]

In logic there is no model-theoretic difference between "more ambiguous" and "weaker" since both mean "admitting more interpretations". You appear to be agreeing with me that Postulate 2 (as worded in the article) must be strengthened before it can be used to prove Proposition 16. The article itself says as much when it points out in Section 7 that postulates 1-4 are "consistent with either infinite or finite space (as in elliptic geometry)". The article is therefore incorrect when it claims without qualification that this and later propositions can be proved from the postulates listed in the article. --Vaughan Pratt (talk) 19:02, 25 August 2013 (UTC)[]
I thought that I was fairly clear, but it does look like I have to spell it out. Postulate 2 had only one meaning for Euclid and for two thousand years this was the way it was interpreted. Interpretation 1 is a fairly modern point of view due to Riemann. You are not allowed to pick and choose which interpretation you wish to use to make your arguments. If you go with interpretation 1 you do not get Euclidean geometry nor Elliptic geometry (without making other changes). With interpretation 2 you get Euclidean geometry and not Elliptic geometry. You are being selective in pointing out what section 7 says by leaving out the condition that says in effect, "When interpreted as a basis for physical space, ...". This statement shows that what follows is not mathematics, which does not involve itself with such interpretations of axioms. This was probably written by a physicist and is only consistent with interpretation 1. The parenthetical remark is clearly false. While this discussion has been fun, it is now beyond the realm of discussing how to improve this article, and so is no longer appropriate for this talk page. If you have a reliable secondary source for your point of view we can discuss its merits, otherwise I bid you farewell. Bill Cherowitzo (talk) 01:47, 26 August 2013 (UTC)[]
You're perfectly clear, but that wasn't my complaint, which is that the article is not at all clear on this point of interpreting Postulate 2.
It is not "my point of view" that Postulate 2 as stated admits multiple interpretations not all of which suffice for Proposition 16, it's a matter of established historical record, as you've pointed out yourself. Furthermore the article says nothing at all about this, an obvious shortcoming of the article which could therefore benefit from the appropriate improvement.
I would however be very interested to see a reliable source for "Postulate 2 had only one meaning for Euclid and for two thousand years this was the way it was interpreted." I'm not sure what you're saying here. Are you claiming that Postulate 2 admits only two interpretations (I can think of others, one of which I've already mentioned above), and that Euclid picked the one that makes his proof sound? Euclid's proof of Proposition 16 is clearly unsound as it stands, since nowhere does it use any details of Postulate 2 that might make it sound, no matter how interpreted. (Whereas one does not need to "pick and choose" an interpretation in order to make this proof-theoretic argument, one does however need to pick one in order to prove Proposition 16.). It follows that Euclid simply overlooked this problem altogether. The prima facie evidence for this is that if he'd noticed the problem his proof would look different. --Vaughan Pratt (talk) 00:02, 27 August 2013 (UTC)[]

Misunderstanding or nonsense?[edit]

"Einstein's theory of general relativity shows that the true geometry of spacetime is not Euclidean geometry"

(A) How (exactly) can a theory (i.e. something theoretically) prove or disprove something which is part of physical reality? (B) If there exists a true geometry, could there be some other?

What did the author want to tell me? Maybe that some (i.e. one of may, not the) noneuclidian geometry (which is not(!) a Einsteinian but, if so, a Minkowski-Geometry) is much more suitable for the scientific description of space and time while presenting a uniform approach. Maybe. — Preceding unsigned comment added by (talk) 17:45, 1 January 2015 (UTC)[]

Broader concequences of euclidean geometry / euclids elements outside geometry[edit]

The concequences of Euclidean geometry / Euclid's Elements were mutch broader than just geometry. For example it had also influence on philosophy via the so called geometrical method used by for instance Hobbes and Spinoza.

I myself wrota a bit about this at hyperbolic geometry#Philosophical consequences but am wondering should it also he something similar somewhere at euclidean geometry or at the eulid's Elements page.

But then where, and how much and i am not really a specialist in this. WillemienH (talk) 05:48, 19 April 2015 (UTC)[]

Euclid's "line" = curve or line?[edit]

Please weigh in here: Talk:Line (geometry)#Line = fr:droite (not fr:ligne). Thanks. Fgnievinski (talk) 04:34, 30 June 2015 (UTC)[]

Pasch's axiom[edit]

There should be a distinct article for Plane geometry because of Pasch's axiom among others. Thank you. (talk) 20:35, 25 January 2016 (UTC)[]

Your request is not clear. There are many two dimensional geometries, the Euclidean plane being one of them, but what do you mean by "plane geometry"? Bill Cherowitzo (talk) 23:34, 25 January 2016 (UTC)[]
Most mathematicians understand what "plane geometry" refers to. If no one is willing to write the article, the redirect should not exist. (talk) 23:45, 25 January 2016 (UTC)[]
Essentially it is just geometry over R2. Plane (mathematics) (talk) 00:43, 26 January 2016 (UTC)[]
That is Euclidean geometry studied by means of coordinates over the reals. Perhaps you are thinking that Euclidean geometry is the geometry given by Euclid's five postulates. This is not the case (but you have to dig fairly deeply into this article to discover that), those axioms are not sufficient to prove all the theorems in Euclid's Elements. A complete set of axioms needed to do this, such as Hilbert's axioms has over twenty items. There are other sets of axioms that also give Euclidean geometry, each with particular strengths and weaknesses (see Foundations of geometry). Birkhoff's system is the simplest and gets the reals involved immediately. However, this system is a bit deceptive since you are assuming all the properties of the reals and this makes things easy to prove because that is a very powerful assumption. Pasch's axiom, for instance, is easy to prove given the properties of the real numbers, but if you don't have that then it must be taken as an axiom, as Hilbert does. The redirect seems appropriate to me, since most people who use the term "plane geometry" are thinking about the Euclidean plane, R2. What you may be thinking about is studying this geometric object analytically (via coordinates) versus studying it synthetically (via axioms). Note that there is also a disambiguation page Plane geometry (disambiguation) that will lead you to pages with other meanings of "plane geometry". Bill Cherowitzo (talk) 04:11, 26 January 2016 (UTC)[]
I am thinking to link plane geometry to plane geometry (disambiguation) (or rename the latter) and make of plane geometry more a kind of referal/general information article on 2 dimensional geometry (mostly linking) to other articles. WillemienH (talk) 09:48, 26 January 2016 (UTC)[]
The disambiguation page needs some serious work. (talk) 16:29, 26 January 2016 (UTC)[]

Why are the Axioms not mentioned[edit]

Under the Axioms heading, the statements of the axioms are not mentioned. Instead, its given as a to-do list under "Let the following be postulated:". For example '1)To draw a straight line from any point to any point' is not a statement as the line can be straight or curved. Since Euclid, himself did not define straight line concretely a better explanation here would make the subject clear what he meant by straight line.

But this is exactly what Euclid said. As you say, he did not define straight line, he only defined line and then proceeded to talk only about straight lines (until he gets to circles). A discussion of what he meant would have to rely on statements by other Greek geometers and philosophers, which is what Heath does. Recall that Wikipedia is not a textbook, these explanations are better left to the sources that Wikipedia points to.--Bill Cherowitzo (talk) 17:47, 2 July 2018 (UTC)[]

Edit-warring to add non-standard guidance[edit]

An IP added the following guidance for editors:

These axioms are generally agreed to be insufficient for modern levels of rigor (see § Axiomatic formulations)

I reverted the IP on the grounds: "Unnecessary pointing to another section". Then another user reverted me on the grounds that "There is good reason to put this statement here; pointing to a section at the end of a long article that could easily be overlooked". I reverted back giving the reason: "This is a POV reason. It is also not standard in Wikipedia articles to refer to other sections, as if readers are too stupid to find them by themselves." An article is not a book or a paper. This type of guidance is not done on Wikipedia articles. I would have no objection if the disputed edit These axioms are generally agreed to be insufficient for modern levels of rigor. is properly sourced and stands on its own by a suitable reference or references, and if the part pointing to another section is dropped. Dr. K. 04:25, 2 March 2019 (UTC)[]

I am convinced that the article is definitely more informative to the general reader when the removed clause is reinserted, and I think that the suggested place is also fine. As far as I noticed, there is nowhere in the article a remark, accessible at this level, that states this fact, which is obvious to the erudite. The currently removed statement, factually correct, just fails to be a citation, so could be labeled as requiring such. However, the removal degrades the article. I can't help to be surprised that a one-time reestablishing of this IP-inserted clause after removal caused a formal warning for edit warring at the TP of an editor with 10,000 edits. Purgy (talk) 13:57, 2 March 2019 (UTC)[]
Your reply does not address my concerns as expressed in my original post. Specifically, the statement I removed:

These axioms are generally agreed to be insufficient for modern levels of rigor (see § Axiomatic formulations)

was not supported by the section it quoted and was WP:SYNTH. Also I repeat, "This is a POV reason. It is also not standard in Wikipedia articles to refer to other sections, as if readers are too stupid to find them by themselves." An article is not a book or a paper. This type of guidance is not done on Wikipedia articles. and I stand by that remark. It is clumsy and bookish to treat a Wikipedia article like a research paper or treatise. You comment: As far as I noticed, there is nowhere in the article a remark, accessible at this level, that states this fact, which is obvious to the erudite. is misplaced. As I explained in my current post, I reverted an edit which was not consistent with the section it referred to, therefore it was misleading and unsourced and had to be removed. In Wikipedia we go by verifiable and sourced statements not by this type of clumsy referrals to sections that do not apply. So your argument to erudition is misplaced because this is not how we verify facts on Wikipedia. As far as the edit-warring warning, restoring an IP's edit so quickly by a named account may indicate that the named account forgot to login the first time, so I did not take any chances. Dr. K. 21:40, 2 March 2019 (UTC)[]
Since I believe that any reply I considered appropriate to your style of leading this discussion would not help to improve the state of this article, I herewith stop at my discretion any further commenting to this matter. Purgy (talk) 08:20, 3 March 2019 (UTC)[]
I was taken aback by these claims of edit-warring when all I did was a single revert of what I considered a very poor edit. The reason given for that edit was rather ludicrous, "linking within an article is to be avoided" flies in the face of Wikipedia reality as a simple cursory review of articles would show. As to the substance of my revert; it is well known and can be found in any reliable source on the axiomatics of geometry that Euclid's treatment is logically inadequate to support our modern concept of Euclidean geometry. This in no way diminishes the value of Euclid's contribution to the field, but our standards of rigor have changed over the years and this contribution can no longer be considered fundamental. While pedagogically useful as an introduction to axiomatics, due to the simplicity of its statements, modern geometry has to go beyond Euclid's treatment to reach logical bedrock. The only POV issue here is that any attempt to deny or hide this would be considered a serious distortion of our understanding of the subject. I will expand upon this in the article. --Bill Cherowitzo (talk) 19:41, 2 March 2019 (UTC)[]
Hold your horses. There is no reason to engage in an apologia for Euclid. Noone asked you to do so and it is therefore useless and condescending. As far as your statement The only POV issue here is that any attempt to deny or hide this would be considered a serious distortion of our understanding of the subject. I will expand upon this in the article. is a failure of WP:AGF and indicates that you did not read my original post above: I would have no objection if the disputed edit These axioms are generally agreed to be insufficient for modern levels of rigor. is properly sourced and stands on its own by a suitable reference or references, and if the part pointing to another section is dropped. - something that you actually did by supplying a book reference and dropping the quotation of the in-article section. Sloppily accusing me of an attempt to deny or hide this, although the only thing I explicitly asked of you was to provide a standard reference for your edit instead of a clumsy reference to a section inside the article, is a failure of WP:AGF and WP:NPA. So stop doing that. Finally, you did not provide a page number for your reference. I syggest you do that. Dr. K. 21:56, 2 March 2019 (UTC)[]
Oh really! Let's see, you accused me of edit warring because I reverted to an IP's contribution assuming that it was possible that I was the IP. You also accused me of not reading your post but at the same time mentioned that I made an edit in accordance to what was in your post. Now these appear to me to be real violations of WP:AGF. You brought up the POV issue in linking to another section and I retorted with what I considered to be the POV issue. You took this as a personal attack and I am sorry that you saw it that way. The only way, in my view, that that could be a personal attack is if it was truly your motivation. And speaking of that link, "It is also not standard in Wikipedia articles to refer to other sections, as if readers are too stupid to find them by themselves." An article is not a book or a paper. This type of guidance is not done on Wikipedia articles." is total nonsense. See the documentation in Template:Section link which gives this option, clearly making it an acceptable linkage. Finally, I always include page numbers in my citations and I did so in this one (p. 8).--Bill Cherowitzo (talk) 23:39, 2 March 2019 (UTC)[]
I am running out of time for now, so I will reply in more detail to your other points above, but for now, let me take issue with your statement: Finally, I always include page numbers in my citations and I did so in this one (p. 8). Nope. This is your edit when you added the reference. Click on that link and tell me where page 8 can be found. If not, you have to acknowledge that my remark that you did not include a page number in your edit was correct. Dr. K. 00:21, 3 March 2019 (UTC)[]
????? Just before the ISBN number, which is where the citation template always puts it. --Bill Cherowitzo (talk) 00:43, 3 March 2019 (UTC)[]
Your annoying edit-summary notwithstanding, I don't know how I could have missed it. I think the pipe sign "|" got mixed with the last "l" in "Prentice Hall" and so I didn't notice the separate "page=" field when I was checking the diff. So I stand corrected. I also saw the rest of your original reply, and I think your arguments there fall within the spectrum of AGF, so I will not reply to them further, although the remark attempt to deny or hide this is still iffy despite your explanation, but I will let it go. Dr. K. 03:02, 3 March 2019 (UTC)[]
As far as your comment: total nonsense. See the documentation in Template:Section link which gives this option, clearly making it an acceptable linkage. Read the template documentation again. Especially where it says:

This template is intended primarily for use in hatnotes rather than in running body text.

This clearly exposes your use of the template as clumsy and unusual, something that I already told you multiple times. So, who is spouting nonsense now? Dr. K. 00:27, 3 March 2019 (UTC)[]

Nomination of Portal:Geometry for deletion[edit]

A discussion is taking place as to whether Portal:Geometry is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The page will be discussed at Wikipedia:Miscellany for deletion/Portal:Geometry until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the page during the discussion, including to improve the page to address concerns raised in the discussion. However, do not remove the deletion notice from the top of the page. North America1000 23:26, 2 June 2019 (UTC)[]

Parallel postulate[edit]

At the time I write this, a section titled "Parallel postulate" includes the following sentence:

The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.

So apparently spherical geometry is just a lie?!

Find such a parallel to the original line, in spherical geometry, and I will give you one of my lungs.

Why is such errant, provable nonsense in an encyclopedia? One might think that mathematics would be a more reliable subject than most for Wikipedia to cover, given that when it is wrong, we can prove it wrong. Yet reality appears to go the opposite way. Whatever can't get past a mathematician goes onto Wikipedia instead.

Even if it were correct, it would be impossible to learn mathematics from this source, because when page A links to B links to C links to A again, one must read A, B & C repeatedly just in order to get the gist of A. When it is actually wrong, on top of that, it is worse than useless.

Shut down Portal:Geometry, they say? Shut down maths on Wikipedia! Wrong is wrong and you can't fix it. Please, let it die. — Preceding unsigned comment added by (talk) 22:29, 22 August 2019 (UTC)[]

Perhaps you should pay a little more attention to what is written. It is clearly stated that the parallel postulate is limited to a plane, as in "In a plane, ...". Spherical geometry is not a plane geometry, so your tirade above is quite misplaced. --Bill Cherowitzo (talk) 19:53, 23 August 2019 (UTC)[]

Section "Constructive approaches and pedagogy"[edit]

I have removed this section, which is written like an essay, and is an WP:original synthesis of WP:fringe theories. For example the author uses "axiomatic proof" for what everybody calls a proof and "Analytic proof" for something that is related to experimentation, and is definitely not a proof, as being "non-deductive". This is the most visible error of this section, but almost every sentence contains an historical error or an opinion that is not shared by the main stream of mathematics. D.Lazard (talk) 16:05, 10 August 2020 (UTC)[]