Talk:Euclidean geometry/Archive 1

plane versus solid

I disagree that Euclidean geometry refers primarily to plane geometry. Certainly I have never thought of it that way. Euclid also did not restrict him to plane geometry in his "Elements". So I hope you don't mind, but I changed that bit. mike40033 07:18, 12 Mar 2004 (UTC)

I would sugest at least

Euclidean geometry often means geometry in the plane

for me (I'm russian) Euclidean geometry means plane geometry if not stated otherwise, maybe for all of you it is different, but then it is not clear why the article on Euclidean geometry explains what plane geometry is...? Tosha 12:43, 29 Jul 2004 (UTC)

I've reworded everything so it's not necessary to make any statement about what it's generally taken to mean.--Bcrowell 19:00, 4 March 2006 (UTC)Hassan. Abdullahi.

merging with non-Eucldean?

Should this article be merged with Non-euclidean geometry? -- The Anome

No, I don't think so. This article ought to discuss Euclidean geometry. At the moment it doesn't actually say much about Euclidean geometry, and instead spends too much time discussing non-euclidean geometry, which is already discussed in Non-euclidean geometry. So it needs a lot of work, and some of it should be moved to Non-euclidean geometry, but it should remain a separate article. --Zundark, 2001 Dec 22

I agree with Zundark that it's not a good idea to merge the two articles. I also agree that the article is somewhat unbalanced by the large amount of discussion of non-Euclidean geometry. However, I disagree with Zundark about the best solution to the problem. I think the solution is simply to add more material about Euclidean geometry. The article would be incomplete if it didn't explain Euclidean geometry's relationship to other forms of geometry. I think it would also be a good idea to rework the article at some point so that all the fancy stuff (non-Euclidean geometry, Godel's theorem, higher-dimensional spaces) comes as late as possible in the article. However, I think it would be a waste of time to do that right now; we should do that after more material on Euclidean geometry itself is added. One obvious thing to add would be a sort of "greatest hits" list of important theorems in Euclidean geometry. It would also be cool to show at least one example of a nontrivial compass-and-straightedge construction.--Bcrowell 19:00, 4 March 2006 (UTC)

Oh, it does have a greatest-hits list at the bottom, but it doesn't tell us what these theorems say, or how important they are. It would be nice to have an example of a complete proof in Euclidean geometry, and I think a good example would be Book 1, Proposition 5, known as the "pons asinorum" or "bridge of fools."--Bcrowell 23:38, 4 March 2006 (UTC)Hassan. Abdullahi.

Is not enough!

In deed,the article is well constructed showing us the basics of Euclidian Geometry but i think is a little short for those who are intrested to learn more about this subject.I hope to find out more in the future!

Wikipedia can be edited by anyone. Give it a go.--THobern 21:54, 18 June 2007 (UTC)

statement of postulates

The five postulates are:

1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and radius.
4. That all right angles equal one another.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

only two perpendicular line can make a 90 degre angle.

Source: http://s13a.math.aca.mmu.ac.uk/Geometry/M23Geom/Euclid/Euclidbook1.html
looxix 21:51 Feb 23, 2003 (UTC)Hassan. Abdullahi

Parallel postulate

The fifth postulate is equivalent to parallel postulate, which can be phrased as follows

• Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

I don't believe this is true, nor the correct way or naming things. The "parallel postulate" to be completely accurate should always refer to Euclid's axiom as he stated it. I believe what is being called the "parallel postulate" here is actually what is rightly called "Playfair's axiom", although to be perfectly honest, there is even some doubt in my mind as to whether this term is meant to mean "exactly one line" or "at most one line" (Playfair used the former phrase, Legendre the latter, so there is some confusion for me, or if it matters.) Revolver 06:30, 20 Mar 2004 (UTC)

The term "equivalent" here is being used, correctly, in a certain strict formal sense. It says that either form of the axiom can be used to prove the other form. The two statements are not obviously equivalent to someone who doesn't know the proof, but that's not what the article is asserting.--Bcrowell 19:00, 4 March 2006 (UTC)

Non intuitiveness of the fifth postulate

From the article:

To the ancients, the parallel postulate seemed less obvious than 
the others; verifying it physically would require us to inspect 
two lines to check that they never intersected, even at some very 
distant point, and this inspection could potentially take an 
infinite amount of time.

This sounds dubious to me. First Euclid's fifth postulate was not formulated in terms of parallel lines. Second I don't think that there is any evidence that the ancients thought that way. And finally why doesn't this apply to the second postulate? Doesn't it take infinite time to examine that a straight line can be indefinitely extended? Nikos.ap 13:15, 12 June 2007 (UTC)

completeness

Currently, the article includes:

 As Godel proved, all axiomatic systems -- excepting the very simplest -- 
 are either incomplete or contradict themselves, and this is no exception.

It seems to me that a sufficiently simple axiomatization of Euclidean geometry might actually be complete. I don't see any way to embed the natural numbers in Euclidean geometry, which is the usual way to verify that Gödel's theorem applies. -- Carl Witty

Carl Witty is correct, and the original remark in the article was incorrect. Godel's theorem doesn't apply here. The article now states it correctly: Euclidean geometry has been proved to be consistent and complete. I've provided a footnote with a reference to a book that discusses this in depth.--Bcrowell 19:00, 4 March 2006 (UTC)Hassan. Abdullahi

construction

Isn't there some work on showing how to make computations with ruler-and-compass constructions, providing you have a pre-existing "program"? The Anome 19:36 21 May 2003 (UTC)

It's:

• Simon Plouffe.The Computation of Certain Numbers Using a Ruler and Compass. Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.3

The Anome 19:56 21 May 2003 (UTC)

removed part

I just removed new subsection, it was correct but irrelevent, might go somewhere else... Tosha 00:00, 30 Mar 2004 (UTC)

Thanks for stating your points of view regarding the correctness and relevence (sic) of the characterizations of physical spaces in terms of Euclidean geometry succinctly, yet explicitly and separately.

Thanks also for the generous scope of your suggestions where else within this encyclopedic representation of what's considered correct (at least: rather than "any where else but ...") this topic might be addressed instead.

Being left to narrow this considerable selection down, perhaps (at least) to

- the discussion of Euclidean planes and spaces by the author of A Modern View of Geometry, W. H. Freeman (1961),

- derivations and statements of certain expressions (such as that attributed to one Tartaglia) which some do seem to find noteworthy after all,

- considerations rather less frivolous than [[Talk:Why 10 dimensions]], or

- being an [[Wiktionary:Also-ran]] to what appears already established,

the choice appears nevertheless daunting ...

Regards, Frank W ~@) R 03:44, 30 Mar 2004 (UTC).

grammar

Some grammatical errors are in this article 209.155.121.101 13:46, 22 December 2005 (UTC)

I've tried to clear these up.--Bcrowell 19:01, 4 March 2006 (UTC)Hassan.Abdullahi

Spelling

Should it be "Euclidean geometry" or "euclidean geometry"?

Inactive link

The link titled "In English" under "The Elements" (http://aleph0.clarku.edu/~djoyce/java/elements/toc.html) is inactive. Someone restore it or just remove it.

Pizzadeliveryboy 08:40, 11 August 2006 (UTC)

GA Re-Review and In-line citations

Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:55, 26 September 2006 (UTC)

Non-Euclidean geometries?

If you change the fifth postulate while keeping the other four, you get the hyperbolic, elliptical, and absolute geometries. But what sort of geometry do you get if you change the fourth postulate while keeping the others? --Carnildo 07:21, 12 December 2006 (UTC)

I'm not aware of anyone ever attempting that. The original rational for trying alternatives to the fifth one was a feeling that the other four were intuitively compelling, but that was not to the same degree (of course this was in the context of the assumption of the time, that the system was both an axiomatic schema and an accurate description of physical space). DaveApter (talk) 15:32, 16 October 2008 (UTC)

As a description of physical reality

I think this section should somehow be reworked. It is mostly about GR and seems to repeatedly overstate things. The article states: "no possible physical test (that) can do any better than a beam of light", but this failed long before GR. Newtonian gravity would cause light to bend. The theories differ in how much the light bends. Light has always been known to bend through glass. Even saying the light was "bent by the Sun's gravity" is not consistent. It really highlights that there IS a way to consider something straighter than light - that being the unbent path one might have thought the light could have taken. So it seems absurd that we might have to "reject the entire notion of physical tests of the axioms of geometry" because light bends. Also the picture caption starts with the word "proof". This word has very special meaning for this page and should not be used so casually. Using the phrase "shows that the true geometry" seems to be using common words that tend to be stronger even than "proves". I think that the article should somehow say that GR POSITS a curved space. Ned Phipps 23:01, 29 December 2006 (UTC)

Light has always been known to bend through glass. The article is talking about light in a vacuum. It really highlights that there IS a way to consider something straighter than light - that being the unbent path one might have thought the light could have taken. Your idea doesn't work, because the concept "could have taken" is undefined. What the article says about GR is completely standard; check any textbook on GR.--75.83.140.254 01:39, 20 January 2007 (UTC)

Actually, the concept "could have taken" is /very/ well defined and is the meaning of having measured the amount of bending (or GR's apparent bending) and having seen the that amount is double the amount that Newton's gravity predicted (light bending within Euclidean space /in a vacuum/). The text reading "is bent by the Sun's gravity" raises, all by itself, the concept that something is considered straighter. I am quite aware of standard GR and that GR says the parabolic path that a baseball takes is apparent only. Finding words to describe /that/ bending as fundamental would be much more descriptive of GR. It is the extreme wording that I object to. "reject the entire notion of physical tests of the axioms of geometry". Physical tests can never rise to the level of /proof/ that is so special to Euclidean Geometry. They should never be completely accepted or rejected.Ned Phipps 02:09, 25 January 2007 (UTC)

"reject the entire notion of physical tests of the axioms of geometry". Physical tests can never rise to the level of /proof/ that is so special to Euclidean Geometry. The text you're quoting seems to agree with your statement. The text reading "is bent by the Sun's gravity" raises, all by itself, the concept that something is considered straighter. If a geodesic passes from point A to point B, through the sun's gravitational field, then there is no other geodesic from A to B that is more straight than that one. The bending being referred to occurs for an observer who has chosen a particular set of coordinates (which are asymptotically flat); a different observer could choose a different set of coordinates in which the ray of light was not considered to be bent, but in those coordinates spacetime would not be asymptotically flat. Obviously this is not an article on general relativity, so it's not practical to go into great detail on this sort of thing. Any short discussion that doesn't use precise, technical terminology and notation is going to be subject to incorrect interpretations.--75.83.140.254 01:35, 29 January 2007 (UTC)

Several different issues seem to be being conflated here. Whether a theorem does or does not follow from the axioms is a matter of mathematical truth, whereas whether the physical space we inhabit is accurately described by the axioms of any particular geometry is a matter for empirical verification. DaveApter 09:38, 18 May 2007 (UTC)

I think that the terminology should be reversed. Something like:

With GR the concept of a path of an object bending is discarded. Instead, objects go straight (geodesics) and the space, itself, is considered curved.

Using the path of light as the example of what bends is very misleading since that is essentially straight whereas the path of a baseball is considered a bend in one and not the other.

My other concern has to do with the fact that experimental verification of this theory is only in the very first order deviations from "flatness" in space and time. Hardly a "proof". There's quite a difference between a physicists theory that has supporting evidence a mathematical proof as introduced by Euclid and which make Euclidean Geometry so special.64.161.207.162 00:10, 1 February 2007 (UTC)

Einstein himself deals elegantly with the jusification for discarding Euclidean geometry as a description of space-time in his 1938 book The evolution of Physics pp 222-234. It is not that one or the other is "True". It is possible to model the universe either way (or in any number of other ways). The strength of General Relativity is in the simplicity and elegance of its underlying postulates, whilst still providing the most accurate match yet obtained for observations made in the real universe. DaveApter 13:16, 20 March 2007 (UTC)

Hmm, the statements about the standard model and GR are all in the class of 'sort-of-true'. When I have some time I will try to rewrite the entire paragraph. The problem is that, for instance, the standard model is not *based* on Euclidean geometry. There is a mathematical trick which could be interpreted as that (which goes under the name 'Wick rotation'), but it can be shown that this is needed to make the theory well defined (more precisely, to define the right causal propagator). I would say that there are far more fundamental problems than choosing the geometry. And even then, the geometry should be a result of the theory, not an input (see background independence). —Preceding unsigned comment added by 130.225.212.4 (talk) 08:33, 27 May 2008 (UTC)

reversion of extensive anonymous revisions

I am reverting the edits by anonymous user 75.83.140.254 .

I would request that anyone contemplating such extensive edits be logged in, and prepared to discuss the merits of the changes with other editors to reach consensus. DaveApter 14:34, 25 January 2007 (UTC)

I'm available via my talk page, and I'm ready to discuss anything you want to discuss. How about discussing edits on their merits, rather than carelessly undoing other people's work? Reverting.--75.83.140.254 01:16, 29 January 2007 (UTC)

Considering that the major portion of your edit (apart from a few minor wording changes) was the wholesale removal of a substantial section of the article which other editors considered relevant and interesting, I respectfully suggest that you take your own advice. DaveApter 13:05, 20 March 2007 (UTC)

figure of Euclid

this is of course not a true depiction of the historical Euclid, whose identity is contested as it is. I think this should be mentioned in the figure caption. --128.139.226.37 12:55, 23 April 2007 (UTC)

Parabolic and elliptical?

The recent edits by Ray Chason [[1]] seem to me to have replaced two statements that were factually accurate with two that were not. I just thought I'd check before reverting them? Any comments? DaveApter 16:40, 22 May 2007 (UTC) I have gone ahead and corrected them. DaveApter 12:52, 24 May 2007 (UTC)

I think you are right. I believe the edit you mention above thought that we are talking about a body in space rather than a body launched from Earth. I tried to clarify things, I hope I did not confuse anything. Oleg Alexandrov (talk) 01:27, 25 May 2007 (UTC)

I agree that your wording has clarified it. Thanks. DaveApter 09:10, 28 May 2007 (UTC)

Euclid's axioms v.s. postulates

I thought that Euclid's axioms were separate from his 5 postulates. And consisted of his basic definitions for lines, points, and circles. Lonjers 06:30, 8 June 2007 (UTC)

Gravitational theory

I have attempted to rewrite this section with several goals in mind.

1. Clearly separate Kepler orbits from projectile trajectories.
2. Identify the role of assumptions and experimental limits.
3. Make the whole thing relevant by leading up to curved space.

One fine technical point: It is traditional when discussing the ellipse/parabola/hyperbola trichotomy to speak in terms of eccentricity; the concept of intersections with infinity, while anachronistically modern (compared to Kepler), seems to more directly clarify the connection to orbits.

Perhaps the revision is a little better than its predecessors, if not ideal. I would hope that future edits retain some mention of the curvature of space. --KSmrq^T 18:49, 16 June 2007 (UTC)

With respect, I think that two of your statements are factually inaccurate, and I have re-formulated them. I don't know that comets ever move in a parabolic trajectory, but I am sure that if they did it would be one that fell into the sun rather than going off to infinity. If you can provide a demonstration of your assertion, I would be interested to see it.

I don't follow why you would want to "Clearly separate Kepler orbits from projectile trajectories". The fact that both Kepler's and Galileo's curve-fitting to respectively celestial and terrestrial movements produced conic sections is likely to have been an important clue leading Newton to postulate a universal set of equations of motion and gravitation which gave a common explanation of both. DaveApter 09:32, 6 August 2007 (UTC)

I have had to back out your revision, because it was wrong. A periodic comet is in an elliptical orbit; it can never escape the Sun's gravitational pull. A one-time comet is almost certainly following a hyperbolic path, and will head out into space never to return. The parabola is the transition between these two possibilities; there is no reason at all such an orbit should cause the comet to fall into the Sun. In terms of eccentricity, we have 0 ≤ e < 1 for an ellipse, 1 < e < ∞ for a hyperbola, and e = 1 for a parabola. This is all very well-known stuff. See, for example Encyclopædia Britannica, The Columbia Encyclopedia, an image, and to work out data yourself, this site. Or, to quote from an astronomy textbook:

Because the orbits of most comets appear to be nearly parabolic, the question arises whether all comets are members of the solar system or whether some might be accidental intruders from interstellar space. The evidence is conclusive, however, that comets have always been members of the solar system.

The parabola is the orbit that marks the transition between ellipses (bound to the Sun) and hyperbolas (unbound). If comets were intruders from interstellar space, their orbits should nearly all be markedly hyperbolic. Most comets are simply on ellipses of very great eccentricity (nearly equal to 1.0). Those few comets that do appear to have had slightly hyperbolic orbits are believed to have approached the Sun on long ellipses that were perturbed by Jupiter or another planet. … The aphelia of new comets are usually at about 50,000 AU from the Sun.

—— Exploration of the Universe, 6th edition (ISBN 978-0-03-034584-5)

We decided to avoid projectile trajectories for a variety of good reasons. The consensus was that they weren't worth the trouble. They require careful explaining, and even then readers tend to be confused. (For example, many people "know" that projectiles move on parabolic paths. The full truth is more complicated, and a pain to get across.) And since this is an article on geometry, not physics, we all agreed comets were the right choice. If you would like to learn more about orbits and trajectories, see the textbook I cited. --KSmrq^T 10:38, 6 August 2007 (UTC)

Who are "We", and where was this consensus arrived at? What were the "good reasons" for avoiding projectile trajectories? Why did you decide to restrict this discussion to comets? In what way do the orbits of comets differentiate themselves from those of other orbiting bodies? DaveApter 20:35, 6 August 2007 (UTC)

If the edit history and my memory serve, JRSpriggs (talk · contribs), Oleg Alexandrov (talk · contribs), and I (at least) all tried our hand at editing the gravitation section at the time of my remark above. I believe the discussions took place on a user talk page, probably Oleg's.

Again, this article is about geometry. If you want more material on orbits of different kinds, this is the wrong place. Nor is this talk page the best place for me to teach you the theory.

But, as a courtesy and to save time, I'll point out some of the reasons to avoid projectiles. First, near ground level on planet Earth an honest trajectory is significantly influenced by air, which is a viscous fluid. Second, we are often told that the trajectory in a vacuum will be a parabola, but that's problematic. We must assume the effects of inverse square graviational field dropoff are negligible; but that contradicts the fact that a suborbital ballistic trajectory is actually an ellipse. Third, the bound trajectories most people encounter in daily life are not orbits in the usual sense; the ellipses intersect the surface. Fourth, for accurate tracking of satellites in Earth orbit Kepler ellipses are too crude; the usual models include perturbations like the non-spherical shape of the Earth. Given that you, yourself, were already confused about the orbits of comets, which are much simpler, it is hopeless and ill-advised to try to wade through the complications of ballistic projectile trajectories.

What about other bodies, say asteroids or moons or other planets? Again we have exposition problems. With comets we naturally see all three kinds of non-degenerate conics (though most are ellipses); not so with the bound orbits of these other bodies, especially planets. As well, the conic sections only appear in the two-body approximation; asteroids and moons often involve more bodies. For example, there are gaps in the asteroid belt because of orbital resonances with Jupiter; some asteroids are found at Lagrange points; and two of Saturn's moons (Janus and Epimetheus) swap orbits every four years. In an article on orbital mechanics it could be delightful and appropriate to explore these things; they are out of place in an article devoted to "Euclidean geometry".

I very much encourage you to learn more about these topics; the book I cited is a good source, and you will find a great deal of material online, including within Wikipedia. But let's not discuss this further on a geometry talk page. --KSmrq^T 09:54, 8 August 2007 (UTC)

Thank you, KSmrq. I would agree that many of the more arcane points you make may be appropriate to an article on astronomy or celestial mechanics, but not to this one that is about geometry. However, I do feel that there are some aspects of this topic that are relevant to this article, and I will expand on this below.

Furthermore, the restriction of the discussion to comets seems to detract from the relevance to the subject of geometry. Let us start by setting out the points on which we agree:

1) The conic-section shapes of orbits and trajectories are only approximations to what is observed in the real world (even on Newtonian assumptions). Reasons for this include (a) The fact that the bodies are not point masses, (b) the irregularity in shape and density of real bodies, (c) the gravitational interference from other bodies, and (d) where relevant, the effect of atmospheric viscous drag. Agreed.

On the other hand, Kepler had plotted the orbits of the known planets as being ellipses, to a high degree of accuracy.

2) A body moving tangentially to a larger body at greater than escape velocity will follow a hyperbolic trajectory (subject to the above limits to the approximation), and disappear into deep space never to return. Agreed.

The paths of comets had been observed to be compatible with being either hyperbolic, parabolic or elongated elliptical - but with insufficient precision in the 17th century to distinguish which.

3) I agree that I was incorrect in my statement above that an orbiting body moving at sub-escape speeds would move in a parabolic trajectory. I also now see that my original characterisation of the differences in velocity in relation to the shape was confused.

4) I agree that strictly speaking a terrestrial trajectory (in vacuo) is an ellipse rather than a parabola, due to the variation of gravitational force with altitude and to the change in direction to the center of the earth with horizontal movement. But for practical purposes the parabola is an accurate approximation (except for cases such as intercontinental ballistic missiles).

Galileo had demonstrated by purely Euclidean arguments that a body moving without resistance in a uniform gravitational field would describe a parabola.

Newton used Euclidean arguments to demonstrate to contemporaries the explanatory power of his laws to predict Kepler's observations.

It seems to me that the significance all this to an article on geometry relates to the deep and mysterious inter-relationship between the abstract world of mathematics and the working of the material universe, to the fact that purely geometrical reasoning can provide proofs that we would derive analytically nowadays, and to the role that the availability of these tools had in the development of Newton's laws of motion and gravitation and the unification of celestial with terrestrial mechanics. DaveApter 10:11, 14 August 2007 (UTC)

Hear, hear. I don't see why "the full truth" is a pain to get across. Neglecting vicissitudes (friction, viscosity, truncation of the parabola by collision with a window or the ground, etc.) is a standard assumption, as is the distinction between local behavior (throwing a baseball) and global behavior (orbiting satellites). It suffices to say that baseballs follow parabolic orbits because gravity in the small is essentially flat while satellites follow conical orbits because gravity in the large is essentially spherical. Both small and large are just approximations; when they are very good approximations geometry as a model of reality can be very helpful. --Vaughan Pratt 21:58, 11 September 2007 (UTC)

What is this article about?

I came to this article because in terms of the good article criteria it is one of the weakest math GAs: indeed an attempt was made to "boldly" delist it recently, but this caused a bit of a stir, so the delistment was temporarily overturned.

I have been hesitant to return to the issue, because this is a great subject (well I would say that), and the article is full of lots of interesting stuff, thanks to the dedicated work of the editors involved. Nevertheless, I can see why it found itself in the delistment firing line, as it is woefully undersourced, and seems, on the face of it, to contain quite a bit of original analysis. Even though the GA criteria are more relaxed about citations now, this article states not just mathematical facts, but historical and analytical material that really needs citation. If I were to tag it, it would look like several fact bombs had exploded: not a pretty sight :-)

But forget about GA: if that process has proven demonstrably incapable of assessing mathematics articles, then who gives a fig? I should be happy to delist the article, make it a Bplus on the mathematics scale, and forget about it.

Except, on reading the article more carefully, I find there are other more important problems with it.

There is a lot of nice stuff here, indeed, but rather too little of it is actually about Euclidean geometry. There are extensive digressions on the relations between Euclidean geometry and logic, and between Euclidean geometry and physics. What is an article on Euclidean geometry doing talking about mereology, Godel's theorem, general relativity and grand unified theories? Why does it not even mention inner product spaces or the Euclidean group? Felix Klein must be rolling in his grave. Triangles are hardly mentioned except to make logical points. Conic sections are introduced in two sentences (the second of which analyses them from the point of view of projective, not Euclidean, geometry) and the rest of the paragraph is all about physics, not geometry. All very well, but does it really belong here? Geometry guy 16:34, 30 September 2007 (UTC)

Considering that general relativity shows that physical space is noneuclidean, it does seem pretty relevant to me. I think the real problem is not that the article discusses these other issues but that it fails to discuss enough about the core subject. That is, I think it needs addition of core Euclidean geometry material more than it needs deletion of extraneous material. Euclidean geometry is a very important topic, and the article is actually quite short.--76.167.77.165 (talk) 02:51, 9 February 2009 (UTC)

Okay, I take that back. It really did need a lot of cutting. Going back over the article, I see a huge amount of accumulated cruft, didactic digressions, unsourced stuff that seems like original research, etc. I've cut a bunch, and also reorganized it quite a bit. The basic structure is now that it gives the basic structure of Euclidean plane geometry, followed by a century-by-century historical summary of later developments. However, I still think it's in depserate need of *additions* of material on the core topic.--76.167.77.165 (talk) 03:51, 9 February 2009 (UTC)

GA Review

For what it is worth, here is what a GA review would look like based on the current article, in my opinion.

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS):

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): c (OR):

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   Fair representation without bias:

5. It is stable.

   No edit wars etc.:

6. It is illustrated by images, where possible and appropriate.

   a (images are tagged and non-free images have fair use rationales): b (appropriate use with suitable captions):

7. Overall:

   Pass/Fail:

The issues are: the lead contains material that is not elaborated in the article; the lack of sources and citations; and the lack of breadth, the off-topic digressions, and the bias towards axiomatic and applied points of view, which I have already discussed. I won't delist for a while. If I get time, I will try and fix some of the breadth issues, but semester is just starting, so I thought I would raise the above issues here, in the hope that someone else will have more time than me... Geometry guy 16:34, 30 September 2007 (UTC)

I've now replaced the GA with Bplus. Geometry guy 18:48, 6 October 2007 (UTC)

I think it's quite a bit better now than it was in October 2007. I've renominated it for GA.--76.167.77.165 (talk) 00:45, 24 February 2009 (UTC)

Thanks for stepping in and improving the article. It is back on my watchlist, and I will help where I can. Geometry guy 09:37, 24 February 2009 (UTC)

Fourth postulate: equal, not congruent

I have amended the fourth postulate to read this way, changing congruent to equal:

All right angles are equal.

In standard usage congruence never applies to angles or sides, only to figures. This is all very straightforward and clear, since the standard term equal works perfectly well for angles and sides that are simply measured numerically. See Congruence (geometry); see also major British and American dictionaries: SOED, and M-W Collegiate (Congruent "2: superposable so as to be coincident throughout"; there is no "throughout" for sides or angles, since they are not compound as geometrical figures are).

The original Greek uses the word ‡ἴσος (or strictly, its feminine plural form ἴσας; see Elements, p. 6, where the Greek is given and the translation is with equal). This word plainly means "equal". It needs to mean that, since Euclid immediately goes on to say things like this: "11. An obtuse angle is greater than a right-angle." Relations like greater than make sense in the context of relations like equality, but not of compounded qualities like congruence.

– Noetica^♬♩ Talk 21:47, 10 December 2007 (UTC)

Equivalence

There's a contradiction between this article and the one on the parallel postulate. Please contribute to the discussion at Talk:Parallel postulate#Equivalence. Joriki (talk) 15:14, 9 January 2008 (UTC)

I made some comments there. I think the issue is basically muddy because of the ambiguity of Euclid's statement of his postulates.--76.167.77.165 (talk) 19:10, 27 February 2009 (UTC)

Parabolic Geometry

I am wondering what was wrong with the Parabolic Geometry page. It seemed like a legit disambiguation to me. Bob the Wikipedian, the Tree of Life WikiDragon (talk) 20:10, 15 April 2008 (UTC)

Whoops. Looks like I accidentally created a freak accident talk page. I'll try to clean up this mess by moving it to where I intended to post it.Bob the Wikipedian, the Tree of Life WikiDragon (talk) 20:13, 15 April 2008 (UTC)

The above two posts were moved from a freak accident page that should have been here. Bob the Wikipedian, the Tree of Life WikiDragon (talk) 20:13, 15 April 2008 (UTC)

I also think that the Parabolic geometry should have its own page, while this can have several meaning in mathematics. Euclidean geometry is only one of the meanings. A quite large group of mathematicians (see, e.g. some articles of Andreas Cap in Vienna, Mike Eastwood in Australia, Jan Slovak in Czech Republic, moreover, a monography with title "Parabolic geometries" will be soon published in Springer) is working in a special case of Cartan geometry that nowadays is called "parabolic geometry", but this does, surprisingly, NOT include the euclidean geometry. The word parabolic comes from the fact that it is modeled on a pair (G,P), where P is a parabolic subgroup -- see the meaning of parabolic in the page Borel subgroup. If its possible, return the Parabolic geometry disambiguation page.Franp9am (talk) 13:19, 19 April 2008 (UTC)

Possible copyright violation

The section Conic sections and gravitational theory reads as though it was copy-n-pasted from a larger work. Could somebody please verify that this section doesn't violate copyright?—Tetracube (talk) 20:41, 5 December 2008 (UTC)

That section is no longer in the article. It was off topic in any case, IMO.--76.167.77.165 (talk) 07:16, 27 February 2009 (UTC)