# Talk:Euclid's Elements

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Okay, so who thinks this should redirect to Euclid? Can we have a show of hands, please? Actually, I think it might be nice to concentrate on Euclid's life in his article, even though we don't know much about it. I think this story may be a lie ;) but this page manages to talk about it for a few paragraphs before getting to the mathematical details, so it could work. Then we could put the actual maths in this article. What do you think? -- Oliver P. 00:21 15 Jun 2003 (UTC)

I've expanded the material on this page quite a bit. Looking at the material in the entry on "Euclid," most of it seems to be duplicated here. Unless we can find some more significant biographical information (are other sources than MacTutor available?), I agree that the articles should be merged. JPB 06:16 6 Jul 2003 (UTC)

Well, this article could go through the books one by one, and summarise what is proven in each one. That would quite nice for this article, but would be a bit overwhelming if put into in the article on Euclid himself, which concentrates more on the overall significance of his work. -- Oliver P. 06:22 6 Jul 2003 (UTC)

The usual practice in classics is to not to separate the author from work, unless there are lots of works, or we have an actual author bio, Cicero for instance. My OCD gives his life two lines, but has a bunch of content for his many lost works. Elements gets most of a paragraph, though this being the OCD, is mostly a discussion of the different manuscripts. I think in this case the Elements are worth a separate article, at least if somebody (me, I suppose :-) ) gets to work and adds the non-Elements info about Euclid. Separate articles for each book of Elements seems a little overboard though. Stan 13:09 6 Jul 2003 (UTC)

Euclid and Greek philosophers made a distinct between axiom and postulate. In fact, the Elements includes 5 of each. Should the five axioms be included here? An axiom is an assumption about everything. A postulate is an assumption about a particular science/area of study or an assumption that is not 'obvious'. gbeehler 11:28 6 Nov 03 (UTC)

??? They are included here at the top of the article, aren't they? MrJones 11:43, 6 Nov 2003 (UTC)
No. The 5 postulates are stated and then called postulates and axioms??? For the record, the axioms given are: (A1) Things which are equal to the same thing are equal to each other. (A2) If equals be added to equals, the wholes are equal. (A3) If equals be subtracted from equals, the remainders are equal. (A4) Things which coincide with one another are equal to one another. (A5) The whole is greater than the part. I think it is important to state these because (1) the complete set was used as a foundation to the argument that the 5th postulate MUST be derivable from the other 9 (2) Saccheri tried to do just that and instead discovered non-Euclidian geometries -- which are logically consistent (3) this lead to the development of axiomatics. gbeehler 17:08, 6 Nov 2003 (UTC)

When hooking up the topic of euclid i was first satisfied with the euclid wiki, till i found out, that there are parts init where the person euclid is doubted in beeing the inventor. So either make the content exact or make a flashy button on top of the euclid wiki. Or as i suggest merge them. (Tales23 (talk) 18:01, 12 January 2009 (UTC))

I removed this sentence because it's false:

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

In fact, Hilbert's axioms for Euclidean geometry are complete. This was proven by Tarski. I'll add info about this to the page when I get a chance. -- Walt Pohl 14:39, 20 Mar 2004 (UTC)

I think the correct statement is something like, "As Godel proved, all axiomatic systems, sufficiently strong enough to express the arithmetic (addition, multiplication) of the natural numbers are either incomplete or contradict themselves." Certainly, it's very easy to come up with trivial axiomatic systems with only a couple axioms that are easy to verify as complete and consistent. —Preceding unsigned comment added by 128.111.88.229 (talkcontribs)

Godel's is more subtle than that: It is possible in Any consistent, axiomatic system to formulate questions that cannot be answered. —Preceding unsigned comment added by 69.195.36.213 (talkcontribs)
Actually the part about the axioms needing to be sufficiently strong to express arithmetic is required (as is the requirement that the axioms be able to be generated algorithmically - else you could use Godel to disprove the existence of God). Godel's proof works by constructing a code (comprised of Godel numbers) which can be used to state theorems in arithmetic. Each statement has a number associated to it. The gag is that you can then make statements in this code about other statements by referring to them by their Godel number. Godel then proves that there is a statement which can be made in this code which refers to another such statement (itself essentially) by Godel number, which is known to be either true or false, but which could not be established to be so within the system itself. Hopefully this makes clear why Godel's proof isn't directly applicable to an article on Euclid's elements. -goodwillhart

## gobbledygook

This is gobbledygook: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. —Preceding unsigned comment added by 69.195.36.213 (talkcontribs)

That depends on what you mean by "gobbledygook". It's not nonsense, but it could be made clearer. Rory 13:08, Sep 5, 2004 (UTC)
Could we make it clearer? I've puzzled over it several times, and never quite figured it out. Brutannica 02:10, 8 Sep 2004 (UTC)
It says that if you take two line segments (A, B) and draw another line segment (C) so that it crosses A nd B, and if that makes two acute angles (θ, φ) on the same side of C, then A and B, extended further, will meet on that side of C.
___      |C
A  ___   |
___|
|__
|θ /__
| /   ___
|/       ...
|           ...
|\       ...
| \   ___
|φ \__
|__
___|
B  ___   |
___      |

Rory 11:45, Sep 8, 2004 (UTC)
Oh. Thanks! Could I modify the postulate then? At least let me change "less than two right angles" with "acute angles." Brutannica 03:03, 9 Sep 2004 (UTC)
Be bold! Rory 11:03, Sep 9, 2004 (UTC)
Hang on a sec. Are those postulates the original, translated ones? If so, then it wouldn't be right to alter them... instead, I should put the explanation after it in parentheses. Brutannica 00:08, 10 Sep 2004 (UTC)
If they are translations they should say who translated them and they should be quoted. As long as we're just stating the postulates, rather than quoting, I think it's fair to put them in as plain language as we can. Rory 12:18, Sep 10, 2004 (UTC)
O.K.... Brutannica 04:15, 11 Sep 2004 (UTC)

## Book XIII not authentic?

I don't think the following sentence is correct, although I have left it in the article:

It is strongly suspected that book XIII was added to the others at a later date.

Was the author of that statement perhaps thinking of the so-called "Book XIV"? - dcljr 09:37, 14 Aug 2004 (UTC)

The information in Book XIII was certainly known to Euclid -- it had been demonstrated decades earlier by Theaetetus of Athens. (The current Wiki page on Theaetetus is just a stubbish entry on Plato's dialogue with him.) --Crunchy Frog 18:15, 18 Aug 2004 (UTC)

There is more to Euclid than just the Elements. He wrote 4 other works we still have today and is credited with 4 more works which have been lost. All of this can be discussed in the Euclid article. The Elements, OTOH, is an almost neverending source of topics for dicsussion. For possible ideas, see this paper I wrote in college on the subject. See also my comment on Talk:Euclid. - dcljr 10:01, 14 Aug 2004 (UTC)

## Carriage of The Elements to Wikipedia?

Although there are many other online sources for the Elements, do people feel that there would be anything to gain from a setup of the propositions on Wikipedia? I intend to work through the Elements someday, using Heath's Dover edition. I would scan the diagrams and present the narrative of the propositions in a contemporary vernacular; this all presupposes that we've internalized the two-millenium debate over The Elements and can link to appropriate articles. Refitting the narratives of Euclid's results in modern prose is also extremely presumptious, and likely unnecessary, but that's why I ask you people first. -Cory

Perhaps Wikibooks or Wikisource would be a better place to put this. - dcljr (talk) 23:10, 23 Jun 2005 (UTC)

## Overstating the case?

Do you think this sentence is right?

That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy.

I think it may be a bit over the top. What do you think? Spondoolicks 11:20, 24 August 2005 (UTC)

I think it's fine as it is. The next sentence explains why: Einstein's space-time. I'm not sure about a direct link to philosophy, but certainly there is an indirect one through relativity. - dcljr (talk) 23:02, 24 August 2005 (UTC)

## Some possible errors

There appear to be some possible sources of errror in this article. I don't believe the statement that the Elements is second only to the Bible in number of editions. Firstly it is a pretty worthless comparison. Secondly I believe it is also false. What about Thomas Akempis' The Imitation Of Christ? I think there are 3000 editions of that. Shakespeare's plays? Perhaps even Pilgrim's progress has more than 1000 editions.

I doubt the content of book 13 is believed to have been added later. Where does this come from?

The opinion about Euclid's elements being responsible for the West advancing beyond the East seems culturally inappropriate. Is Wikipedia only written for Americans and Europeans with overinflated cultural egos?

Also, was Heiberg's work a translation from the Greek or was it a definitive Greek edition? I'm not sure since I am too lazy to go to the library and check, but I thought it was the latter. Someone should definitely check this.

I suspect some of the information about manuscripts/translations/commentaries on/of Euclid's Elements has possibly been confused with information regarding Archimedes' works. There is a lot of unscholarly information on the internet regarding this work, and I suspect that someone is going to have to go to an actual library and check out quite a few of the claims in this article, rather than cut and paste from some of the terrible media articles and other sources of innaccurate information present on the net. -goodwillhart

I think the information about the number of editions being second only to the Bible is correct, or at least attested by reputable sources. I still believe the remark about Book 13 is wrong (see also my comment above), so I'm going to remove it. I know Heiberg's edition is considered a "definitive" Greek edition; I'm pretty sure it didn't contain a translation. As for editions and commentaries, I believe the article to consult for this is the Dictionary of Scientific Biography entry on Euclid by Ivor Bulmer-Thomas. - dcljr (talk) 17:43, 20 October 2005 (UTC)
Well, like I said, Akempis' "On The Imitation of Christ" has over 3000 editions. So something needs to change in the article. The information in the article is probably based on oft quoted (and probably incomplete) statistics from Gutenberg which is probably almost 100 years old. It just isn't correct any more. And what is the point anyhow? How is the number of editions of a book a measure of anything. Unless you define what an edition is, it is a meaningless hyperbole. More useful information would be: the number of languages it has been translated into, the number of translations into English, or even better than all these, the estimated number of volumes printed. Some of these measures would put it well behind many other famous documents, demonstrating the worthlessness of these comparisons. Perhaps if someone could find an actual source (and by that I mean the original source) of the information, one could write that it had a "greater number of editions printed than any other book except the Bible, as measured in the year ....", for, whatever might have been true then, certainly isn't now. Incidentally, what constitutes an edition of the Bible? Does it have to be a translation, or can it be a paraphrase. Does it have to be the whole Bible? Does a red letter vs black letter edition of the same translation count as two or one edition. What if you print a hardcover edition, then a softcover one? What is an edition? Are separate print runs counted as separate editions? If you restrict the definition to distinct translations of the whole work that were published in significant quantities, I suspect the comparison made becomes utterly meaningless. Allow anything as an edition, and the Bible is WAAAAAAY ahead. In related news, the news media is responsible for more errors in print than occur in any other distributed body of work, other than Wikipedia (oh and apparently Encyclopaedia Britanica). -goodwillhart

To call the Elements adequate is just plain not doing the book justice. —Preceding unsigned comment added by 68.77.28.102 (talkcontribs)

## Removed later axiomizations

Certainly Euclid's elements could have been written differently, but whether it could have been written better is debatable at best.

This section implies that proposition 1.4 is logically flawed and that this was discovered in the 19th century. This proposition "proves" that if two triangles have two sides and their included angle equal, the triangles are equal. Euclid uses the method of superposition to prove this.

This proposition is indeed problematic, and it could have been stated as an axiom. As noted in the Heath book on Euclid, this controversy is at least as old as the 16th century. But whatever the merits of including this proposition as an axiom or a definition or whatever, there are some very important aesthetic reasons for Euclid's method. This should not be so flippantly dismissed as a flaw or mistake.

There is another thereom discussed in this section but it is not clear which one it is. —Preceding unsigned comment added by Mark Wolfe (talkcontribs)

## Parallel postulate

Thee is an error in section "Parallel postulate". At the end of the section it reads:

• ... the "real" space in which we live can be non-Euclidean (for example, around black holes and neutron stars).

While it should say:

If no one objects for a week I'm going o fix it. Jim 07:36, 24 November 2006 (UTC)

Done. Jim 16:19, 13 December 2006 (UTC)

The following sentence is based on four assumptions: “the real space in which we live is non-Euclidean and that's why there is a gravitational field in it.” Assumption One is that there is a real space which can be differentiated from an unreal space. Assumption Two is that space is an object that is located in space. Assumption Three is that space has a convex or concave curvature, instead of being flat. Assumption four is based on the Principle of sufficient reason. It is that the reason why there is a gravitational field in real space is that the space is either convex or concave. Are there any non–authoritative proofs that these assumptions are true or are they simply accepted by convention?Lestrade (talk) 22:03, 6 July 2012 (UTC)Lestrade

## bible publications

please cite the part that refers to it only being second in publications to the bible

## Criticism section should be improved

The criticism section is not adequate. "Criticism" of a lack of rigorism from a book written more than two thousand years ago is extremely pedantic, and I would say inadequate, considering that this level of mathematical and axiomatic rigorism only came about in the 19th century with improvements in mathematics by Cauchy and Weierstrauss. It is about the same as criticizing egyptians for using chariots instead of cars two thousand years ago. In fact what is outstanding is the excess of rigorism in the book. This is not a criticism section, but a commentary one.

## 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 21:51, 10 December 2007 (UTC)

## The very original of The Elements

In what form was The Elements originally? Was it gouged on many stone plates? Or in some other way? As far as I know, printing was only invented in the 15th century in Europe, and papyrus, let alone paper, was unkown to ancient Greeks. RokasT (talk) 20:02, 5 January 2008 (UTC)

## Thales theorem

In the section "contents of the books" it is claimed that book 6 contains Thales theorem. Ain't Thales theorem proposition 31 in book 3? Dafer45 (talk) 17:59, 27 March 2008 (UTC)

Is there any reason to believe that Euclid was referring to the specific "Royal Road", to which this article links, rather than to a metaphorical "special path" to make the learning easier for Ptolemy?Czrisher (talk) 21:00, 8 July 2008 (UTC)

## Editio Princeps

There seems to be an inconsistency. The introduction says that the book was first published in type in 1482. The "Editions" section lists the Editio princeps as being published in 1533. Can someone clarify? Rknasc (talk) 14:44, 29 September 2008 (UTC)

## Euclid's Elements

I am a new user of Wikipedia. I am researching school textbooks based on Euclid's Elements. So not academic works on the text of The Elements but classroom textbooks of School Geometry based on The Elements. I am interested in books from 19th centuary onwards. I have nothing so far and would appreciate some titles. This research idea came out of once reading Lewis Carrol 'Euclid and his modern rivals' and the often qouted fact that "Euclid's Elements were the Geometry textbook for 2000 years" I want to find out who wrote the schoolroom versions of Euclid and, if any are still in print, to study from them.78.147.178.174 (talk) 14:46, 21 November 2008 (UTC)

## Language

Which variant of Greek was it written in? —Preceding unsigned comment added by 92.112.44.252 (talk) 21:15, 22 November 2008 (UTC)

## Transcription and Historicization

I changed the article on Euclid considerably as well.

As I said in the talk page on the Euclid article, These are very essential additions to article, and I hope that people will look at them objectively and use this as an opportunity to make good, unbiased articles. Wikipedia gives us the opportunity to question history and rewrite it. As Orwell put it, "Who controls the present controls the past, who controls the past controls the future." We should look more deeply into Euclid and Elements, as to how it was written and transcribed, and the philosophies it propounded, as well as the social and political reasons behind any transcription.

NittyG (talk) 12:27, 10 January 2009 (UTC)

Wikipedia most certainly does not give us the opportunity to rewrite history. Doing so would be engaging in original research, which is prohibited. It does give us the opportunity to update articles in Wikipedia when reliable sources have rewritten history. Finell (Talk) 01:56, 13 January 2009 (UTC)

### Change "Euclid's Elements" to "Elements"

The book is called Elements, not Euclid's Elements. At most we should mention in the first sentence, "Also known as "Euclid's Elements". I would like to change the article title, but I should wait a while, and there should be a discussion ahead of time.

NittyG (talk) 12:38, 10 January 2009 (UTC)

No. As a title to this article, "Elements" is an insufficient title. This and related articles make clear that the title of the book is Elements. Finell (Talk) 01:46, 13 January 2009 (UTC)
Yes change the name as Elements is not Euclids work. --Tales23 (talk) 20:15, 15 January 2009 (UTC)

### Euclid and Alexandira

The intro tells me that euclid lived and worked in Alexandria, where is the reference to it? Is it the known that Theon and Hypatia of Alexandria wrote and edited Euclids work. But where does it states that euclid was there aswell? (Tales23 (talk) 04:30, 13 January 2009 (UTC))

Encyclopaedia Brittannica says "flourished, c300BC, Alexandria, Egypt"; the OED says "A mathematician of Alexandria who flourished about 300 B.C."; the Encylopedia Americana says, "EUCLID OF ALEXANDRIA, Greek mathematician : fl. about 300 BC, taught geometry at Alexandria in the reign of Ptolemy I (323-285 BC)"; a couple of history of maths books I have agree. Does anyone say that he didn't? Myrvin (talk) 13:51, 12 July 2009 (UTC)

## Images of Euclid Elements

Here are 2 scans of elements and 1 of euclid optics http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html --Tales23 (talk) 00:26, 19 January 2009 (UTC)

## Euclidean geometry as opposed to the Elements

There are separate articles on Euclid's Elements and Euclidean geometry. I've been working on the latter. I've deleted a little material from Elements that was actually about later work in geometry, not the Elements, and that therefore IMO belonged in the geometry article (where it was already discussed more fully).--76.167.77.165 (talk) 23:22, 14 February 2009 (UTC) I've swapped a bunch more material back and forth between the two articles, so that the one is about the math, the other about Euclid's book.--76.167.77.165 (talk) 00:01, 16 February 2009 (UTC)

### Hyperbole

Calling it "the most successful[3][4] and influential[5] textbook ever written" is quite some statement. Success and influence cannot be quantified. Going for "one of the most" would be OK, but that is overused on wikipedia. The Origin of Species and a few others spring to mind as contenders...

## Influence

The influence of Euclid's Elements extended beyond western culture. C.V.Raman, the Indian Physicist, described it the most remarkable works of all time. He added, " ... The pages of Euclid are like the opening bars of the music of the grand opera of Nature's great drama. So to say, they lift the veil and show to our vision a glimpse of a vast world of natural knowledge awaiting study." 89.241.245.58 (talk) 20:16, 29 April 2010 (UTC)

## missing axioms

I'm planning to put back some of this material or at least a pointer to it, plus some later logical analysis that I thought I remembered from the article (if I can find it). It's pretty important to the topic since the search for a proof of the fifth postulate led to the development of non-Euclidean geometry. 67.117.130.143 (talk) 18:38, 10 December 2010 (UTC)

For possible use:
• http://halshs.archives-ouvertes.fr/docs/00/46/57/80/PDF/Hilbert_on_Diagrams.pdf
• Avigad. J., Dean, E. and Mumma, J., (2009), A Formal System for Euclid’s Elements, The Review of Symbolic Logic, 2, 3. [from above] LTU discussion
• Manders, K., (2008b), The Euclidean Diagram, in (Mancosu, 2008, p. 112-183). [likewise]
• Mancosu, P., (2008), The Philosophy of Mathematical Practice, ed., Oxford University Press. [again].
• E. T. Dean, "In defense of Euclidean proof"[1]
• J. Mumma. Proofs, pictures, and euclid. Synthese, 175:255–287, 2010 [from here] See also: [2]
67.117.130.143 (talk) 05:33, 17 December 2010 (UTC)

## Criticism

Some years ago this article had contained a section "Criticism", which was removed mainly because of absence of citations, as far as I understand. Besides in another section of this discussion a user had written that the word "criticism" is very inappropriate in this case. OK, let it be named not "Criticism", but for example "Defects"; and let it be a subsection of the section "History". But, as for me, it is important for such a (sub)section to exist in this article. Without it this article does not explain why now the Elements are not used as an important textbook. Noticing defects of the Elements was an important step in the history of mathematics and, of course, in the history of usage of the treatise. I don't want to write the section myself because I don't like to look for proper citations; I hope that somebody else will agree with me and will write the section. --D.M. from Ukraine (talk) 22:57, 17 March 2012 (UTC)

I added some of the old material which is covered by Heath's commentaries. The section needs to be expanded on more missing axioms and suggested new axioms. Also please change the name to something more appropiate than Criticism as I couldn't come up with anything.
213.103.210.34 (talk) 22:55, 24 June 2012 (UTC)
Axiom 4 is considered to be a justification for the method of superposition
213.103.210.34 (talk) 14:54, 6 July 2012 (UTC)

Would a reference to Schopenhauer's criticism of the proofs of the parallel postulate be socially aceptable? Schopenhauer was quite the intellectual pariah.Lestrade (talk) 15:15, 6 July 2012 (UTC)Lestrade

I am not sure this would be appropriate for this article as it is, as I understand it from the wikipedia page, philosophical in nature rather than mathematical.
Mm32pc (talk) 10:39, 9 July 2012 (UTC)

## Note

In the article on Adelard of Bath, it is said that he made the first full translation. This implies that there was an earlier, partial one.

## An argumentative insert bordering on vandalism

The below two sentences are opaque, argumentative and well below the overall high quality of this Wikipedia article.

However, Proclus' papyrus manuscript has been copied to a paper which became available in Europe after 800 years. Therefore, the originality of Proclus' work might have been altered since it was copied by hand.[citation needed]

First of all, a mere technical point... Unlike the displayed certainty of this above insert, we don't actually know on what kind of writing material Proclus wrote down his commentary. It's no doubt very likely that it was papyrus but it's at least conceivable that it could also have been parchment. The bottom line is that we don't know and even more importantly it doesn't matter one iota, so why mention it? Obviously, this was only a way to introduce the argumentative insert/edit quoted above.

So, to the second and main point... Yes indeed, we don't have Proclus' actual original handwritten manuscript. What a surprise - NOT! What's the point of mentioning this when we don't have the original hand writings of a single ancient author (except for maybe the carbonized hand writings of Philodemus in Herculaneum). Anyone with even the most rudimentary understanding of manuscripts and textual transmission throughout the ages would have taken this for granted instead of wasting digital ink on underscoring the obvious.

So, why does he mention this at all? Well, the editor in question makes his intent clear in the next sentence where he puts into question the veracity of Proclus' writings on Euclid. This is a very dubious style of editing, clearly not a neutral point of view by any standard and IMHO it borders on vandalism. The veracity of Proclus' writings cannot be put into question merely on the basis of it being copied throughout the Middle Ages in monastic scriptoria. That's how not only to Proclus' writings but all ancient literature has come down to us, including Euclid's Elements itself. Therefore, this insert should be erased. Currently, it has a "citation needed" tag, but had the editor based his intervention on something more substantial than merely discovering the fact that Proclus' writings like all manuscripts has a textual tradition, then he surely would have written something more serious to begin with. It should be up to him to put in the effort to convince the rest of us that he should be taken seriously, there's no need to leave this kindergarten-level edit/insert for months in the introductory section of the article.

Abvgd (talk) 09:53, 4 January 2014 (UTC)

## error since 2008 in apocrypha section?

Away back in 2008 [https://en.wikipedia.org/w/index.php?title=Euclid%27s_Elements&diff=229786827&oldid=227836296 this edit was made:

• from: the ratio being $\sqrt{\tfrac{10}{3(5-\sqrt{5})}}$
• to: the ratio being $\sqrt{\tfrac{10}{3(5-\sqrt{5})}} = \sqrt{\tfrac{5+\sqrt{5}}{6}}$

This is not an equation (the two sides are not equal)... so I essentially reverted the 2008 edit.Jytdog (talk) 18:05, 11 June 2014 (UTC)

Why do you believe that the two expressions are not equal? Their equivalence is an elementary exercise in rationalizing a denominator. Bill Cherowitzo (talk) 00:05, 12 June 2014 (UTC)
duh, sorry (facepalm) Was going too fast earlier today. thanks for correcting. Jytdog (talk) 00:29, 12 June 2014 (UTC)