Talk:Euclid's Elements
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[edit] Initial comments
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)
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 (talk • contribs)
- 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 (talk • contribs)
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- 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
[edit] 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 (talk • contribs)
- 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)
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- 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 ___ |
___ |
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- 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)
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- 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)
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- O.K.... Brutannica 04:15, 11 Sep 2004 (UTC)
[edit] 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)
[edit] Why this page should not be redirected or merged
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)
[edit] 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
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- Perhaps Wikibooks or Wikisource would be a better place to put this. - dcljr (talk) 23:10, 23 Jun 2005 (UTC)
[edit] 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)
[edit] 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)
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- 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
[edit] Calling Euclid adequate
To call the Elements adequate is just plain not doing the book justice. —Preceding unsigned comment added by 68.77.28.102 (talk • contribs)
[edit] 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 (talk • contribs)
[edit] 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:
- ... the real space in which we live is non-Euclidean and that's why there is a gravitational field in it.
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)
[edit] bible publications
please cite the part that refers to it only being second in publications to the bible
[edit] 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.
[edit] 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)
[edit] 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)
[edit] 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)
