Talk:Euclid's Elements

I think it is a significant flaw that the lead does NOT explain WHAT it is that Elements does. Consider this: "Euclid (fl. 300 BCE) placed at the head of his Elements a series of ‘definitions’ (e.g., “A point is that which has no part”) and ‘common notions’ (e.g., “If equals be added to equals, the sums are equal”), and five ‘requests’. Supposedly these items conveyed all of the information needed for inferring the theorems and solving the problems of geometry, but as a matter of fact they do not. However, the requests (aitemata)—usually called ‘postulates’ in English—must at any rate be granted or Euclid's proofs will not go through."*

That quote seems to me to enormously improve understanding of what the Elements are so well-known for, and yet you'd be hard-pressed to show that there's any indication of the importance of the definitions, common notions, and requests in the lead as a basis for a rigorous system of mathematical or logical deduction which influenced so many. I also note that there's a subtle implication of "request" not present with the words 'postulate' and 'axiom'. (that is, the possibility of denying the request; which, it could be argued ,foreshadows non-Euclidean maths.) So I wonder if the more unusual, yet more faithful, translation is worthy of note?

• excerpted from (Torretti, Roberto, "Nineteenth Century Geometry", The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2021/entries/geometry-19th/>. ) 174.131.48.89 (talk) 00:55, 8 August 2022 (UTC)

Under the editions section does anyone know what the translation subheading means? Does it mean anything other than Greek and that the section preceding it is meant to only include Greek texts? Rhodydog (talk) 17:47, 25 November 2023 (UTC)

The part outside of "translations" are Greek versions. I think I misunderstood the intention here when I added Theon of Alexandria's version a bit ago; I believe the original intention was to only include printed publications (not manuscripts).

Personally I think this section should be split into a separate article describing both manuscripts, print editions, and translations (titled something like List of editions of Euclid's Elements or Transmission and publication of Euclid's Elements), with a much more selective list kept at this page; the current huge list is kind of distracting, and is still quite incomplete so can only get more distracting if it grows. Maybe we could take inspiration from Bible § Versions and translations which splits off long lists into (among other pages) List of English Bible translations. –jacobolus (t) 18:50, 25 November 2023 (UTC)

I agree, a separate article for the editions would be useful so that the main article can focus on the elements itself, its content, history etc. The most historically important editions, could be mentioned in the main article if need be with a brief commentary but more detail kept in the editions article. I see what you mean by modeling the structure after the bible: versions and translations. I'm ok with this proposed change Rhodydog (talk) 04:08, 29 November 2023 (UTC)

I pared down the list of editions by quite a bit, for English there are quite a lot, but I think Heath's is largely still the "standard" english translation and it's passed well into the public domain so we probably don't need many more unless they're doing something interesting like Byrne. Probably we could make the old removed list into a separate article, but maybe it should go into user/draft space first is someone wants to work on it? ideally it would be in a table format, I think trying to scan a long chronological bulleted list for language is rather difficult. Most translations probably aren't notable enough to mention in the prose of the article, I think probably Arabic and Latin are the most important ones to discuss for any Greek text, as they provide good lead-ins to discussions of who was reading the text throughout history, and perhaps the Chinese translation of Euclid should be discussed as well for the same reasons.

Manuscripts are probably similar, our current coverage could be expanded a bit in this article we probably want to talk about the various redactions by Theon and other later commentators in more detail, and a bit about the history of the text/the famous Arethas manuscripts, but the stemma is pretty complex and there are hundreds of manuscripts so we probably would need to cover most of it in a separate article if we wanted to be exhaustive. Psychastes (talk) 16:35, 3 May 2025 (UTC)

The manuscript history seems pretty daunting for any non-expert to clearly summarize. Maybe if someone can find a clear survey paper it would be possible to crib from that. –jacobolus (t) 20:18, 3 May 2025 (UTC)

I've removed the term "error" from the Criticism section because I believe it is misleading based on the content of the section as it stands now. If we add any discussion of actual errors in reasoning or propositions that are used before they're proved (which do exist in the extant text) then we can certainly use "error" for that, but criticisms of the form "Euclid forgot to officially declare that a number is less than or equal to itself" is not the sort of thing most people who are not formalist mathematicians would consider an error. Psychastes (talk) 17:41, 3 May 2025 (UTC)

@Psychastes, can you please stop converting paragraphs to bullet lists, and ideally convert the lists you made back to paragraphs so I don't have to try to do it? The list format is generally inferior where prose can be used instead, including the cases you just converted. (Cf. MOS:PROSE.) –jacobolus (t) 20:08, 3 May 2025 (UTC)

yeah, sure thing, good point; I'll change it back now Psychastes (talk) 20:10, 3 May 2025 (UTC)

@David Eppstein recently changed the § References section from the current columns + reverse-indent style (for the past 6 years, originally set by @Waynejayes) to use a no columns + bullet list style instead, under the edit summary "repeat sentence from lead later and source; convert more sources to consistent sfn style". I reverted the part of this edit which changed the reference list style based on: (1) the change seemed unexplained, unmentioned in the edit summary, which only described apparently unrelated changes; (2) there wasn't (to me) an obvious reason for the change, since both styles seem acceptable, and the previous one had been there for years with no apparent objection; and (3) I personally prefer the use of columns for saving some amount of vertical space and making the information density a bit higher.

In general my preference is to leave stable style choices alone (those that have been some particular way for several years) just to avoid style change for change's sake or to suit one or another person's preference. However, I don't really feel strongly about whether a reverse-indent style or bullet style is used for lists like this, so I would be happy to live with whatever the consensus of other editors is. I would argue for keeping the columns though, even if we switch to bullets. [...] –jacobolus (t) 21:31, 25 July 2025 (UTC)

I don't care about this specific formatting issue. I was merely cleaning up some other cruft in the referencing templates that we didn't need to have any more (specifically, hardcoded columnization) and happened not to recognize this parameter as non-cruft. If someone thinks strongly that indented lists are preferable to bulleted lists for this style of referencing with short footnotes pointing to a longer bibliography, or that bulleted lists are preferable to indented lists, then we can discuss that, but either way is ok with me.

[...]

So anyway I don't disagree with your decision to reverse this change, in isolation, [...] —David Eppstein (talk) 22:57, 25 July 2025 (UTC)

Moved to user talk:jacobolus § Meta-editing discussion between jacobolus and David Eppstein

I moved one line of my original comment, parts of David's comment (indicated by "[...]") and my entire reply to my talk page, which seems like a more appropriate venue. Hopefully that's okay. –jacobolus (t) 05:51, 28 July 2025 (UTC)

It's quite confusing to have footnotes followed by a section about editions of the book. This section should probably ideally be rewritten in prose, and if we're going to have a bullet list it could be moved out of the article. But in any event, to me it seems clearly to be part of the content of the article, rather than an appendix.

In particular, it's confusing to have short footnotes citing commentary from e.g. Heath's edition with the reference pointing to the "editions" section, and other short footnotes citing other sources pointing to the separate "references" section. If anyone ever tries to follow the footnotes on a printed copy of the article, they're going to be left quite confused indeed.

I think it would be better to just duplicate any reference that we want to cite from a shortened footnote into the references section, if necessary. There won't be more than a few of these, and the duplication is not a serious problem. That would also make it easier to e.g. cite the 2nd edition of Heath's book, for which the internet archive has an excellent scan and which is presumably corrected / more authoritative, while still describing the first edition in the list of editions, since describing the first edition is important article content. The useful features of an edition to discuss in the "editions" section, which might include some editorial commentary about each edition, etc., are different than the appropriate material to include about references, which should be focused on stuff like ISBNs and DOIs to help readers verify claims.

@Psychastes made this change to the order in special:diff/1288529675 back in May. Paring the list down was an improvement for this article (I think a list belongs in a separate page, along the lines of List of English Bible translations, where it might be fine to include more entries if anyone wants), but re-ordering the page to put the section out of the content area was in my opinion a dis-improvement.

Can we move that section back? –jacobolus (t) 02:00, 27 July 2025 (UTC)

Nominator: David Eppstein (talk · contribs) 01:52, 19 September 2025 (UTC)

GA review

This review is transcluded from Talk:Euclid's Elements/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: DoctorWhoFan91 (talk · contribs) 18:59, 19 September 2025 (UTC)

Will add initial remarks soon- article looks really good though. HSLover/DWF (talk) 18:59, 19 September 2025 (UTC)

Reception

• What is an "ostraca"?
  •  Done. It's the plural of ostracon, a pottery fragment on which someone scratched writing. Here, it seems to be notes for the construction of a building maybe?
• Johannes de Tinemue,[65] possibly also known as John of Tynemouth - what?
  •  Done I just wrote it as John of Tynemouth. I'm not sure we need to be so careful to clarify that we don't know whether it was really John of Tynemouth or someone else also called John of Tynemouth. I saw a joke like this once about those crank theories about Shakespeare: did you know that the plays of William Shakespeare were not actually written by William Shakespeare, but by someone else, also called William Shakespeare? —David Eppstein (talk) 02:48, 21 September 2025 (UTC)
• The Elements is still considered a masterpiece in the application of logic to mathematics.- Again, where are the citations?
  • "Again"? Anyway, I simply removed the entire paragraph containing this WP:PEACOCK and largely information-free claim. The rest of the paragraph was misplaced and redundant with later material in the criticisms section. —David Eppstein (talk) 08:03, 21 September 2025 (UTC)

  "Again"- because there are a bunch of places, most/all of which are mentioned in the review which mentions, with only one/none citations for this. HSLover/DWF (talk) 08:55, 21 September 2025 (UTC)

• "that two circles with centers at the distance of their radius will intersect in two points."- what does this mean?
  • It means that if you draw File:Byrne 35 diagram 1.png (two circles having the same line segment as their radius, the vesica piscis, the figure James Joyce was so worked up about) then the two circles will actually cross each other instead of somehow mysteriously not crossing. It's obvious but even obvious things require a proof or an assumption. Probably it would help to add an illustration here; I just have to find the right one. —David Eppstein (talk) 02:48, 21 September 2025 (UTC)
• "degenerate cases."- small inline explanation
  • Significantly reworded here, I hope more clearly. —David Eppstein (talk) 08:03, 21 September 2025 (UTC)
• "first-order theory"-same
  • Removed instead, to tighten this section and because it's more of an advanced topic than readers need to worry about here. First-order basically means that the variables are only the things themselves and not sets of things, which is subsumed in the point about this system only using points as variables. —David Eppstein (talk) 08:03, 21 September 2025 (UTC)
• " certain important concepts such as the cross ratio."-what is the cross ratio, why is it important?
  • Add a parenthesis for why it is important. —David Eppstein (talk) 08:03, 21 September 2025 (UTC)
• Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."[47]- Make that less wikivoice
  •  Done —David Eppstein (talk) 20:49, 21 September 2025 (UTC)

Additionally, wherever the article says the Elements is great, use multiple citations(would be easy, half these citations probably say the Elements is great), not one- yes, the Elements is great, but we need to cite it. Will do the source check later when the changes have been made, @David Eppstein:. HSLover/DWF (talk) 12:13, 20 September 2025 (UTC)

•  Done. At least, I don't think there are any vacuous "Elements is great" statements, but I double-cited the comparatives at the start of the methodology, reception, and non-Euclidean geometry sections. I am about to leave for some travels so I may not be very responsive for the next few days. —David Eppstein (talk) 20:49, 21 September 2025 (UTC)

  PS the article also has improved sourcing in Books X and XII, and new paragraphs in Book II, at the start of "medieval", at the end of "renaissance", and at the start of "axiomatics" (renamed and refocused from "criticism"). —David Eppstein (talk) 00:12, 22 September 2025 (UTC)

Checked the added text-fine too. Doing a spot check of 15 refs.

• [1]:checks out
• [11]:checks out, says it's mathematical in general, and not just geometrical
• [20]:checks out, mentions and names them
• [31]: checks out, describes book III
• [41]: checks out
• [53]:checks out
• [61]: checks out (almost to the point of being verbatim)
• [72]: checks out, describes the various manuscripts
• [82]: checks out
• [89]: checks out
• [100]: checks out
• [111]: checks out, the author also says this
• [121]: checks out, gives the example and a lot of other things too
• [129]: checks out, matches description
• [135]: checks out, description correct

Spot check shows everything good, passing article, great work, @David Eppstein:. HSLover/DWF (talk) 18:22, 22 September 2025 (UTC)

Thanks! The near-verbatim source is a little worrisome, though; I found and removed some other old close paraphrasing but maybe I missed some. Will check. —David Eppstein (talk) 13:26, 23 September 2025 (UTC)

Yeah, kinda an issue, but it's just a rather technical line, so there isn't much change one can do to it, so I just let it be. HSLover/DWF (talk) 17:42, 23 September 2025 (UTC)

The source is PD but we do still need to either mark things as quotes or use our own words. Anyway, reworded a little further away. —David Eppstein (talk) 18:49, 23 September 2025 (UTC)

GA review (see here for what the criteria are, and here for what they are not)
It is reasonably well written. a (prose, spelling, and grammar): b (MoS for lead, layout, word choice, fiction, and lists): It is factually accurate and verifiable, as shown by a source spot-check. a (references): {{GAList/check|}y} b (citations to reliable sources): c (OR): d (copyvio and plagiarism): It is broad in its coverage. a (major aspects): b (focused): It follows the neutral point of view policy. Fair representation without bias: It is stable. No edit wars, etc.: 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):
Overall: Pass/Fail:
· · ·

David Eppstein, thanks for making a lot of excellent progress on this article recently. (And thanks to Psychastes and others for earlier efforts.) The basic style now looks more like a Wikipedia article, e.g. content is now mostly sourced.

The article still has significant issues in my opinion, warranting substantial further work. We should ideally hope for something that could pass a review from a professional mathematical historian, and I don't think we're there yet. I don't think the article currently gives a complete or neutral presentation of the content, context, influence, or ongoing scholarly (historical or mathematical) discussion about the book. Some of the summary currently here seems anachronistic, and the broad effect is, I think, somewhat misleading. (To be clear: these are not recent problems with this or related articles, they are not easy problems to tackle, and there has been recent improvement.) The most basic problem, I think, is that modern readers aren't given enough essential context to make sense of a lot of the article.

Some specific criticisms:

Numbers: this article talks about numbers several times, including mentioning "a binary operation from numbers to numbers", "irrational numbers", "number theory", "square numbers", etc., and implicitly discussing numbers in e.g. "multiplication was treated geometrically". But nowhere do we explain that the word "number" meant only counting numbers (1, 2, 3, ...); that geometrical "multiplication" was not multiplication per se, but rather the construction of a rectangle with two given sides (which were not taken to be "numbers"), along with some relevant geometrical facts such as that two rectangles with the same height and different bases had areas proportional to the bases; that the "irrational numbers" involved were also not considered "numbers" but rather (what we now call) line segments, compared by superposing one on the other and seeing which one extended beyond the other.

Magnitudes, ratio, incommensurability, etc.: this article mentions "magnitudes" several times:

• "The common notions exclusively concern the comparison of magnitudes",
• "Book V, which is independent of the previous four books, concerns ratios of magnitudes and the comparison of ratios."
• "Of the Elements, book X is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes."
• "Christopher Zeeman has argued that Book V's focus on the behavior of ratios under the addition of magnitudes, and its consequent failure to define ratios of ratios, ..."

And mentions commensurability a few times, first in the lead and also:

• "X (on incommensurability)"
• "Book X proves the irrationality of the square roots of non-square integers such as ⁠${\displaystyle {\sqrt {2}}}$⁠ and classifies the square roots of incommensurable lines into thirteen disjoint categories."

But if someone doesn't know what a "magnitude" means in this context, all of these descriptions are incomprehensible. Even worse, if a reader clicks through on the wikilink to Ratio, they are presented with a lead section that is confusing and misleading ("a ratio shows how many times one number contains another") and a later section about Euclid's concept of ratio that is incoherent / wrong, with no mention of magnitude anywhere in the article. If they instead click through to Commensurability (mathematics), they are presented with a lead discussing only a different (modern) topic presented without sources, and then later with a description of Euclid's concept which is misleading and confusing.

In my opinion it is absolutely essential to explicitly explain, in this article, what Euclid's concepts are of: magnitude, ratio, proportion, "commensurable", "rational", etc., along with an explicit description of how these concepts differ from related modern concepts. Our best current coverage of this topic seems to be at Irrational number § Ancient Greece, but it's still pretty limited. Ideally we'd have a whole separate article dedicated to this topic (and/or an article about specifically Book V of the Elements), since there are probably thousands of references discussing it over centuries, and it is of ongoing scholarly interest by historians and mathematicians. Since there is no such article, the burden on this article, itself, is much steeper, if we don't want to leave readers grossly misled.

Geometric definitions, postulates, common notions: The definitions of various objects such as points, lines, angles, circles, parallel lines, etc. found in the Elements has been a persistent topic of discussion ever since. We don't mention any of these definitions or discuss the topic at all. Similar to the previous point, ideally we'd have a whole article about Book I and/or about Euclid's definitions per se, but since we don't, there is a burden on this article to cover the topic. One of the few bits of concrete content we do include in this article is a list of postulates and common notions (in a floating table), but as-is this is not really accessible, since these definitions are confusing and mysterious to non-epxerts, and we don't make any effort to unpack them (e.g. we don't explain what it means "to produce a finite straight line continuously in a straight line", something especially likely to confuse if someone doesn't know that for Euclid a "line" can be curved, and, whether straight or curved, is always finite).

In particular it seems to me essential to discuss the difference between points, straight lines, circles, etc. as physical objects or pictures vs. as idealized mathematical concepts, and put some amount of discussion into the reasons that step by step constructions and proofs are useful formality.

We should explicitly describe Euclid's concept of "equality" of lines, surfaces, etc., and explain how they relate to modern concepts like numerical measures of length and area, or congruence. We should discuss the more modern concept of infinite lines and its relation to Euclid's concept. We should mention how Euclid's concept of angles differs from later trigonometrical ones.

Geometric constructions: All we say about this topic currently is "Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge. His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition." We don't explain what it means to "construct" with compass and straightedge (e.g. how that differs from just drawing a picture), explain what kinds of figures are so constructible, etc. We don't talk about how compass and straightedge constructions might differ from other kinds of constructions.

"Deductive" mathematics: We state without explanation that "Euclid's Elements is the oldest extant large-scale deductive treatment of mathematics", and several times call it an "axiomatic system", but nowhere do we explain what "deductive mathematics" means, what an "axiomatic system" is (and the wikilinked Axiomatic system frankly doesn't help much), or explain how this differs from other kinds of mathematics. We don't define any of the words axiom, proof, deduction, etc.

Analysis and synthesis: Nowhere to we mention the concepts of geometric analysis and synthesis, which at least some authors have considered to be a basic feature of Greek mathematics as found in the Elements.

Various sets of axioms: Related to the previous point, one of the aspects I personally find impressive about the first few books of the Elements is that they take some care to not to use more assumptions than necessary to prove each statement. So a number of propositions only rely on affine relationships between parallelograms and polygons, and are fairly cleanly separated from anything requiring a concept of circles or right angles. See below my comments on "Post-Euclidean mathematics".

The Data. The Data is a companion book to the Elements, written by the same author and often used together by later scholars, but our current article only mentions it briefly in passing: "No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach the types of problems encountered in the first four books of the Elements." Our linked article about the Data was created as a "weak stub" in 2006, and hasn't really evolved since. I think it's worth more explicitly describing the relation between these books here (as well as at that article).

Images/diagrams. The Elements included a diagram with every one of its numerous propositions, and we currently include only ~2 of them, neither of which is clearly presented or explained at all. We make the briefest mention that propositions are "presented alongside mathematical proofs and diagrams", but we don't make it clear that, in the Elements, there is always a diagram with every proposition. We don't discuss the topic of diagrams or diagrammatic reasoning, let alone analyze any specific diagrams. One noteworthy topic, for example, is the way that diagrams included in manuscripts/published editions of the Elements have changed over time.

We mention that "Hippocrates of Chios [...] may have originated the use of letters to refer to figures" (who says, on what basis?) We also briefly state that "Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters." We don't give even a single example or any description of what a figure labeled by letters looks like. We don't provide any discussion of the features of this labeling system, or its relation to later approaches.

Form of propositions. Our current article gives very limited concept of what propositions in the Elements actually look like. We describe them abstractly at a high level in § Euclid's method and style of presentation, but we don't give a single example proposition and its proof, so readers are left mystified unless they either already know the content or go read about it elsewhere.

The purpose of the book: Our article doesn't currently explain how the Elements was used. It says a lot about how it is influential, etc., but we don't discuss what its role is in the context of other Greek mathematics, let alone later mathematics. It would be nice if we would give some explicit examples of contemporaneous or later works, describing how they referenced results from the Elements, how they built on its results, whether they ever substituted alternative sources, etc.

Relation to other cultures: Greek mathematics is commonly distinguished from that of Egyptians, Babylonians, or other ancient civilizations. Euclid's elements is commonly contrasted with contemporary or later Hellenistic work in a practical mathematics tradition, e.g. that of Heron. What was unique or novel about the approach exemplified by the Elements, and how did it relate to earlier or alternative contemporary approaches?

Relation to Greek society and other fields: How did Greek mathematics like the Elements relate to other parts of Greek society and technology. For example, much of the "pre-Euclidean" mathematics came out of work by philosophers or their disciples: what is the relation between Euclid-style mathematics and the philosophy of the Pythagoreans, Zeno, Plato, Aristotle, etc.? The tools of geometric construction – the compass and straightedge – come from practical trades like masonry and metalwork: what is the relation between mathematics and those trades? A significant part of the motivation of some sections of the Elements (as well as much of the later motivation for studying it) has been speculated to be in service of astronomy, but we don't even mention the word astronomy.

Apollonius, Archimedes, et al.: We don't currently mention the topic of conic sections or Apollonius' treatise, and make no mention of Archimedes. Both of these mathematicians, often considered the two greatest of their time, were rough contemporaries of Euclid. How did their mathematics relate to Euclid's? Did they make use of the Elements? We also don't discuss Euclid's own other works. We don't mention Pappus of Alexandria who commented on the Elements. We mention Proclus, and even quote his historical remark, but we don't otherwise describe any of his commentary.

Use as a textbook: We state that the Elements was the "most successful textbook ever written", that it "may have been based on an earlier textbook", that "It is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit", that it "was the dominant mathematical textbook in the Medieval Islamic world and Western Europe". But we don't give any indication about how it was taught, who taught it, to what kind of students, what purpose they used it for, how it might have fit into a school program alongside other textbooks / works, what kind of competitors it might have had as a textbook at various times and places, when and whether it was relevant to researchers as well as students, etc. If there are questions that we just don't know about (e.g. how someone would have taught from this), we should perhaps explicitly say so.

Physical format of the book: We mention that the Elements was originally written on papyrus. To be explicit, it was a scroll. We don't mention, but later it was usually bound as a codex. It might be worth giving some explicit discussion about the original and later physical format of the book, which may give some insight into its design.

Influence: We state that the Elements "exerted a great deal of influence", that its "axiomatic approach and constructive methods were widely influential", that "in historical context, it has proven enormously influential in many areas of science", and that its "geometrical system [...] long dominated the field". But the only evidence given of this influence is that there were many translations and editions, along with a brief mention that there were "scholia, or annotations to the text [...] gradually accumulated over time as opinions varied [...]". We don't make any effort to describe the way the Elements was taught, referred to, built upon, and we don't give any examples or discussion about the content of commentary.

Medieval Islam: We currently have one short paragraph about 3 translations into Arabic from ~800–900. (A recent improvement over the previous 1–2 sentences that mentioned 1 translation.) We don't say anything at all about the content of those translations, commentaries in Arabic, influences on other mathematical or scientific from the Islamic golden age, etc., but implicitly treat these translations only as a stepping stone to translations into Latin. This seems Eurocentric and misleading. (Unfortunately there's a real gap in scholarship about medieval Islamic mathematics including translations and commentaries on the Elements, but I think there's enough available that we can say more than we do, and perhaps even make some explicit comment about the gaps in scholarship.)

Post-Euclidean mathematics: We imply that Euclid's mathematics differed from modern mathematics in various ways, but we don't really discuss anything about the history of that transition. For example, we don't make any mention of analytic geometry (let alone anything later) and how it is methodologically different from Euclid's mathematics. We do briefly discuss the parallel postulate and non-Euclidean geometry, but we don't really unpack the way different choices of axioms leads to different systems – I think we could do much better to talk about the various alternative kinds of geometry that can result from alternative axioms.

We also don't give any discussion here of the broadening concept of "number", the change in the concept of ratio, the change from treating points and compass/straightedge as fundamental to basing geometry on sets of numbers etc.

Summary of the text: Our summary of the text has improved recently – thanks David and others – but is still fairly bare bones. In "summary style", the content currently here would be closer to sufficient if we also had a separate article about each of the books (or perhaps each few books, grouped together) with more detail. As a stand-alone summary of the content, though, I am not sure it will satisfy readers. It's more like a very high level table of contents than a summary, mainly mentioning a list of things without explaining what they are, why they matter, how they relate to each-other, how they related to later works, etc.

Textual interpolations and changes: Our article currently doesn't describe the long history of textual changes to the Elements, beyond the single sentence "Later editors such as Theon often interpolated their own proofs of these cases" and the mention in § Apocryphal books that books XIV and XV were added later. This doesn't even begin to cover this topic, about which quite a lot has been written. More generally, our article doesn't really address the topic of authorship and to what extent we know (or don't know) when and how various parts of the text originated. It would be helpful to get concrete here.

Criticism: Our section about criticism is nowhere close to complete or neutral. This is a difficult topic to cover, because there has been so much criticism of such a wide variety that it's hard to know where to start or how to organize it, but our current text seems like a substantially misleading summary of that corpus. The mentions we make of Beeson et al. and Zeeman seem like "undue" emphasis on arbitrarily chosen recent examples in the context of an otherwise extremely limited discussion.

(I could probably make quite a few more topics and say more about these, but I have to run and don't want to delay.)

All the best, –jacobolus (t) 03:47, 20 September 2025 (UTC)

I just wanted to say this is all FA criteria level, and not GA. HSLover/DWF (talk) 06:47, 20 September 2025 (UTC)

I am with Jacobolus in not really understanding FA criteria, at least (GA I think I get). But part of the trouble here (especially re the distinction between types of numbers) is the tension between explaining content in understandable modern terms vs sticking with our attempted understanding of what the ancient point of view was. But also, this is a topic on which the literature is huge (there are entire books on Euclid's Elements that we are not referencing although for me at least in the case of Berlinski that is a deliberate choice because dubiously reliable) so the difficulty is less making sure we cover all that has been said and more choosing what to omit.

In criticism, Beeson and Zeeman were chosen not because they are representative, but because they are distinctive. The literature goes on and on and on about the omitted topological continuity lemmas in Prop. I (which we should and do mention) but Beeson was the source where I found the different information about problems with degenerate cases so that's what I cited for that point. Zeeman similarly: he made a different point than the bulk of the literature and I thought it might be worth including. I do happen to think the computationally verified proofs of Beeson are very important, more than the quibbles about the rigor of Euclid's proofs. —David Eppstein (talk) 07:12, 20 September 2025 (UTC)

Computer-checked proofs of propositions from the Elements seems like an interesting topic; it just doesn't seem supportable for it to dominate a section here titled "criticism" in a section about "modern" mathematics presumably spanning the past ~200 years; I'd consider trimming it down and re-organizing. For instance, it could go in section about computers, which could also discuss the use of interactive geometry software. Or maybe there's some other way of slicing topics up.

Zeeman's paper has been cited 4 times. It's an interesting enough point, but again, the implication of making a paragraph about it here is that this is a significant part of the past couple centuries scholarly commentary on Euclid. –jacobolus (t) 07:47, 20 September 2025 (UTC)

Moving Zeeman's paper to the section about Book V is even more undue weight in my opinion, especially in its current state. The article now implies that Zeeman's paper is essential to the textual summary. I'll try to write a somewhat more complete summary of Book V when I get a chance. The content of Zeeman's paper, per se, would probably fit better in a dedicated article about that topic than here (unless we plan to dramatically expand the scope of this article). –jacobolus (t) 06:28, 24 September 2025 (UTC)

Moving this material to the section about Book V is merely a statement that it is much more relevant to the section about Book V than the section about axiomaties, and sufficiently WP:DUE to mention somewhere. The implication you think you see is not in the text. As for a significant expansion of the book V section, which is currently one of the longer ones: please do pay attention to WP:BALANCE and to WP:GACR 3b. It might make more sense for you to make a separate and standalone article about the Eudoxian theory of proportion, focused more on the content of that theory and less on its specific presentation within the Elements. —David Eppstein (talk) 06:34, 24 September 2025 (UTC)

I don't think it's "DUE to mention somewhere" unless we actually cover the aspects of the topic discussed in typical secondary sources, which I think we currently are skimping on. In the current version it seems like we're just cherrypicking a favorite recent paper which has not been broadly cited and is not mentioned in secondary literature. (Edit: In case it isn't clear: I don't have anything against Zeeman's paper, and I think the paper makes an interesting point.) –jacobolus (t) 06:38, 24 September 2025 (UTC)

I have explained my reason for including it above. "Favorite" is both inaccurate and unduly ad-hominem. It is merely a source that popped up in Google Scholar when I searched for references about Book V specifically. —David Eppstein (talk) 15:32, 24 September 2025 (UTC)

You're misunderstanding me and not addressing the point. I'm not saying that you are trying to promote the paper, but rather that you are substituting your own judgment for the published surveys of the best reliable sources. A strategy of «run a search, skim looking for distinctive points of view, and then relay those while skipping a summary of the bulk of the literature» does a disservice to readers by giving a misleading impression of the topic and what has been written about it. To quote Wikipedia's policy again, "Neutrality requires that mainspace articles and pages fairly represent all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint in those sources." This authoring strategy is almost the opposite of that policy; instead what we're doing is representing viewpoints in proportion to how interesting particular Wikipedians thought they seemed. –jacobolus (t) 16:15, 24 September 2025 (UTC)

Every Wikipedia editor uses judgement in selecting sources. You are using your own judgement in judging this one unworthy and in suggesting instead a paper whose Elements-specific content concerns minor points of the wording of translations of proposition 2. Google Scholar lists some 130k sources that mention Euclid's Elements and doubtless misses some. I don't know what model you have of Wikipedia editing that makes it possible to avoid judgement in selecting sources for a topic like that, nor what model you have for what you think you are doing in this discussion that somehow avoids yourself making judgements. —David Eppstein (talk) 17:33, 24 September 2025 (UTC)

You are, again, misunderstanding me (now on multiple levels). (1) I wasn't "suggesting instead" anything, only pointing to a couple papers you might enjoy reading; I don't think these particular papers are any kind of priority for coverage here. More importantly (2) You are exactly right, there are tens to hundreds of thousands of sources. There are even thousands of sources specifically discussing Book V, perhaps dozens to hundreds of which are focused exclusively on that topic. We should pick some well regarded survey sources about the topic written by professional historians, and our coverage should reflect the topics and viewpoints described in those sources, in rough proportion to the attention given to them, in accordance with fundamental Wikipedia policy. Obviously it is impossible to characterize or summarizing that material without any editorial judgment, but the goal should be to strive for neutral and accurate presentation of that literature. What we shouldn't do is skim through a handful of whatever popped up on Google scholar, and describe a few of those that catch our eye, irrespective of their impact on the literature, ignoring Wikipedia policy. –jacobolus (t) 18:06, 24 September 2025 (UTC)

You are free to use whatever process you judge likely to work for yourself of how to put together material for a Wikipedia article. Mine is something closer to finding sources that present a diversity of ideas, filter out the ones that appear to be fringy (like, in this case, the idea presented in a recent Veritasium video that Euclid's missing topological axioms were not really missing because diagrammatic reasoning could replace them with equal rigor, for which I searched for and did not find mainstream sourcing with an adequate standard of reliability and internal mathematical rigor), and if there are too many then choose which to keep and which to cut based on how heavily they are represented in the literature. But you are writing as if survey articles are free from bias and we can just robotically follow them. If that were the case, all we would ever need to do is find a survey article and copy it verbatim, or if disallowed by copyright then just point to it and tell readers to go there instead. I happen to disagree. For instance, in many cases, academic survey authors will focus on the points of the subject that current academics find interesting, completely skipping over the points that are so obvious as to not make for an interesting academic debate. But Wikipedia articles should state the obvious. —David Eppstein (talk) 18:16, 24 September 2025 (UTC)

We shouldn't "copy [a single survey source] verbatim", that would be plagiarism and also likely not stylistically appropriate. But we also shouldn't «ignore all of the secondary/tertiary literature because everyone has biases and we're better at stating the obvious than the historians» or whatever. But again: the fundamental problem here is that we aren't stating the obvious about this topic. We're skipping right over basic coverage. –jacobolus (t) 18:28, 24 September 2025 (UTC)

Sometimes when I don't state the obvious it is because it is so obvious that I don't think to state it. Sometimes it is because it is so obvious that the published sources don't think to state it and so I don't find it in sources as something to say. And sometimes it is because stating the obvious in tedious long-winded detail (like, say, elaborating the exact content of every proposition rather than trying to mention only the ones of some continued significance) would derail the article.

You might observe, for instance, that even Artmann's book, usually a good source on the detailed content of each book, completely gives up after the first few propositions of Book X and skips the rest. —David Eppstein (talk) 18:31, 24 September 2025 (UTC)

Book X is hard material, written in a style unfamiliar to modern readers, and it would quickly get out of scope to try to give full coverage of its >100 propositions here. But there are multiple whole recent books, and plenty of other papers, chapters, etc., about why Book X was written, what it might be for, how it fits in with the rest of the project, how it relates to modern concepts, etc.; Book X (or the related concepts) would be a worthy topic for a full article (we don't do a good job at e.g. Commensurability (mathematics)). At this article, we could probably try to give some kind of explanation and summary instead of just stating that it's hard and then mentioning a few bits out of context. The gloss of commensurability already here is a good start. (Disclaimer: I have never personally tried to work carefully through too much of Books X–XIII. I wouldn't really feel qualified to write a summary without a lot more effort.) –jacobolus (t) 19:01, 24 September 2025 (UTC)

If you really think that the way I am trying to cover the content of these books is "a few bits out of context", you have missed the point. It is intended to be selected highlights of the parts of the books that our sources have noted as particularly significant or that make a specific connection to other encyclopedia topics. For Book X, the parts that the literature appears to highlight as significant (in terms of mathematics in general, not so much as part of the flow of that book) are the irrationality of the square roots and Euclid's formula for Pythagorean triples. The parts that are there for other reasons are explanations of terms that are different from what readers might think (what commensurability means geometrically and "irrational" meaning something different than now) and the part about the classification of irrationals (which as far as I can tell seems to be important within the book itself but less so for later mathematics; this was included before I started my revisions but I added a source). If there are other parts of the book whose meaning and significance are actually discussed in detail by sources, I don't know what those parts and those sources might be, despite searching for more. I don't think you do, either, so I don't know where your judgement that those sources must somehow exist somewhere is coming from. —David Eppstein (talk) 20:01, 24 September 2025 (UTC)

For example, the book you cited by Knorr (1975) is mostly about this topic, as is Fowler's (1999) book The Mathematics of Plato's Academy. I have only read parts of both books and skimmed some other parts, along with reading some of Knorr's and Fowler's papers along the same lines, and some reviews and responses. As I said, I think it would take me quite a bit of effort to write a good summary, and I don't feel immediately prepared to do it. –jacobolus (t) 21:18, 24 September 2025 (UTC)

Another very recent source (which I have not read beyond the abstract) is: Wagner & Netz (2023), "Between music and geometry: a proposal for the early intended application of Euclid's Elements Book X", British Journal for the History of Mathematics, 38. –jacobolus (t) 21:31, 24 September 2025 (UTC)

A relatively recent book entirely about Book X (which I have also not read) is: Taisbak (1982), Coloured quadrangles: a guide to the tenth book of Euclid's Elements, Copenhagen: Museum Tusculanum. –jacobolus (t) 21:34, 24 September 2025 (UTC)

One more source (in French) is Vol 3 of Vitrac's annotated translation of the Elements (1998), which consists entirely of translation of Book X and commentary about it. I'm having trouble finding any scan online, and I'm not sure if there are inter-library loan copies available to my local public library (some universities nearby have copies, I'm not sure if I can borrow them though). I'm considering whether it's worth trying to buy copies of Vitrac's 4 volume Elements from France. –jacobolus (t) 18:27, 30 September 2025 (UTC)

Also, the Zeeman paper is not "unworthy". But everything we mention here should be given due weight and contextualized so that we don't give readers a confusing or misleading impression. The Zeeman paper is not, per se, really the point here, but just an example; the fundamental problem is that we aren't successfully summarizing the basic topic, with the result that the discussion of the Zeeman paper, in current position and with current text, is a confusing non sequitur. –jacobolus (t) 18:15, 24 September 2025 (UTC)

I agree, balance is important, but so is explaining enough so that readers actually come away learning the basics of the topic. I think it's worth generally expanding our textual summary of most of the books, and giving a bit more context and discussion instead of just a list of the content. But in particular (and easily defensible by "proportion to their prominence in reliable sources"; all of these topics are drowning in centuries of commentary, down to the present):

• I think we'd benefit from splitting the current section about Book I into two parts, one about the definitions, postulates, and axioms at the beginning of I, discussing how they establish "Euclidean (plane) geometry", and a separate section about the propositions in Book I; while in the Elements they are joined into one "Book", they are of somewhat different character, and there is a lot of secondary literature tackling them somewhat separately or even independently. Another alternative could be to make two sub-sections, but I think that would be overall less helpful to readers.
• The section about book II should be moderately expanded, more explicitly pointing out that this is thought to come from the Pythagoreans, and mentioning that its subject is usually called the "application of areas" (ideally that wouldn't be a red link; an article at that title would be a good place to park a fit a more complete and neutral discussion about the controversy over Greek "geometric algebra" than we currently have anywhere on Wikipedia, that could give fair account to all sides).
• The section about book V should be moderately expanded, mentioning that it is about a "theory of proportions" (seems to be a more common keyword in the academic literature than the also found theory of ratio; ideally that also wouldn't be a red link, and could discuss various what is known and speculated about the historical development of this theory; while we're at it we could probably make significant improvements to ratio).
• The section about Book VI should be moderately expanded, with some discussion about why it relies on Book V. We should e.g. mention here that the Pythagorean theorem is proved a second time.

–jacobolus (t) 06:59, 24 September 2025 (UTC)

Independent of what is decided about other points, I think having an article dedicated to the theory of proportions could be a good idea, so I've started a draft using text from this article. Stepwise Continuous Dysfunction (talk) 19:02, 24 September 2025 (UTC)

Anyhow, if you're looking for distinctive modern comments on the Elements, you might enjoy, e.g.:

• JSTOR 2320742, which points out that most of the content of the Elements can be applied if the continuum of lengths is replaced with an arbitrary field (such as a finite field), mostly leaving Euclid's original axioms and discarding Hilbert's axioms of continuity.
• doi:10.1007/BF03024252, which defends Euclid's original approach to some geometric propositions (despite centuries of detractors) as having significant advantages in the context of computational geometry.

–jacobolus (t) 06:40, 24 September 2025 (UTC)

Personally I'm more interested in what makes an article "good" than what meets "good article criteria" (these two have limited overlap in my experience, with the latter mostly focused on relatively trivial stuff spelling and footnote placement), but the latter criteria do ask for articles to be "broad in [..] coverage", "neutral", "illustrated". These three criteria are not clearly defined and as a result reviews (and badged articles) are extremely inconsistent in quality. Personally I think the article has problems with all three of broad coverage, neutrality, and illustration, but again, I'm not really too concerned with the badges, so please feel free to settle the badges somewhere else, and we can try to stick to concrete content discussion here. –jacobolus (t) 07:39, 20 September 2025 (UTC)

I only said this here, because you came to the GANR page- I know good quality and good article criteria can be different- but I'm only here for the latter. You and DE can stick to "concrete content discussion" here- while me and DE stick to GAN criteria on the review page. HSLover/DWF (talk) 07:46, 20 September 2025 (UTC)

One source (which I had not looked up before now, but probably should have) that could be useful for thinking about the kind of content and scope of an article that would be appropriate for Wikipedia is the Dictionary of Scientific Biography. Their article about Euclid by Ivor Bulmer-Thomas and second article about the transmission of the Elements by John Murdoch, can be found in Vol. 4, pp. 414–459. There's also a nice bibliography there. –jacobolus (t) 08:28, 20 September 2025 (UTC)

Making a section to park miscellaneous comments on recent changes. –jacobolus (t) 00:14, 22 September 2025 (UTC)

• "This was laborious and expensive so manuscripts were often confined to wealthy estates and religious institutions, but they also formed the "set texts" on which grammar school instruction was based."
  This seems somewhat confusing and problematic as an intro to a section about medieval transmission/"reception" of the Elements. Grammar schools were a specifically English institution, and the source is discussing education and books in England from the 14th century on, with no mention of Euclid or even of mathematics. But the discussion immediately jumps to 8th century Baghdad. I don't think it's useful to try to summarize the education systems of Byzantine late antiquity, the Islamic Golden age, and every part of Europe from the 12th–15th centuries by making reference to largely unrelated particular English traditions. Surely we can find a source about mathematical manuscripts, per se, and covering their use by Byzantine, Islamic, and early European institutions or people. –jacobolus (t) 00:14, 22 September 2025 (UTC)

Background->1st paragraph, last sentence. Says “Euler” instead of Euclid 2A0E:1D47:8682:AE00:C469:1CFE:BEB3:D4D8 (talk) 13:20, 22 September 2025 (UTC)

@Brandmeister, making a controversial title move of a currently front-page-featured "good article" without discussion is extremely disruptive. Please never do anything like that again. @David Eppstein, can you revert the move pending discussion? –jacobolus (t) 17:14, 24 October 2025 (UTC)

Done. My contribution to the pending discussion: the existing title is well justified under WP:NATDAB. We don't need a disambiguated title when a naturally disambiguated title works well. The edit-summary invocations of WP:NCMANUSCRIPT and WP:TITLEDAB are unconvincing: NCMANUSCRIPT is about physical objects (manuscripts) rather than their contents, and TITLEDAB immediately points to the next section, NATDAB, for how to do the disambiguation, and says nothing about avoiding natural disambiguation in this specific case. —David Eppstein (talk) 17:54, 24 October 2025 (UTC)

Moreover, the name "Euclid's Elements" is extremely widely used as a primary name for this treatise. Google Scholar returns ~15k results for the exact phrase "Euclid's Elements", with another ~4k results for "Elements of Euclid", ~1k for "Euclidean elements". The name "Euclid's Elements", per se, is used as the title, or part of the title, of many modern translations, and directly included in the title of hundreds if not thousands of scholarly papers. Frankly the name "Euclid's Elements" is overall clearer as a title than just "Elements" would be, in a modern context. –jacobolus (t) 00:39, 25 October 2025 (UTC)

I agree. A glance at the titles used in § Selected editions finds "The Elements of Geometrie", "The elements of Euclid", "The first six books of the elements of Euclid" (twice), and "The Thirteen Books of Euclid's Elements". Even among the non-English titles, titles containing Euclid's name are more common than titles not containing his name, and there is no title listed that translates to "Elements" by itself except maybe the Chinese one. So an argument for the move from WP:COMMONNAME also seems dubious. —David Eppstein (talk) 01:55, 25 October 2025 (UTC)

Per WP:NCBOOKS and long-standing practice we don't usually put author's name ahead of title and the same generally applies to manuscripts: Republic (Plato), Academica (Cicero), On Sizes and Distances (Hipparchus), etc. Where disambiguation is needed, as in Elements, it's put in parenthesis per WP:TITLEDAB. The article itself begins with "The Elements", rather than "Euclid's Elements". Brandmeister ^talk 07:39, 25 October 2025 (UTC)

Plato's Republic would probably be a better title for that article, though it's fine either way. Cicero's Academica could plausibly also be titled like that, though the current one is probably better. Neither book On Sizes and Distances would make sense to title with the author's name. This collection of examples is completely unconvincing. –jacobolus (t) 10:33, 25 October 2025 (UTC)

The majority of works by ancient Greek writers is titled that way, with author in parenthetical disambiguation per WP:PARENDIS. This also makes sense, because it separates a title from an author. To an unfamiliar reader "Euclid's Elements" may imply that Euclid is part of the work's title. BRAND ^talk 10:49, 25 October 2025 (UTC)

There's no reason to apply a weird Wikipedian punctuation convention in the face of common usage, and no one is actually hurt by getting the impression that "Euclid is part of the work's title". For all practical purposes, "Euclid" is "part of the work's title" in common parlance. That is, the identifying string by which the work is known typically includes "Euclid" except when that can be inferred from context. Stepwise Continuous Dysfunction (talk) 16:48, 25 October 2025 (UTC)

It is not merely common parlance. Euclid's name literally is part of the title in most published editions. —David Eppstein (talk) 17:02, 25 October 2025 (UTC)

Is there any reason that § Non-Euclidean geometry doesn't mention Giovanni Girolamo Saccheri's 1733 Latin: Euclides ab Omni Naevo Vindicatus, lit. 'Euclid Freed from All Flaws'? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:19, 21 November 2025 (UTC)

Because WP:Summary style means that we briefly summarize the main article on the topic rather than replicating it in its entirety. I think Saccheri is included in the summary sentence "Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded." The fact that Saccheri's failure to prove the fifth postulate by contradiction was later considered by others to be an exploration of the properties of non-Euclidean geometry is anachronistic; that's not what Saccheri was intending to do. —David Eppstein (talk) 17:54, 21 November 2025 (UTC)

There's more about this topic at Parallel postulate § History and Non-Euclidean geometry § History. –jacobolus (t) 18:20, 21 November 2025 (UTC)

The fifth postulate is translated as

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 straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles

This is clearly wrong, since spherical geometry satisfies this postulate. I believe that for Euclid, there was an "f and only" if clause that was not understood by the translator. I suggest to change the formulation as

That, if a straight line is falling on two straight then the interior angles on the same side are less than two right angles if and only if the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

See also Talk:Geometry#5th Postulate. D.Lazard (talk) 18:38, 21 November 2025 (UTC)

I agree. Please go ahead. —David Eppstein (talk) 19:10, 21 November 2025 (UTC)

 Done

For the record, in my above post, instead of "This is clearly wrong, since spherical geometry satisfies this postulate", I shoukd have written "This is clearly wrong, since spherical geometry satisfies this postulate and the four others" D.Lazard (talk) 11:48, 22 November 2025 (UTC)

No, you absolutely should not change a translation sourced to Heath without another source, substituting your own made up version, especially if you don't yourself read Ancient Greek. That's incredibly misleading to readers. –jacobolus (t) 17:22, 22 November 2025 (UTC)

Yes I do not know Ancient Greek. But I know that Euclid knew the existence of parallell linees (lines that do not intersect). It is clear that Euclid considered that the fifth postulate implied the existence of parallel lines, as otherwise he would have introduced this existence as a 6th postulate, or he would have provided a proof. We know that such a proof cannot exist since the five postulate that was presented in the article apply to some non-Euclidean geometries and in particular to spherical geometry.

On the other hand, one do not need to know Ancient Greek language to know that the mathematical language of that time did not allowed a clear distinction between "if" and "if and only if". This why the quoted translations must be talen with care.

In any case, it is incredibly misleading to present as a truth a mathematical fact that has been proven wrong: the fact that the five postulates, as presented before imply the existence of parallel lines. D.Lazard (talk) 22:59, 22 November 2025 (UTC)

Text that purports to be a translation should not be rewritten based on what a non-translator thinks the author might have meant. It's misleading original research, in violation of multiple core Wikipedia policies. This is a complicated and subtle question about which a wide range of serious scholars have devoted significant effort and written a wide variety of commentaries. You recent edit must unquestionably be reverted, but feel free to look for reliable sources discussing the topic and neutrally summarize their content. If you think Euclid's text is wrong, or poorly translated, find a reliable source that directly says so and add relevant discussion based on that. Alternately, it would be fine to find a different comparably authoritative translation to Heath's and substitute those translations every place where we quote Euclid. But telling readers that we are quoting Heath's translation and then instead presenting a Wikipedian's speculative revisions is completely unacceptable. –jacobolus (t) 04:18, 23 November 2025 (UTC)

As one example of a reliable source, your speculative revisions and analysis here is profoundly inconsistent with the discussion in the relevant chapters of David Henderson & Daina Taimiņa's book Experiencing Geometry: Euclidean and Non-Euclidean with History. If there are serious scholars of Greek geometry who believe that these two (among others, perhaps including Heath) entirely misinterpreted Euclid's meaning, then there should be a reliable source somewhere directly stating as much, and we can directly present both claims and let readers make up their own minds about it. Even if there aren't reliable sources for that position, we can certainly find sources and give further detail about the differences needed for axiomatic systems for the plane vs. sphere. –jacobolus (t) 04:35, 23 November 2025 (UTC)

I consulted the translation that I has under hands, whitch is not fully reliable since it is often difficult to distinguish the translated text from translator's comments. Here is what can be said.

• The previous version of the 5th postulate was correct.
• Its converse, that I added (namely that the lines do not intersect if the interior angles equal two right angles) is th.18–prop.27 (for the numbering of my translation)
• The proof of this theorem results from th.9–prop.16 (an external angle of a triangle is greater than each of the opposite internal angles), which itself results from th.1–prop.4 (side-angle-side criterion of congruence). The proof of the latter uses that "two lines do not enclose any region", which is equivalent to the modern "two straight lines have at most one intersection point".
• The assertion "two lines do not enclose any region" is presented in my translation as a 12th common notion.

The fundamental fact that distinguishes Euclidean geometry fron spherical geometry is "two lines do not enclose any region" or equvalently "two lines have at most one intersection point". Indeed, two meridians intersect at both poles and enclose one, and even two, longitude intervals.

I ignore where and how this "12th common notion" appear in the translations on which the article is based, but it seems essential to mention this in the article.

So, I'll revert my edit, and add some clarification, posssibly in explanatory footnotes. D.Lazard (talk) 11:29, 24 November 2025 (UTC)

Thanks! –jacobolus (t) 22:27, 24 November 2025 (UTC)

The article actually stated the fifth postulate twice, once in the table and once in the main text. The version in the text did not have a specific source. It looked like someone had started from Heath and then changed some of the words around, maybe to "modernize" the vocabulary. I think it's simplest if both are just taken from Heath. If we do make alterations, like changing "produced" to "extended", then we should somehow specify that the blockquoted text is modified from Heath, with volume and page numbers for the original translation. Stepwise Continuous Dysfunction (talk) 23:49, 24 November 2025 (UTC)

Even in English the word if is ambiguous, leading to the neologism iff for if and only if in modern mathematical literature. More importantly, some of the proofs in The Elements rely on assumed axioms; the listed common notions and postulates are not adequate. I'm not sure how much the article can say about incorrect proofs without crossing into TMI. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:39, 24 November 2025 (UTC)