Talk:Viète's formula/GA1

GA Review

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Reviewer: Amitchell125 (talk · contribs) 08:37, 1 July 2021 (UTC)

Happy to review this article.

Review

Lead section/Infobox

Who was François Viète? I would amend François Viète to ‘the French mathematician François Viète’. That would not be a very accurate description of who Viète was. I shortened the lead description and added more about him in the "significance" section. —David Eppstein (talk) 20:37, 4 July 2021 (UTC) At present the lead section is not “an introduction to the article and a summary of its most important contents” (see MOS:LEAD). Also, according to the MOS, “significant information should not appear in the lead if it is not covered in the remainder of the article”—your lead explains what the formula is and when it was first published, whilst the main text of the article does not appear to include some of the lead's information. Moved first publication out of lead and added more summary of later material to lead. —David Eppstein (talk) 20:37, 4 July 2021 (UTC) I would enlarge the image to at least x2 to help it look a bit more readable, it’s currently 1.35 (minor point). See MOS:IMGSIZE "Lead images should usually use upright=1.35 at most". —David Eppstein (talk) 19:37, 4 July 2021 (UTC) Understood. 'I would' means it's a suggestion on my part. The image at the top of the article is worth getting right (it provides "a visual association for the topic, and allow readers to quickly assess if they have arrived at the right page", according to MOS:LEAD). If you're unhappy about my suggestion for improving the readability of what is in the end a textual image (see MOS:TEXTASIMAGES for the guidance on this), it's probably better for me to say the image needs to be replaced or removed. Amitchell125 (talk) 20:09, 4 July 2021 (UTC) Removal? What an obtuse way of handling a minor legibility issue. Click on the image if you want to read it (and can read Latin). Otherwise, it's there to provide an impression of historical context and more specifically to show how Viète formatted his formula, because it is rather different from modern formatting of mathematical formulas. —David Eppstein (talk) 20:39, 4 July 2021 (UTC)

1 Significance

• The link to approximating should link to both this word and π.
  • Done. —David Eppstein (talk) 20:49, 4 July 2021 (UTC)
• methods for approximating π to (in principle) arbitrary accuracy had long been known – seems to be in need of copy editing to improve the English’.
  • The grammar seems ok to me. "In principle" is used adverbially, to modify "arbitrary accuracy". "Approximating to arbitrary accuracy" is the idiomatic choice of preposition to use. "Methods for approximating π had long been known" is a normal past-perfect sentence. What do you see as dubious here? —David Eppstein (talk) 20:47, 4 July 2021 (UTC)

In an article that stands out for its eloquence and good use of plain English, this sentence brings me up short, probably due to the "(in principle) arbitrary accuracy" bit. Perhaps simply 'By around 2000 BC, methods for approximating π had been found by the Egyptians and the Babylonians.' (using Beckmann, p.12)? Amitchell125 (talk) 08:38, 5 July 2021 (UTC)

But that loses an important piece of meaning. You can approximate π by saying it's 3 — that's a method for approximating π, but not one with arbitrary accuracy. And on the other hand, we can today easily find thousands of digits of π, but in Viète's time nine digits was difficult, so the ability to calculate to arbitrary accuracy came with an increasing cost to that accuracy, meaning that for practical purposes the accuracy was limited even though theoretically it wasn't. All of that is packed into the current sentence and missing from your replacement. —David Eppstein (talk) 16:24, 6 July 2021 (UTC)

• Who were Archimedes and Ludolph van Ceulen?
  • Mathematicians who did the things described here, about whom going into more detail would be off-topic. See WP:GACR #3b. —David Eppstein (talk) 20:05, 4 July 2021 (UTC)
• has been noted – by anyone in particular?
  • Rewritten to name and linked the authors of these opinions. —David Eppstein (talk) 21:34, 4 July 2021 (UTC)
• a motivating example – could you explain motivating?
  • It is the usual meaning of the English word: an example that motivates a reader who sees it (or maybe more accurately the author of the book cited) to look more deeply and connect this topic with a different one. More specifically, Kac goes through a derivation of the formula in which a certain system of orthogonal polynomials related to binary notation appears. He notices that, at some point in the derivation, one has a sum of products that is equal to a product of sums of the same formulas. This is a strange thing to happen (you can't normally swap the operations of sums and products like that), and Kac asks whether it's a coincidence or an example of a more general phenomenon. The more general phenomenon that he finds to explain this is statistical independence. To say all this in any more detail in the article itself would be rather off-topic, and not really helpful to readers trying to learn about the formula itself; see WP:GACR #3b. —David Eppstein (talk) 20:44, 4 July 2021 (UTC)

Understood, and thanks for your detailed reply. Amitchell125 (talk) 09:20, 5 July 2021 (UTC)

• Link circumference; polygon; Viète (his full name should be linked); explicit formula (see Explicit formula to select the correct link); wave propagation rates (presumably Velocity factor); integrals.
  • The first instances of perimeter, circumference, polygon, Viète, and integral are now linked. Too much repetion of those links, to the first instance in each section, might violate WP:OVERLINK. None of the disambiguation links in explicit formula (disambiguation) goes to a relevant article; the terms are used with their colloquial English meanings. Rewrote the sentence on wave propagation to try to be less technical; I think the correct wikilink there is actually dispersion relation. —David Eppstein (talk) 21:41, 4 July 2021 (UTC)
• used by Archimedes to find the approximation (+ formula) needs a citation.
  • Added. —David Eppstein (talk) 00:42, 5 July 2021 (UTC)
• Is However necessary?
  • Probably not. Removed. —David Eppstein (talk) 00:38, 5 July 2021 (UTC)
• closely related method – I think it needs to be clear which method is being referred to here, as two are referred to in the paragraph.
  • Clarified. —David Eppstein (talk) 00:38, 5 July 2021 (UTC)
• I would link infinite product and π, even though they are linked in the lead (see [[MOS:DL]).
  • Infinite product re-linked. I'm not entirely convinced that π needs even a single link, per MOS:OL "understood by most readers in context": is there any reader of this article who doesn't know what π is?

Agreed, happy if {{pi} wasn't linked at all. Amitchell125 (talk) 09:39, 5 July 2021 (UTC)

Well, I think it does need to be linked in the lead, but not later. —David Eppstein (talk) 16:44, 6 July 2021 (UTC)

• Copy edit the first formula of European mathematics representing a number to something like ‘the earliest formula in European mathematics to represent a number’.
  • Copyedited, but differently than the suggestion. —David Eppstein (talk) 00:38, 5 July 2021 (UTC)

2 Interpretation and convergence

Move the link for limit to where it first occurs in the previous section. It's now linked in the lead. —David Eppstein (talk) 22:17, 4 July 2021 (UTC) I’m being ignorant here, but why is the top expression in this section the other way round from the expression in the lead? I think it's one of these things like adjective order (why do we say big yellow house, and not yellow big house) that is intuitive to people who use these kinds of expression but hard to explain. In terms of the actual mathematical meaning, I think there is no difference. But in terms of usage and intent, equality of two expressions is usually used with the expression we would like to evaluate on the left and with something easier to calculate and understand on the right. So Viète's formula itself has the 2/pi on the left, because pi is an unknown quantity that is being made more understandable by expressing it in terms of known quantities and operations (2, square roots, and products). But in this section, we write the limit on the left and the evaluation of that limit on the right, because by now we know what pi is, so the limit is the more complicated thing and 2/pi is its value. It would be correct to write (complicated product of nested roots) = lim... = 2/pi because the limit is how we take that complicated product and turn it into a rigorously defined mathematical value; we are explaining the complicated product by turning it into a limit. Similarly, we are explaining the limit by stating what its value is, the number 2/pi. —David Eppstein (talk) 22:15, 4 July 2021 (UTC) Understood. Amitchell125 (talk) 10:04, 5 July 2021 (UTC) Who was Ferdinand Rudio? The first person to prove the existence of the limit, as stated in the article. The name is Wikilinked, so anyone who wants more can find out. I don't think we need to state here that his nationality changed from Nassau to Prussia when he was 10; see WP:GACR #3b. —David Eppstein (talk) 20:01, 4 July 2021 (UTC) there is a linear relation between the number of terms and the number of digits seems to repeat the information that follows, and so can be removed as redundant. Huh. Usually in these mathematical nominations I have to argue that WP:TECHNICAL allows good articles to have mathematical content that is beyond what non-mathematical readers can handle (when it is necessary to cover the material). Here, I think WP:TECHNICAL needs to be invoked in the other direction. The first clause, "there is a linear relation between the number of terms and the number of digits", is intended as the less-technical version of the information, aimed at readers for whom the second clause, "the product of the first n terms in the limit gives an expression for π that is accurate to approximately 0.6n digits", would be too technical to read or interpret easily. Removing it would increase the technicality of the article, for no good reason. A little redundancy is not always a bad thing. Sometimes repeating things in a different way can help readers understand what they're reading. And WP:GACR 1a says that we should try to do that, when we can. I did rewrite the first clause, though, to try to make it even simpler. —David Eppstein (talk) 20:54, 4 July 2021 (UTC) I agree with you here, although I have been mercilessly mauled by others for including anything that resembles redundant text. Happy to concur. Amitchell125 (talk) 10:19, 5 July 2021 (UTC) I would include that the Wallis product was published in 1656 by the English clergyman John Wallis (citation - https://www.jstor.org/stable/25759727) Again, the term is Wikilinked and the important part, that despite being a later result this product converges more slowly, is stated here. See WP:GACR #3b. —David Eppstein (talk) 20:01, 4 July 2021 (UTC) My interest in history (as well as maths) may have become evident here. Amitchell125 (talk) 12:26, 5 July 2021 (UTC) I would add a colon after a limit expression. Would you add the colon if the expression were not displayed on a separate line? Do you normally add colons in the middle of sentences with that grammar? Like, in the sentence "America may be interpreted as the name of a country, the United States of America", would you replace the comma by a colon? Mathematical sentence punctuation is not that different from non-mathematical sentence punctuation. —David Eppstein (talk) 20:01, 4 July 2021 (UTC) Now why did it say that? My bad... Amitchell125 (talk) 12:39, 5 July 2021 (UTC) More comments to follow. Amitchell125 (talk) 19:35, 2 July 2021 (UTC)

3 Related formulas

• more than a century later is not verified by the text? (Beckmann, p.95: “almost two centuries later”).
  • In what sense do you think that it is incorrect to say "more than a century" for a length of time that a source describes as "almost two centuries"? $200-\varepsilon >100$ for all sufficiently small values of $\varepsilon$. We should not be following the exact wording of the sources; that would be plagiarism. —David Eppstein (talk) 19:51, 4 July 2021 (UTC)

Perhaps the exact year Euler obtained π (1735 I believe) would be better.Amitchell125 (talk) 12:12, 5 July 2021 (UTC)

I'm not convinced that is the right reference. What in there looks to you like the same formula? See also [1] (unfortunately not a reliable source) where the contributors try to track down this formula to Euler, find instead a different more complicated product formula, and don't find the original formula. So because we don't know the date but we do know when Euler was active, we need to be vague. Because of the stackexchange failure to properly attribute this formula to Euler, I changed it to say "that has often been attributed to Leonhard Euler, more than a century later" (with the restoration of the Beckmann footnote, as Beckmann makes this attribution), rather than saying more definitively that it is by Euler and on what date. —David Eppstein (talk) 16:43, 6 July 2021 (UTC)

• Then, expressing each term of the product on the right as a function of earlier terms using the half-angle formula – this text and the following expression are not verified by Beckmann on pp. 94–95.
  • I sourced it to Morrison instead. (Kac also uses the same derivation but without as much detailed justification for each step.) —David Eppstein (talk) 01:29, 5 July 2021 (UTC)

4 Derivation

Move the link for perimeter to where it first occurs in the main text of the article. Done. —David Eppstein (talk) 19:48, 4 July 2021 (UTC) Thus is a redundant word here. I don't think it is. It suggests that the fact that the series telescopes follows from the previous part of the explanation, in which each term is interpreted as a ratio of two areas. Without "thus", those two parts of the derivation would come across as unrelated assertions. —David Eppstein (talk) 19:48, 4 July 2021 (UTC) Mmmm OK. Amitchell125 (talk) 11:18, 5 July 2021 (UTC) one may prove - ‘one can prove’ sounds better imo. Reworded; I don't think "one" was necessary at all here. —David Eppstein (talk) 19:48, 4 July 2021 (UTC) The image is quite difficult to interpret as the hexadecagon and the circle are so close to each other. I’ve provided an alternative image here. I don't think making three polygons instead of four and changing the red circle to dashed black significantly improves the legibility of the figure. What does help is making it bigger. I tried expanding it to a 1.5 scale factor (it's not the lead image, so that's still within the MOS guidelines). —David Eppstein (talk) 19:48, 4 July 2021 (UTC) Looks better. Amitchell125 (talk) 11:33, 5 July 2021 (UTC)

5 References

Boulder, CO – should read ‘Boulder, Colorado’. As the referencing style needs to be consistent, all locations (and not just some) should be included. Consider using this external link to ensure your ISBN numbers are formatted in the same way. Ref 1 (Beckmann) - consider including this url. Ref 6 (Plofker) - I would replace the url with this. Ref 10 (Cullerne et al); 18 (Levin); 19 (Osler 2007) could all do with a {{subscription required}} template. Ref 11 (Kac) - consider including this url. Ref 14 (Osler 2007) - consider including this url. Ref 20 (Stolarsky) is a dead link. I have been reliably informed, when I brought this up in my own GA reviews of other articles, that there is nothing about consistency of referencing style in the GA criteria; the references must merely be laid out in a reference section as the layout guidelines suggest. Nevertheless, I have added locations and ISBN hyphens for the other book publications. As for the suggestions to add archive urls in place of Google Book urls: for books that are free online, I think archive is generally preferable, for multiple reasons, but this does not apply to any of the books cited here. For books that are non-free and for which only a limited preview is available (all the ones here), I don't see the point of needless churn in replacing one non-free source by another. I think {{subscription required}} is a pointless waste of reader attention. And I think researchgate is a dubious low-quality web scraper only one step removed from forbidden pirate sites like sci-hub. I updated the Stolarsky url. —David Eppstein (talk) 19:34, 4 July 2021 (UTC) Your reliable informant and I may not totally agree, as MOS:FNNR states "Editors may use any citation method they choose, but it should be consistent within an article", which I understand to cover all referencing in an article. Nominators and reviewers do seem to have mixed feelings about the Internet Archive and other downloadable sources—I personally find it invaluable, especially as I have been denied access to local libraries for so long—but I usually recommend urls where i find them, as a number of nominators are happy to use them. Your opinion about access templates isn't unique, but it's unusual in my experience. As a reviewer, I find it useful, but I understand your point. Amitchell125 (talk) 10:52, 5 July 2021 (UTC)

6 External links

Is this worth including as a link? A derivation for the formula can be found here followed here. I found them useful to get my head around the algebra. Worth including? I would prefer not to include these links. Web calculators for simple formulas are almost always bad links to include. And YouTube links are usually discouraged; I'm not convinced these ones offer enough of an additional resource beyond what's in the article to get beyond WP:ELNO #1. —David Eppstein (talk) 19:40, 4 July 2021 (UTC)

On hold

I'm putting the article on hold for a week until 10 July to allow time for the issues raised to be addressed. Regards, Amitchell125 (talk) 19:55, 2 July 2021 (UTC)

• @Amitchell125: I believe I have addressed all of your comments above (implementing many but not all of them). Please take another look. —David Eppstein (talk) 01:30, 5 July 2021 (UTC)

Almost done, thanks for all you've done so far. Amitchell125 (talk) 12:46, 5 July 2021 (UTC)