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== Propose for the change of Description ==

As we know π is an [[Irrational number]] meaning it cant be expressed in a ratio.Therefore I propose to change to change the article description to "Irrational Number, approximately 3.14" [[User:Uncharted 430|Uncharted 430]] ([[User talk:Uncharted 430|talk]]) 06:49, 23 December 2024 (UTC)

:It's a [[real number]]. It's a [[complex number]] with imaginary part 0. It's an [[irrational number]]. It's a [[transcendental number]]. But see [[WP:SDNOTDEF]]: the short description is not the place to give as precise a definition as we can. Short descriptions are mainly used for disambiguating searches on mobile devices: when you search for a term like pi, you will get a lot of results, and it's helpful to have a brief and easy-to-read blurb that clarifies which topic each search term is about. So the current description "Number, approximately 3.14" does that, because it tells you it's the article about the number and not say the article about the Greek letter or [[Pi (film)]] or any number of other topics with the letters pi in them. Putting the word "irrational" doesn't make it easier to tell which search result you want. It might be confusing to non-mathematicians who don't know about irrational numbers. It might cause pedants to tell us that well actually it's transcendental and we should say that instead, and then be even more confusing to non-mathematicians. And merely for the reason that it makes the short description less short, it makes it less quick to read and find among the other short descriptions of all the other pi search results. So I don't think this would be an improvement. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 07:01, 23 December 2024 (UTC)
::@[[User:David Eppstein|David Eppstein]] Thanks for your efforts.But shouldnt it be at the least categorized as [[Irrational number]] [[User:Uncharted 430|Uncharted 430]] ([[User talk:Uncharted 430|talk]]) 07:04, 23 December 2024 (UTC)
:::In the categories at the bottom of the page? No. There, it is categorized as a real transcendental number. The way Wikipedia's category system works, when an article would belong to both a more general category and a more specific subcategory, we only put it into the more specific subcategory. [[:Category:Real transcendental numbers]] is a subcategory of [[:Category:Irrational numbers]], so articles in [[:Category:Real transcendental numbers]] should not be placed in [[:Category:Irrational numbers]]. Instead, people looking at the irrational number category should explore all its subcategories to find the articles in them. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 07:09, 23 December 2024 (UTC)
::::@[[User:David Eppstein|David Eppstein]] Thanks.Much appreciated [[User:Uncharted 430|Uncharted 430]] ([[User talk:Uncharted 430|talk]]) 07:11, 23 December 2024 (UTC)

== First sentence of the article already contains potential weasel word. ==

> The number "Pi" is a mathematical ''constant'' that is the ratio of a circle's circumference to its diameter ... 

The opening sentence already implies a statement that would require extraordinary proof to stand.

Has there been conclusive astronomy / satellite observation of the farthest / oldest reaches of our Universe, to confirm the '''constancy of Pi thoughout the cosmic ages''', from Big Bang to this day?
[Note that significant effort was made in satellite astronomy just to confirm the constancy of Speed of Light and the Fine Structure Constant - indeed finding them to be constant to the full extent of achiveable instrument accuracy.]

After all, '''Pi is fundamentally an empirical ratio, as there was no circle in the heads of philosophers and mathematicians before they saw one in the dust of Babylon and Athens.'''

Thus the abstraction of Pi as a constant ratio of an abstract circle's diameter vs circumference can only follow after the circle as a pyhsical reality is confirmed to be based on a constant Pi throughout the cosmic ages.
The burden to provide that confirmation / proof is on those who make the claim of constancy. [[Special:Contributions/94.21.237.182|94.21.237.182]] ([[User talk:94.21.237.182|talk]]) 13:47, 31 December 2024 (UTC)

: Pi can be defined without reference to geometry, as the last paragraph of the lede mentions.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 14:50, 31 December 2024 (UTC)
:{{mvar|π}} is not dependent on physical reality. Our discussion of circumferences and diameters of circles in this context is related to abstract [[Euclidean space]]. It is incorrect to say that "{{mvar|π}} is fundamentally an empirical ratio". –[[user:jacobolus|jacobolus]]&nbsp;[[user_talk:jacobolus|(t)]] 16:58, 31 December 2024 (UTC)

:There are no weasel words there. {{pi}} is a mathematical object in analysis and Euclidean geometry. Its utility in physics is a separate issue. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 20:00, 31 December 2024 (UTC)
:Certainly no weasel words, but arguably a lack of context. As it says in the subsection "Definition":
::''This definition of π implicitly makes use of flat (Euclidean) geometry''
:However, I strongly believe this is exactly as it should be. Lay readers will generally understand the first sentence quite easily, and including e.g. "in Euclidean geometry" in that sentence would make it less, not more, understandable. [[User:Nø|Nø]] ([[User talk:Nø|talk]]) 10:49, 1 January 2025 (UTC)
::: I do feel that the current lede is to some extent undue weight on Euclidean geometry and history, and does not present a definition of pi that is acceptable by modern standards. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 11:58, 1 January 2025 (UTC)
::::To the vast majority of humanity, and throughout most of pi's history, pi is not an infinite series, or some other advanced maths object - it is the ratio in a circle. And this is not ''wrong'', just because other definitions make more sense in rigorous presentations. One could say the circle thing is the [[WP:PRIMARYTOPIC]], so I don't think it's undue weight in the lead. [[User:Nø|Nø]] ([[User talk:Nø|talk]]) 13:55, 1 January 2025 (UTC)
::::: I mean, one could say the same thing about [[table salt]], but the very first sentence to that article at least says that it's [[sodium chloride]], even though those words would mean nothing to the vast majority of humanity throughout the ages.  So this is a bad argument for excluding important scientific detail from the lede.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 14:18, 1 January 2025 (UTC)
::::: To clarify, other definitions are more convenient in some contexts, but they are not more rigorous. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 15:36, 1 January 2025 (UTC)
:::: In what sense is defining {{pi}} via Euclidean geometry not acceptable by modern standards? Other definitions may be more convenient and more usefull, but the old definition is still acceptable and sometimes even convenient. This is not an issue of accuracy or rigor, but one of style, and the wording should reflect [[WP:MOS|MOS]]. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 15:36, 1 January 2025 (UTC)
::::: The circumference of a circle is not a primitive of Euclidean geometry.  It ''requires'' analysis.  I think this dovetails with a point that the OP was trying to make, namely that if one defined {{pi}} in this way, there would be a question of ''measurement'', a question which analysis solves, but not in an elementary way familiar to most people. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 15:47, 1 January 2025 (UTC)
::::::Any kind of definition of π requires analysis. But the definition in terms of circles is much more widespread, familiar, accessible, and usefully explanatory than definitions directly in terms of infinite series or what have you. –[[user:jacobolus|jacobolus]]&nbsp;[[user_talk:jacobolus|(t)]] 18:46, 1 January 2025 (UTC)
::::::: My overall feeling is that the lede sweeps this completely under the rug in the name of accessibility, which is undue weight.  I propose adding the following clause to the second sentence of the lede: "The number {{pi}} appears in many formulae across [[mathematics]] and [[physics]], ''and [[#Definition|can be formally defined]] from first principles without reference to circles and geometry.''" [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:51, 1 January 2025 (UTC)
:::::::: All of these other supposed "first principles" definitions are, ultimately, alternative interesting methods for measuring circles, can be explained geometrically, and historically came about because people were interested in measuring circles. So I think your proposed sentence would be substantially misleading to a less-technical audience. –[[user:jacobolus|jacobolus]]&nbsp;[[user_talk:jacobolus|(t)]] 18:56, 1 January 2025 (UTC)
::::::::: In that case, something should be added to the lede that describes that the measurement of circles is in fact a difficult mathematical problem.   [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:07, 1 January 2025 (UTC)
::::::::::That sounds like a good idea. –[[user:jacobolus|jacobolus]]&nbsp;[[user_talk:jacobolus|(t)]] 19:19, 1 January 2025 (UTC)
::::::::::I added such a "something" before reading this thread. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 19:21, 1 January 2025 (UTC)
::::::::::: Nice, I'm very happy with that.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 20:00, 1 January 2025 (UTC)
::::::::::::I am also quite happy with the current version.
::::::::::::Last paragraph in the lead starts:
:::::::::::::''Because its definition relates to the circle, π is found in many formulae''
::::::::::::Arguably, as we've already said it can be defined in many ways, we can't talk about "its definition" (singular). Would
:::::::::::::''Because it relates to the circle, π is found in many formulae''
::::::::::::be better? [[User:Nø|Nø]] ([[User talk:Nø|talk]]) 15:26, 3 January 2025 (UTC)

::::::::::::: Done. [[User:Murray Langton|Murray Langton]] ([[User talk:Murray Langton|talk]]) 19:29, 3 January 2025 (UTC)
:Very good remark it's also quite true that the circle is only an object in some defined theory and there you're right about the definition not satisfying the need of exactness (a/a=1) implicating relating real objects in an assembled or declared known value to output an accepted truth and a ratio, an unknown such as the curve of the stated circle and its related diameter and center can tell you in this sentence. 
:Ik Kouyaté [[Special:Contributions/41.242.89.175|41.242.89.175]] ([[User talk:41.242.89.175|talk]]) 18:52, 19 February 2025 (UTC)

== Flat in lede ==

Due to the use of [[spherical trigonometry]] in navigation, I believe that the lead should mention that the classical definition applies only to a flat space. That could be handled simply by adding ''Euclidean'' to the text or by a footnote. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 10:02, 6 January 2025 (UTC)

:I believe that most readers of this article do not know what are a flat and non-flat space. So, your suggestion would make the article too [[WP:TECHNICAL]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:27, 6 January 2025 (UTC)
:I tend to agree with [[user:D.Lazard]]. However, if anyone insist on changing it, I believe "plane" is better than "flat". [[User:Nø|Nø]] ([[User talk:Nø|talk]]) 13:29, 6 January 2025 (UTC)
::There are planes in [[elliptic geometry]] and [[hyperbolic geometry]]. How about ''Euclidean''? -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 14:23, 6 January 2025 (UTC)
:::This seems like a totally unnecessary pedantic point for the lead here. When the word "circle" is used without qualification it is implied that we are talking about a planar circle. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 18:00, 6 January 2025 (UTC)
:Perhaps there is some way to state that any circle drawn in the ''plane'' has circumference and diameter in the same ratio.  E.g., around the first sentence "In the [[Euclidean plane]], any circle drawn has its [[circumference]] and diameter in the same ratio. This ratio is the mathematical constant {{pi}}." [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 14:18, 6 January 2025 (UTC)
::I like that wording. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 14:23, 6 January 2025 (UTC)
::In my opinion this sentence belongs to first paragraph of section {{alink|Definition}}. As everybody knows or believe to know what is a circle and what is {{pi}}, giving technical details may only be confusing for some readers. [[MOS:LEAD]] says "The lead should stand on its own as a concise overview of the article's topic". There is no need to introduce these details in the lead. 
::Also, if the sentence is used to rewrite the first paragraph of {{alink|Definition}}, "drawn" must be removed, since not all circles are drawn, and the sentence would be clearer in plural with "all" replacing "any". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:05, 6 January 2025 (UTC)
:::
:::How about starting the lead with {{tqq|The number '''{{pi}}''' ({{IPAc-en|p|aɪ}}; spelled out as "'''pi'''") is a [[mathematical constant]]. In the [[Euclidean plane]] it is the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]], approximately equal to 3.14159. }} and in {{alink||Definition}} something like {{tqq|{{pi}} was claasicaly defined in [[Euclidean geometry]]as the [[ratio]] of a [[circle]]'s [[circumference]] {{math|''C''}} to its [[diameter]] {{math|''d''}}:}}. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 17:50, 6 January 2025 (UTC)
::::Again, this is not an improvement to introduce in the lead a technical term (Euclidean plane) that is unknown to many readers, and is of no help to understand the sentence. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 18:00, 6 January 2025 (UTC)
::::I also think adding "In the Euclidean plane" to the lead seems unnecessary and (slightly) distracting. Mentioning this later is fine. As "classically defined in Euclidean geometry" may be historically misleading, how about just "defined in plane geometry"? –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 18:11, 6 January 2025 (UTC)
:::::Why "plane geometry" and not "geometry"? Circle are also defined is space geometry, for example as intersections of spheres. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 18:16, 6 January 2025 (UTC)
::::::Also, saying that it ''is'' the ratio of circumference to diameter (the current version) is better than saying that it ''was classically defined as'' the same ratio, to avoid issues such as whether "the" classical definition involves circumference or area, diameter or radius, etc., or whether there is even a single classical definition consistent across cultures. These issues make no difference to the value of pi, or what it is, but they become significant when we say that one specific choice is the classical definition. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 18:21, 6 January 2025 (UTC)
:::::How about putting it in a footnote? -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 13:29, 8 January 2025 (UTC)
::::::Even in footnotes, pedantry is not useful. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:53, 8 January 2025 (UTC)
:::::::It's not pedantry. Things should be as simple as possible, but no simpler. Again, navigation. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 16:15, 8 January 2025 (UTC)
::::::::It is pedantry. Readers have no trouble understanding that unqualified "circle" means a circle in Euclidean space. Elaboration about other types of circles belongs later in the article, not in the lead. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 20:34, 8 January 2025 (UTC)
:::::::::Agreed. If you know that non-Euclidean geometry exists, you know enough mathematics not to be confused by the first two sentences of the Wikipedia article on pi. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 20:47, 8 January 2025 (UTC)
::::I share the sentiment that bringing in "Euclidean plane" would be opaque and pedantic. The audience of this article is, in Americentric terms, [[middle school]] and up. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 20:31, 6 January 2025 (UTC)

== Contradiction? ==

From the section [[Pi#Adoption_of_the_symbol_π|Adoption of the symbol {{pi}}]]:

{{Blockquote
|text=In the earliest usages, the [[Pi (letter)|Greek letter {{Pi}}]] was used to denote the [[semiperimeter]] (''semiperipheria'' in Latin) of a circle and was combined in ratios with [[delta (letter)|{{mvar|δ}}]] (for [[diameter]] or semidiameter) or [[rho|{{mvar|ρ}}]] (for [[radius]]) to form circle constants. (Before then, mathematicians sometimes used letters such as {{mvar|c}} or {{mvar|p}} instead.) The first recorded use is [[William Oughtred|Oughtred's]] {{nobr|"<math>\delta . \pi</math>"}}, to express the ratio of periphery and diameter in the 1647 and later editions of {{lang|la|Clavis Mathematicae|italic=yes}}.
}}

So in the very first usage, did {{pi}} mean semiperimeter or perimeter? This paragraph seems to contradict itself. Does the article contain an error, or am I missing something here? [[User:ISaveNewspapers|ISaveNewspapers]] ([[User talk:ISaveNewspapers|talk]]) 06:48, 12 February 2025 (UTC)

:Yes, the article is confused. The bit about Oughtred is dubious: he did not clearly identify {{pi}} to mean either "perimeter" or "semiperimeter"; what he writes [https://books.google.co.jp/books?id=KTgPAAAAQAAJ&pg=PP3&redir_esc=y#v=onepage&q&f=false reference] is:
:<blockquote>{{delta}}.{{pi}} :: semidiameter.semiperiferia</blockquote>
:...which is to say: "delta is to pi as the (semi)diameter is to the (semi)periphery", where adding or removing both "semi" prefixes makes no difference. Obviously the letters {{delta}} and {{pi}} refer to what follows "semi-" in each case. [[User:Imaginatorium|Imaginatorium]] ([[User talk:Imaginatorium|talk]]) 09:41, 13 February 2025 (UTC)
::Indeed, he consistently uses {{tmath|\tfrac \pi \delta}} in the following pages as a single quantity, clearly shorthand for the unitless and constant numerical quotient of the periphery divided by the diameter of any arbitrary circle. The initial introduction ({{tmath|\! \delta . \pi :: \operatorname{semidiameter} . \operatorname{semiperiphery}\!\!\! }}) just provides a definition/explanation for the subsequent use of {{tmath|\tfrac \pi \delta}}. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 09:46, 13 February 2025 (UTC)
:::⁠⁠One must not forget, that, at that time, a length was not a number, and only zero-dimensional quantities (that is ratios of two quantities of the same nature} were considered as numbers. So, there was no "irrational numbers" but only "irrational ratios". So, I believe that, for Oughtred, {{pi}} and ''{{delta}}'' did not have individual values, and {{tmath|\frac\pi\delta}} was simply a notation for what is now called {{pi}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:42, 13 February 2025 (UTC)
:::: I don't think the point here was whether it was "rational" or not (though I don't read Latin, so maybe I'm missing something). But I agree, the combined symbol {{tmath| \tfrac \pi \delta}} is being used as a single number. As you say, ratios and numbers were considered to be different types of objects. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:28, 13 February 2025 (UTC)
:::I suggest to rewrite the paragraph as 
::::{{tq|In the earliest usages, the [[Greek letters]] {{pi}} and ''δ'' were combined into the fraction {{tmath|\tfrac \pi \delta}} for denoting the ratios [[semiperimeter]] to [[semidiameter]] and perimeter to diameter, which are presently denoted as {{pi}}.<ref name=firstPi>{{Cite book |url=https://books.google.com/books?id=KTgPAAAAQAAJ&pg=PP3 |title=Theorematum in libris Archimedis de sphaera et cylindro declarario |last=Oughtred |first=William |date=1652 |publisher=Excudebat L. Lichfield, Veneunt apud T. Robinson |language=la |quote={{math|''δ''.''π''}} :: semidiameter. semiperipheria}}</ref><ref name="Cajori-2007">{{Cite book |url=https://books.google.com/books?id=bT5suOONXlgC&pg=PA9 |title=A History of Mathematical Notations: Vol. II |last=Cajori |first=Florian |date=2007 |publisher=Cosimo, Inc. |isbn=978-1-60206-714-1 |pages=8–13 |language=en |quote=the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented {{math|3.14159...}} by {{math|''δ'':''π''}}, as did Oughtred more than a century earlier}}</ref><ref>{{wikicite |ref={{harvid|Schepler|1950}} |reference = Schepler, H.C. (1950) "The Chronology of Pi" ''Mathematics Magazine''. '''23'''.{{br}}Part 1. Jan/Feb. (3): 165–170. [[doi (identifier)|doi]]:[https://doi.org/10.2307/3029284 10.2307/3029284].{{br}}Part 2. Mar/Apr. (4): 216-228. [[doi (identifier)|doi]]:[https://doi.org/10.2307/3029832 10.2307/3029832].{{br}}Part 3. May/Jun. (5): 279-283. [[doi (identifier)|doi]]:[https://doi.org/10.2307/3029000 10.2307/3029000].}}{{br}}See p. 220: [[William Oughtred]] used the letter {{pi}} to represent the periphery (that is, the circumference) of a circle.</ref><ref name="Smith-1958">{{Cite book |url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA312 |title=History of Mathematics |last=Smith |first=David E. |date=1958 |publisher=Courier Corporation |isbn=978-0-486-20430-7 |page=312 |language=en}}</ref><ref>{{Cite journal|last=Archibald|first=R.C.|date=1921|title=Historical Notes on the Relation {{math|1=''e''<sup>−(''π''/2)</sup> = ''i''<sup>''i''</sup>}}|jstor=2972388|journal=The American Mathematical Monthly|volume=28|issue=3|pages=116–121|doi=10.2307/2972388|quote=It is noticeable that these letters are ''never'' used separately, that is, {{pi}} is ''not'' used for 'Semiperipheria'}}</ref> (Before then, mathematicians sometimes used letters such as ''c'' or ''p'' instead.{{sfn|Arndt|Haenel|2006|p=166}}) The first recorded use is in [[William Oughtred|Oughtred's]] ''Clavis Mathematicae'' (1647).<ref>See, for example, {{cite book |url=https://archive.org/details/bub_gb_ddMxgr27tNkC |title=Clavis Mathematicæ |last=Oughtred |first=William |date=1648 |publisher=Thomas Harper |location=London |page=[https://archive.org/details/bub_gb_ddMxgr27tNkC/page/n220 69] |language=la|trans-title=The key to mathematics}} (English translation: {{Cite book |url=https://books.google.com/books?id=S50yAQAAMAAJ&pg=PA99 |title=Key of the Mathematics |last=Oughtred |first=William |date=1694 |publisher=J. Salusbury |language=en}})</ref>{{sfn|Arndt|Haenel|2006|p=166}} [[Isaac Barrow|Barrow]] likewise used {{nobr|"<math display=inline>\frac \pi \delta</math>"}} to represent the constant {{math|3.14...}},<ref>{{Cite book|chapter-url=https://archive.org/stream/mathematicalwor00whewgoog#page/n405/mode/1up |title=The mathematical works of Isaac Barrow |last=Barrow |first=Isaac |date=1860 |publisher=Cambridge University press |others=Harvard University |editor-last=Whewell|editor-first=William |pages=381 |language=la |chapter=Lecture XXIV}}</ref> while [[David Gregory (mathematician)|Gregory]] instead used {{nobr|"<math display=inline>\frac \pi \rho</math>"}} to represent {{math|6.28...&thinsp;}}.<ref>{{Cite journal |last=Gregorius |first=David |date=1695 |title=Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae |jstor=102382 |journal=Philosophical Transactions |language=la |volume=19 |issue=231 |pages=637–652 |doi=10.1098/rstl.1695.0114 |bibcode=1695RSPT...19..637G|doi-access=free|url=https://archive.org/download/crossref-pre-1909-scholarly-works/10.1098%252Frstl.1684.0084.zip/10.1098%252Frstl.1695.0114.pdf}}</ref><ref name="Smith-1958" />}}
:::As the current version is contradicted by Archibald's reference, I'll implement boldly this version, leaving to others to improve wording and remove unreliable references (nine references seem too much for such a short paragraph. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:13, 13 February 2025 (UTC)
::::Instead of "in the earliest usages", I would directly say "In Oughtred's ''Clavis Mathematicae'' (1647), the Greek letters ..." –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:30, 13 February 2025 (UTC)
:::::{{done}} with some more ce. I removed also a reference that clearly misinterpreted the primary source, and is contradicted by the other references. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:31, 14 February 2025 (UTC)
{{talkref}}

== Historical questions ==

{{u|Jacobolus}} provided above a [https://books.google.co.jp/books?id=KTgPAAAAQAAJ&pg=PP3&redir_esc=y#v=onepage&q&f=false facsimile] of an Ougthred's book. This facsimile contains the formula {{tmath|\tfrac \pi \delta R\mathrm q}} for the area of a circle. As {{tmath|\mathrm q}} is an abbreviation of "quadratus", meaning "squared", this becomes exactly the modern formula, with {{tmath|\tfrac \pi \delta}} in place of {{pi}} and {{tmath|R\mathrm q}} instead of {{tmath|R^2}}.

However, AFAIK, at Ougthred's time, this formula was not interpreted as the result of two successive numerical operations, and was interpreted as "the ratio of the area of a circle to the area of the square built on a radius is the same as the ratio of a circle circumference to a diameter". This property was certainly known to Archimedes, and certainly much earlier. Indeed, it is a corollary of Archimedes' method for computing {{pi}} with inscribed and circumscribed regular polygons, since the area of such a polygon is the product of its perimeter with the distance from the center to a side (radius of the inscribed circle). 

This set several interesting questions, whose answers deserve to be added to the article, if one finds sources for the answers. 

The first one is when and by whom the above ratio equality was first stated.

This ratio equality implies immediately, for modern mathematicians, that [[squaring the circle]] is equivalent with [[ruler and compass construction|constructing]] a line segment whose length is that of the circumference of a circle. It would be interesting to know when and by whom this equivalence has been explicitly recognized. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:50, 16 February 2025 (UTC)

:(Note, it was [[User:Imaginatorium|Imaginatorium]] who linked the book scan above.) –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 18:18, 16 February 2025 (UTC)

== Popular culture for pi? (and tau) ==

I understand the context - including the use of pi and tau in a programming language is not popular culture, whether specified as "computer culture" or not.  It's a mathematical principle, in the same vein as SIN, COS and TAN. [[User:Chaheel Riens|Chaheel Riens]] ([[User talk:Chaheel Riens|talk]]) 11:31, 25 February 2025 (UTC)

:I suppose not—I dislike those sections very much generally. Thanks for checking, I think you should snip it actually. Apologies. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 11:33, 25 February 2025 (UTC)

::I will merge the "in computer culture" to the "in pop culture" section. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 20:15, 25 February 2025 (UTC)

== The number '''{{pi}}''' ==

"The number '''{{pi}}'''" - is possible to state - but how to explain why {{pi}} isn't listed in the sequence of numbers? [[User:Onemillionthtree|Onemillionthtree]] ([[User talk:Onemillionthtree|talk]]) 18:16, 14 March 2025 (UTC)

:What is "the sequence of numbers"? [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:18, 14 March 2025 (UTC)
::The known numbers by the definition of number (not letters): 0-1-2-3-4-5-6-7-8-9 [[User:Onemillionthtree|Onemillionthtree]] ([[User talk:Onemillionthtree|talk]]) 18:24, 14 March 2025 (UTC)
:::If {{pi}} is not a number, then neither are the symbols you have listed. They are [[numeral]]s (names for numbers).  But in any case, not every number is an integer.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:26, 14 March 2025 (UTC)
::::People could think: "The table {{pi}}" but this hasn't happened. If {{pi}} were used as abbreviation of "table" and everyone accepted it - Greece would still use {{pi}} as a letter. "[[Pi]]" which then continues "The number {{pi}}" except in the many other articles (I haven't looked through all search returns) the greek letters spelled out as searches show returns: "letters". "The number {{pi}}" is an auto-thought situation with {{pi}} of a confusion of repeat reinforcement of the association/meaning {{pi}} designates a number (in maths). I'm not trying to "[[divide and conquer]]" that in your meaning "{{pi}} is a number is wrong"; the article should be moved to "Pi (number)". [[User:Onemillionthtree|Onemillionthtree]] ([[User talk:Onemillionthtree|talk]]) 18:42, 14 March 2025 (UTC)
::::::It's not at all clear what you're talking about. There is a way to request a move of the page if you think the [[WP:PRIMARYTOPIC]] should be the Greek letter (which resides at [[pi (letter)]]). [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:48, 14 March 2025 (UTC)
:::::::[[alpha]], [[beta]], [[gamma]], [[delta]], [[epsilon]], [[zeta]], [[eta]], [[theta]], [[iota]], [[kappa]], [[lambda]], [[mu]], [[nu]], [[xi]], [[omicron]], [[pi]], [[rho]], [[sigma]], [[tau]] [[upsilon]], [[phi]], [[chi]], [[psi]], [[omega]]; except "pi" are all "letters" or multi-pages. [[User:Onemillionthtree|Onemillionthtree]] ([[User talk:Onemillionthtree|talk]]) 18:50, 14 March 2025 (UTC)
:::::::is anomalous with "pi number" in the group. [[User:Onemillionthtree|Onemillionthtree]] ([[User talk:Onemillionthtree|talk]]) 18:52, 14 March 2025 (UTC)
:::It's still unclear what you are talking about, but it's worth noting {{xt|here isn't any "consensus" exisiting}} is plainly wrong. Stability of how things are written in articles like these (e.g. articles that have been highly deliberated upon, this is a [[WP:featured article|featured article]]) is explicitly a form of [[WP:consensus|consensus]]. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 18:29, 14 March 2025 (UTC)
::Since the creation of {{pi}} as a letter it then after is known as letter - though the meaning of {{pi}} could be viewed as superceded by the mathematical understanding - this is only because in England (English) any place where [[Greek language| Greek]] isn't spoken and written - the first understanding is that {{pi}} is a number because the letter isn't used in many (any, rarely) other situations in those countries; but Greece is the place which first described the form: {{pi}} which is as a letter - since then no-one thought it was necessary to disagree with the Grecian decision. [[User:Onemillionthtree|Onemillionthtree]] ([[User talk:Onemillionthtree|talk]]) 18:32, 14 March 2025 (UTC)
:::If I can be blunt: your arguments are very confused and hard to understand, to the point I am not sure it is possible to give you actual answers without it taking hours. I would recommend refraining from editing articles in the manner you have previously, unless you are carefully reflecting what is stated by reliable sources. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 18:35, 14 March 2025 (UTC)

== "before Christ" ==

Stating that a certain number of digits of {{pi}} were known "before Christ" strikes me as a bit of a non sequitur, and also a profoundly pro-western bias in a paragraph largely devoted to Chinese, Indian, Egyptian, and Babylonian knowledge.  After all, Jesus Christ had no impact whatsoever on the subject of this article.  Surely this is an object lesson in why the CE/BCE system is preferable in an article like this?  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 16:39, 17 March 2025 (UTC)

:There is no reason why [[WP:DATEVAR]] here should be exceptionalized imo. I tried to phrase it differently also, but my attempt something like {{xt| Approximations prior to the middle of the 1st millennium were ...}} didn't quite work out. I really disagree with the characterization as a non-sequitur: this phrasing is simply neither obscure in meaning nor in actual attestation.  <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 16:46, 17 March 2025 (UTC)

:: It is though. "BC" is not usually mentally expanded to "before Christ", except insofar as it refers to an arbitrary point for the year zero. In contrast, spelling out "before Christ" does not quite feel the same. I agree that an MOS exception could be made, it seems more satisfactory to convert BC to BCE throughout.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 16:53, 17 March 2025 (UTC)
:::No, I'm saying I see no reason why this article would be what is an exceptional case according to [[MOS:DATEVAR]], i.e. where consensus to change systems would be warranted. I do try to be intellectually honest and avoid tendentiousness when I say I hold strongly that the MOS should say {{xt|The default calendar eras are Anno Domini and Common Era}} and generally mean it, or it should say something else.
:::I agree that the phrasing {{xt|does not quite feel the same}} in a minor sense, but I don't think that in itself is a good reason to ponder the above. I think it may be best to try and write around it, as per my attempt above. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 17:09, 17 March 2025 (UTC)
::::Sorry, I'm not trying to be tendentious, and sincere apologies if that was inferred. I'm just wondering if there's a way to be more inclusive and less jarring in the phrasing. I think using common era throughout the article is a good way forward, just to be clear.  I don't have really stong opinions, fwiw. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:01, 17 March 2025 (UTC)
:::::Not at all—I'm merely saying that I always feel bad insisting and reiterating on my position here. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 18:02, 17 March 2025 (UTC)
:::::I would prefer BCE and CE, but that is a change requiring consaensus. In no event should BC or AD be spelled out. I see no problem with antiquity. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 12:30, 18 March 2025 (UTC)
::::::The problem with antiquity is that we are saying that no better-than-two-digit approximation was known in that time, and for "BC" it is true but for "antiquity" it is false, because a four-digit approximation ws known to Ptolemy in 150 AD, still antiquity. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:57, 18 March 2025 (UTC)
:::::::: I've been bold. It's clear that the problem is less "before Christ" than basically an anachronistic hodge-podge attempting to sort out some kind of historical priority. For one thing, "antiquity" usually refers to [[classical antiquity]] (not, e.g., much earlier stuff from Babylon or Egypt, nor geographically removed things from China or India). I'm not entirely happy with the present form of the first paragraph of the history section, but I think the current version may make forward edits more clear.  "Before Christ" is a totally arbitrary and anachronistic marker that has nothing to do with the subject of the article, in any case. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 17:03, 18 March 2025 (UTC)
::::::::: I think Tito's version is an improvement, but still sorta weird.  I had not been aware that "antiquity" was a specific time period; I thought it just meant "a really long time ago".  Maybe we could go with something like "before the second century AD", which still avoids naming a specific date but is more easily understood. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:22, 18 March 2025 (UTC)
::::::::::Perhaps something like "ancient approximations were accurate to only one or two decimal places, but in the second century AD Ptolemy ..."? That would leave the slight weirdness, to my eyes, of "decimal places", a peculiarity of modern competitive computation and memory feats (usually we'd talk about knowing something to 1% or one part in 30 or some such) but maybe we're stuck with that. Certainly we shouldn't use "before Christ", as if Jesus was a leading mathematician, and it would be contrary to [[WP:ERA]] {{tq|Use either the BC–AD or the BCE–CE notation}}. [[User:NebY|NebY]] ([[User talk:NebY|talk]]) 21:09, 18 March 2025 (UTC)
::::::::::"Before Christ" (or anything making the same claim) also has the problem that any approximation given by Ptolemy was likely known a century or more prior. We just don't know for sure because nearly all of Ptolemy's own sources have been lost. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 21:13, 18 March 2025 (UTC)
:::::::::::I like NebY's version.  We can write it to clarify that Ptolemy may have been passing on earlier knowledge. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:14, 18 March 2025 (UTC)
::::::::::::Decimal is also anachronistic because Ptolemy expressed the approximation in [[sexagesimal]]. (The value he used is 3;8,30 in sexagesimal, 377/120 ≈ 3.141666.) —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:28, 18 March 2025 (UTC)
::::::::::::: I don't see that as fundamental.  We're using "two decimal places" as a measure of accuracy, not a specification of the representation. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:36, 18 March 2025 (UTC)
::::::::::::Thanks; I've produced a phrasing that refrains from saying Ptolemy originates it. I've included "surviving" because we have so little relevant material from ancient history; there might have been better approximations somewhere, we just don't know. This phrasing does without mentions of antiquity; we shouldn't put up stumbling blocks for our general readers, making them pause to puzzle out which antiquity is classical and when the middle of it was. [[User:NebY|NebY]] ([[User talk:NebY|talk]]) 13:48, 19 March 2025 (UTC)
::::::::::::::: Looks good. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 14:30, 19 March 2025 (UTC)
::::::::::::::: OK, so I was bold again, and I think with {{ping|NebY}}'s critical improvement, this is now looking very reasonable. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 23:26, 21 March 2025 (UTC)

== Error in the sexagesimal value? ==

Should this line:    The first five sexagesimal (base 60) digits are 3;8,29,44,0,47    (with a semicolon)
be:                  The first five sexagesimal (base 60) digits are 3.8,29,44,0,47    (with a radix point) [[Special:Contributions/66.170.45.75|66.170.45.75]] ([[User talk:66.170.45.75|talk]]) 18:07, 17 March 2025 (UTC)

:Never mind.  I should have read-up more on Sexagesimal.  https://en.wikipedia.org/wiki/Sexagesimal [[Special:Contributions/66.170.45.75|66.170.45.75]] ([[User talk:66.170.45.75|talk]]) 18:18, 17 March 2025 (UTC)

== Article mispell ==

Why is it "formulae" instead of formulas or formula? My first language isnt english, but can you explain me that if its not a error? [[User:Noloblox|Noloblox]] ([[User talk:Noloblox|talk]]) 14:23, 25 March 2025 (UTC)

:Does your preferred English dictionary not give formulae as a plural of formula? Mine do. [[User:NebY|NebY]] ([[User talk:NebY|talk]]) 14:28, 25 March 2025 (UTC)
::Well, i dont have any dictionary. I learned english from myself, so thanks you to tell me it. [[User:Noloblox|Noloblox]] ([[User talk:Noloblox|talk]]) 11:00, 26 March 2025 (UTC)
:''Formula'' is a Latin word, whose Latin plural is ''formulae''. In English both ''formulae'' and ''formulas'' are common. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:42, 25 March 2025 (UTC)

== On WP main page for  pi day 2026? ==

This article was [[Wikipedia:Today's featured article/July 22, 2012|on the WP main page 13 years ago]], when it [[WP:Featured_article_candidates/Pi/archive1|first obtained FA status]].   I'm planning to nominate this article to be on the [[WP:TFA|front page of WP]] for [[pi day]], next year.  I wanted to do it this year, but got too busy. The [[WP:Today's featured article/requests|requests don't open]] until a few months before, so that would be about Dec 2025.  Between now and then I may touch-up the article a bit to make sure it is still FA quality.   It is already in pretty good shape, but I can see a few small improvements:   removing the footnote superscripts from the Lead section; moving a couple of really minor facts from the body into footnotes; touching up picture locations; ensuring that citation formats are uniform, section [[Pi#Projective_geometry]] probably needs more evidence that it has a special relation to pi (beyond pi merely appearing in some formulae);  section [[Pi#Total_curvature]] has no cites at all;  etc.   If anyone has any suggestions, let me know. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 15:53, 28 March 2025 (UTC)

== Better sources for  "Total curvature" and "Projective geometry"? ==

The section [[pi#Role and characterizations in mathematics]] is pretty good, but  there are a couple of subsections within it that seem only tangentially related to pi.  A subsection should only be created for a given subject if the subject has a special relationship to pi.  Merely using pi or depending on pi is not sufficient.  To be included, the subject should have a particularly noteworthy relationship with pi; or it the subject should define pi in a novel way; or  the relationship with pi is documented in sources that are ''about'' pi;  or pi appears in the subject in a surprising way (and is noted by sources).   

A couple of  subsections do not appear to  meet the standard of a strong relationship with pi:    [[pi#Total curvature]] and [[pi#Projective geometry]].  The former has no citations, and the latter has a single cite that is not particular to pi (source is:  Ovsienko, V.; Tabachnikov, S. (2004). "Section 1.3". Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge Tracts in Mathematics. Cambridge University Press. ISBN 978-0-521-83186-4.  ).

Can anyone provide sources that demonstrate a unique & strong relationship between those two subjects and pi?  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 19:24, 31 March 2025 (UTC)

:For total curvature, Arroyo Garay & Mencia "When is a periodic function the curvature of a closed plane curve?" (''Amer. Math. Monthly'') looks very relevant. It sources the material quoted here and a converse fact that a periodic function is the curvature function of a closed curve (possibly with different period) iff its integral over the period is a rational multiple of 2{{pi}} (and that the period is indeed different if it is not an integer multiple). Another textbook source is Prasalov's ''Differential Geometry'' where this is one of the earliest topics covered and a whole section is devoted to it (Section 1.3, "The Total Curvature of a Closed Plane Curve", pp. 7-10). Prasalov names the theorem that the total curvature of a simple closed curve is &pm;2{{pi}} the "Umlaufsatz" and credits it to [[Heinz Hopf]] (1933). The article on [[total curvature]] is badly sourced, but the lack of sources here is an artifact of the way its lead is excerpted to form the content here, and does not indicate that there is any lack of sourcing for this material in the literature. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 20:24, 31 March 2025 (UTC)
::The excerpt template is in my opinion a terrible misfeature, an inappropriate use of [[transclusion]]. (Transclusion is excellent for many other purposes on Wikipedia.) We'd be better off copy/pasting the content here and then modifying the content to be relevant for each context (lead of [[Total curvature]] vs. subsection here) separately, including adding sources as appropriate here. The content appropriate for a lead section is almost never an appropriate summary for a subsection, and typical Wikipedia sourcing conventions differ between lead vs. body of articles. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 22:06, 31 March 2025 (UTC)
:::I agree. In fact probably we should just go ahead and do this here. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 22:16, 31 March 2025 (UTC)
::::After sources are identified, the next question (also for   Projective Geometry) is:  do the sources suggest that the relationship to π is sufficiently unique/important to include in the π article?  Phrased another way: there are thousands of formulas that include π, what is the threshold rule to include a formula in the "Role and characterizations in mathematics" section?  Do  sources state that the relationship to π is especially important?  

::::For example: regarding total curvature: the fact that the integral of the curvature is a multiple of π seems kind of intuitive.   Do any sources say that the occurrence of π is especially significant?  Do any sources state that π's presence is surprising?   Or is the presence of π simply an artifact of the fact that the paths are looping around a point?  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 22:35, 31 March 2025 (UTC)
:::::It looks like the Total curvature section was added by [[user:fgnievinski]] in 2023, so I posted a note on their talk page asking for their input. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 23:16, 31 March 2025 (UTC)
:::::: [[Template:Excerpt]] served its purpose, which was to draw attention to a remarkable relationship between the two topics. Now that someone cares more about it, a custom text will get written. However, it'd be a pity if the main article is not improved with the new sources. I might be inclined to merge the improvements back into the main article. And in case the two texts are similar enough, [[template:excerpt]] can always be restored, to avoid future duplicate work. We're here to improve the encyclopedia, not just this specific article. [[User:Fgnievinski|fgnievinski]] ([[User talk:Fgnievinski|talk]]) 23:35, 31 March 2025 (UTC)
:::::::The {{t|excerpt}} template should, quite frankly, never be used for any purpose on Wikipedia. It is strictly inferior to simple copy/paste in every case. Among other problems, it breaks the link between article history and the article's actual content, making it impossible to read specific timestamped static versions of Wikipedia articles. It should ''certainly'' not be used to make texts that were intentionally changed to more appropriately match their separate contexts the same, forcing changes from one place into another where they are inappropriate. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 23:52, 31 March 2025 (UTC)
::::::::[[WP:TFDHOWTO]]. [[User:Fgnievinski|fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:09, 1 April 2025 (UTC)
:::::::::I agree it would be a good idea to just eliminate the template and avoid the associated problems, but it surely has some dedicated promoters who are active in community discussions, and Wikipedia processes are notoriously incapable of changing anything with nontrivial adoption and motivated defenders, so this sounds like a lot of effort on a big political fight for a low chance of success. All I can do is remove it in articles I am paying attention to, where we can have a concrete discussion in each case, and hope that other editors in other areas of Wikipedia do the same so that the template doesn't infect the whole project too broadly. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 19:47, 1 April 2025 (UTC)
:::::::@[[User:Fgnievinski|Fgnievinski]] - You wrote "...a remarkable relationship between the two topics.."  Can you provide a source that discusses the relationship, and specifies how it is remarkable?    Otherwise, it is just another equation (out of tens of thousands)  that has pi in it.  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 00:06, 1 April 2025 (UTC)
::::::::I think the "remarkable relationship" is that if you travel around a closed curve, the direction you are pointing ends up doing a whole number of turns. This seems relatively inevitable and unremarkable to me, but may still be worth mentioning in this article. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 19:49, 1 April 2025 (UTC)
:::::::::I guess it's a stronger statement about radians being the natural unit of angles, as the result is not just N. [[User:Fgnievinski|fgnievinski]] ([[User talk:Fgnievinski|talk]]) 20:41, 1 April 2025 (UTC)
::::::::::Thanks for the reply.  A couple of questions:  (1) do you have any sources that remark on the relationshiop (vs just your own thoughts)?  (2)  What is so special about pi in relation to total curvature, it seems like a trivial and unremarkable relationship. </br>
::::::::::My feeling is that the appearance of pi here is nothing special ... just one of thousands of formulas related to circles & loops around the origin.  Curvature = 1/r.  Total integral around a circle = 1/r * 2*pi*r = 2 * pi.  If we include total curvature equation in this article, why not include thousands of others?  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 15:54, 2 April 2025 (UTC)
:::::::::::@Fgnievinski - Pinging on this ... did you find any sources that talk about the relationship between pi and total curvature? Any given editor might find  dozens of equations (that include pi)  fascinating, but it doesn't seem wise to include them in this article unless some source says that the  relationship is somehow notable or speical. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 00:35, 6 April 2025 (UTC)
:I took a quick crack at clarifying the description of total curvature, but feel free to copyedit or rewrite if you have a better one. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 04:32, 1 April 2025 (UTC)

Trying to get closure on whether or not these two subsections are appropriate for this article:    Personally, I don't think either have a very special relationship to pi.  But it looks like editor D. Eppstein found a source (based on his comments above).    What about the Projective Geometry subsection? I don't see a source identified that states that   Projective Geometry  has a special/unique relationship to pi. I don't think we should rely on the say-so of the editor that added the section -  it is best to find a source that says "Wow! look at this amazing, unexpected occurrence of pi"  or "Here is very unique appearance of pi that surprised the great mathematician J. Smith". Best would be a book/article ''about'' pi that identifies Projective Geometry (or at least that piece described in the subsection) as having a special relationship to pi. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 03:33, 19 April 2025 (UTC)

:I am worried that the "projective geometry" section might secretly just be an in-joke, a [[mathematics made difficult]] way of writing that the sin and cos functions have half-period {{pi}}. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 07:27, 19 April 2025 (UTC)
::I think the point here is that {{pi}} rather than 2{{pi}} appears in the spectrum of the Laplace operator on <math>RP^1</math>, which is why I think that this "mathematics made difficult" section serves a useful role. In a way, it's not that sine and cosine have some half-period, but that the most primitive kind of one-dimension geometry implicates {{pi}} rather than 2{{pi}}. I.e., {{pi}} is a natural and necessary quantity in one-dimensional projective geometry. I could see this being dealt with in a different form in the article (e.g., alongside the discussion of Landau), but the inescapability of {{pi}} even in *projective* geometry, seems worth noting.  (Not sure about sourcing, to which I defer to others.) [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 20:22, 19 April 2025 (UTC)
:::If it were trying to be expository rather than obfuscatory, it would describe the space of functions <math>p\sin(x+q)</math> directly rather than only indirectly describing it as the solution to a certain second-order differential equation. And it would observe that these functions have roots that repeat with period {{pi}} rather than indirectly defining the subfamily of these functions that have roots at a particular value <math>t</math> without ever saying that these are the functions <math>p\sin(x-t)</math>, and then indirectly stating that the mapping from <math>t</math> to this family is periodic with period {{pi}} (because adding {{pi}} to the offset negates the parameter <math>p</math>). —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 20:54, 19 April 2025 (UTC)
:::: That seems reasonable, although perhaps bringing tangent into the mix. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 20:59, 19 April 2025 (UTC)
:::::I know less than zero about Projective Geometry, so have no opinion about how special the relationship is to pi.  I'm interested in this mostly from the [[WP:RS]], [[WP:V]], and [[WP:OR]]  angle.   It's kinda cringe-inducing when an editor adds a subsection into this article simply because they ''think'' that the relationship is special.  So, we need to find a quote from a source where the source says ''"This is a remarkable situation where pi unexpectedly appears"'' or ''"Here is an unusual formula that can be used  to define pi in a unique way"'' or something like that.  If no source can be found, maybe the Projective Geometry section should be pruned (until a source is located).  Otherwise, the article could bloat over time.  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 21:35, 19 April 2025 (UTC)
::::::I think the projective geometry section is trivial and should be removed. But I think your argument is bad. What you are looking for is not the sort of thing people write even when it is true. Good expository writing, the sort we should use as a reliable source, would show us that pi appears in some situation, and leave it to the reader to infer that it is significant that it appears because otherwise why would it have been described in that way, rather than telling the reader what to believe. [[Show, don't tell]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:47, 19 April 2025 (UTC)
:::Isn’t this just another way of saying that if we associate pairs of antipodal points on the circle we only end up with a half-circle’s worth of unique pairs? –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 04:29, 20 April 2025 (UTC)
::::I think so, yes. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 05:46, 20 April 2025 (UTC)
::::I think not.  A circle requires a metric, but there is no metric here.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 12:21, 20 April 2025 (UTC)
:::::Yes there is, the metric on the one-dimensional space of parameters t, in which this paragraph follows trivially from the fact that the absolute value of the sine repeats with with half the period of the sine itself. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:22, 20 April 2025 (UTC)
:::::: There is a natural one-dimensional metric, but not a two dimensional metric. There is no circle: a plane figure in a two-dimensional Euclidean space. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:13, 20 April 2025 (UTC)
:::::::"There is no circle" is false. This passage explicitly considers (in obfuscatory language) the one-dimensional real projective line as being the circle modulo its central symmetry. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:28, 20 April 2025 (UTC)
:::::::: Does it? The projective line is the quotient of the vector space V, not a circle inside V.  There is a natural symplectic form on V, but I'm not sure about a metric.  Maybe <math>v^2 + (v')^2</math>?  I guess the point is that the "circle" isn't obfuscated so much as it is emergent or unnecessary. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 12:11, 21 April 2025 (UTC)
:I think the section about total curvature can be removed, as it speaks more of N than pi. 
:By the way: have you tried searching for "popularization of mathematics + pi"? It may turn up the kind of tertiary sources you're after. [[User:Fgnievinski|fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:08, 20 April 2025 (UTC)

== Uniform convention for page ranges in citations... per FA ==

The article is FA. FA requires uniform cite formats. That includes how page ranges are presented.  The article has a mixture: some end in period, some do not.  Either way is fine, but using periods tends to be more useful & readable. Also, template  <nowiki>{{sfn}}  </nowiki>is used in the article, and that ''always'' ends page ranges with a period.  Unless anyone objects ([[User:Jacobolus]]?) my plan is to change all to use periods.  I'm guessing it is about 20 cites that need the change.  Again, this article needs to satisfy the FA requirements.  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 00:03, 1 April 2025 (UTC)

:@[[User:Noleander|Noleander]] If I look back a few years (e.g. when this article was made an FA, [[special:permalink/495660335]]), almost none of the short footnotes used periods. Over time the number with periods increased moderately but were still a minority. Usually it's easiest to leave the prevailing version, per [[WP:STYLEVAR]], so I'd go for removing all of the periods (in {{t|sfn}} you can add <code>|ps=none</code>), but I don't personally care much either way. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 01:26, 1 April 2025 (UTC)
::Thanks for replying.  The FA reviewers are pretty picky these days.  What was passable in 2012 often won't fly today.  As far as I know, every single article that has achieved FA status in recent years had periods after all page ranges in all citations.   If you don't object, let's use periods for this article.  As I mentioned above, I'm going to try to get this pi article on the WP main page on pi day in 2026; to make that go smoothly,  the article shouldn't have any issues that would raise eyebrows at  [[WP:TFA]].  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 01:47, 1 April 2025 (UTC)
:::{{tq|i=yes|every single article that has achieved FA status in recent years had periods after all page ranges in all citations}} – so in other words to pass FA now requires deliberately violating the Wikipedia Manual of Style's recommendations about style variations? Bizarre. (Did you take a look at the version of this article from 2012? I'd say it's pretty consistent, and seems totally fine; inconsistent styles have slowly crept in over time afterwards.) –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 01:53, 1 April 2025 (UTC)
::::Yes, I took a look at the [https://en.wikipedia.org/w/index.php?title=Pi&oldid=495660335 the 2012 version of this pi article that achieved FA status] ... in fact, [[Wikipedia:Featured_article_candidates/Pi/archive1|I wrote it]].  The page range formatting in that version is ugly and inconsistent, based on current FA standards. FA articles are expected to  keep up with current FA standards.  Indeed, there is a whole process for that at [[WP:Featured article review]], and articles that were once FA are routinely demoted if they fail to meet current standards.  I'm happy to do all the work required to make the citations in this pi article consistent, I'm not asking anyone to do any work.  I'm not doing it randomly: it is expressly (as I mentioned above) to make sure that the article meets current FA standards by March 2026, so it the article can be on the WP main page on pi day.  I agree with you that once an article adopts a particular convention (imiperial vs metric;  UK english vs American; etc) it should not be changed arbitrarily thereafter.   But the cite format changes I'm proposing are not arbitrary: they are to ensure that the article meets the requirements to appear on the WP front page next year.  Is it okay with you if we go with "periods at end of page range" approach?   [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 15:46, 2 April 2025 (UTC)
:::::Just to be clear, the part you find "ugly" is just the decision not to end short footnotes with a final period? Can you explain more about which part is inconsistent? Is there somewhere in the featured article criteria or manual of style where it says this is a requirement? Is there a relevant RFC somewhere?
:::::Again, I don't care strongly about this, and looking at the article history it does seem like you were the person who established the previous style so I'm happy to defer to you about it, but treating this as a requirement seems quite arbitrary and contrary to [[WP:STYLEVAR]]. [[Wikipedia:Citing sources]] doesn't say anything about it. Short footnotes lacking a final period are common on Wikipedia, and don't seem to cause any problem. Is this just some weird FA reviewer pet peeve that somehow became the FA gospel? –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 23:35, 2 April 2025 (UTC)
::::::The "ugliness" I perceive is the fact that the page ranges sometimes end in period, and some do not.  The fact that they follow a pattern of "short" vs "long" cites is not significant:  I've never seen an FA review that applied different rules for "short" vs  "long" cites.  Also, the usage of the <nowiki>{{sfn}}</nowiki> template has been steadily growing  over the years, so why burden future editors of this article with the obligation to include a "no period" tag?   
::::::The FA reviewers do have certain standards that may not be found ''explicitly'' in writing in WP MOS .. but many fall under the umbrella of the concrete, written rule:  "citations must use a consistent style".  Some cites have author-link tag, but others don't? That's a problem. Some page ranges use hyphen, but other use n-dash? That's a problem.
::::::If you agree that it would be cool to have this article on the WP front page for pi day next year, then it is prudent to go through the article now (before proposing at TFA) and do an informal Featured Article Review (FAR) ourselves: fix any issues that may prevent it from appearing on the front page.  Otherwise, the people at TFA may say ''"gee, its been 13 years since it got promoted to FA, it has probably fallen into disrepair.  We should put it through the FAR review process and make sure it is up to today's FA standards."''  Isn't it better if we perform that task now, 11 months before pi day 2026, rather than do it the month before? 
::::::Since you don't care strongly about the page # issue, can we put the issue to bed? Our time is probably better spent evaluating the [[ pi#Role and characterizations in mathematics]] section and ensuring that the recently added subsections have sources that demonstrate the the subsections are sufficiently noteworthy for this article. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 20:19, 3 April 2025 (UTC)
:::::::{{tq|i=yes|"The fact that they follow a pattern of "short" vs "long" cites is not significant"}} – Sure it is. One is a full citation formatted using our ugly but ubiquitous "CS 1" style which for better or worse includes piles of gratuitous periods everywhere as terminators for each piece of metadata, and the other is basically a sentence fragment.
:::::::{{tq|i=yes|"Some cites have author-link tag, but others don't? That's a problem." }} – Seriously? With all due respect, anyone worrying that not every citation's author has a wikilink attached to it needs to get a life.
:::::::{{tq|i=yes|"Some page ranges use hyphen..."}} – this is outright incorrect typography and we have a bot that aggressively fixes it.
:::::::Anyway, it seems like there are no complaints so by all means put periods at the end of these fragment footnotes if you want. If you do, please put the periods outside of the template rather than in the {{code|1=pages=X–Y}} parameter, where they don't belong. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 21:04, 3 April 2025 (UTC)
::::::::It is fully consistent to do any one of:
::::::::*Never use author-link tags
::::::::*Use author-link tags for every bluelinkable author
::::::::*Use author-link tags for only the first occurrence of every bluelinkable author
::::::::*Use author-link tags for every bluelinkable author and for some redlinked authors without bluelinks who are deemed to be likely to be notable
::::::::None of these should be problematic with respect to FA status. For consistency, though, we should choose one of these and stick to it. My usual preference is to link every bluelinkable author. That entails someone checking that we haven't missed any of them. Noleander seems the most insistent on this point and therefore the obvious volunteer to do this check. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:36, 3 April 2025 (UTC)
:::::::::Sure, I'd be happy to do that. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 23:37, 5 April 2025 (UTC)
::::::::I ensured that all the harvnb templates have final period outside the template. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 00:01, 6 April 2025 (UTC)

== Another (notable?) occurence of π in number theory ==

The truth of [[Riemann hypothesis]] is widely known to imply the "best" or "tightest" bound on the difference between π(x) (not about pi number itself, but this is the [[prime-counting function]]) and li(x) (the [[logarithmic integral|log integral]]), the formula for which REALLY INVOLVES the 1/(8π) factor [see ''Von Koch'' (1901)]. Can the aforementioned stuff be placed somewhere in this article or not? (I have doubted whether this occurence of π is notable enough or not) 
[[Special:Contributions/2001:EE0:4F0E:1A70:417D:FECF:A620:337F|2001:EE0:4F0E:1A70:417D:FECF:A620:337F]] ([[User talk:2001:EE0:4F0E:1A70:417D:FECF:A620:337F|talk]]) 05:40, 1 April 2025 (UTC)

== Improved formula box layout .. no content change ==

In section [[Pi#Computer_era_and_iterative_algorithms]] there is a box containing an algorithm, the layout was wonky (spanned full page & was way above the associated body text).   I improved it here:  https://en.wikipedia.org/w/index.php?title=Pi&diff=1284169011&oldid=1284166727   Content of body/text/box is unchanged.   Let me know if there are any concerns. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 00:18, 6 April 2025 (UTC)

== Moved cites from lead down into body ==

The lead had about 6 or 7 superscript cites in it.  There was no rhyme or reason as to which sentences had cites, and the vast majority of other sentences, which did not have cites.  I moved the cites down into the body, adjacent to the body sentences on the  associated topics. There was one peculiar cite (in the lead) that was a blog that I removed as it seemed unreliable and I could not determine which body sentence it was most appropriate for (it appeared to be emphasizing that pi was known known to over a trillion digits, which is already amply sourced in other cites).  The lead has no cites now.  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 00:23, 6 April 2025 (UTC)

== Footnote with five redundant sources? ==

The body has the sentence:  {{green|Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including  with which he computed 20 digits of π in one hour}}.  That sentence  has a footnote which begins {{green|Sandifer ... }}.  That footnote is currently number [82].  That footnote is bulleted and has five distinct sources.   Bulleted cites are discouraged in FA articles, but they are acceptable when used in moderation. Footnote [82] is not moderate.  I haven't started scrutinizing the five individual sources yet, but it  seems like that could be pared down to one or two. If anyone wants to keep all five, speak up. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 22:49, 6 April 2025 (UTC)

:Surely you should check that the sources actually source what they are claimed to source rather than bean-counting.
:We need sources at this point in the article for four separate claims:
:*"popularized this series in his 1755 differential calculus textbook"
:*"later used it with Machin-like formulae"
:*the specific formula pi/4=5 arctan 1/7 + ...
:*"computed 20 digits of π in one hour".
:If it takes six sources for all of this (one secondary source for each of these four claims and two original publications by Euler retained for historical reference) then it takes six sources. If these claims are not in any of the five sources given, on the other hand, then we have a problem. Whether they are formatted as a single bulleted footnote or as six separate footnotes should be irrelevant.
:The Sandifer source does not contain the 1755 date, and does not mention 20 digits in one hour. Its quote "Euler has a whole repertoire of such formula" will suffice for "later used it with Machin-like formulae" (p2), because the formulae described at this point of Sandifer are indeed Machin-like. It does not give the specific formula in exactly the form we use, but does give an equivalent formula with the denominator cleared (p6).
:The first Hwang source, "Some Observations on the Method of Arctangents for the Calculation of π" (name mis-ordered; Hwang is the surname), now removed, does source the specific formula in the form we use, credited to Euler, and does also source the "20 digits in one hour" claim for Euler's use of this formula. Its only mention of 1755 is for a related arctangent formula, not this material.
:The second Hwang source, "An elementary derivation of Euler’s series for the arctangent function", again mentions the 1755 arctangent formula but is otherwise irrelevant.
:I conclude that the removal of the first Hwang source was a mistake; we need both that source and Sandifer. The removal of the second Hwang source was correct. And we still need a secondary source for "popularized this series in his 1755 differential calculus textbook". —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:04, 7 April 2025 (UTC)
::Of course the sources should be checked before deletion, as I said ''"I haven't started scrutinizing the five individual sources '''yet''', ..."  ''  My post above was asking for input on the general principle of paring down sources when they are redundant or not needed.   Did someone delete some of the sources?  They appear to be all intact. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 02:25, 7 April 2025 (UTC)
:::@[[User:Noleander|Noleander]] someone else removed the two Hwang sources but they're back now —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 04:23, 7 April 2025 (UTC)
::::Ah, thanks.  Based on your research I suppose all those [82] sources should remain. I may tidy their appearance  a bit cuz I recently got dinged in a feature article nomination because my I used bullets in a couple of my citations. [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 05:47, 7 April 2025 (UTC)
:::::We just use separate footnotes elsewhere like [22][23], so maybe we should do the same here. But I think "An elementary derivation of Euler’s series for the arctangent function" can be removed even if the others can't. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:44, 7 April 2025 (UTC)
::::::Hwang (2005) isn't necessary for verifying the claim in the article, but is a nice accessible derivation of this formula for any reader curious about mathematical, rather than historical, details. Maybe it would be helpful to add a line of prose to the footnote saying that directly. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:29, 7 April 2025 (UTC)
:::::My recommendation is to just put {{t|pb}} between multiple sources in one footnote. It would probably be a good idea to consolidate some of the other consecutive footnotes in this article, as there's really not much benefit to putting 3 footnotes in a row when none of them is used elsewhere. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:18, 7 April 2025 (UTC)
::It may be hard to find a source directly saying "popularized", but if you look around most people credit this to Euler, sometimes citing his book specifically (e.g. [https://www.jstor.org/stable/2299198 Bennett 1925], {{tq|i=yes|"Another familiar series for arctangent x is the following: ... given by Euler in 1755."}}; [https://www.jstor.org/stable/1403789 Dodge 1996] {{tq|i=yes|"Finally in 1755, using arctan formula, Euler found one that converged much faster than any other: ..."}}; Arndt & Haenel 2001, {{tq|i=yes|"But Euler discovered an elegant way to reduce the the amount of work. He derived the identity ... (5.26) [Euler, 1755], and inserted the arguments of (5.25) into it."}}; Hwang 2004, {{tq|i=yes|"If this identity is used together with Euler's series for the arctangent, discovered in 1755..."}}). You have to go digging to find out that Newton came up with the same formula 90 years prior. Roy (2021) says, p. 214: {{tqb|Newton developed his profound ideas on interpolation and finite differences starting in the mid-1670s. In the early 1680s, he applied the method of differences to infinite series and in June/July of 1684, he wrote two short treatises on the topic....  The first chapter of the second treatise dealt with infinite series in a manner similar to that of his early works of 1669 and 1671. However, the second chapter employed the entirely new idea of applying finite differences to derive an important transformation of infinite series, often called Euler’s transformation.... Newton noted one remarkable special case of his transformation: {{tmath|1=\textstyle \tan^{-1} t = \ldots}} (10.4)}} p. 215: {{tqb|Newton's transformation (10.4) for the arctangent series is obviously important, so it is not surprising that others rediscovered it, since Newton's paper did not appear in print until 1970. In August 1704, Jakob Bernoulli communicated the {{tmath|1= t = 1}} case of (10.4) to Leibniz as a recent discovery of Jean Christophe Fatio de Duillier. Jakob Hermann, a student of Bernoulli, found a proof for this and sent it to Leibniz in January 1705. This proof is identical with that of Newton's when specialized to {{tmath|1= t = 1}}. Johann Bernoulli, and probably others, succeeded in deriving the general form (10.4). Bernoulli, in fact, applied the general form in his 1742 notes on series and thereby derived a remarkable series for {{tmath|\pi^2}} found earlier by Takebe Katahiro by a different technique. In 1717, the French mathematician Pierre Rémond de Montmort (1678–1719) rediscovered Newton's more general transformation (10.3) with a different motivation and method of proof.}} later p. 408 {{tqb|Bernoulli gave no further details in his letter, but in 1742 he offered an explanation in the fourth volume of his ''Opera Omnia''. His method was to divide Newton's transformed series for {{tmath|\arctan t}}, (10.4), written up in the ''De Computo'', by {{tmath|\textstyle 1 + t^2}} and then integrate. Recall that Newton had not published his work, so the alternative series for {{tmath|\arctan t}} in powers of {{tmath|\tfrac{t}{1+t^2} }} was Bernoulli’s rediscovery....}} –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:54, 7 April 2025 (UTC)
:: Instead of "Euler popularized", it might be more accurate to say something like "This series is now often credited to Leonhard Euler, who included it in his 1755 differential calculus textbook ...". –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 18:19, 7 April 2025 (UTC)
:: I'm not sure where the 20 digits in under an hour claim originates. It can also be found in Borwein & Borwein 1987, p. 340. Edit: an earlier source is [https://archive.org/details/diequadraturdes01beutgoog/page/n52/mode/2up Beutel 1913, pp. 46–47]. While we're here, I wonder if it's worth explicitly pointing out, in the article text, that Euler was able to do this because {{tmath|1=\textstyle 7^2 + 1 = 50 = \tfrac12(100)}} which facilitated doing computations with decimal arithmetic. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 22:03, 7 April 2025 (UTC)
:I think it is usually worth linking both directly to original sources and to a secondary source about it, whenever possible. There's really no harm in adding links to original sources. Some readers may be curious to see what the historical sources look like and find them to be helpful – at least, I know I find such sources to be helpful, and I wish Wikipedia articles included them much more often. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:17, 7 April 2025 (UTC)

== "[[:Draft:Three-point-one-four-one-five-nine]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
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The redirect <span class="plainlinks">[//en.wikipedia.org/w/index.php?title=Draft:Three-point-one-four-one-five-nine&redirect=no Draft:Three-point-one-four-one-five-nine]</span> has been listed at [[Wikipedia:Redirects for discussion|redirects for discussion]] to determine whether its use and function meets the [[Wikipedia:Redirect|redirect guidelines]]. Readers of this page are welcome to comment on this redirect at '''{{section link|1=Wikipedia:Redirects for discussion/Log/2025 April 28#Various draftspace redirects}}''' until a consensus is reached. <!-- Template:RFDNote --> [[User:Xoontor|Xoontor]] ([[User talk:Xoontor|talk]]) 15:09, 28 April 2025 (UTC)

== Pi article on WP main page: request submitted for [[pi day]] 2026 ==

FYI:  I nominated this [[pi]] article to appear on the main page of WP, on [[pi day]],  March 14, 2026.  The request is at  [[WP:Today%27s_featured_article/requests/pending]].  There is no guarantee that it will end up appearing, because (a) the TFA coordinators may receive other nominations that are more significant, or recently achieved FA status; (b) Another FA article with special attachment to March 14 may get nominated for that day; or (c) the TFA coordinators may think that [[pi day]] is too USA-centric, since most of the  world uses "14-3" rather than "3-14" to designate March 14th.  [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 18:08, 6 May 2025 (UTC)

:You can just propose [[May Day|31 April]] as a fallback day for Europeans. ;) –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 19:18, 6 May 2025 (UTC)
::😂   [[User:Noleander|Noleander]] ([[User talk:Noleander|talk]]) 19:38, 6 May 2025 (UTC)