Talk:Pythagorean theorem

Why Zimba proof was deleted?

Why short Zimba trigonometric proof (main idea) from this revision https://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&oldid=1149322678#Jason_Zimba_trigonometric_proof%5B25%5D was deleted? @David Eppstein Kamil Kielczewski (talk) 19:14, 11 April 2023 (UTC)

There are literally hundreds of proofs of the theorem, maybe thousands. Picking out and including only one of these, sourced only to its primary publication, makes no sense, because there is no clear selection criterion for it that would not also cause us to also include hundreds of other proofs. We should only include proofs with significant historical recognition, not recent flash-in-the-pan media hype and even more not primary sourced but otherwise non-notable proofs vaguely connected to recent flash-in-the-pan media hype. —David Eppstein (talk) 19:39, 11 April 2023 (UTC)

I agree with David Eppstein; the Zimba proof is insufficiently noteworthy. —Quantling (talk | contribs) 19:49, 11 April 2023 (UTC)

The criterion for it is that is very short, simple and use only calculations without involving geometry (in direct way) like other proofs. So it can be very useful especially for people who hat not goot geometrical intuition (so we are dealing here with usability for a wider audience)

In the other side, for historical point of view, this is also first known trigonometrical proof. Kamil Kielczewski (talk) 20:02, 11 April 2023 (UTC)

The claims of being especially simple or of being the first non-circular trigonometric proof need secondary sources. We cannot make those claims based only on the original primary publication. —David Eppstein (talk) 20:06, 11 April 2023 (UTC)

The information about "first non-circular trigonometric proof" was not included into deleted proof (in the same way like "primality" (in some way) of the some other proofs on this page).

Simplicity is obvious because tricky part is only adding zero by: x-(x-y) (and use some old known formulas) - I doubt anyone will describe such obvious things in an article. Kamil Kielczewski (talk) 20:19, 11 April 2023 (UTC)

This is missing the point. Arguments here for why it's a good proof are not what is needed to justify its inclusion. If nobody has written secondary sources singling it out as a good proof, we cannot include it. —David Eppstein (talk) 20:26, 11 April 2023 (UTC)

Ok, here is secondary source which mention that this is first trigonometric proof:

"OTHER TRIGONOMETRIC PROOFS ON PYTHAGORAS THEOREM", N. Luzia, 2015, https://arxiv.org/pdf/1502.06628.pdf Kamil Kielczewski (talk) 20:37, 11 April 2023 (UTC)

That is not reliably published. And it has no depth in its coverage of the Zimba publication. —David Eppstein (talk) 20:54, 11 April 2023 (UTC)

Zimba should be included, here is a published reference of it https://www.tandfonline.com/doi/full/10.1080/00255572.2025.2607284 ~2026-19457-39 (talk) 19:38, 28 March 2026 (UTC)

It was the first proof of PT to use trig. to get there. And this has been confirmed by whatever mathematical societies matter in the US. It was an achievement recognized by many academic bodies, so I'm pretty sure that if it wasn't accurate, it would have come to light by now. Honestly, half of your reasoning sounds petty and bitter. 46.33.96.32 (talk) 16:15, 1 November 2024 (UTC)

The Zimba proof relies on the angle-addition formula for sines. However with that formula and γ = α + β, the result is more immediate: one can insert sin α = cos β = a/c, cos α = sin β = b/c, and sin γ = 1 into sin γ = sin α cos β + sin β cos α to give 1 = (a/c)² + (b/c)². —Quantling (talk | contribs) 20:26, 12 April 2023 (UTC)

Pythagorean theorem dates from more than 1,000 years; trigonometry date from more than 500 years. Since them, hundred of great mathematicians have studied their relationship. So it is very unlikely that something really new can be found on this subject. So, for mentioning Zimba's proof, one requires a secondary source that attests that this is really new. This is really unlikely that this will ever occur for the following reason. The fundational principle on which is based trigonometry is that the trigonometric ratios depend only on one acute angle of a right triangle, and do not depend on the size of the riangle. This is directly used in the proofs of § Proof using similar triangles and § Trigonometric proof using Einstein's construction. Any other trigonometric proof must use this foundational principle. All the proofs suggested in this talk page use this foundational principle and some other trigonometric properties. This makes them definively less interesting and less elegant than the proofs that are already there. So, they have a low encyclopedic value and do not deserve to be mentioned. D.Lazard (talk) 21:39, 12 April 2023 (UTC)

in your proof you use a,b,c (from geometry object - triangle) - but Zimba use only two arbitrary angles x and y (without involving geometry in direct way like you). Kamil Kielczewski (talk) 09:49, 14 April 2023 (UTC)

If you don't want a, b and c, the shorter-than-Zimba proof gets even shorter. With the angle-addition formula for sines and γ = α + β, the result is immediate: one can insert cos β = sin α, sin β = cos α, and sin γ = 1 into sin γ = sin α cos β + sin β cos α to give 1 = sin² α + cos² α. —Quantling (talk | contribs) 13:35, 14 April 2023 (UTC)

$\sin \gamma =1$ cannot be used because the trigonometric definition of sine as ration of opposite side to hypotenuse does not apply, namely, you cannot have two right angles inside a right triangle! Zimba was careful to note that trigonometric functions of angles $0$ or ${\frac {\pi }{2}}$ cannot be directly used. Danko Georgiev (talk) 11:36, 15 April 2023 (UTC)

In your proof, you assume that sin α = cos β and sin γ = sin(α + β)=1 - I'm not sure that this assumptions are independent of Pythagorean theorem - you also didn't explain where you got these assumptions from? (from geometry - triangle?). Zimba assumptions was weaker than your - he use arbitrary x and y angles and assume only that 0 < y < x < pi/2. (so he did not have to refer to any geometrical figure). This is why Zimba proof is quite interesting and qualitative different from other proofs. Kamil Kielczewski (talk) 14:48, 14 April 2023 (UTC)

As I see it, the opposite of α is the adjacent of β (and vice-versa) when they are from a right triangle, so sin α = cos β and cos α = sin β follow immediately from the definitions that Zimba gives for sin and cos. Zimba uses that α (well, "x" in his notation) is from a right-triangle when he argues that sin² α + cos² α = 1 leads to (a/c)² + (b/c)². (In contrast, instead of β = π/2 − α, Zimba uses an unrelated angle "y".)

I see that Zimba argues that sin and cos as he defines them are defined only on the open interval (0, π/2), but not at 0 or at π/2. I'm not sure why he couldn't have simply specified the value of those functions at those points and then shown that the subtraction formulas still work when one or more of their inputs are in this expanded domain. Perhaps he considered that less elegant than the approach he did take.

I am curious. Does Zimba claim to be the first to observe that the angle-subtraction formulas for sine and cosine can be proved without assuming the Pythagorean theorem? Does Zimba claim to be the first to observe that the subtraction formulas can be used to prove sin² α + cos² α = 1? Does Zimba claim to be the first to put these two thoughts together? Does Zimba claim that his approach is distinct from previous approaches because he avoided using sin and cos at 0 and π/2? —Quantling (talk | contribs) 16:22, 14 April 2023 (UTC)

You use sin, cos and γ, α, β with asumption sin α = cos β and sin γ = sin(α + β)=1

He use sin, cos and angles x,y with asumption 0 < y < x < pi/2.

I think that if your sin/cos funtions are the same as Zimba sin/cos functions (at least in (0,pi/2)) then whe shoud not refer to they definitions when we compare proofs - because you both uses same functions.

Zimba only shows that functions sin/cos can be defined independent of Pythagorean theorem, to be sure that using them in proofs is allowed.

But back to the proofs themselves - his proof is just pure symbolic and base only on sin/cos properties (substraction formulas) (which is somehow beautiful), your proof (I supose) need to relate to some triangle.

I'm not sure that Zimba was first - but if not, then should exists similar results before him. But so far I haven't found any Kamil Kielczewski (talk) 17:32, 14 April 2023 (UTC)

Yes, we'd need a secondary source to make any claim that a proof was 'first'. We can't rely on what editors happen to have found themselves. MrOllie (talk) 17:53, 14 April 2023 (UTC)

Yep, but deleted proof (here) not contains information that it was first. Kamil Kielczewski (talk) 18:02, 14 April 2023 (UTC)

You claimed it was first further up this page. But the text in the article itself presented no indication that it is noteworthy - Which is why it got deleted. Subjective claims about simplicity and simplifying things on the talk page might be a fun diversion, but the only way a mention could stay in the article is with good support from secondary sourcing - and not in the form of self-published arxiv stuff. - MrOllie (talk) 18:06, 14 April 2023 (UTC)

@Kamil Kielczewski: Regarding "his proof is just pure symbolic ... your proof (I supose) need to relate to some triangle." He uses triangles, but I suppose that you mean right triangles. Yes, agreed, he gets all the way to sin² x + cos² x = 1 without referring to a right triangle, though he needs a right triangle for the next step, to get to (a/c)² + (b/c)² = 1. @MrOllie: Agreed! —Quantling (talk | contribs) 18:15, 14 April 2023 (UTC)

yep, agree Kamil Kielczewski (talk) 18:37, 14 April 2023 (UTC)

@D.Lazard @MrOllie I found a solution to this impasse.

Currently in the article in the Algebraic proofs section there is a proof based on this source - so you consider this source to be reliable.

Well, Zimba's proof has also been included in this source which you found reliable (because you allowed this source to be used on this page for many years) here.

In both proofs in this source there is information about who is considered to be the first author of the proof (12th century Hindu mathematician Bhaskara, and Jason Zimba) - although in both proofs on Wikipedia this information is not provided.

Therefore, it can be consistently assumed that information about Zimba's proof is based on reliable sources (unless you have double standards) Kamil Kielczewski (talk) 08:14, 15 April 2023 (UTC)

The fact that a proof is sourced from a unreliable source does not means that there are not reliable sources for this proof. In fact, the Cut-the-knot page for the algebraic proof refers to several older sources (one is almost 2,000 years old). On the other hand, the Cut-the-knot page for Zimba's article refers only to Zimba's article.

Also, comparing Zimba's proof with that of § Trigonometric proof using Einstein's construction, I cannot see any advantage of Zimba's proof: both use the definition of sine and cosine given in Trigonometric ratios and similarity of right triangles. The latter is simple and direct, while Zimba's proof requires an elaborated geometrical construction and the proof of an auxiliary trigonometric formula.

Also, the last sentence of Zimba's introduction suggest that his aim is to prove the Pythagorean trigonometric identity without using the Pythagorean theorem, rather that proving the Pythagorean theorem without using Pythagorean trigonometric identity. This suggests that his article is not primarily about a proof of the Pythagorean theorem. In any case, § Trigonometric proof using Einstein's construction can be easily modified for proving both simultaneously.

These are technical reason for not including Zimba's proof, but, again, the main reason for not including it is that inclusion requires WP:Notability, and Zimba's article is not notable enough for being mentioned. D.Lazard (talk) 10:02, 15 April 2023 (UTC)

Zimba should be included. It's original if we view this task as a natural one in functional equations, as in this new reference: https://www.tandfonline.com/doi/full/10.1080/00255572.2025.2607284.

It's certainly more interesting or original than Einstein's. Einstein is included because of idol-worshipping. ~2026-19333-53 (talk) 14:18, 28 March 2026 (UTC)

The definition of trigonometric functions given in Trigonometric ratios is standard from centuries on, and is independent from Pythagorean theorem. So, Zimba's definition has nothing new. As Pythagorean theorem is about right triangles, it is impossible to provide a proof that does not involve any right triangle. The trigonometric proof given in the article does not require subtraction formula or any other trigonometric identity. D.Lazard (talk) 18:20, 14 April 2023 (UTC)

I'm surprised by what you write - can you provide a link (or explain it) to a trigonometric proof which not require any other trigonometric identity? Kamil Kielczewski (talk) 18:35, 14 April 2023 (UTC)

Look at § Trigonometric proof using Einstein's construction. D.Lazard (talk) 10:05, 15 April 2023 (UTC)

standard from centuries on, – To be precise, this definition dates from about the middle of the 18th century, and became standard somewhere around the middle of the 19th century. –jacobolus (t) 18:46, 14 April 2023 (UTC)

New trigonometric proof

This video by polymathematic demonstrates a trigonometric proof of the Pythagorean theorem recently discovered by Calcea Johnson and Ne'Kiya Jackson, two high school students at St. Mary's Academy in New Orleans, who recently presented it at the (2023?) Spring Southeastern Sectional Meeting of the American Mathematical Society. They used a pure (mostly) trigonometric proof, using what they call a "waffle cone" geometric construction to arrive at the equation a² + b² = 2ab / sin (2a) = c². It would be nice to add this to the article, in the "Trigonometric Proofs" section. (I'm not sure how to present this proof myself.) — Loadmaster (talk) 22:57, 23 April 2023 (UTC)

See multiple long discussions above, starting at § Proof using trigonometry —David Eppstein (talk) 07:16, 24 April 2023 (UTC)

Archived discussion is here. — Loadmaster (talk) 23:20, 30 October 2024 (UTC)

Font style for named points

Moved from User talk:Jacobolus

Hi, in a recent edit to [Pythagorean Theorem] you [jacobolus] stated it would be better to standardize on LaTeX rather than math templates, but at the very least named points and variables should use italics.

According to MOS:MATH#Graphs and diagrams,

There is no general agreement on what fonts to use in graphs and diagrams. In geometrical diagrams points are normally labelled using upper case letters, sides with lower case and angles with lower case Greek letters.

Recent[when?] geometry books tend to use an italic serif font in diagrams as in for a point. This allows easy use in LaTeX markup. However, older books tend to use upright letters as in and many diagrams in Wikipedia use sans-serif upright A instead. Graphs in books tend to use LaTeX conventions, but yet again there are wide variations.

For ease of reference diagrams and graphs should use the same conventions as the text that refers to them. If there is a better illustration with a different convention, though, the better illustration should normally be used.

I read that bold statement to be transitive: the text that refers to the diagrams should try to match the diagrams. It's easier to change text to match the diagram, than it is for most people to create and upload corrected diagrams with italics.

As far as "it would be better to standardize on LaTeX...", that's strictly editors' opinion, and MOS:FORMULA doe not prescribe that in any way.  — sbb (talk) 19:37, 10 November 2025 (UTC)

The font chosen for mathematical symbols, including symbolic names of points in geometric diagrams, should be consistent through the article and should be distinct from running prose. Italic letters do a better job at that than roman letters, and are correspondingly much more common across Wikipedia. As you say, this is partly because LaTeX, used for tricky mathematical expressions sitewide (including this page) default to using italic symbols.

In my opinion it's not essential that we rush to switch every diagram to the same letter style, and I don't expect readers to be confused between seeing A, A, A, or A in a diagram label and A or ⁠$A$⁠ in the text. Beyond inconsistent labels, the diagrams are extremely inconsistent in all aspects of their style, including size, line styles, color palette, etc., and readers can understand that they were made by various authors with various aesthetic preferences/skills using various tools. However, it is quite confusing to switch back and forth between different font styles for math embedded in prose from one paragraph to another within the article based on whichever font each diagram author happened to use.

You are right that using LaTeX vs. math templates is a matter of personal preference. There was previously a mix of LaTeX math, bare wiki-markup, and math templates (including both roman and italic fonts) from one section of this article to another. Standardizing the style from section to section seems like a reasonable goal. You opted to convert the LaTeX examples to math templates wherever possible. This does lead to inconsistency from one line to the next when block math expressions use LaTeX and inline ones don't, but it is allowed by MOS:FORMULA, so I won't try to revert those changes. It's a reasonable enough standard style to choose. However, in my opinion switching italics to roman letters throughout the article, or using a mix of roman and italic fonts for the same symbols within the article body, is quite harmful. –jacobolus (t) 20:55, 10 November 2025 (UTC)

To reply specifically to the quoted passage from MOS:MATH#Graphs and diagrams; I believe you are misreading. As I understand it, the point of this passage is to advise editors: if you make a new diagram for an encyclopedia article, you should try to match the prevailing style chosen for the mathematical symbols in the article. However, if you are choosing between existing diagrams made by someone else to use as an illustration for some section of an article, choose based on the quality of the diagram overall rather than just which font each one uses. –jacobolus (t) 21:19, 10 November 2025 (UTC)

If an article mixes {{math}} and <math> for inline math expressions then I find it acceptable to change the article to choose one over the other. However, when an article starts as (at least almost) exclusively one of them, a wholesale switch to the other one too easily leads to pointless edit warring, please avoid that.

If one sees naked plain text, such as ''x'', rather than plain text in math font, such as {{mvar|x}}, I find it acceptable to change the former to the latter.

The diagrams should adhere to the article text's convention when feasible. Not the other way around. —Quantling (talk | contribs) 21:26, 10 November 2025 (UTC)

Some of the diagrams have italicized points (A, B, etc.) and some upright (A, B, etc.). The paragraphs describing each diagram are separated by headings or subheadings; it's very clear which text applied to which diagram. However, it would be even clearer for the text to more accurately reflect the diagram it's describing. When possible, ideally the diagrams should be updated to be consistent article-wide; however, until such time, it's much easier to make the local text formatting consistent with the nearby diagrams.  — sbb (talk) 21:47, 10 November 2025 (UTC)

Picking different fonts on different articles is okay. Switching fonts for the same symbols from one paragraph to another within the same document is, in my opinion, absurd. It's unexpected and mystifying for readers, is something that never happens in any other kind of source, and is not supported by the wikipedia manual of style (cf. MOS:VAR, MOS:CONSISTENT, MOS:DATEVAR). Using different fonts between text and diagrams is not ideal, but is also common (inside and outside of Wikipedia) and not that hard for readers to make sense of. –jacobolus (t) 22:11, 10 November 2025 (UTC)

Italic mathematical symbols and exponents in a hatnote

This article includes the hatnote which formerly said:

In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB² denotes the square of the length AB and not the product A × B².

I changed this to:

{{hatnote|In this section, and as usual in geometry, a "word" of two capital letters,
such as {{mvar|AB}} denotes the length of the [[line segment]] defined by the points
labeled with the letters, and not a multiplication. So, ''{{math|''AB''{{sup|2}}}}''
denotes the square of the length {{mvar|AB}} and not the product
''{{math|''A'' × ''B''{{sup|2}}}}''.}}

Which does not italicize the multiplication sign or the numerical exponent. In my opinion this is better for two reasons. Most importantly: mathematical expressions should obey the rules of mathematical typography rather than the font style of the surrounding prose (including in a hatnote). But also: the expressions used in the article which we are referring to don't italicize mathematical symbols or exponents, so it is more helpful to readers to use the same style.

User:Sbb seems to have a serious problem with this (it's not clear on what grounds) and has reverted my change 3 times in a row with edit summaries "fixing {{math}} font formatting in hatnote", "Not broken: entire phrase is in a hatnote; it's okay to italicize the exponent when the entire paragraph is italic explanatory", and finally "Please stop randomly reverting every improvement". Does anyone else prefer italicized exponents? –jacobolus (t) 21:36, 10 November 2025 (UTC)

Without getting into the heart of the dispute. My understanding of math typography is that the exponents should not be in italic unless they are variables. A.Cython (talk) 21:41, 10 November 2025 (UTC)

No, italicized exponents are incorrect, bad, and wrong, Sbb's edits are incorrect, bad, and wrong, and reverting them was appropriate. In mathematical formulas, italicization is part of the semantics; it should not be reversed merely for consistency with the surrounding text.

One alternative might be to use LaTeX-math rather than template-math for the formulas in this hatnote; that would automatically give them their mathematically-correct italicization without the nested-double-single-quote hacks. —David Eppstein (talk) 21:43, 10 November 2025 (UTC)

Jacobolus's suggestion to "manually recode / manually typeset" an entire explanatory hatnote just so that two exponents' aren't italicized is ridiculous. It's a hatnote. The semantic meaning of the hatnote markup and that it's emphasis and italicized for emphatic, exceptional reasons.  — sbb (talk) 21:51, 10 November 2025 (UTC)

I wasn't recommending that, but only proposing it as an alternative if the current markup is considered unacceptable for some inscrutable reason. Also, you leaving a big warning banner on my talk page is in poor taste. Please keep further discussion about the topic on this page where it is relevant. –jacobolus (t) 22:00, 10 November 2025 (UTC)

I left a banner on your talk page because I'm required to, in order to make a 3RR dispute complaint. You have reverted my edits 4 times in 24 hours. I'd prefer to leave this part of the conversation on your talk page, and not here. But you dragged it here. I find your actions equally in poor taste.  — sbb (talk) 22:30, 10 November 2025 (UTC)

You reverted my edit 3 times, and I reverted your revert 3 times. Is your goal a mutual double ban or something? Or if you prefer we can revert the article to the stable version from yesterday pending discussion, and decide on changes via formal RFC. –jacobolus (t) 22:34, 10 November 2025 (UTC)

Again, this isn't the forum for this. Please.  — sbb (talk) 00:50, 11 November 2025 (UTC)

You're right... but this is merely explanatory text in a hatnote (which is italic by default), describing that in this section/article, AB² means the square of the geometric side AB and not A×B². That's it, an entire out of normal context explanatory hatnote (which is what hatnotes are for). It gets rendered as ...and not A×B² simply because the entire thing is inside the hatnote. That's it. No other italicizing of exponents.

The alternative suggestion to jump through weird formatting hoops just to not get an italicized ² (in a hatnote, mind you) is frankly, tedious and missing the larger semantic point.  — sbb (talk) 21:59, 10 November 2025 (UTC)

Weird formatting hoops are more or less fine. Semantically incorrect mathematical typography is in my opinion not acceptable. –jacobolus (t) 22:02, 10 November 2025 (UTC)

sbb, your comment here gets the mathematics wrong again in a different way. Italicized variable names are different from upright variable names. It is not ok to switch back and forth between the two styles for the same variable. Jumping through formatting hoops is better than wrong notation. —David Eppstein (talk) 22:26, 10 November 2025 (UTC)

I argue that it's not the wrong notation, because it's in a hatnote. I'm trying to make those variables consistent, and not have a mish-mash of italics, upright roman, and italic or upright san serif.

Overly strict holding to non-italics exponents in an italic emphatic explanatory paragraph is the non-sensical approach, IMO.  — sbb (talk) 22:33, 10 November 2025 (UTC)

Mathematical typography often seems non-sensical to newcomers, but it has its own logical internal structure. –jacobolus (t) 22:35, 10 November 2025 (UTC)

Update: apparently my workaround is misunderstood by the mediawiki parser which wraps the <i> around the wrong thing and leads to a linter error. So I tried an alternative workaround:

In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB² denotes the square of the length AB and not the product A × B².

This is even uglier markup, but hopefully now not broken. –jacobolus (t) 22:29, 10 November 2025 (UTC)

Just so that everybody can see what we're talking about clearly, below is a table of my (sbb) edit and result, and jacobolus's (latest proposed) edit and result. IMO, the simpler wikitext syntax more than makes up for the minimal gains of having a non-italicized exponent in an explanatory hatnote where everything is italicized anyways.

sbb version	jacobolus version
{{hatnote|1= In this section, and as usual in geometry, a "word" of two capital letters, such as {{math|AB}} denotes the length of the [[line segment]] defined by the points labeled with the letters, and not a multiplication. So, {{math|AB{{sup|2}}}} denotes the square of the length {{math|AB}} and not the product {{math|A × B{{sup|2}}}}.}}	{{hatnote|1= In this section, and as usual in geometry, a "word" of two capital letters, such as {{mvar|AB}} denotes the length of the [[line segment]] defined by the points labeled with the letters, and not a multiplication. So, <span style="font-style:normal">{{math|{{mvar|AB}}{{sup|2}}}}</span> denotes the square of the length {{mvar|AB}} and not the product <span style="font-style:normal">{{math|{{mvar|A}} × {{mvar|B}}{{sup|2}}}}</span>.}}
In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB2 denotes the square of the length AB and not the product A × B2.	In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB2 denotes the square of the length AB and not the product A × B2.

 — sbb (talk) 01:15, 11 November 2025 (UTC)

It does not matter how elegant the code is, the end result is what matters, and the end result should not have the exponents in italic if they are not variables. My 2 cents.A.Cython (talk) 01:21, 11 November 2025 (UTC)

I'm amenable to some simpler version of the markup if someone has a better solution. The point is not just to make the markup gratuitously illegible, but to fix the incorrect typography of the original. For clarity: the original hatnote was added by D.Lazard in 2023, which at some point was slightly modified. Here are the original version and the version from yesterday:

D.Lazard 2023 version	Yesterday version
{{hatnote|In this section, and as usual in geometry, a "word" of two capital letters, such as {{mvar|AB}} denotes the length of the [[line segment]] defined by the points labeled with the letters, and not a multiplication. So, {{math|AB{{sup|''2''}}}} denotes the square of the length {{mvar|AB}} and not the product <math>A\times B^2.</math>}}	{{hatnote|In this section, and as usual in geometry, a "word" of two capital letters, such as {{mvar|AB}} denotes the length of the [[line segment]] defined by the points labeled with the letters, and not a multiplication. So, {{math|''AB''{{sup|2}}}} denotes the square of the length {{mvar|AB}} and not the product <math>A\times B^2.</math>}}
In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB2 denotes the square of the length AB and not the product $A\times B^{2}.$	In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB2 denotes the square of the length AB and not the product $A\times B^{2}.$

The former has the digit '2' correctly non-italicized, but is a bit confusing because it achieves that end by wrapping it in an <i> tag. –jacobolus (t) 02:01, 11 November 2025 (UTC)

If I would have written this hatnote yesterday instead of two years ago, I would have used {{tmath}} everywhere instead of a mix of {{mvar}}, {{math}} and <math>. Hopefully, this would have avoided this discussion. D.Lazard (talk) 02:31, 11 November 2025 (UTC)

This article was just today switched from a random mishmash of math typesetting methods to using math/mvar templates for inline expressions and LaTeX for block expressions. Switching to LaTeX in running text would also solve the problem. –jacobolus (t) 02:58, 11 November 2025 (UTC)

{{math}} everywhere is exactly what I wrote, but it kept getting reverted, because that's apparently unacceptable because it italicizes the exponent. Again, in an italicized paragraph. I'm really surprised this is such a sticking point.  — sbb (talk) 03:58, 11 November 2025 (UTC)

(Notice: {{tmath}} is a LaTeX template, not the same as {{math}}.) –jacobolus (t) 04:22, 11 November 2025 (UTC)

(Dammit, small text with poor readers, reading dark mode at night. I misread {{tmath}} in D.Lazard's msg as {{math}}. Nevermind)  — sbb (talk) 18:01, 11 November 2025 (UTC)

Ptolemy’s theorem and law of cosines

We could mention that Pythagoras’s theorem is a special case of Ptolemy’s theorem, which we prove independently of Pythagoras itself. It’s also proven by the law of cosines which, despite normally being proved by using Pythagoras, can also be proved by constructing 3 triangle altitudes and calculating c^2-a^2-b^2 (as shown in our Wikipedia article https://en.wikipedia.org/wiki/Law_of_cosines#from three altitudes). Overlordnat1 (talk) 13:39, 18 December 2025 (UTC)

Semi-protected edit request on 24 January 2026

Remove the link on the word conjecture to the wikipedia page of the same name The word conjecture in the following sentance (located under "Other proofs of the theorum" and within that, under "Proof using similar triangles"): "One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development at that time." does not refer to a maths conjecture, which is the article that it links to. Rather it refers to a conjecture in the dictionary definiton of the word. Blob2030 (talk) 13:05, 24 January 2026 (UTC)

Good catch. I've made the change. Will Orrick (talk) 13:43, 24 January 2026 (UTC)

Intersecting Chords Theorem

Pythagoras’s theorem can be used to prove the intersecting chords theorem by expressing the radius in two different ways by using Pythagoras’s theorem and equating the expressions, so this is another consequence we could mention. Overlordnat1 (talk) 13:48, 27 January 2026 (UTC)

Four-part history

@Will Orrick @David Eppstein et al. — the article text currently includes The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system., but this looks like WP:SYNTH to my eyes. If we can't find a source to back this "four parts" analysis, I'd like to remove it from the article. —Quantling (talk | contribs) 19:18, 24 March 2026 (UTC)

To accurately state "the Pythagorean theorem was known to X culture" we need to be unambiguous about what aspect of the theorem was known to each culture. To do otherwise would lead to (and has led to) endless arguments about how culture X was first because they happened to mention a specific right triangle once, but no, culture Y was first because they were the only ancient culture to use axiomatic deductive reasoning to prove things, etc. etc. Or, alternatively, we would have a haphazard list of historical achievements with no organizing principle. How do you propose to achieve this necessary unambiguity and organization, without a summary statement like this? (I'm not convinced that "knowledge of the relationships among adjacent angles" makes sense as a description of one of the ways the theorem was known, but that's a different issue.) —David Eppstein (talk) 20:31, 24 March 2026 (UTC)

This statement looks like a summary to me, not a synthesis. The phrasing could potentially be improved (I am also not sold on "knowledge of the relationships among adjacent angles"), but I don't see a need to remove it completely. Stepwise Continuous Dysfunction (talk) 22:08, 24 March 2026 (UTC)

What if we don't count them but say "includes"? The history of the development of the theorem includes knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system. That way it doesn't sound like we are reporting that these four categories constitute the generally accepted way to fully partition the development of the theorem. —Quantling (talk | contribs) 01:28, 25 March 2026 (UTC)

I like the idea of not counting them, but am a little unsatisfied by this choice because it leaves in the problematic "knowledge of the relationships among adjacent angles" (what is this supposed to mean?) and its first point doesn't distinguish between knowing about the 3–4–5 triangle and knowing Euclid's formula (two very different levels of sophistication).

If I were synthesizing the levels of knowledge myself, they would be something like

1. Measurements of specific right triangles such as the 1–1–√2 and 3–4–5 triangles
2. Method for computing the diagonal of an arbitrary rectangle or the hypotenuse of an arbitrary right triangle
3. Systematic generation of Pythagorean triples
4. Deductive proof of the correctness of these methods

(with the Egyptians being at the first stage, the Babylonians at the second and third, and the later Greeks at the fourth). —David Eppstein (talk) 01:55, 25 March 2026 (UTC)

To me it seems more like somebody's theory than a fair summary of the material presented in the article. There seems to be agreement already that the third stage doesn't belong, but I have objections to all four stages. I see no evidence that any culture knew of Pythagorean triples but not the general relationship among the sides of a right triangle. Old Babylonian tablets show evidence of both knowledge of the general rule and knowledge of numerous triples (and, apparently, a scheme for generating them). It's hard to date these tablets relative to one another, but there's no evidence that one of these things was known before the other. In India, the various Sulba Sutras discuss the general rule, the special case of the isosceles right triangle, and particular triples all within one short section of text. Euclid's Elements contains proofs of both the general theorem and the rule for generating triples. In China we have a visual proof of the theorem for the 3-4-5 triangle, but it seems unreasonable to say that this proof is not completely general. This source is later than the others, although parts of it (but not, I believe, the proof) are thought to be based on much earlier material.

The one case where this is arguable is Egypt, but the current treatment in the article is unsatisfactory. The mention of the Berlin Papyrus is fine, although inconclusive, but the reference to Plutarch is questionable. I tried a quick search to find a scholarly source assessing the passage in Moralia V. Most of the material that came up was of doubtful quality. The only reputable material I was able to find was at the Kiwi Hellenist blog (so not usable usable as a source for our article, but perhaps informative), which has this to say,

"This isn’t mathematics, it’s numerology and wordplay. Worse still, ‘the Egyptians’ is a spurious reference: we know of no Egyptian interest in right-angled triangles. We do know that Pythagorean teachings like reincarnation were often spuriously traced back to faux Egyptian mysticism. Other allusions to the above set of allegories can be found in Aristotle and other writers, and those sources confirm that the allegories originate in Pythagoreanism, not in Egypt (see Burkert 1972: 32-4, 40, 429)."

This is not to say I am confident that ancient Egypt had no knowledge of the Pythagorean theorem or Pythagorean triples, but only that none of the very limited surviving evidence supports that they did.

My reading of all this is that in every example we have of a culture with knowledge of the theorem, we are looking at a snapshot at a point in time where knowledge was already fairly advanced. If there were early stages, we don't know what they looked like.

Long ago the article used to contain speculations about use of Pythagorean triples by henge builders in northwestern France and Britain. These may have been based on something in van der Waerden's book Geometry and Algebra in Ancient Civilizations, but got removed at some point based on their not being taken seriously by most historians. It could be that Stage 1 was meant to refer to that.

To interject my personal opinion, or perhaps my personal confusion, I really don't understand what Stage 1 is supposed to look like. Is the claim that some culture was aware that triangles with sides 3, 4, 5 are right (or at least close to right) but that they hadn't noticed the relation $3^{2}+4^{2}=5^{2}$? Or that they had noticed the relation but didn't realize an analogous statement held for other right triangles? Or something else? The claim the knowledge of Pythagorean triples predated knowledge of the general rule has always puzzled me. I really think a reference is needed explaining and supporting this idea.

I'm also troubled by Stage 4. Why are we being so coy? The criterion is obviously designed so that only the Greeks pass, but it's made to sound as if it's some objective criterion according to some theory of historical development of mathematical ideas. Obviously the deductive system developed in Euclid's Elements is unique in the early history of mathematics and influenced much that came after. But that doesn't have anything in particular to do with the Pythagorean theorem. To me the real question is whether people cared about why the rule was true and were able to demonstrate it to their own satisfaction. My opinion is that the Chinese proof counts and that the Babylonian cut-and-paste geometry methods (inferred from the language used to describe their algorithms) strongly imply that they had a systematic understanding of why the rule holds. I know that Karine Chemla has studied both the cases of India and China, but I have not delved into her work sufficiently to be able to understand her ideas.

Finally, the opening sentence of the first paragraph, "There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof," also bothers me. If there really is scholarly debate, we should be able to locate it and cite it. If there are no sources to be found, then perhaps there is no debate. Perhaps most serious historians feel that knowledge is too fragmentary even to discuss the issue. I guess this isn't entirely true, but it's probably the case that relatively little has been written about this by professional historians. Will Orrick (talk) 03:00, 25 March 2026 (UTC)

Ok, now that is starting to sound like original research. We should omit the mention of Egyptian knowledge of the 3-4-5 triangle in Plutarch because some blog thinks it's too poetic?? We also do have evidence of the Egyptians stretching rope to measure land; what I have not found evidence for is the oft-repeated theory that they stretched 3-4-5 triangles to make right angles. That sort of use (without the theory behind it) is what stage 1 is supposed to look like. (It is obvious that they did know about constructing right angles somehow but we don't have enough evidence to make that connection from the sources we have so far, and we certainly don't have evidence that the Egyptians went beyond stage 1.) —David Eppstein (talk) 05:15, 25 March 2026 (UTC)

A footnote from Knorr has: "Egyptian papyri attest that the Egyptians, perhaps as far back as 2000 B.C., recognized the property of the triple ⁠$3$⁠, ⁠$4$⁠, ⁠$5$⁠ that ⁠$3^{2}+4^{2}=5^{2}$⁠. But historians tend to doubt that they recognized these numbers as forming the sides of a right triangle. Cf. T. L. Heath, Euclid, I, p. 352. The opinion that the Egyptians possessed forms of the 'Pythagorean theorem' on right triangles is argued by M. Cantor. It is, in fact, a view proposed by Plutarch. Cf. K. Bretschneider, Die Geometrie und die Geometer vor Euklides, 1870, p.80." (Link to Bretschneider if anyone reads German...). –jacobolus (t) 05:35, 25 March 2026 (UTC)

Maor's Pythagorean Theorem book gives a few relevant quotations about Egyptian knowledge:

In 90% of all the books [on the history of mathematics], one finds the statement the Egyptians knew the right triangle of sides 3, 4 and 5, and that they used it for laying out right angles. How much value has this statement? None!
—Bartel Leendert van der Waerden.

There is no indication that the Egyptians had any notion even of the Pythagorean Theorem, despite some unfounded stories about "harpedonaptai" [rope stretchers], who supposedly constructed right triangles with the aid of a string with 3 + 4 + 5 = 12 knots.
—Dirk Jan Struik.

There seems to be no evidence that they knew that the triangle (3, 4, 5) is right-angled; indeed, according to the latest authority (T. Eric Peet, The Rhind Mathematical Papyrus, 1923), nothing in Egyptian mathematics suggests that the Egyptians were acquainted with this or any special cases of the Pythagorean theorem.
—Sir Thomas Little Heath.

–jacobolus (t) 05:41, 25 March 2026 (UTC)

Well, yes, that's what I meant about not having evidence of 3-4-5 rope stretching. We have evidence of (straight) rope stretching and we have unscholarly popular texts claiming that the Egyptians used 3-4-5 rope stretching, but the scholarly works do not give these claims much credence. Still, I think the prominence and relative antiquity of Plutarch's explicit claim that they knew of this triangle makes it worth at least a brief mention. —David Eppstein (talk) 05:59, 25 March 2026 (UTC)

@David Eppstein What's with the aggressive tone? I never said that the mention of Plutarch should be removed. Also I acknowledged that a blog post cannot be considered a reliable source for Wikipedia purposes. Nevertheless, Peter Gainsford is a professional classicist, and his views can be used to inform the discussion on this Talk page. He cites Burkert's book, which certainly is a reliable source (although, of course, no single source is definitive, particularly on contentious topics). Unfortunately I no longer have access to that book, so I am unable to determine whether and how it supports Gainsford's argument. What I do know is that historians of Ancient Greece tend to be skeptical in general of Greek attributions of various bits of Greek mathematical knowledge to Egypt.

What I did say in my comment is that the reference to Plutarch is "questionable," which it is. We, at the very least, owe the reader a warning that many historians don't seem to consider the Plutarch passage to be reliable history (as evidenced, for example, by the quotations provided by jacobolus). Right now the article states, "According to Plutarch, the ancient Egyptians did know about the 3:4:5 right triangle, identifying its sides with Osiris, Isis, and Horus respectively." Among the problems with this are the word "ancient". Ancient Egypt spanned 3000 years of history and Plutarch lived at the very end of that. I'm a rank amateur, so I have no idea which Egyptians Plutarch would have had in mind when he mentioned "the Egyptians", but a professional classicist might know. I think that paraphrasing an ancient primary source without providing additional context has the potential to mislead readers. Another problem is "did know". Readers will understand this as a counterpoint to "but the problem does not mention a triangle" from the preceding sentence about the Berlin Papyrus, and may infer that the 3-4-5 triangle was known during the Middle Kingdom period, 2000 years before Plutarch. This is a wild jump, but we are encouraging readers to make it by not providing adequate disclaimers.

All this is a thin thread on which to hang a theory of "Stage 1" cultures. If we suppose, despite all the misgivings, that some Ancient Egyptians did know the 3-4-5 right triangle, it could well be that they also knew the full theorem and the evidence just hasn't come down to us. But let's suppose they just knew the 3-4-5 triangle and nothing more. Isn't it more plausible to see this as a stage of degeneration rather than a precursor stage? Maybe the 3-4-5 triangle filtered into Egypt via surveyors or craftsmen from some other culture that did know the full theory, perhaps Mesopotamia. Obviously this theory has been manufactured from whole cloth, but that's equally true of the "Stage 1" theory.

I appreciate that Stepwise Continuous Dysfunction took the initiative to edit the article, but why did that have to happen before a consensus was reached? I see that there's now a new stage, "calculations involving specific right triangles". Again, I know of no evidence that any culture went through such a stage on the path to developing the Pythagorean theorem. My opinion is that, as it stands, the first paragraph is original research, not even synthesis. Really it's just plain editorializing. Disjointed is bad, but disjointed is better than having an introductory paragraph that's not based on anything solid. Will Orrick (talk) 15:56, 25 March 2026 (UTC)

"Calculations involving specific right triangles" was my attempt to summarize stuff like the Babylonian method for finding the hypotenuse of an isosceles right triangle (equivalently, the diagonal of a square). I think this is a fair summary, but of course I am not absolutely committed to that phrasing. I did not think that the language of the list carried an implication that the topic areas listed were stages that had to be progressed through in strict chronological order. However, I can see why someone might have gotten that impression from it, so I have made a further small modification to dilute that implication. Stepwise Continuous Dysfunction (talk) 19:43, 25 March 2026 (UTC)

I agree that in summarizing these aspects of the topic it is not necessary to refer to them as chronologically-sequenced stages, and recent edits by SCD reflect that. As for "calculations involving specific right triangles", see for example YBC 7289. —David Eppstein (talk) 20:05, 25 March 2026 (UTC)

I think it's anachronistic/misleading to distinguish calculations of a "specific" triangle vs. presenting an algorithm which works in general by way of a concrete example. –jacobolus (t) 21:05, 25 March 2026 (UTC)

True, though the "in general" here means, e.g., "for isosceles right triangles of any size". That's still a "specific right triangle", in that it is "the" 1–1–√2 triangle. Alternate phrasing suggestions welcome! Stepwise Continuous Dysfunction (talk) 02:26, 26 March 2026 (UTC)

No, there are Old Babylonian tablets describing the method of finding the diagonal from two sides of a rectangle or finding one side from the diagonal and the other side using an algorithm equivalent to the Pythagorean Theorem. For example:

The breadth is 2 cubits, the height 0;40 (cubits). What is its diagonal? You: square 0;10, the width. You will see 0;01 40. Square 0;40, the length. You will see 0;26 40. Add it to 0;01 40. You will see 0;28 20. What is the square root? the square root is 0; 41 ... ... . The diagonal.

See doi:10.2307/1359891. –jacobolus (t) 04:14, 26 March 2026 (UTC)

Yes, we have other evidence that the Babylonians had a general method. But if we only had YBC 7289 as evidence, we would only be able to say that they knew about certain specific right triangles (the isosceles right triangle with side length ${\tfrac {1}{2}}$). —David Eppstein (talk) 05:40, 26 March 2026 (UTC)

I don't understand your point of your hypothetical counterfactual. –jacobolus (t) 18:14, 26 March 2026 (UTC)

Being able to calculate the metric properties of a specific right triangle such as the isosceles right triangle is an aspect of the Pythagorean theorem that could be (and has been) used as evidence for some level of knowledge of the theorem. But it is not evidence of having a general method. Individual calculations and a general method are two different things.

To take an unrelated example: we can formulate the concept of the Shannon capacity of a graph. We can calculate it in one specific example, a 5-cycle. But we do not have a general method and do not even know its value for the next nontrivial case, the 7-cycle. So our knowledge of this topic includes calculating specific examples (or one specific example) but not a general method. —David Eppstein (talk) 19:19, 26 March 2026 (UTC)

Okay, but there was clearly knowledge of a general method by ~1700 BC by Old Babylonian mathematicians, with multiple relevant documents produced in different cities. So we shouldn't give an impression that there's any clear evidence about a historical development starting with a few specific examples and moving toward a general theory, since we don't actually have evidence like that. –jacobolus (t) 19:54, 26 March 2026 (UTC)

"Being able to calculate the metric properties of a specific right triangle such as the isosceles right triangle is an aspect of the Pythagorean theorem that could be (and has been) used ...".

Maybe I don't know what you mean by "metric properties", but I'd really like to see a reference for "and has been". To my mind, and I've given this a lot of thought, there's the Meno argument, which gives you the isosceles right triangle, and there's the full theorem. I can't think of any other special case that can be done independently of the full theorem. Will Orrick (talk) 19:55, 26 March 2026 (UTC)

Re a single calculation being used as evidence of Pythagorean knowledge: See for example hdl:2451/63930 which includes the text "YBC 7289, which graphically testifies to Babylonian knowledge of the Pythagorean theorem". (The existence of evidence of more advanced Babylonian knowledge is irrelevant to this example.) Theaetetus is another interesting example: at a certain point the Greeks knew that the non-integer square roots of integers up to 17 were irrational (and therefore for instance that the 1-4 right triangle has an irrational side), prior to developing a general method that went beyond that. —David Eppstein (talk) 20:01, 26 March 2026 (UTC)

I'm not persuaded by Thaetetus. I think it's fair to say that nobody knows why Theodorus stopped at 17. But more importantly, there's no explicit connection made in what Plato writes with right triangles; he only talks about squares of certain areas:

"Theodorus here was illustrating something about powers to us, showing that a three-foot square and a five-foot square do not have a common measure with respect to a single foot. He dealt with each in this way until he reached seventeen, and then he stopped for some reason. Now, since the number of powers appeared unlimited, it occurred to us to try to comprehend them into a unity by which we could refer to all these powers."

It's the incommensurability proof that needs generalizing, not the determination of the length of an unmentioned hypotenuse.

Here's the main point: I think we're all agreed that we can't present a theory of discovery of the Pythagorean theorem since we don't have a reliable source for that. Some of us think the first paragraph should just be deleted; some of us want to keep it as a summary of what follows. I don't think it's fair that that summary include hypothetical situations that we have no evidence actually occurred. (And I don't count a single caption on an artifact at an exhibition curated by experts who were fully aware of all the evidence for knowledge of the full theorem, going back to the 1940s. We can't know how they would have presented things if the artifact were an isolated piece.) Will Orrick (talk) 21:33, 26 March 2026 (UTC)

I am not strongly persuaded by the connection between Theaetetus and right triangles but it does exist in the literature: see Spiral of Theodorus and its sources. —David Eppstein (talk) 21:55, 26 March 2026 (UTC)

Another problematic article. It cites a popular book in support of the idiotic claim, "It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus." Next sentence: "Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him." Of course this is what many historians have surmised, but it is stated as a bald fact.

We don't know whether Theodorus ever used the spiral, but if he did, I don't see how it is evidence for knowledge of special cases of the Pythagorean theorem without the general case. How would that work? The idea that Theodorus knew that a right triangle with legs 1 and 1 has hypotenuse ${\sqrt {2}},$ that a right triangle with legs 1 and ${\sqrt {2}}$ has hypotenuse ${\sqrt {3}},$ and everything up to the statement that a right triangle with legs 1 and 4 has hypotenuse ${\sqrt {17}},$ but got stuck on the right triangle with legs 1 and ${\sqrt {17}}$ because he didn't know the general Pythagorean theorem is...ridiculous.

If you think Theodorus used the spiral, you're going to be pretty confident he knew the Pythagorean theorem Will Orrick (talk) 23:57, 26 March 2026 (UTC)

It's unclear whether you were aiming this rant at me, at those other people who theorized a connection between the passage in Theaetetus about Theodorus, and the spiral of Theodorus, or the authors of our article on the spiral, but I agree with you that the spiral returning to itself seems an unlikely and dubious reason for getting stuck at 17. That's what I meant about "not strongly persuaded".

I am more likely to side with Knorr, who in The Evolution of the Euclidean Elements (pp 181ff) found a plausible argument that handles all the cases less than 17 and then gets stuck at 17. But this argument also involves the measurements of right triangles, with one integer side and integer hypotenuse, and with the remaining side involving the square root you are trying to show irrational. So there's still a connection to this article.

As for "pretty confident he knew the Pythagorean theorem": I think you're missing the point. Yes, I'm pretty confident Theodorus knew the Pythagorean theorem. It was not an example of not knowing the theorem. It was merely an example of a phenomenon where sometimes one can calculate examples but not know the general result that extends them. It happens to be somewhat connected to the Pythagorean theorem but the unknown general result is the irrationality of square roots of non-square integers, not the Pythagorean theorem. It seems reasonable to believe that, similarly, some much earlier mathematician might have known that the 3-4-5 triangle was right but not known the more general pattern. If we believe Plutarch then this might have been the state of knowledge in ancient Egypt. —David Eppstein (talk) 06:50, 27 March 2026 (UTC)

It has even been suggested that the 3-4-5 triangle was the only case that Pythagoras knew or cared about . Stepwise Continuous Dysfunction (talk) 08:15, 27 March 2026 (UTC)

"...I agree with you that the spiral returning to itself seems an unlikely and dubious reason for getting stuck at 17." I do happen to think this, but I never said anything about it; it's irrelevant to our discussion of the history of the Pythagorean theorem.

"... I'm pretty confident Theodorus knew the Pythagorean theorem. It was not an example of not knowing the theorem." I now see that we appear to be on the same page.

@Stepwise Continuous Dysfunction The Stanford Encyclopedia Pythagoras article is a good one. If it is true that his only concern was the 3-4-5 right triangle, then I would say this is likely a case of what I called "degeneration" in an earlier comment, a suggestion that is hinted at in the subsequent text:

"Modern scholarship has shown, moreover, that long before Pythagoras the Babylonians were aware of the basic Pythagorean rule and could generate Pythagorean triples (integers that satisfy the Pythagorean rule such as 3, 4 and 5), although they never formulated the theorem in explicit form or proved it (Høyrup 1999, 401–2, 405; cf. Robson 2001). Thus, it is likely that Pythagoras and other Greeks first encountered the truth of the theorem as a Babylonian arithmetical technique (Høyrup 1999, 402; Burkert 1972a,429). It is possible, then, that Pythagoras just passed on to the Greeks a truth that he learned from the East."

I'm clearly not winning any adherents to my views, so, not having consensus to do so, I will refrain from editing the history section of the article. I will, however, try to find time to write down what I think ought to be done to fix the profound problems in that section, almost every paragraph of which is flawed. Others can decide what to do with it. Will Orrick (talk) 15:57, 29 March 2026 (UTC)

That falls under "understanding the relationship among the sides of a right triangle". To put it anachronistically, they were applying the Pythagorean theorem without (so far as we know) being able to prove it. Stepwise Continuous Dysfunction (talk) 20:24, 26 March 2026 (UTC)

Why do you think they thought of it as applying the Pythagorean theorem, rather than calculating the diagonal of a square?

Incidentally, there's another aspect of knowledge of the Pythagorean theorem that nowadays we teach to elementary school children, long before they see axiomatic deduction, but was not formulated until 1731 (according to Maor pp 133–134): its use as a method for calculating distances between points in coordinate geometry. Obviously, this depends on having a concept of coordinate geometry, invented roughly 100 years earlier by Descartes, so this would not have been understood in ancient cultures. But it is history, so maybe it should be mentioned in the history section? —David Eppstein (talk) 20:32, 26 March 2026 (UTC)

I don't think the indentation on this comment was an error. Stepwise Continuous Dysfunction was replaying to the earlier comment of jacobolus containing the Robson quote, not to your comment. Will Orrick (talk) 20:54, 26 March 2026 (UTC)

Yes, I was saying that the procedure of which Robson gives an example is what we, anachronistically, would call applying the Pythagorean theorem. Stepwise Continuous Dysfunction (talk) 20:58, 26 March 2026 (UTC)

I think the conclusion they were unable to prove it is premature. (Unless you're one of those people who think nothing was proved until Zermelo wrote down his axioms.) They did not write down proofs, but their algorithms appear to have been closely tied to geometrical operations that can be understood as justifying the steps. See in particular the work of Jens Høyrup. Will Orrick (talk) 21:10, 26 March 2026 (UTC)

I think you are misunderstanding. There is no "conclusion they were unable to prove it". There is merely silence on things we have no evidence for. —David Eppstein (talk) 21:57, 26 March 2026 (UTC)

I think we tend to harbor many biases about what ancient people were capable of. The phrase "...without (so far as we know) being able to prove it" is not so neutral compared with other ways the same thought might have been phrased. Will Orrick (talk) 23:21, 26 March 2026 (UTC)

Yes, we have biases about what ancient people were capable of. We also tend to think of ancient mathematics in modern terms. This can lead to thinking that an ancient mathematical technique is cumbersome, when really we're comparing it to the wrong thing and not using it as its practitioners did. Or it can lead to thinking that people in the past must have known that X follows from Y, when that implication is really only obvious in a more recent context. Stepwise Continuous Dysfunction (talk) 00:14, 27 March 2026 (UTC)

All good points. I'm just trying to get away from the attitude that the ancients were like the least competent undergraduate who tries to survive a STEM course by memorizing all the formulas without understanding. Undoubtedly some students in Babylonian scribal schools adopted that procedure, but somebody had to come up with the algorithms. The idea that that person did so by guesswork, with no attempt at understanding is, I guess, possible, but not the most likely hypothesis. Will Orrick (talk) 00:31, 27 March 2026 (UTC)

"depends on having a concept of coordinate geometry, invented roughly 100 years earlier by Descartes" this is also a historically misleading summary, both about what people did thousands of years before Descartes, and also about what Descartes himself did. –jacobolus (t) 22:34, 26 March 2026 (UTC)

"Einstein" proof

It's probably worth mentioning that the attribution of the proof we credit to Einstein is somewhat speculative. The only direct thing we have from Einstein himself is the following:

I remember that an uncle told me the Pythagorean Theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in “proving” this theorem on the basis of the similarity of triangles.... For anyone who experiences [these feelings] for the first time, it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time to be possible in geometry.

Here's Strogatz:

Einstein, unfortunately, left no such record of his childhood proof. In his Saturday Review essay, he described it in general terms, mentioning only that it relied on “the similarity of triangles.” The consensus among Einstein’s biographers is that he probably discovered, on his own, a standard textbook proof in which similar triangles (meaning triangles that are like photographic reductions or enlargements of one another) do indeed play a starring role. Walter Isaacson, Jeremy Bernstein, and Banesh Hoffman all come to this deflating conclusion, and each of them describes the steps that Einstein would have followed as he unwittingly reinvented a well-known proof.

Twenty-four years ago, however, an alternative contender for the lost proof emerged. In his book Fractals, Chaos, Power Laws, the physicist Manfred Schroeder presented a breathtakingly simple proof of the Pythagorean theorem whose provenance he traced to Einstein. Schroeder wrote that the proof had been shown to him by a friend of his, the chemical physicist Shneior Lifson, of the Weizmann Institute, in Rehovot, Israel, who heard it from the physicist Ernst Straus, one of Einstein’s former assistants, who heard it from Einstein himself. Though we cannot be sure the following proof is Einstein’s, anyone who knows his work will recognize the lion by his claw.

–jacobolus (t) 16:03, 28 March 2026 (UTC)

Fractals, Chaos, Power Laws originally came out in 1991. The proof was circulating without attribution to Einstein prior to that. It could have been discovered independently, or it could have spread through scientist gossip through someone like Straus. (It appeals to physicists' affection for scaling arguments.) Stepwise Continuous Dysfunction (talk) 22:12, 28 March 2026 (UTC)

The proof by dissection and scaling is also given in the Project Mathematics! episode about the Pythagorean theorem, which came out in 1988. Caltech put the show on YouTube in 2017; the relevant bit is near the end of the episode (an earlier episode in the series covered scaling and similarity). Stepwise Continuous Dysfunction (talk) 22:29, 28 March 2026 (UTC)

My impression was that the proof is considerably older than Einstein, but it's plausible Einstein also came up with this argument (perhaps at a young age, or perhaps much later, with a different proof in his youth). But we probably shouldn't present the attribution as certain when it's somewhat closer to speculative gossip. –jacobolus (t) 00:51, 29 March 2026 (UTC)

Semi-protected edit request on 11 June 2026

Change "Pythagorean theorum" to "Baudhyayana Pythagorean Theorum". ~2026-34511-62 (talk) 07:57, 11 June 2026 (UTC)

No, Wikipedia's rule is to use the most common name in English textbooks, independently, in case of theorems, of the name of the first discoverer. D.Lazard (talk) 08:52, 11 June 2026 (UTC)