# Talk:Fermat's right triangle theorem

This review is transcluded from Talk:Fermat's right triangle theorem/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: RoySmith (talk · contribs) 21:05, 24 March 2021 (UTC)

I'm starting this review.  My plan is to do two major passes through the article, first for prose, the second to verify the references.  In general, all my comments will be suggestions which you can accept or reject as you see fit. -- RoySmith (talk) 21:05, 24 March 2021 (UTC)


### Checklist
GA review (see here for what the criteria are, and here for what they are not)
It is reasonably well written.
a (prose, spelling, and grammar):  b (MoS for lead, layout, word choice, fiction, and lists): 

It is factually accurate and verifiable.
a (reference section):  b (citations to reliable sources):  c (OR):  d (copyvio and plagiarism): 

It is broad in its coverage.
a (major aspects):  b (focused): 

It follows the neutral point of view policy.
Fair representation without bias: 

It is stable.
No edit wars, etc.: 

It is illustrated by images and other media, where possible and appropriate.
a (images are tagged and non-free content have fair use rationales):  b (appropriate use with suitable captions): 

Overall:
Pass/Fail: 


### Prose
#### Lead section
Regarding the ordering of the 6 formulation bullet-points, if this is commonly known as "Fermat's right triangle theorem", it seems odd that forulation is not the first bullet.  Is there some logic to why they're in that order?
I think maybe I intended them to match the order they were covered in the article body? But they didn't even do that. Anyway I reordered to put the triangles first, and added a little more prose to keep it from being quite so bullety. —David Eppstein (talk) 06:27, 25 March 2021 (UTC)
I like the new presentation, especially teasing apart the geometric forms from the algebraic ones.  But, was Fermat talking about rationals, or was his work only talking about integers and people later generalized that to rationals? -- RoySmith (talk) 13:39, 25 March 2021 (UTC)
Regarding the accompanying figure, why the circles?  I haven't yet read the rest of the article, so maybe that's explained later on, but at this point I'm just left wondering about them.  The caption doesn't refer to them at all, hence the mystery.
The intent was to show graphically that the two a's were equal (both radii of one circle), and that the two triangles were right (one of them inscribed in the other circle with its hypotenuse as diameter, the other having one side as diameter of the circle and the other tangent to the circle). But I think you're right, it was more confusing than helpful. I've uploaded a new version of the figure without the circles. —David Eppstein (talk) 06:27, 25 March 2021 (UTC)
Yeah, the new figure works better.  FWIW, it was immediately obvious that the two sides "a" were radii of one of the circles, but I couldn't figure out what the other circle was showing.  In any case, resolved now. -- RoySmith (talk) 13:42, 25 March 2021 (UTC)


#### Squares in arithmetic progression
"In 1225, Fibonacci was challenged to find".  Who challenged him?  I have a Monty Python-esqe mental image of some rogue leaping out of the shadows, sword drawn, demanding a proof.  The sentence also parses ambiguously.  I initially read it as listing several properties the triples should have: 1) they are equally spaced, 2) they form an arithmetic progression,  and then started getting parse failures.  I think (but I'm not sure), "which" is better than "that" here, but could also be left out completely.  So, something like, "...for triples of equally spaced square numbers (i.e. an arithmetic progression), and for the spacing between these numbers (which he called a congruum)", although I'm not sure that's what you're trying to say.
Expanded to add more of the history, and split up the confusing sentence.David Eppstein (talk) 06:49, 25 March 2021 (UTC)
Yeah, that's better.  Nit: after the equations, where it picks up "That is, by the Pythagorean theorem", I think conventionally we treat that as part of the same sentence, so don't capitalize "That". -- RoySmith (talk) 13:46, 25 March 2021 (UTC)
(pausing here, I'll pick it up later, but this may be slow going)

"they would form two integer-sided right triangles in which the pair {\displaystyle (d,b)}(d,b) gives one leg and the hypotenuse of the smaller triangle and the same pair also forms the two legs of the larger triangle."  Are a, b, c, d here the same a, b, c, d in the figure at the top?  I think not, and that's really confusing.  If I'm understanding this correctly, this is (a, c) in the figure.
You're right, the figure didn't match the text. It was easier to relabel the figure than to figure out how to make the text match the old figure. —David Eppstein (talk) 06:49, 25 March 2021 (UTC)
You refer to "one leg and the hypotenuse".  From the examples here, it looks like it's always the shorter leg.  Is it always the shorter leg?  If so, why not just say "the shorter leg" and remove the ambiguity.  Or is this a hedge against a 45-45-90 triangle?
I don't see why it would always be the shorter leg. Why couldn't it be the longer one? It's the shorter one in the figure, but that's only because I had to pick one possibility. —David Eppstein (talk) 06:49, 25 March 2021 (UTC)
Interesting.  Yeah, I think you're right, it doesn't have to be the shorter one.  There's nothing in the algebraic presentations that lead me to think it is, but the diagram makes it appear that way to me.  Maybe a note clarifying that it doesn't have to be the specific case drawn in the diagram?  Or maybe that's not necessary?  Your call. -- RoySmith (talk) 13:51, 25 March 2021 (UTC)


#### Areas of right triangles
"Fermat was inspired not by Fibonacci but by an edition of Diophantus published by Claude Gaspar Bachet de Méziriac".  Add the dates, "not by Fibonacci's 1225 treatment, but by the 15xx (16xx?) edition of..."  Looping back to the lead, the first date mentioned in the article is 1225, but it took me a bit to sort out that it's referring to what Fibonacci did 350 years prior to Fermat.  The lead should mention up front when Fermat published his theorem.
Ok, clarified. —David Eppstein (talk) 07:14, 25 March 2021 (UTC)
David Eppstein, The way it's written now, it could still be that Fermat was a contemporary of Fibonacci, and the theory wasn't published until 300 years after his death.  Is it known when he actually did the work, so you can say something like, "formulated in 16xx but not published until 1670 among the posthumous works..." -- RoySmith (talk) 13:55, 25 March 2021 (UTC)


#### Fermat's proof
"he wrote a proof in his copy of Bachet's Diophantus, which his son discovered and published posthumously."  Clarify that you're talking about Fermat's son, not Bachet's son.
Done. Actually, while clarifying the history above, I realized that we mentioned Bachet in two different places in the article, a little unnecessarily, so I removed the second mention and just referred to the book as "Diophantus's Arithmetica" the second time. But it still now says Fermat's son. —David Eppstein (talk) 07:14, 25 March 2021 (UTC)
this gives his son's name as Clement-Samuel. -- RoySmith (talk) 13:57, 25 March 2021 (UTC)
That does it for my comments on the prose.  I'll come back and do another pass for the other GA criteria, but probably not today.


#### Notes section
"The fact that there can be no two right triangles that share two of their sides".  We may be back into snot-nosed kid territory, but this is only true if the triangles are non-overlapping.  I'm not sure, but it's possible it also only applies in a plane.  Which brings me to wondering if somewhere in the lead you should mention that everything here only applies to plane geometry.
I don't see how they could share their (whole) sides without coinciding rather than merely overlapping, in which case they are not two. But yes, probably we should specify the Euclidean plane, because the non-Euclidean geometries also have right triangles and I have no idea what happens with respect to rationality of sides and areas in those geometries. Done. —David Eppstein (talk) 07:14, 25 March 2021 (UTC)


#### References
No issues with non-WP:RS.


### Wrapping up
I made a few more comments above.  You can act on them as you see fit, but I'm fine with this as it is and I'll go ahead and pass the review.  -- RoySmith (talk) 14:03, 25 March 2021 (UTC)