-1

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one?

I was looking for the tag "circumference", supplied the newer synonym (in this field) perimeter. Apparently this tag is not related, since it was removed.

I will link the Wikipedia article on arc length from suggestions, even though I don't know if it fits; it certainly doesn't answer my question.

https://en.wikipedia.org/wiki/Arc_length

The proof is likely related to isoperimetry as per suggestion, but again I'm asking for proofs.

https://en.wikipedia.org/wiki/Isoperimetric_inequality

I heavily suspect any intuitive proof to not be analytic, since mine isn't. If someone made a simpler intuitive proof, I want to know it and who to credit; if someone made an analytic proof, I will use it as advanced course reference.

Henrik Erlandsson
  • 105
  • "Oh, and "circumference" needs to be on the tag list." - Not particularly. The circles tag would suffice. Also I don't really see how this ties to fractals or surface integrals necessarily so I'll remove those tags. – PrincessEev May 04 '19 at 21:57
  • Also it seems like this has already been discussed on MSE: https://math.stackexchange.com/questions/4808/why-does-a-circle-enclose-the-largest-area – PrincessEev May 04 '19 at 21:58
  • 1
    Sounds like you want the tag [tag:arc-length]. Not sure which of the current tags I would remove to add it, though. – Eric Wofsey May 04 '19 at 22:00
  • We shouldn't have to have a tag for every single defintion; that reduces efficiency and increases clutter, that is my point. In what discussion of circumferences would discussion of circles in general be irrelevant? It's tied to circles, so the circles tag suffices. – PrincessEev May 04 '19 at 22:01
  • Eevee, I don't see the first proof credit or the most intuitive proof credit in this duplicate, which is what I'm asking. Unless asking who proved it is prohibited here.

    If you're not familiar with the relation between an enclosure's integral and its area it just means you're not familiar with it.

    Remove the fractal tag if you want, but it started out as a progressive approximation of the enclosure. If that's the proofs we have, it's related.

     – Henrik Erlandsson May 04 '19 at 22:04
  • There, removed the request for the tag and linked Wikipedia's article on arc length @EricWofsey. Obviously for there to be an area we must speak of a curve that encloses it; a subset. Both concave and convex curves enclose an area, which is why the duplicate provided doesn't suffice as answer, although the intuitive answer is the curve must be convex; a subset of enclosing curves and a further subset of curves as per the WP article which is why I don't know if it applies well. – Henrik Erlandsson May 04 '19 at 22:16
  • 2
    Your question is more about the history of mathematics than about mathematics. "Most intuitive" is not well defined. The wikipedia page you find unsatisfactory is likely to provide the closest thing you will find to an answer. Finally, your comments to @EeveeTrainer are rude. That's unacceptable on this site. – Ethan Bolker May 04 '19 at 22:22
  • 3
    I believe this is an isoperimetric problem: https://en.m.wikipedia.org/wiki/Isoperimetric_inequality –  May 04 '19 at 22:27
  • @ChrisCuster Indeed. That's the link I thought the OP added. – Ethan Bolker May 04 '19 at 22:29
  • Perimeter will do. Editing the question. But what I asked was who wrote the proofs that a circular perimeter yields the maximum enclosed area. Why is this so hard? The duplicate does not answer my question. – Henrik Erlandsson May 04 '19 at 22:32
  • Dido was a queen. It's interesting to find a reference to Asimov in this. I added some background to really spell it out. I'm looking for analytic proof; there may be none through history; the simplest intuitive one is second prize and I'll likely have to settle for that. So who made the one you like best? – Henrik Erlandsson May 04 '19 at 23:07
  • Please check Dido's problem in calculus of variations(tag) ..it is a textbook isoperimetric problem It poses and answers what is DE of shape such that.. – Narasimham May 04 '19 at 23:11
  • @Narasimham The previous Dido comment was removed, but again see the question as stated. You say textbook problem, I just want names of the ones providing the proofs. If they are from antiquity and famous, it should be all the easier to answer my question with a link to the proof. – Henrik Erlandsson May 05 '19 at 00:04
  • Do you know to use Euler Lagrange theorem of variational calculus? – Narasimham May 05 '19 at 00:39
  • @Narasimham Spencer answered, providing two links that will at least let me post an answer to the question, proof for the relation of circumference (as Wolfram alpha confirms is a valid equivalent to perimeter) to area withstanding. Perhaps. I will do my best. – Henrik Erlandsson May 05 '19 at 01:58

2 Answers2

2

The historical information your request can be found here :

http://mathworld.wolfram.com/IsoperimetricProblem.html

A proof using the calculus of variation can be found here :

https://mathematicalgarden.wordpress.com/2008/12/21/the-problem-of-dido/

Spencer
  • 12,271
  • This should at the very least immediately allow me to put the 'integrals' tag removed by Eevee Trainer back in, thanks. – Henrik Erlandsson May 05 '19 at 01:04
  • I had the suspicion analytic proof might not be available, and the late synthetic proof by Steiner seems to strengthen that. I also wanted some much more intuitive reasoning for "the why", but even though that may be available, this would be two questions in one, so I will drop that. After adaptation of my question to fit the set of questions that can be asked here, I will accept this answer. Could you describe the links? Along the lines of, "Steiner provided the first synthetic proofs in 1841, and this blog article connects the parts." Thanks. – Henrik Erlandsson May 17 '19 at 14:33
0

OK. Since you persist I give a short abbreviated well known Dido's isoperimetric problem. The object and constraint functions are

$$ \int y dx -R \sqrt{1+y^{'2}} dx$$ where $R$ is a constant Lagrangian multiplier. The Lagrangian $F(x,y,y^{'})$ is given in square brackets

Euler-Lagrange equation of Variational calculus is:

$$ F-y^{' }\frac{\partial F}{\partial y^{'}}= const. c$$

$$[y-R\sqrt{1+y^{'2}}]-y^{'}\frac{-R y^{'}}{\sqrt{1+y^{'2}}}= c $$

$$y- \frac{R}{1+y^{'2}}= c$$ since $ \sec \phi=\sqrt{1+y^{'2}} $ the DE required is

$$ \frac{y-c}{ \cos \phi}=R $$

which represents all circles with center shifted up/down on x-axis by an amount $c.$ Note that the Lagrange multiplier $R$ has an arbitrary magnitude which is nothing but the radius of a circle thus defined.

Historical

Newton's work on Variational approach in fact predated Euler and Lagrange. But the definition of variation and its formalism ( Lagrange included proof with freedom from direct geometric representations) and rigorous structuring of Euler-Lagrange of variational definitions gained acceptance and adoption. Newton first used the method to find optimal shape of a projectile (travelling at supersonic speed ?)that offers minimum aerodynamic resistance to flight. The rocket shape has cusps at $\pm 30^{\circ}$ to flight path.

Aerodynamic Newton Prob Ref

Trivia about Aerodynamic Newton's Problem .. a edges of pages written/published in Newton's own handwriting after Principia's publication got burnt after his cat tipped a lighted candle, in Motte's Principia translation iirc..

Narasimham
  • 40,495