Have we discovered or invented maths?

[selfAdjoint]

Yeah, I have been looking into this and found out that you can only approximate the arc length of an ellipse. There is an infinite series but I assume it does not converge. The strange thing is that if you look at an ellipse it certainly has an arc length so why can't we find a closed solution?

The form of the integral you get when you set up the problem of the arc length of an ellipse was not one the 17th century mathematicians could solve, and it remained that way for a century. In the early 19th century Abel and Jacobi inverted the integral and discovered the elliptic functions. so called from their relation to this very problem. They are new functions, different from the exponentials and circular and hyperbolic functions known up to that time. The theory of the elliptic functions was very big in the 19th and 20th century, and they still come into a lot of research. So you can express the arc length of an ellipse in terms of these functions but not in terms of more traditional functions.

 

[mathwonk]

there is a fascinating exposition of this in chapter one, volume one, of the book "topics in complex analysis" by C. L. Siegel. An Italian count named Fagnano showed that although he could not calculate the length of arc of an ellipse from a beginning point to a later point, he could do something interesting: given one later point, although he could not calculate the arclength that far, he COULD find another point whose arclength was twice as great as the first (unknown) arclength.

so he could not calculate arclengths, but he could double arclengths.

this led eventually to the theory of elliptic functions and to the group law on an elliptic curve,

i.e. euler showed how to generalize this and find, given two points, another point such that the arc length to that third point was the sum of the arclengths of the first two, points, although he could not calculate any of the lengths explicitly!

this is an incredible story. and i thought with your talent for noticing nice things you just might see something.

 

[selfAdjoint]

Yes I worked through Fagnano's original paper once upon a time, it's very short. A really neat result. After Euler, Legendre spent years discovering all kinds of relationships in the elliptic integrals and he wrote a treatise on them, in two dense volumes. It was these properties that Legendre had explicated, which Abel and Jacobi used to develop the identities of their elliptic functions. It was remarked that Legendre could have discovered them but didn't. The point is the analogy between the arcsine integral
u = \int_0^x \frac{dt}{\sqrt{1 - t^2}}
which defines u as the inverse sine of x, and x = sin u, and the elliptic integral
u = \int_0^x\frac{dt}{\sqrt{1 - t^2}\sqrt{1 - k^2t^2}}
which defines u as the inverse of an elliptic function: x = sn u depending on a parameter k.

Lengendre was so focussed on the integrals that he couldn't see to invert them. Commenting on this later, Jacobi suggested "Man muss immer umkehren", or as I clumsily translate it, "Ever evert".

 

[mathwonk]

have you ever read galois works? apparently in the same letter in which galois wrote out the theory of groups and applications to solving polynomials equations, he also wrote out a theory of abelian integrals. for some reason this is essentially never mentioned in the literature but seems to predate riemanns work on the topic by some 25-30 years. on the other hand i do not recall galois using the idea of topology, i.e. connectivity of surfaces, to study these integrals as riemann did.

thus galois did know there was a number p asociated to an abelian integral or algebraic curve, i.e. a number p which counted the number of holomorphic differentials on the curve, but i think he did not link it with the number 2p of loop cuts needed to dissect the surface as riemann did. any comments? (i do not have access to galois here now.)

so, if this is right, the genus p precedes riemann, but riemann first linked the genus to the topology of the surface.

 

[selfAdjoint]

I can't help but adding two facts about elliptic function. One, they have two periods (the circular functions like sine and cosine of course have one, from 1 to 2pi). In the complex plane where they are most fully realized these are two complex numbers, think of them as vectors of different length in different directions. Then they can be regarded as two sides of a parallelogram, and this basic cell will be replicated - tiled - infinitely many time across the complex plane with the same values being taken at corresponding points in each iteration. Or if you prefer you can define them as single valued on a torus (their Riemann surface), and remarkably, their properties can be derived just from the fact that they can be so defined. Well maybe not so remarkable, given that the properties of the sine and cosine can be derived from their relation to the circle.

Two, they have LOTS of applications in engineering and physics.

 

[mathwonk]

this brings me back to an interesting fact about the torus i.e.; the formulas for its area and volume. i.e. take a look at the volume formula for a torus and also for its area, and notice how the area is related to the volume as a partial derivative.

there are also similar relations for the area and volume of a cylinder. maybe there is something like that for an ellipse, (but probably not).

in the interest of generalizing elliptic funtions, there are some interesting functions of several variables called theta functions, that have amazing properties discovered by riemann.