Harristotelian Logic

Remarkably then, a parametric rational solution for circles, ellipses and hyperbolas has been known since antiquity as according to at least on web source the following result was known to Fermat:

Given x^2 - Dy^2 = 1, in rationals:

y = 2t/(D - t^2)

and

x = (D + t^2)/(D - t^2)

showing it more traditionally versus the way that results from my own re-derivation.

And you get a parameterization for hyperbolas with integer D>0, for circles with D=-1, and for ellipses with D<-1.

So for D>0, you get hyperbolas. For D<0, you get ellipses, including the circle at the D=-1 value.

Here's a link for those wishing to see an established source give the circle one:
See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17

So you can categorize those conic sections entirely with one number: the D number.

I'd guess it's related to the eccentricity.

For practical purposes I'd think that these rational solutions could lead to some rather fast computations, like for astronomical orbits, as well as maybe show some physics from the D numbers that result in nature.

The weird thing is the math people had these equations for hundreds of years, but they look at x^2 - Dy^2 = 1 as a Diophantine equation—integers only.

Fermat himself according to the source I read dismissed the parameterization as of no interest to him as he could generate solutions so easily with it!!! He was looking for integer solutions only. But hey, he WAS a lawyer who just did math for fun!!!

Isn't history great?