#### Note
1. The Dandelin Spheres

While it is easy to prove
these facts analytically given the standard form equation for a general ellipse,

the proof never reveals
why that they should be true. They were so much of a mystery that 200 years
after Kepler,
in the 19th century, a Belgian geometer named Germinal
Dandelin gave a remarkable geometric proof of the focus-locus property of
ellipses, using a construction that has come to be known as the *Dandelin
spheres*.

The
main picture of his construction is shown below, and you may learn more about
his argument in Eric Weisstein's article at mathworld.