Lemma

The last Dimensions movie is a proof, which is straightforward enough: it rests
entirely on Euclid's Elements, in particular Pythagoras'Theorem and Thales.
One interesting twist: it makes use of a lemma, which is an intermediate
finding or theorem, which needs to be established for the actual proof to proceed.

                                                                     *  *  *

What is to be proven: that stereographic projection (from the north pole of the sphere)
will transfer a circle image form the sphere to the ground plane. Here, B Riemann's
likeness.We are in the realm of Geometry, this says nothing about what happens
to the proportions within the image. The quite famous picture of Riemann has been
 modelled onto a sphere image. A quick look convinces one that projection incurs
its own  form of distortion. Earth maps cannot compensate for this inescapable reality;
whatever projection model one chooses, the original longitude and latitude remain to
give the viewer a sense of what is going on.

*  *  *

Cutting s sphere creates a circle.

 Everything on the circle belongs to the sphere.

That sphere can be continued to a cone.

(Our lemma) ... which is symmetrical.

Every projection traces a circle.

 That projection can be carried to the South Plane.

We are now familiar with Riemann's Sphere.