Using some simple trigonometry:
If theta is half of the angle between the two tangent lines, then the scale between adjoining radii for shrinking circles, is = (1-sin(theta)) / (1 + sin(theta)).
For the next largest circle, the formula is the reciprocal.

There are also Steiner chains, Pappus chains, and Ford circles.
- Brian

Using sine of the angle to calculate the location of the first radius did cause some rounding, like a radius of 10 going to 9.999999... or so, which I did not care for :-)