Among the relevant hemispheres,
we will distinguish those whose hats
intersect the region we are trying to cut up,
i.e. whose radius is 
,
and refer to them as
babies.
Each hemisphere spends three steps as a baby:
If the radius is a,
the hemisphere will be a baby when
,
when
,
and when
.
During each of these steps,
the piece of the (x,y)-plane
that we associate to the hemisphere will consist of the base of the
hemisphere, and no more.
By deflecting the tubes radially,
we can arrange things so that the tubes stay more or less vertical,
and so that the horizontal cross-section of every tube never decreases,
and increases by a definite factor between the beginning of the first step
and the end of the third.
(Look back at Figure 9.)
This takes care of the babies.