In the proof we have just gone through,
we neglected to determine precise values for various
``universal constants.''
As a result,
while we now know that there is a universal lower bound
for 
,
and hence a universal upper bound 
for 
,
we do not know concrete values for these bounds.
With sufficient patience, we could work out precise bounds.
However, the resulting bounds are liable to be exceedingly poor.
To get good bounds by the cutting method,
a keener knife and a steadier hand will be needed.