Understanding Key space
I'm not sure exactly what you're not getting.

You understand that the keyspace for a simple substitution cipher is $26!$; do you understand that $26! = 1 \times 2 \times 3 \times 4 \times ... \times 24 \times 25 \times 26$, correct?

So, we have $26! = 403291461126605635584000000$, correct?

We also have $2^{88} = 309485009821345068724781056$, correct?

Would you agree that these are "roughly" the same (where roughly in this case means within 30%)?


  keys = 26^4 = 2^18


What might be confusing you is the equal sign; obviously $26^4 = 456976$ is not exactly $2^{18} = 262144$; I believe the idea the author is trying to convey is that they're close, well, at least close enough for his analysis.  I'd personally use $\approx$ here...


  (though should it not be 26!^4?)


Actually, for Engima, 26 is correct; each rotor can be set in one of 26 positions; hence each rotor can do one of 26 specific permutations; it can't do an arbitrary 26! permutation.