C (1,3,5) to F (4,6,1) is a rising fourth.
That's because F is four notes up from C - C,D,E,F.
C (1,3,5) to G (5,7,2) is a falling fourth.
That's because G is four notes down from C - C,B,A,G.
It's probably easier to think about these if you play the simplest triads at first - e.g. (using your notation) 5,7,9 for G. That way it's clear that your whole chord is going up or down by a certain number of steps in the scale. That's because octaves are important; harmonically 1=8=15=..., 2=9=16=... etc.
From the excerpt I was able to view on Amazon, it looks as if the section has two main conclusions:
- You can get from any diatonic chord to any other using an interval of 2,3 or 4 (since a rising 5th is equivalent do a falling 4th)
- You can reach every diatonic chord using only one of those three intervals - 2nds: 1,2,3,4,5,6,7,1 - 3rds: 1,3,5,7,2,4,6,1 - 4ths: 1,4,7,3,6,2,5,1
(Ignore the word "diatonic" for now if you don't know what it means -- I only put it in to satisfy pedants)
There's nothing there about having to make chord sequences that stick to one interval or another. Experimenting with those cycles just gives you a chance to get a feel for how the chords relate to each other, and how the transition from one to the other sounds.