Q:

What is a root movement by fourth in chord progression

I'm reading music theory for computer musician and in chapter 18 they talk about chord progression and root movement by fourth, second and third.
So it makes sense for the second and third, but for the fourth, they say the chord progression is 1, 3, 5 then 4, 6, 1 (still makes sense so far), but then they say it ends with 5, 7, 2.
I'm confused as to why they call it fourth if it jumps to 4 one time and then jumps to 1 the second time.
Note: The numbers are notes from C on the keyboard. I'm using numbers to illustrate why it makes sense that it jumps to 4, but then it doesn't make sense that it jumps to 5.

A:

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.

