5.34A preview of secondary dominants

In C major, E major is technically not diatonic because it contains:

G♯

But:

E7 → Am

works beautifully.

Why?

Because E is the dominant of A.

So we temporarily treat Am as a destination:

E7 → Am

Then continue:

Am → Dm → G7 → C

This gives:

E7 → Am → Dm → G7 → C

Notice the roots:

E → A → D → G → C

A chain of fifths.

This is exactly the sort of structure your original chord diagram was showing.