Somehow, the "original proof" seems simpler.

> The first rigorous proof that π is irrational is from Johann Heinrich Lambert in 1761. He proved that if x≠0 is rational, then tan(x) must be irrational. Since tan(π/4)=1 is rational, then π must be irrational.

Of course, the proof for (tan(x) is irrational for rational x != 0) might be complicated, but at least it's easy to see why this proves that PI is irrational.

This one, on the other hand, I can't even follow in principle what it's trying to say.

That's of course because I don't understand integrals and differentials.