For a general number field $K$, Dirichlet's unit theorem states that the unit group of the ring of integers of $K$ is a finitely generated group of rank $r_1+r_2-1$.

It seems that standard algebraic number theory textbooks usually prove this theorem by using Minkowski's theory or using the Blichfeldt-Minkowski Lemma to show that the group of norm 1 ideles modulo $K^\times$ is compact. Since both Minkowski and Blichfeldt were born significantly later than Dirichlet, I wonder that Dirichlet's proof may be different from the above two?

I should say that, for real quadratic fields, or equivalently, Pell's equation $x^2-dy^2=1$, I have saw an argument in Dirichlet's book "Lectures on number theory" where the author first shows that, for the irrational number $\sqrt{d}$ there are infinitely many rational numbers $\frac{p}{q}$ such that $$ |\frac{p}{q} -\sqrt{d}| \leq \frac{1}{q^2}. $$ Using this result, one can easily produce infinitely many principal ideals with bounded norms of the ring of integers $\mathbb{Q}(\sqrt{d})$ whence produce a nontrivial unit. Done!

To me this argument is distinct from the Minkowski's method in some sense and I am unable to generalize the above argument to prove the Dirichlet's unit theorem for general number fields. So I wonder that how Dirichlet proved his theorem for general number fields. Thanks.