  A new proof is given for the correctness of the powers of two descent method
for computing discrete logarithms. The result is slightly stronger than the
original work, but more importantly we provide a unified geometric argument,
eliminating the need to analyse all possible subgroups of
$\mathrm{PGL}_2(\mathbb F_q)$. Our approach sheds new light on the role of
$\mathrm{PGL}_2$, in the hope to eventually lead to a complete proof that
discrete logarithms can be computed in quasi-polynomial time in finite fields
of fixed characteristic.
