  We analyze a proof of Bruck to obtain an explicit rate of asymptotic
regularity for Ces\`aro means in uniformly convex Banach spaces. Our rate will
only depend on a norm bound and a modulus $\eta$ of uniform convexity. One
ingredient for the proof by Bruck is a result of Pisier, which shows that every
uniformly convex (in fact every uniformly nonsquare) Banach space has some
Rademacher type $q>1$ with a suitable constant $C_q$. We explicitly determine
$q$ and $C_q$, which only depend on the single value $\eta(1)$ of our modulus.
Beyond these specific results, we summarize how work of Bruck has inspired
developments in the proof mining program, which applies tools from logic to
obtain results in various areas of mathematics.
