  We consider the Euler equations on $\mathbb{T}^d$ with analytic data and
prove lower bounds for the radius of spatial analyticity $\epsilon(t)$ of the
solution using a new method based on inductive estimates in standard Sobolev
spaces. Our results are consistent with similar previous results proved by
Kukavica and Vicol, but give a more precise dependence of $\epsilon(t)$ on the
radius of analyticity of the initial datum.
