  We improve the lower bound for the classical exponent of approximation
$w_{n}^{\ast}(\xi)$ connected to Wirsing's famous problem of approximation to
real numbers by algebraic numbers of degree at most $n$. Our bound exceeds
$n/\sqrt{3}\approx 0.5773n$ and thus provides a reasonable qualitative
improvement to previous bounds of order $n/2+O(1)$. We further establish new
relations between several classical exponents of approximation.
