  Let $A$ and $A'$ be two Artin groups of spherical type, and let
$A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$
(resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$
and, up to permutation of the indices, $A_i$ and $A'_i$ are commensurable for
every $i$. We prove that, if two Artin groups of spherical type are
commensurable, then they have the same rank. For a fixed $n$, we give a
complete classification of the irreducible Artin groups of rank $n$ that are
commensurable with the group of type $A_n$. Note that it will remain 6 pairs of
groups to compare to get the complete classification of Artin groups of
spherical type up to commensurability.
