A. W. Reid [R, Theorem 2.1] showed that if $\Gamma_1$ and $\Gamma_2$ are arithmetic lattices in $G = PGL_2 ({\mathbb R})$ or in $PGL_2 ({\mathbb {C}})$ which give rise to isospectral manifolds, then $\Gamma_1$ and $\Gamma_2$ are commensurable (after conjugation). We show that for $d \geq 3$ and ${\mathcal{S}} = PGL_d ({\mathbb {R}})/PO_d ({\mathbb {R}})$ or for ${\mathcal{S}} = PGL_d ({\mathbb {C}})/PU_d ({\mathbb {C}})$ , the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by ${\mathcal{S}}$ . The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants