Let $\kappa$ be a regular cardinal, and let $B$ be a subalgebra of an interval algebra of size $\kappa$. The existence of a chain or an antichain of size $\kappa$ in $\mathscr{B}$ is due to M. Rubin (see [7]). We show that if the density of $B$ is countable, then the same conclusion holds without this assumption on $\kappa$. Next we also show that this is the best possible result by showing that it is consistent with $2^{\aleph_0} = \aleph_{\omega_1}$ that there is a boolean algebra $B$ of size $\aleph_{\omega_1}$ such that length$(B) = \aleph_{\omega_1}$ is not attained and the incomparability of $B$ is less than $\aleph_{\omega_1}$. Notice that $B$ is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see, e.g. [1] and [2].