And if the Romans found that the longbow was outclassed then they would have copied the Mongolian bow. I would imagine that a 13th Century Roman legion would have included a cohort or two of bowman supporting heavy infantry. The auxillaries would be heavy cavalry (Frankish) (and maybe some light cavalry). By the way the reason the Romans were such good road builders was to enable the legions to get from point to another in a hurry.
yes, I know this is all speculation but it's fun.