Assume that Heron of Alexandria makes something useful out of his experiments. He had all the componens of a useful steam engine down, including pistons and crankshafts. Of course, it would be some time before anything truly productive could be made from his work.
Lets assume, by AD 200-300 Rome has locomotives and steamships (as well as other industrial machines that don't matter for the sake of this conversation). How far out could the Romans reasonably project their power?
Or, if you don't like the timing of that scenario, just work with fundamentals: Romans, locomotives, steamships.