Other mistake: When Anne and Peter are looking at the stars, Anne points to a star and Peter knows what star she's pointing to. It is impossible to point to a star and have someone else know exactly what star you're pointing to. Try it.
Factual error: It appeared that, during the engagement with the Serapis, every hit by either ship's guns resulted in an explosion and fire. Unfortunately, explosive shells wouldn't be used on shipboard (aside from bomb vessels) until the next century.