The Romans - S2-E4
Factual error: Nero seems to be middle-aged, but the historical Nero was 27 at the time Rome burned.
The Dalek Invasion of Earth - S2-E2
Factual error: The Doctor states that The Daleks invaded Earth because it is the only planet with a magnetic core. In fact, every planet in the solar system has a magnetic core.
The Romans - S2-E4
Factual error: The geographical references in this story are somewhat suspect. Assissium is the modern town of Assissi, some 70 miles north of Rome and the same distance east from the sea. The Great Fire of Rome took place in July 64AD.
Factual error: In episode one, Abraham Lincoln seems to have affected a southern accent and put on some weight.
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★