Final answer:
The capital of British Canada burned down by the United States in 1813 was York, now known as Toronto. It was part of the War of 1812, and this American action led to the later British burning of Washington, D.C., including the White House and Capitol.
Step-by-step explanation:
The Burning of York
The capital of British Canada that was burned down in 1813 by United States forces was York, which is present-day Toronto, Ontario. This event occurred during the War of 1812, a conflict between the United States and the British Empire, with Canada being a British colony at the time. The burning of York was a retaliation by American forces in response to earlier British attacks and was part of the broader hostilities of the war. It is important to note that the burning of Washington, D.C. by the British in 1814 was a separate incident, where public buildings including the White House and the Capitol were set ablaze as a reprisal for the American attack on York.