re: Did The Battle of Trenton save America?

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Just speculating, but had we remained under British rule, and part of the Commonwealth, would we be better or worse off?

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Worse.

We'd have not expanded as far West, as the La Purchase was from France (the Brits didn't get along with them then), and almost certainly the Texas and California parts would have stayed Spanish/Mexican.

We also would not have industrialized to the point that we did. It would have been more raw materials for the English factories, with maybe some New England or similar factories at most.

So we'd have been a lot smaller, more rural, with a much more powerful Mexico building up on the Western half of North America.

The Mexicans probably do the La Purchase to get the port of NOLA and control of the Mississippi River, moving Eastward in the process. The Brits had the Eastern Seaboard anyway, so without a Westward expansion, they wouldn't have been interested.

By the time the 20th Century rolled around, we'd have been similar to Argentina but English-speaking, while Mexico would be developing into a world power (maybe not as strong as WE are, but greater than they ever were).