Why would the british want to control new york city?

it held precious military supplies for the americans.
it was the center of all military operations.
it was seen as a key strategic location.
it had thousands of troops they could hold as prisoners.

But I would say in the moment, in 1863, when the nation was completely divided, Lincoln was there to honor the Union dead, those who were supporting the principles and the values of the United States, which is why he makes reference in the very beginning of the Gettysburg Address back to the Declaration of Independence

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