Southern states began seceding from the union and forming the Confederate States of America. James Buchanan, although at the end of his term, was president at the time and he correctly pointed out that there was no constitutional justification for using the force of arms to compel a state to remain in the union or force them back into it.
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Battle of Fort Sumter is credited with being the start of the civil war. Fort Sumter was a union occupied fort in South Carolina, a state which had seceded from the union and joined the Confederacy. Union troops refused to leave Confederate territory despite being given warnings and a deadline to withdraw. When the deadline passed, the Confederacy took the fort but there wasn't a single death as a result of the battle.
Lincoln, newly elected President of the Union, used the battle of Fort Sumter as justification to do what his predecessor, Buchanan, knew was unconstitutional, Lincoln raised and army for the purpose of invading the southern states and forcing them back into the Union.
Much like George W. Bush declaring that he had to abandon free market principles in order to "save" the free market system, Lincoln abandoned the Unions Constitutional principles in order to "save" the Union.
As for the suspension of Habeus Corpus, such a declaration, reserved for rebellions within Union states, would only apply to the Union, not to the newly formed Confederate states. The Confederate states had already created their own government, had written their own constitution, and had formulated their own laws.
After seceding from the Union, the Confederacy became a distinctly separate nation, just as Canada is a distinctly separate nation from the United States, so were the Confederate States. Laws passed in the Union were not binding on the Confederate states any more than they would have been binding on the nation of Canada or other nations of the world.
Those who believe the ends justify the means will gladly overlook Lincoln's Constitutional transgressions but it nevertheless opened the door for future presidents to ignore the Constitution - so long as the "end" was seen as justification for the means.
Cue the drooling progbots who will now begin, fallaciously, attacking me as being pro-slavery for daring to speak the truth.