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The three G's, God, Glory, and Gold. They could spread their beliefs and society (God) to whomever they encountered, making them "less savage," they would gain glory just for conquering new lands and expanding their empires, and they could achieve riches by taking the lands they found for their own personal use and resource collection.
The Confederacy? The interest seems to be the act of slavery and the war Lincoln is talking about must be the Civil War, which was taking place while Lincoln was addressing his second inaugural. Lincoln was reasoning that the Confederacy was fighting to keep slavery, which is the interest the "insurgents" have. Lincoln also did not want slavery to continue to spread, but he was fine with having slavery, "restrict the territorial enlargement if [slavery]".