Final answer:
Lincoln inspires hope in the Gettysburg Address by tying the sacrifices made during the Civil War to the enduring and rejuvenating promise of democracy and freedom, and by portraying a future where the Union is strengthened and committed to its founding principles.
Step-by-step explanation:
Abraham Lincoln, in his Gettysburg Address, intended to inspire hope by focusing on the principle of a government of the people, by the people, for the people and the promise of a new birth of freedom. In the context of the Civil War, Lincoln's words aimed to remind the nation of the democratic ideals upon which it was founded and to look forward to a future where those ideals would not only endure but become stronger.
He does this by assuring the audience that the sacrifices of those who died at Gettysburg were not in vain, setting forth a vision where the living commit to the 'great task remaining before us', which is to ensure these dead shall not have died in vain. This vision is presented as a continuum of the founding fathers' original goals, reinforcing the dedication to the ideals of liberty and equality and establishing a sense that the Union would emerge from the war renewed and more committed to its founding principles.