[dancing spawn of ungoliant]

RotK, The Last Debate:

"'We must walk open-eyed into that trap, with courage, but small hope for ourselves. For, my lords, it may well prove that we ourselves shall perish utterly in a black battle far from the living lands; so that even if Barad-dûr be thrown down, we shall not live to see a new age. But this, I deem, is our duty. And better so than to perish nonetheless - as we surely shall, if we sit here - and know as we die that no new age shall be.'

They were silent for a while. At length Aragorn spoke. 'As I have begun, so I will go on. We come now to the very brink, where hope and despair are akin. To waver is to fall. Let none reject the councels of Gandalf, whose long labours against Sauron come at last to their test.'"


I understand it thusly: Gandalf has suggested making a diversion which will practically be a suicide. That's the despair part. But by doing this, they help Frodo to accomplish his task and save Middle-earth, so there's some hope, too.

At this point, it seems to Aragorn and his lot that there isn't one without the other; hope and despair are now related. If they are afraid to sacrifice themselves for a greater good, even though it's possible that the diversion won't even work, there sure is no hope for defeating Sauron. However, if they are ready to put their lives at stake, there's a chance to succeed.

Edit: cross-posted.