How can hope and despair be akin?

[HerenIstarion]

hope and despair are directed toward the future

Verily, but not solely. More often than not they both look back to the past, but despair more so - I hope (amdir) things will go well as they always did go well in the past, I despair for everything is already lost (in the past, even if this 'past' was minute ago), and there is nothing to look forward to - hope (amdir) for any more.

Estel is in present - I believe that all His designs are for my good (and for my joy), and I do the 'believing' now. Both Amdir and despair need reason or cause (coming, again, from the past), Estel does not - there is no reason to believe, one just does or does not. I agree with davem - Estel can not be akin to despair, the very belief excludes possibility of despair, whilst Amdir surely can.

PS Almost forgot about 'akin' - they are akin as they cross at one point - Frodo is Aragorn's reason to hope or to despair. Hope-estel makes him fight, but he can't help going on hoping (amdir) and despairing, for here he has cause and reason - frail thread (as Gandalf told Theoden) on which the fate is suspended:

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 now reject the counsels of Gandalf, whose long labours against Sauron come at last to their test

There are all three here - estel, amdir and despair. The way of action (for hope and despair here are not expressions of idle state, but of the action) chosen at the Council of Elrond is the mix of hope and despair - for every step they take through the book is indeed 'walking on the brink'. And estel (Hope in Eru) is expressed through Gandalf - 'let none now reject...'

PPS I've been a bit hasty yesterday - I did not intend to say that the meaning Aragorn may be putting into the phrase comes down only to my PS. As in other matters, Tolkien is like Shrek and onions - many-layered. For two, Aragorn may indeed be 'being clever' - meaning that highly paradoxical turn of phrase is intended to wind up his listeners, make them feel urgent importance of matters being discussed.