This is 21.414:

γήθησέν τ᾽ ἄρ᾽ ἔπειτα πολύτλας δῖος Ὀδυσσεύς

then rejoiced the long-suffering divine Odysseus

_{Homer. Odyssey 21.414. My translation.}

A. T. Murray (1919) translates the line as "Then glad at heart was the much-enduring, goodly Odysseus". So Fitzgerald means that Odysseus laughed for joy (not for amusement) because he believed that Zeus would favour him in the impending battle with the suitors.

I think that this is a fairly unexceptional use of “laugh”, which can be used to describe several emotions: the OED lists ”joy, mirth, amusement, or derision”.

This is 21.414:

γήθησέν τ᾽ ἄρ᾽ ἔπειτα πολύτλας δῖος Ὀδυσσεύς

then rejoiced the long-suffering divine Odysseus

_{Homer. Odyssey 21.414. My translation.}

A. T. Murray (1919) translates the line as "Then glad at heart was the much-enduring, goodly Odysseus". So Fitzgerald means that Odysseus laughed for joy (not for amusement) because he believed that Zeus would favour him in the impending battle with the suitors.

I think that this is a fairly unexceptional use of “laugh”, which can be used to describe several emotions: the OED lists ”joy, mirth, amusement, or derision”.

The line itself is formulaic—there is an identical line at 13.353, and very similar lines at 7.329, 8.199, 13.250, 18.281 and 24.504.