Actually, I did one, just for my own benefit. You aren't going to like it.
They first see the eagle on the 8th day from Lorien. They are much too far from
Lorien by that point for Legolas to see an eagle flying from Khazad-Dum to
Lorien. In addition, Gandalf says that he arrived "and found you but lately
gone." 8 days is not "but lately gone." It's long gone.
Also, Legolas says that it's a hunting eagle, wheeling and descending. He
doesn't see it carrying anything, let alone a naked wizard.
They see it again on the 11th and 12th days out.
On the 15th day out, they meet Gandalf. He tells them that the eagle they have
seen is "Gwaihir the Windlord, who rescued me from Orthanc. I sent him before
me to watch the River and gather tidings."
By the way, who was it who said that finding the brooch slowed the Hunters
down, so that the hobbits arrived at Fangorn first? And then who was it who
argued that the Orcs shouldn't be gaining
distance?
