Book II, Chapter 9: "The Great River" The Fellowship travels south down the Anduin. Near the end of the fourth day, Sam thinks he sees "a log with eyes" following them. He tells Frodo he suspects the "log" is actually Gollum. That night, Frodo sees Gollum's eyes; Aragorn tells Frodo that he, too, has been worried about the creature's presence.
On the eighth night of their journey, the Fellowship is attacked by Orcs shooting arrows from the Anduin's banks. Further down the River, a large, black, winged creature flies overhead, and Frodo feels a pain in his shoulder. Legolas shoots the creature out of the sky, but it falls into darkness and is not seen. (We learn in Book III that such creatures are the winged steeds of the Nazgûl, or Ringwraiths, who appear in a different form after the flood Elrond commanded at the end of Book I.)
Aragorn decides to take the Fellowship past the Rapids of Rauros to Amon Hen, the ancient place of coronation for high kings, where he will then decide a further course. Boromir objects, wishing to proceed immediately to Minas Tirith; he reluctantly consents, but tells Aragorn he will go no further than Amon Hen: "There I shall turn to my home, alone if my help has not earned the reward of any companionship."
On the tenth day, they reach Amon Hen, passing two great and ancient statues of Isildur and his brother Anárion, Aragorn's ancestors. "They could go no further without choice between the east-way and the west."