The end of The Fellowship of the Ring finds the Company as far down the Great River, Anduin, as they can travel by boat: the Falls of Rauros. This area has much history dealing with Númenor and interesting things take place here.
(Above: Amon Hen (left), Tol Brandir (middle), Amon Lhaw (right))
At this point, the Anduin is wide and forms a large lake called Nen Hithoel (meaning “Mist-cool Water”) above the Falls. The area, as shown in the picture above, is separated into three hills. The western hill is called Amon Hen, the Hill of Seeing, and the eastern hill is Amon Lhaw, the Hill of Hearing. The island-peak in the midst of the water is called Tol Brandir, or the Tindrock in the Common Tongue. The Company enters Nen Hithoel and they beach their three elven boats boats on the western shore of the River, on a strip of grass called Parth Galen (“Green Sward”).
Upon both Amon Lhaw and Amon Hen, the Kings of old built a seat: the Seat of Hearing and the Seat of Seeing respectively. It is said that no living beast has ever set foot on Tol Brandir, though in the book birds are noted circling it. In both the movie and book Frodo passes many ruins of ancient statues and monuments on the slope of Amon Hen as he ventures alone to contemplate the remainder of the journey. Eventually he arrives at the top of the Hill and finds the Seat of Seeing. At this point, due to his fear of Boromir’s advance on him, he is wearing the Ring.
He climbs onto the Seat and he is given strange visions. It is not described whether the visions are due to him wearing the Ring, simply an old kind of magic in the Seat, or some combination of both. Regardless, in the book (not the movie), Frodo is given visions of many places in Middle-earth as he looks around. He can see Minas Tirith in Gondor, the horses and spearmen in Rohan, the great delta at the end of the Anduin, and finally he sees Minas Morgul and the land of Mordor beyond. He sees Mount Doom, his ultimate destination, and eventually is shone the Dark Tower, Barad-dûr itself. Suddenly, he is aware of an Eye in the tower and that it’s gaze is looking to find him. He feels the eye searching Amon Lhaw, moving across Tol Brandir, and struggles with himself to remove the Ring as the Eye moves to Amon Hen where he is.
He is able to take the Ring off fortunately, if only just in time. He has made his mind up as to how the Quest must proceed, and the Breaking of the Fellowship takes place soon after.