Aragorn traces Frodo's path to Amon Hen and back again.
When he hears the sound of Boromir's horn and the cries of the orcs, Aragorn runs to help Boromir, calling the name "Elendil! Elendil!" (3.1.6).
He finds his human comrade in a glade about a mile from Parth Galen. Boromir looks like he's just resting, but his body has arrows poking out all over. Yikes.
There are stacks of dead orcs around him. Well done, Boromir.
After Boromir confesses to Aragorn that he tried to take the Ring from Frodo, he apologizes, and tells Aragorn that Merry and Pippin have been taken by orcs. He thinks they're still alive.
Poor Boromir thinks he has failed, but Aragorn tells him just the opposite. He also promises that Minas Tirith, the main stronghold of Gondor, shall not fall, if he has anything to say about it.
Boromir smiles and dies.
A despairing Aragorn laments that the Company has failed, and it's all his fault.
Legolas and Gimli arrive to find Aragorn weeping and clasping Boromir's lifeless hand.
Gimli wants to follow the orcs and save their hobbit friends, but Aragorn is concerned about abandoning the Ring-bearer.
Aragorn, Legolas, and Gimli examine the bodies of the dead orcs to find clues of where they were headed, which leads them to discover that the orcs are not from Mordor. Interesting… very interesting.
As it turns out, some of the orcs have an "S" on their chests: "S" for Saruman. This prompts Aragorn to realize that evil is afoot in Isengard, and that Saruman may have learned of Gandalf's death. Not good.
Legolas, meanwhile, tells Aragorn they only found two boats on the shore.
They lay Boromir in a boat with his weapons and send him down the Anduin, which flows towards his homeland of Gondor. Aragorn and the elf sing a song to mark Boromir's passing.
Aragorn looks at the banks of the river and realizes that Frodo and Sam left together, which is sad. But he's comforted by the fact that Frodo has Sam with him.
That's when he decides to follow Gimli's plan. The three of them should follow the orcs to save Merry and Pippin. He does not want to abandon them to death.
As they set out at a fast pace to catch up with the orcs, Aragorn is clearly feeling pumped: he cries out, "We will make such a chase as shall be accounted a marvel among the Three Kindreds: Elves, Dwarves and Men. Forth the Three Hunters!" (3.1.64).