Name and describe the three men who rule after Caesar's assassination in Shakespeare's "Julius Caesar."
I'm assuming you mean after the defeat of Brutus and Cassius also, in which case is was Octavian Caesar (JC's great-nephew and adopted heir), Marc Antony, and Lepidus.
Lepidus was by far the politically weakest member of this group. As the play chronicles, Antony was ready to get rid of him almost immediately after winning. He was originally included in the triumvirate almost soley because he commanded a large army and had loads of money. Lepidus lost his power when most of his armies deserted him in favor of Octavious.
Marc Antony suffered from a bitter bout of jealous after not receiving a large portion of Caesar's will. Antony was angry about not receiving more power than Octavius, and eventually started a battle to earn that power. Antony's army was defeated five years after Lepidus was removed, leaving OC as the sole remaining leader. Antony, of course, committed suicide with Cleopatra.
Octavius was destined for greatness when he was named heir to Julius Caesar. Following the events of the play, he would become the first emporer of Rome, not quite what JC had imagined but close enough. After defeating Antony's army, OC united and organized provinces in Egypt, Syria, Judea, and Asia before returning home to Rome and being named Augustus Caesar. This is the Caesar Augustus that we all know from the bible who called for the census that brought Joseph and Mary to Bethlehem.
The three men who rule Rome after Caesar's death are Antony, Lepidus, and Octavius Caesar.
Antony was a friend of Julius Caesar's and was dismayed over his death; however, he was smart enough not to tell the conspirators that immediately after the assassination! He knew that he needed to bide his time and use his funeral speech for Caesar as the way to turn the people against those that murdered Caesar. Antony is smart, loyal, and not given to rash acts of passionate nature - he uses his common sense.
Lepidus is actually a rather minor character in the play's action. According to Octavius, Lepidus is a "tried and valiant soldier" (4.1), to which Antony replies, "So is my horse." Antony expresses concern over whether Lepidus is really up to the task of ruling Rome:
"This is a slight, unmeritable man,
Meet to be sent on errands." (4.1)
Octavius Caesar is the heir to Julius Caesar (his great-nephew)because JC did not have children. Octavius is very opinionated, as mentioned above in dealing with Antony's thoughts on Lepidus. In addition, Octavius decided to follow his own military strategies at the Battle of Philippi, rather than take orders from Antony, which lets us know that these two are not going to rule Rome peaceably together.
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