Shakespeare's Tempest is an interesting play; it's one of the few that is hard to categorize, and it is one that seems to be completely without background source material, unlike most of his works, which drew from myth and older stories.
As he usually does in his plays, Shakespeare explores multiple themes through multiple plotlines and character development. Some of the themes in Tempest are:
1. Kinship and relationships as expressed through Prospero's relationship with his brother, his daughter, and his servants.
2. Uses and abuses of power, again as expressed through Prospero, who has traded the power of governmental authority for the power of words, through books and magic. Shakespeare also addresses how Prospero's power is expressed differently through his servants Ariel and Caliban and his power over the elements.
3. The consequences of knowledge as expressed through the use of magic, which Prospero gives up at the end.
4. The idea of fate vs. human intervention, since though Prospero seems to be able to control everything, he cannot control the way people's emotions and actions will affect the outcome of his machinations.
There are other themes that you may find in your own reading, but these should be enough to get you started.
There are several major themes in Shakespeare's The Tempest. The first is questioning of legitimacy of political leadership. One of the great ironies of the play is that Prospero is plotting to regain his dukedom from Antonio, whom he criticizes as an usurper. Prospero is the older brother and argues that as the oldest male son of the previous Duke he is a legitimate ruler. Yet, Caliban is the sole male son of Sycorax and thus, by Prospero's own standards, the legitimate ruler of the island. Yet Prospero has no problem with not only usurping his rule but also enslaving him. In certain ways, one could argue that this is a microcosm of European colonialism, in which Europeans would argue for legitimacy as a concept applying to their own rulers but quite blithely depose or murder rulers in Africa and the Americas and claim their lands.
Another major theme in the play is that of illusion versus reality. Prospero's magic is makes things seem other than they actually are. The island itself is a realm of magic and illusion. This is analogous to the work of Shakespeare, the playwright, and his actors, whose "magical" art is to create an illusion for the audience. In a sense, the aim of the characters in the play is to dispel illusions and return to reality while the object of the audience is to enjoy the illusions while they last and only return to reality at the end of the play.
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