How does Shakespeare's The Tempest explore the theme of discovery?

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Not only does the play celebrate the wonder of the discovery of a "brave new world," it shows the characters in the play making discoveries about themselves and others.

For example, at the end of the play Caliban discovers that Stephano is not a god. In a moment of insight that perhaps reveals he has more depths than Prospero perceives, Caliban refers to himself as a “thrice-double ass” for his mistake.

Alonso discovers a sense of remorse over what he has done to Prospero, even if Antonio does not.

Prospero has been deeply shocked and hurt by the treachery of Antonio. He had thought he could he could trust his brother, if nobody else, and the fact that his own close kin turned on him made his sense of injury all the greater. Nevertheless, by the end of the play, Prospero discovers he can forgive his enemies, even if neither Antonio nor Sebastian feel remorse. Prospero states:

The rarer action is
In virtue than in vengeance.

In other words, it would be commonplace to take revenge on his enemies, having them in his power, but he is choosing the more unusual path of "virtue" by showing mercy and forgiveness. This ends the play on a positive note, and sounds a theme important to Shakespeare, which is that mercy is an all-important trait, especially in powerful people. What we discover inside, Shakespeare implies, is more important than all the new worlds we might find.

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The physical discovery of the island by Prospero is important, as it sets in train the events of the play. It also allows Shakespeare to explore other aspects of the theme of exploration. For instance, once he’s landed on the island, Prospero discovers the power of magic, which he uses to reassert the control over other people and events that he lost when he was deposed by his brother as Duke of Milan.

The magical island location also allows a number of characters to discover something new about themselves. While Antonio and Sebastian are intimidated by the island and its strange, barren landscape, Gonzalo views it in a much more positive light. And in due course, Antonio himself follows suit, eventually coming to recognize the wrongs that he’s committed after a period of self-discovery.

Miranda, the product of a sheltered upbringing, discovers the power of love and affection. In doing so, she learns an awful lot about herself and who she really is. By the same token, Ferdinand discovers his own true self in the intoxicating, all-consuming love he develops for Miranda. Once more, the island setting, with its strangeness and air of magic, has acted as a catalyst for discovery, transforming Ferdinand’s understanding of himself, his emotions, and the world around him.

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In a sense, the theme of discovery is at the heart of Shakespeare's The Tempest. On the one hand, you have the shipwrecked crew stranded by the storm at the beginning of the play. Much of the rest of the play is concerned with the stranded characters discovering the strange and exotic island, complete with magic, spirits, and strange creatures. Likewise, another major component of the play focuses on Miranda's discovery of human society. Stranded on the island with her father and few other companions, Miranda has little knowledge of civilization or human nature. However, with the arrival of the shipwreck (and especially Ferdinand, her future husband), Miranda gradually discovers the intricacies of human society and relationships. 

Though you certainly don't need to make this connection, it's possible to link the play's theme of discovery to colonization. In some ways, Prospero can be seen as the prototype for the European who discovers and colonizes native populations, as he arrives on the island and presses Caliban into his service. Thus, though the theme of discovery is often exciting and fantastical within the context of the play, it also comments on the burgeoning European tendency to discover and exploit new lands.  

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