How does Prospero describe his magic?
Shakespeare refers to Prospero's magic as “the liberal arts” that Prospero learned through “secret studies” (The Tempest 17). This scholarly approach to magic elevates it beyond the typical notion that magical ability is innate and is proof that Prospero's magic is scientifically grounded.
Prospero takes power into his own hands and uses magic to manipulate and coerce changes not only in the elements but in the subjects around him including his own daughter. By manipulating the weather, Prospero is able to assume respect, hope as well as an element of fear from the people around him.
He is a master of a learned art which enables him to use magical equipment like books, a staff or a magic garment. Moreover he commands natural spirits to play out a mimesis, which makes nothing on the island what it seems to be.
Prospero tells Ariel to go release the men, and now alone on stage, delivers his famous soliloquy in which he gives up magic. He says he will perform his last task and then break his staff and drown his magic book. Ariel now enters with Alonso and his companions, who have been charmed and obediently stand in a circle.
“For I am all the subjects that you have, Which first was mine own king.” “As you from crimes would pardoned be, Let your indulgence set me free.” “The rarer action is In virtue than in vengeance.” “My high charms work, And these, mine enemies, are all knit up In their distractions.
Like the tempest, Prospero's books are a symbol of his power. “Remember / First to possess his books,” Caliban says to Stephano and Trinculo, “for without them / He's but a sot” (III.
Prospero has power over Ariel, who is a spirit that he enslaves after saving him from the witch of the island. He uses this debt to control Ariel to do his bidding. He also uses force by threatening Ariel with his knowledge of magic to remind him how powerful he is.
In The Tempest, power and control are dominant themes. Many of the characters are locked into a power struggle for their freedom and for control of the island, forcing some characters (both good and evil) to abuse their power. For example: Prospero enslaves and treats Caliban badly.
By isolating Ferdinand, Prospero instills feelings of helplessness and abandonment in him. He is able to manipulate him into accepting any help he can get. This results in Ferdinand listening to Ariel's song about his supposed dead father and following him to meet Miranda.
The pursuit of knowledge gets Prospero into trouble in the first place. By neglecting everyday matters when he was duke, he gave his brother a chance to rise up against him. His possession and use of magical knowledge renders him extremely powerful and not entirely sympathetic.
Why did Prospero give up his magic power short answer?
Prospero decides to give up magic, because magic (and studying magic) is what caused him to end up on the island in the first place. His choice shows that he thinks because he's so eager to learn new things and to become wiser, it ultimately lead to his downfall, and lead to his throne being taken away from him.
Prospero can be seen as an authoritarian dictator. He has colonised and taken over Caliban's island; he manipulates his daughter and her young suitor; he is cruel in his treatment of Caliban, and also Ferdinand, and keeps Ariel in a state of servitude by using draconian threats.

Prospero is one of Shakespeare's more complicated characters. He shows himself to be all at once kind, cruel, vindictive and forgiving.
Prospero decides to give up magic, because magic (and studying magic) is what caused him to end up on the island in the first place. His choice shows that he thinks because he's so eager to learn new things and to become wiser, it ultimately lead to his downfall, and lead to his throne being taken away from him.
Magic in the play is mainly used to get characters to fall in love with people they would not usually pursue. In this way, their foolish romantic behavior appears even sillier because it often has an unlikely object, such as when Titania falls in love with Bottom, who has the head of an ass.