As happens to young couples in most of Shakespeare's comedies, and in one notable tragedy, Romeo and Juliet, Ferdinand and Miranda fall deeply in love at first sight in The Tempest.
Miranda sees Ferdinand from a distance, and is the first to fall in love.
MIRANDA. I might call him
A thing divine, for nothing natural
I ever saw so noble. (1.2.491-493)
When they're closer together, Ferdinand instantly succumbs to Miranda's ethereal beauty and wants to stay with her forever.
FERDINAND. Most sure, the goddess
On whom these airs attend! Vouchsafe my
May know if you remain upon this island,
And that you will some good instruction give
How I my bear me here. (1.2.498-503)
Prospero arranged for Ariel to bring Ferdinand to Miranda, and Prospero is delighted with the result, which he believes will help him to recover his dukedom in Milan.
PROSPERO. ... At the first sight
They have changed eyes. (1.2.523-524
Ferdinand proposes marriage even faster than Juliet proposed to Romeo in the balcony scene. (Romeo and Juliet, 2.2.148-154)
FERDINAND. O, if a virgin,
And your affection not gone forth, I'll make you
The Queen of Naples. (1.2. 531-533)
That's all that Prospero wants to hear for now, and he decides to slow down the romance.
PROSPERO. [to Ferdinand] Soft, sir! One word more.
[Aside] They are both in either's powers; but this swift
I must uneasy make, lest too light winning
Make the prize light. (1.2.534-538)
Prospero can't resist controlling Miranda and Ferdinand's relationship the same way he controls everything and everyone else on the island, but their love for one another exists beyond Prospero's sorcery. Ferdinand and Miranda overcomes every test and condition that Prospero imposes on them.
Ferdinand and Miranda's quiet, loving relationship is a stark contrast to the noisy, contentious relationships between and among all of the other characters on the island. Their relationship represents calm in the midst of the tempest of intrigue and revenge swirling around them.
This is clearly exemplified in act 5. Prospero is trying to resolve the many complex and confusing issues of the play. Prospero leads Alonso into Prospero's home, where they discover Ferdinand and Miranda, quietly playing chess, oblivious to the chaos outside.
Ferdinand and Miranda are apparently having their first argument, which, compared to every other argument on the island, is respectful, loving, remarkably civil, and wholly lacking in merciless, vengeful intent.
[Here Prospero discovers Ferdinand and Miranda playing at chess]
MIRANDA. Sweet lord, you play me false.
FERDINAND. No, my dear'st love,
I would not for the world.
MIRANDA. Yes, for a score of kingdoms you should wrangle,
And I would call it fair play. (193-197)
Ferdinand and Miranda's love is a reminder to everyone else of their own temporarily lost humanity. Prospero renounces his sorcery, freedom is restored to servants and masters alike, order returns to the island, and happiness reigns.