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"A phenomenon solved by modern philosophy and astronomy. The Laputians' great improvements in the latter. The king's method of suppressing insurrections."
Gulliver then launches into a long description of how exactly Laputa functions: first of all, the island has a crater in the center of it that collects rain water, which is why rain doesn't just fall off it.
At the center of the island is a deep canyon with a giant lodestone, a naturally occurring magnet, in the middle of it.
The King uses this lodestone to raise and drop the island and to keep it moving in relation to the Earth's own magnetic poles.
The movement of Laputa has limits: it can't go beyond the king's own dominions, in other words, the islands that he controls at sea level. It also can't rise higher than four miles above the Earth.
It is the job of the King's astronomers to do the actual manipulation of the lodestone at his orders.
They also spend a lot of time discovering things about the solar system and the stars.
The only thing that limits the King's control of the Earth below him is that all of his cabinet members have estates on the islands below Laputa, so they find the idea of dominating the islands under them to be pretty risky for their own families.
At the same time, the King still has two methods for keeping his authority over the lower islands without absolutely enslaving them:
(1) if any of them refuse to pay tribute, he can make his island float directly overhead, blocking their sunlight and rain, until they give in;
and (2) if they continue to refuse to obey him, the King can drop his island directly on their heads.
The King has rarely ordered this kind of total destruction because (a) his ministers have their homes down below, and (b) his own people would revolt against him.
Well, and there's one more reason why the King doesn't do this: secretly, he worries that the power of his magnet might not be strong enough to lift the island again if it comes crashing to earth.
Laputa also has a law that neither the King nor his two eldest sons, nor the queen (while she can still have children) are allowed to leave the island.