How we cite our quotes:
This diversion is only practiced by those persons who are candidates for great employments, and high favour at court. They are trained in this art from their youth, and are not always of noble birth, or liberal education. When a great office is vacant, either by death or disgrace (which often happens,) five or six of those candidates petition the emperor to entertain his majesty and the court with a dance on the rope; and whoever jumps the highest, without falling, succeeds in the office. (1.3.2)
The Emperor of Lilliput uses this rope dance to test "candidates for great employments." The person who can jump the highest wins a government job. So, basically, if you can impress the Emperor by doing something completely unrelated to government, bam! You can get a government job. How might this rope dance work as a larger metaphor for politics at court?
This open bold declaration of mine was so opposite to the schemes and politics of his imperial majesty, that [the Emperor of Lilliput] could never forgive me. He mentioned it in a very artful manner at council, where I was told that some of the wisest appeared, at least by their silence, to be of my opinion; but others, who were my secret enemies, could not forbear some expressions which, by a side-wind, reflected on me. [...] Of so little weight are the greatest services to princes, when put into the balance with a refusal to gratify their passions. (1.5.5)
Even though Gulliver manages single-handedly to stop the Blefuscu naval fleet from invading Lilliput (that's the "greatest service to princes" he's talking about), he still becomes the Emperor's enemy. Why? Because Gulliver refuses to use his great strength to enslave the Blefuscudians, even though the Emperor of Lilliput asks him to. This is the problem with politics in Lilliput: keeping the Emperor happy often means lying, flattery, or hypocrisy. And acting according to a pretty basic moral rule – do not make people slaves – makes the Emperor angry. Politics and morality do no seem to be compatible.
When I had for some time entertained their excellencies, to their infinite satisfaction and surprise, I desired they would do me the honour to present my most humble respects to the emperor their master, the renown of whose virtues had so justly filled the whole world with admiration, and whose royal person I resolved to attend, before I returned to my own country. Accordingly, the next time I had the honour to see our emperor, I desired his general license to wait on the Blefuscudian monarch, which he was pleased to grant me, as I could perceive, in a very cold manner; but could not guess the reason, till I had a whisper from a certain person, "that Flimnap and Bolgolam had represented my intercourse with those ambassadors as a mark of disaffection;" from which I am sure my heart was wholly free. And this was the first time I began to conceive some imperfect idea of courts and ministers. (1.5.7)
Gulliver sucks up to Lilliput's enemy emperor, the Emperor of Blefuscu. Somehow, it doesn't seem to occur to him that this will make him look bad to Lilliput's own emperor. So, he's pretty naive. But what really gets to Gulliver is the discovery that his enemies of court have been secretly trying to turn the Emperor against him. They use Gulliver's friendly words to the Blefuscu Emperor as ammunition to prove to the Lilliput Emperor that Gulliver is not loyal to him. Again, we have to say, Gulliver claims to be shocked by these activities, which suggests that he is the most naive guy in the world. This makes his final, total mistrust of all people at the end of Part 4 all the more stark.