In the essay "A Modest Proposal," what are direct examples of Swift's reliance on authority to lend weight in his own modest proposal?
Although your question references Swift's proposal, let me be picky and reframe the question as the speaker's proposal (or the essayist's proposal, or even the Proposer's proposal). We need to make it clear that the speaker in the essay is not the same person as the author and social critic Jonathan Swift. The distinction helps us understand that the essay is satire, and that Swift doesn't actually think Irish society should solve its problems by eating their own babies!
This speaker does rely on others' authority throughout the piece, which adds to its persuasive power. Let's look at some examples, in order from the beginning to the end of the essay.
1. "As I have been informed by a principal gentleman in the county of Cavan, who protested to me, that he never knew above one or two instances under the age of six, even in a part of the kingdom so renowned for the quickest proficiency in that art."
Here, the speaker defers to a "gentleman" to add authority to the statement that kids under six are barely good for anything, not even making a living by stealing.
2. "I am assured by our merchants, that a boy or a girl before twelve years old, is no saleable commodity..."
Above, the speaker calls on the authority of merchants--specialists who know what sells and what doesn't--in order to make a more believable statement about how old kids have to be before they're of any value in the slave trade.
3. "I have been assured by a very knowing American of my acquaintance in London, that a young healthy child well nursed, is, at a year old, a most delicious nourishing and wholesome food..."
By mentioning the American living in London, the speaker invents an authority on cannibalism, one with both the wild exoticism of being an American and the more grounded quality of living in nearby England. This same friend is mentioned a bit later on as further authority for the idea that cannibalism is acceptable:
"...my American acquaintance assured me from frequent experience, that their flesh was generally tough and lean..."
(By now, I hope it's clear that most of these instances of borrowed authority add to the dark humor of the essay!)
4. "...for we are told by a grave author, an eminent French physician, that fish being a prolifick dyet, there are more children born in Roman Catholick countries about nine months after Lent, the markets will be more glutted than usual..."
Above, the reference to the French man, who's not just a physician but an "eminent" one, lends authority to speculations about market fluctuations in the availability of fish compared to children as a source of food. As you can see in this example in particular, the speaker often uses the "reliance on authority" trick as a way of finding something sane to say about whatever awful point he's just brought up!
5. "A very worthy person, a true lover of his country, and whose virtues I highly esteem, was lately pleased, in discoursing on this matter, to offer a refinement upon my scheme."
Though he's on shakier territory here by inventing some vague patriot, the speaker is using the borrowed authority of a typical citizen to assert that the proposal is a great idea and that it'll go over well with people who have great morals. (Again, I hope these examples offer some humor!)
6. "...the famous Salmanaazor, a native of the island Formosa, who came from thence to London, above twenty years ago, and in conversation told my friend, that in his country, when any young person happened to be put to death, the executioner sold the carcass to persons of quality, as a prime dainty..."
Above is an authority from another country (Taiwan, which used to be called Formosa) whom the speaker relies on to further lend weight to cannibalism as an acceptable practice.
Of course, it's worth considering which of the examples above are stronger, more effective instances of relying on authority. You might point out, for example, that the eminent French physician from the fourth example above is a more effective authority on whom the speaker relies than, say, the random unnamed guy from the fifth example.