How does Jonathan Swift use ethos, logos, and pathos in "A Modest Proposal"?

Because “A Modest Proposal” is such a deeply ironic work, the use of ethos, pathos and logos differs markedly from their normal usage in, for instance, a political speech. All these rhetorical devices are typically used to persuade the audience to agree with the speaker or writer. Here, they are all used to horrify the reader by the callous inhumanity of the argument. This is the principal reason why logos is the most prevalent device. The cool appraisal of what a child is worth considered as a piece of meat is intended to provoke outrage at the heartlessness of such a calculation. This effect is exacerbated by the references to women as “breeders” and the constant comparisons with sheep and cattle.
Swift’s calculations are a particularly effective way of using logos to convey human tragedy more effectively than pathos could, by including a mass of human misery in apparently dispassionate statistics:

I again subtract fifty thousand for those women who miscarry, or whose children die by accident or disease within the year. There only remains one hundred and twenty thousand children of poor parents annually born.

Of the apparent pathos in the essay, only the pathos in the opening is not entirely ironized, since the sight of beggars on the streets is indeed melancholy, though not for the reasons Swift is to profess. His solution is the principal source of irony, since it is worse than the problem:

It is a melancholy object to those who walk through this great town or travel in the country, when they see the streets, the roads, and cabin doors, crowded with beggars of the female sex, followed by three, four, or six children, all in rags and importuning every passenger for an alms.

Ethos is also used ironically in the final paragraph, when the author protests his own disinterest, as though there are so few legitimate objections to his proposal that this is the most serious of which he can conceive and he must dispel it by insisting on the purity of his own motives:

I profess, in the sincerity of my heart, that I have not the least personal interest in endeavoring to promote this necessary work, having no other motive than the public good of my country, by advancing our trade, providing for infants, relieving the poor, and giving some pleasure to the rich. I have no children by which I can propose to get a single penny; the youngest being nine years old, and my wife past child-bearing.

The speaker employs ethos, attempting to establish his own credibility with his audience, when he talks about being "assured by our merchants" as to the value of young children or being "assured by a very knowing American" who has, apparently, experienced a system very like the one the speaker proposes. He makes it sound as though he has consulted experts who support his idea, lending him credibility.
The speaker employs pathos, appealing to his audience's emotions, at the start of the text when he says,

It is a melancholy object to those, who walk through this great town, or travel in the country, when they see the streets, the roads and cabin doors crowded with beggars of the female sex, followed by three, four, or six children, all in rags, and importuning every passenger for an alms. These mothers instead of being able to work for their honest livelihood, are forced to employ all their time in strolling to beg sustenance for their helpless infants.

He attempts to draw an incredibly sympathetic picture of the poor Irish peasants who are forced to beg for their sustenance, especially the women and children, dressed "all in rags" and seeking kind souls who are willing to spare a few pennies. In this way, he tries to win over his audience's hearts so that they will be more open to listen to his proposal.
The speaker employs logos when he begins to make his mathematical calculations:

The number of souls in this kingdom being usually reckoned one million and a half, of these I calculate there may be about two hundred thousand couple whose wives are breeders; from which number I subtract thirty thousand couple, who are able to maintain their own children, . . . but this being granted, there will remain an hundred and seventy thousand breeders. I again subtract fifty thousand, for those women who miscarry, or whose children die by accident or disease within the year. There only remain an hundred and twenty thousand children of poor parents annually born. The question therefore is, How this number shall be reared, and provided for? which, as I have already said, under the present situation of affairs, is utterly impossible by all the methods hitherto proposed.

He attempts to present very logical calculations about how the community cannot go on the way it is with the sheer number of children born to the poor Irish. This way, his proposal will seem all the more logical to his audience.

One of the strengths of Swift's satire is his relentless use of logos. Swift's narrator employs one statistic after another to demonstrate the efficacy of his "proposal." He points out that there are over one hundred thousand children born in Ireland that could be profitably sold for food. This, he computes, will have a dramatic effect on the economy of the kingdom:

Whereas the Maintainance of an hundred thousand Children, from two Years old, and upwards, cannot be computed at less than Ten Shillings a piece per Annum, the Nation's Stock will be thereby encreased fifty thousand pounds per Annum, besides the profit of a new Dish, introduced to the Tables of all Gentlemen of Fortune in the Kingdom, who have any refinement in Taste, and the Money will circulate among our selves, the Goods being entirely of our own Growth and Manufacture.

In many ways the appeal to logos in the essay is exactly what Swift is satirizing. He is claiming that attacking the social problems of a society in an overly rational way (without considering morality) can lead mankind to make decisions with terrible consequences. However, Swift also appeals to pathos, citing the "melancholy sight" of beggars and orphans on street corners and the "horrid practice of women murdering their Bastard children" for want of means to support them. He uses ethos when he points out, sardonically, that he has no financial interest in the "proposal" because his children are past the age where they are edible.