Swift's The Lady's Dressing Room (1732) is a typical, extended Swift scatalogical (toilet humor) satire on an ostensibly upper-class woman's (Celia) dressing room and beautification techniques. The satire, however, is so successful that most readers would not want to go within a hundred feet of Celia's dressing room.
The poem is immediately set up with mock-heroic overtones with the names of Celia and Strephon, both classical characters and, of course, Celia is actually described as "the Goddess." After Celia leaves her dressing room, Strephon decides to examine its contents, and from the beginning of the third stanza, the inventory becomes increasingly unsavory and grotesque.
Swift tells us, for example, that Celia's "dirty Smock" has stained arm-pits: her comb is so dirty that no brush could clean it; Celia, that dainty Goddess, wears gloves to smooth her hands made of her most recent lapdog's skin; and the oil she uses on her skin is either puppy piss from "Tripsey's whelp" or oil rendered from the puppy's body. The litany of the grotesque bodily fluids goes from worse to much worse when we get to "the scrapings of her Teeth and Gums,/A nasty Compound of all Hues. . . ."
When Strephon looks into Celia's chest--which is characterized as "Pandora's Box," the box that contains all the evils in the world--he's hit with "Vapours." But, as he feels the bottom of the chest, he finds "such vile Machine" that his horrified reaction is that "she better learn to keep/Those Secrets of the hoary deep!" This is undoubtedly a reference to what in polite society is called a "sex toy," and even in the context of this poem, is a bit startling.
Strephon is so horrified by what he finds during his search that he can no longer see the charms in Celia or other women, and Swift notes that Strephon needs to see women as Swift does--look at the result and forget how the result is obtained.
Lady Mary Wortley Montagu's is a much more light-hearted, generally non-scatalogical (until the last two lines) poem not based on mock heroics. Montagu essentially has "Doctor Swift," emphasizing that he's a cleric, meeting his "dearest Betty," who happens to be a prostitute and whom Montagu describes as "this dull hard hearted creature" who accepts four Pounds (a significant sum) for her companionship.
In addition to satirizing Swift himself, she manages to get a dig in at Alexander Pope, one of Swift's friends and part of his literary circle, who wrote a blistering satire against Montagu as "Sappho." According to Montagu, Pope's writing is "so much rhyme and little reason."
When Swift attempts to engage in sex with Betty, and apparently cannot, he blames his inability on Betty's "damned close stool so near my nose," reminding the reader that a chamberpot was kept right under the bed for convenience, and in an echo of Swift's satire on Celia, Montagu has Swift cry out, "Your dirty smock, and stinking toes/Would make a Hercules as tame/As any beau. . . ."
After demanding his money Swift and threatening to describe Betty's dressing room, Betty responds, "I'm glad you'll write/You'll furnish paper when I shite," directly comparing his poetry to toilet paper.
Both poems accomplish their satiric goals admirably.