Dedication Summary

To Mrs. Arabella Fermor

Madam,
It will be in vain to deny that I have some Regard for this Piece, since I Dedicate it to You. Yet You may bear me Witness, it was intended only to divert a few young Ladies, who have good Sense and Good Humour enough, to laugh not only at their Sex's little unguarded Follies, but at their own. But as it was communicated with the Air of a Secret, it soon found its Way into the World. An imperfect copy having been offer'd to a Bookseller, You had the Good-Nature for my Sake to consent to the Publication of one more correct: This I was forc'd to before I had executed half my Design, for the Machinery was entirely wanting to compleat it. The Machinery, Madam, is a Term invented by the Criticks, to signify that Part which the Deities, Angels, or Daemons, are made to act in a Poem: For the ancient Poets are in one respect like many modern Ladies; Let an Action be never so trivial in it self, they always make it appear of the utmost Importance. These Machines I determin'd to raise on a very new and odd Foundation, the Rosicrucian Doctrine of Spirits.
I know how disagreeable it is to make use of hard Words before a Lady; but 'tis so much the Concern of a Poet to have his Works understood, and particularly by your Sex, that You must give me leave to explain two or three difficult Terms.
The Rosicrucians are a People I must bring You acquainted with. The best Account I know of them is in a French Book call'd Le Comte de Gabalis, which both in its Title and Size is so like a Novel, that many of the Fair Sex have read it for one by Mistake. According to these Gentlemen, the four Elements are inhabited by Spirits, which they call Sylphs, Gnomes, Nymphs, and Salamanders. The Gnomes, or Daemons of Earth, delight in Mischief; but the Sylphs, whose Habitation is in the Air, are the best-condition'd Creatures imaginable. For they say, any Mortals may enjoy the most intimate Familiarities with these gentle Spirits, upon a Condition very easie to all true Adepts, an inviolate Preservation of Chastity. As to the following Canto's, all the Passages of them are as Fabulous, as the Vision at the Beginning, or the Transformation at the End; (Except the Loss of your Hair, which I always mention with Reverence.) The Human Persons are as Fictitious as the Airy ones; and the Character of Belinda, as it is now manag'd, resembles You in nothing but in Beauty. If this Poem had as many graces as there are in Your Person, or in Your Mind, yet I could never hope it should pass thro' the World half so Uncensured as You have done. But let its Fortune be what it will, mine is happy enough, to have given me this Occasion of assuring You that I am, with the truest Esteem,
Madam,
Your Most Obedient
Humble Servant.
A. Pope.

• If you ever wind up reading a lot of 18th-century British poetry, you'll notice that many of the longer poems feature a letter of dedication, usually to a famous, powerful, rich, or important person (ideally, for the poet, a person who has all four of those things going on at once) at the very beginning of the work. Poets would often do this as a way to associate their work with that powerful or famous person, kind of like the way Nike named a style of basketball shoe the Air Jordan. Call it a form of literary endorsement.
  
• The Rape of the Lock is no exception. While Arabella Fermor isn't famous at this point or politically powerful, here Pope has to work his way through a potentially complicated situation: he needs to make sure that Arabella is okay with him taking her story public. And even if she's not okay with it, he needs to make sure his readers know that he at least tried to make her okay with it. He's going to publish it either way.
• You might notice that the letter addresses Arabella as "Mrs."; she's not actually married at this point, though. People in the 18th century addressed all respectable women as "Mrs", which was shorthand for "Mistress."
• First, Pope explains to her his reasons for publishing the original poem (as opposed to just writing it out and giving it to all of the parties concerned): according to him, he was forced to take it public, as copies were leaking out and it would have been published anyway, so why not make sure the official version is the one that gets out? (Don't be fooled by Pope's disclaimer here, though. He knows this poem is top-notch and wants to take public credit for it).
• Pope then explains some of the more obscure bells and whistles he's added to the poem's original story. That's the "Machinery" he describes in the second and third paragraphs.
• Notice here his two-sided, "compliment" (NOT) to women readers in general, and Arabella in particular, when he says "I know how disagreeable it is to make use of hard Words before a Lady; but 'tis so much the Concern of a Poet to have his Works understood, and particularly by your Sex, that You must give me leave to explain two or three difficult Terms."
• If this sounds condescending to you, you're right. It is. Pope may be gallant and polite here, but like most 18th-century men of the educated class, he had a sadly low opinion of women's intellect and abilities. We're still two hundred and twenty-six years away from women getting the vote, and just over two hundred and fifty years away from Title IX, folks.
• "Machinery," as Pope next explains to Arabella, is a fancy word for the supernatural elements in an epic poem. For Homer and Virgil, these would be the old Greek gods who meddled in all of the battlefield action; for Pope, it's the Sylphs and Gnomes he describes here. The Rosicrucians were a medieval secret society that practiced alchemy and dabbled in Middle Eastern philosophy. 
• That book Pope mentions with the French title? It's Le Comte de Gabalis ("the Count Gabalis"), a screwball comedy written in the 1670s about occult beliefs and mystical spirits, with some history and philosophy thrown in. The Rosicrucians loved it. 
• The final two paragraphs of this opening letter dedicate the poem to Arabella, and are also a fancy version of the "all characters in this work are fictitious" disclaimer you'll often find at the beginning or end of a movie. 
• You might also notice that a lot of the words in the letter (and throughout the entire poem) appear to be randomly capitalized or italicized, or misspelled. While sometimes the capitalization or italicization is meaningful (as when Pope personifies a noun, or wants to emphasize a word), for the most part it is random. 
• Here's the deal: in the early 18th century, the English language wasn't yet fully standardized. There were no official, comprehensive dictionaries or guides to correct or proper grammar and spelling. (We know, you might now be wishing you lived back then, right?) At this point too few people knew how to read and write to even bother. But over the next hundred years, literacy exploded, and as more folk became literate, language became more regulated and codified.