Talk about a fantasy woman. To Frederick, at least. And in fact, Flaubert based Madame Arnoux on an older woman he was in love with—a lady named Elisa Schlésinger. And you guessed it—she was married, too. So as far-fetched as the whole scenario might seem, it's based our author himself.
From the moment Frederick sees Marie on the deck of boat in Paris, he's never able to let go completely of the idea of possessing her. Let's check out the love-at-first-sight moment to see what all the fuss is about:
What he then saw was like an apparition. She was seated in the middle of a bench all alone, or, at any rate, he could see no one, dazzled as he was by her eyes. At the moment when he was passing, she raised her head; his shoulders bent involuntarily; and, when he had seated himself, some distance away, on the same side, he glanced towards her.
She wore a wide straw hat with red ribbons which fluttered in the wind behind her. Her black tresses, twining around the edges of her large brows, descended very low, and seemed amorously to press the oval of her face...
Never before had he seen more lustrous dark skin, a more seductive figure, or more delicately shaped fingers than those through which the sunlight gleamed. He stared with amazement at her work-basket, as if it were something extraordinary. What was her name, her place of residence, her life, her past? He longed to become familiar with the furniture of her apartment, all the dresses that she had worn, the people whom she visited; and the desire of physical possession yielded to a deeper yearning, a painful curiosity that knew no bounds. (1.1.20-22)
Our guy is obsessed with her shoes, the hem of her dress, her shawl, her hair, the fur trim on her coat—the list goes on. She doesn't even seem like a full human, really. She's more like a sum of parts that fascinate him: her perfectly oval face in the lamplight, her black eyes, her shoulders, and her generally exotic appearance. And since Frederick is out of a job, he dedicates himself pretty much full time to thinking about her.
The point? Almost everything we know about this woman comes through Frederick's gaze. So can we trust it? Can we understand a woman based on what a googly-eyed man who's totally obsessed says about her?
In spite of Frederick's erotic entanglement with Rosanette and who knows who else, our protagonist only really loves Madame Arnoux. There's something special about this one.
The two not-so-lovebirds enjoy a period of deep friendship, which Frederick hopes to consummate by getting a little rendezvous apartment for them. But sure enough, the lady doesn't show up. Not realizing that her son is deathly ill, Frederick gets back at her by starting a torrid affair with Rosanette. After Madame Arnoux's husband's bad business practices drive them into bankruptcy, she has to move out of Paris, and that's the end of that.
At the end of the novel, when Marie finally gives Frederick the go ahead, he decides he's not interested:
When they came back to the house, Madame Arnoux took off her bonnet. The lamp, placed on a bracket, threw its light on her white hair. Frederick felt as if some one had given him a blow in the middle of the chest. (2.19.49)
What does this rejection—this guttural reaction—say about Frederick and about the quality of his love for Marie Arnoux?
Even as Frederick spends 400 pages consumed by this woman, he never really thinks about why. All of his thoughts come down to how she looks and how, well, motherly and moral she is. Although he's definitely attracted to her in a sexual way, Frederick also sees her as a homebound figure, and she can often be found knitting or teaching her daughter piano. This woman is the embodiment of virtue—and therefore unlike any other woman in the novel.
We could take all sorts of guesses: Does he love her as a substitute for his own groveling mother? Does he love her because she's not like all of the bimbos of the demi-monde? Does he love her because she's just so hot? Flaubert doesn't tell us; that's for sure. And we're guessing Frederick might not have a clue.