Parallel lines appear throughout Sentimental Education and represent the unrequited love Frédéric holds for Madame Arnoux. Parallel lines appear in many different contexts, but they generally mark moments when Frédéric has seen or is thinking about Madame Arnoux. On the boat, when Frédéric first spots Madame Arnoux, he notices that the riverbanks looked like two ribbons. After he first has dinner at the Arnouxes’ home, the lamps on the street are described as shining in two straight lines. On the day he is supposed to meet Madame Arnoux at an apartment he has rented out, he sees student demonstrators marching in two lines. Significantly, parallel lines appear in the climactic scene when Madame Arnoux offers herself to Frédéric: she describes her new home to him, including the “double avenue of chestnuts.” In these moments, Frédéric may have spoken to or come close to connecting physically with Madame Arnoux, but, like parallel lines, their lives never succeed in intersecting.
Parallel lines appear during Frédéric’s interactions with other women as well. At Madame Dambreuse’s home, guests sit on chairs positioned in two straight lines. When Frédéric takes Rosanette to the races, two lines of posts delineate the course. When he prepares to go out with Rosanette on another occasion, he notices that the street lamps were like a double string of pearls. Whether the parallel lines appear when Frédéric is with Madame Arnoux or with other women, the meaning of the image is clear: just as parallel lines can never meet and cross, Frédéric and Madame Arnoux are doomed to remain apart throughout their lives.
Roses appear at two significant points of the novel and in both cases represent the impossibility of love. First, roses play a role in granting Frédéric and Madame Arnoux their first real intimacy. When Frédéric visits the Arnouxes outside of Paris, Monsieur Arnoux leaves Madame Arnoux to go boating with other guests and then gives her a bouquet of roses, which she does not want. Later, sitting with Frédéric in a carriage, Madame Arnoux tosses the roses out the door, an act that only Frédéric witnesses. This is the first secret they share, but the rose incident ultimately leads nowhere—their love is and always will be impossible. Roses appear again in the form of the name of Frédéric’s lover, Rosanette. Although Rosanette is a serious partner for Frédéric, someone who wishes to build a life with him, he maintains his love for Madame Arnoux throughout this affair, which ultimately dooms it. The true love Frédéric might have felt for the son he has with Rosanette is also doomed, since the child dies in infancy. Roses, traditionally symbols of love, instead suggest heartbreak in Sentimental Education.
Madame Arnoux’s white hair, which she exposes when she comes to offer herself to Frédéric, fully reveals to Frédéric the passage of time and represents the true end of a love affair that never really began. Madame Arnoux has been part of Frédéric’s life since he was eighteen years old, remaining the one constant element in a life filled with political unrest, other lovers, social conquests, career pursuits, and travels. Twenty years after he first sees her, his love for her still exists, although at this point it has taken on a life of its own. Madame Arnoux, whom he hasn’t seen in years, is in many ways no longer a woman but a fantasy; he barely knew her when she was physically present, so his love is rooted in his idealized image of her rather than fact. When Madame Arnoux reappears and reveals her white hair, she becomes, suddenly, human.
Fully present and willing to actually consummate Frédéric’s love, the very human Madame Arnoux loses her appeal. Just like that, Frédéric’s feelings for her are reversed, and he waits impatiently for her to leave. Madame Arnoux has committed the ultimate transgression—she has aged, thus changing utterly from the image Frédéric has held in his imagination all these years. The woman he had held as an ideal specimen of femininity has fallen from grace. Far from being the one true object of Frédéric’s eternal desire, Madame Arnoux is now so devoid of sexuality that she kisses him as a mother would. The white hair signifies and reveals Madame Arnoux’s true self, and, therefore, the love affair must end.