In Chapter 13, why has Lucy "cried hot tears" and suffered a great inward turmoil in Villette following Madame Beck's discovery that Dr. John was in the garden with her?

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It is always important to look at such quotes in context to discover what is being suggested by them. We see that, following Lucy Snowe's awarenss of Madame Beck's spying upon her and Dr. John, that she is at first amused by the conclusions that Madame Beck will have no doubt reached regarding her meeting with Dr. John. Lucy says that "the spectacle of a suspicious nature so far misled by its own inventions, tickled me much." However, after this laughter and the amusement that it indicates is finished, note how Lucy moves to sadness and tears:

Yet as the laugh died, a kind of wrath smote me, and then bitterness followed: it was the rock struck, and Meribah's waters gushing out. I never had felt so strange and contradictory an inward tumult as I felt for an hour that evening: soreness and laughter, and fire, and grief, shared my heart between them.

Such contradictory emotions obviously give rise to the "complicated, disquieting thoughts" that Lucy Snowe suffers, which we can infer result from perhaps her own romantic feelings towards Dr. John and the way that they are suspected by Madame Beck through appearances, though in reality Dr. John shows no romantic feelings towards her. We see in this section of the novel a real insight into Lucy's character and her desire to love and be loved, and her sadness at her lonely state in life.