Fire takes on multiple meanings in the play, but it most strongly symbolizes death and the eventual and inevitable end of the human species. Like Thomasina's diagram of heat exchange, as exemplified by Mr. Noakes's steam engine, all will eventually end. As the law of thermodynamics prescribes, we will all eventually burn up. Fire is destruction and death happening over and over again. Septimus burns Lord Byron's letter, unread, a rare and valuable piece of historical literature. Fire is also sexual, the burn that keeps bodies in motion. Septimus observes that Mrs. Chater is in a state of "tropical humidity as would grown orchids in her drawers in January". Thomasina and Valentine wish to describe and analyze the universal laws of heat and destruction. The final scene is the greatest culmination of the fire motif. While Valentine and Hannah discuss the meaning of Thomasina's heat-exchange diagram, Thomasina holds the flame that will eventually cause her own destruction. As Thomasina and Septimus waltz, the audience is aware of Thomasina's fate. We can see the workings and progress of the heat diagram before our eyes.
Sex persists as the anti-academic driving force in Arcadia. Academic knowledge is never separated far from carnal knowledge—academic knowledge somehow equating sexual prowess. For example, when Bernard makes his great discovery he immediately propositions Hannah, indicating how academic knowledge gives Bernard sexual confidence. Sex is also equated with heat, making it the eventual objective and need of all humans. The relationship between Thomasina's theory of heat exchange and sex is clearly articulated by Chloe who tells Valentine that Newton forgot to account for sex in his deterministic universe. Heat, like sex, is unchangeable, persistent, and random.
Mathematics and "Simple English Algebra" is the foundation of Arcadia. The mysteries of math reveal greater truths about humanity and the family as a whole. Mathematics is also a source of pride within the play. Valentine, as a chaos mathematician himself, is reluctant to share Thomasina's theory and fractal with Hannah. Thomasina's algebra and geometry lessons culminate into her genius understanding of the laws of thermodynamics and chaos theory. The laws of thermodynamics dictate the fate of all the characters on stage, and the realization of such fate eventually conclude the play (most tragically, Thomasina's own ironic death by fire). Septimus and Thomasina, along with Gus and Hannah, succumb to the law of thermodynamics by coming together in a waltz. The couples know their mathematical, unstoppable fate and embrace each other in spite of it.
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