There are two sorts of knowledge in Arcadia: the knowledge of love and academic knowledge. These two types of knowledge are in constant conflict throughout the text. It is only the proposition of marriage, the intellectual justification for sex, which allows a resolution between the two forces. The theme of love vs. intellect is touched upon in the first pages of the play. Thomasina interrupts her lesson with Septimus by asking what carnal knowledge is. Sexual knowledge always acts in conflict with intellectual knowledge, and here it gets in the way of the lesson. Thomasina also remarks on the conflict between emotion and intellect in her history lesson. Her question is prompted by Septimus himself who was found having sex with Mrs. Chater in the gazebo the day before. Thomasina describes Cleopatra as making "noodles of our sex" because Cleopatra was weakened by love. Thomasina heralds Queen Elizabeth who would not have been tempted by love to give away land or power. The great Hannah Jarvis is, like Thomasina's Queen Elizabeth, unswayed by romantic passions. She believes, as does Thomasina, that romantic inclinations would destroy or distract her from her work. Hannah refuses warmth or emotion: she refuses a kiss, denies Bernard's propositions, laughs at Valentine's proposal, and brushes off Gus's flirtation.
Nonetheless, Hannah, like Thomasina, Septimus, and Gus all waltz at the conclusion of the play. Hannah cannot refuse emotion or the bashful Gus by the end of the play and is drawn into an uncomfortable and uneasy dance. The conflict between emotion and intellect is resolved because Hannah suddenly understands that the two are inseparable. Hannah is unlike Thomasina, who unconsciously understands this, driven forcefully by the mystery of both.
Sex remains the final mystery of Arcadia. Septimus, in the conclusion of the play, reveals the final sadness and emptiness of an academic life: "When we have found all the mysteries and lost all the meaning, we will be alone, on an empty shore." Septimus implies that the mysteries of mathematics will someday be solved. As if knowing his own fate, Septimus embraces and kisses Thomasina in earnest, finally indulging in the mystery of his attraction and love. Septimus will not go to Thomasina's room, although she asks him, but he is restrained for a reason that remains unknown. Septimus realizes the ultimately unfulfilling nature of academic progress but will only tragically experience the fulfilling nature of love for a brief moment in a waltz and kiss with Thomasina. In the same manner, Hannah Jarvis submits to a dance with Gus. She, like Septimus, has solved her mystery and now looks to Gus for fulfillment and new mysteries.
Septimus describes to Thomasina the path of knowledge, a humanity that drops knowledge and learning as it picks up new ideas and developments. Septimus tells Thomasina she should not be upset at the loss of the library of Alexandria because such discoveries will be had again, in another time and possibly in another language. This story is ironic to the fate of Thomasina's own discoveries that aren't unearthed until 1993 by Valentine. Thomasina's discoveries are made again: chaos theory and thermodynamics are formal concepts by the time her primer is found and analyzed. Arcadia works as a description of humanity's own progression of knowledge. While Thomasina and Septimus make new discoveries, Hannah and Valentine work to find their discoveries. The work of Thomasina and Septimus is lost but later found again.
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.
The Gardens of Sidley Park symbolize the transformation and transition between romanticism and classicism. Mr. Noakes wishes to alter the gardens into the picturesque and thoroughly romantic style and means to tear out the gazebo in favor of a hermitage and drain the lake with a newly improved steam engine. Lady Croom accuses Mr. Noakes of reading too many novels by Radcliff, such as The Castle of Otranto (actually written by Horace Walpole, as Mr. Chater points out), and The Mysteries of Udolpho. Mr. Noakes means to transform the green, lush perfect Englishman's garden into an "eruption of gloomy forest and towering crag," Lady Croom describes it as a haunt of "hobgoblins." As Hannah describes it, the garden is a classical painting imposed on landscape or "untamed nature in the style of Salvatore Rosa everything but vampires". The garden represents romanticism, (for Hannah) a decline from thinking to emotion, and the need for "false emotion" and "cheap thrills."
The modern day characters wear the Regency Clothes or clothes that would be worn to a fancy dress ball in Thomasina's time. Regency Clothes symbolize high society and privilege. The dress not only links the two generations and time periods, but it reveals the hay day of the English aristocratic family. Chloe, Gus, and Valentine wear the outfits to have their pictures taken and dress for the annual dance. The dress reestablishes their power as a family and role in the community, seemingly diminished in modern times.
The Primer is the symbol of learning and academia. Thomasina is the first to use the primer, which once belonged to Septimus; however, at the conclusion of the play, Septimus has taken back his primer. Septimus's use of his the primer once again symbolizes his return to being a student; this time he is a student of Thomasina, who has surpassed his knowledge and teachings.