The play shifts back to the early nineteenth century. It is morning and Thomasina and Septimus sit together in the schoolroom. Septimus is reading a letter that has just arrived, and Jellaby waits. In front of Septimus is The Couch of Eros, Mr. Chater's book, and sheets of notes that Septimus has taken for the review. Like Valentine, Septimus also has a tortoise, Plautus, that currently acts as a paperweight. When Septimus finishes the letter, he folds the paper and puts it in the leaves of The Couch of Eros.
Thomasina is translating a paper in Latin and is having some difficulty. Thomasina tells Septimus that her mother is in love with Lord Byron. Thomasina saw Lord Byron and her mother in the Gazebo laughing together. Thomasina thinks that Septimus is jealous of his friend, Lord Byron. Septimus hands Thomasina her corrected mathematics lesson book, and she is upset that Septimus gave her an "Alpha minus" (A-) on her problems. Thomasina tells Septimus that his equations are only for commonplace manufactured forms. Thomasina wants to create the kind of equations that make nature—an equation to make a flower rather than a circle, cone, or square. Septimus attempts to steer Thomasina back to the history of Cleopatra, but Thomasina tells him she hates Cleopatra. Thomasina hates Cleopatra because she gave away things out of love and weakness. Septimus takes the Latin paper Thomasina has been struggling with and reads it easily. Thomasina breaks into tears, accuses Septimus of cheating and runs out of the room.
Captain Brice enters the room, followed by Mr. Chater, who stands behind Brice. Chater tells Septimus to address Brice when Chater wants to speak with him (Chater). Septimus then talks to Brice as if he is Chater, which eventually infuriates Brice. Chater is still angry with Septimus because he slept with his wife. Lady Croom enters the room and tells Chater that Lord Byron would like a copy of his book. Lady Croom takes The Couch of Eros, which now has three letters in it, and returns to the garden.
Scene four switches, once again, to the present time. Hannah is reading aloud to Valentine, who is holding Lightning, the tortoise. Hannah is reading from Thomasina's portfolio and gives it to Valentine to look at. The pages of Thomasina's book are filled with iterated equations or equations that feed solutions of one equation into the next. Valentine is surprised that Thomasina would be doing this because iteration has only been practiced for the last twenty years. Valentine attempts to explain the significance of iteration to Hannah with little luck. Valentine tells Hannah that if each algorithm fed into itself a thousand times each dot would land in an unexpected place. In other words, the unpredictable results of iteration are like the unpredictability of nature.
Bernard enters with a copy of English Bards and Scotch Reviewers. Inside the book there is a penciled superscription, apparently written by Byron, that mentions Chater and "Eros." Hannah demands that Bernard give her proof that Byron wrote the subscription, and she tells Bernard that he doesn't even know if Byron ever came to Sidley Park. Valentine interjects that Byron did, indeed, come to Sidley Park—Valentine has seen a record of him in the game book. Byron runs off to find Chloe who knows where the game book would be. Valentine asks Hannah if he can keep Thomasina's books so he can work out a diagram of her equations. Hannah asks why no one tried Iteration or feedback before. Valentine explains that there wasn't enough time or pencils to do the equations. Without an electronic calculator, it would have been impossible for Thomasina to continue the equations indefinitely.
One of the central conflicts in Arcadia is the struggle between intellect and emotion. Although by the Chater Proposal, all seems charged by the random and unexpected movement of bodies in heat, there is an academic and intellectual thrust and persistence within the work to solve mathematical and historical mysteries. As in the case of Cleopatra, aptly pointed out by Thomasina, the random motion of bodies (or love) is rival to the pursuit of intellectual knowledge and progress. It would seem that intellectual knowledge and sexual or romantic knowledge cannot coexist; however at the conclusion of the play the audience is left with a shaky compromise.
This compromise is forged carefully. The champion of intellectual knowledge is Hannah Jarvis, who, as the scholar of modern day Sidley Park, wrote her first novel about injustice inflicted on Lady Caroline Lamb by Lord Byron through the couple's relationship. Hannah's purpose at Sidley Park is to write a book that essentially criticizes romanticism and heralds classicism. Hannah believes that the life of the hermit of Sidley Park, the "genius of the place" but also "an idiot in the landscape" is an apt metaphor for the downfall of romanticism. Hannah refuses to believe that Lord Byron would kill Ezra Chater because of "gut instinct." Hannah may be right about Lord Byron and Ezra Chater, but what she does not know is that Septimus Hodge, the hermit of Sidley Park, fell into the hermitage for an arguably romantic reason. Septimus escaped into the hermitage out of lost love, destined and determined to finally prove Thomasina's theory, not because of false knowledge or wasted work. Septimus, like Thomasina, is unable to solve the proof because he lacks the proper technology and, even in a life's work, cannot finish the problem. Hannah rejects sexual knowledge in all forms; she won't be photographed or submit to a kiss, she refuses Valentine, and brushes off Gus's flirtation. It seems that Hannah, at one time, knew love, "Chaps sometimes wanted to marry me," but she has turned away, "I don't know a worse bargain. Available sex against not being allowed to fart in bed."
Mrs. Chater, the subject of the Chater principle, has the greatest amount of carnal knowledge. Although she remains unseen, the "Chater" as she's called, seduces Septimus, Lord Byron, Captain Brice, and her husband. She is the infamous woman who demonstrates the random and unexpected nature of carnal knowledge and "bodies in heat." The Chater has, as expected, no known academic ties or pursuits mentioned in the play.
While the play progresses toward Thomasina's fiery end in one story and toward the solution of her diagram in the other, there is an unexpected urgency toward sexual knowledge. As Valentine explains Thomasina's discovery to Hanna, Thomasina begs Septimus for a waltz and kiss before her seventeenth birthday. The knowledge that both stories come to—the irreversible end of life—leads the characters to find what is finally the most important and vital sort of knowledge. As Septimus suggests, the "improved Newtonian Universe" will die and grow cold, as Valentine interjects, "the heat goes into the mix." Thomasina asks once again for a dance. The knowledge each has found is cold and does not lead to heat, which is the final essential ingredient to life. The compromise is the realization, the knowledge of what knowledge will bring and cannot bring—it is the fear of knowing without heat or knowledge of another.