Septimus Hodge and Thomasina Coverly sit in the front room of an old estate in Derbyshire, England. The house is surrounded by beautiful, traditional, and park-like landscape, lush and green. Thomasina, a curious and rather impetuous girl of thirteen, is the student of Septimus, who is twenty- two. Thomasina asks Septimus what a "carnal embrace" might be. Jellaby, the butler, interrupts the conversation. Jellaby brings a letter to Septimus from Mr. Chater. Septimus reads the letter and tells Jellaby to tell Mr. Chater that he will have to wait until the lesson is finished.
After Jellaby leaves, Thomasina asks Septimus if he thinks it is odd that when one stirs jam in his or her rice pudding in one direction, the jam will not come together again if they swirl the pudding in the other direction. In other words, she asks why one cannot stir things apart. Thomasina's question leads to a discussion about Newton's law of motion. Thomasina believes that if one could stop every atom in motion, a person could write a formula for the future. Noakes enters the room, soon followed by Lady Croom, mistress of the estate, and Captain Edward Brice. Lady Croom is very upset by Noakes's plans for the landscaping of Sidley Park. Lady Croom thinks that Noakes's plans are too modern, Sidley park is beautiful and an "Arcadia" as it is.
The scene has changed to the present day, apparent from the clothing of the characters on stage. The action of Arcadia shifts from the early nineteenth century to the present day. The setting is still Sidley Park, but there have been changes in the surrounding landscape with time. The modern day characters, Hannah, Chloe, and Bernard, sit in the same room as Thomasina and Septimus. Bernard Nightingale, critic, comes to meet Hannah at the estate. Bernard is looking for information on Ezra Chater. Hannah is looking for information on the Sidley Hermit, whose death she attributes to the breakdown of the Romantic Imagination. Bernard tells Hannah he wants to collaborate with her on a project. Apparently, Bernard's copy of Ezra Chater's The Couch of Eros belonged to Lord Byron and inside the book there are three documents that have led Bernard to believe Lord Byron killed Chater in a duel. Bernard believes that Lord Byron slept with Chater's wife, which led Chater to challenge Lord Byron to a duel die by his hand. Because Lord Byron left the US in 1809, soon after Chater published his last known work, Bernard assumes he was fleeing.
The play shifts back to the early nineteenth century. It is morning and Thomasina and Septimus sit together in the schoolroom. Thomasina tells Septimus that his equations are only for commonplace manufactured forms. Thomasina wants to create the kind of equations that make nature, such as an equation to make a flower rather than a circle, cone, or square. Captain Brice enters the room, followed by Mr. Chater who stands behind Brice. Chater is still angry with Septimus because he slept with his wife.
Scene four switches, once again, to the present time. 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. In Scene five, Bernard practices his lecture for Valentine, Chloe, and Gus. Bernard is practicing the speech he will give to introduce his new, groundbreaking theory that Lord Byron killed Ezra Chater over a woman. As Bernard begins his dramatic oration, Hannah excitedly enters the room to talk to Valentine. Hannah is still not convinced that Byron killed Chater or even wrote the letters found inCouch of Eros to Chater.
Scene seven switches constantly between time periods with no obvious divisions. The scene begins with a discussion between Chloe and Valentine. Valentine tells Chloe that the universe is deterministic; one might be able to predict everything to come if he had a computer large enough. Chloe interjects that the formula wouldn't work because of sex—people might fancy people who weren't part of the plan or proper formula. Thomasina is now sixteen. Septimus gives Thomasina an essay from the Scientific Academy in Paris that is like Thomasina's own work—the scientist has found a contradiction in Newton's theory of determinism. Thomasina tells Septimus that she was right; the problem with determinism is likely hidden in the author's observations about the action of bodies in heat. Thomasina understands the second law of thermodynamics (which states that heat is irreversible).
Bernard enters the room, followed by Hannah, who is carrying a garden book. Inside the garden book there contains an entry from October 1st that proves that Chater the poet was the same Chater that was killed by a monkey bite in Martinique in 1810; thus, Bernard's theory is destroyed—Byron did not kill Ezra Chater. Septimus enters with an oil lamp and carries Thomasina's primer. Thomasina enters secretively, barefoot in her nightgown, and holding a candlestick. It is the night before Thomasina's seventeenth birthday, and she wants Septimus to teach her how to waltz. Valentine stumbles in and tells Hannah that Thomasina's diagram is of heat exchange. Septimus and Thomasina discuss Thomasina's diagrams, too, while she tries to get him to dance. The music heard from inside the house changes to a waltz, and Thomasina and Septimus begin to dance. Septimus kisses Thomasina on the mouth, and the couple begins to dance again. Septimus sends Thomasina up to bed with a lit candle. Gus enters and gives Hannah a picture Thomasina drew of Septimus, and Plautus that proves Septimus is the hermit. Gus and Hannah begin to dance.