Arcadia is set in Sidley Park, an English country house in Derbyshire, and takes place in both 1809/1812 and the present day (1993 in the original production). The activities of two modern scholars and the house's current residents are juxtaposed with those of the people who lived there in the earlier period.

In 1809, Thomasina Coverly, the daughter of the house, is a precocious teenager with ideas about mathematics, nature and physics well ahead of her time. She studies with her tutor Septimus Hodge, a friend of Lord Byron (an unseen guest in the house). In the present, writer Hannah Jarvis and literature professor Bernard Nightingale converge on the house: she is investigating a hermit who once lived on the grounds; he is researching a mysterious chapter in the life of Byron. As their studies unfold – with the help of Valentine Coverly, a post-graduate student in mathematical biology – the truth about what happened in Thomasina's time is gradually revealed.

The play's set features a large table, used by the characters in both past and present. Props are not removed when the play switches time period; books, coffee mugs, quill pens, portfolios, and laptop computers appear together, blurring past and present. An ancient but still living tortoise also appears in every scene, symbolising long-suffering endurance and the continuity of existence.

Scene 1 (Act 1)

The play opens on 10 April 1809, in a garden-front room of the house. Septimus Hodge is trying to distract 13-year-old Thomasina from her curiosity about "carnal embrace" by challenging her to prove Fermat's Last Theorem; he also wants to focus on reading the poem "The Couch of Eros" by Ezra Chater, a guest at the house. Thomasina starts asking why jam mixed in rice pudding can never be unstirred, which leads her on to the topic of determinism and to a beginning theory about chaotic shapes in nature. This is interrupted by Chater himself, who is angry that his wife was caught in the aforementioned "carnal embrace" with Septimus; he has come to demand a duel. Septimus tries to defuse the situation by heaping praise on "The Couch of Eros". The tactic works, because Chater does not know it was Septimus who savaged an earlier work of his, "The Maid of Turkey". Then landscape architect Richard Noakes enters, shortly accompanied by Captain Brice and Lady Croom; the three discuss proposed modifications to the gardens, while Thomasina sketches an imaginary hermit on Noakes's technical drawing of the garden.

Scene 2

The setting shifts to the present day. Hannah Jarvis is researching the house, the garden, and specifically the hermit, for a study of hermits and the Romantic imagination. Bernard Nightingale enters with Chloe Coverly; she conceals his identity from Hannah, as he had given Hannah's last book a poor review. Chloe's brother, Valentine, is gathering data on the population biology of the grouse in the surrounding grounds, using the house's "game books". When Chloe accidentally reveals Bernard's identity, Hannah reacts angrily; but she agrees to share her research material, which enables him to propose the theory that one of the 1809 inhabitants, Ezra Chater, was killed in a duel by Lord Byron. Bernard notes that records of Chater the poet disappeared after 1809, and the only other Chater of record is a botanist.

Scene 3

The third scene returns to the earlier time frame; Septimus is again tutoring Thomasina, this time in translating Latin. Again their focus diverts, this time to the destruction of the Alexandrian Library, which upsets Thomasina. She mourns the loss of the knowledge stored there, and Septimus responds that all that was lost will eventually turn up again. They are again interrupted by Chater, who succeeds in challenging Septimus to the duel, having learned (from Lord Byron off-stage), that Septimus wrote the damning review.

Scene 4

Hannah rediscovers Thomasina's primer containing her ideas on iteration and chaos theory; this recalls Septimus' assertion that what was lost is eventually rediscovered. Valentine reacts with interest to the notes, as his own research centres on similar concepts.

Scene 5 (Act 2)

Still in the present, Bernard gives Hannah, Valentine, and Chloe a preview of his lecture theorising that Lord Byron shot and killed Chater in a duel. When Hannah and Valentine challenge his logic, Bernard launches into a diatribe about the irrelevance of science, then departs for his lecture (and a promotional media appearance) in London. Hannah begins to suspect that the hermit of Sidley Park – who was reportedly obsessed with algebraic computations about the heat death of the universe, the theory suggested in Thomasina's diagram – could have been Septimus.

Scene 6

Returning to 1809, we learn that the duel never occurred. Instead, the Chaters left for the West Indies with Captain Brice, Mr. Chater serving as the expedition's botanist and Mrs. Chater as the captain's secret paramour. Lord Byron has also left the country. Septimus has gone rabbit hunting for Thomasina, who favours rabbit pie; he returns to find Lady Croom searching for him. She has found two letters Septimus wrote in case he should die in the duel: one to Thomasina, is about rice pudding, and the other is a love letter addressed to herself. Lady Croom invites Septimus to an amorous rendezvous.

Scene 7

The final scene takes place in both 1812 and the present, the actions running concurrently. Some present-day characters are in fancy dress for a party, so that both casts appear similarly attired. Chloe reads a newspaper report on the Byron murder theory and then talks about determinism with Valentine, echoing the discussion between Septimus and Thomasina. Chloe, however, believes that sex is the force disrupting the universe's ordered plan. Valentine, using his computer to extrapolate Thomasina's ideas, relates them to the concept of entropy; he wonders whether Thomasina or Septimus was the genius behind the theories. Hannah and Valentine mention that Thomasina died in a fire on the eve of her seventeenth birthday.

Meanwhile, Thomasina asks Septimus to teach her to dance, for her forthcoming birthday party. Lady Croom enters, complaining to Noakes about the noise of his steam engine; Thomasina notes that the machine obeys the laws of entropy (which have not yet been formalized), which describe the universe as winding down. In the present, Bernard arrives and is met by Hannah, who has found a letter detailing the facts of Chater's death – this discovery totally discredits his theory and vindicates Lord Byron's reputation. While Septimus awaits appropriate music for Thomasina's dance lesson, he examines the sketch she made to illustrate the irreversibility of heat; his action mirrors that of Hannah and Valentine, who also pondered the same diagram. Bernard is caught in a compromising position with Chloe, and is asked to depart.

Eventually a waltz starts, and Septimus dances with Thomasina, their relationship increasingly complicated by hints of romance. Gus (Valentine and Chloe's younger brother, who has been silent for the entire play) hands another of Thomasina's drawings to a surprised Hannah. It depicts Septimus and the tortoise, confirming her suspicion that the hermit, who had a tortoise called Plautus, was actually Septimus. After Thomasina's tragic death, he apparently became a hermit; accepting her challenge to the laws of the universe as propounded by Newton, he worked for the rest of his life to apply "honest English algebra" to the question of the universe's future.

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