Septimus Hodge and Thomasina Coverly sit in the front room of an old estate in Derbyshire, England. The house is surrounded by beautiful, traditional park-like landscape, which is lush and green. Thomasina, a curious and rather impetuous girl of thirteen, is the student of Septimus, who is twenty-two. Each is working on separate problems when Thomasina asks Septimus what "carnal embrace" might be. Thomasina overheard Jellaby, a servant at the estate, telling the cook that Mrs. Chater, wife of the poet Ezra Chater, had been found in carnal embrace in the gazebo. Jellaby had heard the story from Mr. Noakes, gardener of the estate, who had actually witnessed the event. Septimus tells Thomasina that the act of "carnal embrace" is throwing ones arms around a side of beef.
Thomasina, quite perceptive, tells Septimus that a gazebo is not a "meat larder" and asks if carnal embrace is kissing. Thomasina demands that Septimus tells her the truth, and so Septimus gives her the true scientific meaning: the insertion of the male genital into the female. Uncomfortable with this disclosure, Septimus quickly returns to work. Thomasina pesters Septimus to tell her more about sexual intercourse. 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 into swirls in one direction, the jam will not come together again if they swirl the pudding in the opposite 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.
Mr. Chater suddenly swings the door to the room open. Septimus bids Thomasina to leave the room. Chater accuses Septimus of "insulting" his wife in the gazebo. Septimus tells Chater that he is wrong and that he made love to Mrs. Chater in the gazebo the day before at Mrs. Chater's request. Chater challenges Septimus to a duel, but Septimus declines. Septimus tells Chater that he cannot shoot him because there are only two or three first rank poets living, Chater apparently one of them. Septimus distracts Mr. Chater by complementing him on his new poem, "The Couch of Eros," and tells Chater he will write a good review of the work. Chater, flattered, forgives Septimus for his indiscretion and even offers to sign Septimus's copy of "The Couch of Eros." Septimus only means to distract Chater.
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 sound of hunting fire outside the window precedes Lady Croom's exit. Lady Croom, in the style of a grand general, orders Noakes, Brice, and Chater to follow her. As Mr. Chater leaves, he shakes Septimus's hand in friendship. Thomasina and Septimus are again alone. Thomasina remarks that she has grown up with the sound of hunting guns and that her father's life is recorded in the game book by the game he has shot. Thomasina delivers a secret note to Septimus from Mrs. Chater.
It has been suggested that one of Tom Stoppard's favorite ideas is "all men desire to know." This seems particularly evident in Arcadia, a play obsessed with knowledge of many kinds. The characters in Arcadia seek three different sorts of knowledge: mathematical knowledge, historical knowledge and sexual knowledge.
The play opens with the problem (quite literally) of mathematical knowledge. Septimus has given Thomasina the challenge of finding a proof for Fermat's Last Theorem (more to keep her occupied than in hopes of her solving it). At the time the play was written Fermat's Last Theorem was, indeed, a great mathematical task. Thomasina proposes her own original solution to the theorem: Fermat's marginal note was an eternally tormenting joke to drive posterity mad. It is ironic that in real life, shortly after the play opened, Andrew Wiles announced a proof of Fermat's theorem that has, after subsequent amendments, been accepted as correct. But the quest for mathematical knowledge persists within the play. Thomasina is the genius girl who can miraculously understand the foundations of thermodynamics and chaos theory a century before their formal definition.
Thomasina's algebra lesson is interrupted by her own search for another type of knowledge. Thomasina asks Septimus what "carnal embrace" is. Septimus's characteristically witty reply, that it is the act of throwing one's arms around a side of beef, does not deter Thomasina from her desire to know about sex. Chloe, Thomasina's modern counterpart, has less desire for formal, mathematical, or book knowledge but craves sexual knowledge. For Thomasina, the desire for sexual knowledge is a juvenile curiosity&emdash more a means to marriage and a first waltz. On the other hand, for the modern hormonal Chloe, sex is real sex; Chloe persuades Bernard to go up into the library stacks with her for what may be real sex. Until Thomasina is sixteen, she only desires the waltz and kiss. While Thomasina asks Septimus to come to her room after they waltz in the conclusion of the book, he refuses, and she is content.
Thomasina studies history with disdain and boredom. As she tells Septimus, she is bored with and hates Cleopatra. Thomasina abhors Cleopatra's weakness for men and sex, as she complains Cleopatra makes "noodles of our sex." Thomasina has seemingly distinguished between sex that is exciting and sex that weakens women and destroys knowledge and progress. Thomasina, herself, seeks sexual knowledge and mathematical knowledge but does not sacrifice one for the other. Historical knowledge is also sought after more urgently in the present. In scenes depicting modern-day Sidley Park, historical knowledge is rewarded by great fame and possibly sexual prowess. The modern characters value historical knowledge foremost. Bernard, of course, lusts after historical knowledge most of all, intent on receiving any and all fame it may bring. Hannah, with more reserve, also looks among the books of Sidley Park for a glimpse into the past and writes bestsellers on her findings.
The intertwining past and present of Sidley Park provides commentary on the progression of knowledge or quest for knowledge in modern times. The modern day characters are concerned with the workings and findings of the past, while Thomasina and Septimus work to make new discoveries. The quest of all of the scholars thus forms a sort of loop; what is undervalued in one generation is greatly revered in the next. The state of inquiry revolves and evolves from an interest in the future to that of the past. And, like Septimus's apt description of humanity's quest for knowledge, the modern day continues to pick up what has been lost in the past, while simultaneously finding new ideas and formulas.