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Thomasina asks her tutor, Septimus, about a mysterious bit of gossip she has overheard: that Mrs. Chater was discovered in carnal embrace in the gazebo.
Thomasina does not believe Septimus's initial explanation that carnal embrace is throwing one's arms around a side of beef. Thomasina gets him to give her the proper definition, which disgusts her.
Thomasina remarks on her recent scientific observations to Septimus, but he is not impressed.
Thomasina shares another new idea: that if you knew where all the atoms in the universe are, where they were going at a single point in time, and were good enough at math, you could write a formula for the future.
Septimus tells her about Fermat's note, written in the margin of one of his books, about the great proof Fermat discovered but doesn't have room to describe. Thomasina says she knows what Fermat meant: he was making a joke to drive people crazy looking for a nonexistent proof.
After a bunch of business with Chater and Lady Croom, Septimus is talking about carnal embrace and everyone else is talking about landscape. Thomasina manages to put the conversation back on the road to making sense.
Thomasina draws a picture of a hermit to go with the hermitage in Mr. Noakes's landscape book.
Thomasina gives Septimus a letter from Mrs. Chater.
The next day, Thomasina painfully works on translating a Latin assignment that Septimus has given her.
Thomasina tells Septimus that she saw her mother with Lord Byron in the gazebo. She is convinced that Lady Croom is in love with Lord Byron.
Thomasina also tells Septimus that at breakfast, Lord Byron let slip to Mr. Chater that Septimus was the author of a scathing review of Chater's poem, "The Maid of Turkey."
Thomasina protests against both the less-than-perfect grade she receives from Septimus on her math homework and his limited conception of the possibilities of geometry.
Thomasina decides to plot the graph of an apple leaf to try to figure out its equation from the shape.
Septimus tells Thomasina to get back to her Latin translation about Cleopatra. Thomasina protests Cleopatra's lovesick stupidity.
Thomasina turns furious when Septimus fluently translates her assignment, showing that he already knew the English version (because it's from Shakespeare!).
Three years later, Thomasina chases her brother Augustus, who is threatening to reveal a secret she told him.
Thomasina starts her drawing lesson with Septimus.
Thomasina is disappointed that Septimus gave her a zero on her homework, but once she explains what she was trying to do he takes it back to look it over further.
Thomasina tells Septimus of her crush on Lord Byron, and Septimus tries to squash her hopes.
Thomasina explains to Septimus what she told Augustus: that Septimus kissed her to seal his promise to teach her how to waltz.
Septimus tells Thomasina about a French book he's reading that discusses some of the same scientific ideas that she's been talking about, and she borrows the book to read.
Thomasina draws a diagram to show that Mr. Noakes's new steam engine will never put out as much energy as it takes in.
Thomasina also draws a portrait of Septimus and his tortoise Plautus.
That evening, Thomasina sneaks into the schoolroom where Septimus is working and kisses him.
She explains: it's the second kiss she promised for when he has taught her how to waltz, paid in advance.
Thomasina waits for the music to be right for waltzing while Septimus reads her homework.
Thomasina talks with Septimus about her work and the end of the world it predicts, and then they waltz.
Septimus and Thomasina kiss.
Septimus lights Thomasina's candle for her, warning her to be careful with the flame, and she invites him to come to her room.