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Back to the present day. On the table: The tortoise (Lightning again, though he looks the same as Plautus), Septimus's portfolio, a math textbook, a paper with a diagram on it, and Thomasina's math notebook (all, we find out, newly rediscovered by Bernard, except for the tortoise).
Hannah is reading aloud to Valentine from a note written by Thomasina in the math textbook, which bears a strong resemblance to Fermat's original note promising a proof of his theorem that Thomasina and Septimus were discussing earlier. Thomasina's promised-but-never-fulfilled proof is for what she calls the New Geometry of Irregular Forms.
The piano in the next room is still being played as Valentine tries to explain to Hannah what Thomasina had been doing.
Time for a math lesson (we'll try to keep it brief): Thomasina has been graphing the same formula on different scales (so if, for example, the first time through the formula she used 1-10 as her starting values, on the next page she'd use 5.0-5.9, and then 5.50-5.59, and so on, zooming in on smaller numbers each time).
Except, Valentine goes on to say, it's not quite that – because instead of just going in order, using first 1, and then 2, and so on, she's using the result she gets each time as the number she starts with her next time through the formula. Valentine calls this an "iterated algorithm," but really it just means that she's doing something over and over, using the ending point each time as the starting point for the next repetition.
Why is this surprising? Because it's the same thing Valentine is doing with his grouse data, and since it's a fairly recent development, seeing it in a nineteenth-century schoolgirl's notebook is like finding E=mc2 in a cave painting.
Hannah keeps pressing Valentine for more information, though he seems kind of cranky.
Valentine does, however, explain a bit further the similarities between his work and Thomasina's.
Valentine is looking at population changes (and it's time for another math lesson): this year you have x number of goldfish, next year you have y number of goldfish, and x turns into y by stuff happening (goldfish getting eaten, having babies, being stolen by aliens, etc.). According to Valentine, that stuff is math – if you look at patterns over a long enough time span, there's a formula that describes how x turns into y.
And finally we find out what's going on with the grouse: Valentine is using the information from Sidley Park's game shooting records to track local grouse populations over long periods of time in order to try to figure out the algorithm – the mathematical rule – that describes changes in their numbers.
Hannah is still trying to figure out what Thomasina was doing from the skeptical Valentine, but does get him to admit that her technique could be used to draw, oh, say, the apple leaf that just happens to be on the table.
Valentine finally cheers up: it turns out what really excites him is math, and specifically the parts of it we still don't know.
Bernard enters: he's made a discovery. In the copy of Byron's English Bards and Scotch Reviewers in the Sidley Park library, there's a handwritten verse describing Chater – though it's not, as Hannah is quick to point out, in Byron's handwriting.
Hannah, however, has a discovery of her own: an 1810 letter from Lady Croom mentioning her husband's marriage to Mrs. Chater – meaning that Mr. Chater was dead by 1810.
Bernard is even more convinced that Byron killed Chater in a duel, but Hannah thinks he's jumping to conclusions – there's no evidence that Byron ever visited Sidley Park.
Valentine steps in and says that yes, actually, there is – Byron is mentioned in one of the game books.
An ecstatic Bernard thunders out on his new quest, while Hannah and Valentine return to their previous conversation about Thomasina's mathematics; Gus comes in to join them (though he still doesn't speak).
Valentine explains that the reason no one picked up on his grouse method before is that the technology wasn't there: what takes a few minutes with a computer would take decades to do by hand.
Hannah, speaking loudly in her excitement, asks Valentine if that's the only problem: having enough time.
Gus runs out of the room, and Valentine says that he hates people shouting.
Valentine finally replies to Hannah: it's not just having enough time that's the issue, it's actually wanting to do it in the first place, because only a madman would take up such a project.
After Valentine leaves, Hannah picks up the copy of the magazine that contains the essay about the crazy Sidley Park hermit she discussed with Bernard earlier.
A change in light suggests dawn, and the sound of a pistol shot rings out. What's going on? Too bad, it's time for intermission!