The action takes place in a single room of a country manor(Stoppard plays with the Unities, but uses them heavily). The odd-numbered scenes take place in 1809, the even-numbered in the present. The characters of the present day are trying to determine from records what transpired in the earlier era. It's a common literary device, usually for comic effect, to watch people draw wrong conclusions about events the audience is in on. We know what actually did happen; we laugh at the bumblers misinterpreting the signs.
This theme of working backward from the data is supported by even the sound effects: gunshots in the distance (who or what has been shot?), piano for four hands offstage (who is playing, and why does it grow more intense, then break off?), and a steam engine (for what purpose?). Even the recursive mathematics uses it.
Asimov used the device at least twice. It is central to A Canticle For Liebowitz. Carol Kendall uses it in The Gammage Cup, and a host of detective parodies get mileage out of sleuths being too clever by half in their interpretations. Heck, I even used it myself in a novel I wrote years ago (unpublished, don't try to google for it).
Stoppard had earlier used it to brilliant effect in "After Magritte," my all-time favorite one-act.
Thelma: For some reason my mind keeps returning to that one-legged footballer we passed in the car... What position do you suppose he plays? I mean, what fantastic pluck! What never-say-die spirit, you know what I mean? Bloody unfair on the rest of the team, mind you - you'd think the decent thing to do would have been to hang up his boot.I digress. Any excuse to quote from that play. Back to "Arcadia."
Harris: It wasn't a football, it was a turtle.
Thelma: A turtle?
Harris: Or a large tortoise.
Harris: He was carrying a tortoise.
Thelma: You must be blind.
Harris: It was he who was blind.
Thelma: And how do you explain the West Bromwich Albion football shirt?
Harris: Pyjamas. He was wearing pyjamas.
Thelma: Pyjamas...I suppose he was hopping in his sleep. Yes, I can see it now - a bad dream - he leaps to his foot, grabs his tortoise and feels his way into the street -
Either one of the plays would be interesting alone. Woven together, they take on a universality that highlights one of his main themes.
The comic word-play and sight gags are there as in Stoppard's earlier work - the same tortoise is present in both eras - but he is after bigger and more serious game. The teenage girl in 1809 is gradually intuiting mathematics that will not be discovered for a hundred and fifty years,* setting down diagrams and equations in her copy book, which the later characters have access to. Stoppard gets the math right, by the way, as he got the physics correct in "Jumpers." That in itself is unusual in a modern playwright. The maths are a springboard to both jokes and arguments about determinism, randomness, and inevitability. Stoppard, in his usual risque way, concludes that sex is what undermines determinism, with clever references to the thermodynamics of bodies in heat.
The audience is ahead of the present-day characters in understanding the connections to the past, but even we do not have the full weight descend upon us until the very end. We also solve as we go.
I am always certain that I am missing much of what Stoppard sneaks in as jokes or illustrations. I picked up the subtle reference of the dahlias in print, but they are not mentioned in the dialogue, and not a person in a thousand would both recognize the flower and the mathematical connection. I picked it up only by chance, even with the hint. How many more have I missed?
* Iterated algorithms, for those who care. You don't have to previously understand these to follow the plot.