Is the play a comedy or a tragedy?
There are certainly tragic elements in Arcadia. Thomasina is a precocious genius just starting to discover her theories and fall in love with Septimus, when she dies in a fire on the eve of her seventeenth birthday. Septimus, full of grief, spends the rest of his life as the Sidley hermit, trying to solve her equations and bring meaning to her demise. Furthermore, the overarching reality is that the universe will grow cold and all within will grow cold and die. These are unavoidable truths, but they are not the whole story. There is wit, verve, irony, sweetness, and a sense of optimism that pervades the text. The play ends on a hopeful note, even amid our awareness that Thomasina will die. The tone and the themes preclude this work from being a tragedy, and place it more in the vein of a comedy.
What is the importance of the setting?
While not necessarily uncommon for the setting of a play to remain the same the entire time, Stoppard's choice to have each scene set in the same Sidley Park drawing room is more important than it initially seems. By having the same setting and the same props throughout, it reinforces the themes of the fluidity of time, the similarities of past and present, and the ties between the characters. At one point the modern characters even wear the costumes of those from 1809, reinforcing the notion of slippages in time. Furthermore, the fact that many of the characters - Mrs. Chater, Byron, the Crooms - are not seen onstage, and that the gardens/gazebo/hermitage are left to our imagination rather than a central setting for the action asserts the centrality of ideas, discussion, research, speculation, and imagination rather than concreteness and literalness.
Why does Stoppard end the play as he does?
By the end of the play Bernard's theories have been proven wrong, Thomasina approaches the eve of her death, Hannah has discovered the Sidley Park hermit is Septimus, and a general sense of frustration pervades. However, Stoppard ends with the lovely, fluid waltz of Thomasina and Septimus (and the more awkward but nonetheless charming waltz of Gus and Hannah) to give his audience a sense of hope and optimism even when we all know the universe is growing cold. Love matters, the pursuit of knowledge matters, and taking a moment to dance matters. The waltz may be a dance that is very time-based, but it is also a universal gesture and one that reveals the unity of people and ideas across great divides.
What purpose does the Septimus/Chater/Mrs. Chater/Byron story serve within the context of the play?
The play deals with some very big questions, and to some extent the love triangle/quadrangle we have here seems to be almost irrelevant. However, there are points to be aware of. Mrs. Chater's "tumidity," or heat, is part of the idea that sex is chaos, and the world is full of disorder. The duel forms the basis of Bernard's erroneous theories, which reinforce the need for the right kind of scholarly pursuit and the right way to think about the past. It gives us a way to think about cause and effect, the messiness of relationships, and the way to act when making claims about the motivations, desires, and behavior of those before us.
What role does truth play in the novel?
Characters claim they are in search of truth: Thomasina's quest to solve Fermat's last theorem, to translate Latin, to map a leaf, to understand how the universe works; Bernard's quest to figure out if Byron killed Chater; Hannah's exploration into the identity of the hermit and the reason for his residence; Valentine's study of the grouse population. However, the journey toward truth is a hard one, filled with many pitfalls, false starts, and opportunities to misinterpret and misunderstand. One must be aware that the full truth may not be known, or known yet, or may be lost. Truth is difficult to grasp, subject to multiple interpretations, and can sometimes be disastrous in its unfolding.