counting-board

 Respond to *one* of the following topics. In all cases, your responses should be clearly organized and should make specific reference (with citation) to the primary texts. You should address along the way each of the questions contained in your chosen topic, but your essay should not just be a bullet-point response, but rather a reflection on the topic that synthesizes your thoughts on the various individual questions. You may want to (and in some cases will have to) read beyond our assigned passages. Do not hesitate to take a stance, nor to raise a question that you find puzzling and might want to think about more. 600 words is an appropriate length. (1) Familiarize yourself with some of the basic methods of the counting-board (you might wish to play with toothpicks or some such), and then study the counting board implementation of the root extraction methods in The Nine Chapters. (You will have to look up the Shen-Crossley-Lun book in the library: please just scan or photocopy pages you wish to read, so that the book remains available for other students.) Come up with your own examples of quadratic equations, and explain their solutions on the counting-board—your explanation should include diagrams representing the procedure, which you may draw by hand. Also give in parallel a symbolic “algebraic” explanation of the procedure. Compare these two approaches at a practical and/or conceptual level. (2) In Quadrature of Parabolas, Archimedes begins with yet another proof (that we did not discuss), of the quadrature of parabolas, culminating in Proposition 17 of the book. Like the argument in The Method, this proof uses physical principles like the law of the lever. Study this argument, and compare and contrast it with the argument in The Method. Discuss carefully how each argument works with—or works around—concepts of the infinite that arise when studying non-rectilinear areas. (3) Read through the definitions and postulates in Sphere and Cylinder I. Carefully go over all of the ingredients in Archimedes’ proof of the calculation of the surface area of a sphere, and illustrate the use of the various postulates. Some of these postulates are particularly interesting, because they are from a modern perspective interesting mathematical results capable of proof via analytic methods. Read the section of the commentary of Eutocius (also translated in Netz’s edition of Sphere and Cylinder I, to which we have unlimited ebook access via the university library) on Archimedes’ definition and postulates in Sphere and Cylinder I: how does Eutocius understand the role of the postulates? In your view, how does Archimedes understand the role of the postulates? (4) Come up with your own topic. Whatever it is, you should state it clearly, and your essay should involve detailed reading of at least one primary text.