Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:       This would be a good week to begin to think about your Book Review, perhaps even deciding on which book you plan to review.
  A prominent 20th century British mathematician described the Greeks as `fellows of another college'. Their accomplishments and methods remain valid and inspiring to many, even after over 2 millenia. This illustrates a major feature of mathematics, which sets it apart from other human endeavors: Mathematics is additive. While topics may go in or out of style and what is important may change over the generations, the discoveries of the past remain valid and they form the foundation of current work.
      I believe that Stillwell starts with the Greeks because of their modern outlook, they were the first to use the deductive method and require that mathematical truths be proven. They also used abstraction (think Platonic ideals, or potentially infinite Euclidean lines, or...) There is an enormous amount written about Greek accomplishments in mathematics and culture. Stillwell presents a selection of their more elementary accomplishments.
      The discussion of curves in Chapter 2 of Stillwell is informed by advances made nearly two millenia after the Greeks; namely René Déscartes' cartesian coordinates in the first half of the 17th century.
• Reading:
  • Chapters 1 and 2 of Stillwell.
    In your readings of Stillwell, I'd like it if you could include reading the exercises, as well as the text.
        The historical notes at the end of the chapter provide some interesting color about the lives of the protagonists; they are likewise worth reading.
  • While we appear to have a great understanding of Greek mathematicians and their mathematics, this knowledge has not come to us as easily as, say our knowledge of 19th century physicists. To appreciate this, please read the St. Andrews page How do we know about Greek mathematics?.
  • In Plato's dialog Meno type "inquisitive" into your web browser's text search window to get quickly to the scene with the slave boy. Read that part of the dialog.
• Assignment: Due Monday, January 27. (HW 3)   It is OK to discuss this among yourselves.
        Some of these problems will require you to find other material than just what is in the readings.
  To hand in: We are using E-campus for homework submission.
  1. Who was the British mathematician who made the remark at the top of the page? What do you think he meant by this?
  2. Greek mathematics deeply affected at least two US presidents. Research and write about the influence of Euclid and Pythagoras on US Presidents (There is one US president for each of these Greek mathematicians). For each, there is at least one very interesting detail/story. Find it, and describe it.
  3. What were the three geometric problems of antiquity? (Sometimes called the Three Classical Problems). For one, describe it origins (at least the best that you can find out). Were any solved by the Greeks (in any fashion)?
  4. Do Exercise 1.3.4 in Stillwell. Note that t is the vertical coordinate of the point where the longish secant meets the vertical axis. How is this related to the `world's sneakiest substitution' from Calculus?
  5. Do Exercise 1.4.2 of Stillwell. This is one of the easiest proofs of the Pythagorean Theorem, and is often presented in elementary geometry classes.
  6. Do Exercises 2.4.1 and 2.4.2 from Stillwell.
  7. Do Exercises 2.5.1 and 2.5.2 from Stillwell.