Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: I am travelling again this week. I tell you this not to make excuses for my not responding to reasonable questions, but to give you some insight into what it is that a research mathematician does. On Tuesday, I will be at the East Texas Mathematics Teacher's Circle run by Prof. Jane H. Long who is also an Associate Director for the National Association of Math Circles. Then I travel to Oklahoma State University to work with Dr. Michael DiPasquale on a project involving Semi-Algebraic Splines. There, I run an activity at a local High School on "Dissecting Donuts", talk to their math club on "Balls and Boxes: Common shapes in uncommon dimensions". I also give two research talks in the mathematics department at seminars there. I almost forgot, I give a research talk in a seminar at Texas A&M on Monday. While this is busy, it is less so than last week, where I never got back to my hotel room before 10 PM, after leaving it at 7 AM.
      Do think about your second paper.
      Our readings this week concentrate on the Calculus, continuing from last week. Chapter 10 in Stillwell's book covers many of the topics that are taught in the Calculus, as well as quite a lot about formulas involving infinite processes, including infinite sums and infinite products. There is also an historical perspective on the Calculus, which you see starts before Newton, and continues until the 18th century, which we will eventually get to. A less scholarly, but more entertaining perspective on some figures from this period is provided by our friend Bell. For those of us who teach or wll teach calculus, this additional material should be helpful in giving us, if not our students, some perspective on the material and the times of its development.
• Reading:
  • Chapter 10 in Stillwell's book.
  • Chapters six and seven in Bells's "Men of Mathematics" on Newton and Leibnitz. Some of this is entertaining.
  • J. V. Grabiner, "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine 56 (1983) 195-206.
    I met Judy Grabiner in graduate school (her husband was a mathematical friend of my then Ph.D. advisor, Dr. Graham Allan at Cambridge), and later her son was a postdoc with me at the MSRI in 1997. I have found her articles on history of mathematics to be interesting reading, and I hope that you will like this one from 35 years ago.
  • Watch the entertaining Numberphile video on 1+2+3+.... I will walk you through a correct derivation of this later.
  • Look up some sources about logarithms. What were they used for? Who invented them?
    Up until the 1970s logarithms were commonly used as a shortcut to multiplication, because the logarithm turns multiplication (hard) into addition (easy). (For those who have had modern algebra, this is because the logarithm function is a group isomorphism from the multiplicative group of the positive real numbers to the additive group of the real numbers.)
    When I was a student in high school, they had just stopped teaching logarithms as a tool for calculation. Perhaps more interesting is that our perceptions, specifically hearing and sight, are on a logarithmic scale. You might want to think about that as you look up material and do the homework problem on the distribution of data.
• Assignment: Due Monday, March 12. (HW 9)
  I will have a shorter assignment the next two weeks, and will discuss with Mehrzad about being relaxed about due dates; several people are travelling in March, (Spring Breaks) and many have an important holiday at the end of the month.
  To hand in: Email a .pdf to Mehrzad Monzavi Mehrzad@math.tamu.edu.
  1. Exercise 10.5.1 on page 192. This shows a limitation of power series, which is clear if you look at the graph of the function y=(x)^1/2.
  2. Everybody loves the Fibonacci sequences (from a problem of Fibonacci about rabbits; it is interesting to look up the source.) Do the three problems in Section 10.6 in Stillwell about this interesting sequence.
  3. Do Exercise 10.7.3. For this, use the formula for ζ(1-s) given at the bottom on page 196. You have to understand that Γ(n) = (n-1)!, when n is an integer, such as 2, as well as the remarkable formula for ζ(2), as discussed at the start of Section 10.4.
     Write a couple of sentences about at least one thing that is completely wrong with the Numberphile video. (For the critical thinkers, this is easy.)
     This should not be hard mathematically, but may cause some conceptual reorientation.
  4. I have adapted a project from Prof. Fulling about Benford's law, and the logarithmic scale. Please do the exercise on that page.