Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:
The exercise last week of trying to show the fundamental property of logarithms (log(a)+log(b)=log(ab)) using exhaustion is, I believe worthwhile. It suggests that the Greeks could have discovered logarithms and their properties. It also introduces us to the idea that algebraic properties of functions may be determined by transforming integrals, which reappears in this book, from the sine function to the much more exotic elliptic functions that were a focus (and glory of) of 19th century mathematics.

This week, I will go a bit easy on you, as you working on your Book Reviews. We will read about the mathematics between the Greeks and the Renaissance in Europe. Stillwell concentrates on a small slice—the number-theoretic work in Asia (China and India). Next week, we will jump into the mathematics developed in Europe that forms the basis for the mathematics of the past 500 years or so.

Stillwell skips two important developments:

• The origins and rise of the Indo-Arabic number system that we all use today,
• How that number system and the algorithms for multiplication made their way from the Islamic world of the 10th century (give or take a couple of hundred years) to the medieval world.


The readings from St. Andrews fill in some of these gaps, and give a whole lot more insight on other mathematical developments from Asia.
I learned learned a lot about these topics from two books I own, Berggrens' Episodes in the Mathematics of Medieval Islam and Swetz's Capitalism and Arithmetic: The New Math of the 15th Century.
• Reading:
  • Chapter 5 of Stillwell on Number Theory in Asia (China and India).
  • St. Andrews pages on Indian Mathematics. You can skip the two sub-pages on the number π—while fascinating, it is a bit off topic this week. As Math history aficinados, do read them sometime.
    The last link on that page, by Ian G. Pearce, is an extensive treatise on Indian Mathematics, in which the author argues that Indian Mathematics is wrongly maligned. You can skip this—I did, even though I read many hours of other sources from this period when I prepared this week.
  • With the same advice about skipping π, read the St. Andrews pages on Islamic Mathematics.
• Assignment: Due Monday, February 10. (HW 5)
  To hand in: We are using E-campus for homework submission.
  1. Do exercises 5.4.5 and 5.4.6 in Stillwell. It helps to recall the definitions. Write e^x for the usual exponential function from Calculus, where Euler's number e is approximately 2.7182818284590452353602874713527. Then the hyperbolic cosine, cosh(x) of the real number x is cosh(x):=(e^x+e^-x)/2. Similarly, the hyperbolic sine, sinh(x) of the real number x is sinh(x):=(e^x-e^-x)/2. These functions satisfy (cosh(x))²- (sinh(x))² = 1 (check this!), and many other identities similar to the classical trigonometric identities.
     The reason for this is that the hyperbolic functions are almost the same as the trigonometric functions, when viewed properly: cos(θ):=(e^iθ+e^-iθ)/2, and similarly sin(θ):=(e^iθ-e^-iθ)/2/i. Here i is the complex imaginary unit, i² = -1.
  2. I found the treatment of rational triangles and the proof of Herons's formula in the exercises to Section 5.6 in Stillwell to be interesting. Please do all four exercises. (I have a soft spot for Heron's formula, having learned it in middle school, it is the one bit of my mathematical knowledge that I have used in a practical consultation. I can tell the story on Piazza if anyone asks.)
  3. The Islamic mathematician Thabit gave a marvelous generalization of the Pythagorean Theorem. Its statement is problem 2 on Allen's page. Do this problem. By `similarity', he means use similar triangles. I realize that the figure may pose a problem.