Updates to initial announcements

Alternative on Egypt and Mesopotamia

If your textbook has not arrived yet -- and even if it has -- you might want to read Don Allen's material on Egypt and Babylon. I think it has some advantages over Katz's treatment.

Supplement to the lecture

When I wrote "Lecture 1" I forgot that Egypt and Babylon were in Week 2 of my old syllabus. Here is some substantive (mathematical!) material I should have included.

False position, then and now

Some of the solution methods in the original Egyptian and Babylonian sources are called "false position" by modern commentators. Roughly speaking, this means making an initial guess for the answer, then using some ensuing calculations to correct or improve the result. What does "method of false position" mean today? Pages 52-53 of the (highly readable and informative) book by F. S. Acton, Numerical Methods that Work (Harper and Row, 1970) describes a "false position" algorithm for finding zeros of functions, an alternative to "Newton's method" as described in all calculus textbooks. Newton's method gets into the calculus books because it uses calculus, whereas false position requires only algebra and perhaps geometry (in the sense that the theory behind the method becomes clearer if you draw a graph). In many problems, however, the false-position method is better. The point is that Newton is unstable; if the derivative of the function f is zero, or nearly zero, near the root we are seeking (a place where f(x)=0), then the Newton algorithm, which involves dividing by f '(xn) in the process of finding a better approximation, xn+1, may send the sequence of approximations shooting off into some faraway, irrelevant region. Suppose, on the other hand, that you have two approximations, xn-1 and xn. Then you can construct the line through the two corresponding points on the graph of f and look at its intersection with the horizontal axis to determine xn+1. In other words, one is constructing a secant line to the graph, rather than the tangent line of Newton's method. (Acton gives the resulting formula for xn+1, which is easier for you to derive than for me to type in HTML.) If f is continuous and the two points lie on opposite sides of the horizontal axis, then one is guaranteed that a root exists between them and moving to xn+1 will bring one closer to it. The false position algorithm ensures the sign change condition by using xn-2 instead of xn-1 when necessary. Keep this background in mind as you read about the ancient instances of "false position". Are they special cases of the modern concept, or just vaguely analogous?

Pi in the Bible

Dr. Allen's link to the article with this title seems to have broken, but I recall that we had some discussion of it in 2003. The issue concerns a verse in Kings (I think) describing Solomon's temple: A certain vessel was 10 cubits across and 30 cubits around. The question is: Did the Hebrews really think that pi = 3? (Katz notes that the Babylonians sometimes used that value.) My opinion is that probably they were not quite that naive. They could understand what an approximation is (even if the best Babylonian mathematicians did not specify approximations carefully as a modern person would) and still be perfectly happy with one decimal place of accuracy for most purposes. The author of the chronicles was writing a history, not an engineering plan. It was like a modern journalist describing 215 yards as "about as long as two football fields". By the way, I have read that the Egyptians sometimes used pi = (square root of 10), but I don't think Katz mentions that.

Teresi

The laudatory review of Teresi's book (linked to Allen's page) might be supplemented by the more critical (and more substantive) review by Anthony Grafton, American Scientist 91 (March-April 2003) 169-171. His main criticism is the same as Allen's: Teresi exaggerates the extent to which the scientific discoveries of non-Western civilizations have been ignored in the past.

Vista

If you haven't done so already, explore our class page at http://elearning.tamu.edu . There are 3 facilities installed:

How to submit written assignments

Timing and placement

Formats

Communicating mathematical symbolism electronically is a scandalous headache. Different students will have different software and expertise available, so I do not aspire to impose uniformity.

  1. Paper mail and Fax should be avoided unless necessary. Occasionally you may need to send a hand-written diagram or (less likely in this course) a long hand-written calculation.
  2. E-mail attachments will probably be the most common mode, whether you are mailing things to me or installing them in Vista Discussion folders. Various formats are possible. I am very much a TEX partisan, but I understand that many students will need to use something like Microsoft Word.
  3. For various reasons, it may sometimes be better to install your paper on a personal Web page and e-mail the URL to me (and, when appropriate, the class).