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:
- Gradebook (empty so far).
- Discussion lists. So far I have created 3, and 3 of you have put
your bios in the appropriate one. Let's try to keep them organized;
that is, please exercise some judgment about
whether to post a message as a new one or a reply to a related old one.
- Chat Room. It's up to you (collectively) to see whether this is
useful. You might use discussion messages or e-mail to arrange with other
students (or me) to log on at the same time.
By the way, local students are welcome to show up at my office hours;
see my home page for times.
How to submit written assignments
Timing and placement
- Contributions for general consumption (such as your little
autobiography) should be posted in the Vista
Discussions right away.
- Weekly homework
and major papers (book review, etc.) should be sent to
me
privately.
Do not post them in Vista before the due date.
On or after the due date they may be posted; should they be?
Probably other students are not interested in your routine
homework papers, but the book reviews will be of general
interest.
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.
- 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.
- 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.
- I've been advised that a good method is to use Word or some
other word processor with an equation editor, and to export the
file in .rtf format (more portable than .doc or whatever the
processor's proprietary format is).
- If you can produce a .pdf file, that is probably the most
widely readable, apart from HTML.
- You can make an HTML file. The mathematical typesetting
available within HTML is primitive but often adequate (note the
subscript on "TeX" above). There are editors that make the
process easier (but I am largely ignorant of them).
- Sometimes (especially in informal e-mail) it is enough to
write plain ASCII text with the mathematics in
pidgin TeX.
- 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).