"Lecture" for Week 2
Egypt and Mesopotamia
I have little to add to Dr. Allen's material, but here are a few comments.
This pertains to
last week's reading.
I noted a number of misprints and omissions.
- Most obviously, in the example on page 5, n = 5 (not 7) and
q = n - 1 = 4.
- The main formula in Theorem 1 is missing a term "+ 1"
(clearly visible in the Maple code).
- The theorem and the Maple code are both correct, but to make them
precisely consistent one should let i and j in the theorem
from 1 to n instead of from 0 to n - 1, and delete the "+ 1"
from the subscripts on the left of the main formula.
The result of this change
is just to move the first row and column to be the last row and column.
- If you have sharp eyes, you noted that the Chinese magic square
on page 3
and Dürer's square on page 4 have the extra property that each
also sums to the magic number!
The algorithm in Theorem 1 does not have this property;
it is understood that we are not including it in our definition of a magic
square (although some people do).
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
better approximation, xn+1, may send the sequence of
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
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
algorithm ensures the sign change condition by using xn-2
instead of xn-1
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
Pi in the Bible
Dr. Allen's link to the article with this title is broken.
The issue concerns the Bible
verse I Kings 7:23 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?
My opinion is that probably they were not
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, the Egyptians sometimes used pi = (square
root of 10), which is a rather good approximation.
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.
Still more about course procedures and resources
I have added some things to the "announcements" list on our course page.
Most urgent is the instructions for the book review.
Next most important is a style guide from the Mathematical Association
of America for bibliographies in mathematical papers. I don't really
care what system you use for references as long as it is consistent and
provides all the needed and standard information, but the MAA guide
is an excellent default to adopt. Especially valuable is the advice
on how to cite web pages. You've seen the final two items before, but
I put them there for the record.
If you got the PDF file of the first anthropology article last week,
you probably noticed that the article right after it is by the authors
of the second article and can be taken as a rebuttal of the first one.
The reference is
I've added it to the list in the first lecture, for posterity.
P. Pica, C. Lemer, V. Izard, and S. Dehaene,
"Exact and Approximate Arithmetic in an Amazonian Indigene Group,"
Science 306 (15 October 2004) 499-503.
Misprints and all that
I apologize for the condition of the on-line textbook.
There are typos, grammatical errors, etc. that are beyond
my power to correct.
I will list them (as I did above) when they are serious.