HM/MA 004, Assignment 5, due Friday, 11 October 2002, 10:00 AM
(Note on collaboration: If you do the homework in pairs
or groups, please list the name(s) of your collaborator(s)
on what you each turn in.)
- The new ways of talking about curves. (10 points)
The symptom of an ellipse with major axis AB can be verbally
described in the style of classical mathematics as follows: ``For any point
C on
the ellipse, the square on the perpendicular dropped from C to the point
D on AB has a
constant ratio to the rectangle on the distances AD and
BD.'' Rewrite this description as an algebraic equation for the ellipse,
in the
"coordinate geometry" notation of either Fermat or Descartes.
- The new ways of solving problems about curves. (12 points)
Use Fermat's method of "adequality" to find the tangent to
this ellipse at C. (Extend the axis AB out
beyond A, and draw the tangent to
C intersecting the extended axis at E: we then want to find
the location of E,
that is, the distance DE. Label some point O
on CE near point C,
and drop the perpendicular from O to the point I
on AB. If we say that DA = d,
BD = b, DE = a, and DI = e,
then use your equation for the ellipse and
Fermat's adequality technique to find an expression for
a in terms of b and
d. (Hint: use the relation DC/OI = DE/IE from the
similar right triangles!)
- Are the new ways better? (10 points)
Write a paragraph explaining why such methods of
solution were considered more "general" than classical geometric ones.
- Paper outline. (25 points)
Write an approximately one-page (or one and a half if you use big double
spacing!) outline of your midterm project due next week. Keep in mind
the stated goal of the project: to go beyond our readings in exploring
one or more of the mathematical developments we've discussed. Be frank
about the remaining holes in your current progress: I would rather see
a detailed, specific outline with a few explicit admissions of "I don't
know what I think about this point yet" or "I haven't yet found the
information I need to answer this question" than a carefully smooth
and vague general overview that doesn't risk going into detail. List
at least three published sources you are using in your research.
(Actual publication acts as a quality filter that isn't yet
available for information on Web sites. You can of course reference
an electronic transcription of a previously published text, but I
want the publication citation too.)