HM/MA 004, Assignment 2, due Friday, 27 September 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.)
- (10 points)
- Do Exercise 3.1 on page 108 of
Mathematical Expeditions: namely, give a proof
for Hippocrates' quadrature of a semicircular lune based on the
construction in Figure 3.1, on page 98.
- Compare this
construction with the slightly different one used in the proof
of the same quadrature on page 50 of Fauvel and Gray: if the
semicircular lunes in the two figures are identical, what is the relation
between the right triangles? Prove it.
- (10 points) In Defs. 18--20
of Book XI of Euclid's Elements, a cone is defined as
generated by the hypotenuse of a right triangle being turned
about one of its legs; Euclid and other early geometers considered
a conic section to be produced only by some plane cutting the cone
perpendicular to this hypotenuse.
- Why did they call a parabola a "section of a right-angled
cone"? What was a "section of an acute-angled cone", and a
"section of an obtuse-angled cone", and why?
- If you want to cut any of the three kinds of conic section
from any cone, as Apollonius and other later geometers assumed, how
do you have to hold the cutting plane for each of them?
- (10 points) Do the first bit of Exercise 3.11 on page
117 of Mathematical Expeditions: namely, give an
algebraic proof of Archimedes' result in Proposition 23 of the
Quadrature of the Parabola (page 113) about the sum of an arbitrary
number of areas, each of which is one-quarter of the previous one.
(Start out with some numerical examples if you like, in order to
convince yourself that you actually believe it.)
- (10 points) What do you think of classical Greek
geometry? What are its most important characteristics and goals?
What things about it seem particularly striking to you?