310 10. EUCLIDEAN GEOMETRY
drawn to the other reference line. The commentary on this problem by Pappus, who
mentioned that Apollonius had left a great deal unfinished in this area, inspired
Fermat and Descartes to take up the implied challenge and solve the problem com-
pletely. Descartes offered his success in solving the locus problem to any number
of lines as proof of the value of his geometric methods.
Questions and problems
10.1. Show how it would be possible to compute the distance from the center of
a square pyramid to the tip of its shadow without entering the pyramid, after first
driving a stake into the ground at the point where the shadow tip was located at
the moment when vertical poles cast shadows equal to their length.
10.2. Describe a mechanical device to draw the quadratrix of Hippias. You need
a smaller circle of radius 2/ð times the radius that is rotating, so that you can
use it to wind up a string attached to the moving line; or conversely, you need the
rotating radius to be ð/2 times the radius of the circle pulling the line. How could
you get such a pair of circles?
10.3. Prove that the problem of constructing a rectangle of prescribed area on part
of a given base a in such a way that the defect is a square is precisely the problem of
finding two numbers given their sum and product (the two numbers are the lengths
of the sides of the rectangle). Similarly, prove that the problem of application with
square excess is precisely the problem of finding two numbers (lengths) given their
difference and product.
10.4. Show that the problem of application with square excess has a solution for
any given area and any given base. What restrictions are needed on the area and
base in order for the problem of application with square defect to have a solution?
10.5. Use an argument similar to the argument in Chapter 8 showing that the side
and diagonal of a pentagon are incommensurable to show that the side and diagonal
of a square are incommensurable. That is, show that the Euclidean algorithm, when
applied to the diagonal and side of a square, requires only two steps to produce the
side and diagonal of a smaller square, and hence can never produce an equal pair.
To do so, refer to Fig. 24.
In this figure AB = BC, angle ABC is a right angle, AD is the bisector of angle
CAB, and DE is drawn perpendicular to AC. Prove that BD = DE, DE = EC,
and AB = AE. Then show that the Euclidean algorithm starting with the pair
{AC, AB) leads first to the pair {AB, EC) = {BC, BD), and then to the pair
{CD, BD) = {CD, DE), and these last two are the diagonal and side of a square.
10.6. It was stated above that Thales might have used the Pythagorean theorem in
order to calculate the distance from the center of the Great Pyramid to the tip of its
shadow. How could this distance be computed without the Pythagorean theorem?
10.7. State the paradoxes of Zeno in your own words and tell how you would have
advised the Pythagoreans to modify their system in order to avoid these paradoxes.
10.8. Do we share any of the Pythagorean mysticism about geometric shapes that
Proclus mentioned? Think of the way in which we refer to an honorable person
as upright, or speak of getting a square deal, while a person who cheats is said
to be crooked. Are there other geometric images in our speech that have ethical
connotations?
