Eudoxus of Cnidos

Exercises

1. Suppose x is a number whose cube is equal to its negative:

Show that x = 0 by employing a double reductio ad absurdum proof.  That is, show that x can't be positive, or negative.
2. An arbelos (Greek for "cobbler's knife") is a figure bounded by circular arcs, like a lune.  Given a semicircle with diameter AB, choose a point D on the diameter and draw semicircles on diameters AD and DB.  The arbelos is the region inside the original semicircle that is outside the smaller semicircles:

Use Eudoxus' result on the areas of circles to show that the arbelos has the same area as the circle drawn on diameter CD where C is the point where the perpendicular to AB at D strikes the large circle.