After learning knowledge about constructible numbers in the last chapter, we are now ready to tackle the so-called "three classical problems" in ancient Greek geometry.  They are
  • Doubling a cube
  • Trisecting an angle
  • Squaring a circle
These problems were extremely influential in the development of Greek geometry.  It turns out that none of the above can be done by Euclidean constructions.  We will use the "main theorem" that will be introduced in the next page to prove that it is impossible to double a cube or trisect an angle by straightedge and compass only.  As for squaring a circle, a more advanced theorem will be needed.