Altophobia (or Acrophobia) is the technical term for the fear of heights. An altophobic person may panic or get vertigo if they look down from a high point. In extreme cases, it could render them unable to use most furniture.
Ironically, an alto (or contralto to non-USians) is the lowest singing voice a normal female can have.
Sources:
-http://en.wikipedia.org/wiki/Acrophobia