The island was called Ilha Formosa, or Beautiful Island, by the Portugese explorers in the 16th century.
While Taipei is large and crammed with crazy monuments to past dictators, the countryside is definately spectacular and beautiful. My visit was five days in August 1983.

On trip I can heartily recommend, if you don't get car-sick easily, is the trip down the East Coast and then up Toroko Gorge and across the middle of the country.
One sees nice views of miles of lonely beaches with pounding surf. (Sorry, I did not get a good picture).
Then it is a long slow trip up the gorge. The road seems too narrow for two way traffic but vehicles do manage to pass.
When the passenger looks up the route seems impossible for bus travel; looking down to the distant bottom it is hard to believe that one is on the same road. At the top the route passes over the highest Asian road pass east of Tibet.