Which small area of mathematics had fully developed already and thus no more research in this area? Which small area of mathematics had fully developed already and thus no more research in this area? For example, no more PHD research in Euclidean Geometry anymore.
 A: Some additional examples that may be of interest: 


*

*Spherical trigonometry.

*Quaternion analysis (in the 19th century a rather hot and active area of research).

*Synthethic projective geometry (in the tradition of Steiner and Poncelet).

*Analytical theory of generalized continued fractions.

*Geometry of straight-line linkages (popular from the 1870s to the 1890s); mathematical kinematics in general.


Personally, I'd say quaternion analysis is the most spectacular example of an area that was once very fashionable and now all but forgotten.
All these things are relative, of course. It would be very hard, I think, to find mathematical disciplines in which the amount of active research is exactly zero.  
A: Here are some "fully developed areas" (and not having PhDs awarded in), which once upon a time were active research areas:


*

*Squaring the circle and trisecting the angle using compass and ruler. 

*Proving the 5th postulate from the rest of Euclidean axioms. 

*Proving Fermat's last theorem using elementary algebra/number theory. 

*More recent: constructing homotopy 3-spheres which are not homeomorphic to the standard 3-sphere. 
(Of course, areas described in items 1, 2 and 3 are very still active among math cranks.)   
I can go on with examples, but I think, you got my point by now. 
A: No one area in mathematics can be considered as fully developed. This is because some day, some one will come up with some brilliant ideas that will
extend our research to a new dimension; or even disprove what we have accepted as must-be-true.
