I read about Mobius developing Barycentric and homogeneous coordinates, and I read about homogeneous coordinates and what they are and I'm totally on board with taking a line from the origin and seeing where it intersects with a given plane (e.g. the Z=1 plane) and that I can use that point to define the line and vice-versa.
I'm told this was really helpful to the mathematicians of 1830 onwards, but I can't find out why. So why was it so useful? Were there specific problems they were working on that suddenly became much easier? What could they do with homogeneous coordinates that they couldn't do before, apart from classifying lines by their point of intersection with a plane?