TITLE: "Epicycles" (Ptolemy style) in math theory?
QUESTION [17 upvotes]: By analogy:
The epicycles of Ptolemy explained the known facts in the sun system and in this sense were not "wrong". But they distracted from a better insight. From another viewpoint, everything fell neatly into place.
Do you have examples from pure math? Again, you have a theory, it's OK in the sense that it gives the answers you want, but it's hopelessly contrieved, and suddenly someone sees the stuff from another viewpoint and everything becomes ridiculously simple? 
It's quite probable a comparable question with other wording has already been asked, a link then would suffice. :-)
(QUICK EDIT as long as it's hot: While I already upped an answer with nice examples, I also would be interested which "current" theories you see as epicyclic and in need of a Kopernikus/Kepler/Newton. Remember, soft question, a lot of esthetics is involved in the judgment.)

REPLY [2 votes]: Hamilton's quaternions made physical calculations a good deal easier, but vector calculus and vector algebra have proven to be far superior tools in mathematical physics.