TITLE: Extremely messy proofs
QUESTION [142 upvotes]: Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery.  What I am not being taught, largely, is what came before these slick, short proofs.  What did mathematicians do before so-and-so proved such-and-such lemma?  Where, in other words, are the tedious, long proofs that we can look to as examples of the horrible mess we are escaping?  What insights helped mathematicians escape those messes?  
Right now I am particularly interested in examples from measure theory.  What did people do before, for example, Dynkin's lemma or Caratheodory's extension theorem?  Or were these tools available from near the start?
An answer should include both some indication of how tedious and long the old approach was and how much slicker and shorter the modern approach is.  Ideally, it should also discuss how the transition between the two happened.
(If you prefer the old approach to the modern approach, for example for pedagogical reasons, that would also be interesting to hear about.)

REPLY [7 votes]: The Quadratic Equation formula:   Al-Khawarizmi (c 800 AD) did not have negative numbers, nor zeros, and also did not possess the needed algebraic notation. Therefore he had to devote 6 chapters of his treatise on algebra ("Hisab al-jabr w'al-muqabala") to different types of quadratics, and the rules for solving each one of them.
See "Quadratic, cubic and quartic equations", MacTutor History of Mathematics archive.