A different proof of the insolubility of the quintic?
This is the content of the Abel-Ruffini theorem (whose proof predates Galois')
http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
A topological proof may be found in this paper.
https://www.tmna.ncu.pl/static/files/v16n2-02.pdf