  This article surveys many standard results about the braid group with
emphasis on simplifying the usual algebraic proofs.
  We use van der Waerden's trick to illuminate the Artin-Magnus proof of the
classic presentation of the algebraic mapping-class group of a punctured disc.
  We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy,
and, hence, recover the Dehornoy right-ordering of the braid group.
  We then turn to the Birman-Hilden theorem concerning braid-group actions on
free products of cyclic groups, and the consequences derived by Perron-Vannier,
and the connections with the Wada representations. We recall the very simple
Crisp-Paris proof of the Birman-Hilden theorem that uses the Larue-Shpilrain
technique. Studying ends of free groups permits a deeper understanding of the
braid group; this gives us a generalization of the Birman-Hilden theorem.
Studying Jordan curves in the punctured disc permits a still deeper
understanding of the braid group; this gave Larue, in his PhD thesis,
correspondingly deeper results, and, in an appendix, we recall the essence of
Larue's thesis, giving simpler combinatorial proofs.
