The Collaborative International Dictionary of English v.0.48:

Mesmerism \Mes"mer*ism\, n. [From Mesmer, who first brought it
   into notice at Vienna, about 1775: cf. F. mesm['e]risme.]
   An earlier name for hypnosis or hypnotism, the art of
   inducing an extraordinary or abnormal state of the nervous
   system, in which the actor claims to control the actions, and
   communicate directly with the mind, of the recipient. It is
   believed to be a state between sleep and wakefulness, in
   which a person is more susceptible to suggestion than when
   awake. See Animal magnetism, under Magnetism.
   [1913 Webster]

WordNet (r) 3.0 (2006):

mesmerism
    n 1: the act of inducing hypnosis [syn: hypnotism,
         mesmerism, suggestion]

The Devil's Dictionary (1881-1906):

MESMERISM, n.  Hypnotism before it wore good clothes, kept a carriage
and asked Incredulity to dinner.