When I received the invitation to open this discussion my first feeling was of diffidence, for, the interior of the earth being necessarily inaccessible to direct observation, the solution of the problems connected with it has principally been left to mathematical research, and this must remain the final court of appeal. In these circumstances it seemed verging on presumptuousness to address an audience consisting so largely of mathematicians in inauguration of a discussion on the interior of the earth. Second thoughts showed that there was much to he said against this view, for, though mathematics is the court of appeal, it can only decide on the facts placed before it by the sciences of observation, and so the discussion seems profitably prefaced by a statement of the leading facts which have been collected, and those conclusions which are so directly derived from them as to have almost the validity of observation.