When a student of mathematics commences the study of a subject which involves the assimilation of what are, to him, fundamentally new ideas, his progress is, as a rule, slow at first. And, even after he has become accustomed to these ideas, he may still require a long course of laborious practice, before he can attain to that mastery of the method which would enable him to use it as a powerful aid to research. Thus students, familiar with geometrical methods, when first commencing the study of Cartesian analysis, require much practice before they can call up mentally the geometrical figure corresponding to a given equation. And, the more general the new method is, the greater is the difficulty felt to be. So, in Hamilton's system of quaternions, the difficulty of assimilation is greater than it is in the Cartesian analysis. And it seems as if it were for this reason that, in recent years, attempts have been made, by men of known mathematical ability, to smooth the paths.