The study of mathematics is apt to commence
in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour
of its methods, all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid
the stars are weighed and the billions of molecules in drop of water are counted. yet, like the ghost of Hamlet's father,
this great science eludes the efforts of our mental weapons to grasp it - " 'Tis here, 'tis there, ''tis gone "-and what we
do see does not suggest the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our gross methods.
" A show of violence," if ever excusable, may surely be " offered " to the trivial results which occupy the pages of some
elementary mathematical treatises.
The reason for this failure of the science
to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical
procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate
learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception.
Without a doubt, technical facility is a first requisite for valuable mental activity: we shall fail to appreciate the rhythm
of Milton, or the passion of Shelley, so long as we find it necessary to spell the words and are not quite certain of the
forms of the individual letters. In this sense there is no royal road to learning. But it is equally an error to confine attention
to technical processes, excluding consideration of general ideas. Here lies the road to pedantry.