objection would I conceive be its extreme speciality. But this is not
an objection that can be brought against Modern Geometry as a whole:
in regard to any particular parts of it which may appear open to such
an objection, the question is whether they are or are not, for their
own sakes, or their bearing upon other parts of the science to which
they belong, worthy of being entered upon and pursued.

But admitting (as I do not) that Pure Mathematics are only to be
studied with a view to Natural and Physical Science, the question
still arises how are they best to be studied in that view. I assume
and admit that as to a large part of Modern Geometry and of the Theory
of Numbers, there is no present probability that these will find any
physical applications. But among the remaining parts of Pure
Mathematics we have the theory of Elliptic Functions and of the
Jacobian and Abelian Functions, and the theory of Differential
Equations, including of course Partial Differential Equations. Now
taking for instance the problem of three bodies--unless this is to be
gone on with by the mere improvement in detail of the present
approximate methods--it is at least conceivable that the future
treatment of it will be in the direction of the problem of two fixed
centres, by means of elliptic functions, &c.; and that the discovery
will be made not by searching for it directly with the mathematical
resources now at our command, but by "prospecting" for it in the field
of these functions. Even improvements in the existing methods are more
likely to arise from a study of differential equations in general than
from a special one of the equations of the particular problem: the
materials for such improvements which exist in the writings of
Hamilton, Jacobi, Bertrand, and Bour, have certainly so arisen. And
the like remarks would apply to the physical problems which depend on
Partial Differential Equations.

I think that the course of mathematical study at the University is
likely to be a better one if regulated with a view to the cultivation
of Science, as if for its own sake, rather than directly upon
considerations of what is educationally best (I mean that the best
educational course will be so obtained), and that we have thus a
justification for a thorough study of Pure Mathematics. In my own
limited experience of examinations, the fault which I find with the
men is a want of analytical power, and that whatever else may have
been in defect