As to Partial Differential Equations, they are "high" as being an
inverse problem, and perhaps the most difficult inverse problem that
has been dealt with. In regard to the limitation of them to the second
order, whatever other reasons exist for it, there is also the reason
that the theory to this order is as yet so incomplete that there is no
inducement to go beyond it; there could hardly be a more valuable step
than anything which would give a notion of the form of the general
integral of a Partial Differential Equation of the second order.
I cannot but differ from you _in toto_ as to the educational value of
Analytical Geometry, or I would rather say of Modern Geometry
generally. It appears to me that in the Physical Sciences depending on
Partial Differential Equations, there is scarcely anything that a
student can do for himself:--he finds the integral of the ordinary
equation for Sound--if he wishes to go a step further and integrate
the non-linear equation (dy/dx) squared(d squaredy/dt squared) = a squared(d squaredy/dx squared) he is simply
unable to do so; and so in other cases there is nothing that he can
add to what he finds in his books. Whereas Geometry (of course to an
intelligent student) is a real inductive and deductive science of
inexhaustible extent, in which he can experiment for himself--the very
tracing of a curve from its equation (and still more the consideration
of the cases belonging to different values of the parameters) is the
construction of a theory to bind together the facts--and the selection
of a curve or surface proper for the verification of any general
theorem is the selection of an experiment in proof or disproof of a
theory.
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