Mathematics, or as he
termed it "Useless Algebra," should be curtailed, there was a smart
and interesting correspondence between him and Prof. Cayley, who was
the great exponent and advocate of Pure Mathematics at Cambridge. Both
of them were men of the highest mathematical powers, but diametrically
opposed in their views of the use of Mathematics. Airy regarded
mathematics as simply a useful machine for the solution of practical
problems and arriving at practical results. He had a great respect for
Pure Mathematics and all the processes of algebra, so far as they
aided him to solve his problems and to arrive at useful results; but
he had a positive aversion to mathematical investigations, however
skilful and elaborate, for which no immediate practical value could be
claimed. Cayley on the contrary regarded mathematics as a useful
exercise for the mind, apart from any immediate practical object, and
he considered that the general command of mathematics gained by
handling abstruse mathematical investigations (though barren in
themselves) would be valuable for whatever purpose mathematics might
be required: he also thought it likely that his researches and
advances in the field of Pure Mathematics might facilitate the
solution of physical problems and tend to the progress of the
practical sciences. Their different views on this subject will be
seen from the letters that follow:

ROYAL OBSERVATORY, GREENWICH,
LONDON, S.E.
_1867, Nov. 8_.

MY DEAR SIR,

I think it best to put in writing the purport of what I have said, or
have intended to say, in reference to the Mathematical Studies in the
University.

First, I will remark on the study of Partial Differential Equations.
I do not know that one branch of Pure Mathematics can be considered
higher than another, except in the utility of the power which it
gives. Measured thus, the Partial Differential Equations are very
useful and therefore stand very high, as far as the Second Order.
They apply, to that point, in the most important way, to the great
problems of nature concerning _time_, and _infinite division of
matter_, and _space_: and are worthy of the most careful study. Beyond
that Order they apply to nothing. It was for the purpose of limiting
the study to the Second Order, and at the same time working it
carefully, philo