ed by the
preceding general observations. The rapid scientific advances of the
past century have increased the mathematical needs very rapidly. The
advances in college mathematics which have been made possible by the
improvements of the secondary schools have scarcely kept up with the
growth of these needs, so that the current mathematical needs cannot
be as fully provided for by the modern college as the recognized
mathematical needs of the eighteenth century were provided for by the
colleges of those days.

There appears to be no upper limit to the amount of useful
mathematics, and hence the aim of the college must be to supply the
mathematical needs of the students to the greatest possible extent
under the circumstances. In order to supply these needs in the most
economical manner, it seems necessary that some of them should be
supplied before they are fully appreciated on the part of the student.
The first steps in many scientific subjects do not call for
mathematical considerations and the student frequently does not go
beyond these first steps in his college days, but he needs to go much
further later in life. College mathematics should prepare for life
rather than for college days only, and hence arises the desirability
of deeper mathematical penetration than appears directly necessary for
college work.

=Advanced work in college mathematics=

Another reason for more advanced mathematics than seems to be directly
needed by the student is that the more advanced subjects in
mathematics are a kind of applied mathematics relative to the more
elementary ones, and the former subjects serve to throw much light on
the latter. In other words, the student who desires to understand an
elementary subject completely should study more advanced subjects
which are connected therewith, since such a study is usually more
effective than the repeated review of the elementary subject. In
particular, many students secure a better understanding of algebra
during their course in calculus than during the course in algebra
itself, and a course in differential equations will throw new light on
the course in calculus. Hence college mathematics usually aims to
cover a rather wide range of subjects in a comparatively short time.

Since mathematics is largely the language of advanced science,
especially of astronomy, physics, and engineering, one of the
prominent aims of college mathematics should be to keep in close touch
with the other