An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

PREFACE.

THE method of Limits is generally allowed to
be the best and most natural basis upon which to
found the principles of the Differential Calculus; in
the following pages this method is exclusively adopted,
no use whatever being made of series in the demonstration of fundamental propositions. The following
is an outline of the work, which is by no means
offered to the reader as a complete treatise on the
subject, but merely as an exposition of its more prominent and useful principles.
In Chap. I, certain terms, afterwards to be used,
are defined and explained. In Chap. II the nature
of a Limiting Value is fully set forth, and the important distinction (which ought never to be overlooked) between an actual and a limiting value is
pointed out and illustrated by examples. Chap. III
contains a set of Lemmas, which are necessary in
order to render the use made of limiting values in
the Differential Calculus perfectly legitimate; and
here I have endeavoured to confine myself to what
seems really essential. In Chap. IV certain important
limiting values are obtained. Chap. V contains the
Rules for Differentiation, in the demonstration of
a