An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

PREF CE
The subject-matter for the following pages has its origin
in- various sources well known to the general mnathematical
reader- vet it is felt that the view of the differential and integral calculus presented in the sequel generally does not come to
be recognized by the student sufficiently early, in his course.
The view referred to is suggested in a number of excellent
works upon infinitesimal calculus, but the writer's aim is to treat
the subject of this paper as a chapter in algebra, preparatory and
supplementary to a course in differential and integral calculus,
and not as a part of that course. The historical order of development has been followed in general outline, though the exercises
are set in modern notation. The historical order of development
of a science is usually (not always; cf. Hist. logarithms) the
easiest to follow'; moreover any other method withholds history
and often fails to give a sltfficienlt account of the science involved. This stands to reason for the actual development is likely to be the result of necessity. The reader is referred to several
excellent articles and books for the history, which, after reading
this paper, should be intelligible to the beginner.
The object iis not merely to gain the historical view of the
infinitesimal analysis but to prepare the student for the solution
of problems in applied mathematics. The processes of differentiation and integration are acquired without much difficulty;
but to see the integral with facility in a. problem in analytic mechanics er physics requires clear notions as to sums and limits
of sums. Such notions are of much more importance 'to the
physicist and engineer than the more elaborate methods for in