An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

PREFACE.

The present work owes its origin to the special circumstances of
the Professorship which I hold at the Polytechnikum. Having to give
instruction in the Differential and Integral Calculus to classes of
technical students, who require a knowledge of Analysis chiefly as an
instrument for the solution of mechanical problems, I have had to
adopt a strictly limited conception of a function in order that the proofs
of the general theorems might be simplified. But I could not avoid feeling
it all the more due to such of my hearers as desire to devote themselves
to the study of Mathematics, that I should put within their reach a
supplement to my lectures, giving greater prominence to the systematic
parts of Analysis.
Such a supplement may not, I think, be altogether superfluous
even for a larger circle of readers, since most treatises on Higher
Analysis (one exception is the recent work of Dr. Lipschitz), are so
occupied with its practical applications that they enter but inadequately
into any discussion of the principles on which Analysis is founded.
The student first realises the necessity of discussing these fundamental
problems, when he comes to study treatises introductory to the Theory
of Complex Functions, where much that he had probably become
accustomed to regard as established by Analysis appears once more
called into question. That the scientific discussion of its principles
should thus be severed from the practical applications of Analysis has
no justification in the nature of the subject, and any such severance
is quite unsuitable in teaching it. I cannot indeed claim to have
wholly avoided this in the following Essay. Even in the necessary
division of its contents into four Books a separation is apparent
which is based upon the fact that in the Theory of Real Functions
the data are much more detailed than in that of Complex Functions.
But besides the purely didactic aim, my guiding wish has been that
my work might contribute towards laying the foundations on which
the Differential and Integral Calculus may some day come to be
treated with perfect unity of system.
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