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.

First Book.

Real numbers and functions of real numbers.
The conceptions of space and of number are the subject matter of
mathematical investigations. These investigations accordingly diverge
into two main branches: Geometry and Analysis. It thus appears that
Mathematics are of fundamental importance to all our knowledge of
Nature: for our representations of space contain the simplest properties
which are common to all things in the surrounding world; and accurate comparison or measurement of quantities leads always to concrete
numbers of the units employed: in order to understand the result, we
require a knowledge of numbers and of their combinations.
Nature in its phenomena is perpetually exhibiting change; the
simplest changes we perceive externally are changes of place, motions.
The representation of motion is necessarily combined with that of continuity, i. e. of an uninterrupted connexion in space and of an uninterrupted sequence in time. To describe thoroughly the phenomena
of motion is to assign every circumstance in numbers of concrete
units: so that if the series of numbers is also to enable us to describe
motion, it must contain a continuous series of quantities. Thus the
first problem of Analysis is: to develope the conception and the properties of the continuous series of numbers.
First Chapter.
Rational numbers.
1. The natural series of numbers, which arises by adding on a
thing to others in counting, advances always by unity; each number
is defined by the preceding number and by unity. This series of integers starting from unity can be continued on indefinitely. Now as each
several number is a sum of repeatedly added units, such a sum of units
can be composed of different given numbers. This arithmetical operation, merely a continuted reckoning up of groups of units, is called
Addition; it embraces all other operations, from it all others arise.
The fundamental proposition for addition is: the sum of given numbers
HA RNA.CK, Calculus. 1