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.

Third Chapter.
Conception of a variable quantity and calculation with variables,
specially with infinite quantities.
9. The totality of rational and irrational numbers forms the continuous series of numbers. To represent a continuum as a
whole we necessarily require an intuitive image. We naturally take
the right line, the simplest representation of a continuum in space.
Having assumed a unit of length, the points of a right line are
determined by assigning their distances from a fixed zero-point, with
the + or - sign, according as the point considered is right or left
of that origin. The distance of each point can be expressed by a
rational or irrational number. For, what is characteristic of a point
is that it bounds a definite length from the origin; this length
measured by the assumed unit is a number which can be expressed
either in a finite form, or as is taught in E uelid's elements, with
arbitrary approximation by continued subdivision of the unit; and this is
the very idea of obtaining a series defining the distance. Thus to each
point of the continuum belongs one and only one number, our general
conception of number is never in default. And conversely: to each
number belongs one and also only one point, since each number
determines a length to be constructed, each length an end point.
Thus according as a point moves upon a right line, its distance from
the origin travels through the continuous series of numbers. The information we have now acquired can also be thus stated: Rational numbers
enable us to express as approximately as can be desired every value
of the continuous series of numbers.
In order to contemplate the conception of all numbers possible
within an interval, that is, between two fixed values, we introduce into
analysis the representation of variability apart from the image of a
definite motion on the right line. This representation was first employed in the most general manner by Newton, the way for it having
been prepared by Geometry, specially from the time of Descartes.*)
A quantity is said to be vairiable, when it is able to assumze different
numerical values. As in purely arithmetical investigations we no longer
consider what are the things given in number, so in the conception
x) (1596 —1650).