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.

16

1 Conception of a variable quantity.

Bk. I. ch. III.

of "variable quantity" we have also to free ourselves entirely from
considering what this quantity represents. The distance of a movable
point, the temperature, the tension of vapor, in a word, everything
measurable in nature can enter into calculations as variable quantity.
A quantity is said to be continuous or continuously variable within an
interval i.e. between two numbers, when there is no numerical value
between two numbers xO and xO + - however close together which it
does not assume as it changes from xo to xo + 6. Thus the statement, a quantity changes continuously from the value a to the value
b, amounts to this, the quantity travels through all numbers between
a and b and there is no break in the sequence of the numbers.
But now since when we want to fix the variable, we are not always able to assign a value which it assumes in a closed form but
only as the limiting value of a series, we often use a phrase drawing
attention to this change towards a determinate value whether rational or
irrational. We say: the continuous variable approximates infinitely to
a determinate value b or converges to b, when the series of numbers
through which it passes has this limiting value; that is, for any
number 6 however small, some place must be assignable in the series
from which onwards all values of the continuous variable differ from
b by less than 6. We shall always denote by the word "infinite", a
c o n t i n u ous change towards a determinate limiting value.
In particular a variable becomes "infinitely small", when as it
varies continuously there is no condition which prevents its absolute amount
becoming less than any assignable number, i.e. when it has the limiting
value zero.
The variable is called 'infinitely great" and its limiting value
written with the sign + oo, when as it varies continuously there is
no condition which prevents its absolute amount becoming greater
than any assignable number. From the continuous series of numbers
through which such a variable ultimately passes, discontinuous series
of numbers can be selected according to some law, (for instance if
we only pay attention to integers,) but they no longer satisfy the two
properties already recognised as necessary for series defining numbers.
Nevertheless certain calculations can be effected with these series of
numbers, on which account we introduce them as a distinct numerical
conception by the following more accurately stated definition:
A variable becomes in a determinate manner positively or negatively
infinitely great, when the series of' numbers through which it travels,
has the properties, first that the values after a certain place are only
positive or only negative, and, second that a place can be found in the
series after which all values are greater in amount than an assigned
number however great.