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.

~ 5.

Powers and roots

7

Hence manifestly the roots of positive rational numbers cannot
universally be extracted by means of rational numbers only, but evolution in like manner with the problems of subtraction and of division
introduces a new conception into the theory of numbers. Employing
Euclid's method of inclusion within limits, this conception is expounded as follows:
If o be any rational number, then in the unlimited series of
fractions
n 2 3.
0, n > o r (>),
there must be a consecutive pair ac= g and 3 = e+l such that
an <~ < a.<
The difference 3 - a is -. Now if a' be another number greater than
a, fractions with the greater denominator a' can be inserted between
the fractions a and p, this may be indicated by
oL 1 + I1+ ~+ t
i Ct G ~' G ' G P ~
In this series there must be two consecutive values ac and P1 such that
x" < a < 1,
where a, > a or at least is equal to a, in case the required value occurs
in the first interval, and /j < / or at most is equal to (f, in case the
last interval should have to be taken. The difference t(1 - ac is
always less than -, for this is the difference between the first and the
last value of the series, intervening terms must have smaller differences.
Continuing this procedure with a new denominator a" > a', we obtain two new fractions a2 and (32 having a difference smaller than
-, and so on. This will result finally either in a number being found
whose bth power is equal to 7; in which case the nature of b is
such, that it has a rational nth root which can be expressed by one
of the denominators a, a, ",..., or, if not, the two series, that of
the lower limits: a < ac < a2.. < c,... < a,+v... and that
of the upper limits: P > P1 > 3... >. > (t... go on
indefinitely.
These two series have the following properties: Although the
series of the a however far it is carried contains only increasing numbers, and that of the p only decreasing ones, and this is not affected
if as is possible equalities occur among them, still each a is smaller