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.

~.6. 7.

Calculation with numbers in general.

11

fractions of unity, is subordinate to this definition, which holds good
for number in general.
Similarly it follows on applying multiplication that
a t, a1 31, a'9- 2 7 ' * *.t.
form a new series which likewise possesses the required properties.
For when the amounts of a, and A,, are at the most increased or
diminished by 6, the product a+v,,+,v is also at the most only
altered by the amount ac 6 + Pfi 6 + 62 which we can make as small
as we please by a suitable choice of A, since there are superior limits
which a and f3 never exceed, and 6 is arbitrarily small. Thus
(II) Lir (a) (I,,) = LimL (C,. 1 C ).
We find by division the new series
a, o) oI ~2 A,
P' i' P ' ( "'
Excluding the case that the limiting value of the series of divisors
vanishes, and so that of the,/ falls below any assignable value, there
is a superior value which these quotients are certain never to exceed;
and if a, and i,, are altered by 6, then the difference
abs [.~ _, - -= abs (_+ _ ) ]
is a number whose amount is arbitrarily small. Therefore
~(III) L ) - Li m ((7).
Similarly from (II) we find for involution with a positive integer
exponent
(IV) (Lim (ac))n = Lim (ac,),
and thence for extracting the root of a positive number, i.e. of a
number whose defining series from a certain place contains only positive terms;
(V) i/Lim (,) - Lim (j^y).
As before, these last two equations amount merely to the statements,
that a rational or irrational number can be qtacnl proximic raised
to a power, or have a root extracted, by performing these respective
operations upon a number in the defining series quianz proxinw equal
to it in value.
It is convenient to base our introduction of powers with integer
negative exponents, as Newton did, on the equation
An 1
An An-r